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Методы оптимизации в машинном и глубоком обучении. От простого к сложному

В данной статье представлен обзор различных популярных (и не только) оптимизаторов, которые применяются в машинном и глубоком обучении, в частности для обучения нейронных сетей. Мы рассмотрим их основную идею и ключевые особенности, переходя от простых к более сложным концепциям. Помимо этого, в самом конце вы сможете найти большое количество дополнительных источников для более детального ознакомления с материалом.

Ноутбук с данным материалом можно загрузить на GitHub (rus).

Содержание

Пару слов про функции потерь
Классический градиентный спуск
Более быстрые и точные оптимизаторы
Обучение и визуализация оптимизаторов
Нарушение работы адаптивных методов со скользящим средним
Стратегии изменения скорости обучения
Работа с большими мини-пакетами
Проксимальные методы
Методы второго порядка
Дополнительные источники

Пару слов про функции потерь

Функции потерь можно разделить на выпуклые и невыпуклые. Первый вариант чаще всего встречается в классическом машинном обучении из-за относительной простоты моделей, а также из-за того, что у выпуклых функций локальный минимум по определению является и глобальным (но не обязательно единственным). Невыпуклые функции, в свою очередь, чаще всего встречаются в нейронных сетях из-за их высокой сложности и в данном случае поиск глобального минимума является трудной задачей, поэтому на практике здесь также используются методы из случая с выпуклыми функциями из-за их способности хорошо аппроксимировать стационарные точки в невыпуклом случае. Проще говоря, методы для выпуклого случая могут хорошо сходиться в локальном оптимуме для невыпуклого случая за приемлемое время, значение которого близко к глобальному.

Классический градиентный спуск

Начнём с того, что если градиент — это вектор наибыстрейшего возрастания функции, то антиградиент — вектор наибыстрейшего убывания, и именно при движении в данную сторону будет расположена минимальная ошибка модели. Тогда градиентный спуск можно определить как численный метод итеративной оптимизации для нахождения весов (коэффициентов) модели путём минимизации её ошибки, представленной в виде функции потерь. Пример работы данного алгоритма на линейной и логистической регрессий с нуля можно посмотреть здесь и здесь.

Градиентный спуск для выпуклого и невыпуклого случая

Алгоритм строится следующим образом:

1) изначально происходит инициализация весов с нулевыми значениями;
2) далее на основе установленных весов модель делает прогноз;
3) на основе полученного прогноза рассчитывается градиент ошибки, после чего происходит обновление весов в сторону антиградиента функции потерь;
4) шаги 2-3 повторяются до тех пор, пока градиент не станет нулевым, то есть пока алгоритм не сойдётся в минимуме (как правило, в данном случае используется критерий останова, например, пока разность градиентов на текущей и предыдущей итерациях не станет меньше заранее установленного порогового значения).

Если для вычисления градиентов функции потерь используется полный обучающий набор, то такой градиентный спуск называется пакетным. Не смотря на то, что пакетный градиентный спуск (batch gradient descent) практически всегда хорошо масштабируется в отношении количества признаков, данный алгоритм работает очень медленно на больших наборах данных и требует значительных затрат в виде дополнительной памяти для хранения всех градиентов.

Для увеличения производительности на больших датасетах применяются 2 подхода:

1) Мини-пакетный градиентный спуск (mini-batch gradient descent), когда на каждом шаге вычисление градиентов происходит на небольших случайных поднаборах (мини-пакетах). Обычно размер мини-пакета берётся в виде и может достигать нескольких десятков тысяч образцов: выбор размера зависит от особенностей задачи и применяемой модели.
2) Стохастический градиентный спуск (stochastic gradient descent), когда на каждом шаге из обучающего набора берётся лишь один образец. Очевидно, что такой вариант должен работать гораздо быстрее предыдущего, но с другой стороны, из-за ещё более стохастический природы данный алгоритм менее стабилен и может потребоваться больше итераций для сходимости, а также mini-batch дает прирост в производительности из-за аппаратной оптимизации матричных вычислений на GPU. Стоит также отметить, что на данный момент под стохастическим градиентный спуском очень часто подразумевается mini-batch.

В отличие от пакетного градиентного спуска, в данных случаях функция потерь будет снижаться скачками вверх-вниз, приближаясь к минимуму, но так и не достигнув его. В случае с мини-пакетным градиентным спуском функция потерь будет расположена чуть ближе к минимуму за счёт меньшего размера скачков. С другой стороны, в отличие от стохастического градиентного спуска, ему может быть труднее пройти локальные минимумы в невыпуклом случае.

Более быстрые и точные оптимизаторы

Помимо того, что градиентный спуск в своём классическом представлении плохо подходит для невыпуклых случаев из-за вышеописанных проблем, может также возникнуть проблема с "седловыми точками", когда на поверхности функции потерь в виде седла имеются точки, в которых значение функции максимально по одним направлениям и минимально по другим, то есть в данных точках будет практически нулевое снижение градиентов и они ошибочно могут быть приняты моделью за оптимум. Поэтому для быстрого поиска оптимального решения используются более интересные стохастические (mini-batch) оптимизации, про самые популярные из которых сейчас и пойдёт речь.

Momentum

Предположим, что со склона с кочками катится мяч, который сначала будет двигаться медленно, но по мере накопления кинетической энергии его скорость будет расти, что ему поможет преодолеть небольшие кочки и остановить своё движение в более глубокой. Именно данный принцип лежит в основе моментной оптимизации. Momentum принимает во внимание значения предыдущих градиентов, которые используются для определения направления и скорости перемещения в пространстве параметров v_t (называется вектором первого момента, поскольку масса в данном случае равна 1), а для предотвращения её быстрого увеличения используется коэффициент ослабления импульса $\beta$ , который отвечает за силу влияния предыдущего градиента на текущий, то есть выступает в качестве механизма трения мяча о поверхность: значение 0 означает высокое трение, а 1 его отсутствие. Обычно значение $\beta$ устанавливается 0.9.

Алгоритм обновления параметров моментной оптимизации

$v_t \leftarrow \beta v_{t-1} - \alpha dw_{t-1}$

$w_t \leftarrow w_{t-1} + v_t$

Momentum способен решить сразу две проблемы: плохую обусловленность матрицы Гессе и дисперсию стохастического градиента. С другой стороны, из-за вычисления градиента функции потерь в локальной позиции модель может колебаться вокруг локальных минимумов, что в свою очередь увеличивает время схождения.

Хотя во многих статьях обозначается как , а как , на данный момент нет единого стандарта и в разных статьях можно встретить для них разные обозначения, но для унификации во всех примерах будут использоваться следующие обозначения: — это первый момент или ещё по-другому экспоненциально взвешенное скользящее среднее градиентов, а — второй момент или экспоненциально взвешенное скользящее среднее квадратов градиентов, однако в случае AdaGrad это просто квадрат градиентов.

Nesterov momentum

Для решения вышеописанной проблемы применяется небольшая оптимизация, суть которой заключается в вычислении градиентов не в локальной позиции, а чуть дальше в направлении момента. Другими словами, функция потерь будет снижаться не в текущей позиции, а в которой якобы должна оказаться в будущем, следуя направлению импульса.

Алгоритм обновления параметров моментной оптимизации Нестерова

$v_t \leftarrow \beta v_{t-1} - \alpha d(w_{t-1} + \beta v_{t-1} )$

$w_t \leftarrow w_{t-1} + v_t$

Схема Momentum vs Nesterov momentum

Такой подход позволяет сойтись гораздо быстрее и чуть ближе к оптимуму, однако он всё ещё может быть чувствительным в отношении одних направлений пространства параметров и нечувствительным в других из-за трудностей подбора оптимальной скорости обучения. Ниже будут представлены оптимизации градиентного спуска с адаптивной скоростью обучения.

AdaGrad

Adaptive Gradient является одной из первых оптимизаций градиентного спуска с адаптивным шагом, в которой снижается градиент вдоль самых крутых направлений. Проще говоря, на каждом шаге происходит обновление параметров модели с учётом значений как предыдущих градиентов, так и их квадратов.

Алгоритм обновления параметров AdaGrad

$s_t \leftarrow s_{t-1} + dw_{t-1}^2$

$w_t \leftarrow w_{t-1} - \frac{\alpha}{\sqrt{s_t + \epsilon}} dw_{t-1}$

Где $\epsilon = 1e-8$ — сглаживающий параметр для избегания деления на ноль.

Такой простой подход позволяет алгоритму быстро начать обучение, а затем замедлить его, когда градиент становится меньше или меняет знак. Однако на практике AdaGrad хорошо справляется с выпуклыми случаями, а вот при обучении нейронных сетей нередко останавливается слишком рано из-за быстрого снижения скорости обучения, так и не достигнув глобального оптимума.

RMSProp

RMSProp (Root Mean Square Propagation) — модификация AdaGrad, адаптированная для лучшей работы в невыпуклом случае. Основная идея заключается в изменении способа агрегирования градиента на экспоненциально взвешенное скользящее среднее. Другими словами, вместо накопления всех квадратов градиента с начала обучения, накапливаются квадраты градиента только из самых последних итераций.

Алгоритм обновления параметров RMSProp

$s_t \leftarrow \beta s_{t-1} + (1 - \beta) dw_{t-1}^2$

$w_t \leftarrow w_{t-1} - \frac{\alpha}{\sqrt s_t + \epsilon} dw_{t-1}$

В целом, RMSProp является крайне эффективной оптимизацией, но не самой лучшей. Например, могут возникнуть трудности обучения в многомерных пространствах из-за проблем с масштабированием градиентов в разных направлениях.

Adam и его модификации

Adam (Adaptive Moment Estimation) объединяет в себе концепции Momentum и RMSProp, практически являясь серебряной пулей в задачах стохастической оптимизации. Как можно заметить, расчёт моментов в Adam очень схож с Momentum и RMSProp за исключением того, что к ним применяется поправка на смещение. В целом, благодаря таким улучшениям Adam сходится быстрее и лучше предшественников, а также более устойчив к подбору гиперпараметров, что делает его более стабильным решением в большинстве случаев.

Алгоритм обновления параметров Adam

$v_t \leftarrow \beta_1 v_{t-1} + (1 - \beta_1) dw_{t-1}$

$s_t \leftarrow \beta_2 s_{t-1} + (1 - \beta_2) dw_{t-1}^2$

$w_t \leftarrow w_{t-1} - \alpha \frac{v_t}{\sqrt s_t + \epsilon}$

Поправка на смещение:

$v_t \leftarrow \frac{v_t}{1 - \beta_1^t}$

$s_t \leftarrow \frac{s_t}{1 - \beta_2^t}$

Где $\beta_1 = 0.9, \ \ \beta_2 = 0.99$ используются по умолчанию.

С другой стороны, данный алгоритм более сложен в анализе и может быть чувствителен к шуму, что может привести к проблемам со сходимостью, а также он требует больших объемов памяти, поскольку хранит не только параметры модели, но а также информацию о градиентах и накопленном импульсе. Помимо этого, у Adam и всех предыдущих алгоритмов есть ещё один неприятный недостаток: если продолжить обучение модели после прерывания в некоторой точке, придётся восстанавливать не только веса модели из данной точки, но а также и всю информацию о накопленных параметрах.

Однако время не стоит на месте и были разработаны модификации Adam, призванные устранить в той или иной степени недостатки, описанные выше. К наиболее популярным и интересным модификациям можно отнести следующие:

AdaMax использует экспоненциально взвешенную норму бесконечности для обновления вектора второго момента, то есть скорость обучения изменяется, основываясь максимальной величине градиента в истории:

$s_t \leftarrow max(\beta_2 s_{t-1} |dw_{t-1}|)$

Nadam (Nesterov-accelerated Adaptive Moment Estimation) использует другую поправку на смещение для вектора первого момента:

$\hat v_t \leftarrow \frac{(1 - \beta_1^t) dw_{t-1}}{1 - \Pi_{i=1}^{t} \beta_1^i} + \frac{\beta_1^{t+1} v_t}{1 - \Pi_{i=1}^{t+1} \beta_1^i}$

AdamW добавляет L2-регуляризацию к функции потерь и при обновлении весов:

$$w_t \leftarrow w_{t-1} - \frac{\alpha v_t}{\sqrt s_t + \epsilon} + \lambda w_{t-1}$

Yogi обновляет вектор второго момента с учётом разности второго момента и квадрата градиента:

$s_t \leftarrow s_{t-1} - (1 - \beta_2) sign(s_{t-1} - dw_{t-1}^2) dw_{t-1}^2$

Adan (Adaptive Nesterov Momentum) использует модифицированный New Nesterov Momentum (NME позволяет избежать дополнительных затрат на вычисление градиента в точке экстраполяции) для оценки первого и второго моментов, что позволяет значительно ускорить сходимость и найти приближённую точку первого порядка с заданной точностью ϵ. Схема работы алгоритма выглядит следующим образом:

Обучение и визуализация оптимизаторов

Для наглядности рассмотрим небольшой пример работы различных оптимизаторов на данных Boston Housing. Сначала мы создадим простую нейросеть в Pytorch и обучим её на всех тренировочных данных, на которых после же оценим снижение потерь оптимизаторов на каждой итерации. Также для любителей Keras мы обучим простую нейросеть на мини-пакетах, но уже оценим снижение потерь на тестовом наборе.

Загрузка и подготовка датасета

import pandas as pd
import matplotlib.pyplot as plt
from sklearn.preprocessing import StandardScaler
from sklearn.model_selection import train_test_split

df_path = "/content/drive/MyDrive/BostonHousing.csv"
boston = pd.read_csv(df_path)
print(boston)


        crim    zn  indus  chas    nox     rm   age     dis  rad  tax  \
0    0.00632  18.0   2.31     0  0.538  6.575  65.2  4.0900    1  296   
1    0.02731   0.0   7.07     0  0.469  6.421  78.9  4.9671    2  242   
2    0.02729   0.0   7.07     0  0.469  7.185  61.1  4.9671    2  242   
3    0.03237   0.0   2.18     0  0.458  6.998  45.8  6.0622    3  222   
4    0.06905   0.0   2.18     0  0.458  7.147  54.2  6.0622    3  222   
..       ...   ...    ...   ...    ...    ...   ...     ...  ...  ...   
501  0.06263   0.0  11.93     0  0.573  6.593  69.1  2.4786    1  273   
502  0.04527   0.0  11.93     0  0.573  6.120  76.7  2.2875    1  273   
503  0.06076   0.0  11.93     0  0.573  6.976  91.0  2.1675    1  273   
504  0.10959   0.0  11.93     0  0.573  6.794  89.3  2.3889    1  273   
505  0.04741   0.0  11.93     0  0.573  6.030  80.8  2.5050    1  273   

     ptratio       b  lstat  medv  
0       15.3  396.90   4.98  24.0  
1       17.8  396.90   9.14  21.6  
2       17.8  392.83   4.03  34.7  
3       18.7  394.63   2.94  33.4  
4       18.7  396.90   5.33  36.2  
..       ...     ...    ...   ...  
501     21.0  391.99   9.67  22.4  
502     21.0  396.90   9.08  20.6  
503     21.0  396.90   5.64  23.9  
504     21.0  393.45   6.48  22.0  
505     21.0  396.90   7.88  11.9  

[506 rows x 14 columns]

print(boston.isna().sum())


crim       0
zn         0
indus      0
chas       0
nox        0
rm         5
age        0
dis        0
rad        0
tax        0
ptratio    0
b          0
lstat      0
medv       0
dtype: int64

boston.dropna(inplace=True)

X, y = boston.iloc[:, :-1].values, boston.iloc[:, -1].values
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=0)

X_train_s = StandardScaler().fit_transform(X_train)
X_test_s = StandardScaler().fit_transform(X_test)

Параметры нейросети

nn_params = {'in_features': X.shape[1], 'h1': 16, 'h2': 8, 'out': 1}

Сравнение оптимизаторов Pytorch на тренировочных данных

from torch import nn, optim, FloatTensor, relu, manual_seed


class TorchNN(nn.Module):
    def __init__(self, in_features, h1, h2, out):
        super(TorchNN, self).__init__()
        self.fc1 = nn.Linear(in_features, h1)
        self.fc2 = nn.Linear(h1, h2)
        self.fc3 = nn.Linear(h2, out)

    def forward(self, x):
        x = relu(self.fc1(x))
        x = relu(self.fc2(x))
        x = self.fc3(x)

        return x


def torch_losses(data, model, optim_cls, optim_params, num_epochs=100):
    losses = []
    X_train, y_train = data
    criterion = nn.MSELoss()
    optimizer = optim_cls(model.parameters(), **optim_params)

    for epoch in range(num_epochs):
        y_pred = model.forward(X_train)
        loss = criterion(y_pred, y_train)
        losses.append(loss.item())

        optimizer.zero_grad()
        loss.backward()
        optimizer.step()

    return losses


manual_seed(0)
X_train_s_tensor = FloatTensor(X_train_s)
y_train_tensor = FloatTensor(y_train).view(-1, 1)

torch_optimizers = [
    (optim.SGD, 'SGD', {'lr': 0.001}),
    (optim.SGD, 'Momentum', {'lr': 0.001, 'momentum': 0.9}),
    (optim.SGD, 'Nesterov Momentum', {'lr': 0.001, 'momentum': 0.9, 'nesterov': True}),
    (optim.Adagrad, 'Adagrad', {'lr': 0.01}),
    (optim.RMSprop, 'RMSprop', {'lr': 0.01}),
    (optim.Adam, 'Adam', {'lr': 0.01}),
    (optim.Adamax, 'Adamax', {'lr': 0.01}),
    (optim.AdamW, 'AdamW', {'lr': 0.01}),
    (optim.NAdam, 'NAdam', {'lr': 0.01})
    ]

# Visualization
plt.figure(figsize=(14, 6))

for optim_cls, optim_name, optim_params in torch_optimizers:
    nn_model = TorchNN(**nn_params)
    t_losses = torch_losses((X_train_s_tensor, y_train_tensor), nn_model, optim_cls, optim_params)
    plt.plot(range(len(t_losses)), t_losses, label=f'{optim_name}')

plt.title('Pytorch optimizers comparison on train data')
plt.xlabel('Epoch')
plt.ylabel('Loss')
plt.legend()
plt.show()

Сравнение оптимизаторов Keras на тестовых данных

from keras import optimizers
from keras.layers import Dense
from keras.models import Sequential
from keras.utils import set_random_seed


def create_keras_nn(optimizer, in_features, h1, h2, out):
    model = Sequential()
    model.add(Dense(input_shape=(in_features,), units=h1, activation='relu'))
    model.add(Dense(input_shape=(h1,), units=h2, activation='relu'))
    model.add(Dense(units=out))
    model.compile(optimizer=optimizer, loss='mse', metrics=['mse'])

    return model


def keras_losses(train_data, test_data, optimizers, nn_params, num_epochs=10):
    losses = []
    X_train, y_train = train_data
    X_test, y_test = test_data

    for name, optimizer in optimizers:
        model = create_keras_nn(optimizer, **nn_params)
        model_history = model.fit(X_train, y_train, batch_size=32, epochs=num_epochs,
                                  verbose=0, validation_data=(X_test, y_test))
        losses.append((name, model_history.history['val_loss']))

    return losses


set_random_seed(0)

keras_optimizers = [
    ('SGD', optimizers.SGD(learning_rate=0.001)),
    ('Momentum', optimizers.SGD(learning_rate=0.001, momentum=0.9)),
    ('Nesterov Momentum', optimizers.SGD(learning_rate=0.001, momentum=0.9, nesterov=True)),
    ('Adagrad', optimizers.Adagrad(learning_rate=0.01)),
    ('RMSprop', optimizers.RMSprop(learning_rate=0.01)),
    ('Adam', optimizers.Adam(learning_rate=0.01)),
    ('Adamax', optimizers.Adamax(learning_rate=0.01)),
    ('AdamW', optimizers.AdamW(learning_rate=0.01)),
    ('Nadam', optimizers.Nadam(learning_rate=0.01))
    ]


plt.figure(figsize=(14, 6))

k_losses = keras_losses((X_train_s, y_train), (X_test_s, y_test), keras_optimizers, nn_params)

for name, k_loss in k_losses:
    plt.plot(k_loss, label=name + ' test Loss')

plt.title('Keras optimizers comparison on test data')
plt.xlabel('Epoch')
plt.ylabel('Loss')
plt.legend()
plt.show()

Промежуточные выводы

Поскольку SDG, Momentum и Nesterov momentum являются менее стабильными алгоритмами и больше всего зависят от начальных значений learning rate, для них была задана меньшая скорость обучения, чем для остальных методов. На первом графике видно, что даже в таком случае эти алгоритмы немного подвержены колебаниям, хотя и быстро сошлись. Второй же график хорошо демонстрирует как обучение на мини-пакетах значительно ускоряет сходимость алгоритмов. В случае с AdaGrad возникла классическая проблема резкого снижения скорости обучения, которую в данном случае можно легко решить, установив изначально более высокое значение. А вот RMSprop хорошо сработал в обоих случаях и даже лучше Adam-подобных оптимизаторов.

Отсюда следует интересный вывод, что адаптивные методы могут сходиться быстрее предшественников далеко не в каждой ситуации, но зато они гораздо стабильнее минимизируют функцию потерь. Конкретно в данном случае это означает, что если мы установим одинаковые произвольные значения learning rate для всех оптимизаторов, то неадаптивные методы зачастую будут снижаться с очень большими рывками и могут не сойтись вообще, в то время как адаптивные методы будут более плавно сходиться, хоть и медленно.

Стоит добавить, что данный пример лишь в очередной раз доказывает, что нет универсального оптимизатора, который являлся бы лучшим выбором для всех задач машинного обучения, поскольку многое зависит от самых разных факторов, включая природу данных, сложность модели и цели обучения. Однако это не отменяет тот факт, что в случае со сложными ландшафтами потерь адаптивные методы практически всегда будут превосходить простые стохастические оптимизаторы не только в плане стабильности, но и конечно же в скорости. Поэтому в большинстве ситуаций RMSprop и особенно семейство оптимизаторов Adam можно рассматривать в качестве хорошего стартового решения.

Нарушение работы адаптивных методов со скользящим средним

Поскольку работа всех диагональных адаптивных методов в той или иной степени основана на использовании покоординатного learning rate, то в случае, когда последовательность s_t перестаёт монотонно не убывать, алгоритм не может нормально сойтись к локальным минимумам. Как правило, такая ситуация означает приближение к критическим точкам и происходит при уменьшении $dw_{t-1}^2$ в сравнении с предыдущей накопленной историей с точностью до некоторой константы.

Данную проблему можно устранить путём наделения таких алгоритмов долговременной памятью о прошлых градиентах для исключения отрицательных регуляризаторов подобно тому, как это реализовано в AMSGrad. Проще говоря, выполняется условие $s_t >= s_{t-1}$ , что возможно благодаря простому трюку:

$\hat s_0 = 0, \ \hat s_t = max(\hat s_{t-1}, s_t) \ and \ \hat S_t = \ diag(\hat s_t)$

Однако стоит иметь в виду, что в таком случае требуются дополнительные расходы памяти, что может быть особенно проблематично для моделей с разреженными параметрами. Решить данную проблему можно за счёт использования непостоянных $\beta_1^t$ и $\beta_2^t$ , что было предложено в алгоритме AdamNC:

$v_t \leftarrow \beta_1^t v_{t-1} + (1 - \beta_1^t) dw_{t-1} \\ s_t \leftarrow \beta_2^t s_{t-1} + (1 - \beta_2^t) dw_{t-1}^2$

Помимо того, что такой подход не требует изменения структуры оригинального Adam, он также позволяет достичь хороших показателей сходимости.

Однако в работе Tran Thi Phuong и Le Trieu Phong было показано, что проблема в доказательстве сходимости AMSGrad заключается в обработке гиперпараметров, рассматривая их как равные, в то время как это не совсем верно. Так авторами был представлен контрпример в контексте простой задачи выпуклой оптимизации, который показывает эту проблему. Алгоритм должен сходиться согласно следующей лемме:

$R(T) \leq \sum_{i=1}^{d} \sum_{t=1}^{T} \frac{\sqrt{\hat{s}_{t,i}}}{2\alpha_t(1 - \beta_1^t)} ((w_{t,i} - w^*_{i})^2 - (w_{t+1,i} - w^*_{i})^2) + \\ + \sum_{i=1}^{d} \sum_{t=1}^{T} \frac{\alpha_t}{1 - \beta_1} \frac{v^2_{t,i}}{\sqrt{\hat{s}_{t,i}}} + \sum_{i=1}^{d} \sum_{t=2}^{T} \frac{\beta_1^t \sqrt{\hat{s}_{t-1,i}}}{2\alpha_{t-1}(1 - \beta_1)} (w_{t,i} - w^*_{i})^2$

Однако проблема заключается в равенстве, которое на самом деле может быть как положительным, так и отрицательным:

$(w_{t,i} - w^*_{i})^2 - (w_{t+1,i} - w^*_{i})^2$

Поэтому авторы представили новое доказательство сходимости для AMSGrad, а также алгоритм AdamX на его основе, который учитывает различия в гиперпараметрах и позволяет избежать проблемы выше. Решение выглядит следующим образом:

$\hat s_1 = s_1, \ \hat s_t = max \left(\frac{(1 - \beta_1^t)^2}{(1 - \beta_1^{t-1})^2} \hat s_{t-1}, s_t \right) \ \ if \ \ t \geq 2, \ and \ \hat S_t = diag(\hat s_t)$

Но стоит иметь в виду, что как и в случае с AMSGrad, AdamX также требует дополнительных расходов памяти.

Стратегии изменения скорости обучения

Как можно было заметить, скорость обучения является одним из ключевых параметров в градиентном спуске, которая может оказывать значительное влияние на сходимость и конечную ошибку модели. Помимо вышеописанных стратегий сокращения скорости обучения, ещё одними из самых популярных являются следующие:

1) $\alpha = \frac{1}{1 + decay \_ rate * t} \alpha_0 \ - \$ плавное затухание;
2) $\alpha = k^t \alpha_0 \ - \$ экспоненциальное затухание;
3) $\alpha = \frac{k}{\sqrt{t}} \alpha_0$ , где $k \ -$ константа, а $t \ -$ число итераций;
4) дискретное сокращение, когда после заданного количества итераций шаг уменьшается с определённым размером.

Не смотря на то, что на сегодняшний день в алгоритмах на основе градиентного спуска используется автоматическое снижение скорости обучения как в библиотеке sckit-learn, так и в библиотеках глубокого обучения, подбор learning rate всё ещё нужно выполнять аккуратно: алгоритм может преждевременно выйти на плато либо вовсе разойтись. Пример приведён на графике ниже.

Другим подходом в изменении скорости обучения является её резкое увеличение, чтобы "вытолкнуть" модель из возможных локальных минимумов, в которых она может застрять в процессе обучения. Такой подход называется Warm Restart и включает в себя циклическое изменение скорости обучения: начиная с высокого значения, скорость постепенно снижается по определённому правилу, а затем после достижения минимального значения она снова повышается. Это особенно полезно, когда модель сходится к неоптимальному решению из-за плохой инициализации или когда пространство параметров содержит множество локальных минимумов.

Одним из популярных методов, использующий такой подход, является CosineAnnealingWarmRestarts, в котором скорость обучения регулируется на основе "косинусного отжига" с перезапусками, возвращаясь к своему начальному значению:

$η_t = η_{min} + \frac{1}{2}(η_{max} - η_{min})(1 + \cos(\frac{T_{cur}}{T_{i}}\pi))$

Где:

$\eta_t$ — скорость обучения в момент времени ;
$\eta_{\text{max}}$ — начальная скорость обучения;
$\eta_{\text{min}}$ — минимальная скорость обучения;
$T_{\text{cur}}$ — количество эпох с момента последнего перезапуска;
а — количество эпох между двумя перезапусками.

С полным списком "планировщиков" скорости обучения можно ознакомиться здесь и здесь.

Работа с большими мини-пакетами

Рассмотрим ситуацию, когда нам необходимо обучить сложную нейронную сеть на больших данных с использованием нескольких графических процессоров (GPU). Одно из самых простых решений будет заключаться в вычислении стохастических градиентов для каждого батча отдельно на разных GPU с их последующим усреднением по всем GPU на каждом этапе обновления параметров модели.

По сути, такой подход равносилен обучению с одним огромным батчем и на первый взгляд может показаться оптимальным, но не всё так просто, поскольку это может привести к такому явлению как generalization gap, при котором возникает ухудшение обобщающей способности модели (иногда значительное). Почему так происходит? При использовании батча большого размера оптимизатор начинает лучше распознавать ландшафт функции потерь для определённой выборки, что увеличивает шансы скатиться в узкие локальные минимумы с низкой обобщающей способностью. Тогда даже при незначительном сдвиге ландшафта (при так называемом distributional shift, когда происходит переход от тренировочной выборки к тестовой) значение функции потерь может резко увеличиться. Проще говоря, большие батчи могут приводить к переобучению.

С другой стороны широкие локальные минимумы более устойчивы к изменениям в ландшафте функции потерь и способны обеспечить лучшую обобщающую способность, а для того чтобы в них попадать можно использовать оптимизаторы с более точным динамическим подбором learning rate, про которые далее пойдёт речь. Однако перед этим стоит добавить, что использование маленьких батчей в оптимизации также является не самой хорошей идеей, поскольку это может привести к более шумным оценкам градиента и в конечном счёте к недообучению модели.

LARS

Первым на очереди будет Layer-wise Adaptive Rate Scaling (LARS), использующий momentum в качестве базового алгоритма. Его основная идея заключается в подборе скорости обучения не для каждого нейрона или всей нейросети, а для каждого слоя отдельно. Проще говоря, обновление весов происходит с учётом локального learning rate, рассчитанного как отношение нормы весов к сумме нормы весов и градиентов. Работа алгоритма показана ниже:

Не смотря на то, что такой метод позволяет использовать батчи большого размера для обучения без потери качества, он плохо справляется со сложными моделями, где каждый слой имеет важное значение для общей производительности, например, с моделями внимания, такими как BERT. Это связано с тем, что если в одном из слоёв сети наблюдается нестабильное снижение градиентов, то к этому слою будет применяться меньшая скорость обучения, а поскольку LARS стремится синхронизировать обновления по всем слоям, то скорости обучения для более "стабильных" слоёв также будут снижены для предотвращения расхождения в обучении. Следовательно, более "стабильные" слои будут обновляться не так быстро, как могли бы, что в итоге значительно замедлит обучение всей модели и окажет негативное влияние на конечный результат.

LAMB

В ответ на это был разработан более продвинутый оптимизатор, известный как Layer-wise Adaptive Moments Based optimizer (LAMB), который можно рассматривать как применение LARS к оптимизатору Adam. Основное различие заключается в использовании двойной стратегии нормализации:

1) Нормализация каждого измерения (или параметра модели) относительно квадратного корня из второго момента, используемого в Adam. Это означает, что каждый параметр модели обновляется с учётом его собственной истории изменений, что помогает более точно настраивать веса модели.
2) Применение послойной нормализации. Другими словами, обновления параметров модели происходят с учётом масштаба каждого слоя в нейронной сети. Это помогает предотвратить слишком большие изменения в весах слоёв и избежать нестабильности в процессе обучения.

Алгоритм работает следующим образом:

Такой подход позволил значительно улучшить процесс обучения модели BERT даже на батчах большого размера без потери точности. Так команде Google Brain удалось сократить время обучения с 3 дней до всего лишь 76 минут, увеличив размер пакета до предела памяти модуля TPUv3. Более того, как и другие оптимизаторы, LAMB имеет ряд модификаций, способных не только ускорить обучение (как в NVLAMB благодаря предварительной нормализации градиентов), но и сделать его ещё более стабильным в плане сходимости (как в N-LAMB и NN-LAMB).

Прокcимальные методы

Проксимальные методы — это класс алгоритмов, предназначенных для решения задач выпуклой оптимизации негладких функций. Основная идея таких методов заключается в замене исходной сложной задачи оптимизации на серию более простых подзадач, аппроксимируя новое решение к текущему. Другими словами, данные методы представляют собой обобщение проективных градиентных методов.

Метод проксимальной минимизации

Может быть полезен в тех случаях, когда трудно минимизировать функцию потерь , но легко (или хотя бы проще) минимизировать её регуляризованную версию, то есть плюс квадратичная величина. Данный алгоритм имеет множество разных интерпретаций в зависимости от целей использования. Наиболее популярными являются следующие:

1) Регуляризация Тихонова позволяет находиться $w_{k+1}$ недалеко от и ускоряет сходимость, особенно при использовании метода сопряжённого градиента. Это обусловлено тем, что наличие регуляризации обеспечивает сильную выпуклость задачи:

$w_{k+1} = \text{argmin} \left\{ f(w) + \frac{1}{2\alpha_k} \| w - w_k \|^2 \right\} = prox_{\alpha_k f}(w_k)$

2) Градиентный поток обеспечивает сходимость в точке минимума из любой начальной точки при $t \to \infty$ :

$\frac {d}{dt} w(t) = - \nabla f(w(t))$

Решив данное дифференциальное уравнение с помощью прямой схемы Эйлера, получим:

$w_{k+1} = w_k - \alpha \nabla f(w_k)$

Однако такое решение менее стабильно, чем с использованием обратной схемы Эйлера. Тогда задача приобретает вид:

$\frac{w_{k+1} - w_k}{\alpha} = - \nabla f(w_{k+1})$

Поскольку уравнение представлено в неявном виде, а его левая часть является по сути градиентом функции g(w^*) в точке $w_{k+1}$ , тогда для выпуклого случая:

$\nabla (f(w^*) + g(w^*))(w_{k+1}) = 0 \Rightarrow \\ \Rightarrow w_{k+1} = \text{argmin} \left\{ f(w^*) + \frac{1}{2\alpha_k} \| w^* - w_k \|^2 \right\} = prox_{\alpha_k f}(w_k)$

Также стоит отметить, что идея градиентного потока может быть обобщена на случаи, когда не дифференцируется с помощью субградиентного дифференциального включения:

$\frac {d}{dt} w(t) \in - \partial f(w(t))$

3) Итеративное уточнение основано на идее асимптотически исчезающей регуляризации Тихонова и используется для решения линейных уравнений вида с использованием факторизации Холецкого для $X + \frac{1}{\alpha} I$ .

Рассмотрим задачу минимизации квадратичной функции:

$f(w) = \frac{1}{2} w^T X w - y^T w$

где $X \in S_+^n$ (множество симметричных положительных полуопределённых x матриц). Такая задача эквивалентна решению системы линейных уравнений Xw = y . Когда несингулярно, то единственным решением является $w = X^{-1} y$ . Такую же проблему можно заметить в методе наименьших квадратов. Проксимальный оператор для при w_k можно выразить аналитически:

$prox_{\alpha_k f}(w_k) = \text{argmin} \left\{ \frac{1}{2} w^T X w - y^T w + \frac{1}{2} \| w - w_k \|^2 \right\} = \\ = \left(X + \frac{1}{\alpha} I \right)^{-1} \left(y + \frac{1}{\alpha} w_k \right)$

Переписав данное выражение, получим:

$w_{k+1} = w_k + \left(X + \frac{1}{\alpha} I \right)^{-1} (y - X w_k)$

Проксимальный градиентный метод (PGM)

Применяется для решения проблемы вида $min \rightarrow f(w) + g(w)$ , где является гладкой (дифференцируемой), а негладкой функцией, для которой существует быстрый проксимальный оператор. Тогда, выполнив градиентный шаг по и проксимальный по , получим итеративный процесс вида:

$w_{k+1} = prox_{\alpha_k g} (w_k - \alpha_k \nabla f(w_k))$

Ещё такой процесс называется forward-backward splitting, где forward относится к шагу градиента, а backward — к проксимальному шагу.

PGM также имеет различные интерпретации:

1) Majorization-minimization используется для решения проблемы вида:

$min \rightarrow q_{\alpha}(w, w_k) = \frac{1}{2} \| w - (w_k \ - \alpha_k \nabla f(w_k)) \|^2 + \alpha g(w)$

А процесс обновления $w_{k+1}$ выглядит следующим образом:

$w_{k+1} = \text{argmin} \ q_{\alpha}(w, w_k)$

2) Fixed point iteration основано на идее, что существует фиксированная точка , которая является решением $min \rightarrow f(w) + g(w)$ лишь в том случае, когда она является точкой оператора forward-backward:

$(I + \alpha \partial g)^{-1} (I - \alpha \nabla f)$

Тогда решение приобретает вид:

$w^* = (I + \alpha \partial g)^{-1} (I - \alpha \nabla f) (w^*) = prox_{\alpha g} (w^* - \alpha_k \nabla f(w^*))$

3) Forward-backward интегрирование градиентного потока представляет собой метод численного интегрирования дифференциального уравнения градиентного потока, в котором используется прямой Шаг Эйлера для дифференцируемой части и обратный шаг Эйлера для возможно недифференцируемой части . Если исходить предположения, что является также дифференцируемой, то систему градиентного потока можно представить в виде:

$\frac {d}{dt} w(t) = - \nabla f(w(t)) - \nabla g(w(t))$

Тогда получим процесс обновления, называемый forward-backward splitting:

$w_{k+1} = (I + \alpha \nabla g)^{-1} (I - \alpha \nabla f) w_k$

Также стоит добавить, что существует вариант ускоренного проксимального градиентного метода, который включает в себя этап экстраполяции:

$z_{k+1} = w_k + p_k (w_k - w_{k-1}) \\ w_{k+1} = prox_{\alpha_k g} (z_{k+1} - \alpha_k \nabla f(z_{k+1}))$

где p_k — параметр экстраполяции, а одна из самых простых схем его расчёта выглядит следующим образом:

$p_k = \frac {k}{k + 3}$

ISTA (Iterative Shrinkage(Soft)-Thresholding Algorithm)

Данный метод можно рассматривать как обновление проксимального градиента, применяемое к L1-регуляризованной задаче наименьших квадратов:

$\|Xw - y\|^2 + \lambda \|w\| \rightarrow min$

Тогда прокс-оператор для l1-нормы будет выглядеть следующим образом:

$prox_{\lambda \alpha} (w) = argmin \left\{\frac{1}{2 \alpha} \|w - w^* \|^2 + \lambda \|w^* \| \right\} = \\ = argmin \left\{\sum_{i=1}^d \left[ \frac{1}{2 \alpha} (w_i - w_i^*) + \lambda |w_i^*| \right] \right\} = T_{\lambda \alpha} (w)$

где $T_{\lambda \alpha} (w)$ — оператор с плавным установлением порога, который для одномерной задачи является решением со следующим правилом:

$T_{\lambda \alpha} (w_i) = \begin{cases} w_i - \lambda \alpha, \ \ w_i \geq \lambda \alpha \\ 0, \ \ \ \ \ \ \ \ \ \ \ \ \ |w_i| \leq \lambda \alpha \\ w_i + \lambda \alpha, \ \ w_i \leq - \lambda \alpha \\ \end{cases}$

Отсюда получим градиентный шаг обновления:

$w_{k+1} = T_{\lambda \alpha_k} (w_k - \alpha_k X^T (Xw_k- y))$

Как можно заметить, данный метод позволяет получить разреженную модель, поскольку часть координат будет зануляться в случае $|w_i| \leq \alpha$ , что особенно полезно в задачах компьютерного зрения и обработки сигналов. Однако ISTA в чистом виде имеет низкую скорости сходимости, поэтому на практике используются его различные модификации. Например, FISTA (от слова Fast), основанное на ускорении Нестерова, где используется дополнительный шаг экстраполяции:

$t_{k+1} = \frac{1 + \sqrt{1 + 4t_k^2}}{2} \\ z_{k+1} = w_k + \frac{t_k - 1}{t_{k+1}} (w_k - w_{k-1}) \\ w_{k+1} = prox_{\lambda \alpha} (z_{k+1}) = T_{\lambda \alpha_k}(z_{k+1} - \alpha_k \nabla f(z_{k+1}))$

Такой подход позволил достичь скорости сходимости O(1/k^2) вместо O(1/k) как в ISTA. К слову, на сегодняшний день FISTA также имеет ряд модификаций, делающих его работу более стабильной и быстрой в ряде случаев.

Методы второго порядка

Методы оптимизации второго порядка — это алгоритмы, которые используют информацию о вторых производных (гессиане) функции потерь для более быстрого и точного нахождения точек минимума с очень неровным и сложным ландшафтом потерь.

Вторая производная характеризует скорость изменения градиента функции потерь за счёт измерения её кривизны, то есть позволяет оценить ожидаемое улучшение от шага градиентного спуска. На примере квадратичной функции это будет означать следующее:

1) Если кривизна отрицательная (отрицательное значение второй производной), то функция потерь снижается быстрее, чем ожидалось на основе прогноза градиента. Это может привести к более быстрому достижению минимума, но также увеличивает риск его перепрыгнуть при использовании больших шагов.
2) Если кривизна нулевая (нулевое значение второй производной), то градиент точно указывает на изменение функции потерь. В этом случае, градиентный спуск будет эффективно минимизировать функцию потерь, поскольку нет ни ускорения, ни замедления в изменении её значений.
3) Если кривизна положительная (положительное значение второй производной), то функция потерь снижается медленнее, чем ожидалось на основе прогноза градиента. Это говорит о том, что при использовании слишком больших шагов можно не только замедлить процесс сходимости, но и увеличить значение функции потерь, отдаляясь от оптимального решения.

Метод Ньютона

Использует разложение в ряд Тейлора второго порядка для приближения функции f(w) в окрестностях точки :

$f(w + t) \approx f(w) + \nabla f(w)t + \frac{1}{2} \nabla ^ 2 (w) t^2$

Приравняв к нулю $\nabla f(w + t)$ , получим оптимальное решение, то есть направление спуска:

$0 = \frac{d f(w + t)}{dt} = \nabla f(w) + \nabla ^ 2 f(w) \Rightarrow \\ \Rightarrow t = - \frac {\nabla f(w)}{\nabla ^ 2 f(w)} = - H^{-1}(w) \nabla f(w)$

Тогда получим итеративный процесс обновления параметров следующего вида:

$w_{k+1} = w_k + t_k = w_k - \alpha_k H^{-1} (w_k) \nabla f (w_k)$

Если шаг $\alpha_k = 1$ , то это классический метод Ньютона, а при другом размере шага $\alpha_k \in (0, 1)$ получим дэмпированный (damped) метод Ньютона.

Такой подход обеспечивает квадратичную скорость сходимости в окрестностях оптимума за счёт более правильного оценивания кривизны линий уровня функции потерь, что делает его также устойчивым к плохо обусловленным задачам (задачам, в которых небольшое изменение в начальных данных может привести к значительным ошибкам в решении).

Однако метод Ньютона применим только в случае положительно определённой матрицы , иначе говоря, если не все собственные значения гессиана положительны вблизи седловой точки, то обновление параметров может произойти в неверном направлении. Одно из самых простых решений данной проблемы заключается в регуляризации гессиана через прибавление константы $\lambda$ :

$w_{k+1} = w_k + t = w_k - \alpha_k [H^{-1} (w_k) + \lambda I] \nabla f (w_k)$

Помимо этого, у метода Ньютона есть ещё 2 серьёзных недостатка, которые делают его неприменимым к нейронным сетям с большим количеством параметров:

1) Чувствительность к выбору начальной точки. Квадратичная скорость сходимости обеспечивается лишь в близи оптимума, поэтому если начальная точка расположена далеко от него, то алгоритм будет сходиться медленней и может не сойтись вообще. Отсюда следует необходимость в подборе шага $\alpha_k$ как и в случае с градиентным спуском. Частично данную проблему можно решить также с применением различных видов регуляризацией, например, такой как кубической.
2) Высокая требовательность к вычислительным ресурсам. Поскольку число элементов гессиана равно квадрату числа параметров , затраты памяти для их хранения составят , а вычислительная сложность алгоритма будет .

Далее будут рассмотрены альтернативы, использующие некоторые преимущества метода Ньютона, но при этом не требующие таких больших вычислительных затрат.

Метод сопряжённых градиентов

В отличие от метода наискорейшего спуска, в данном методе используется спуск в сопряжённых направлениях. Это означает, что в отличие от метода наискорейшего спуска, где направление спуска всегда совпадает с антиградиентом текущей точки и может привести к зигзагообразным траекториям, в данном случае направления спуска выбираются таким образом, чтобы они были сопряжёнными (ортогональными) относительно гессиана функции потерь, то есть текущее направление выбирается с учётом предыдущего значения:

$d_k = \nabla f(w) + \beta_k d_{t-1}$

Направления d_k и $d_{k-1}$ являются сопряжёнными, если выполняется условие $d_k^T H d_{k-1}=0$ , а коэффициент $\beta_k$ определяет какую часть направления $d_{k-1}$ следует прибавить к текущему направлению поиска.

Найти сопряжённые направления можно, просто получив собственные вектора для $\beta_k$ , однако это вычислительно неэффективно, поэтому для расчёта $\beta_k$ обычно используются следующие методы:

1) Метод Флетчера-Ривса:

$\beta_k = \frac{\nabla f(w_k)^T \nabla f(w_k)}{\nabla f(w_{k-1})^T \nabla f(w_{k-1})}$

2) Метод Полака-Рибьера:

$\beta_k = \frac{(\nabla f(w_k) - \nabla f(w_{k-1}))^T \nabla f(w_k)}{\nabla f(w_{k-1})^T \nabla f(w_{k-1})}$

Отсюда следует, что для квадратичной поверхности сопряжённость направлений гарантирует поиск минимума не более чем за поисков в -мерном пространстве. Если же функция потерь выходит за рамки простой квадратичной формы и становится более сложной, то не всегда возможно обеспечить сохранение минимума вдоль ранее определённых сопряжённых направлений. Поэтому в нелинейном случае сопряжённых градиентов периодически выполняется процедура рестарта, когда алгоритм заново начинает поиск вдоль направления неизменённого градиента, что в итоге позволяет лучше адаптироваться к изменениям в ландшафте функции потерь и более эффективно находить точки минимума.

BFGS (Broyden–Fletcher–Goldfarb–Shanno)

Является одним из самых известных квазиньютоновских методов и основан на аппроксимации обратного гессиана матрицей , которая итеративно уточняется в ходе обновлений низкого ранга. Обозначив приближение гессиана симметричной положительно определённой матрицей B_k размера $d \ \text{x} \ d$ , получим функцию потерь аналогичную в методе Ньютона:

$m(w_{k+1}) \approx f(w_{k+1}) + \nabla f(w_{k+1})t + \frac{1}{2} \nabla ^ 2 (w_{k+1}) t^2$

Основываясь на информации из последнего шага, $B_{k+1}$ должен удовлетворять условию, при котором градиент $m(w_{k+1})$ соответствует градиенту на двух последних итерациях w_k и $w_{k+1}$ :

$\nabla m(w_k - w_{k+1}) = \nabla m(- \alpha_k d_k) = \nabla f(w_{k+1}) - B_{k+1} \alpha_k d_k = \nabla f(w_k)$

Отсюда получаем уравнение секущей, которое ещё известно как квазиньютоновское условие:

$B_{k+1} \alpha_k d_k = \nabla f(w_{k+1}) - \nabla f(w_k)$

Перепишем данное уравнение в более удобной форме, сделав несколько упрощений:

$B_{k+1} s_k = y_k \\ s_k = (w_{k+1} - w_k) = \alpha_k d_k \\ y_k = \nabla f(w_{k+1}) - \nabla f(w_k)$

Согласно данному уравнению, матрица $B_{k+1}$ может преобразовать s_k в y_k только в случае, если они будут удовлетворять условию кривизны:

Когда выполняется данное условие, уравнение секущей всегда имеет решение, а для однозначного определения вводится дополнительное условие, что среди всех симметричных матриц, удовлетворяющих уравнению секущей, $B_{k+1}$ будет расположено наиболее близко к текущей матрице B_k . Другими словами, мы получим задачу вида:

$min \rightarrow \|B - B_k \|$

Которую можно решить достаточно просто с использованием взвешенной нормы Фробениуса:

$\|A \|_W = \|W^{1/2} A W^{1/2} \|_F$

Где $\|\cdot \|_F$ определяется через $\|C \|^2_F = \sum_{i=1}^n \sum_{j=1}^n c^2_{ij}$ , а матрица весов может быть выбрана в качестве любой при выполнении условия W y_k = s_k . Определим как обратный усреднённый гессиан $\overline G_k^{-1}$ , где:

$\overline G_k = \int_{0}^{1} \nabla^2 f(w_k + \alpha_k d_k) d \lambda$

С помощью теоремы Тейлора можно показать, что:

$y_k = \overline G_k \alpha_k d_k = \overline G_k s_k$

При таком выборе матрицы взвешенная норма Фробениуса является безразмерной, что является желательным свойством, поскольку позволяет избежать зависимости от единиц измерения при решении задачи минимизации. Тогда получим решение, которое называется методом DFP (Davidon-Fletcher-Powell):

$B_{k+1} = (I - \frac{y_k s_k^T}{y_k^T s_k}) B_K (I - \frac{s_k y_k^T}{y_k^T s_k}) + \frac{y_k y_k^T}{y_k^T s_k}$

Поскольку в квазиньютоновских методах нас больше интересует обратный гессиан, перепишем данную формулу в более удобном виде. Для этого определим $M=B_k^{-1}$ и воспользуемся обобщением формулы Шермана–Моррисона–Вудбери:

$B^{-1} = A^{-1} - A^{-1} U (I + V^T A^{-1} U) V^T A^{-1}$

В таком случае решение DFP приобретает следующий вид:

$M_{k+1} = M_k - \frac{M_k y_k y_k^T M_k}{y_k^T M_k y_k} + \frac{s_k s_k^T}{y_k^T s_k}$

В этом и заключается фундаментальная идея квазиньютоновского обновления: вместо того, чтобы пересчитывать приближённые гессианы (или обратные гессианы) с нуля на каждой итерации, мы применяем простую модификацию, которая объединяет самую последнюю наблюдаемую информацию о функции потерь с информацией из текущего приближения гессиана.

Однако формулу обновления из алгоритма DFP можно сделать ещё более эффективной, проделав аналогичные процедуры с M_k вместо B_k . В итоге получим обновление BFGS:

$M_{k+1} = (I - \frac{s_k y_k^T}{y_k^T s_k}) M_k (I - \frac{y_k s_k^T}{y_k^T s_k}) + \frac{s_k s_k^T}{y_k^T s_k}$

Мы также можем вывести версию алгоритма BFGS, которая работает с аппроксимацией гессиана B_k , а не M_k , если снова воспользуемся обобщением формулы Шермана–Моррисона–Вудбери:

$B_{k+1} = B_k - \frac{B_k s_k s_k^T B_k}{s_k^T B_k s_k} + \frac{y_k y_k^T}{y_k^T s_k}$

Тогда направление спуска будет выглядеть следующим образом:

$d_k = - M_k \nabla f(w_k)$

А параметры будут обновляться по следующему правилу:

$w_{k+1} = w_k + \alpha_k d_k$

Подобно методу сопряжённых градиентов, в алгоритме BFGS создаётся серия линейных поисков на основе информации о кривизне функции, однако важное отличие заключается в том, что в данном случае достижение точки, максимально приближённой к истинному минимуму вдоль заданного направления, не так критично. Это делает данный алгоритм более эффективным, поскольку не требуется тратить дополнительное время на уточнение результатов каждого линейного поиска, однако для хранения матрицы требуется O(d^2) памяти из-за чего данный алгоритм также малопригоден для моделей с большим количеством параметров.

SR1 (Symmetric Rank-1)

Представляет собой более простое обновление ранга 1, которое сохраняет симметрию матрицы и удовлетворяет уравнению секущей, но при этом исчезают гарантии, что обновлённая матрица B_k (или M_k ) будет положительно определённой. Тем не менее, данный метод может быть полезен, когда стандартные предположения BFGS не выполняются или когда необходимо использовать методы определения доверительной области.

Симметричное обновление ранга 1, которое также известно как формула Бройдена, имеет общий вид:

$B_{k+1} = B_k + \sigma v v^T$

Где $\sigma \in \{-1, 1 \}$ , а также выбирается вместе с $B_{k+1}$ таким образом, чтобы удовлетворять уравнению секущей. Подставляя обновление выше в данное уравнение, получим:

$y_k = B_k s_k + [\sigma v v^T] s_k$

Поскольку член в скобках является скалярным, должно быть кратно y_k - B_k s_k для некоторого скалярного значения $\delta$ :

$y_k - B_k s_k = \sigma \delta^2 [s_k^T (y_k - B_k s_k)] (y_k - B_k s_k)$

Данное уравнение выполняется только в том случае, когда:

$\sigma = sign [s_k^T (y_k - B_k s_k)] \\ \delta = \pm |s_k^T (y_k - B_k s_k)|^{-1/2}$

Следовательно, формула обновления для B_k приобретает следующий вид:

$B_{k+1} = B_k + \frac{(y_k - B_k s_k) (y_k - B_k s_k)^T}{(y_k - B_k s_k)^T s_k}$

Применив обобщение формулы Шермана–Моррисона–Вудбери, получим соответствующее обновление для обратной аппроксимации гессиана M_k :

$M_{k+1} = M_k + \frac{(s_k - M_k y_k) (s_k - M_k y_k)^T}{(s_k - M_k y_k)^T y_k}$

Не смотря на менее стабильную работу в сравнении с DFP и BFGS, преимущество SR1 заключается в создании неопределённых аппроксимаций гессиана, что позволяет получить больше информации о локальной кривизне функции потерь и лучше исследовать пространство для поиска направлений, которые могут привести к лучшему решению.

L-BFGS (Limited-memory BFGS)

Данная модификация использует только последние (обычно от 3 до 20) пар векторов $\{s_i, y_i \}$ и начальное приближение M_0 для аппроксимации M_k , что позволяет иметь линейные затраты O(md) не только на память, но и на итерацию. Чтобы лучше разобраться как это происходит, сначала перепишем формулы обновления BFGS в упрощённом виде:

$w_{k+1} = w_k + \alpha_k M_k \nabla f (w_k) \\ M_{k+1} = V_k^T M_k V_k + \rho_k s_k s_k^T \\ V_k = I - \rho_k y_k s_k^T, \ \rho_k = \frac{1}{y_k^T s_k}$

Тогда приближение M_k можно представить в следующем виде:

$\begin{align} M_k &= (V_{k-1}^{T} ... V_{k-m}^{T}) M_0 (V_{k-m} ... V_{k-1}) + \\ &+ \rho_{k-m} (V_{k-1}^T ... V_{k-m+1}^{T}) s_{k-m} s_{k-m}^T (V_{k-m+1} ... V_{k-1}) + \\ &+ \ ... \\ &+ \rho_{k-1} s_{k-1} s_{k-1}^T \end{align}$

Из этого выражения можно вывести рекурсивный алгоритм для эффективного вычисления произведения $M_k \nabla f(w_k)$ , который называется L-BFGS two-loop recursion:

$\begin{align*} & q \leftarrow \nabla f(w_k) \\ & \textbf{for } i \ = \ k - 1, \ k - 2, ..., \ k - m \\ & \hspace{1.3cm} \alpha_i \leftarrow \rho_i s_i^T q \\ & \hspace{1.3cm} q \leftarrow q - \alpha_i y_i \\ \\ & r \leftarrow M_0 q \\ & \textbf{for } i \ = \ k - m, \ k - m + 1, ..., \ k - 1 \\ & \hspace{1.3cm} \beta \leftarrow \rho_i y_i^T r \\ & \hspace{1.3cm} r \leftarrow r + s_i (\alpha_i - \beta) \\ \\ & M_k \nabla f(w_k) = r \end{align*}$

Преимущество данной рекурсии заключается в том, что умножение на исходную матрицу M_0 изолировано от остальных вычислений, что позволяет свободно выбирать эту матрицу и изменять её между итерациями. Методом выбора M_0 , показавшим себя хорошо на практике, является установка $M_0 = \gamma_k I$ , где:

$\gamma_k = \frac{s_{k-1}^T y_{k-1}}{y_{k-1}^T y_{k-1}}$

В данном случае $\gamma_k$ позволяет произвести оценку размера истинного гессиана по последнему направлению поиска. Это гарантирует хорошую масштабируемость направления поиска d_k , в результате чего на большинстве итераций $\alpha_k = 1$ . Теперь можно сделать формальное определение алгоритма L-BFGS следующим образом:

$\begin{align*} & \text{Choose starting point } w_0 \\ & m > 0, \ k \leftarrow 0 \\ & \textbf{repeat until convergence} \\ & \hspace{1.3cm} M_0 = \gamma_k I \\ & \hspace{1.3cm} d_k = - M_k \nabla f(w_k) \hspace{0.4cm} \text{from algorithm above} \\ \\ & \hspace{1.3cm} w_{k+1} = w_k + \alpha_k d_k, \hspace{0.3cm} \text{where} \ \alpha_k \ \text{satisfies the Wolfe's conditions} \\ & \hspace{1.3cm} r \leftarrow r + s_i (\alpha_i - \beta) \\ & \hspace{1.3cm} \textbf{if } k > m: \\ & \hspace{2.6cm} \text{Discard the vector pair } \{s_{k-m}, y_{k-m} \} \ \text{from storage} \\ & \hspace{1.3cm} s_k \leftarrow w_{k+1} - w_k \\ & \hspace{1.3cm} y_k = \nabla f(w_{k+1}) - \nabla f(w_k) \\ & \hspace{1.3cm} k \leftarrow k + 1 \end{align*}$

Shampoo

Вышеописанные методы всё ещё уступают в скорости и потреблении памяти обычным градиентным методам, особенно при обучении больших глубоких нейросетей. Отчасти это связано с тем, что они требуют точного, а не стохастического вычисления градиентов. В идеале нам хотелось бы иметь такой алгоритм, который совмещает в себе точность алгоритмов второго порядка и скорость алгоритмов первого порядка. В попытке приблизиться к этой цели появился алгоритм Shampoo, который применяется для стохастической оптимизации не только в матричных, но и в тензорных пространствах. Рассмотрим его основную идею.

Начнём с того, что данный алгоритм относится к так называемым предобусловленным методам (в контексте машинного обучения предобуславливание можно рассматривать как процесс изменения условий задачи таким образом, чтобы алгоритмы могли работать с ней более точно и эффективно). Вместо использования обратного гессиана явным образом в Shampoo используется произведение градиентов для формирования предобусловленной матрицы, которая выполняет роль, схожую с гессианом, храня информацию о кривизне поверхности функции потерь. По сути, данный алгоритм представляет собой многомерный аналог AdaGrad, а для матричного случая его принцип работы выглядит следующим образом:

На каждой итерации из матриц L_t размера x и R_t размера x , содержащих информацию о втором моменте из накопленных градиентов, формируются две предобусловленные матрицы, которые потом умножаются на градиентную матрицу слева направо соответственно. Каждая из предобусловленных матриц работает только с одним измерением градиента, сокращаясь по остальным, что позволяет неплохо ускорить процесс обучения. Объём пространства, который Shampoo использует в данном случае, составляет O(m^2 + n^2) вместо O(m^2 n^2) в сравнении с полноматричными методами. Более того, поскольку предобуславливание включает в себя обращение матрицы (и чаще всего спектральное разложение), объём вычислений для построения левого и правого предобуславливателей также сокращается с O(m^3 n^3) до O(m^3 + n^3) .

Однако shampoo всё ещё требует больших вычислительных затрат, что особенно выделяется при работе с полносвязными и эмбэддинг слоями. Основная сложность возникает при вычислении p-х корней матриц и $R_t^{-1/4}$ , которые были реализованы с использованием спектрального разложения (SVD), что может занимать много времени. Поэтому для ускоренной работы Shampoo было разработано несколько улучшений:

1) Вычисление предобуславливателей раз в несколько сотен шагов. Поскольку в большинстве случаев структура функции потерь не изменяется существенно с каждым шагом, то и особого эффекта на снижение точности произойти не должно.
2) Использование эффективных итерационных методов для вычисления p-х корней вместо SVD, например, применение модификаций на основе метода Шура-Ньютона.
3) Использование одного из предобуславливателей $L_t^{-1/2p}$ или $R_t^{-1/2q}$ , где .
4) Применение параллельного обучения на GPU.

Стоит отметить, что данные улучшения без проблем можно использовать для Shampoo в его общей форме, который применим к тензорам произвольной размерности. В фреймворках глубокого обучения реализована именно тензорная форма данного алгоритма, а его принцип работы выглядит следующим образом:

Проследив историю развития методов второго порядка, становится очевидно, что с появлением таких алгоритмов как Shampoo, их применение становится всё более привлекательным с практической точки зрения. На мой скромный взгляд, в будущем их популярность будет расти и они будут играть значимую роль не только в глубоком обучении, но и в других областях инженерии и науки.

Дополнительные источники

Статьи:

«Adam: A Method for Stochastic Optimization», Diederik P. Kingma, Jimmy Ba;
«Decoupled Weight Decay Regularization», Ilya Loshchilov, Frank Hutter;
«Incorporating Nesterov Momentum into Adam», Timothy Dozat;
«Adaptive Methods for Nonconvex Optimization», Manzil Zaheer, Sashank J. Reddi, Devendra Sachan, Satyen Kale, Sanjiv Kumar;
«Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models», Xingyu Xie, Pan Zhou, Huan Li, Zhouchen Lin, Shuicheng Yan;
«On the Convergence of Adam and Beyond», Sashank J. Reddi, Satyen Kale, Sanjiv Kumar;
«On the Convergence Proof of AMSGrad and a New Version», Tran Thi Phuong, Le Trieu Phong;
«SGDR: Stochastic Gradient Descent with Warm Restarts», Ilya Loshchilov, Frank Hutter;
«Optimization for deep learning: theory and algorithms», Ruoyu Sun;
«Large Batch Training of Convolutional Networks», Yang You, Igor Gitman, Boris Ginsburg;
«Large Batch Optimization for Deep Learning: Training BERT in 76 minutes», Yang You, Jing Li, Sashank Reddi, Jonathan Hseu, Sanjiv Kumar, Srinadh Bhojanapalli, Xiaodan Song, James Demmel, Kurt Keutzer, Cho-Jui Hsieh;
«Proximal Algorithms», Neal Parikh, Stephen Boyd;
«A Fast Iterative Shrinkage-Thresholding Algorithm», for Linear Inverse Problems, Amir Beck, and Marc Teboulle;
«Improving Fast Iterative Shrinkage-Thresholding Algorithm: Faster, Smarter and Greedier», Jingwei Liang, Tao Luo, Carola-Bibiane Schönlieb;
«Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice», Jeffrey Pennington, Samuel S. Schoenholz, Surya Ganguli;
«Numerical Optimization», Jorge Nocedal, Stephen J. Wright;
«A regularized limited-memory BFGS method for unconstrained minimization problems», Shinji SUGIMOTO and Nobuo YAMASHITA;
«Shampoo: Preconditioned Stochastic Tensor Optimization», Vineet Gupta, Tomer Koren, Yoram Singer;
«Scalable Second Order Optimization for Deep Learning», Rohan Anil, Vineet Gupta, Tomer Koren, Kevin Regan, Yoram Singer;
«A Schur-Newton method for the matrix p'th root and its inverse», C.-H. Guo and N. J. Higham.

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10 мая 2024 г.

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Firstly, the importance of verifying the authenticity of online communications, especially those about financial matters. Secondly, the potential for recovery exists even in cases where it seems hopeless, thanks to innovative services like TechY Force Cyber Retrieval. Lastly, the cryptocurrency community needs to be more aware of these risks and the available solutions to combat them. My experience serves as a warning to others to be cautious of online impersonators and never to underestimate the potential for recovery in the face of theft. It also highlights the critical role that professional retrieval services can play in securing your digital assets. In conclusion, while the cryptocurrency space offers unparalleled opportunities, it also presents unique challenges, and being informed and vigilant is key to navigating this landscape safely. W.h.a.t.s.A.p.p.. +.15.6.1.7.2.6.3.6.9.7. M.a.i.l T.e.c.h.y.f.o.r.c.e.c.y.b.e.r.r.e.t.r.i.e.v.a.l.@.c.o.n.s.u.l.t.a.n.t.c.o.m. T.e.l.e.g.r.a.m +.15.6.1.7.2.6.3.6.9.7
15.11.25 12:47 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $184,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez a crypto recovery agent. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
15.11.25 14:39 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.11.25 14:39 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.11.25 15:31 MATT PHILLIP

I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $184,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez a crypto recovery agent. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.
15.11.25 15:52 [email protected]

A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.
16.11.25 14:43 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.11.25 14:44 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
16.11.25 20:38 [email protected]

A scam cost me $72,000 in USDT. It shook me up. USDT is a stablecoin linked to the dollar. Its value stays even. I believed I found a safe path to build my wealth. At the start, all seemed fine. My account grew to $120,000 in profits. But when I tried to withdraw, the site locked me out. No way to get in. No money left. Fear took over. I felt stuck and alone. These frauds hit crypto investors often. They lure with fast riches. Then they steal your cash and disappear. Billions vanish each year from such schemes. I looked for aid in every spot. Online boards. Help chats. None helped. Then a buddy offered support. He had dealt with the same issue once. He mentioned Sylvester Bryant. My friend praised his expertise. I contacted him at once. His email is [email protected]. Sylvester Bryant changed everything. He heard my tale with no blame. His crew jumped in quickly. They checked all scam details first. One by one, they followed my lost USDT trail. They used software to track the blockchain. That is the open log of coin transfers. It reveals fund paths. Scammers try to cover their steps. Bryant's team went far. They reached out to related platforms and services. Each day brought progress. No easy ways. They shared updates with me always. Each message and talk stayed open and true. Finally, they got back every dollar. My $52,000 returned whole. The effort needed drive and resolve. Bryant's fairness shone through. He added no secret costs. Only fair pay for the job. My worry faded. I relaxed once more. Nights grew calm. My faith in recovery grew strong. If fraud stole your funds, move fast. Contact Sylvester Bryant. He treats such cases with skill. Email at [email protected]. Or use WhatsApp at +1 512 577 7957 or +44 7428 662701. Do not delay. Reclaim what is yours.
17.11.25 03:24 johnny231

INFO@THEBARRYCYBERINVESTIGATIONSDOTCOM is one of the best cyber hackers that i have actually met and had an encounter with, i was suspecting my partner was cheating on me for some time now but i was not sure of my assumptions so i had to contact BARRY CYBER INVESTIGATIONS to help me out with my suspicion. During the cause of their investigation they intercepted his text messages, social media(facebook, twittwer, snapchat whatsapp, instagram),also call logs as well as pictures and videos(deleted files also) they found out my spouse was cheating on me for over 3 years and was already even sending nudes out as well as money to anonymous wallets,so i deciced to file for a divorce and then when i did that i came to the understanding that most of the cryptocurrency we had invested in forex by him was already gone. BARRY CYBER INVESTIGATIONS helped me out through out the cause of my divorce with my spouse they also helped me in retrieving some of the cryptocurrency back, as if that was not enough i decided to introduce them to another of my friend who had lost her most of her savings on a bad crytpo investment and as a result of that it affected her credit score, BARRY CYBER INVESTIGATIONS helped her recover some of the funds back and helped her build her credit score, i have never seen anything like this in my life and to top it off they are very professional and they have intergrity to it you can contact them also on their whatsapp +1814-488-3301. for any hacking or pi jobs you can contact them and i assure you nothing but the best out of the job
17.11.25 11:26 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.11.25 11:27 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19.11.25 01:56 VERONICAFREDDIE809

Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I got connected to the best female expert AGENT Jasmine Lopez,,( [email protected] ) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.
19.11.25 08:11 JuneWatkins

I’m June Watkins from California. I never thought I’d lose my life savings in Bitcoin. One wrong click, a fake wallet update, and $187,000 vanished in seconds. I cried for days, felt stupid, ashamed, and completely hopeless. But God wouldn’t let me stay silent or defeated. A friend sent me a simple message: “Contact Mbcoin Recovery Group, they specialize in this.” I was skeptical (there are so many scammers), but something in my spirit said “try.” I reached out to Mbcoin Recovery Group through their official site and within minutes their team responded with kindness and clarity. They walked with me step by step, and stayed in constant contact. Three days later, I watched in tears as every single Bitcoin returned to my wallet, 100% recovered. God turned my mess into a message and my shame into a testimony! If you’ve lost crypto and feel it’s gone forever, don’t give up. I’m living proof that recovery is possible. Thank you, Mbcoin Recovery Group, and thank You, Jesus, for never leaving me stranded. contact: (https://mbcoinrecoverygrou.wixsite.com/mb-coin-recovery) (Email: [email protected]) (Call Number: +1 346 954-1564)
19.11.25 08:26 elizabethmadison

My name is Elizabeth Madison currently living in New York. There was a time I felt completely broken. I had trusted a fraudulent bitcoin investment organization, who turned out to be a fraudster. I sent money, believing their sweet words and promises on the interest rate I will get back in return, only to realize later that I’ve been scammed. On the day of withdrawal there was no money in my account. The pain hit deep. I couldn’t sleep, I kept asking myself how I could have been so careless, meanwhile my mom was battling with a stroke and the expenses were too much. For days, I cried and blamed myself. The betrayal, the disappointment and my mom's health issues all of this stress made me want to give up on life. But one day, I decided that sitting in pain wouldn’t solve anything. I picked myself up and chose to fight for what I lost then I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how he helped people recover their funds from online fraud. I emailed all the transactions and paperwork I had with the fraudulent organization and they helped me recover all my lost money in just five days. If you have ever fallen victim to scammers, contact GREAT WHIP RECOVERY CYBER SERVICES to help you recover every penny you have lost. (Text +1(406)2729101) (Email [email protected])
19.11.25 08:27 elizabethmadison

My name is Elizabeth Madison currently living in New York. There was a time I felt completely broken. I had trusted a fraudulent bitcoin investment organization, who turned out to be a fraudster. I sent money, believing their sweet words and promises on the interest rate I will get back in return, only to realize later that I’ve been scammed. On the day of withdrawal there was no money in my account. The pain hit deep. I couldn’t sleep, I kept asking myself how I could have been so careless, meanwhile my mom was battling with a stroke and the expenses were too much. For days, I cried and blamed myself. The betrayal, the disappointment and my mom's health issues all of this stress made me want to give up on life. But one day, I decided that sitting in pain wouldn’t solve anything. I picked myself up and chose to fight for what I lost then I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how he helped people recover their funds from online fraud. I emailed all the transactions and paperwork I had with the fraudulent organization and they helped me recover all my lost money in just five days. If you have ever fallen victim to scammers, contact GREAT WHIP RECOVERY CYBER SERVICES to help you recover every penny you have lost. (Text +1(406)2729101) Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site (Email [email protected])
19.11.25 16:30 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
19.11.25 16:30 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
20.11.25 15:55 mariotuttle94

HIRE THE BEST HACKER ONLINE FOR CRYPTO BITCOIN SCAM RECOVERY / iFORCE HACKER RECOVERY After a security breach, my husband lost $133,000 in Bitcoin. We sought help from a professional cybersecurity team iForce Hacker Recovery they guided us through each step of the recovery process. Their expertise allowed them to trace the compromised funds and help us understand how the breach occurred. The experience brought us clarity, restored a sense of stability, and reminded us of the importance of strong digital asset and security practices. Website: ht tps:/ /iforcehackers. c om WhatsApp: +1 240-803-3706 Email: iforcehk @ consultant. c om
21.11.25 10:56 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
21.11.25 10:56 marcushenderson624

Bitcoin Recovery Testimonial After falling victim to a cryptocurrency scam group, I lost $354,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to recover my stolen funds and I came across a lot of testimonials online about Capital Crypto Recovery, an agent who helps in recovery of lost bitcoin funds, I contacted Capital Crypto Recover Service, and with their expertise, they successfully traced and recovered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and recovery protocols. They are trusted and very reliable with a 100% successful rate record Recovery bitcoin, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypto. Contact: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.11.25 04:41 VERONICAFREDDIE809

Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I got connected to the best female expert AGENT Jasmine Lopez,,( [email protected] ) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.
22.11.25 22:04 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.11.25 22:04 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
22.11.25 22:05 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.11.25 03:34 Matt Kegan

SolidBlock Forensics are absolutely the best Crypto forensics team, they're swift to action and accurate
23.11.25 09:54 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.11.25 18:01 mosbygerry

I recently had the opportunity to work with a skilled programmer who specialized in recovering crypto assets, and the results were nothing short of impressive. The experience not only helped me regain control of my investments but also provided valuable insight into the intricacies of cryptocurrency technology and cybersecurity. The journey began when I attempted to withdraw $183,000 from an investment firm, only to encounter a series of challenges that made it impossible for me to access my funds. Despite seeking assistance from individuals claiming to be Bitcoin miners, I was unable to recover my investments. The situation was further complicated by the fact that all my deposits were made using various cryptocurrencies that are difficult to trace. However, I persisted in my pursuit of recovery, driven by the determination to reclaim my losses. It was during this time that I discovered TechY Force Cyber Retrieval, a team of experts with a proven track record of successfully recovering crypto assets. With their assistance, I was finally able to recover my investments, and in doing so, gained a deeper understanding of the complex mechanisms that underpin cryptocurrency transactions. The experience taught me that with the right expertise and guidance, even the most seemingly insurmountable challenges can be overcome. I feel a sense of obligation to share my positive experience with others who may have fallen victim to cryptocurrency scams or are struggling to recover their investments. If you find yourself in a similar situation, I highly recommend seeking the assistance of a trustworthy and skilled programmer, such as those at TechY Force Cyber Retrieval. WhatsApp (+1561726 3697) or (+1561726 3697). Their expertise and dedication to helping individuals recover their crypto assets are truly commendable, and I have no hesitation in endorsing their services to anyone in need. By sharing my story, I hope to provide a beacon of hope for those who may have lost faith in their ability to recover their investments and to emphasize the importance of seeking professional help when navigating the complex world of cryptocurrency.
24.11.25 11:43 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.11.25 11:43 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
24.11.25 16:34 Mundo

I wired 120k in crypto to the wrong wallet. One dumb slip-up, and poof gone. That hit me hard. Lost everything I had built up. Crypto moves on the blockchain. It's like a public record book. Once you send, that's it. No take-backs. Banks can fix wire mistakes. Not here. Transfers stick forever. a buddy tipped me off right away. Meet Sylvester Bryant. Guy's a pro at pulling back lost crypto. Handles cases others can't touch, he spots scammer moves cold. Follows money down secret paths. Mixers. Fake trades. Hidden swaps. You name it, he tracks it. this happens to tons of folks. Fat-finger a key. Miss one digit in the address. Boom. Billions vanish like that each year. I panicked. Figured my stash was toast for good. Bryant flipped the script. He jumps on hard jobs quick. Digs deep. Cracks the trail. Got my funds back safe. You're in the same boat? Don't sit there. Hit him up today. Email [email protected]. WhatsApp +1 512 577 7957. Or +44 7428 662701. Time's your enemy here. Scammers spend fast. Chains churn non-stop. Move now. Grab your cash back home.
25.11.25 05:15 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
25.11.25 13:31 mickaelroques52

CRYPTO TRACING AND INVESTIGATION EXPERT: HOW TO RECOVER STOLEN CRYPTO_HIRE RAPID DIGITAL RECOVERY
25.11.25 13:31 mickaelroques52

I’ve always considered myself a careful person when it comes to money, but even the most cautious people can be fooled. A few months ago, I invested some of my Bitcoin into what I believed was a legitimate platform. Everything seemed right, professional website, live chat support and even convincing testimonials. I thought I had done my homework. But when I tried to withdraw my funds, everything fell apart. My account was blocked, the so-called support team disappeared and I realized I had been scammed. The shock was overwhelming. I couldn’t believe I had fallen for it. That Bitcoin represented years of savings and sacrifices and it felt like everything had been stolen from me in seconds. I didn’t sleep for days and I was angry at myself for trusting the wrong people. In my desperation, I started searching for solutions and came across Rapid Digital Recovery. At first, I thought it was just another promise that would lead nowhere. But after speaking with them, I realized this was different. They were professional, clear and understanding. They explained exactly how they track stolen funds through blockchain forensics and what steps would be taken in my case. I gave them all the transaction details and they immediately got to work. What impressed me most was their transparency, they gave me updates regularly and kept me involved in the process. After weeks of investigation, they achieved what I thought was impossible: they recovered my stolen Bitcoin and safely returned it to my wallet. The relief I felt that day is indescribable. I went from feeling hopeless and broken to feeling like I had been given a second chance. I am forever grateful to Rapid Digital Recovery. They didn’t just recover my money, they restored my peace of mind. If you’re reading this because you’ve been scammed, please know you’re not alone and that recovery is possible. I’m living proof that with the right help, you can get your funds back... Contact Info Below WhatSapp: + 1 414 807 1485 Email: rapiddigitalrecovery (@) execs. com Telegram: + 1 680 5881 631
26.11.25 18:18 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.11.25 18:20 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
26.11.25 19:13 James Robert

I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have access to my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site
26.11.25 19:13 James Robert

I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have access to my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site
26.11.25 19:13 James Robert

I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have access to my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site
27.11.25 10:56 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.11.25 10:56 harristhomas7376

"In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.11.25 20:04 deborah113

Scammed Crypto Asset Recovery Solution Hire iFORCE HACKER RECOVERY When I traded online, I lost both my investment money and the anticipated gains. Before permitting any withdrawals, the site kept requesting more money, and soon I recognized I had been duped. It was really hard to deal with the loss after their customer service ceased responding. I saw a Facebook testimonial about how iForce Hacker Recovery assisted a victim of fraud in getting back the bitcoin she had transferred to con artists. I contacted iForce Hacker Recovery, submitted all relevant case paperwork, and meticulously followed the guidelines. I'm relieved that I was eventually able to get my money back, including the gains that were initially displayed on my account. I'm sharing my story to let others who have been conned know that you can recover your money. WhatsApp: +1 240-803-3706 Email: iforcehk @ consultant. c om Website: ht tps:/ /iforcehackers. c om
27.11.25 23:48 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
27.11.25 23:48 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.11.25 00:08 VERONICAFREDDIE809

Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I found someone Agent Jasmine Lopez ([email protected])WhatsApp at +44 736-644-5035. ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it.
28.11.25 11:15 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.11.25 11:15 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
28.11.25 11:43 elizabethmadison

My name is Elizabeth Madison currently living in New York. There was a time I felt completely broken. I had trusted a fraudulent bitcoin investment organization, who turned out to be a fraudster. I sent money, believing their sweet words and promises on the interest rate I will get back in return, only to realize later that I’ve been scammed. On the day of withdrawal there was no money in my account. The pain hit deep. I couldn’t sleep, I kept asking myself how I could have been so careless, meanwhile my mom was battling with a stroke and the expenses were too much. For days, I cried and blamed myself. The betrayal, the disappointment and my mom's health issues all of this stress made me want to give up on life. But one day, I decided that sitting in pain wouldn’t solve anything. I picked myself up and chose to fight for what I lost then I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how he helped people recover their funds from online fraud. I emailed all the transactions and paperwork I had with the fraudulent organization and they helped me recover all my lost money in just five days. If you have ever fallen victim to scammers, contact GREAT WHIP RECOVERY CYBER SERVICES to help you recover every penny you have lost. (Text +1(406)2729101) Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site (Email [email protected])
28.11.25 11:43 elizabethmadison

My name is Elizabeth Madison currently living in New York. There was a time I felt completely broken. I had trusted a fraudulent bitcoin investment organization, who turned out to be a fraudster. I sent money, believing their sweet words and promises on the interest rate I will get back in return, only to realize later that I’ve been scammed. On the day of withdrawal there was no money in my account. The pain hit deep. I couldn’t sleep, I kept asking myself how I could have been so careless, meanwhile my mom was battling with a stroke and the expenses were too much. For days, I cried and blamed myself. The betrayal, the disappointment and my mom's health issues all of this stress made me want to give up on life. But one day, I decided that sitting in pain wouldn’t solve anything. I picked myself up and chose to fight for what I lost then I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how he helped people recover their funds from online fraud. I emailed all the transactions and paperwork I had with the fraudulent organization and they helped me recover all my lost money in just five days. If you have ever fallen victim to scammers, contact GREAT WHIP RECOVERY CYBER SERVICES to help you recover every penny you have lost. (Text +1(406)2729101) Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site (Email [email protected])
29.11.25 12:35 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.11.25 12:35 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
29.11.25 12:35 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
30.11.25 20:37 robertalfred175

CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.12.25 12:27 Thomas Muller

YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK WHATSAPP +1(208)713-0697 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with, website https://greatwhiprecoveryc.wixsite.com/greatwhip-site text +1(406)2729101 email [email protected]
01.12.25 12:27 Thomas Muller

YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK WHATSAPP +1(208)713-0697 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with, website https://greatwhiprecoveryc.wixsite.com/greatwhip-site text +1(406)2729101 email [email protected]
01.12.25 23:45 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.12.25 23:45 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
02.12.25 02:21 donald121

In 2025 alone, hackers stole over $1.5 billion in digital assets from users worldwide. That's a wake-up call for anyone holding crypto. Theft hits hard because once funds move, they're tough to get back. Common ways it happens include phishing emails that trick you into giving up keys, big exchange breaches, or malware sneaking into your wallet. Marie guide walks you through steps to recover stolen cryptocurrency. You'll learn quick actions to stop more loss, how to trace funds, and ways to fight back legally. Plus, tips to avoid this mess next time. reach her (infocyberrecoveryinc@gmail com and whatsapp:+1 7127594675)
02.12.25 15:05 Matt Kegan

Reach out to SolidBlock Forensics if you want to get back your coins from fake crypto investment or your wallet was compromised and all your coins gone. SolidBlock Forensics provide deep ethical analysis and investigation that enables them to trace these schemes, and recover all your funds. Their services are professional and reliable.
03.12.25 09:22 tyrelldavis1

I still recall the day I fell victim to an online scam, losing a substantial amount of money to a cunning fraudster. The feeling of helplessness and despair that followed was overwhelming, and I thought I had lost all hope of ever recovering my stolen funds. However, after months of searching for a solution, I stumbled upon a beacon of hope - GRAYWARE TECH SERVICE, a highly reputable and exceptionally skilled investigative and recovery firm. Their team of expert cybersecurity professionals specializes in tracking and retrieving money lost to internet fraud, and I was impressed by their unwavering dedication to helping victims like me. With their extensive knowledge and cutting-edge technology, they were able to navigate the complex world of online finance and identify the culprits behind my loss. What struck me most about GRAYWARE TECH SERVICE was their unparalleled expertise and exceptional customer service. They took the time to understand my situation, provided me with regular updates, and kept me informed throughout the entire recovery process. Their transparency and professionalism were truly reassuring, and I felt confident that I had finally found a reliable partner to help me recover my stolen money. Thanks to GRAYWARE TECH SERVICE, I was able to recover a significant portion of my lost funds, and I am forever grateful for their assistance. Their success in retrieving my money not only restored my financial stability but also restored my faith in the ability of authorities to combat online fraud. If you have fallen victim to internet scams, I highly recommend reaching out to GRAYWARE TECH SERVICE - their expertise and dedication to recovering stolen funds are unparalleled, and they may be your only hope for retrieving what is rightfully yours. You can reach them on whatsapp+18582759508 web at ( https://graywaretechservice.com/ ) also on Mail: ([email protected]
03.12.25 21:01 VERONICAFREDDIE809

Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I found someone Agent Jasmine Lopez ([email protected]) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.
03.12.25 22:17 Tonerdomark

I lost $300,000 in USDC to a phishing scam. Scammers tricked me with a fake wallet link. They drained my account fast. I felt hopeless. No way to get it back. Then Sylvester stepped in. His skills traced the funds. He recovered every bit. USDC is a stablecoin tied to the dollar. Phishing scams hit hard in crypto. They fool you with urgent emails or sites. Billions vanish each year this way. Sylvester knows blockchain tracks. He used tools to follow the trail. I got my money back in weeks. Skills like his turn loss to win. Don't wait if scammed. Contact Mr. Sylvester now. Email: yt7cracker@gmail. com. WhatsApp only: + 1 512 577 7957 or + 44 7428 662701. He helped me. He can help you.
04.12.25 01:37 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.12.25 01:37 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.12.25 04:35 Tonerdomark

I lost $300,000 in USDC to a phishing scam. Scammers tricked me with a fake wallet link. They drained my account fast. I felt hopeless. No way to get it back. Then Sylvester stepped in. His skills traced the funds. He recovered every bit. USDC is a stablecoin tied to the dollar. Phishing scams hit hard in crypto. They fool you with urgent emails or sites. Billions vanish each year this way. Sylvester knows blockchain tracks. He used tools to follow the trail. I got my money back in weeks. Skills like his turn loss to win. Don't wait if scammed. Contact Mr. Sylvester now. Email: [email protected]. WhatsApp only: + 1 512 577 7957 or + 44 7428 662701. He helped me. He can help you.
04.12.25 10:32 Tonerdomark

I lost $300,000 in USDC to a phishing scam. Scammers tricked me with a fake wallet link. They drained my account fast. I felt hopeless. No way to get it back. Then Sylvester stepped in. His skills traced the funds. He recovered every bit. USDC is a stablecoin tied to the dollar. Phishing scams hit hard in crypto. They fool you with urgent emails or sites. Billions vanish each year this way. Sylvester knows blockchain tracks. He used tools to follow the trail. I got my money back in weeks. Skills like his turn loss to win. Don't wait if scammed. Contact Mr. Sylvester now. Email: [email protected]. WhatsApp only: + 1 512 577 7957 or + 44 7428 662701. He helped me. He can help you.
04.12.25 18:25 smithhazael

Hire Proficient Expert Consultant For any form of lost crypto "A man in Indonesia tragically took his own life after losing his family's savings to a scam. The shame and blame were too much to bear. It's heartbreaking to think he might still be alive if he knew help existed. "PROFICIENT EXPERT CONSULTANTS, I worked alongside PROFICIENT EXPERT CONSULTANTS when I lost my funds to an investment platform on Telegram. PROFICIENT EXPERT CONSULTANTS did a praiseworthy job, tracked and successfully recovered all my lost funds a total of $770,000 within 48hours after contacting them, with their verse experience in recovery issues and top tier skills they were able to transfer back all my funds into my account, to top it up I had full access to my account and immediately converted it to cash, they handled my case with professionalism and empathy and successfully recovered all my lost funds, with so many good reviews about PROFICIENT EXPERT CONSULTANTS, I’m glad I followed my instincts after reading all the reviews and I was able to recovery everything I thought I had lost, don’t commit suicide if in any case you are caught in the same situation, contact: Proficientexpert@consultant. com Telegram: @ PROFICIENTEXPERT, the reliable experts in recovery.
04.12.25 21:45 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
04.12.25 21:45 elizabethrush89

God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
05.12.25 08:35 into11

The digital world of cryptocurrency offers big chances, but it also hides tricky scams. Losing your crypto to fraud feels awful. It can leave you feeling lost and violated. This guide tells you what to do right away if a crypto scam has hit you. These steps can help you get funds back or stop more trouble. Knowing what to do fast can change everything,reach marie ([email protected] and whatsapp:+1 7127594675)
05.12.25 08:48 Tonerdomark

SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]
06.12.25 01:44 Tonerdomark

SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]
06.12.25 01:48 Tonerdomark

SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]
06.12.25 10:36 Thomas Muller

YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK WHATSAPP +1(208)713-0697 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with, website https://greatwhiprecoveryc.wixsite.com/greatwhip-site text +1(406)2729101 email [email protected]
06.12.25 10:36 Thomas Muller

YOU CAN REACH OUT TO GREAT WHIP RECOVERY CYBER SERVICES FOR HELP TO RECOVER YOUR STOLEN BTC OR ETH BACK WHATSAPP +1(208)713-0697 I once fell victim to online investment scheme that cost me a devastating €254,000. I’m Thomas Muller from Berlin, Germany. The person I trusted turned out to be a fraud, and the moment I realized I’d been deceived, my entire world stopped. I immediately began searching for legitimate ways to recover my funds and hold the scammer accountable. During my search, I came across several testimonies of how Great Whip Recovery Cyber Services helped some people recover money they lost to cyber fraud, I contacted Great Whip Recovery Cyber Service team and provided all the evidence I had. Within about 36 hours, the experts traced the digital trail left by the fraudster, the individual was eventually tracked down and I recovered all my money back. You can contact them with, website https://greatwhiprecoveryc.wixsite.com/greatwhip-site text +1(406)2729101 email [email protected]
06.12.25 10:39 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
06.12.25 10:42 michaeldavenport238

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR Call Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
07.12.25 08:43 Tonerdomark

SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]
08.12.25 02:17 liam

I recently fell a victim of cryptocurrency investment and mining scam, I lost almost all my life savings to BTC scammers. I almost gave up because the amount of crypto I lost was too much. So I spoke to a friend who told me about ANTHONYDAVIESTECH company. I Contacted them through their email and i provided them with the necessary information they requested from me and they told me to be patient and wait to see the outcome of their job. I was shocked after two days my Bitcoin was returned to my Wallet. All thanks to them for their genius work. I Contacted them via Email: anthonydaviestech @ gmail . com all thanks to my friend who saved my life
08.12.25 09:07 Tonerdomark

SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = [email protected]
09.12.25 00:18 swanky

For a long time, I had heard tales of individuals striking it rich through cryptocurrency investments, but I had little knowledge of how the system operated. The potential for financial gain piqued my interest, and I decided to dive in and invest. To help me navigate this complex landscape, I joined a group of online traders who promised to guide me through the investment process. Their confidence and expertise made me feel reassured about my decision.After spending some time learning from them and observing their trading strategies, I felt compelled to invest a substantial amount of money to which i lost, now in search of recovering my funds i got referred to anthonydavies on telegram a funds recovery specialist with his team help i was able to get back $300000 of my usdc back. you can reach him via anthonydaviestech AT gmail dot com
09.12.25 01:01 Tonerdomark

SYLVESTER BRYANT WAS A PROFESSIONAL/ RELIABLE HACKER AND HIGHLY RECOMMENDED I’m very excited to speak about him as a Bitcoin Recovery agent, this cyber security company was able to assist me in recovering my stolen funds in cryptocurrency. I’m truly amazed by their excellent service and professional work. I never thought I could get back my funds until I approached them with my problems and provided all the necessary information. It took them time to recover my funds and I was amazed. Without any doubt, I highly recommend Sylvester for your BITCOIN, USDC, USDT, ETH Recovery, for all Cryptocurrency recovery, digital funds recovery, hacking Related issues, contact Sylvester Bryant professional services waapp only= +1 512 577 7957 or + 44 7428 662701 EMAIL = Yt7CRACKER@gmail. com
09.12.25 05:44 swanky

For a long time, I had heard tales of individuals striking it rich through cryptocurrency investments, but I had little knowledge of how the system operated. The potential for financial gain piqued my interest, and I decided to dive in and invest. To help me navigate this complex landscape, I joined a group of online traders who promised to guide me through the investment process. Their confidence and expertise made me feel reassured about my decision.After spending some time learning from them and observing their trading strategies, I felt compelled to invest a substantial amount of money to which i lost, now in search of recovering my funds i got referred to anthonydavies on telegram a funds recovery specialist with his team help i was able to get back $300000 of my usdc back. you can reach him via anthonydaviestech AT gmail dot com
09.12.25 10:24 lane3215

It is distressing to lose USDT to a bitcoin wallet hack. Although challenging, recovering stolen USDT is feasible. Your chances increase if you move swiftly and strategically. Marie can help you with reporting the theft, recovering USDT, and taking immediate action. You can reach her via mail at [email protected], WhatsApp at +1 7127594675.
09.12.25 10:25 lane3215

It is distressing to lose USDT to a bitcoin wallet hack. Although challenging, recovering stolen USDT is feasible. Your chances increase if you move swiftly and strategically. Marie can help you with reporting the theft, recovering USDT, and taking immediate action. You can reach her via mail at [email protected], WhatsApp at +1 7127594675.
09.12.25 14:16 Matt Kegan

Grateful i came across SolidBlock Forensics. After investing in crypto trade and couldn't make withdrawals, it dawned on me something was wrong. They kept on asking for taxes, fees for maintenance, and more money for admin reasons. But being represented by SolidBlock Forensics, i was able to file reports, and finally, received all my investments with returns. Its great to know we have professionals that handle such issues and get the job done.

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Н Новости

Методы оптимизации в машинном и глубоком обучении. От простого к сложному

Содержание

Пару слов про функции потерь

Классический градиентный спуск

Более быстрые и точные оптимизаторы

Momentum

Nesterov momentum

AdaGrad

RMSProp

Adam и его модификации

Обучение и визуализация оптимизаторов

Промежуточные выводы

Нарушение работы адаптивных методов со скользящим средним

Стратегии изменения скорости обучения

Работа с большими мини-пакетами

LARS

LAMB

Прокcимальные методы

Метод проксимальной минимизации

Проксимальный градиентный метод (PGM)

ISTA (Iterative Shrinkage(Soft)-Thresholding Algorithm)

Методы второго порядка

Метод Ньютона

Метод сопряжённых градиентов

BFGS (Broyden–Fletcher–Goldfarb–Shanno)

SR1 (Symmetric Rank-1)

L-BFGS (Limited-memory BFGS)

Shampoo

Дополнительные источники

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Н Новости

Методы оптимизации в машинном и глубоком обучении. От простого к сложному

Содержание

Пару слов про функции потерь

Классический градиентный спуск

Более быстрые и точные оптимизаторы

Momentum

Nesterov momentum

AdaGrad

RMSProp

Adam и его модификации

Обучение и визуализация оптимизаторов

Промежуточные выводы

Нарушение работы адаптивных методов со скользящим средним

Стратегии изменения скорости обучения

Работа с большими мини-пакетами

LARS

LAMB

Прокcимальные методы

Метод проксимальной минимизации

Проксимальный градиентный метод (PGM)

ISTA (Iterative Shrinkage(Soft)-Thresholding Algorithm)

Методы второго порядка

Метод Ньютона

Метод сопряжённых градиентов

BFGS (Broyden–Fletcher–Goldfarb–Shanno)

SR1 (Symmetric Rank-1)

L-BFGS (Limited-memory BFGS)

Shampoo

Дополнительные источники

Похожие новости

Беларусь заблокировала OKX, Bybit и другие криптобиржи: виной — «ненадлежащая реклама»

В Польше повторно рассмотрят криптовалютный законопроект

Как прикрутить AI к табуретке

Норвегия откладывает цифровую крону

CryptoQuant: $112K для биткоина — реально, если ФРС снизит ставки и институты вернутся в ETF

Glassnode: без пробоя $95K биткоин останется в ловушке медвежьего давления

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