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[Перевод] Законы масштабирования нейронных языковых моделей

Инвестиции, открытие новых законов... Сотни лет прошло, а ничего не меняется...

Эта статья от 23 января 2020 года не так известна, как "Всё, что вам нужно - это внимание". Но, думаю, впоследствии она войдет в новейшую техноисторию как аналог трёх законов Ньютона для LLM (сами авторы статьи сравнивают открытые ими принципы с уравнениями для идеального газа). Возможно, именно благодаря аргументам большой группы специалистов OpenAI, изложенным в этой статье, инвесторы поверили, что GPT-1 имеет будущее, нужно только на порядки больше параметров, оборудования, данных и миллиарды долларов инвестиций. И всё заверте...

Законы масштабирования нейронных языковых моделей

Jared Kaplan *, Johns Hopkins University, OpenAI
Sam McCandlish *, OpenAI
Tom Henighan, OpenAI
Tom B. Brown, OpenAI
Benjamin Chess, OpenAI
Rewon Child, OpenAI
Scott Gray, OpenAI
Alec Radford, OpenAI,
Jeffrey Wu, OpenAI
Dario Amodei, OpenAI

(* Равный вклад. Авторы: Джаред Каплан (Jared Kaplan) и Сэм Маккэндлиш (Sam McCandlish) возглавляли исследование. Том Хениган (Tom Henighan) участвовал в LTSM-экспериментах. Том Браун (Tom B. Brown), Рон Чайлд (Rewon Child), Скотт Грей (Scott Gray) и Алек Рэдфорд (Alec Radford) разработали оптимизированную реализацию трансформатора. Джефри Ву (Jeffrey Wu), Бенджамин Чесс (Benjamin Chess) и Алек Рэдфорд (Alec Radford) разработали наборы текстовых данных. Дарио Амодей (Dario Amodei) обеспечивал руководство на протяжении всего проекта)

Мы изучаем эмпирические законы масштабирования для оценки производительности языковых моделей на основе кросс-энтропийных потерь. Потери масштабируются по степенному закону в зависимости от размера модели, размера набора данных и объема вычислений, используемых для обучения, причем некоторые тенденции охватывают более семи порядков соответствующих величин. Другие детали архитектуры, такие как ширина или глубина сети, оказывают минимальное влияние в широком диапазоне. Простые уравнения управляют зависимостью переобучения от размера модели/набора данных и зависимостью скорости обучения от размера модели. Эти отношения позволяют нам определить оптимальное распределение выделенного бюджета на вычисления. Более крупные модели значительно более эффективны с точки зрения выборки, так что наиболее эффективное с точки зрения вычислений обучение включает обучение очень больших моделей на относительно скромном объеме данных и остановку значительно раньше сходимости.

Содержание

1 Введение
2 Предыстория и методы
3 Эмпирические результаты и базовые степенные законы
4 Исследование предела бесконечных данных и переобучения
5 Законы масштабирования с размером модели и временем обучения
6 Оптимальное распределение бюджета вычислений
7 Связанные работы
8 Обсуждение
A Краткое содержание степенных законов
B Эмпирическая модель эффективной границы вычислений
C Предостережения
D Дополнительные графики

Введение

Язык предоставляет естественную область для изучения искусственного интеллекта, поскольку подавляющее большинство задач рассуждений может быть эффективно выражено и оценено на языке, а тексты, собранные со всего мира дают нам богатые данные для обучения без учителя с помощью генеративного моделирования. Глубокое обучение недавно достигло значительного прогресса в языковом моделировании, причем современные модели [18, 18, 19, 19, 19] приближаются к человеческому уровню производительности на многих конкретных задачах [19], включая составление по запросу связных образцов текста, содержащих несколько абзацев [24].

Еще пять лет назад в научной статье отмечалось, что языковые модели могут делать "составление по запросу связных образцов текста, содержащих несколько абзацев"! - прим.переводчика.

Можно ожидать, что производительность языкового моделирования зависит от архитектуры модели, размера нейронных моделей, вычислительной мощности, используемой для их обучения, и данных, доступных для этого процесса обучения. В этой работе мы эмпирически исследуем зависимость потерь языкового моделирования от всех этих факторов, сосредоточившись на архитектуре Трансформера [17, 18]. Широкие пределы производительности на языковых задачах позволяют нам изучать тенденции на более чем семи порядках масштаба используемых величин.

На протяжении всего исследования мы будем наблюдать точные степенные законы масштабирования для производительности как функции времени обучения, длины контекста, размера набора данных, размера модели и бюджета вычислений.

1.1 Краткое содержание

Наши ключевые выводы для языковых моделей на основе Transformer-а следующие:

Производительность сильно зависит от масштаба, и слабо — от формы модели: Производительность модели наиболее сильно зависит от масштаба, представляющего собой совокупность трех факторов: (1) количество параметров модели N (исключая эмбеддинги), (2) размер набора данных D и (3) объем вычислений C, используемых для обучения. В разумных пределах производительность очень слабо зависит от других архитектурных гиперпараметров, таких как глубина и ширина нейросети. (Раздел 3)

Плавные степенные законы: Производительность имеет степенную зависимость от каждого из трех факторов масштаба N, D, C, когда они не ограничены двумя другими, с тенденциями, охватывающими более шести порядков величины (см. Рисунок 1). Мы не наблюдаем признаков отклонения от этих тенденций на верхнем конце графика, хотя производительность должна в конечном итоге "выровняться" перед достижением нулевых потерь. (Раздел 3)

Универсальность переобучения: Производительность улучшается предсказуемо, пока мы масштабируем N и D вместе, но начинает улучшение начинает замедляться, если один фактор - либо N, либо D, фиксирован, а другой увеличивается. Штраф за производительность (performance penalty) предсказуемо зависит от соотношения N^0.74 / D, что означает, что каждый раз, когда мы увеличиваем размер модели в 8 раз, нам нужно увеличить данные только примерно в 5 раз, чтобы избежать штрафа. (Раздел 4)

Универсальность обучения: Кривые обучения следуют предсказуемым степенным законам, параметры которых практически не зависят от размера модели. Экстраполируя начальную часть кривой обучения, мы можем примерно предсказать потери, которые будут достигнуты, если обучать намного дольше. (Раздел 5)

Перенос улучшается с тестовой производительностью: Когда мы оцениваем модели на тексте с другим распределением, чем то, на котором они обучались, результаты сильно коррелируют с результатами на обучающем валидационном наборе с примерно постоянным смещением в потерях — другими словами, перенос на другое распределение влечет постоянный штраф, но в остальном улучшается примерно в соответствии с производительностью на обучающем наборе. (Раздел 3.2.2)

Эффективность выборки: Крупные модели более эффективны с точки зрения выборки, достигая того же уровня производительности с меньшим количеством шагов оптимизации (Рисунок 2) и используя меньше точек данных (Рисунок 4).

Сходимость неэффективна: При работе с фиксированным бюджетом вычислений C, но без других ограничений на размер модели N или доступные данные D, мы достигаем оптимальной производительности, обучая очень большие модели и останавливаясь значительно раньше сходимости (см. Рисунок 3). Максимально эффективное с точки зрения вычислений обучение будет, таким образом, намного более эффективным с точки зрения выборки, чем можно было бы ожидать на основе обучения небольших моделей до сходимости, с требованиями к данным, растущими очень медленно, как D ∼ C^0.27, с увеличением объема вычислений для обучения. (Раздел 6)

Оптимальный размер батча: Идеальный размер батча для обучения этих моделей примерно пропорционален степенному закону от потерь и продолжает определяться измерением шума градиента [18]; он составляет примерно 1-2 миллиона токенов при сходимости для самых больших моделей, которые мы можем обучить. (Раздел 5.1)

Вместе эти результаты показывают, что производительность языкового моделирования улучшается плавно и предсказуемо по мере соответствующего масштабирования размера модели, данных и вычислений. Мы ожидаем, что более крупные языковые модели будут работать лучше и будут более эффективными с точки зрения выборки, чем текущие модели.

Рисунок 1: Производительность языкового моделирования плавно улучшается по мере увеличения размера модели (Parameters), размера набора данных в токенах (Dataset size) и объема вычислений в PF-днях (Compute), используемых для обучения. Для оптимальной производительности все три фактора должны масштабироваться вместе. Эмпирическая производительность имеет степенную зависимость от каждого отдельного фактора, когда он не ограничен двумя другими.

Здесь и далее объемы вычислений указываются в PF-днях, где один PF-день = $10^{15} \times 24 \times 3600 = 8.64 \times 10^{19}$ операций с плавающей запятой (FLOPS) - прим. переводчика

Здесь мы показываем прогноз при использовании достаточно небольшого размера батча. Смотрите рисунок 13 для сравнения с чисто эмпирическими данными.

Рисунок 2: Мы демонстрируем серию экспериментов по обучению языковых моделей, где размер моделей варьируется от 10³ до 10⁹ параметров (за исключением эмбеддингов).

Рисунок 3. По мере увеличения доступных вычислительных ресурсов мы можем выбирать, как их распределить: на обучение более крупных моделей, использование больших размеров батчей и увеличение количества шагов обучения. Мы иллюстрируем это на примере увеличения вычислительных ресурсов в миллиард раз. Для оптимально эффективного с точки зрения вычислений обучения большая часть увеличения ресурсов должна быть направлена на увеличение размера модели. Для предотвращения повторного использования данных требуется относительно небольшое увеличение объема данных. Из увеличения объема данных большая часть может быть использована для повышения параллелизма за счет увеличения размера батчей, при этом требуется лишь очень небольшое увеличение времени последовательного обучения. (Multiplicative Contribution - Мультипликативный вклад)

1.2 Краткое содержание законов масштабирования

Тестовые потери трансформера, обученного для авторегрессионного моделирования языка, могут быть предсказаны с использованием степенного закона, когда производительность ограничена только одним из следующих факторов: количеством параметров без учета эмбеддингов N, размером набора данных D или оптимально распределенным бюджетом вычислений C_min (см. Рисунок 1):

1. Для моделей с ограниченным количеством параметров, обученных до сходимости на достаточно больших наборах данных (N_cне включает параметры эмбеддинга):

$L(N)=(N_c/N)^{α_N}; \quad αN∼0.076, \quad Nc∼8.8×10^{13} \quad \text{(1.1)}$

2. Для больших моделей, обученных на ограниченном наборе данных с ранней остановкой:

$L(D)=(D_c/D)^{α_D}; \quad α_D∼0.095, \quad D_c∼5.4×10^{13} (токенов) \quad \text{(1.2)}$

3. При обучении с ограниченным объемом вычислений, достаточно большим набором данных, оптимально подобранным размером модели и достаточно малым размером батча (оптимальное использование вычислений):

$L(C_{\min}) = \left(\frac{C_c^{\min}}{C_{\min}}\right)^{\alpha_C^{\min}}; \quad \alpha_C^{\min} \sim 0.050, \quad C_c^{\min} \sim 3.1 \times 10^8 \text{ (PF-days)} \quad \text{(1.3)}$

Мы также наблюдаем эмпирическую степенную зависимость с вычислительными ресурсами C, используемыми для обучения (Рисунок 1), при фиксированном размере батча, но для прогнозирования следует использовать зависимость с C_min⁡. Они связаны уравнением (5.5).

Эти соотношения сохраняются на протяжении восьми порядков величины для C_min, шести порядков величины для N и более двух порядков величины для D. Они очень слабо зависят от формы модели и других гиперпараметров трансформера (глубина, ширина, количество голов внимания), с конкретными числовыми значениями, связанными с обучающим набором [19]. Степенные законы $\alpha_{N}$ , $\alpha_{D}$ , $\alpha_{C}^{\min}$ определяют степень улучшения производительности, которую можно ожидать при масштабировании N, D или $C_{\min}$ ; например, удвоение количества параметров приводит к уменьшению потерь на коэффициент $2^{-\alpha_{N}} = 0.95$ . Точные числовые значения $N_{c}$ , $C_{c}^{\min}$ и $D_{c}$ зависят от размера словаря и токенизации и, следовательно, не имеют фундаментального значения.

Критический размер батча, который определяет компромисс между скоростью и эффективностью для параллелизма данных [18], также примерно подчиняется степенному закону от L:

$B_{\text{crit}}(L) = \frac{B_{*}}{L^{1/\alpha_{B}}}, \quad B_{*} \sim 2 \cdot 10^{8} \text{ токенов}, \quad \alpha_{B} \sim 0.21 \quad (1.4)$

Уравнения (1) и (2) вместе предполагают, что по мере увеличения размера модели мы должны увеличивать размер набора данных сублинейно в соответствии с $D \propto N^{\frac{\alpha_{N}}{\alpha_{D}}} \sim N^{0.74}$ . Фактически, мы обнаруживаем, что существует единое уравнение, объединяющее (1) и (2), которое управляет одновременной зависимостью от N и D и определяет степень переобучения:

$L(N, D) = \left[\left(\frac{N_{c}}{N}\right)^{\frac{\alpha_{N}}{\alpha_{D}}} + \frac{D_{c}}{D}\right]^{\alpha_{D}} \quad \quad(1.5)$

с аппроксимациями, показанными слева на Рисунке 4. Мы предполагаем, что эта функциональная форма также может параметризовать обученную логарифмически-подобную модель для других задач генеративного моделирования.

При обучении заданной модели на конечное количество шагов обновления параметров S в пределе бесконечных данных, после начального переходного периода кривые обучения могут быть точно аппроксимированы (см. справа на Рисунке 4):

$L(N, S) = \left(\frac{N_{c}}{N}\right)^{\alpha_{N}} + \left(\frac{S_{c}}{S_{\min}(S)}\right)^{\alpha_{S}} \quad \quad(1.6)$

где $S_{c} \approx 2.1 \times 10^{3}$ и $\alpha_{S} \approx 0.76$ , а $S_{\min}(S)$ — минимальное возможное количество шагов оптимизации (обновлений параметров), оцененное с использованием уравнения (5).

Рисунок 4.Слева: Тестовая ошибка при ранней остановке L(N,D) изменяется предсказуемо в зависимости от размера набора данных D и размера модели N согласно уравнению (1.5). Справа: После начального переходного периода кривые обучения для всех размеров моделей N могут быть аппроксимированы уравнением (1.6), которое параметризовано через S_min⁡ — количество шагов при обучении с большим размером батча (подробности в разделе 5.1).

При обучении с фиксированным бюджетом вычислений C, но без других ограничений, уравнение (6) приводит к заключению, что оптимальный размер модели N, оптимальный размер батча B, оптимальное количество шагов S и размер набора данных D должны расти как:

$N \propto C^{\alpha_{C}^{\min}/\alpha_{N}}, \quad B \propto C^{\alpha_{C}^{\min}/\alpha_{B}}, \quad S \propto C^{\alpha_{C}^{\min}/\alpha_{S}}, \quad D = B \cdot S \quad \quad(1.7)$

с параметром:

$\alpha_{C}^{\min} = 1 / \left(1/\alpha_{S} + 1/\alpha_{B} + 1/\alpha_{N}\right) \quad \quad(1.7)$

что близко соответствует эмпирически оптимальным результатам $N \propto C_{\min}^{0.73}$ , $B \propto C_{\min}^{0.24}$ и $S \propto C_{\min}^{0.03}$ . По мере увеличения бюджета вычислений C его следует тратить в основном на более крупные модели, без значительного увеличения времени обучения или размера набора данных (см. Рисунок 3). Это также означает, что по мере увеличения размера моделей они становятся все более эффективными с точки зрения выборки. На практике исследователи обычно обучают меньшие модели дольше, чем это было бы оптимально с точки зрения вычислений, из-за ограничений аппаратного обеспечения. Оптимальная производительность зависит от общего объема вычислений как степенной закон (см. уравнение (1.3)).

Мы предоставляем некоторую базовую теоретическую мотивацию для уравнения (1.5), анализ соответствия кривых обучения и их последствий для времени обучения, а также разбивку наших результатов по токенам. Также мы проводим краткие сравнения с LSTM и рекуррентными трансформерами [18].

1.3 Обозначения

Мы используем следующие обозначения:

L — кросс-энтропийные потери в натах (nats). Обычно они усредняются по токенам в контексте, но в некоторых случаях мы указываем потери для конкретных токенов в контексте.
N — количество параметров модели, исключая все словарные и позиционные встраивания (эмбеддинги)
$C \approx 6NBS$ — оценка общего объема вычислений для обучения без учета встраиваний, где B — размер батча, а S — количество шагов обучения (т.е. обновлений параметров). Мы указываем числовые значения в PF-днях, где один PF-день = $10^{15} \times 24 \times 3600 = 8.64 \times 10^{19}$ операций с плавающей запятой.
D — размер набора данных в токенах.
$B_{\text{crit}}$ — критический размер батча [18], определенный и обсуждаемый в разделе 5.1. Обучение с критическим размером батча примерно обеспечивает оптимальный компромисс между временем и эффективностью вычислений.
$C_{\min}$ — оценка минимального объема вычислений без учета эмбеддингов, необходимого для достижения заданного значения потерь. Это объем вычислений, который был бы использован, если бы модель обучалась с размером батча значительно меньше критического.
$S_{\min}$ — оценка минимального количества шагов обучения, необходимого для достижения заданного значения потерь. Это также количество шагов обучения, которое было бы использовано, если бы модель обучалась с размером батча значительно больше критического.
$\alpha_{X}$ — показатели степенных законов для масштабирования потерь как $L(X) \propto 1/X^{\alpha_{X}}$ , где X может быть любым из $N, D, C, S, B, C^{\min}$ .

2. Предыстория и методы

Мы обучаем языковые модели на WebText2, расширенной версии набора данных WebText [19], токенизированного с использованием байт-парного кодирования (BPE) с размером словаря $n_{\text{vocab}} = 50257$ . Мы оптимизируем autoregressive log-likelihood, авторегрессионное логарифмическое правдоподобие (т.е. cross-entropy loss , кросс-энтропийные потери), усредненное по контексту из 1024 токенов, что также является нашей основной метрикой производительности. Мы записываем потери на тестовом распределении WebText2 и на выборке других текстовых распределений. В основном мы обучаем декодер-трансформеры [18, 18, 17], хотя также обучаем модели LSTM и универсальные трансформеры [18] для сравнения.

2.1 Масштабирование параметров и вычислений для трансформеров

Мы параметризуем архитектуру трансформеров с использованием гиперпараметров $n_{\text{layer}}$ (количество слоев), $d_{\text{model}}$ (размерность residual stream), $d_{\text{ff}}$ (размерность intermediate feed-forward слоя), $d_{\text{attn}}$ (размерность attention output) и $n_{\text{heads}}$ (количество голов внимания на слой). Мы включаем $n_{\text{ctx}}$ токенов во входной контекст, где $n_{\text{ctx}} = 1024$ , если не указано иное.

Residual stream (остаточный поток) в трансформерах (прим. переводчика)

Residual stream (остаточный поток) в трансформерах — это концепция, связанная с архитектурой Transformer и использованием остаточных связей (residual connections). Остаточный поток относится к основному пути, по которому данные передаются через слои модели, с добавлением входных данных на каждом этапе.

Residual stream — это "поток данных", который проходит через все слои трансформера, накапливая информацию на каждом шаге. На каждом слое к residual stream добавляются новые преобразования (например, self-attention или feed-forward), но исходные данные (или их промежуточные представления) сохраняются благодаря остаточным связям. Это можно представить как "основной путь", по которому данные передаются через модель, с постепенным обогащением информации. Либо можно сравнить с рекой, которая течёт через всю модель. На каждом слое в неё впадают "притоки" (результаты преобразований), но основное русло (residual stream) сохраняет свою целостность и продолжает нести информацию дальше.

Intermediate feed-forward layer (промежуточный полносвязный слой) в трансформерах (прим. переводчика)

Intermediate feed-forward layer (промежуточный полносвязный слой) в трансформерах расположен между слоями внимания (self-attention).

После того как данные проходят через механизм self-attention, они поступают в промежуточный feed-forward слой. Этот слой обычно состоит из двух линейных преобразований (полносвязных слоёв) с нелинейной функцией активации (например, ReLU) между ними.

Промежуточный feed-forward слой применяет нелинейное преобразование к каждому токену независимо. Это позволяет модели извлекать более сложные и абстрактные признаки из данных. В отличие от механизма self-attention, который работает с глобальными зависимостями между токенами, слой feed-forward фокусируется на локальных преобразованиях.

Attention output (выход механизма внимания) в трансформерах (прим. переводчика)

Attention output (выход механизма внимания) в трансформерах — это результат работы механизма self-attention (самовнимания). Выход представляет собой взвешенную комбинацию всех токенов в последовательности, где веса определяются их важностью относительно друг друга.

Attention output позволяет каждому токену "видеть" контекст всей последовательности, что особенно полезно для задач, где важно учитывать глобальные зависимости (например, машинный перевод или генерация текста). Механизм внимания, в свою очередь, позволяет модели динамически выбирать, на какие части входных данных "смотреть" в зависимости от задачи.

Мы используем N для обозначения размера модели, который мы определяем как количество параметров без учета эмбеддингов:

$N \approx 2d_{\text{model}}n_{\text{layer}}\left(2d_{\text{attn}} + d_{\text{ff}}\right) = 12n_{\text{layer}}d_{\text{model}}^{2} \quad \quad(2.1)$ $\quad \text{при стандартных} \quad d_{\text{attn}} = d_{\text{ff}}/4 = d_{\text{model}}$

где мы исключили смещения (biases) и другие второстепенные члены. Наши модели также имеют $n_{\text{vocab}}d_{\text{model}}$ параметров в матрице эмбеддингов и используют $n_{\text{ctx}}d_{\text{model}}$ параметров для позиционных эмбеддингов, но мы не включаем их при обсуждении "размера модели" N; мы увидим, что это дает значительно более четкие законы масштабирования.

Вычисление прямого прохода трансформера включает примерно

$C_{\text{forward}} \approx 2N + 2n_{\text{layer}}n_{\text{ctx}}d_{\text{model}} \quad \quad(2.2)$

операций сложения-умножения, где коэффициент 2 возникает из-за операции умножения-накопления, используемой в матричном умножении. Более подробный подсчет параметров и вычислений на операцию приведен в Таблице 1.

Операция	Параметры	FLOPs на токен
Embedding (встраивание)	$(n_{vocab}+n_{ctx})d_{model}$	$4d_{\text{model}}$
Attention: QKV (запросы, ключи, значения)	$n_{\text{layer}} d_{\text{model}} 3d_{\text{attn}}$	$2n_{\text{layer}} d_{\text{model}} 3d_{\text{attn}}$
Attention: Mask (маска)	$\text{—}$	$2n_{\text{layer}} n_{\text{ctx}} d_{\text{attn}}$
Attention: Project (проекция)	$n_{\text{layer}} d_{\text{attn}} d_{\text{model}}$	$2n_{\text{layer}} d_{\text{attn}} d_{\text{embed}}$
Feedforward (прямое распространение)	$n_{\text{layer}} 2d_{\text{model}} d_{\text{ff}}$	$2n_{\text{layer}} 2d_{\text{model}} d_{\text{ff}}$
De-embed (обратное встраивание)	$\text{—}$	$2d_{\text{model}} n_{\text{vocab}}$
Итого (без эмбеддингов)	N = $2d_{\text{model}} n_{\text{layer}} (2d_{\text{attn}} + d_{\text{ff}})$	C_forward= $2N + 2n_{\text{layer}} n_{\text{ctx}} d_{\text{attn}}$

Таблица 1. Количество параметров и оценка вычислительных операций (прямой проход) для модели трасформера. Малозначимые члены, такие как нелинейности, смещения (biases) и нормализация слоёв, опущены.

Для контекстов и моделей с $d_{\text{model}} > n_{\text{ctx}}/12$ контекстно-зависимые вычислительные затраты на токен составляют относительно небольшую долю от общего объема вычислений. Поскольку мы в основном изучаем модели, где $d_{\text{model}} \gg n_{\text{ctx}}/12$ , мы не включаем контекстно-зависимые члены в нашу оценку вычислений для обучения. Учитывая обратный проход (примерно вдвое больше вычислений, чем прямой проход), мы определяем оценку вычислений без учета эмбеддингов как $C \approx 6N$ операций с плавающей запятой на токен обучения.

2.2 Процедуры обучения

Если не указано иное, мы обучаем модели с оптимизатором Adam [13] в течение фиксированных $2.5 \times 10^{5}$ шагов с размером батча 512 последовательностей по 1024 токена. Из-за ограничений памяти наши самые большие модели (более 1 миллиарда параметров) обучались с использованием Adafactor [17]. Мы экспериментировали с различными скоростями обучения и графиками, это показано в Приложении D.6. Мы обнаружили, что результаты на этапе сходимости в значительной степени не зависят от графика скорости обучения. Если не указано иное, все обучающие запуски, включенные в наши данные, использовали график скорости обучения с линейным прогревом (linear warmup) в течение 3000 шагов, за которым следовало косинусное затухание до нуля.

2.3 Наборы данных

Мы обучаем наши модели на расширенной версии набора данных WebText, описанного в [14]. Оригинальный набор данных WebText представлял собой веб-скрапинг внешних ссылок с Reddit до декабря 2017 года, которые получили как минимум 3 кармы. Во второй версии, WebText2, мы добавили внешние ссылки Reddit за период с января по октябрь 2018 года, также с минимальным порогом в 3 кармы. Порог кармы служил эвристикой для определения того, нашел ли кто-то ссылку интересной или полезной. Текст новых ссылок был извлечен с использованием библиотеки Newspaper3k на Python. В целом, набор данных состоит из 20,3 млн. документов, содержащих 96 ГБ текста и $1.62 \times 10^{10}$ слов (определенных с помощью утилиты `wc`). Затем мы применяем обратимый токенизатор, описанный в [14], который дает $2.29 \times 10^{10}$ токенов. Мы резервируем $6.6 \times 10^{8}$ из этих токенов для использования в качестве тестового набора, а также тестируем на аналогично подготовленных выборках из Books Corpus [22], Common Crawl [18], английской Википедии и коллекции публично доступных интернет-книг.

3. Эмпирические результаты и основные степенные законы

Чтобы охарактеризовать масштабирование языковых моделей, мы обучили широкий спектр моделей, варьируя различные факторы, включая:

Размер модели (от 768 до 1,5 миллиарда параметров, исключая эмбеддинги)
Размер набора данных (от 22 миллионов до 23 миллиардов токенов)
Архитектуру (включая глубину, ширину, количество голов внимания и размер промежуточного слоя)
Длину контекста (1024 для большинства экспериментов, хотя мы также экспериментировали с более короткими контекстами)
Размер батча (2¹⁹ для большинства экспериментов, но мы также варьировали его для измерения критического размера батча)

В этом разделе мы представим данные вместе с эмпирически обоснованными аппроксимациями, отложив теоретический анализ на последующие разделы.

3.1 Приблизительная независимость от формы трансформера и гиперпараметров

Производительность трансформера очень слабо зависит от параметров формы, таких как количество слоев n_layer, количество голов внимания n_heads и размер промежуточного слоя d_ff, при условии, что общее количество параметров N (исключая эмбеддинги) остается фиксированным. Чтобы установить эти результаты, мы обучали модели фиксированного размера, варьируя один гиперпараметр. Это было проще всего сделать для случая с количеством голов внимания n_heads. При варьировании количества слоев n_layer мы одновременно изменяли размер модели d_model, чтобы сохранить N ≈ 12 n_layerd_model² фиксированным. Аналогично, чтобы варьировать d_ff при фиксированном размере модели, мы также изменяли параметр d_model, как того требует подсчет параметров в Таблице 1. Независимость от количества слоев n_layer может быть объяснена тем, что глубокие трансформеры ведут себя как ансамбли более мелких моделей, как это было предложено для ResNet [38]. Результаты показаны на Рисунке 5.

Рисунок 5. Производительность очень слабо зависит от формы модели, когда общее количество неэмбеддинговых параметров N остается фиксированным. Ошибка изменяется всего на несколько процентов при широком диапазоне форм. Небольшие различия в количестве параметров компенсируются использованием аппроксимации L(N) в качестве базовой линии. В частности, соотношение сторон может варьироваться в 40 раз, лишь незначительно влияя на производительность; модель с (n_layer,d_model) = (6; 4288) достигает ошибки, которая отличается менее чем на 3% от модели (48;1600), использованной в [19].

3.2 Производительность в зависимости от количества параметров NN (исключая эмбеддинги)

На Рисунке 6 мы демонстрируем производительность широкого спектра моделей, начиная с небольших моделей с архитектурой (n_layer,d_model)=(2;128) и заканчивая моделями с миллиардами параметров, варьирующимися по архитектуре от (6; 4288) до (207; 768). Здесь мы обучали модели до почти полной сходимости на полном наборе данных WebText2 и не наблюдали переобучения (за исключением, возможно, самых больших моделей).

Рисунок 6.Слева: Когда мы включаем параметры эмбеддингов, производительность, по-видимому, сильно зависит от количества слоев в дополнение к количеству параметров.Справа: Когда мы исключаем параметры эмбеддингов, производительность моделей с разной глубиной сходится к единой тенденции. Только модели с менее чем 2 слоями или с экстремальными соотношениями глубины к ширине значительно отклоняются от этой тенденции.

Как показано на Рисунке 1, мы наблюдаем устойчивую тенденцию в зависимости от количества параметров NN, которую можно аппроксимировать первым членом уравнения (1.5):

$L(N) \approx \left(\frac{N_c}{N}\right)^{\alpha_N} \quad \quad(3.1)$

Чтобы наблюдать эти тенденции, крайне важно изучать производительность как функцию N; если вместо этого использовать общее количество параметров (включая параметры эмбеддингов), тенденция несколько затушевывается (см. Рисунок 6). Это говорит о том, что матрицу эмбеддингов можно сделать меньше без ущерба для производительности, как это было показано в недавних работах [19].

Хотя эти модели были обучены на наборе данных WebText2, их тестовая ошибка на различных других наборах данных также подчиняется степенному закону в зависимости от N с почти идентичным показателем степени, как показано на Рисунке 8.

3.2.1 Сравнение с LSTM и универсальными трансформерами

На Рисунке 7 мы сравниваем производительность LSTM и трансформеров в зависимости от количества параметров N (исключая эмбеддинги). LSTM обучались на том же наборе данных и с той же длиной контекста. Мы видим из этих графиков, что LSTM показывают такую же производительность, как и трансформеры, для токенов, появляющихся в начале контекста, но не могут сравниться с производительностью трансформеров для более поздних токенов. Мы представляем степенные зависимости между производительностью и позицией в контексте в Приложении D.5, где все более высокие степени для более крупных моделей предполагают улучшенную способность быстро распознавать паттерны.

Рисунок 7

Мы также сравниваем производительность стандартных трансформеров с рекуррентными трансформерами [18] на Рисунке 17 в приложении. Эти модели повторно используют параметры и поэтому показывают немного лучшую производительность в зависимости от NN, но за счет дополнительных вычислений на параметр.

3.2.2 Обобщение на различных распределениях данных

Мы также тестировали наши модели на наборе дополнительных текстовых данных. Тестовая ошибка на этих наборах данных в зависимости от размера модели показана на Рисунке 8; во всех случаях модели обучались только на наборе данных WebText2. Мы видим, что ошибка на других распределениях данных плавно улучшается с увеличением размера модели, параллельно с улучшением на WebText2. Мы обнаруживаем, что обобщение почти исключительно зависит от ошибки на валидационной выборке в рамках распределения обучения и не зависит от продолжительности обучения или близости к сходимости. Мы также не наблюдаем зависимости от глубины модели (см. Приложение D.8).

Рисунок 8.Слева: Производительность обобщения на других распределениях данных плавно улучшается с увеличением размера модели, с небольшим и очень медленно растущим отклонением от распределения обучения WebText2.Справа: Производительность обобщения зависит только от производительности на распределении обучения и не зависит от фазы обучения. Мы сравниваем обобщение сходившихся моделей (точки) с обобщением одной крупной модели (пунктирные кривые) в процессе ее обучения.

3.3 Производительность в зависимости от размера набора данных и вычислений

Мы демонстрируем эмпирические тенденции для тестовой ошибки в зависимости от размера набора данных DD (в токенах) и объема вычислений CC на Рисунке 1.

Для тенденции с DD мы обучали модель с (n_layer, n_embd)=(36; 1280) на фиксированных подмножествах набора данных WebText2. Мы останавливали обучение, как только тестовая ошибка переставала уменьшаться. Мы видим, что результирующие тестовые ошибки можно аппроксимировать простым степенным законом:

$L(D) \approx \left(\frac{D_c}{D}\right)^{\alpha_D} \quad \quad(3.2)$

в зависимости от размера набора данных. Данные и аппроксимация представлены на Рисунке 1.

Общий объем вычислений, использованных во время обучения, можно оценить как C=6NBS , где B — размер батча, S — количество шагов обновления параметров, а коэффициент 6 учитывает прямое и обратное прохождение. Таким образом, для заданного значения C мы можем сканировать все модели с различными N, чтобы найти модель с наилучшей производительностью на шаге $S=\frac{C}{6BS}$ . Обратите внимание, что в этих результатах размер батча B остается фиксированным для всех моделей, что означает, что эти эмпирические результаты не являются действительно оптимальными. Мы учтем это в последующих разделах, используя скорректированный C_min⁡ для получения более четких тенденций.

Результат представлен в виде жирной черной линии на левом графике Рисунка 1. Его можно аппроксимировать следующим образом:

$L(C) \approx \left(\frac{C_c}{C}\right)^{\alpha_C} \quad \quad(3.3)$

На рисунке также показаны изображения отдельных кривых обучения, чтобы прояснить, когда отдельные модели являются оптимальными. Мы более подробно изучим оптимальное распределение вычислений позже. Данные настоятельно утверждают, что эффективность использования данных улучшается с увеличением размера модели, и мы также иллюстрируем это напрямую на Рисунке 19 в приложении.

4. Исследование предела бесконечных данных и переобучения

В разделе 3 мы обнаружили несколько базовых степенных законов для производительности языковых моделей. Здесь мы изучим производительность модели размером N, обученной на наборе данных с D токенами, варьируя N и D одновременно. Мы эмпирически продемонстрируем, что при оптимальном обученнии тестовая ошибка соответствует степенному закону, заданному уравнением (1.5). Это дает рекомендации о том, сколько данных нам нужно для обучения моделей увеличивающегося размера, сохраняя переобучение под контролем.

4.1 Предложенное уравнение L (N, D)

Мы выбрали параметризацию (1.5) (повторено здесь для удобства):

$L(N, D) = \left[\left(\frac{N_{c}}{N}\right)^{\frac{\alpha_{N}}{\alpha_{D}}} + \frac{D_{c}}{D}\right]^{\alpha_{D}} \quad \quad(4.1)$

используя три принципа:

Изменения в размере словаря или токенизации ожидаемо изменяют ошибку на общий множитель. Параметризация L(N,D) (и всех моделей с возможностью расчета ошибки) должна естественно допускать такое масштабирование.
Фиксируя DD и устремляя , общая ошибка должна стремиться к L(D). И наоборот, фиксируя N и устремляя , ошибка должна стремиться к L(N).
L (N,D) должна быть аналитической при , то есть иметь разложение в ряд по степеням с целыми показателями. Теоретическое обоснование этого принципа значительно слабее, чем для первых двух.

Наш выбор L (N,D) удовлетворяет первому требованию, так как мы можем масштабировать N_c и D_c при изменении словаря. Это также подразумевает, что значения N_c и D_c не имеют фундаментального значения.

Поскольку мы останавливаем обучение рано, когда тестовая ошибка перестает улучшаться, и оптимизируем все модели одинаково, мы ожидаем, что более крупные модели всегда будут показывать лучшие результаты, чем меньшие модели. Однако при фиксированном конечном D мы также не ожидаем, что какая-либо модель сможет приблизиться к наилучшей возможной ошибке (то есть к энтропии текста). Аналогично, модель фиксированного размера будет ограничена своей емкостью. Эти соображения мотивируют наш второй принцип. Заметим, что знание L(N) при бесконечном D и L(D) при бесконечном N полностью определяет все параметры в L (N,D).

Третий принцип более спекулятивен. Есть простая и общая причина, по которой можно ожидать, что переобучение будет масштабироваться как ∝1/D при очень больших D. Переобучение должно быть связано с дисперсией или отношением сигнал/шум в наборе данных [17], и это масштабируется как 1/D . Это ожидание должно выполняться для любой гладкой функции потерь, так как мы ожидаем, что можем разложить ошибку в ряд вблизи предела D→∞ . Однако этот аргумент предполагает, что поправки 1/D доминируют над другими источниками дисперсии, такими как конечный размер батча и другие ограничения на эффективность оптимизации. Без эмпирического подтверждения мы не были бы уверены в его применимости.

Третий принцип объясняет асимметрию между ролями N и D в уравнении (5). Возможны очень похожие симметричные выражения, но они не будут иметь разложения по 1/D с целыми степенями и потребуют введения дополнительного параметра.

Например, можно было бы использовать:

$L(N, D) = \left[\left(\frac{N_c}{N}\right)^{\alpha_N} + \left(\frac{D_c}{D}\right)^{\alpha_D}\right]^{\beta}$

но такое выражение не имеет разложения по степеням 1/D .

В любом случае, мы увидим, что наше уравнение для L (N,D) хорошо соответствует данным, что является наиболее важным обоснованием нашего выбора L (N,D).

4.2 Результаты

Рисунок 9. Ранняя остановленная тестовая ошибка L(N,D) предсказуемо зависит от размера набора данных D и размера модели N согласно уравнению (1.5). Слева: Для больших D производительность следует прямому степенному закону от N. Для меньшего фиксированного D производительность перестает улучшаться с увеличением N, и модель начинает переобучаться. (Обратное также верно, см. Рисунок 4.) Справа: Степень переобучения преимущественно зависит от соотношения $N^{(α_N/α_D)}/D$ , как предсказано в уравнении (4.3). Линия представляет нашу аппроксимацию этого уравнения.

Мы регуляризуем все наши модели с помощью 10% dropout и останавливаем обучение, как только тестовая ошибка перестает уменьшаться. Результаты представлены на Рисунке 9, включая аппроксимацию для четырех параметров α_N,α_D,N_c,D_c в уравнении (5):

Параметр
Значение	0.076	0.103	6.4 × 10¹³	1.8 × 10¹³

Таблица 2. Значения для L(N, D)

Мы получаем отличное соответствие, за исключением случаев, когда набор данных был уменьшен в 1024 раза, до примерно 2×10⁷ токенов. С таким маленьким набором данных эпоха состоит всего из 40 шагов обновления параметров. Возможно, такой крошечный набор данных представляет собой другой режим для языкового моделирования, так как переобучение происходит очень рано в обучении (см. Рисунок 16). Также обратите внимание, что параметры немного отличаются от полученных в разделе 3, так как здесь мы аппроксимируем полное L(N,D), а не только L(N,∞) или L(∞,D).

Чтобы исследовать границы предела бесконечных данных, мы можем напрямую изучить степень переобучения. Для всех, кроме самых больших моделей, мы не видим признаков переобучения при обучении на полном наборе данных WebText2 с 22 миллиардами токенов, поэтому мы можем считать его представителем D=∞. Таким образом, мы можем сравнить конечное D с пределом бесконечных данных, определив:

$\delta L(N, D) \equiv \frac{L(N, D)}{L(N, \infty)} - 1 \quad \quad(4.2)$

и изучая его как функцию N и D. Фактически, мы видим эмпирически, что зависит только от определенной комбинации N и D, как показано на Рисунке 16. Это следует из степенного закона уравнения (1.5), которое подразумевает:

$\delta L \approx \left(1 + \left(\frac{N}{N_c}\right)^{\frac{\alpha_N}{\alpha_D}} \frac{D_c}{D}\right)^{\alpha_D} - 1 \quad \quad(4.3)$

Заметим, что при больших D эта формула также имеет разложение в ряд по степеням 1/D .

Мы оцениваем, что вариация ошибки при разных случайных начальных значениях составляет примерно 0.02, что означает, что для избежания переобучения при обучении до этого порога сходимости нам требуется:

$D \gtrsim (5 \times 10^3) N^{0.74} \quad \quad(4.4)$

С этим соотношением модели с менее чем 10⁹ параметров могут быть обучены с минимальным переобучением на наборе данных WebText2 с 22 миллиардами токенов, но наши самые большие модели столкнутся с некоторым умеренным переобучением. В общем, это соотношение показывает, что размер набора данных может расти сублинейно с увеличением размера модели, избегая переобучения. Однако это обычно не является максимально эффективным использованием вычислений. Мы также подчеркиваем, что мы не оптимизировали регуляризацию (например, вероятность dropout) при варьировании размера набора данных и модели.

5. Степенные законы масштабирования в зависимости от размера модели и времени обучения

В этом разделе мы продемонстрируем, что простой степенной закон хорошо описывает зависимость ошибки от размера модели N и времени обучения. Сначала мы объясним, как использовать результаты работы [18] для определения универсального шага обучения S_min⁡, который учитывает тот факт, что большинство наших моделей не обучались с оптимальным размером батча. Затем мы покажем, что зависимость ошибки от размера модели и времени обучения можно аппроксимировать с помощью уравнения (1.6). Позже мы используем эти результаты для прогнозирования оптимального распределения вычислительных ресурсов между размером модели и временем обучения, а затем подтвердим это предсказание.

5.1 Коррекция для обучения при Bcrit(L)

Простая эмпирическая теория для зависимости обучения от размера батча была разработана в работе [18] (см. также [18], [19]). В ней утверждалось, что существует критический размер батча B_crit для обучения; для B, изменяющимся вплоть до B_crit , размер батча можно увеличивать с минимальным ухудшением вычислительной эффективности, тогда как для $B>B_{crit}$ увеличение B приводит к уменьшению отдачи. Также утверждалось, что шкала шума градиента дает простое предсказание для B_crit , и что она не зависит напрямую от размера модели, за исключением значения ошибки, которое было достигнуто. Эти результаты можно использовать для прогнозирования того, как время обучения и вычисления будут изменяться в зависимости от размера батча. Чтобы максимально эффективно использовать как время обучения, так и вычисления, лучше всего обучать с размером батча $B≈B_{crit}$ . Обучение при $B≫B_{crit}$ минимизирует количество шагов обучения, тогда как $B≪B_{crit}$ минимизирует использование вычислений.

Было показано, что для широкого спектра задач нейронных сетей количество шагов обучения S и количество обработанных примеров данных E=BS удовлетворяют простому соотношению:

$\left(\frac{S}{S_{\min}} - 1\right)\left(\frac{E}{E_{\min}} - 1\right) = 1 \quad \quad(5.1)$

при обучении до любого фиксированного значения ошибки L. Здесь S_min — минимальное количество шагов, необходимое для достижения L, а E_min⁡ — минимальное количество примеров данных, которые должны быть обработаны.

Мы демонстрируем это соотношение для трансформеров на Рисунке 18 в приложении. Это соотношение определяет критический размер батча:

$B_{\text{crit}}(L) \equiv \frac{E_{\min}}{S_{\min}} \quad \quad(5.2)$

который является функцией целевого значения ошибки. Обучение с критическим размером батча обеспечивает примерно оптимальный компромисс между временем и вычислительной эффективностью, требуя 2S_min шагов обучения и обработки $E=2E_{min}$ экземпляров данных.

На Рисунке 10 мы построили график критического размера батча и шкалы шума градиента как функции ошибки обучения для двух разных моделей. Мы видим, что B_crit(L) не зависит от размера модели и зависит только от ошибки L.

(Хотя критический размер батча примерно соответствует шкале шума градиента, для всех наших последующих анализов мы используем прямые измерения B_crit из Рисунков 18 и 10.)

Рисунок 10. Критический размер батча B_crit следует степенному закону от ошибки по мере улучшения производительности и не зависит напрямую от размера модели. Мы обнаружили, что критический размер батча примерно удваивается при каждом уменьшении ошибки на 13%. B_crit измеряется эмпирически на основе данных, показанных на Рисунке 18, но он также приблизительно предсказывается шкалой шума градиента, как в [18].

Таким образом, предсказания работы [18] продолжают выполняться для языковых моделей на основе трансформеров. Критический размер батча можно аппроксимировать степенным законом от ошибки:

$B_{\text{crit}}(L) \approx \frac{B_*}{L^{1/\alpha_B}} \quad \quad(5.3)$

где B_∗≈2×10^8 и α_B ≈ 0.21 .

Мы выбрали эту параметризацию для B_crit(L), потому что, когда ошибка приближается к своему минимальному значению L_min⁡, шкала шума градиента должна расходиться, и мы ожидаем, что B_crit будет следовать этой шкале. Мы не знаем L_min⁡, так как не видим признаков того, что наши модели приближаются к ней, но L_min⁡>0, поскольку энтропия естественного языка не равна нулю. Поскольку, по-видимому, L_min⁡ намного меньше значений L, которые мы достигли, мы использовали параметризацию, в которой B_crit расходится при L→0 .

Мы будем использовать B_crit(L) для оценки соотношения между количеством шагов обучения S при обучении с размером батча B=2¹⁹ токенов и количеством шагов обучения при $B≫B_{crit}$ . Это просто:

$S_{\min}(S) \equiv \frac{S}{1 + B_{\text{crit}}(L)/B} \quad (мин.шагов- при: B≫B_{crit}) \quad \quad(5.4)$

для любого заданного целевого значения ошибки L. Это также определяет критическое значение вычислений, необходимых для обучения до L с моделью размера N, если бы мы обучали при $B≪B_{crit}(L)$ . Это:

$C_{\min}(C) \equiv \frac{C}{1 + B/B_{\text{crit}}(L)} \quad (мин.вычислений- при: B≪B_{crit}) \quad \quad(5.5)$

где C=6NBS оценивает объем вычислений (исключая эмбеддинги), использованных при размере батча B.

5.2 Результаты для L(N,Smin⁡) и производительность в зависимости от размера модели и объема вычислений

Теперь мы будем использовать S_min⁡, определенное в уравнении (5.4), чтобы получить простую и универсальную аппроксимацию зависимости ошибки от размера модели и времени обучения в пределе бесконечных данных. Мы будем аппроксимировать стабильные, оптимизированные с помощью Adam кривые обучения с использованием уравнения (1.6), повторенного здесь для удобства:

$L(N, S_{\min}) = \left(\frac{N_c}{N}\right)^{\alpha_N} + \left(\frac{S_c}{S_{\min}}\right)^{\alpha_S} \quad \quad(5.6)$

для ошибки. Мы включаем все шаги обучения после периода разогрева графика скорости обучения и находим аппроксимацию данных с параметрами:

Параметр
Значение	0.077	0.76	6.5 × 10¹³	2.1 × 10³

Таблица 3. Значения для L(N,S)

С этими параметрами мы получаем аппроксимации кривых обучения на Рисунке 4. Хотя аппроксимации не идеальны, мы считаем их довольно убедительными, учитывая простоту уравнения (5.6).

Данные и аппроксимации можно визуализировать иным, более интересным способом, как показано на Рисунке 11. Там мы изучаем тестовую ошибку как функцию размера модели, фиксируя либо общий объем вычислений C, использованных при обучении, либо количество шагов S. Для аппроксимаций мы используем уравнения (5.5) и (5.4) вместе с параметрами выше и уравнением (5.6).

Рисунок 11. Когда мы фиксируем либо общий объем вычислений, либо количество шагов обучения, производительность следует L(N,S) из уравнения (5.6). Каждому значению бюджета вычислений соответствует оптимальный размер модели, который максимизирует производительность. Неудивительно, что аппроксимация на малых S не идеальна, так как степенной закон для кривых обучения нарушается на самых ранних этапах обучения.

Степенной закон зависимости ошибки от S_min⁡ отражает взаимодействие динамики оптимизатора и ландшафта ошибки. Поскольку аппроксимации лучше всего работают на поздних этапах обучения, когда ошибка может быть приблизительно квадратичной, степенной закон должен предоставлять информацию о спектре гессиана ошибки. Его универсальность предполагает, что плотность собственных значений гессиана примерно не зависит от размера модели.

5.3 Нижняя граница для шага ранней остановки

Результаты для L (N,S_min⁡) можно использовать для вывода нижней границы (и грубой оценки) шага, на котором следует остановить обучение, когда обучение ограничено данными. Это мотивировано идеей, что кривые обучения для конечного и бесконечного D для данной модели будут очень похожи, пока мы не достигнем $S_{min}⁡≈S_{stop}$ . Таким образом, переобучение должно быть пропорционально поправке от простого завершения обучения на S_stop. Это будет недооценивать S_stop, потому что на самом деле тестовая ошибка будет уменьшаться медленнее, когда у нас есть конечное D, и поэтому нам потребуется больше шагов обучения для достижения оптимальной тестовой ошибки при конечном D. Эта логика приводит к неравенству:

$S_{\text{stop}}(N, D) \gtrsim \frac{S_c}{[L(N, D) - L(N, \infty)]^{1/\alpha_S}} \quad \quad(5.7)$

где L(N,∞) — сходимая ошибка, оцененная с бесконечными доступными данными. Данное неравенство и его сравнение с эмпирическими данными показаны на Рисунке 16 в приложении. На этом рисунке значения S_stop и L(N,D) являются эмпирическими (хотя S_stop скорректирован для имитации обучения при $B≫B_{crit}$ ), а L(N,∞) вычисляется из аппроксимации L(N,D) , оцененной при D=∞ .

6. Оптимальное распределение вычислительного бюджета

Мы показали эмпирическую тенденцию изменения производительности в зависимости от объема вычислений, использованных во время обучения, в правом верхнем углу Рисунка 1. Однако этот результат предполагал обучение с фиксированным размером батча B, тогда как мы знаем, что на самом деле можно обучать более эффективно, используя размер батча B_crit, обсуждаемый в разделе 5.1.

(Можно спросить, почему мы изначально не обучали с B_crit. Причина в том, что он зависит не только от модели, но и от целевого значения ошибки, которое мы хотим достичь, и поэтому является постоянно движущейся целью).

Большие и малые значения ошибки могли быть достигнуты с меньшим количеством примеров или шагов соответственно, и корректировка этой составляющей неэффективности обучения путем стандартизации до критического размера батча приводит к более четким и предсказуемым тенденциям.

В этом разделе мы исправим это упущение. Намного важнее то, что мы используем результаты раздела 5, чтобы определить оптимальное распределение вычислений между размером модели N и количеством данных, обработанных во время обучения, а именно $2B_{crit}S_{min}$ ⁡. Мы определим это распределение как эмпирически, так и теоретически, используя уравнение для L(N,S_min⁡), и покажем, что эти методы согласуются.

6.1 Оптимальная производительность и распределения

Давайте сначала изучим ошибку как функцию оптимально распределенных вычислений из уравнения (5.5). Результат показан на Рисунке 13 вместе с аппроксимацией степенным законом. Мы видим, что по сравнению с графиком вычислений на Рисунке 1, новая аппроксимация с C_min несколько улучшена.

Учитывая L(C_min⁡), естественно спросить, какой оптимальный размер модели N(C_min⁡) обеспечивает минимальную ошибку при заданном объеме вычислений. Оптимальный размер модели показан на Рисунке 14. Мы наблюдаем, что N(C_min⁡) можно очень хорошо аппроксимировать степенным законом:

$N(C_{\min}) \propto (C_{\min})^{0.73} \quad \quad(6.1)$

На Рисунке 12 мы показываем эффект обучения моделей субоптимальных размеров (см. Приложение B.4).

Рисунок 12. Слева: При фиксированном бюджете вычислений оптимален определенный размер модели, хотя модели несколько большего или меньшего размера могут быть обучены с минимальными дополнительными вычислениями. Справа: Модели, превышающие размер, оптимальный для вычислений, требуют меньше шагов для обучения, что позволяет потенциально ускорить обучение, если доступен достаточный дополнительный параллелизм. Обратите внимание, что это уравнение не следует использовать для очень больших моделей, так как оно справедливо только в степенной области кривой обучения после начальных переходных эффектов.

Рисунок 13. При корректировке производительности для симуляции обучения значительно ниже критического размера батча мы обнаруживаем несколько измененный степенной закон для L(C_min⁡) по сравнению с полностью эмпирическими результатами. Заметный выступ на уровне 10⁻⁵ PF-дней отмечает переход от однослойных к двухслойным сетям; мы исключаем однослойные сети из степенных аппроксимаций. Именно тенденция L(C_min⁡) , как мы ожидаем, обеспечивает надежную экстраполяцию для больших объемов вычислений.

По определению $Cmin⁡≡6NB_{crit}S$ , и поэтому мы можем использовать N(C_min⁡) для получения дополнительных результатов. В частности, поскольку предыдущие аппроксимации показывают, что $B∝L^{−4.8}$ и $L∝C^{−0.05}_{min}⁡$ , мы можем заключить, что $B_{crit}∝C^{0.24}_{min}$ . Это приводит нас к выводу, что оптимальное количество шагов будет расти очень медленно с увеличением вычислений, как:

$S_{\min} \propto (C_{\min})^{0.03} \quad \quad(6.2)$

что соответствует эмпирическим результатам на Рисунке 14. Фактически измеренный показатель настолько мал, что наши результаты могут быть даже согласованы с показателем, равным нулю.

Рисунок 14. Слева: Каждому значению бюджета вычислений C_min⁡ соответствует оптимальный размер модели N. Оптимальный размер модели растет очень быстро с увеличением C_min⁡, увеличиваясь в 5 раз при каждом 10-кратном увеличении вычислений. Количество обработанных примеров данных составляет оставшуюся часть увеличения, возрастая относительно скромно — всего в 2 раза. Справа: Количество шагов оптимизации, скорректированное с учетом размера батча, также растет очень медленно или вообще не растет, что означает, что большая часть увеличения обработанных экземпляров данных может быть использована для увеличения размера батча.

Таким образом, мы заключаем, что при масштабировании языкового моделирования с оптимальным распределением вычислений мы должны преимущественно увеличивать размер модели N, одновременно увеличивая размер батча через $B∝B_{crit}$ с незначительным увеличением количества последовательных шагов. Поскольку эффективное использование вычислений требует относительно небольшого количества шагов оптимизации, может потребоваться дополнительная работа по ускорению динамики раннего обучения.

6.2 Предсказания, исходя из L(N,Smin⁡)

Результаты для L(C_min⁡) и распределения могут быть предсказаны из уравнения L(N,S_min⁡), полученного в разделе 5. Учитывая наше уравнение для L(N,S_min⁡), мы можем подставить $S_{min}⁡=C_{min}⁡/(6NB)$ и затем найти минимум ошибки как функции N, фиксируя объем вычислений. Мы подробно выполняем эту процедуру в Приложении B, где также предоставляем некоторые дополнительные предсказания.

Для ошибки как функции объема вычислений мы предсказываем, что:

$L(C_{\min}) = \left(\frac{C_c^{\min}}{C_{\min}}\right)^{\alpha_C^{\min}} \quad \quad(6.3)$

где

$\alpha_C^{\min} \equiv \frac{1}{1/\alpha_S + 1/\alpha_B + 1/\alpha_N} \approx 0.054 \quad \quad(6.4)$

что отлично согласуется с показателем на Рисунке 13. Мы также предсказываем, что:

$N(C_{\min}) \propto (C_{\min})^{\alpha_C^{\min}/\alpha_N} \approx (C_{\min})^{0.71} \quad \quad(6.5)$

что также соответствует масштабированию на Рисунке 14 с точностью до нескольких процентов. Наши степенные законы дают прогностическую основу для оценки производительности языкового моделирования.

6.3 Противоречия и гипотеза

Мы не наблюдаем признаков отклонения от прямолинейных степенных законов при больших значениях вычислений, данных или размера модели. Однако наши тенденции должны в конечном итоге выровняться, поскольку естественный язык имеет ненулевую энтропию.

Действительно, тенденции для эффективного использования вычислений, описанные в этом разделе, уже содержат явное противоречие. На масштабах, на несколько порядков превышающих задокументированные здесь, производительность, предсказанная законом масштабирования L(C_min⁡), снижается ниже ожидаемой, учитывая медленный рост данных с увеличением вычислений. Это подразумевает, что наши степенные законы должны нарушиться до достижения этого момента, но мы предполагаем, что точка пересечения имеет более глубокий смысл: она дает оценку точки, в которой языковые модели на основе трансформеров достигают максимальной производительности.

Поскольку объем данных, используемых для эффективного использования вычислений, растет медленно с увеличением бюджета вычислений, производительность, предсказанная L(C_min⁡) , в конечном итоге достигает нижней границы, установленной степенным законом L(D) (см. Рисунок 15). Давайте рассмотрим это более подробно.

Рисунок 15. Далеко за пределами размеров моделей, которые мы изучаем эмпирически, мы обнаруживаем противоречие между нашими уравнениями для L(C_min⁡) и L(D) из-за медленного роста данных, необходимых для эффективного использования вычислений. Точка пересечения обозначает момент, до которого наши предсказания остаются справедливыми. Расположение этой точки сильно зависит от точных значений показателей степенных законов, полученных в наших аппроксимациях.

Чтобы контролировать переобучение, результаты раздела 4 подразумевают, что мы должны масштабировать размер набора данных как:

$D ∝ N^{0.74} ∝ C^{0.54}_{min} \quad \quad(6.6)$

где мы использовали N(C_min⁡) для эффективного использования вычислительного ресурса из Рисунка 14.

Давайте сравним это с требованиями к данным для эффективного использования вычислительного ресурса. Если мы обучаем с критическим размером батча (то есть $C=2C_{min}$ ⁡) и никогда не повторно используем данные во время обучения, мы обнаруживаем, что использование данных растет с вычислениями как:

$D(C_{\min}) = \frac{2C_{\min}}{6N(C_{\min})} \approx \left(4 \times 10^{10} \text{ токенов}\right) (C_{\min}/\text{PF-Day})^{0.26} \quad \quad(6.7)$

Это максимальная скорость, с которой размер набора данных может продуктивно расти с увеличением вычислений, так как это означает, что мы обучаем только на одной эпохе. Но это увеличивает размер набора данных намного медленнее, чем в уравнении (6.6). Это, по-видимому, подразумевает, что эффективное использование вычислений в конечном итоге столкнется с проблемой переобучения, даже если процесс обучения никогда не использует данные повторно!

Согласно Рисунку 1, мы ожидаем, что когда мы ограничены размером набора данных (то есть переобучением), ошибка должна масштабироваться как $L(D)∝D^{−0.095}$ . Это подразумевает, что ошибка будет масштабироваться с вычислениями как $L(D(C_{min}⁡))∝C_{min}^{⁡−0.03}$ , когда мы ограничены имеющимися данными. Снова мы сталкиваемся с противоречием, так как это в конечном итоге пересекается с нашим предсказанием для L(C_min⁡) из Рисунка 13, где мы нашли масштабирование $L(C_{min}⁡)∝C_{min}^{⁡−0.050}$ .

Точка пересечения L(D(C_min⁡)) и L(C_min⁡) находится при:

$C^* \sim 10^4 \text{ PF-дней}, \quad N^* \sim 10^{12} \text{ параметров}, \quad$ $D^* \sim 10^{12} \text{ токенов}, \quad L^* \sim 1.7 \text{ нат/токен} \quad \quad(6.8)$

хотя численные значения сильно неопределенны, варьируясь на порядок в любую сторону в зависимости от точных значений показателей степенных законов. Наиболее очевидная интерпретация заключается в том, что наши степенные законы нарушаются до или при достижении этой точки, которая все еще находится на много порядков выше как по вычислениям, так и по размеру модели.

Можно также предположить, что эта точка пересечения имеет более глубокий смысл. Если мы не можем увеличить размер модели за пределы N^∗ без качественно других требований к данным, возможно, это означает, что, достигнув C_min⁡^∗ и N^∗ , мы извлекли всю надежную информацию, доступную в данных естественного языка. В этой интерпретации L^∗ даст приблизительную оценку энтропии на токен естественного языка. В этом сценарии мы ожидаем, что тенденция ошибки выровняется на уровне или до L^∗.

(Определяя слова с помощью утилиты wc, получим следующее: набор данных WebText2 содержит 1.4 токена на слово и 4.3 символа на токен).

Мы можем предположить функциональную форму L(C_min⁡) по мере ее выравнивания, рассматривая версию нашего набора данных для обучения с добавленным шумом. Например, мы могли бы добавить случайную строку токенов к каждому контексту, показанному модели, чтобы искусственно увеличить ошибку на постоянный аддитивный фактор. Тогда расстояние от уровня шума $L−L_{noise}$ будет более значимой метрикой производительности, и даже небольшое уменьшение этого расстояния может представлять значительное улучшение качественной производительности. Поскольку искусственный шум одинаково влияет на все наши тенденции, критическая точка 6.8 не изменится (за исключением абсолютного значения L^∗) и может быть значимой, даже если она происходит после выравнивания.

7. Связанные работы

Степенные законы появляются из самых разных источников [18]. Степенные зависимости от размера модели и набора данных в задачах оценки плотности [6] и в моделях случайных лесов [12] могут быть связаны с нашими результатами. Эти модели предполагают, что показатели степенных законов могут иметь очень приблизительную интерпретацию как обратное количество релевантных признаков в данных.

Некоторые ранние работы [1], [Good1] обнаружили степенные зависимости между производительностью и размером набора данных. Более поздние работы [17], [19] также исследовали зависимость между размером модели и размером данных; их работа, возможно, наиболее близка к нашей в литературе.

(После завершения этой работы также появилась статья [19], в которой делаются аналогичные предсказания о зависимости ошибки как от размера модели, так и от размера набора данных).

Однако обратите внимание, что [17] обнаружили суперлинейную зависимость размера набора данных от размера модели, тогда как мы находим сублинейную зависимость. Есть некоторые параллели между нашими выводами об оптимальном распределении вычислений и [19], включая степенные кривые обучения. EfficientNets [19] также, по-видимому, подчиняются приблизительному степенному закону между точностью и размером модели. Недавняя работа [19] изучает масштабирование как с размером набора данных, так и с размером модели для различных наборов данных и аппроксимирует функциональную форму, похожую на нашу.

EfficientNet [19] предлагает масштабировать глубину и ширину экспоненциально (с разными коэффициентами) для оптимальной производительности моделей изображений, что приводит к степенной зависимости ширины от глубины. Мы обнаруживаем, что для языковых моделей этот показатель должен быть примерно равен единице при масштабировании (так как соотношение ширины и глубины должно оставаться фиксированным). Но что более важно, мы обнаруживаем, что точные архитектурные гиперпараметры не так важны, как общий масштаб языковой модели. В [16] утверждалось, что глубокие модели могут функционировать как ансамбли более мелких моделей, что потенциально может объяснить этот вывод. Более ранние работы [16] сравнивали ширину и глубину и обнаружили, что широкие ResNet могут превосходить глубокие ResNet в задачах классификации изображений. Некоторые исследования фиксируют вычисления на примере данных, которые, как правило, масштабируются пропорционально количеству параметров модели, тогда как мы исследуем масштабирование как с размером модели, так и с количеством вычислений для обучения.

Различные работы [17, 18] исследовали обобщение в сильно перепараметризованных моделях, обнаруживая "переход к застреванию" [19], когда размер модели достигает размера набора данных (это может потребовать обучения на много порядков больше, чем обычно, и, в частности, не использует раннюю остановку). Мы не наблюдаем такого перехода и обнаруживаем, что необходимый объем данных для обучения масштабируется сублинейно с размером модели. Расширения в размере модели, особенно при большой ширине [18, 19], могут предоставить полезную основу для размышлений о некоторых наших зависимостях масштабирования. Наши результаты по оптимизации, такие как форма кривых обучения, могут быть объяснены с помощью модели шумного квадратичного приближения (noisy quadratic model), которая может давать довольно точные предсказания [19] в реалистичных условиях. Чтобы сделать эту связь количественной, потребуется характеристика спектра гессиана [18, 19, 18].

8. Обсуждение

Мы наблюдали последовательные степенные зависимости ошибки логарифмического правдоподобия (log-likelihood loss) языковой модели от количества параметров N (исключая эмбеддинги), размера набора данных D и объема оптимизированных вычислений C_min, как это показано в уравнениях (1.5) и (1.6). В отличии от этих сильных зависимостей, мы обнаружили очень слабую зависимость от многих архитектурных и оптимизационных гиперпараметров. Поскольку зависимости от N, D и C_min являются степенными, с увеличением масштаба наблюдается уменьшение отдачи.

Мы смогли точно смоделировать зависимость ошибки от N и D, а также от N и S, когда эти параметры изменяются одновременно. Мы использовали эти соотношения для вывода масштабирования вычислений, степени переобучения, шага ранней остановки и требований к данным при обучении больших языковых моделей. Таким образом, наши зависимости масштабирования выходят за рамки простого наблюдения и предоставляют прогностическую основу. Можно интерпретировать эти соотношения как аналоги закона идеального газа, который связывает макроскопические свойства газа универсальным образом, независимо от большинства деталей его микроскопических составляющих.

Естественно предположить, что зависимости масштабирования будут применимы к другим задачам генеративного моделирования с функцией потерь максимального правдоподобия, а возможно, и в других условиях. С этой целью будет интересно проверить эти зависимости в других областях, таких как модели изображений, аудио и видео, а также, возможно, для дистилляции случайных сетей. На данный момент мы не знаем, какие из наших результатов зависят от структуры данных естественного языка, а какие являются универсальными. Также было бы интересно найти теоретическую основу, из которой можно вывести зависимости масштабирования: "статистическую механику", лежащую в основе "термодинамики", которую мы наблюдали. Такая теория могла бы позволить вывести другие, более точные предсказания и обеспечить систематическое понимание ограничений степенных законов.

В области естественного языка важно исследовать, приводит ли продолжающееся улучшение ошибки к улучшению в релевантных языковых задачах. Плавные количественные изменения могут скрывать значительные качественные улучшения: "больше — это другое". Например, плавный совокупный рост экономики не дает указаний на конкретные технологические разработки, которые его обеспечивают. Аналогично, плавные улучшения в ошибке языковой модели могут скрывать кажущиеся качественными изменения в возможностях.

Наши результаты упорно утверждают, что более крупные модели будут продолжать показывать лучшие результаты, а также будут гораздо более эффективны в использовании данных, чем оценивалось раньше. Большие модели (т.е. размер модели) могут быть важнее больших данных. В этом контексте дальнейшее исследование параллелизма моделей оправдано. Глубокие модели могут обучаться с использованием конвейерной обработки [18], которая разделяет параметры по глубине между устройствами, но в конечном итоге требует увеличения размера батча при использовании большего количества устройств. Широкие сети, с другой стороны, более подходят для параллелизации [18], поскольку большие слои могут быть разделены между несколькими рабочими слоями с меньшей последовательной зависимостью. Разреженность [19, 17] или ветвление (например, [17]) могут позволить еще быстрее обучать большие сети за счет увеличения параллелизма моделей. А использование методов, таких как [17, 19], которые увеличивают сети по мере их обучения, может позволить оставаться на границе эффективного использования вычислений на протяжении всего процесса обучения.

Благодарности

Мы хотели бы поблагодарить Shan Carter, Paul Christiano, Jack Clark, Ajeya Cotra, Ethan Dyer, Jason Eisner, Danny Hernandez, Jacob Hilton, Brice Menard, Chris Olah и Ilya Sutskever за обсуждения и отзывы на черновики этой работы.

Приложения

A. Свод степенных законов

Для удобства мы предоставляем ниже краткое описание ключевых тенденций, описанных в этой работе.

Параметры модели	Данные	Вычисления	Размер батча	Формула
			Fixed	$L(N) = \left(\frac{N_c}{N}\right)^{\alpha_N}$
	D	Early Stop	Fixed	$L(D) = \left(\frac{D_c}{D}\right)^{\alpha_D}$
Optimal		C	Fixed	$L(C) = \left(\frac{C_c}{C}\right)^{\alpha_C}$ (naive)
$N_{opt}$	D_opt	C_min	$B≪B_{crit}$	$L(C_{\min}) = \left(\frac{C_c^{\min}}{C_{\min}}\right)^{\alpha_C^{\min}}$
	D	Early Stop	Fixed	$L(N, D) = \left[\left(\frac{N_c}{N}\right)^{\frac{\alpha_N}{\alpha_D}} + \frac{D_c}{D}\right]^{\alpha_D}$
		S шагов	B	$L(N, S) = \left(\frac{N_c}{N}\right)^{\alpha_N} + \left(\frac{S_c}{S_{\min}(S)}\right)^{\alpha_S}$

Таблица 4.

Эмпирически подобранные значения для этих тенденций следующие:

Степенной закон	Масштаб (зависит от токенизации)
$\alpha_N = 0.076$	$\quad N_c = 8.8 \times 10^{13} \text{ параметров (неэмбеддинги)}$
$\alpha_D = 0.095$	$\quad D_c = 5.4 \times 10^{13} \text{ токенов}$
$\alpha_C = 0.057$	$\quad C_c = 1.6 \times 10^7 \text{ PF-дней}$
$\alpha_C^{\min} = 0.050$	$\quad C_c^{\min} = 3.1 \times 10^8 \text{ PF-дней}$
$\alpha_B = 0.21$	$\quad B_* = 2.1 \times 10^8 \text{ токенов}$
$\alpha_S = 0.76$	$\quad S_c = 2.1 \times 10^3 \text{ шагов}$

Таблица 5.

Оптимальные параметры для эффективного использования вычислений:

Эффективное значение для вычисления	Степенной закон	Масштаб
$N_{\text{opt}} = N_e \cdot C_{\min}^{p_N}$		$N_e = 1.3 \cdot 10^9 \text{ параметров}$
$B \ll B_{\text{crit}} = \frac{B_*}{L^{1/\alpha_B}} = B_e C_{\min}^{p_B}$		$B_e = 2.0 \cdot 10^6 \text{ токенов}$
$S_{\min} = S_e \cdot C_{\min}^{p_S}$ (нижняя граница)		$S_e = 5.4 \cdot 10^3 \text{ шагов}$
$D_{\text{opt}} = D_e \cdot C_{\min}^{p_D}$ (1 эпоха)		$D_e = 2 \cdot 10^{10} \text{ токенов}$

Таблица 6.

(Приложения B, C и D не приводятся, ввиду их размера и более теоретического характера)

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19 янв. 2025 г.

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Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
22.02.25 18:01 benluna0991

Mark Zuckerberg. That’s the name I was introduced to when I first encountered the cryptocurrency mining platform, WHATS Invest. A person claiming to be Zuckerberg himself reached out to me, saying that he was personally backing the platform to help investors like me earn passive income. At first, I was skeptical—after all, how often do you get a direct connection to one of the world’s most famous tech entrepreneurs? But this individual seemed convincing and assured me that many people were already seeing substantial returns on their investments. He promised me a great opportunity to secure my financial future, so I decided to take the plunge and invest $10,000 into WHATS Invest. They told me that I could expect to see significant returns in just a few months, with payouts of at least $1,500 or more each month. I was excited, believing this would be my way out of financial struggles. However, as time passed, things didn’t go according to plan. Months went by, and I received very little communication. When I finally did receive a payout, it was nowhere near the $1,500 I was promised. Instead, I received just $200, barely 13% of what I had expected. Frustrated, I contacted the support team, but the responses were vague and unhelpful. No clear answers or solutions were offered, and my trust in the platform quickly started to erode. It became painfully clear that I wasn’t going to get anywhere with WHATS Invest, and I began to worry that my $10,000 might be lost for good. That's when I discovered Certified Recovery Services. Desperate to recover my funds, I decided to reach out to them for help. In just 24 hours, they worked tirelessly to recover the majority of my funds, successfully retrieving $8,500 85% of my initial investment. I couldn’t believe how quickly and efficiently they worked to get my money back. I’m extremely grateful for Certified Recovery Servicer's fast and professional service. Without them, I would have been left with a significant loss, and I would have had no idea how to move forward. If you find yourself in a similar situation with WHATS Invest or any other platform that isn’t delivering as promised, I highly recommend reaching out to Certified Recovery Services They were a lifesaver for me, helping me recover nearly all of my funds. It's reassuring to know that trustworthy services like this exist to help people when things go wrong. They also specialize in recovering money lost to online scams, so if you’ve fallen victim to such a scam, don’t hesitate to contact Certified Recovery Services they can help! Here's Their Info Below: WhatsApp: +1(740)258‑1417 mail: [email protected], [email protected] Website info; https://certifiedrecoveryservices.com
23.02.25 22:00 eunice49954

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
24.02.25 06:36 ANDREW DAVIS

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Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
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Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
24.02.25 07:19 maggie4567

Jasmine Lopez specializes in recovering stolen cryptocurrencies, especially ETH/USDT. She has built a strong reputation for helping victims reclaim their lost funds. A personal example highlights her effectiveness: I lost $111,000 and, thanks to her prompt action, I recovered it all within 24 hours. Her dedication and skills eased my financial stress. She is always ready to assist others with similar issues. For help, she can be reached by email at Recoveryfundprovider@gmail. com or contact her through WhatsApp at +44 736 644 5035. Her Insta is recoveryfundprovider.
27.02.25 12:46 monikaguttmacher

Weddings are supposed to be magical, but the months leading up to mine were anything but. Already, wedding planning was a high-stress, sleep-deprived whirlwind: endless details to manage, from venue deposits and guest lists to dress fittings and vendor contracts. But nothing-and I mean, nothing-compared to the panic that washed over me when I realized that somehow, I had lost access to my Bitcoin wallet-with $600,000 inside. It happened in the worst possible way. In between juggling my to-do lists and trying to keep my sanity intact, I lost my seed phrase. I went through my apartment like a tornado, flipping through notebooks, checking every email, every file-nothing. I sat there in stunned silence, heart pounding, trying to process the fact that my entire savings, my security, and my financial future might have just vanished. In utter despair, I vented to my bridesmaid's group chat for some sympathetic words from the girls. Instead, one casually threw out a name that would change everything in a second: "Have you ever heard of Tech Cyber Force Recovery? They recovered Bitcoin for my cousin. You should call them." I had never heard of them before, but at that moment, I would have tried anything. I immediately looked them up, scoured reviews, and found story after story of people just like me—people who thought they had lost everything, only for Tech Cyber Force Recovery to pull off the impossible. That was all the convincing I needed. From the very first call, I knew I was in good hands. Their team was calm, professional, and incredibly knowledgeable. They explained the recovery process in a way that made sense, even through my stress-fogged brain. Every step of the way, they kept me informed, reassured me, and made me feel like this nightmare actually had a solution. And then, just a few days later, I got the message: "We have recovered your Bitcoin." (EMAIL. support @ tech cyber force recovery . com) OR WHATSAPP (+1 56 17 26 36 97) I could hardly believe my eyes: Six. Hundred. Thousand. Dollars. In my hands again. I let out my longest breath ever and almost cried, relieved. It felt like I woke up from a bad dream, but it was real, and Tech Cyber Force Recovery had done it. Because of them, I walked down the aisle not just as a bride, but as someone who had dodged financial catastrophe. Instead of spending my honeymoon stressing over lost funds, I got to actually enjoy it—knowing that my wallet, and my future, were secure. Would I refer to them? In a heartbeat. If you ever find yourself in that situation, please don't freak out, just call Tech Cyber Force Recovery. They really are the real deal.
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07.03.25 15:18 Ariduk

BEST LINK FOR RECOVERY SCAM ON TRADING INVESTMENT AND OTHERS TROUBLESHOOT? Welcome To General Hacking Techniques Service known as DHACKERS. Year 2025 we are active and best in what we do, as we give Solution to every problem concerning Web3 INTERNET activities, we guide you right to a positive fund Recovery e.t.c. Question From Most of Our Client, HOW POSSIBLE AND TIME WILL IT TAKE TO RECOVERY LOST OR SCAM FUND? Our Answer. Yes is 89.9% possible and how long it takes your Fund to be recovered depend on you. The fact is there are lot of fake binary investment companies out their, same a lot fake recovery companies and agents too. CAUTION 1). Make sure you ask one or two questions concerning the service and how they render there recovery services. 2) Do not give out your scammed details to any agent or hacker when you are not yet ready to recover back your fund. 3) do not make any payment when you are not sure of the service if you are to make one. (4) We discovered that most of this fake hacker or agent do give us scam details, playing the victim, because they can afford the service charge. 5) As long you have your scam details with you, you can recover your fund back anytime any-day with the right channel. Contact us from our Front Desk. [email protected] For advice and services, our standby Guru email. [email protected] [email protected] List of Service. ▶️Binary Recovery ▶️Data Recovery ▶️University Result Upgraded ▶️Clear your Internet Blunder and controversy ➡️Increase your Credit Score ➡️Wiping of Criminal Records ➡️Social Media Hack ➡️Blank ATM Card ➡️Load and wipe ➡️Phone Hacking ➡️Private Key Reset etc. For quick response. Email [email protected] Border us with your jobs and allow us give you positive result with our hacking skills. All Right Reserved (c)2025
08.03.25 04:42 andreassenhedda

If you’ve fallen victim to online scammers who have deceived you into investing your hard-earned money through fraudulent Bitcoin schemes, know that you're not alone. Countless individuals have been misled by scammers using various tactics to swindle money through deceptive investment platforms or promises of high returns. These scams often come in the form of fake cryptocurrency investments, Ponzi schemes, or phishing scams designed to steal your Bitcoin and other assets. Unfortunately, many victims suffer not only financial loss but also the psychological toll of realizing they’ve been taken advantage of. In some cases, scammers go even further, threatening legal consequences or using intimidation to keep victims silent. However, all hope is not lost. There are ways to recover your funds and potentially even bring those responsible to justice. One promising resource that can assist in reclaiming lost funds is Lee Ultimate Hacker, a trusted platform designed to help victims of online fraud. This service specializes in helping individuals who have been scammed through cryptocurrency investments or similar schemes. The platform’s expertise can help you navigate the process of recovering your money, giving you a chance to fight back against those who’ve wronged you. To begin the recovery process, it’s important to gather any proof of payment or transaction history involving the scammers. This evidence is crucial in tracing the flow of funds and identifying the scam operation behind it. Once you have collected your documentation, you can reach out to Lee Ultimate Hacker via LEEULTIMATEHACKER @ AOL . COM or wh@tsapp +1 (715) 314 - 9248 for assistance. They offer professional services to investigate fraudulent schemes, track Bitcoin transactions, and work to reverse fraudulent transfers. The recovery process can be complex and requires expert knowledge of blockchain technology and financial investigations, but with the help of a dedicated team, you’ll have a better chance of seeing your funds returned. In addition to helping you recover your assets, Lee Ultimate Hacker also works towards identifying the scammers and reporting them to the appropriate authorities. They understand the urgency of halting these fraudsters before they can victimize others. With their assistance, you not only increase the chances of recovering your funds but also play a part in holding cybercriminals accountable. If you've been affected by Bitcoin scams or other fraudulent online activities, reaching out to a service like Lee Ultimate Hacker can provide hope and a clear path forward. Their team can guide you through the recovery process, offering expert support while you take steps to reclaim what you’ve lost. It’s time to take action and work toward reclaiming your hard-earned money.
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09.03.25 17:22 Wayne707

‎Scammers have ruined the forex trading market, they have deceived many people with their fake promises on high returns. I learnt my lesson the hard way. I only pity people who still consider investing with them. Please try and make your researches, you will definitely come across better and reliable forex trading agencies that would help you yield profit and not ripping off your money. Also, if your money has been ripped-off by these scammers, you can report to a regulated crypto investigative unit who make use of software to get money back. If you encounter with some issue make sure you contact ( [email protected] ) they're recovery expert and a very professional one at that. ‎
10.03.25 18:18 springwilli

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10.03.25 20:19 wendytaylor015

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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]
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12.03.25 05:35 cholevasaca

As the senior teacher at Greenfield Academy, I wanted to share our ordeal with TECH CYBER FORCE RECOVERY after our school was targeted by a malicious virus attack that withdrew a significant amount of money from our bank account. A third-party virus infiltrated our system and accessed our financial accounts, resulting in a USD 50,000 withdrawal. This attack left us in a state of shock and panic, as it posed a serious threat to our school's financial stability. Upon realizing what had happened, we immediately contacted TECH CYBER FORCE RECOVERY for help. Their team responded promptly and began investigating the situation. They worked tirelessly to track down the virus, analyze its behavior, and understand how it had bypassed our security measures. Most importantly, they focused on recovering the funds that had been stolen from our bank account. Thanks to TECH CYBER FORCE RECOVERY's quick and expert intervention, they were able to successfully recover all the funds that had been withdrawn. Their team worked closely with our bank and utilized advanced recovery methods to ensure the full amount was returned to our account. We were incredibly relieved to see the stolen funds restored, and their efforts prevented any further financial loss. I am incredibly grateful for TECH CYBER FORCE RECOVERY's professionalism, expertise, and swift action in helping us recover the stolen funds. Their team not only managed to undo the damage caused by the virus but also ensured that our financial security was restored. I highly recommend their services to any organization dealing with similar cyberattacks, as their team truly went above and beyond to resolve the issue and protect our assets. CONTACTING THEM TECH CYBER FORCE RECOVERY EMAIL. [email protected]
14.03.25 21:40 adelfinalongo

I had the worst experience of my life when the unthinkable happened and my valued Bitcoin wallet disappeared , I was literally lost and was convinced I’m never getting it back , but all thanks to LEE ULTIMATE HACKER the experienced and ethical hacker on the web and PI they were able to retrieve and help me recover my Bitcoin wallet. This highly skilled team of cyber expertise came to my rescue with their deep technical knowledge and cutting-edge tools to trace cryptocurrency and private investigative prowess they were rapid to pin point the exact location of my missing Bitcoin, LEE ULTIMATE HACKER extracted my crypto with modern technology, transparency and guidance on each step they took, keeping me on the loop and reassuring me that all will be well: contact LEE ULTIMATE HACKER via LEEULTIMATEHACKER @ AOL . COM telegram: LEEULTIMATE wh@tsapp +1 (715) 314 - 9248 for all your cryptocurrency problems and you’ll have a prompt and sure solution.
15.03.25 00:29 irenmroma

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15.03.25 14:34 spencerwerner

It wasn’t an easy process, and it required patience, but the team’s dedication, attention to detail, and methodical approach paid off. I felt an overwhelming sense of relief and gratitude. What had once seemed like a permanent loss was now being reversed, thanks to the help of Tech Cyber Force Recovery. Not only did the recovery restore my financial situation, but it also restored my sense of trust and confidence. I had almost given up hope, but now, with my funds recovered, I feel like I can move forward. I’ve learned valuable lessons from this experience, and I’m more cautious about my financial decisions in the future. What began as a desperate search on Red Note turned into a life-changing recovery. Thanks to Tech Cyber Force Recovery, I now feel more hopeful about my financial future, with the knowledge that recovery is possible. telegram (@)techcyberforc texts (+1 5.6.1.7.2.6.3.6.9.7)
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17.03.25 12:20 browne

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17.03.25 13:14 vinnypraise

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17.03.25 13:14 vinnypraise

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17.03.25 15:23 Charlesagrimes

Running the small business was hard enough in itself without having to suffer such stress as loss of access to a Bitcoin wallet. I had put $532,000 into Bitcoin for my business in case of emergencies, but one day, the wallet was nowhere to be seen. I had trusted it to my friend while he managed my financials, little knowing he would prove less than trustworthy. I couldn't envision the gut-wrenching feeling coupled with a great feeling of folly for having placed so much trust in someone else's hands. I could not believe my eyes to see that all of my digital fortune was gone. Literally, every second of my busy day froze in disbelief as I tried to conceive the loss. I remembered the hours I had put into building my business, securing my investments, and planning for the future. Now, I was staring at an empty wallet that once held my safety net. This hit me hard-not only had I lost $532,000, but the betrayal hurt further because it came from someone I considered reliable. The emotional toll was huge, and this financial hit could cripple my business. In the midst of my despair, I knew I had to act fast. I went online, searching everywhere for a solution through which I could get back in control of my finances. That is where I came across Digital resolution services. I knew them for recovery related to lost crypto, but also somehow skeptical. At that point, it looked like I had nothing left to lose. I reached out, and with our very first conversation, I gained the impression that they did understand my situation. Their team was nothing short of extraordinary. They approached my case with professionalism and empathy, meticulously explaining the recovery process in clear, simple terms that alleviated some of my anxiety. They set realistic expectations while promising to do everything possible to retrieve my funds. Over the next several weeks, they worked tirelessly, employing advanced blockchain forensic techniques and sophisticated tracking tools that I’d never even heard of before. It kept me informed with constant updates and gave me hope in this desperate time. Finally, it came-the day Digital resolution services recovered my lost Bitcoin. Relief overwhelmed me, and it wasn't all about the money but control and self-trust in my financial future. This experience taught me the most valuable lesson: never lose control over your own financial security and never put blind trust in anyone. With Digital resolution services, I recovered not only my $532,000 but also learned how to protect my assets better in the future. Contact Digital Resolution Services: Email: digitalresolutionservices (@) myself. c o m WhatsApp: +1 (361) 260-8628 Stay protected Charles agrimes
18.03.25 23:49 lindaspringer311

My name is Linda Springer, and I am part of an ultra-high-net-worth family in the United Kingdom. Managing our wealth has always been a top priority, and I’ve always looked for opportunities to grow and diversify our investments. When I was introduced to an offshore banking investment program called Market Fund, it seemed like the perfect opportunity to enhance our portfolio. The program promised impressive returns, and with professional advisors and seemingly legitimate documentation, I felt confident that it was a secure investment. However, my trust in Market Fund quickly proved to be misplaced. Initially, everything seemed to go smoothly. Reports showed strong growth, and I was reassured by the continuous updates from the program’s advisors. But when I tried to withdraw my funds, everything took a sharp turn. The Market Fund website became unresponsive, and all my attempts to reach the advisors through emails and phone calls went unanswered. It was then that I realized I had fallen victim to a scam, and the £60,000 I had invested was gone. Feeling devastated and unsure of what to do next, I turned to a friend I had met at a club called The Vault. During one of our conversations, Alex, a fellow club-goer, mentioned how Rapid Digital Recovery had helped them recover funds from a similar fraudulent scheme. Intrigued by their success story, I decided to reach out to Rapid Digital Recovery....Whatsapp: +1 4 14 80 71 4 85.. hoping they could help me recover my money as well. From the moment I contacted them, the team at Rapid Digital Recovery took immediate action. They began investigating the scam and used advanced technology to trace the origins of Market Fund. Through their efforts, they uncovered a network of fake websites and shell companies that had been designed to deceive investors like myself. Thanks to their persistence and expertise, Rapid Digital Recovery successfully recovered the £60,000 I had lost.....Email: rapid digital recovery (@) execs. com.. Not only did they restore my financial security, but they also exposed the fraudulent operation behind Market Fund, preventing others from falling victim to the same scam. While the experience was incredibly stressful, it served as an important reminder of the need for due diligence and professional oversight when dealing with offshore investments. I am incredibly grateful to Rapid Digital Recovery for their support, expertise, and determination in recovering my funds. Telegram: h t t p s: // t. me /Rapiddigitalrecovery1
20.03.25 17:21 katherineingram

Dr. Katherine Ingram
20.03.25 17:21 katherineingram

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20.03.25 19:34 ysabelbristol

I am YSABEL, a medical student living with my grandmother. For the past few years, I’ve been diligently saving my grandmother’s money to ensure she has a comfortable and secure future. I invested over $40,000 into an online platform, which promised significant returns. After completing several steps, they told me that to withdraw the money, I needed to make another payment of $73,000. As a medical student, my time is already stretched thin, and managing my grandmother’s finances on top of my studies was overwhelming. That's when a costudent, who had gone through a similar experience, recommended MUYERN TRUST HACKER. Initially, I was skeptical. After all, how could anyone help me recover the money I had already lost to such a sophisticated scam? But after reaching out to them, I quickly realized I was in good hands. From the very first conversation, the team at MUYERN TRUST HACKER was professional, compassionate, and knowledgeable. They understood the gravity of my situation and took the time to thoroughly review all the details of my case. They explained the recovery process clearly, step by step, and assured me that they would work tirelessly to help recover the funds I had lost. Their expertise in dealing with online fraud was evident, and for the first time in weeks, I felt a sense of hope that I could regain control over this situation. The recovery process wasn’t immediate, and there were moments of uncertainty, but MUYERN TRUST HACKER remained dedicated and persistent. As someone who has been working hard to save and protect my grandmother’s money, it was devastating to feel like I might lose everything to a scam. But MUYERN TRUST HACKER gave me the support, expertise, and guidance I needed to recover what was lost. I’m deeply grateful for their help, and I now have a renewed sense of hope that I can protect my grandmother’s future. If you find yourself in a similar situation, feeling trapped and overwhelmed by a fraudulent company, I highly recommend MUYERN TRUST HACKER. (Whats A p p at + 1 (440) (335) (0205) Their team provides real solutions, and their professionalism and dedication were a lifeline for me during a very difficult time. Thanks to them, I feel confident that I can move forward and continue to protect my grandmother’s savings. Alternatively, contact their tele gram also if you need immediate attention at muyerntrusthackertech.
21.03.25 21:19 carmenbeechum643

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22.03.25 14:13 [email protected]

The glow of RGB lights still haunts me. There I was, mid-stream, hyping up a Fortnite squad when an email pretending to be a sponsorship opportunity with the subject line "ENERGY DRINK COLLAB!!! *" appeared on my second monitor. I clicked— big mistake. By the time my chat spammed "*SCAM ALERT" in neon caps, a trojan had already ghosted my Bitcoin wallet, $320,000 gone, poof, like a noob disconnecting mid-game. My facecam caught the exact moment my soul left my body: jaw open, headset tilted, the background of anime posters judging me silently. The VOD blew up. Of course, it did. Pandemonium erupted. Donation alerts became panic emojis. My mods DM'd links to "HOW TO FIX CRYPTO THEFT" amidst banning trolls. My wallet? A barren wasteland. My DMs? A cemetery of "*F"s and crypto-bros pitching recovery scams. Then, a lifeline—a chatter named *xX_Cryptosolution_69 typed, "TRUST GEEKS HACK EXPERT. THEY CLAPPED A HACKER FOR MY DOGE ONCE." Desperate, I Googled them mid-stream, muting to scream into a pillow. TRUST GEEKS HACK EXPERT team responded like NPCs scripted for heroics. “Send us the malware file,” they said. “**And your wallet logs. We’ll handle the rest.” For 12 days, they reverse-engineered the trojan, dissecting its code like speed runners cracking a glitch. The virus, it turned out, was a knockoff ransomware dubbed “Crypto rush” (its dev had left a “HACK THE PLANET!!” Easter egg in the code, cringe). TRUST GEEKS HACK EXPERT squad traced its path, resurrecting private keys from registry fragments and backup clouds I’d forgotten existed. The return stream was record-breaking. I rebooted my rig, wallet restored, and titled the stream "HOW I UNBRICKED $320K (AND MY CAREER)." Chatters donated Bitcoin out of solidarity, and schadenfreude. Even my rival streamer, DrL33tGamer, raided me with 10k viewers. TRUST GEEKS HACK EXPERT? They viewed anonymously and left a sub with the message: "GG EZ. These internet Gandalfs didn't just fix a hack—they authored the greatest plot twist in my online existence. Now, my new website, Stream Vault, runs on a server guarded like Fort Knox, and I vet sponsors like the CIA. That fake energy drink company? Its domain now points to a Rickroll. If your crypto gets pawned by a script kiddie, skip the rage quit. Ping the TRUST GEEKS. They're the ultimate cheat code for catastrophe. Just maybe have a malware scanner in closer proximity than your energy drinks next time. (CONTACT SERVICE ) Email, Trustgeekshackexpert[At]fastservice[Dot]com Telegram, Trustgeekshackexpert Email, [email protected] Website, www://trustgeekshackexpert.com
23.03.25 12:24 maryjul75

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23.03.25 22:01 joaneldnde

The ground trembled like a nervous intern on espresso shots. One minute, I was monitoring my geothermal Bitcoin miners, humming in harmony with Iceland's most unpredictable volcano. Next? An eruption painted the sky gray with ash, raining destruction like an out-of-control blockchain fork. Power cables flickered out. Servers turned into abstract-art pieces. And my wallet with $460,000 worth of mining revenue fried faster than a motherboard in a tidal wave of lava. I was knee-deep in volcanic mud, clutching the charred wallet, wondering if the universe had a vendetta against renewable energy. For weeks, I’d played geothermal gambler, harnessing Earth’s anger to mine crypto. Now, Mother Nature had countered with a literal power move. My wallet’s backups? Corrupted by ash-clogged drives. My cold storage? Warmer than a freshly erupted fissure. Even the volcanologists on my team shrugged. “We predict lava, not ledger errors,” one said, handing me a business card signed at the edges. “Try these Cyber Constable Intelligence. They’ve fixed crypto in weird places.” Cyber Constable Intelligence phoned on the first ring. Cyber Constable Intelligence saved not just crypto. They demonstrated that even the fury of nature cannot surpass human tenacity. My operation now operates robustly, excavating coins with Earth's anger and a backup generator sufficient to run a small glacier. The volcano? Still grumbling. My wallet? Locked inside a fireproof safe, as irony bites sharper than an Icelandic winter. If your crypto somehow gets smothered beneath the pyroclastic ash of life, skip the freak-out. Call the Cybers. They'll dig through lava streams until your cash bubbles up to the surface. Just maybe set up your rigs a few miles closer to the crater next time. If you’re facing a similar problem I highly recommend contacting Cyber Constable Intelligence 📞 WhatsApp:+1 (2 5 2 ) 3 7 8 7 6 1 1 🌐 Website: www cyberconstableintelligence.com 📩 Telegram: https://t.me/cyberconstable
23.03.25 22:18 [email protected]

A few months ago, I stumbled upon a post about a cryptocurrency investment platform that I thought was a good idea for me to invest in crypto, I didn’t realize I was being catfished by the cryptocurrency investment manager who promised me huge returns on my investment. I lost my capital of $170,000 and interest without receiving any profits in return, I was devastated And I realized I had been scammed out of my cryptocurrency. I felt hopeless and didn’t know where to turn. i searched everywhere for a solution and That’s when my friend introduced me to a Crypto recovery platform Trust Geeks Hack Expert. At first, I was skeptical—after all, I had just been scammed—but Trust Geeks Hack Expert professionalism and transparency reassured me. They took the time to analyze my case, track my lost funds, and guide me through the recovery process. To my amazement, they successfully retrieved my stolen crypto! I can’t express how grateful I am to Trust Geeks Hack Expert team. Their expertise and dedication gave me a second chance. If you’ve been a victim of a crypto scam, don’t lose hope—Trust Geeks Hack Expert. is the real deal! for assistance, Email: [email protected] (TeleGram:: Trustgeekshackexpert) & what's A p p +1 7 1 9 4 9 2 2 6 9 3
24.03.25 10:03 Diegolola514gmail.com

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Fast Agent Recovery is a consulting firm that specializes in the recovery of assets from financial fraud. We know how to recover your funds and we have helped thousands of scam victims from around the world to recover their money. If you are a victim of Binary Options fraud, Forex fraud, Bitcoin fraud, Dating scams or one of the many other the online fraudulent practices that permeate the internet then file a complaint on www.fastrecoveryagent,.com to get an assessment if we can help you get your money back too. File a complaint: https://www.fastrecoveryagent.com/
27.03.25 01:52 debrapavon89

The moment my cold storage device flatlined, erasing $385,000 in Bitcoin I’d saved to launch a chain of zero-waste cafés, I became a ghost haunting my own life. I scrolled through forums until my eyes were streaming, trawling through threads filled with such mouthfuls as "irreversible blockchain entropy" and "cryptographic oblivion A Reddit thread finally revealed to me Cyber Constable Intelligence I reached out, the team at Cyber Constable Intelligence took immediate action. They began tracing the fraudulent car auction site and thoroughly investigating the scam. To my relief, after several weeks of hard work, Cyber Constable Intelligence successfully tracked down the scammers and recovered the full $385,000 I had lost. Their hard work allowed me to move on from this unfortunate experience, I highly recommend their services to anyone who has been affected by online fraud Reach out to their Info below WhatsApp: 1 252378-7611 Website info; www.cyberconstableintelligence.com
30.03.25 03:25 orrinharber

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30.03.25 06:22 grace111

I recommend Marie ([email protected] and WhatsApp: +1 7127594675) for recovering lost or stolen bitcoin, USDT, or any other cryptocurrency from fraudulent investment sites because they are very knowledgeable in the industry and will reimburse all of your money. I can say with confidence that she was the only one who was able to credit my account with $52,760 of the money that had gone missing, based on my prior transactions with them. She was the only one who could. Since they are the only ones that can fully return your missing funds to your account without any deductions, I sincerely appreciate their job and am recommending her to you today.
31.03.25 11:00 maryjul75

Fast Agent Recovery is a consulting firm that specializes in the recovery of assets from financial fraud. We know how to recover your funds and we have helped thousands of scam victims from around the world to recover their money. If you are a victim of Binary Options fraud, Forex fraud, Bitcoin fraud, Dating scams or one of the many other the online fraudulent practices that permeate the internet then file a complaint on www.fastrecoveryagent,.com to get an assessment if we can help you get your money back too. File a complaint: https://www.fastrecoveryagent.com/
02.04.25 23:16 frederickhandy

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03.04.25 07:57 messijohn

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03.04.25 10:58 elizabethrush89

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03.04.25 10:58 elizabethrush89

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04.04.25 09:49 mkihn634

When my $870,000 Bitcoin investment was trapped in the crumbling ruins of an offshore exchange, I felt as if my financial future had been buried at sea. It was meant to be my safety net, a shield against life’s uncertainties, but now it seemed like nothing more than a digital ghost. Each news update about the exchange's insolvency hit me like a tidal wave, dragging my hope under. It was a sleepless night, scanning through horror stories in forums as other people were losing everything. I could see my dreams flying out the window. That is until a former client, who had weathered a similar storm, said GRAYWARE TECH SERVICES in a hushed but confident tone. "They are the best there is," he said. "They were like detectives and lawyers.". Desperation drove me to call them, and in the initial discussion, I knew I was dealing with experts who had experienced everything. Their legal experts were well-versed with international financial rules like the back of their hands. They deciphered the network of shell corporations and offshore locations quicker than I could update my email. Their technical team, no less relentless, followed the transaction trails with precision akin to a surgeon. What impressed me most was their thoroughness. Every update came with legal documentation so polished it looked fit for a courtroom. They liaised with authorities across borders, cutting through red tape with the precision of a seasoned diplomat. When obstacles arose, and they did, the team adapted without breaking stride. Their persistence became my anchor. Exactly 27 days after my initial call, I received an email that made my heart skip. My Bitcoin had been recovered and safely transferred to my new secure wallet. I stared at the screen, tears mixing with disbelief and relief. GRAYWARE TECH SERVICES not only retrieved my money but also restored my faith in getting finances. They guided me through the entire process of protecting my assets in this unstable world known as the online space. I now sleep soundly knowing that an impenetrable system shields my backside, thanks to their assistance. Their legal acumen is as sharp as their technological prowess. I owe my financial future to their tireless work. They truly are the guardians of the digital age. You can reach them on web at ( https://graywaretechservices.com/ ) also on Mail: ([email protected]) whatsapp (+18582759508)
06.04.25 02:29 famousmullica

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06.04.25 11:30 maryjul75

As at September this year I got scammed by a fake investment broker, they took my savings, happiness, health, hope, trust and left me in tears and agony. My next thought was capitalized on suicide attempts, I also tried two different hackers who lured me to borrow for my colleagues but yet another scam. I saw a link on my Facebook group which I joined and luckily I came across www.fastrecoveryagent.com, they work in hand with the FBI in the united states. So I reported my case with evidence of payment made to the so-called investment broker, the ic3 investigator's carried out investigation to confirm if I was also tell the truth and after they must have concluded on my case I received back my money. I'm so grateful for your help today, words alone can't show how Happy and Alive I'm ever since I came across your great works. Thanks once more.
07.04.25 11:16 Lindaporche5

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07.04.25 11:16 Lindaporche5

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07.04.25 18:09 rashmiramesh

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07.04.25 18:32 elizabethrush89

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07.04.25 18:32 elizabethrush89

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07.04.25 21:03 elizabethrush89

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08.04.25 03:00 natashaohnson

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10.04.25 23:26 patricialovick86

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10.04.25 23:27 patricialovick86

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11.04.25 00:48 patricialovick86

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11.04.25 03:20 edwardoelliott6

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11.04.25 08:10 Beatrice Gallagher

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13.04.25 01:13 michaeldavenport218

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13.04.25 01:13 michaeldavenport218

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14.04.25 23:44 elizabethrush89

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17.04.25 03:56 tylerkevin

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18.04.25 20:39 michaeldavenport218

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18.04.25 20:39 michaeldavenport218

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19.04.25 02:00 [email protected]

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20.04.25 00:45 khouser

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