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[Перевод] Квантование в картинках: раскрываем тайны сжатия LLM

Большие языковые модели (LLM, Large Language Model), как подсказывает их название, часто отличаются значительными размерами и слишком велики для того, чтобы нормально работать на обычных компьютерах. Масштабы этих моделей могут измеряться миллиардами параметров. Обычно для обеспечений достойной скорости их работы необходимы GPU с серьёзными объёмами видеопамяти (VRAM).

Из-за этого проводится всё больше и больше исследований, посвящённых уменьшению размеров подобных моделей. Исследователи совершенствуют обучение моделей, используют адаптеры, прибегают к другим способам их оптимизации. Один из главных приёмов уменьшения размеров моделей называется квантованием (quantization).

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Здесь мы познакомимся с квантованием в сфере LLM и рассмотрим его с разных сторон. Это позволит глубоко понять суть происходящего. Мы исследуем разные методологии квантования, поговорим о различных вариантах его применения, обсудим принципы, лежащие в его основе.

В этом материале более 50 рисунков, которые помогут вам приблизиться к интуитивному пониманию разных аспектов квантования!

Часть 1. «Проблема» LLM

Как уже говорилось, LLM, большие языковые модели, названы так из-за огромного количества содержащихся в них параметров. В наши дни речь обычно идёт о миллиардах параметров (это, в основном, веса — weight), хранение которых может быть достаточно дорогим удовольствием. В ходе работы модели (этот этап жизненного цикла моделей часто называют «инференсом» — от «inference») создаются значения активации (их нередко называют просто «активациями» — от «activations»), представляющие собой, если не вдаваться в подробности, результат умножения входных данных и весов модели. Для хранения активаций тоже может понадобиться довольно много места.

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В результате перед нами встаёт задача эффективного представления миллиардов чисел. Нужно минимизировать объём памяти, необходимый для хранения каждого конкретного значения.

Начнём с самого начала и поговорим о том, как числовые значения представляются в компьютерных системах, а уже потом обсудим вопрос их оптимизации.

Как числовые значения представляются в компьютерах

Числовые значения часто хранят в формате числа с плавающей запятой (floating point number, float). Это — положительные или отрицательные числа с десятичным разделителем (decimal point).

Такие значения хранятся в виде «битов», или двоичных цифр. Стандарт IEEE-754 описывает то, как именно биты могут представлять числовые значения, играя одну из трёх ролей: бит знака (sign), поле порядка или показателя степени (exponent) и поле значащей части числа (significand) (ещё называемой дробной частью (fraction), или мантиссой (mantissa)).

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Обладая сведениями об этих трёх составных частях двоичного представления числа, можно узнать о том, что это за число, вычислить его десятичное значение:

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Чем больше битов используется для представления некоего значения — тем оно, обычно, точнее:

cc2349fdb7127c2cef0f04e6b2e7a0d3.png

Ограничения памяти

Чем больше битов имеется в нашем распоряжении — тем больше диапазон значений, которые можно представить с помощью этих битов.

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Интервал представимых чисел, которые можно закодировать в значениях некоего формата, называется его динамическим диапазоном (dynamic range), а расстояние между двумя соседними значениями называется точностью (precision).

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Очень приятной особенностью применения битов является следующий факт: используя некий формат представления чисел можно вычислить то, сколько памяти понадобится для хранения некоего числового значения. Так как один байт памяти вмещает 8 бит, можно создать простую формулу, подходящую для анализа большинства форматов представлений чисел с плавающей запятой.

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Учитывайте, что на то, сколько памяти или видеопамяти понадобится во время практического использования модели, влияют и другие факторы — такие, как размер контекста (context size) и архитектура модели.

Представим модель с 70 миллиардами параметров. Большинство моделей, в их исходном виде, используют 32-битные числа типа float (их ещё называют числами одинарной точности, single-precision numbers; их так же называют full-precision numbers). В результате только для того, чтобы загрузить такую модель в память, понадобится 280 Гб памяти.

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Тут возникает вполне понятное желание минимизировать количество битов, представляющих параметры (и не только при использовании, но и при обучении модели!). Но, по мере того, как снижается точность используемых чисел, точность ответов модели тоже обычно снижается.

Итак: нужно уменьшить количество битов, представляющих числа, и при этом не ухудшить точность работы модели… Именно тут нам на помощь приходит квантование!

Часть 2. Введение в квантование

Цель квантования заключается в том, чтобы уменьшить точность параметров модели, превращая числа более высокой разрядности (например — 32-битные числа с плавающей запятой) в числа более низкой разрядности (например — в 8-битные целые числа).

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При уменьшении количества битов, используемых для представления исходных параметров, часто наблюдается некоторая потеря точности (уменьшение шага между соседними числами, granularity).

Для того чтобы наглядно проиллюстрировать этот эффект, можно взять любое изображение и «перерисовать» его, используя всего 8 цветов (оригинал взят отсюда).

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Обратите внимание на то, что увеличенная часть нового изображения выглядит более «зернистой», чем та же часть оригинала, так как теперь мы используем меньше цветов.

Главная задача квантования — снизить количество битов (цветов), необходимых для представления исходных параметров, и при этом как можно лучше сохранить точность этих параметров.

Распространённые типы данных

Для начала рассмотрим распространённые типы данных и поговорим о том, как подействует на модель использование именно их, а не 32-битных чисел с плавающей запятой (FP32).

FP16

Рассмотрим пример, где показан переход от 32-битных к 16-битным числам (к числам с половинной точностью, half-precision numbers, FP16) с плавающей запятой.

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Обратит внимание на то, что диапазон чисел, которые помещаются в значения FP16, существенно меньше, чем в значения FP32.

BF16

Для того чтобы выйти на тот же диапазон значений, который соответствует исходному формату FP32, был создан формат bfloat16 (brain floating point), который представляет собой «урезанный FP32».

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Формат BF16 использует то же количество битов, что и FP16, но с его помощью можно представлять числа, лежащие в более широком диапазоне. BF16 часто используется в моделях глубокого обучения.

INT8

Когда количество битов уменьшают ещё сильнее — приближаются к области целочисленных представлений чисел, выходя из сферы чисел с плавающей запятой. Переход от типа FP32 к INT8 (8-битное целое число, integer), где применяется всего 8 битов, ведёт к тому, что количество битов, используемых ранее, сокращается в четыре раза:

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Целочисленные вычисления, что зависит от аппаратного обеспечения, могут выполняться быстрее, чем вычисления с плавающей запятой. Но это не всегда так. Правда, вычисления, где применяются типы данных с меньшим количеством бит, обычно выполняются быстрее.

В ходе каждого сеанса уменьшения разрядности числа производится отображение, «сжимающее» исходное FP32-представление числа так, чтобы оно, в новой форме, поместилось бы в представление с меньшим количеством битов.

На практике нет нужды в отображении всего диапазона FP32 [-3.4e38, 3.4e38] на диапазон INT8. Достаточно найти способ отображения диапазона имеющихся данных (параметров модели) на INT8.

Обычные методы сжатия/отображения представлены симметричным и асимметричным квантованием, которые являются разновидностями линейного отображения.

Исследуем эти методы квантования, позволяющие перейти от чисел формата FP32 к числам INT8.

Симметричное квантование

При применении симметричного квантования диапазон исходных значений, представленных числами с плавающей запятой, отображается на симметричный диапазон в квантованном пространстве, расположенный вокруг нуля. Обратите внимание на то, как в предыдущих примерах диапазоны значений до и после квантования остаются центрированными вокруг нуля.

Это значит, что новое значение для нуля из пространства значений с плавающей запятой — это тоже ноль в квантованном пространстве.

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Хорошим примером разновидности симметричного квантования является так называемое квантование с использованием максимальных абсолютных значений (absolute maximum (absmax) quantization).

Имея список значений, мы берём самое большое абсолютное значение (α) и используем его, задавая с его помощью границы диапазона при выполнении линейного отображения.

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Обратите внимание на то, что диапазон значений [-127, 127] — это симметричный диапазон. А [-128, 127] — это асимметричный диапазон. Применяемый диапазон значений INT8 зависит от метода квантования.

Так как это — линейное отображение, центрированное вокруг нуля, формула для его выполнения получается довольно-таки простой:

Сначала вычисляем коэффициент масштабирования (s), используя следующие данные:

  • b — это количество байтов, к которому мы хотим привести исходное значение в ходе квантования (8).

  • α — это самое большое абсолютное значение.

Затем используем s для квантования входных данных x:

61c34f97b2795e2aac0db741e9a51a4d.png

Подставляя в формулы имеющиеся значения — получим следующее

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Для восстановления исходных FP32-значений мы можем использовать уже вычисленный коэффициент масштабирования (s). Так мы проводим обратное квантование (деквантование, dequantize) ранее квантованных значений.

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Вот что получается, когда сначала применяют квантование, а потом — деквантование для восстановления исходных значений:

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Можно заметить, что некоторые числа, вроде 3,08 и 3,02, назначаются одному и тому же значению INT8, а именно — числу 36.

При обратном квантовании этих значений для возврата к FP32 они теряют некоторую долю точности и их больше нельзя отличить друг от друга.

Часто это называют ошибкой квантования (quantization error). Вычислить эту ошибку можно, найдя разницу между исходными и деквантованными значениями.

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В целом можно сказать, что чем меньше количество битов — тем больше ошибка квантования, с которой можно столкнуться.

Асимметричное квантование

Асимметричное квантование, в отличие от симметричного, не отличается свойством симметричного расположения значений вокруг нуля. При его применении выполняется сопоставление минимального (β) и максимального (α) значений из диапазона float-чисел, с минимальным и максимальным значениями квантованного диапазона.

Метод, который мы исследуем, называется квантованием относительно нуля (zero-point quantization).

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Заметили, что позиция нуля сдвинута? Потому это и называется «асимметричным квантованием». Минимальное и максимальное значения находятся на разном расстоянии от нуля в диапазоне [-7,59, 10,8].

Из-за того, что позиция нуля сдвинута, для выполнения линейного отображения необходимо вычислить нулевую точку диапазона INT8. Нам, как и ранее, необходимо найти коэффициент масштабирования (s), но теперь в соответствующей формуле используется разность между границами диапазона значений INT8 — [-128, 127].

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Обратите внимание на то, что сейчас вычисления получились немного сложнее. Это — из-за необходимости вычисления нулевой точки (z) в диапазоне INT8, используемой для смещения весов.

Теперь, как и прежде, подставим в формулы реальные значения:

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Для обратного квантования значений из INT8 в FP32 нам понадобится ранее вычисленный коэффициент масштабирования (s) и нулевая точка (z).

Что касается остального — тут всё так же просто и понятно, как раньше:

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Если сопоставить симметричное и асимметричное квантование, поместив рядом схемы, описывающие их, сразу будет видна разница между ними:

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Обратите внимание на то, что для симметричного квантования характерно центрирование значений вокруг нуля, и на то, что при применении асимметричного квантования значения смещены относительно нуля.

Отображение диапазона и обрезка значений

В предыдущих примерах мы исследовали то, как диапазон значений некоего вектора может быть отображён на представление меньшей разрядности. Хотя такой подход и позволяет отобразить весь диапазон значений вектора, с этим подходом сопряжена серьёзная проблема. Это — выбросы (outliers).

Представим, что имеется следующий вектор:

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Видно, что одно из значений гораздо больше остальных. Его и можно счесть выбросом. Если мы будем отображать полный диапазон этого вектора — все маленькие значения сведутся к одному и тому же представлению меньшей разрядности и их нельзя будет различить:

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Тут показаны результаты ранее рассмотренного метода квантования absmax. Так же ведёт себя и асимметричное квантование в том случае, если перед его проведением не прибегнуть к обрезке значений.

Обрезка (clipping, клиппинг) — это удаление некоторых из исходных значений. Обрезка предусматривает установку нового динамического диапазона для исходных значений — такого, чтобы всем выбросам было бы назначено одно и то же значение.

В нижеприведённом примере, если вручную задать динамический диапазон как [-5, 5], то все числа, находящиеся за его пределами, независимо от их значений, будут приведены к значениям -127 или 127:

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Главный плюс обрезки выбросов заключается в значительном уменьшении ошибки квантования для тех значений, которые выбросами не являются. Но ошибка квантования значений-выбросов при этом возрастает.

Калибровка

В рассмотренном примере показан безыскусный метод, где диапазон [-5, 5] был выбран произвольным образом. Процесс выбора подобных диапазонов называется калибровкой (calibration). Его цель заключается в поиске такого диапазона, который включает в себя как можно больше значений, благодаря чему минимизируется ошибка квантования.

Калибровка выполняется по-разному для разных видов параметров моделей.

Веса (и смещения)

Веса (weights) и смещения (biases) LLM можно рассматривать как статические значения, так как они известны до запуска модели. Например, файл Llama 3 размером примерно в 20 Гб состоит, в основном, из весов и смещений модели.

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Так как параметров-смещений гораздо меньше, чем параметров-весов (миллионы против миллиардов), смещения часто хранят в виде чисел более высокой точности (таких, как INT16) и основное внимание уделяют квантованию весов. В случае с весами, которые представлены статическими, заранее известными данными, применяются следующие приёмы калибровки, направленные на выбор диапазона значений, который будет использоваться при квантовании:

  • Самостоятельный выбор процентиля входного диапазона.

  • Оптимизация среднеквадратичной ошибки (mean squared error, MSE) между исходными и квантованными весами.

  • Минимизации энтропии (KL-дивергенции) между исходными и квантованными значениями.

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Если, например, применить метод, предусматривающий выбор перцентиля, это приведёт к обрезке значений, напоминающей то, что мы уже видели в предыдущих примерах.

Значения активации

Данные, которые постоянно обновляются при работе LLM обычно называют значениями активации.

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Обратите внимание на то, что эти значения называются значениями активации из-за того, что их часто «пропускают» через те или иные функции активации, вроде сигмоиды или ReLU.

Активации, в отличие от весов, меняются по мере того, как в модель, в ходе её работы, поступают новые данные. Это усложняет задачу их точного квантования.

Так как активации обновляются после прохождения данных по скрытым слоям, мы можем узнать их значения только во время работы модели, по мере того, как модель обрабатывает поступающие в неё данные.

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Если не вдаваться в подробности, то существует два метода калибровки системы квантования, применяемой для работы с весами и значениями активации:

  • Post-Training Quantization (PTQ).

    • Квантование выполняется после обучения модели.

  • Quantization Aware Training (QAT).

    • Квантование выполняется в процессе обучения или дообучения модели.

Часть 3. Квантование, выполняемое после обучения модели

Один из самых популярных подходов к квантованию — это Post-Training Quantization (PTQ) — квантование, выполняемое после обучения модели. Речь идёт об обработке весов и значений активации.

Квантование весов выполняется с использованием симметричного или асимметричного подходов.

Квантование активаций, однако, требует данных, полученных из работающей модели. Это позволяет узнать потенциальное распределение таких данных, так как до этого возможный диапазон их значений неизвестен.

Для квантования значений активации применяют два основных подхода:

  • Динамическое квантование.

  • Статическое квантование.

Динамическое квантование

После того, как данные прошли через скрытый слой, мы собираем вычисленные значения активации:

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Затем распределения значений активации используются для вычисления нулевой точки (z) и коэффициента масштабирования (s), которые необходимы для квантования.

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Процесс повторяется каждый раз, когда данные проходят через новый слой. В результате у каждого слоя имеются собственные значения z и s, а значит — с каждым из них связана собственная схема квантования.

Статическое квантование

Применение статического квантования, в отличие от динамического, не предусматривает вычисления нулевой точки (z) и коэффициента масштабирования (s) во время реальной работы модели. При статическом квантовании всё это вычисляют заранее.

Для нахождения этих значений используется калибровочный набор данных (calibration dataset). Он передаётся модели не с целью получения от неё, например, ответа на некий вопрос, а с целью сбора потенциальных распределений значений активации.

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После сбора этих значений можно вычислить s и z для выполнения квантования во время реальной работы модели.

А вот когда модель занимается настоящей работой — значения s и z повторно не вычисляются. Они используются в глобальном масштабе и применяются для квантования всех значений активации.

В целом можно отметить, что динамическое квантование обычно точнее статического, так как при его применении выполняется вычисление значений s и z для каждого скрытого слоя. Но такой подход может повысить время, необходимое на проведение вычислений, обеспечивающих работу модели.

Статическое же квантование, в отличие от динамического, не такое точное, но более быстрое, так как при его применении значения s и z уже известны.

Мир 4-битного квантования

Квантование с использованием значений, разрядность которых меньше 8 битов, зарекомендовало себя как сложная задача. Всё дело в том, что с пропаданием каждого бита растёт ошибка квантования. К счастью, существует несколько хитрых способов квантования, предусматривающих использование 6-ти, 4-х и даже 2-х битных значений (хотя уходить ниже 4 битов при использовании этих методов обычно не рекомендуется).

Мы исследуем два подобных метода, которые обычно всплывают на HuggingFace:

  • GPTQ (вся модель работает на GPU).

  • GGUF (можно переносить некоторые слои в оперативную память и обрабатывать их с помощью CPU).

GPTQ

GPTQ — это, вероятно, один из самых известных среди используемых на практике методов 4-битного квантования (Frantar, Elias, et al. «Gptq: Accurate post-training quantization for generative pre-trained transformers.» arXiv preprint arXiv:2210.17323 (2022)).

Он задействует асимметричное квантование и применяется послойно. Каждый слой, перед переходом к следующему, обрабатывается независимо от других:

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Во время этого послойного процесса квантования сначала, на основе весов слоя, выполняется вычисление обратной матрицы Гессиана (Inverse Hessian). Это — матрица вторых производных функции потерь модели. Она показывает нам то, насколько чувствителен выход модели к изменениям каждого из весов.

Проще говоря — она демонстрирует важность (обратную) каждого веса в слое: чем меньше значение, соответствующее весу, тем важнее этот вес.

Веса, которым в матрице Гессиана соответствуют меньшие значения, более важны, так как маленькие изменения в этих весах могут привести к серьёзным изменениям в результатах работы модели.

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Теперь квантуем и деквантуем веса первой строки матрицы весов:

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Этот процесс позволяет вычислить ошибку квантования (q), которую можно взвесить, используя обратную матрицу Гессиана (h_1).

Получается, что мы, принимая во внимание важность веса, находим взвешенную ошибку квантования:

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Далее — перераспределяем взвешенную ошибку квантования по другим весам в строке. Это позволяет поддерживать в стабильном состоянии общий функционал сети и выходные данные, формируемые сетью.

Например, если сделать это для второго веса, для .3 (x_2), то получится, что мы добавляем к нему ошибку квантования (q), умноженную на соответствующее значение из обратной матрицы Гессиана (h_2).

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Можно сделать то же самое и для третьего веса в том же ряду:

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Мы продолжаем выполнять этот процесс перераспределения взвешенной ошибки квантования до тех пор, пока не завершится квантование всех значений.

Этот подход показывает отличные результаты из-за того, что веса обычно связаны друг с другом. Поэтому, когда у некоего веса возникает ошибка квантования, значения других, связанных с ним весов, соответствующим образом корректируются (посредством обратной матрицы Гессиана).

Примечание: авторы этого метода использовали несколько хитрых приёмов для ускорения вычислений и улучшения производительности. Например — это добавление коэффициента демпфирования к матрице Гессиана, это «отложенная пакетная обработка» («lazy batching») данных, это выполнение предварительных вычислений с применением метода Холецкого. Очень рекомендую посмотреть это видео, посвящённое данной теме.

Совет: загляните в репозиторий проекта EXL2 в том случае, если вам нужен метод квантования, ориентированный на оптимизацию производительности и повышение скорости работы сети.

GGUF

GPTQ — это прекрасный метод квантования, который подходит для обработки полных LLM на GPU. Но у GPU не всегда имеется достаточно ресурсов для подобных расчётов. В таких ситуациях можно, вместо GPTQ, прибегнуть к методу GGUF. Он предусматривает перенос слоёв LLM в оперативную память и их обработку на CPU (подробности о GGUF можно найти здесь).

Применение GGUF позволяет, в случаях, когда не хватает видеопамяти, пользоваться и обычными системными ресурсами компьютера, и ресурсами видеоускорителя.

В этот метод постоянно вносят изменения, он может зависеть от разрядности квантования. Но, в целом, в его основе лежат следующие принципы.

Сначала веса слоя разбивают на «суперблоки» («super» block), каждый из которых содержит набор «подблоков» («sub» block). Эти блоки используются для вычисления коэффициента масштабирования (s) и значения «альфа» (α).

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Для квантования «подблока» можно использовать рассмотренный ранее метод absmax. Напомним, что при его применении веса умножают на коэффициент масштабирования (s):

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Коэффициент масштабирования вычисляется с применением информации из «подблоков», но его квантование выполняется с использованием информации из «суперблока», у которого имеется собственный коэффициент масштабирования:

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Такое блочное квантование использует коэффициент масштабирования (s_super) из «суперблока» для квантования коэффициента масштабирования (s_sub) из «подблока».

У каждого из коэффициентов масштабирования может быть собственный уровень квантования, так как при работе с «суперблоком» обычно используются значения более высокой точности, чем при работе с «подблоками».

В качестве примера рассмотрим несколько уровней квантования (2-битное, 4-битное и 6-битное):

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Обратите внимание на то, что, в зависимости от типа квантования, необходимо дополнительное минимальное значение (m) для корректировки нулевой точки. Квантование этих значений выполняется так же, как и квантование коэффициента масштабирования (s).

Взгляните на исходный PR, где имеется обзор всех уровней квантования. Кроме того, обратите внимание на этот PR, где есть подробности о квантовании с использованием матриц значимости.

Часть 4. Обучение моделей с учётом квантования

В 3 части мы говорили о том, как квантовать модель после обучения. Минус этого подхода заключается в том, что при квантовании не учитывается реальный процесс обучения сети.

Именно тут уместно вспомнить об обучении моделей с учётом квантования (Quantization Aware Training, QAT). При таком подходе вместо того, чтобы квантовать модель после обучения (Post-Training Quantization, PTQ), стремятся к тому, чтобы находить соответствующие значения в процессе обучения моделей. Ниже показано, как модель «изучает» параметры квантования во время обратного прохода.

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Применение QAT обычно даёт более точные результаты, чем применение PTQ, так как квантование уже учитывается в ходе обучения модели. Вот как это всё работает.

В процессе обучения модели выполняются так называемые «фиктивные» (fake) операции квантования. Это — когда сначала квантуют FP32-веса, например, в INT4, а потом деквантуют обратно в FP32:

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Это позволяет модели учесть процесс квантования во время обучения, при вычислении функции потерь и при обновлении значений весов.

При применении QAT делается попытка исследования ландшафта функции потерь в поиске «широких» минимумов. Это позволяет уменьшить ошибки квантования, так как работа с «узкими» минимумами обычно приводит к увеличению ошибок квантования.

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Например — представим, что во время обратного прохода не учитывается квантование. Мы, в соответствии с методом градиентного спуска, выбираем вес, которому соответствует наименьшее значение функции потерь. Но, если этот вес попадает в «узкий» минимум, с его квантованием будет сопряжена достаточно большая ошибка.

В противовес такому подходу, в том случае, если при обучении модели учитывается квантование, будет выбрано другое значение веса, соответствующее «широкому» минимуму. Применение этого значения приведёт к тому, что ошибка квантования будет меньше, чем в первом случае.

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В результате, хотя применение PTQ приводит к меньшим потерям при использовании значений высокой точности (например — FP32), применение QAT приводит к меньшим потерям при использовании значений низкой точности (например — INT4). А это — именно то, что нам нужно.

Эпоха 1-битных LLM: BitNet

Переход к четырём битам, как мы уже видели, означает работу со значениями достаточно скромных размеров. А что если пойти ещё дальше?

А дальше нас ждёт BitNet. В этой модели для представления весов используются 1-битные значения. Каждый конкретный вес может равняться либо -1, либо 1.

Достигается это путём внедрения процесса квантования непосредственно в трансформенную архитектуру.

Напомним, что трансформеры — это основа большинства LLM. Применение трансформеров предусматривает выполнение вычислений, в которых используются линейные (linear) слои:

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Эти линейные слои обычно представлены с использований чисел достаточно высокой точности, вроде FP16, и именно в этих слоях и размещено большинство весов модели.

В модели BitNet линейные слои заменяют на сущности, которые разработчики этой модели называют BitLinear-слоями:

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BitLinear-слой работает так же, как и обычный линейный слой — он вычисляет выходные данные, перемножая веса на активации.

Но, в отличие от линейных слоёв, в слое BitLinear веса модели представлены с помощью 1-битных значений, а значения активации — с помощью INT8-чисел:

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В слое BitLinear, как и в ходе работы QAT, проводятся «фиктивные» операции квантования, что позволяет проанализировать эффект, который оказывает на работу модели квантование весов и активаций:

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В публикации о BitNet вместо α используется γ, но, так как в вышеприведённых примерах используется α, здесь мы продолжим эту традицию. Кроме того, обратите внимание на то, что β здесь — это не то же самое значение, которое мы использовали при квантовании относительно нуля. Здесь это — среднее абсолютное значение.

Поговорим о том, как именно работают BitLinear-слои.

Квантование весов

В ходе обучения модели веса хранятся с использованием INT8-чисел, а затем производится их квантование в 1-битные значения с использованием простой стратегии, реализуемой посредством знаковой функции (signum function).

Смысл этого в том, что распределение весов смещается так, чтобы его центром стал бы 0, а после этого всему, что левее нуля, назначется -1, а всему, что правее — 1:

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В дополнение к этому тут запоминается значение β (среднее абсолютное значение), которым пользуются для деквантования.

Квантование значений активации

Для обработки значений активации в BitLinear применяется absmax-квантование. Активации преобразуются из типа FP16 в тип INT8, так как они, для целей умножения матриц (×), должны быть представлены с более высокой точностью.

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Здесь, кроме того, запоминается значение α (самое большое абсолютное значение), которое позже понадобится для деквантования.

Деквантование

Мы запомнили значения α (самое большое абсолютное значение) и β (среднее абсолютное значение), так как они понадобятся нам для деквантования значений активации и возврата их в формат FP16.

Выходные значения активации перемасштабируют с использованием {α, γ}, что позволяет деквантовать их, преобразовав к числам, точность которых соответствует их исходному виду:

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Вот и всё! Эта процедура сравнительно проста, она позволяет описывать модели, используя всего два значения — -1 и 1.

Воспользовавшись этим подходом, авторы заметили, что чем больше становится размер модели — тем меньше оказывается разрыв между эффективностью работы её 1-битного и FP16-варианта.

Но это справедливо только для достаточно больших моделей (>30 миллиардов параметров). Эффективность моделей меньшего размера страдает сильнее.

1,58 бита всем LLM

Модель BitNet 1.58b была выпущена для того, чтобы решить вышеупомянутую проблему, связанную с масштабированием (Ma, Shuming, et al. «The era of 1-bit llms: All large language models are in 1.58 bits.» arXiv preprint arXiv:2402.17764 (2024).).

Суть применяемого в ней нового метода заключается в том, что веса модели теперь, помимо значений -1 и 1, могут принимать и значение 0. При работе с моделью используется тернарный подход к представлению весов. Интересно то, что столь простое решение — добавление к возможным значениям весов обычного ноля — привело к значительным улучшениям по сравнению с BitNet и позволило серьёзно ускорить вычисления.

Мощь ноля

Почему начало использование значения 0 привело к столь серьёзным улучшениям?

Всё дело — в особенностях умножения матриц!

Для начала разберёмся с тем, как умножают матрицы. При вычислении выходов слоя модели мы умножаем матрицу весов на вектор входных значений. Ниже можно видеть первый этап умножения первой строки матрицы весов на входные значения:

26d0f1ad34f418b2fd302e7363a894dc.png

Обратите внимание на то, что то умножение, которым мы тут занимаемся, состоит из двух операций. Сначала перемножают отдельные веса и входные значения, а потом складывают то, что получилось.

Создателям BitNet 1.58b удалось отказаться от операции умножения. Смысл значений, которые могут принимать веса, заключается в следующем:

  • 1: прибавить это значение.

  • 0: это значение не нужно.

  • -1: вычесть это значение.

В результате, если веса модели были подвергнуты квантованию, в ходе которого их преобразовали в 1,58-битный формат, умножение матрицы весов и вектора входных данных будет выглядеть так:

4fa3b4ab7ef81a21c817f27a4628861c.png

Этот подход не только значительно ускоряет вычисления, но ещё и позволяет выполнять фильтрацию признаков (feature filtering).

Записывая в определённые веса значение 0, мы можем просто их игнорировать. А в случае с 1-битным представлением весов такой возможности нет, так как значения при таком подходе либо вычитают, либо складывают, но не игнорируют.

Квантование

Для квантования весов в BitNet 1.58b используется метод absmean — разновидность метода absmax, о котором мы уже говорили.

В ходе работы сжимают распределение весов и используют среднее абсолютное значение (α) для квантования других значений. Каждое значение, в итоге, сводится к одному из трёх вариантов — -1, 0 или 1:

bf7bac099d3aac51aca1cac73911cf15.png

В BitNet 1.58b активации квантуются почти так же, как в BitNet. Единственное различие заключается в том, что для масштабирования этих значений применяется диапазон [-2ᵇ⁻¹, 2ᵇ⁻¹], а не [0, 2ᵇ⁻¹]. Для квантования используется метод absmax.

На этом всё! Для квантования значений в 1,58-битный формат нужно (в основном) прибегнуть к двум приёмам:

  • Надо воспользоваться тернарным представлением весов, добавив 0 к их возможным значениям, которые теперь могут быть представлены числами [-1, 0, 1].

  • Надо воспользоваться методом absmean для квантования весов.

Модель BitNet b1.58 с 13 миллиардами параметров эффективнее в смысле задержек, использования памяти и потребления энергии, чем LLM, использующая формат FP16 с 3 миллиардами параметров.

Как результат — мы получаем легковесные модели, на каждый параметр которых, в среднем, нужно 1,58 бита. Это не только экономит память, необходимую для их хранения, но и повышает эффективность использования вычислительных ресурсов при работе с ними.

Итоги

Вот и кончилось наше путешествие по миру квантования! Надеюсь, этот материал позволил вам лучше понять возможности и перспективы сжатия LLM, помог разобраться в таких вещах, как GPTQ, GGUF и BitNet. Кто знает — до чего дойдёт прогресс в деле уменьшения размеров и повышения эффективности больших языковых моделей?

Если хотите ещё картинок, посвящённых разным аспектам LLM — можете взглянуть на мою книгу «Hands-On Large Language Models».

О, а приходите к нам работать? 🤗 💰

Мы в wunderfund.io занимаемся высокочастотной алготорговлей с 2014 года. Высокочастотная торговля — это непрерывное соревнование лучших программистов и математиков всего мира. Присоединившись к нам, вы станете частью этой увлекательной схватки.

Мы предлагаем интересные и сложные задачи по анализу данных и low latency разработке для увлеченных исследователей и программистов. Гибкий график и никакой бюрократии, решения быстро принимаются и воплощаются в жизнь.

Сейчас мы ищем плюсовиков, питонистов, дата-инженеров и мл-рисерчеров.

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Источник

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    I would recommend NVIDIA TECH HACKERS to anyone that needs this service. I decided to get into crypto investment and I ended up getting my crypto lost to an investor late last year. The guy who was supposed to be managing my account turned out to be a scammer all along. I invested 56,000 USD and at first, my reading and profit margins were looking good. I started getting worried when I couldn’t make withdrawals and realized that I’ve been scammed. I came across some of the testimonials that people said about NVIDIA TECH HACKERS and how helpful he has been in recovering their funds. I immediately contacted him in his mail at [email protected] so I can get his assistance. One week into the recovery process the funds were traced and recovered back from the scammer. I can't appreciate him enough for his professionalism.

  • 17.10.25 20:23 stevekalfman

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  • 17.10.25 21:42 marcushenderson624

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

  • 17.10.25 21:42 marcushenderson624

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  • 17.10.25 21:42 marcushenderson624

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

  • 21.10.25 08:39 debby131

    Given how swiftly the cryptocurrency market moves, losing USDT may be a terrifying and upsetting experience. Whether you experienced a technical problem, a transaction error, or were the victim of fraud, it is important to understand the potential recovery routes. Marie can assist you in determining the specific actions you can take to attempt to regain your lost USDT when you need guidance and clarification. You can reach her via email at [email protected] and WhatsApp at +1 7127594675.

  • 21.10.25 11:45 harristhomas7376

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  • 21.10.25 11:45 harristhomas7376

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  • 22.10.25 04:48 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 22.10.25 07:36 donnacollier

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  • 22.10.25 11:59 elizabethrush89

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  • 22.10.25 11:59 elizabethrush89

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  • 22.10.25 16:31 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 23.10.25 01:52 marcushenderson624

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

  • 23.10.25 01:52 marcushenderson624

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

  • 23.10.25 15:47 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 23.10.25 21:43 patricialovick86

    How To Recover Your Bitcoin Without Falling Victim To Scams: A  Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected]  You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 23.10.25 21:43 patricialovick86

    How To Recover Your Bitcoin Without Falling Victim To Scams: A  Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected]  You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 24.10.25 06:08 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 24.10.25 06:40 elizabethrush89

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

  • 24.10.25 06:40 elizabethrush89

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

  • 24.10.25 06:54 MATT PHILLIP

    I never imagined I’d fall for a crypto romance scam but it happened. Over the course of a few months, I sent nearly $150,000 worth of Bitcoin to someone I genuinely believed I was building a future with. When they disappeared without a trace, I was left heartbroken, humiliated, and financially devastated. For a long time, I didn’t tell anyone. I felt ashamed. But eventually, while searching for answers, I came across a Reddit thread that mentioned Agent Jasmine Lopez. I reached out, not expecting much. To my surprise, she treated me with kindness, not judgment. She used advanced tools like blockchain forensics, IP tracing, and smart contract analysis and with persistence and legal support, she was able to recover nearly 85% of what I lost. I know not everyone gets that kind of outcome, but thanks to [email protected] WhatsApp at +44 736-644-5035, I’ve started to reclaim not just my assets, but my confidence and peace of mind. If you’re going through something similar, you’re not alone and there is hope.

  • 24.10.25 18:18 patricialovick86

    How To Recover Your Bitcoin Without Falling Victim To Scams: A  Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected]  You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 24.10.25 18:18 patricialovick86

    How To Recover Your Bitcoin Without Falling Victim To Scams: A  Testimony Experience With Capital Crypto Recover Services, Contact Telegram: @Capitalcryptorecover Dear Everyone, I would like to take a moment to share my positive experience with Capital Crypto Recover Services. Initially, I was unsure if it would be possible to recover my stolen bitcoins. However, with their expertise and professionalism, I was able to fully recover my funds. Unfortunately, many individuals fall victim to scams in the cryptocurrency space, especially those involving fraudulent investment platforms. However, I advise caution, as not all recovery services are legitimate. I personally lost $273,000 worth of Bitcoin from my Binance account due to a deceptive platform. If you have suffered a similar loss, you may be considering crypto recovery, The Capital Crypto Recover is the most knowledgeable and effective Capital Crypto Recovery Services assisted me in recovering my stolen funds within 24 hours, after getting access to my wallet. Their service was not only prompt but also highly professional and effective, and many recovery services may not be trustworthy. Therefore, I highly recommend Capital Crypto Recover to you. i do always research and see reviews about their service, For assistance finding your misplaced cryptocurrency, get in touch with them, They do their jobs quickly and excellently, Stay safe and vigilant in the crypto world. Contact: [email protected]  You can reach them via email at [email protected] OR Call/Text Number +1 (336)390-6684 his contact website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 25.10.25 00:49 tyleradams

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  • 25.10.25 00:50 stevekalfman

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  • 25.10.25 05:05 victoriabenny463

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  • 25.10.25 10:13 wendytaylor015

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  • 25.10.25 10:13 wendytaylor015

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

  • 26.10.25 01:03 Christopherbelle

    A terrible financial shock hit me hard. I lost $860,000 in a fake crypto scheme. Things looked good until I tried to cash out my earnings. Suddenly, my account vanished into thin air. I told the police many times, but help never showed up. Despair washed over me; I felt totally beaten. Then, I found Sylvester Bryant Intelligence. Right away, they were skilled and honest about the whole thing. They looked deep into my situation. They tracked where the stolen funds went. They fought hard for me every step of the way. To my great surprise, they got my money back. I thought that was impossible. I owe them so much for their hard work and truthfulness. Sylvester Bryant Intelligence restored my peace. If scams took your crypto, contact them now. Contact Info: Yt7cracker@gmail . com WhatsApp: ‪+1 512 577 7957. or +44 7428 662701.

  • 26.10.25 18:09 victoriabenny463

    A few months ago, I fell for what looked like a legitimate online trading platform. The website was sleek, the support team was responsive, and they even showed me fake profit charts that seemed to grow daily. I started small, and after a few test withdrawals worked, I invested more eventually depositing 2 Bitcoin in hopes of long-term returns. But one morning, when I tried to withdraw, everything changed. The platform suddenly locked my account, claiming there were additional fees to release my funds. The support chat stopped responding, and the website started giving error messages. That’s when I realized I’d been scammed. I was devastated. Two Bitcoins wasn’t pocket change, it was my savings. I filed reports online, but everyone told me it was impossible to recover crypto once it’s sent. Then, through a discussion forum, I came across Dexdert Net Recovery, a team specializing in tracing and recovering stolen digital assets. Skeptical but desperate, I reached out. Their response was immediate and professional. They asked for transaction IDs, wallet addresses, screenshots of my correspondence with the fake platform, and any proof of payment I had. Within days, their analysts traced the stolen Bitcoin through multiple wallets using blockchain forensics, They updated me step-by-step, Working with their recovery experts and blockchain partners, Dexdert Net Recovery successfully froze the funds and verified ownership of my wallet. After 2 days of persistence and verification, I finally saw my 2 Bitcoin restored back into my own wallet. I couldn’t believe it. What stood out most wasn’t just that they recovered my funds, but how transparent and supportive the Dexdert team was throughout the process. They educated me on crypto safety, how to spot fake trading sites, and how to secure my digital assets in the future. Dexdert Net Recovery truly lives up to its name. They gave me back not just my Bitcoin, but my peace of mind. Reach Out Information Below Via: Email: ([email protected]) Telegram: (https://t.me/Dexdertprorecovery) WhatsApp: (+1 (859) 609‑4156)

  • 27.10.25 11:15 harristhomas7376

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

  • 27.10.25 11:15 harristhomas7376

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

  • 27.10.25 17:08 raymondgonzales

    Three Cheers for Wizard Hilton Cyber Tech, Bitcoin Savior Extraordinaire In the ever-evolving world of cryptocurrency, where volatility and uncertainty reign supreme, one individual has emerged as a beacon of hope and innovation - Wizard Hilton Cyber Tech, the self-proclaimed "Bitcoin Savior Extraordinaire." This enigmatic figure, with his uncanny ability to navigate the complexities of the digital asset landscape, has captivated the attention of crypto enthusiasts and skeptics alike. Wizard Hilton Cyber Tech journey began as a humble programmer, tinkering with the underlying blockchain technology that powers Bitcoin. But it was his visionary approach and unwavering commitment to the cryptocurrency's potential that propelled him into the limelight. Through a series of strategic investments, groundbreaking algorithmic trading strategies, and a knack for anticipating market shifts, Wizard Hilton Cyber Tech has managed to consistently outperform even the savviest of Wall Street traders. Wizard Hilton Cyber Tech ability to turn even the most tumultuous of Bitcoin price swings into opportunities for substantial gains has earned him the moniker "Wizard," a testament to his unparalleled mastery of the digital currency realm. But Wizard Hilton Cyber Tech impact extends far beyond personal wealth accumulation - Wizard Hilton Cyber Tech has tirelessly advocated for the widespread adoption of Bitcoin, working tirelessly to educate the public, lobby policymakers, and collaborate with industry leaders to overcome the barriers that have long hindered the cryptocurrency's mainstream acceptance. With infectious enthusiasm and boundless energy, Wizard Hilton Cyber Tech has become a true champion of the Bitcoin cause, inspiring a new generation of crypto enthusiasts to embrace the transformative power of this revolutionary technology. As the digital currency landscape continues to evolve, Wizard Hilton Cyber Tech name will undoubtedly be etched in the annals of cryptocurrency history as a visionary, a trailblazer, and, above all, the Bitcoin Savior Extraordinaire. To get your stolen bitcoin back, reach out to Wizard Hilton Cyber Tech via: Email : wizardhiltoncybertech ( @ ) gmail (. ) com WhatsApp number  +18737715701 Good Day.

  • 27.10.25 17:20 fatimanorth

    The Federal Trade Commission said that more than $1 billion had been lost to cryptocurrency scams in 2023. Victims frequently lose everything, including hope, trust, and wealth. When your hard-earned money disappears into digital shadows, you feel trapped. Nowadays, cryptocurrency frauds are very common. You can get help from recovery specialist Marie at [email protected] and via WhatsApp at +1 7127594675.

  • 28.10.25 00:55 elizabethrush89

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  • 28.10.25 00:55 elizabethrush89

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  • 29.10.25 03:43 Christopherbelle

    A sudden money crisis struck me. I dropped $82,000 to a phony crypto scam. All seemed fine until I went to pull out my gains. Then, i was unable to get back my hard earned money. I reported it to the police several times. No aid came my way. Deep sadness hit me; I felt crushed. That's when I learned about Sylvester Bryant Intelligence. They acted quick and fair from the start. They checked my case closely. They followed the path of my lost cash. They pushed strong for me at each turn. In shock, they brought my funds back. I had figured it could not happen. I thank them for their effort and honesty. Sylvester Bryant Intelligence gave me calm again. If a scam stole your crypto, reach out to them today. Contact Info: Yt7cracker@gmail . com WhatsApp: +1 512 577 7957 or +44 7428 662701.

  • 29.10.25 10:56 wendytaylor015

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

  • 29.10.25 10:56 wendytaylor015

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  • 30.10.25 09:49 Christopherbelle

    I suffered a huge money blow when $80,000 slipped away in a bogus crypto plot. It all looked great at first. Then I went to pull out my gains, and poof—my account was gone. I kept telling the cops about it, but they did zip. I felt shattered and lost. That's when I stumbled on Sylvester Bryant Intelligence. Things turned around fast. Right away, they showed clear steps, sharp know-how, and real drive to fix it. They tracked my missing cash step by step and pushed hard till they got it back. I swear, I figured it was a lost cause. Their straight-up work and trust brought back my calm. If a scam stole your crypto, reach out to them now. yt7cracker@gmail . com | WhatsApp: +1 512 577 7957 or +44 7428 662701

  • 30.10.25 12:03 elizabethrush89

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  • 31.10.25 13:25 Lillian Lizzy

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  • 31.10.25 13:25 Lillian Lizzy

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  • 01.11.25 03:27 elizabethrush89

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  • 01.11.25 03:27 elizabethrush89

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  • 01.11.25 15:47 kattygates

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  • 01.11.25 15:49 kattygates

    I met an individual on an international dating app, Bumpy. After talking for a few months, the individual encouraged me to start investing on crypto asset trading platform, btc01.org. I started trading and believed I was making profit until I tried to withdraw some of my money, I was asked to pay extra for tax which I did and they kept asking then I knew I have been scammed, I had to contact HACKWEST AT WRITEME DOT COM who then helped to recover the money, the total money I invest is $198,000 in bitcoin. The website is no longer operational. If anyone is in a similar issue, you can as well contact HACK WEST. They also fixed my credit report.

  • 02.11.25 10:55 Christopherbelle

    I suffered a crushing blow when $45,000 slipped away in a phony crypto scam. It all looked great at first. But when I went to pull out my gains, my account just vanished. I called the cops over and over. Still, no luck. I felt shattered and lost. That’s when I stumbled on Sylvester Bryant Intelligence. Things turned around fast. Right away, they showed real openness, sharp skills, and a drive to fix it. They tracked my missing money step by step. And they pushed hard until they got it back. I honestly didn’t believe it could happen. Their straight-up approach and solid work brought back my calm. If a scam took your crypto, reach out to them now. yt7cracker@gmail . com | WhatsApp: +1 512 577 7957 or +44 7428 662701

  • 02.11.25 10:56 Christopherbelle

    I suffered a crushing blow when $45,000 slipped away in a phony crypto scam. It all looked great at first. But when I went to pull out my gains, my account just vanished. I called the cops over and over. Still, no luck. I felt shattered and lost. That’s when I stumbled on Sylvester Bryant Intelligence. Things turned around fast. Right away, they showed real openness, sharp skills, and a drive to fix it. They tracked my missing money step by step. And they pushed hard until they got it back. I honestly didn’t believe it could happen. Their straight-up approach and solid work brought back my calm. If a scam took your crypto, reach out to them now. yt7cracker@gmail . com | WhatsApp: +1 512 577 7957 or +44 7428 662701

  • 02.11.25 15:23 michaeldavenport238

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  • 02.11.25 15:23 michaeldavenport238

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

  • 04.11.25 04:59 Christopherbelle

    How to Recover Bitcoin Stolen by a Scammer I dropped £75,000 into a fake crypto scheme. Everything seemed perfect right from the start. But when I tried to cash out my profits, the whole account disappeared. I rang the police time and again. They offered no help at all. I felt broken and alone. Then I found Sylvester Bryant Intelligence. My luck changed in no time. They spoke plainly, worked smart, and focused on my case. They followed the trail of my lost cash bit by bit. They fought until they recovered it all. I never thought it was possible. Their honest approach and expertise restored my peace. If a scam has stolen your crypto, contact them now. [email protected] | WhatsApp: +1 512 577 7957 or +44 7428 662701

  • 04.11.25 09:30 robertalfred175

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  • 04.11.25 09:30 robertalfred175

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

  • 05.11.25 02:43 jordan_11

    I made a mistake of not researching about their their investment firm(Binary option), it had a lot of promises which made me fall for all their fake promises , after trying to get my withdrawal out from there, I started having a lot of difficulties. Then I contacted their customer support, they only replied a week after and strangely wanted me to make more deposits just to enable my withdrawal, so i started my research of how I could report this treacherous company to the necessary authorities and also maybe i could get back my investment, The only people that were willing to help me out was Treqora Intel, their customer support replied so fast and immediately directed me to their investigative team which took my up my case and started the tracing and reclaim process with swift, after a week i got an email from Treqora Intel stating that a partial amount had been reclaimed, I was so grateful about their help , then I was asked to decide if I wanted a Partial deposit or wanted all my investment reclaimed and sent to me at once, But I was so desperate at the time so I asked that my partial reclaimed ETH be sent to me first , So the deposit took about few minutes before i got it in my wallet and withdrew it. Then my remaining investment took 2 weeks to get back according to their customer support at Treqora Intel. Immediately They got it, I was contacted again Via email that my reclaimed had been fully reclaimed, so grateful till this day to Treqora Intel for taking their time to help me out with what could have been a very depressing moment to me. And yes no upfront payment until the reclaim has been completed and you've been paid fully. Email:support@treqora. com 📱 WhatsApp: ‪‪‪+1 (7 7 3) 9 7 7 - 7 8 7 7‬ ‬‬ 🌐 Website: Treqora. com.

  • 05.11.25 02:44 jordan_11

    Crypto Recovery Services _Hire Treqora Intel

  • 05.11.25 02:48 jordan_11

    Legitimate Crypto Recovery Companies-CONSULT TREQORA INTEL. After falling victim to a crypt0currency scam group, I lost $35,000 worth of USDT. I thought all hope was lost from the experience of losing my hard-earned money to scammers. I was devastated and believed there was no way to recover my funds. Fortunately, I started searching for help to rec0vermy stolen funds and I came across a lot of testimonials online about TREQORA INTEL , an agent who helps in rec0ver of lost bitc0in funds, I contacted TREQORA INTEL , and with their expertise, they successfully traced and rec0vered my stolen assets. Their team was professional, kept me updated throughout the process, and demonstrated a deep understanding of blockchain transactions and rec0very protocols. They are trusted and very reliable with a 100% successful rate record Rec0very bitc0in, I’m grateful for their help and highly recommend their services to anyone seeking assistance with lost crypt0. Email:support@treqora. com Whats_App: ‪‪‪‪‪+1 (7 7 3 ) 9 7 7 - 7 8 7 7‬ ‬‬ ‬‬, Website: Treqora dot com.

  • 05.11.25 16:20 robertalfred175

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

  • 05.11.25 16:21 robertalfred175

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

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