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Математические основы рекуррентных нейросетей (детские вопросы и ответы, о которых не принято говорить)

Зачем всё это?

Сейчас в сети можно встретить огромное количество разной литературы и курсов, которые предлагают разобраться в основах нейросетей, так зачем же нужна ещё одна подобная статья? И почему именно рекуррентные нейросети?

Что касается второго вопроса, то на него я отвечу чуть позже в этом тексте, а сейчас рассмотрим вопрос полезности этой публикации. До недавнего времени я и сам смотрел бы с сомнением на идею написания такой статьи. Изначально мне хотелось просто проверить некоторые свои идеи. Но пока я приводил в порядок все свои знания по этой теме, стараясь добиться полного понимания, я осознал, что большинство источников в основном просто повторяют друг друга. Некоторые вопросы для меня оставались не прояснёнными. Размышляя над различными моментами я подумал, что возможно эти рассуждения смогут принести пользу ещё кому-то. Так возникло желание не просто упорядочить свои мысли, но и оформить их в виде статьи.

Я постарался написать этот текст таким образом, чтобы он достаточно полно описывал математические выкладки, и при этом не дублировал множество других источников на эти темы. Поэтому часть материала я буду приводить без детального разъяснения, ссылаясь на другие публикации, однако на некоторых спорных моментах, по которым у меня возникли вопросы или которые недостаточно полно описаны в литературе, я буду отдельно заострять внимание и поделюсь всеми моими соображениями на эту тему. При этом, все математические преобразования постараюсь приводить достаточно подробно, даже если они и кажутся мне очевидными.

Также сразу оговорюсь, что эта статья ставит целью разобраться именно в основных концепциях рекуррентных нейросетей. Так что здесь не будет описания и сравнения достоинств и недостатков различных популярных, но более сложных архитектур (GRU, LSTM,...). Моя задача --- лишь заполнить те возможные пробелы в базовых знаниях, которые могли остаться у самостоятельно изучающего нейросети по доступным источникам. Имея надёжный фундамент будет проще разобраться и в сложных вещах.

Учитывая всё вышесказанное, могу сказать, что эта статья предназначена скорее для читателей, которые уже знакомы в какой-то степени с рассматриваемой темой, но хотели бы более ясной картины. Если же для читателя всё описанное тут давным давно понятно, но у него есть много других интересных мыслей, которыми очень хочется поделиться или обсудить некоторые моменты, тогда добро пожаловать в комментарии.

Впрочем никакой сложной математики здесь не будет — знаний правил дифференцирования и основ работы с матрицами будет вполне достаточно, так что статья может быть полезна и для первого ознакомления с нейросетями.

Однако, прежде чем переходить собственно к материалу, небольшое...

Лирическое вступление

Я всегда интересовался темой искусственного интеллекта. Ещё будучи школьником я зачитывался книгами про роботов, про технические, моральные и философские проблемы, которые встанут перед человечеством, если загадка сознания когда-нибудь будет решена и станет возможным создание искусственного разума. Сборник "Твоё электронное Я", подаренный мне на одной из олимпиад, оказал существенное влияние на всё моё мировоззрение.

Однако я никогда не мог погрузится в эту тему полностью и с головой. Каждый раз, когда я начинал изучать ту или иную тему, моё внутреннее чувство говорило мне, что всё-таки это не то, чего я хочу... А хотел я собственно того, что сейчас принято называть термином AGI. Я пытался понять смысл разума и вообще высшей психической и нервной деятельности. В своих поисках я забредал в самые разные области человеческого знания. Все ранние достижения в области машинного обучения, обработки текстов, даже какие-то элементы компьютерной лингвистики не только не отвечали моим представлениям о настоящем искусственном интеллекте, но даже не давали намёка в каком направлении стоит проводить исследования. А без ощущения направления мой интерес к теме постепенно затухал.

С появлением больших языковых моделей интерес снова появился. И хотя, после поверхностного ознакомления с их архитектурой, знакомое чувство "это-опять-не-то" вновь всплыло у меня в голове, но некоторые особенности поведения этих нейросетей навели меня на мысль, что "то-самое" уже где-то рядом. Пожалуй довольно забавно, что заинтересовали меня прежде всего особенности, которые вообще-то скорее можно отнести к багам и от которых обычно хотят избавиться.

Первое, что хотелось бы отметить --- т.н. "галлюцинирование нейросети". Это такие ситуации, когда нейросеть очень уверенно выдаёт ответ, который выглядит для стороннего наблюдателя-непрофессионала вполне правдоподобным, но который тем не менее является абсолютно ложным. Эта особенность нейросетей считается едва ли не самым большим их минусом и препятствием к использованию. Однако для меня это наоборот был фактор, который усилил мой интерес. Я вспомнил себя на экзаменах, когда я не знал ответа на вопрос и... я делал ровно то же самое --- галлюцинировал в меру своих возможностей и кстати зачастую удачно :)

Есть и ещё более убедительный на мой взгляд аргумент в пользу тезиса, что галлюцинирование нейросети --- это сигнал приближения к человеческому разуму. Я говорю о явлении, которое называется Парамнезия, или иначе "ложные воспоминания". На хабре есть статья, посвящённая ложным воспоминаниям их разновидностям и причинам их появления. Не буду останавливаться на этом вопросе более детально, для меня сейчас важен тот факт, что эти воспоминания --- это не просто некоторая сознательная ложь. Для говорящего это самая что ни на есть реальная реальность, и он готов её отстаивать. И это может говорить о некоторых глубинных механизмах работы головного мозга, а аналогия, которая наблюдается у искусственных нейросетей, даёт некоторую надежду на возможность моделирования этого процесса так сказать in vitro.

Сейчас появляется всё больше свидетельств, что галлюцинации --- это естественное следствие усложнения системы и появления у неё зачатков сознания (хотел взять в кавычки, но передумал...). В публикации Галлюцинации LLM --- это не баг указывается, что именно рассуждающие модели (reasoning-models) дают гораздо больший процент галлюцинаций. Так что возможно в скором времени компаниям, занимающимся разработкой достаточно сложных ИИ-продуктов потребуются штатные педагоги, психологи, психотерапевты и психиатры. И вовсе не для выгоревших сотрудников. А вот пошутил я сейчас или нет, станет понятно уже в относительно недалёком будущем :)

Следующее, на что могу обратить внимание --- это некоторые особенности генерации видео. Причём ранние попытки пожалуй даже более показательны. В них в кадре постоянно рождаются, эволюционируют и исчезают различные образы. В качестве примера могу привести совершенно замечательные работы режиссёра, хореографа и дизайнера анимации Джеймса Герда (James Gerde) "танцующие скульптуры богов", в частности Афродиты. Замечу, что сама танцующая фигура (т.е. её форма, динамика движения) --- снята на видео, но всё остальное (сцена, антураж, текстура) --- это всё творчество нейросети и именно там можно в полной мере увидеть визуальные отголоски мыслей, рождённых неживой материей.

Это опять-таки напомнило мне мои собственные ощущения и воспоминания, но ещё более ранние. Когда я был дошкольником, у меня в комнате на стене напротив кровати висела одна фотография. В целом ничего особенного, на фотографии была изображена какая-то городская площадь с гуляющими по ней людьми, в центре стояла стела, по краям --- дома. Типичный городской пейзаж. Однажды я проснулся достаточно рано, когда ещё не полностью рассвело, но основные очертания окружающих предметов уже можно было разглядеть. И вот в таких рассветных сумерках я уставился на эту фотографию. Вскоре я обнаружил, что если смотреть на неё непрерывно, стараясь не двигать зрачками, то фигуры на фотографии начинают оживать, двигаться, потом появляются совершенно новые персонажи, и даже пейзажи и окружение могли меняться. Картинки были неясные, почти ускользающие, особенно если пытаться на них сильно концентрироваться, однако для расслабленного наблюдения они воспринимались как вполне определённые. Более того, небольшим мысленным усилием можно было их направлять и даже в некоторой степени режиссировать сюжет. В итоге передо мной разворачивалось целое действо за которым было очень интересно наблюдать. Всё это могло длиться минут десять. Потом, когда света становилось больше, фотография возвращалась к своему обычному состоянию. Я проводил этот эксперимент несколько раз --- так он меня впечатлил. Так вот, когда я увидел видео сгенерированное нейросетями я вспомнил те образы, возникавшие в моём сознании. Динамика видео сгенерированного искусственной нейросетью очень точно соответствовала моим воспоминаниям.

Кстати, в 2025 году видео с танцующей Афродитой было переделано с использованием новых разработок. Да, видео безусловно стало более качественным, но кажется оно потеряло часть того шарма, который придавали ему эти неясные постоянно возникающие и исчезающие формы. Продолжая аналогию с моими воспоминаниями можно сказать, что в ярком дневном свете фотография обрела чёткие очертания потеряв полудремотную утреннюю расслабленность.

Однако, как я уже говорил, несмотря на некоторый всплеск интереса после знакомства с последними достижениями в области искуственно интеллекта изучение нейросетей основанных на архитектуре трансформера не могло меня полностью удовлетворить. И вот недавно вдруг выяснилось, что мои сомнения уже не только мои. Точно так же в этой технологии начали сомневаться и люди, которые не то, что варятся в этом супе всю жизнь, а буквально являются шеф-поварами на той кухне. Так например Илья Суцкевер, канадо-израильский учёный родом из Нижнего Новгорода и по совместительству со-основатель компании Open AI, недавно заявил, что эпоха масштабирования нейросетевых моделей завершилась и пришло время вернуться к исследованиям. Ему вторит Янн ЛеКун, который ещё более категоричен. По его мнению LLM --- это в принципе тупиковая ветвь развития.

Что ж, я не являюсь гуру в области нейросетей и искусственного интеллекта, но тем не менее решил немного покопаться в этой теме. Архитектура трансформеров начала своё триумфальное шествие после выхода в 2017 году статьи "Attention Is All You Need", написанной учёными из Google. И с этой же статьи началось постепенное угасание популярных ранее рекуррентных нейросетей. Это-то меня и задевало с момента начала моего ознакомления с трансформерами. Мне кажется, что рекуррентные нейросети ещё должны себя проявить. Во всяком случае биологические нейросети в нашей голове можно отнести именно к рекуррентным.

В поиске объекта исследований

Начиная со второй половины прошлого века на восток нашей страны к заливу Петра Великого каждый год по весне съезжалось множество нейрофизиологов изо всех крупных научных центров Советского Союза. Виной тому был вот этот замечательный моллюск:

Это "аплизия" или иначе "морской заяц". Благодаря этому красавцу в 2000 году получил Нобелевскую премию по физиологии и медицине американский нейробиолог австрийского происхождения Эрик Кандель. Причина, по которой аплизии завоевали сердца многих нейробиологов по всему миру, в том, что они максимально удобны для изучения. В своей книге "В поисках памяти" Кандель подробно описывает как именно он пришёл к такому выбору. Нервная система аплизий достаточно проста --- всего 20000 нейронов. Нейроны имеют крупные размеры (некоторые --- до 1 мм в диаметре) и удобны для непосредственного исследования. Клетка практически не повреждается при введении в неё микроэлектродов и сохраняет жизнеспособность ещё 5-10 часов.

Ну что ж, мне для начала экспериментов также нужен подопытный образец, и тоже максимально простой. Этакий аналог апплизий для нейробиологов. И я такого подопытного нашёл --- это совершенно замечательная реализация простейшей рекуррентной нейросети в одном гисте на гитхабе от Андрея Карпаты: min-char-rnn.py. В этой реализации всего около 100 строк кода и при этом она даёт уже довольно интересные результаты из которых можно сделать некоторые предположения и выводы.

Однако, когда я начал разбираться в алгоритме подробнее, я осознал, что несмотря на внешнюю простоту, это не так уж и просто, даже несмотря на то, что какие-то основы теории мной уже были освоены, да и некоторые эксперименты с нейросетями я уже проводил. Но тогда для экспериментов я применял tensorflow, который скрывал большую часть всей внутренней кухни. А программа min-char-rnn принципиально не использует никаких высокоуровневых библиотек. Самое сложное, что там есть --- это numpy для работы с векторами и матрицами. Идеальный подопытный! Кстати "char" в названии программы означает, что текст скармливается в нейросеть символ за символом. Тем самым выводятся за скобки все вопросы связанные с эмбеддингами.

Однако прежде чем переходить к более детальному разбору кода, алгоритма и математики, сделаем ряд замечаний по обозначениям, терминологии и некоторым общим вопросам относительно искусственных нейросетей.

Обозначения

Для обозначения векторов и матриц будем использовать жирный прямой шрифт (x, h, W,...).
Вектора всегда будут представляться в виде столбцов (это соответствует коду min-char-rnn).
Для обозначения транспонированных векторов и матриц будем использовать верхний индекс T, набранный прямым шрифтом (h^T, W^T).
Непосредственно векторную форму записи выражений будем использовать достаточно редко, обычно в самом конце. В промежуточных преобразованиях и вектора и матрицы будем записывать курсивом с явным указанием индексов как x_i, W_ij.
Транспонирование матриц при индексной записи производится просто перестановкой индексов местами: $W^{\mathrm T}_{ij} = W_{ji}$
Для индексации элементов векторов и матриц будем использовать i, j, k.
Для индексации элементов временнОго ряда будем использовать t. Однако в силу того, что эта индексация зачастую играет особую роль, то для того что бы этот индекс сразу бросался в глаза примем следующее правило: индекс временнОго ряда t будем ставить вверху и заключать в квадратные скобки: $h_i^{[t-1]}$ .

Переход между векторной и индексной формами записи

Сделаем также пару замечаний об индексной записи векторов и матриц.

Т.к. по соглашению, все вектора --- это столбцы, то фактически их можно представить, как матрицы размером n x 1, т.е. элементы в индексной записи будут иметь вид x_i₁, где i --- индекс строки (он же индекс элемента), а 1 --- индекс единственного столбца. Такую запись мы использовать явно не будем, но всё равно будем держать её в голове. Что бы понять почему это важно, рассмотрим матричное умножение.

Матричное умножение $\mathbf W \cdot \mathbf h$ в индексной записи имеет вид $\sum\nolimits_{j}{W_{ij}h_{j}}$ . Заметим, что т.к. в индексной записи множители являются скалярами, то они очевидно коммутируют и мы можем переставлять их как нам заблагорассудится. Однако при переводе обратно в векторную форму нужно соблюдать аккуратность. Можно использовать следующее мнемоническое правило --- соседние множители как бы "сцепляются индексами", по которым идёт суммирование. При этом порядок индексов результата определяется порядком индексов по которым не идёт суммирование. Например $\mathbf A \cdot \mathbf B \rightarrow \sum\nolimits_k{A_{ik}B_{kj}} = C_{ij}$ . Здесь порядок индексов у C именно такой --- ij, но никак не ji.

Посмотрим, как работает это правило. Обозначим $\mathbf z = \mathbf W \cdot \mathbf h$ или в индексной записи $z_i = \sum\nolimits_{j}{W_{ij}h_{j}}$ Пользуясь тем, что W_ij и h_j --- скаляры переставим их местами: $z_i = \sum\nolimits_{j}{h_{j}W_{ij}}$ или, вспоминая матричное представление вектора $z_{i1} = \sum\nolimits_{j}{h_{j1}W_{ij}}$ . В таком виде вернуться к векторной форме записи нельзя, поскольку элементы не сцепляются индексами, но индексы мы можем переставить с помощью транспонирования: $z_{i1} = \sum\nolimits_{j}{h^{\mathrm T}_{1j}W^{\mathrm T}_{ji}}$ . В таком виде правая часть уже соответствует матричному произведению, но как видим у результирующего объекта, соответствующего этому произведению индексы будут идти вот в таком порядке: 1i, при том что в левой части порядок индексов другой. Что бы согласовать порядок индексов мы опять же можем воспользоваться операцией транспонирования, причём не важно какую часть будем транспонировать, правую или левую. Пусть будет левая: $z^{\mathrm T}_{1i} = \sum\nolimits_{j}{h^{\mathrm T}_{1j}W^{\mathrm T}_{ji}}$ . Вот теперь уже можно получить и векторную форму: $\mathbf z^{\mathrm T} = \mathbf h^{\mathrm T} \cdot \mathbf W^{\mathrm T}$ . Мы получили известную формулу транспонирования матричного произведения.

Обучение нейросети

Думаю, что все читатели, которые дочитали до этого момента, уже имеют некоторые представления об искуственных нейросетях. С точки зрения математики нейросеть --- это просто функция многих переменных, которые можно разделить на две группы: 1) входные данные и 2) веса. Это разделение для нас принципиально вот по какой причине. При работе с нейросетью можно выделить два режима: 1) режим обучения; и 2) режим использования. Во время режима использования нейросети, её веса --- это по сути константы. Результат работы зависит только от входных данных. Однако в режиме обучения всё как раз наоборот --- мы меняем веса так, что бы добиться от нейросети максимального соответствия идеальному решению на тех же входных данных. Т.е. в этом случае входные данные мы рассматриваем как константы, а веса --- как переменные.

Поскольку нейросеть --- это функция, то её можно продифференцировать. Соответственно если мы рассматриваем режим обучения, а на этом этапе входные данные --- это константы, то производные по входным данным будут равны нулю.

Итак, повторим... в режиме обучения нейросети входные данные --- константа, веса нейросети --- переменные. Зафиксировав входные данные мы хотим изменить веса так, что бы нейросеть стала в некотором смысле "лучше".

Как мы знаем, что бы формализовать понятие "лучше" в смысле работы нейросети, вводят функцию потерь, loss или в используемых далее обозначениях L --- это некоторым образом придуманная скалярная функция от всех данных нейросети, которую мы хотим минимизировать. Если обозначить {x_i --- весь набор входных данных, а {w_j} --- весь набор весов, то

При таком подходе скалярность функции потерь важна, т.к. минимизировать вектор невозможно, поскольку вектора невозможно сравнивать (если сравнивать их длины, то длина вектора --- это уже скаляр).

Предположим, что мы немного изменили веса нейросети на величину $\Delta w_j$ . Тогда новое значение функции ошибки будет $L + \Delta L = L(x_i, w_j + \Delta w_j)$ , которое можно разложить в ряд Тейлора с точностью до бесконечно малых первого порядка:

$L + \Delta L \approx L + \sum\limits_j{\cfrac{\partial L}{\partial w_j}\Delta w_j} = L + (\nabla L \cdot \Delta\mathbf w)$

т.е.

$\Delta L \approx (\nabla L \cdot \Delta\mathbf w)$

где в конце стоит скалярное произведение двух векторов: $\Delta\mathbf w$ --- вектор поправок к весам и вектор $\nabla L$ (читается набла эль), называемый градиентом скалярной функции L.

$\nabla L = \left(\cfrac{\partial L}{\partial w_1}, ..., \cfrac{\partial L}{\partial w_M}\right)$

Для обучения нейросети мы хотим уменьшить значение функции ошибок L и желательно побыстрее, а значит величина $\Delta L$ должна быть отрицательна и максимальна по модулю. Скалярное произведение векторов --- это произведение их длинн на косинус угла между ними, значит оно будет отрицательно и максимально по модулю, когда угол между векторами будет $180^\circ$ , т.е. когда вектора будут противоположно направлены. Иными словами нам выгодно в качестве $\Delta\mathbf w$ взять вектор, направленный противоположно $\nabla L$ : $\Delta\mathbf w = -\xi\nabla L$ (здесь $\xi$ --- просто некоторый числовой параметр, влияющий на стабильность и сходимость алгоритма; как правило имеет достаточно маленькое значение).

Это вкратце суть метода градиентного спуска --- одного из наиболее популярных методов обучения нейросетей (есть и другие методы, но о них как-нибудь в следующий раз). Метод градиентного спуска хорошо известен, много где описан, но я его здесь привёл не только и не столько для полноты картины, сколько из-за того, что хотел бы сделать акцент на терминологии, используемой в литературе в контексте применения этого метода к обучению нейросетей, о чём будет следующий параграф.

Здесь же зафиксируем, что для вычисления поправок к весам нам неплохо бы знать значения частных производных функции потерь по всем весам нейросети или, что то же самое, градиент функнции потерь.

Так градиент или ошибка?

Один из наиболее популярных практических методов применения метода градиентного спуска --- это метод, который я буду называть методом вычисления градиентов в обратном направлении. Тут искушённый читатель закономерно спросит, а почему бы не использовать общепринятое название метод обратного распространения ошибки?... И будет безусловно прав в своём удивлении, т.к. именно таков устоявшийся термин в русскоязычной литературе. Однако я не считаю его удачным... Оригинальный? --- пожалуй да; запоминающийся? --- безусловно; но вот удачный ли он? Тут у меня есть сомнения.

Термин "метод обратного распространения ошибки" говорит нам, что смотрите, вот у нас есть некоторая ошибка, т.е. отклонение результата от желаемого значения. Мы берём эту ошибку и дальше прогоняем по всей нейросети в обратном направлении, некоторым образом изменяя веса нейросети. Такую трактовку можно встретить во множестве мест. Например вот статья на хабре: Фундамент AI: обратное распространение ошибки простыми словами:

Идея обратного распространения заключается в том, чтобы распространять ошибку (потери) в обратном направлении после того, как мы её рассчитали, изменяя веса таким образом, чтобы ошибка немного уменьшилась.

Или википедия: Метод обратного распространения ошибки:

Основная идея этого метода состоит в распространении сигналов ошибки от выходов сети к её входам, в направлении обратном прямому распространению сигналов в обычном режиме работы.

Это только пара примеров из результатов, найденных на первой странице в поиске на Яндексе.

Так что же не так в этом термине?... Чтобы разобраться в этом, давайте возьмём нашего подопытного и посмотрим на его внутренности. Открываем оригинальный код от Андрея Карпаты. Нас интересует функция lossFun --- именно она используется в режиме обучения нейросети и именно в ней вычисляются поправки к весам.

Всю функцию можно разбить на две основные части: строки 36..43 --- вычисление в прямом направлении (прямой проход); и строки 44..58 --- вычисление в обратном направлении (обратный проход), которое по сути и есть то самое обратное распространение ошибки.

В процессе вычисления в прямом направлении вычисляются различные промежуточные значения (подробнее об этом я расскажу дальше в этом тексте), которые сохраняются в массивы для дальнейшего использования. В конце, в строке 43 вычисляется ошибка loss как сумма (по времени) кросс-энтропии между ожидаемым результатом (точным) и фактически полученным.

Казалось бы, ну хорошо, вот мы в процессе прямого прохода вычислили ошибку... отлично, давайте распространять её "обратно" по нейросети. Но если посмотреть на код "обратного распространения ошибки", то мы видим, что эта ошибка в нём вообще никак не используется. Более того, если немного внимательнее посмотреть целиком на весь код, то заметим, что единственная цель вычисления этой ошибки --- это... барабанная дробь... периодически печатать в лог её сглаженные значения! Вот и всё "распространение"...

Так что никакого распространения ошибки не происходит. Рассматриваемый метод --- это просто удобный, наглядный и объективно очень красивый способ вычисления градиентов... но и только.

Термин "обратное распространение ошибки" некорректен и в дальнейшем невольно может приводить к некорректному изложению материала или даже ложным логическим умозаключениям, которые выглядели бы относительно разумно, если отталкиваться только от названия. Простейший пример: допустим мы инициализировали веса нейросети случайным образом и пусть при этом звёзды сложились так, что функция потерь нашей нейросети случайно оказалась в локальном максимуме. Это значит, что вычисленная ошибка будет достаточно велика. Тогда, опираясь на название метода, можно было бы предположить, что поправки к весам будут велики, ведь мы "распространяем" большую ошибку. Но в действительности все поправки будут вообще нулевые! И этот "неожиданный" вывод становится очевидным, как только мы вспомним, что мы не ошибку распространяем, а вычисляем градиенты.

Наверняка для искушённых спецов всё это и не имеет значения, но для новичка, который только погружается в материал может сыграть злую шутку. А возможно и не только для новичка. Взять например книгу Кришнанду Чаудхури "Математика и архитектура глубокого обучения, СПб.: Питер, 2026" (перевод на русский язык ООО «Прогресс книга», 2025). В ней на стр. 340 имеется формула под номером (8.16):

$z^{(l)} = w^{(l)}a^{(l-1)} + b^{(l)}.$

А к этой формуле есть подпись:

Прямое распространение ошибки для произвольного слоя.

Что?... Какое ещё прямое распространение ошибки? Я знаю, что существуют методы автоматического дифференцирования в прямом направлении, например с помощью дуальных чисел, или метод FGD, и я надеялся там почитать именно об этих подходах. Но нет, из формулы да и вообще из общего смысла главы понятно, что речь идёт именно об обычном вычислении на нейросети в прямом направлении. И это явно не единичная опечатка --- там во всём параграфе такое вычисление называется прямым распространением ошибки.

Честно говоря, мне пришлось потратить некоторое время и усилия, что бы осознать все эти тонкости, и что бы пазл сложился окончательно. Не знаю, возникали или возникнут ли подобные вопросы ещё у кого-нибудь, но надеюсь эта моя заметка кому-нибудь принесёт пользу и сделает мир немного понятнее.

К слову, стоит сделать замечание, что в англоязычной литературе используется термин backward propagation или backpropagation, т.е. просто "обратное распространение". И хоть не уточняется, что же именно у них там распространяется, но всё-таки этот термин безусловно гораздо лучше, т.к. не наталкивает на какие-то некорректные выводы. В частности англоязычная википедия даёт следующее определение backpropagation:

In machine learning, backpropagation is a gradient computation method commonly used for training a neural network in computing parameter updates;

Иными словами это именно что метод вычисления градиентов, а не распространение какого-то мифического эфира. Беглый поиск по англоязычной литературе подтвердил, что в этом плане терминология там более корректная. Так например оригинальная версия уже упомянутой выше книги "Математика и архитектура глубокого обучения" использует термины forward propagation и backward propagation, что выглядит вполне приемлемым.

Разумеется не все русскоязычные источники грешат подобного рода неточностями. Здесь я пожалуй не могу пройти мимо главы метод обратного распространения ошибки из учебника по машинному обучению от "Яндекс.Образование". Я дополнительно пересмотрел главу держа в голове то, что изложил выше и должен сказать, что там вся терминология используется вполне корректно. Да, название главы говорит о распространении ошибки, но это как раз нормально. Название должно быть узнаваемым и известным, что бы было проще искать информацию. В самой главе этот термин несколько раз используется, но тоже исключительно как название. Более того, иногда вместо распространения ошибки используют английский термин backward propagation, но часто просто говорят о градиентах и производных.

В общем кажется, что неплохо бы навести порядок в русскоязычной терминологии... Если меня читают люди из "Яндекс.Образование", то может имеет смысл добавить в книгу небольшое разъяснение термина? Просто предложил.

Появление нашего подопытного на свет

Автор программы, которую я выбрал для экспериментов --- Андрей Карпаты (Andrej Karpathy), учёный родом из Чехословакии. В настоящее время он работает в созданной им Eureka Labs, которая позиционируется как школа нового типа, где делается акцент на широкое применение искусственного интеллекта в образовательном процессе. Около 10 лет назад в 2015 году вошёл в группу учёных, основавших Open AI, а примерно за полгода до этого выложил первую версию простейшей рекуррентной нейросети для обработки текста, которую я и выбрал в качестве лабораторного животного.

Собственно код min-char-rnn Карпаты выложил вместе со своей статьёй Необоснованная эффективность рекуррентных нейронных сетей, в которой можно найти описание довольно интересных результатов экспериментов, но прежде чем переходить к экспериментам, мне сперва хотелось добиться полного понимания математики, заложенной в тот код, который действительно привлекал своей минималистичной красотой.

Скажу сразу, что в тот момент сходу разобраться во всём самостоятельно у меня не вышло. Тогда на просторах интернета я нашёл разбор кода этой нейросетки от Eli Bendersky. Кстати, на хабре есть перевод. Проштудировав статью я понял только то, что всё несколько сложнее, чем я ожидал изначально. Тогда мне попалось ещё одно описание, сделанное Michel Kiffel. Забавно, что в нём он ссылается на текст Eli Bendersky и говорит в свою очередь (даю сразу перевод):

Однако, даже прочитав все комментарии Eli Bendersky, у меня осталось больше вопросов, чем ответов. <...> Мне потребовалось некоторое время и несколько страниц математических выкладок, чтобы по-настоящему понять, что происходит.

Значит не у меня одного возникли затруднения с этим кодом. Статья Kiffel'а несколько прояснила ситуацию, но полного понимания всё равно не было.

Тогда я решил проделать все расчёты самостоятельно и получить те формулы, которые используются в коде. Т.е. именно в том самом виде. Вот эти расчёты я и предлагаю сейчас вашему вниманию. Когда в моих рассуждениях я буду доходить до моментов, которые вызвали у меня затруднения при чтении статей Eli Bendersky и Michel Kiffel, я постараюсь сделать на них акцент, рассказав кратко что именно мне было непонятно и почему.

Разбираем подопытного

При разборе кода я буду максимально близко следовать нотации, которую использует Michel Kiffel. Она имеет несколько отличий от той, которая есть в коде. В частности при прямом проходе в коде используются названия переменных {xs,hs,ys,ps}. Для краткости записи формул я опускаю суффикс s, т.е. буду использовать вот такой набор {x,h,y,p}. Суффискс s означает по-видимому 'sequence' в том смысле, что эти переменные представляют собой массивы, которые хранят последовательность соответствующих значений для разных моментов времени. Вообще-то это только моя догадка, которую могу подтвердить тем, что в первой версии кода в обратном проходе тоже были переменные-массивы с аналогичным смыслом, а именно dhs, dys. Но вторым коммитом они из массивов превратились в обычные переменные и потеряли суффикс s.

Также я добавил обозначение z для промежуточного результата прямого прохода нейросети, который получается перед вычисление гиперболического тангенса. Точно такое же обозначение есть у Kiffel'а. В коде это значение вообще никак не фиксируется, но при обратном проходе есть переменная dhraw, которая по смыслу соответствует поправке к этому значению.

Я пропущу описание общей структуры нейросети --- его можно посмотреть например в переводе статьи от Eli Bendersky.

Визуально мы можем представить нашу нейросеть в виде следующей схемы (это несколько модифицированная схема из статьи Eli Bendersky)

Красным цветом на этой схеме отмечены веса и смещения нейросети, т.е. независимые переменные по которым нам надо будет продифференцировать нашу функцию потерь. Зелёным цветом --- промежуточные значения, или зависимые переменные. По ним мы также будем дифференцировать функцию потерь, но только как промежуточный этап вычислений.

Как известно (и о чём можно прочитать например в тех же статьях Bendersky и Kiffel), основная проблема с рекуррентными нейросетями, мешающая применять метод вычисления градиентов в обратном направлении --- это цикличность графа вычислений, которая решается развёртыванием нейросети на несколько итераций. При этом важной деталью является то, что все веса и смещения во всех итерациях строго одинаковы. Метод получил название "Backward Propagation Through Time" или BPTT.

В целом подход понятен и обычно в статьях это просто проговаривают без иллюстраций. Но давайте всё-таки нарисуем картинку для нейросети, развёрнутой на три итерации (в оригинальном коде Андрея Карпаты используется 25 итераций):

Меня здесь интересует прежде всего состояние скрытого слоя нейросети до и после интервала, на котором она развёртывается. Т.е. вот эти самые хвостики, которые на иллюстрации отмечены как $h^{[0]}$ и $h^{[3]}$ . Eli Bendersky также обращает на них внимание. Вот что он пишет (здесь и далее привожу цитаты в переводе @honyaki:

<...> Выходное состояние h на последней итерации на самом деле "никуда не пропадает", и мы предполагаем, что у него нулевой градиент. Входное состояние h для первой итерации тоже равно нулю по аналогичным причинам.

Честно говоря, меня такое объяснение не устраивает. Градиент --- это вычисляемая сущность, с чего это мы вдруг предполагаем, что он равен нулю? Точнее предположить-то мы можем, в смысле выдвинуть гипотезу, но потом её надо будет как-то обосновать. Michel Kiffel в своей публикации этот вопрос мягко обходит стороной просто предлагая свой вывод формулы для вычисления градиента, но по сути из этого вывода можно сделать достаточно простое заключение: этих хвостов не существует! Именно так. Разворачивая рекуррентную нейросеть во времени мы на самом деле получаем совершенно другую нейросеть, которая не рекуррентна, т.е. не содержит циклов и пригодна для применения метода вычисления градиентов в обратном направлении. И единственные её входы --- это набор $x^{[t]}_i$ . Других входов нет. Веса, которые мы получим после обучения этой нейросети мы в дальнейшем используем в нашей исходной рекуррентной нейросети и ожидаем, что она при этом заработает именно так, как нам надо.

Итак в режиме обучения наша нейросеть имеет следующий вид:

Следующая фраза, которая несколько меня смутила, вот эта:

<...> Остаётся вопрос о том, как же нам обновить весовые коэффициенты, ведь мы рассчитываем градиенты для них отдельно на каждой итерации. Мы просто суммируем их.

Нууу... интуитивно в общем-то это понятно, но всё-таки хотелось бы более подробного разъяснения откуда именно появляется это суммирование. Так например зависимость каждого $L^{[\tau]}$ от $\mathbf {Why}$ однозначно и полностью определяется на итерации $\tau$ , но в то же время зависимость от $\mathbf {Whh}$ определяется не одной итерацией, а целым рядом итераций $t \in 1..\tau$ . Можно без вычислений предположить, что тут появится некоторое суммирование. А после этого у нас будет ещё и суммирование всех $L^{[\tau]}$ .

Вывод формул

Давайте наконец-то перейдём собственно к разбору всей математики.

Для прямого прохода на итерации t у нас есть входное значение $\mathbf x^{[t]}$ и предыдущее значение скрытого слоя $\mathbf h^{[t-1]}$ . Для входных значений используется т.н. one-hot кодирование, которое иногда называют one-of-k. Учитывая, что наша нейросеть работает посимвольно, т.е. за одну итерацию обрабатывает ровно один символ, можно описать суть one-hot кодирования в следующем виде. Предположим, что каждый уникальный символ из нашего текста записан в некоторый массив (назовём его алфавит или словарь) и таким образом имеет числовую характеристику --- индекс символа в словаре. Обозначим этот индекс $\alpha$ . Предположим, что размер словаря N. Тогда one-hot представление символа из алфавита --- это массив из N чисел, все из которых равны нулю за исключением одного, который равен 1. Это элемент с индексом, равным индексу символа.

$\mathrm{onehot}(\alpha): \underbrace{ \overbrace{0, 0, 0, 0 \dots 0,}^{\alpha - 1 \mbox{ нулей}} 1, 0, 0 \dots 0}_{N \mbox{ элементов} }$

Такое представление можно рассматривать как распределение вероятности того, что символ соответствует тому или иному элементу словаря. Поскольку вход однозначно определён, то и распределение вероятностей для него имеет вот такой достаточно простой вид.

Думаю написать формулы прямого прохода, опираясь на схему нейросети и код, не составит труда ни для кого. В векторном виде они имеют вид:

$\begin{array}{lcl} \mathbf z^{[t]} &=& \mathbf {Wxh} \cdot\mathbf x^{[t]} + \mathbf {Whh} \cdot\mathbf h^{[t-1]} + \mathbf {bh} \\ \mathbf h^{[t]} &=& \mathrm{th}(\mathbf z^{[t]}) \\ \mathbf y^{[t]} &=& \mathbf {Why} \cdot\mathbf h^{[t]} + \mathbf {by} \\ \mathbf p^{[t]} &=& \mathrm{softmax}(\mathbf y^{[t]}) = \cfrac{\exp(\mathbf y^{[t]})}{\sum\limits_{i}{\exp(y^{[t]}_i)}} \\ L^{[t]} &=& \mathrm{cross\_entropy}(\mathbf {target}^{[t]}, \mathbf p^{[t]}) \\ loss &=& L = \sum\limits_{t}{L^{[t]}} \\ \end{array}$

Здесь только кратко остановимся на смысле последних трёх операций. А точнее даже двух из них, т.к. с самой последней всё и так ясно.

Функция softmax(...) превращает результат в распределение вероятностей в том же самом смысле, что и рассмотренное ранее распределение вероятностей для входных данных. На самом деле, я тут несколько слукавил. Разумеется функция softmax выдаёт такое распределение вероятностей не сама по себе. Это мы хотим получить такое распределение и для этого выбираем операцию, результат которой в принципе можно проинтерпретировать таким образом. А затем выбрав подходящую функцию потерь и минимизировав её мы автоматически получим, что вот эта операция действительно выдаёт нужное нам распределение вероятности. Собственно главное, что требуется от результата softmax --- это 1) количество элементов результата должно быть равно количеству элементов входа; 2) каждый элемент должен находиться в интервале [0..1]; и 3) сумма всех элементов должна быть равна 1. Можно придумать множество функций, удовлетворяющих этому требованию, но именно softmax оказалась наиболее удобной. В частности она наилучшим образом соответствует следующей операции --- кросс-энтропии, поскольку экспонента в softmax в некотором смысле компенсируется логарифмом в кросс-энтропии.

Кросс-энтропия определяет меру сходства двух распределений вероятности и является естественным обобщением понятия энтропии по Шеннону для одного распределения. Формула для энтропии по Шеннону была по сути получена эмпирическим путём исходя из требований непрерывности и удовлетворения определённым граничным условиям. Кросс-энтропия обладает следующими свойствами: 1) она неотрицательна и 2) минимальна только при точном совпадении распределений. Последнее как раз и сводит нашу задачу к задаче минимизации.

Формула для кросс-энтропии имеет следующий вид:

$L^{[t]} = \mathrm{cross\_entropy}(\mathbf {target}^{[t]}, \mathbf p^{[t]}) = -\sum\limits_{i}{{target}^{[t]}_{i} \ln(p^{[t]}_i)}$

причём по соглашению, при вычислении энтропии и кросс-энтропии приято следующее правило: $0 \cdot \ln(0) = 0$ .

Поскольку перед нами стоит задача построить нейросеть, способную предсказывать следующий символ, то целевое распределение для вычисления кросс-энтропии на шаге t ( $\mathbf {target}^{[t]}$ ) --- это распределение, соответствующее следующему символу, т.е. $\mathbf x^{[t+1]}$ . В силу того что, как уже было сказано, для входных данных используется кодирование one-hot, то у вектора $x^{[t+1]}_i$ все элементы равны 0 кроме одного, который равен 1. Используя принятое ранее обозначение, запишем индекс этого элемента как $\alpha^{[t+1]}$ .

Тогда, учитывая вышесказанное, можем записать формулу для кросс-энтропии как:

$L^{[t]} = -\sum\limits_{i}{x^{[t+1]}_i \ln(p^{[t]}_i)} = -\ln(p^{[t]}_{\alpha^{[t+1]}})$

Итак, для обучения нейросети нам требуется минимизировать нашу полную функцию потерь L, настраивая параметры нейросети $\mathbf {Wxh}, \mathbf {Whh}, \mathbf {Why}, \mathbf {bh}, \mathbf {by}$ . Для этого будем использовать метод градиентного спуска, что потребует от нас вычисления производных. Замечу, что можно провести все вычисления сразу в векторном виде используя аппарат матричного дифференцирования, описанный например в учебнике по машинному обучению от Яндекс.Образование, но я здесь не хочу сильно выходить за рамки школьной программы, поэтому проведём все выкладки в индексной записи и только финальный вариант формул будем переводить обратно в векторную запись, учитывая правила, описанные выше.

Перепишем формулы выше явно расписав все операции через индексы.

$\begin{array}{lcl} z^{[t]}_i &=& \sum\limits_{k}\left(Wxh_{ik} \cdot x^{[t]}_k + Whh_{ik} \cdot h^{[t-1]}_k \right) + bh_i \\ h^{[t]}_i &=& \mathrm{th}\left(z^{[t]}_i\right) \\ y^{[t]}_j &=& \sum\limits_{i}\left(Why_{ji} \cdot h^{[t]}_i\right) + by_j \\ p^{[t]}_i &=& \cfrac{\exp\left(y^{[t]}_i\right)}{\sum\limits_{j}{\exp\left(y^{[t]}_j\right)}} \\ L^{[t]} &=& -\ln p^{[t]}_{\alpha^{[t+1]}} \\ L &=& \sum\limits_{t}{L^{[t]}} \\ \end{array}$

И теперь, что бы получить формулы для обратного прохода, начнём дифференцировать функцию потерь. Для вычисления частных производных я буду использовать формализм полного дифференциала. Из формулы полного дифференциала функции многих переменных

$d\left[f(x_1, x_2,...,x_n)\right] = \cfrac{\partial f}{\partial x_1}dx_1 + \cfrac{\partial f}{\partial x_2}dx_2 + ... + \cfrac{\partial f}{\partial x_n}dx_n$

видно, что частные производные можно получить, как как множители при соответствующих дифференциалах переменных. Также этот формализм позволяет применять цепное правило для дифференцирования сложных функций.

Дифференциал полной функции потерь записывается тривиально:

$dL = \sum_{t} dL^{[t]}$

Далее раскрываем дифференциалы для частей функции потерь, полученных на каждой итерации:

$dL^{[t]} = -d\left(\ln\left(p^{[t]}_{\alpha^{[t+1]}}\right)\right) = -\cfrac{dp^{[t]}_{\alpha^{[t+1]}}}{p^{[t]}_{\alpha^{[t+1]}}}$

Для вычисления $dp^{[t]}_{i}$ для краткости записи временно опустим индекс t и введём следующие обозначения: $e_i = \exp\left(y_i\right)$ и $\Sigma = \sum_{j}{\exp(y_j)} = \sum_j{e_j}$ . Дифференциалы для этих выражений запишутся в следующем виде:

$\begin{array}{lcl} de_i &=& e_i\mspace{5mu} dy_i \\ d\Sigma &=& d\left(\sum\limits_j{e_j}\right) = \sum\limits_j{de_j} = \sum\limits_j{e_j\mspace{5mu} dy_j} \end{array}$

Тогда представляя p_i как $p_i=e_i / \Sigma$ , запишем для него дифференциал:

$\begin{array}{lcl} dp_{i} &=& \cfrac{ e_i\mspace{5mu} dy_i \mspace{5mu} \sum\limits_j{e_j - e_i} \mspace{5mu} \sum\limits_j{e_jdy_j} } { \Sigma^2 } = \cfrac{ e_i\mspace{5mu} dy_i \mspace{5mu} \Sigma - e_i \mspace{5mu} \sum\limits_j{e_jdy_j} } { \Sigma^2 } \\ &=& \cfrac{e_i}{\Sigma}dy_i - \sum\limits_j{\cfrac{e_i e_j dy_j}{\Sigma^2}} = p_idy_i - \sum\limits_j{p_ip_jdy_j} \end{array}$

Используя известное свойство символа Кронекера $\delta_{ij}$ , запишем $dy_i = \sum\nolimits_j{\delta_{ij}dy_j}$ , тогда объединяя две операции суммирования в одну, вынося общий множитель p_i и группируя члены суммы по дифференциалам dy_j получим

$\begin{array}{lcl} dp_{i} &=& p_i \sum\limits_j{\left( \delta_{ij} - p_j \right) dy_j}, \end{array}$

или, возвращая индексацию по t

$\begin{array}{lcl} dp^{[t]}_i &=& p^{[t]}_i \sum\limits_j{\left( \delta_{ij} - p^{[t]}_j \right) dy^{[t]}_j}, \end{array}$

Подставляя это выражение в дифференциал $dL^{[t]}$ , получим

$\begin{array}{lcl} dL^{[t]} &=& \sum\limits_j{\left(p^{[t]}_j - \delta_{\alpha^{[t+1]}j} \right) dy^{[t]}_j}, \end{array}$

Таким образом мы можем записать выражение для частной производной по $y^{[t] }_j$ в следующем виде:

$\cfrac{\partial L^{[t]}}{\partial y^{[t]}_j} = p^{[t]}_j - \delta_{\alpha^{[t+1]}j}$

Т.е. эта частная производная равна просто выходному вектору у которого один из элементов уменьшен на 1. Красота этой формулы как раз и определяется тем, насколько хорошо подходят друг другу функции softmax и кросс-энтропия.

Если мы посмотрим на строки 49-50 программы

    dy = np.copy(ps[t])
    dy[targets[t]] -= 1

то увидим, что это как раз и есть выражение для частной производной.

Тут я сделаю небольшой комментарий относительно обозначений. Выражения dWxh, dWhh, dWhy, dbh, dby в коде могут немного сбивать с толку, т.к. это не дифференциалы а условно говоря поправки к соответствующим переменным (с точностью до множителя). Как мы помним из параграфа про метод градиентного спуска, эти поправки (опять же с точностью до множителя) как раз и есть частные производные. Т.е. выражение dWhy в программе --- это именно матрица частных производных $\cfrac{\partial L}{\partial Why_{ij}}$ . По аналогии с вышеперечисленными, в коде используются и dy, dh, dhraw, dhnext, которыми также обозначают соответствующие частные производные.

Замечу ещё, что тут мы подошли к моменту, когда зависимости $L^{[t] }$ от элементов по которым мы дифференцируем становятся более сложными. Поэтому дальше будем рассматривать дифференцирование поэтапно, начиная с самой первой итерации цикла обратного прохода $t = t_{max}$ , потом $t_{max}-1$ , $t_{max}-2$ и т.д

Ведём символ $\delta$ для обозначения частной производной полной функции ошибок по той или иной переменной, накопленной на текущей итерации цикла $t=\tau$ :

$\delta by_j = \sum\limits^{t_{max}}_{t=\tau}{\cfrac{\partial L^{[t]}}{\partial by_j}}$

и аналогичные $\delta Wxh_{ij}$ , $\delta bh_i$ ,... Эти величины в точности соответствуют переменным dby, dWxh, dbh,... в коде на каждой итерации цикла.

Двигаемся дальше. Мы рассмотрели дифференциал $dp^{[t]}_i$ и получили выражение для дифференциала L через дифференциал y, теперь вычислим сам $dy^{[t]}_j$ . Формула для $y^{[t] }_j$ достаточно проста и проблем с дифференцированием ни у кого возникнуть не должно, во всяком случае при использовании индексной записи (хотя Eli Bendersky написал аж целую статью на эту тему...). Дифференцируя $y^{[t] }_j$ получаем:

$\begin{array}{lcl} dy^{[t]}_j &=& \sum\limits_i{\left(dWhy_{ji} \cdot h^{[t]}_i + Why_{ji} \cdot dh^{[t]}_i\right) + dby_j} \\ \end{array}$

Подставляем это выражение в формулу для дифференциала функции потерь, имея в виду, что значение для частной производной по $y^{[t] }_j$ мы уже посчитали ранее:

$\begin{array}{lcl} dL^{[t]} &=& \sum\limits_j{\delta y_j dy^{[t]}_j} = \sum\limits_{ij}{\left(\delta y_j h^{[t]}_i dWhy_{ji} + \delta y_j Why_{ji} dh^{[t]}_i\right)} + \sum\limits_j{\delta y_j dby_j} \\ \end{array}$

Собирая выражения перед дифференциалами переменных получаем следующие частные производные:

$\begin{array}{lcl} \cfrac{\partial L^{[t]}}{\partial Why_{ji}} &=& \delta y_j h^{[t]}_i \\ \cfrac{\partial L^{[t]}}{\partial by_j} &=& \delta y_j \\ \cfrac{\partial L^{[t]}}{\partial h^{[t]}_i} &=& \sum\limits_j{\delta y_j Why_{ji}} \\ \end{array}$

Или, учитывая принятые обозначения для переменных, в которых мы будем накапливать соответствующие частные производные полной функции потерь, запишем:

$\begin{array}{lrl} \delta Why_{ji} &+=& \delta y_j h^{[t]}_i \\ \delta by_j &+=& \delta y_j \\ \delta h_i &+=& \sum\limits_j{\delta y_j Why_{ji}} \\ \end{array}$

Переведём сразу эти выражения в векторную запись, учитывая мнемоническое правило, описанное выше:

$\begin{array}{lrl} \delta \mathbf {Why} &+=& \delta \mathbf y (\mathbf h^{[t]})^{\mathrm T} \\ \delta \mathbf {by} &+=& \delta \mathbf y \\ \delta \mathbf h &+=& \mathbf {Why}^{\mathrm T} \delta \mathbf y \\ \end{array}$

Эти выражения почти точно соответствуют строкам 51-53 программы:

    dWhy += np.dot(dy, hs[t].T)
    dby += dy
    dh = np.dot(Why.T, dy) + dhnext

за исключением того, что для dh вместо инкремента используется добавление dhnext, но вообще говоря dhnext --- это и есть значение dh на предыдущей итерации цикла, как будет видно далее.

В последнем выражении для дифференциала $dL^{[t]}$ мы имеем два дифференциала независимых переменных (dWhy_ji и dby_j) и один дифференциал зависимой переменной ( $dh^{[t]}_i$ ), который надо раскрыть. Дифференциал выражения для $h^{[t] }_i$ (т.е. гиперболического тангенса) можно посмотреть по справочнику:

$dh^{[t]}_i = d\left(\mathrm{th}{(z^{[t]}_i)}\right) = \left(1 - (h^{[t]}_i)^2\right)dz^{[t]}_i$

Соответственно частная производная по $z^{[t]}_i$ :

$\delta z_i = \left(1 - (h^{[t]}_i)^2\right) \delta h^{[t]}_i$

или в векторном виде:

$\delta \mathbf z = \left(1 - \mathbf h \odot \mathbf h \right) \odot \delta \mathbf h$

где $\odot$ --- поэлементное умножение. Это выражение соответствует строке 54 программы:

    dhraw = (1 - hs[t] * hs[t]) * dh

Далее вычисляем дифференциал $z^{[t]}_i$ , выражение для которого тоже достаточно простое для дифференцирования и нужно только вспомнить, что входные данные $x^{[t]}_k$ являются константой:

$dz^{[t]}_i = \sum\limits_{k}{\left( dWxh_{ik} \cdot x^{[t]}_k + dWhh_{ik} \cdot h^{[t-1]}_k + Whh_{ik} \cdot dh^{[t-1]}_k \right)} + dbh_i$

Аналогично всему сделанному ранее, подставляем этот дифференциал в общее выражение для $dL^{[t]}$ и, собирая выражения перед дифференциалами независимых переменных, получаем следующие частные производные (я буду сразу записывать их в инкрементальном виде для получения полных частных производных от общей функции потерь):

$\begin{array}{lcl} \delta bh_{i} &+=& \delta z_i \\ \delta Wxh_{ik} &+=& \delta z_i x^{[t]}_k \\ \delta Whh_{ik} &+=& \delta z_i h^{[t-1]}_k \\ \end{array}$

Или в векторном виде:

$\begin{array}{lcl} \delta \mathbf{bh} &+=& \delta \mathbf z \\ \delta \mathbf{Wxh} &+=& \delta \mathbf z (\mathbf x^{[t]})^{\mathrm T} \\ \delta \mathbf{Whh} &+=& \delta \mathbf z (\mathbf h^{[t-1]})^{\mathrm T} \\ \end{array}$

Эти выражения соответствуют строкам 55-57

    dbh += dhraw
    dWxh += np.dot(dhraw, xs[t].T)
    dWhh += np.dot(dhraw, hs[t-1].T)

и с ними думаю всё очевидно.

Некоторый интерес представляет дифференциал ещё одной зависимой переменной: $dh^{[t-1]}_k$ . Он соответствует состоянию скрытого слоя в момент t-1 и, поэтому должен быть добавлен к соответствующему дифференциалу при дифференцировании $L^{[t-1]}$ , т.е. на следующей итерации цикла. Иными словами это то самое dhnext, которое отсутствовало в наших формулах ранее и было равно нулю на первой итерации цикла. Получим выражение для соответствующей частной производной, подставив дифференциал $dz^{[t]}_i$ в выражение для $dL^{[t]}$ (пишу с сокращениями, опуская все члены, которые уже были рассмотрены):

$\begin{array}{lcl} dL^{[t]} &=& ... + \sum\limits_i{\delta z_i dz^{[t]}_i} + ... \\ &=& ... + \sum\limits_{ik}{\delta z_i Whh_{ik} \cdot dh^{[t-1]}_k} + ... \\ \end{array}$

Выписывая выражение перед $dh^{[t-1]}_k$ получаем значение частной производной, которое надо сохранить в dhnext:

$dhnext_k = \sum\limits_i{\delta z_i Whh_{ik}}$

Или в векторной записи (напомню, что нам надо в правой части получить порядок индексов такой: k1, а значит $Whh_{ik}$ надо транспонировать, что бы переставить индексы и ещё переставить множители, что бы они "сцепились индексами"):

$\mathbf{dhnext} = \mathbf{Whh}^{\mathrm{T}} \cdot \delta\mathbf{z}$

Это последняя строка цикла backward propagation:

    dhnext = np.dot(Whh.T, dhraw)

Выполнив все итерации цикла обратного прохода мы получим в переменных dWxh, dWhh, dWhy, dbh, dby частные производные полной функции потерь по соответствующим весам и смещениям, которые в дальнейшем используются в качестве поправок для изменения этих весов и смещений.

Заключение

На этом я закончу разбор этой программы. Я постарался пройти все математические выкладки, попутно освещая вопросы, которые возникали у меня в голове в процессе самостоятельного вывода всех формул. Буду признателен, если в комментариях вы поделитесь мнением, насколько был интересен такой формат разбора. Не слишком ли подробно описаны некоторые очевидные вещи, или может наоборот какие-то моменты остались неясными.

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7 февр. 2026 г.

28.11.25 11:15 robertalfred175

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04.12.25 21:45 elizabethrush89

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04.12.25 21:45 elizabethrush89

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

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

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06.12.25 01:44 Tonerdomark

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06.12.25 01:48 Tonerdomark

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06.12.25 10:36 Thomas Muller

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06.12.25 10:36 Thomas Muller

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06.12.25 10:39 michaeldavenport238

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06.12.25 10:42 michaeldavenport238

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

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

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08.12.25 09:07 Tonerdomark

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

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

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

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

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

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

Grateful i came across SolidBlock Forensics. After investing in crypto trade and couldn't make withdrawals, it dawned on me something was wrong. They kept on asking for taxes, fees for maintenance, and more money for admin reasons. But being represented by SolidBlock Forensics, i was able to file reports, and finally, received all my investments with returns. Its great to know we have professionals that handle such issues and get the job done.
12.12.25 22:38 swanky

After reading some reviews on how [anthonydaviestech AT gmail dot (c om) helps people recover money and Cryptocurrencies lost to scammers, I decided to contact him to help me recover mine which I lost in february 2025. To my surprise he was able to trace the USDT from the first wallet to all the wallets it has been sent to. He moved them out of those wallets and returned them back to mine and even added extra to me, it felt like magic.
13.12.25 08:22 Natasha Williams

I am Natasha Williams from Dallas. I want to share my testimony to encourage anyone who has ever fallen victim to a scam or fraud. Some time ago, I was defrauded by some fraudulent cryptocurrency investment organization online, I was a victim and I lost a huge amount of money, $382,000. I felt angry, disappointed and helpless but I refused to give up and stay calm. I came across this agency, GREAT WHIP RECOVERY CYBER SERVICES.. who helped people recover their money from scammers and the testimonies I saw were quite amazing. And I decided to contact them. I gathered every piece of evidence, chats, receipts, account details, and messages and reported the case to the agency, GREAT WHIP RECOVERY CYBER SERVICES. After 73hours of follow up and not losing faith, the fraudster was traced and held accountable and I recovered all my money back. I highly recommend, GREAT WHIP RECOVERY CYBER SERVICES agency if you have ever fallen victim to scammers, you can contact them. Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site email: [email protected] Call Line: +1(406)2729101
13.12.25 08:23 Natasha Williams

I am Natasha Williams from Dallas. I want to share my testimony to encourage anyone who has ever fallen victim to a scam or fraud. Some time ago, I was defrauded by some fraudulent cryptocurrency investment organization online, I was a victim and I lost a huge amount of money, $382,000. I felt angry, disappointed and helpless but I refused to give up and stay calm. I came across this agency, GREAT WHIP RECOVERY CYBER SERVICES.. who helped people recover their money from scammers and the testimonies I saw were quite amazing. And I decided to contact them. I gathered every piece of evidence, chats, receipts, account details, and messages and reported the case to the agency, GREAT WHIP RECOVERY CYBER SERVICES. After 73hours of follow up and not losing faith, the fraudster was traced and held accountable and I recovered all my money back. I highly recommend, GREAT WHIP RECOVERY CYBER SERVICES agency if you have ever fallen victim to scammers, you can contact them. Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site email: [email protected] Call Line: +1(406)2729101
13.12.25 08:28 Natasha Williams

I am Natasha Williams from Dallas. I want to share my testimony to encourage anyone who has ever fallen victim to a scam or fraud. Some time ago, I was defrauded by some fraudulent cryptocurrency investment organization online, I was a victim and I lost a huge amount of money, $382,000. I felt angry, disappointed and helpless but I refused to give up and stay calm. I came across this agency, GREAT WHIP RECOVERY CYBER SERVICES.. who helped people recover their money from scammers and the testimonies I saw were quite amazing. And I decided to contact them. I gathered every piece of evidence, chats, receipts, account details, and messages and reported the case to the agency, GREAT WHIP RECOVERY CYBER SERVICES. After 73hours of follow up and not losing faith, the fraudster was traced and held accountable and I recovered all my money back. I highly recommend, GREAT WHIP RECOVERY CYBER SERVICES agency if you have ever fallen victim to scammers, you can contact them. Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site email: [email protected] Call Line: +1(406)2729101
13.12.25 15:29 Anthonymorphy30

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13.12.25 18:52 James Robert

I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have access to my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site or Call +1(406)2729101
13.12.25 18:54 James Robert

I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have access to my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site or Call +1(406)2729101
18.12.25 12:12 Anthony Irvin

I forgot my password 🔑 -- But they helped me recover my entire wallet. Successful Brute-Force Hack... On January 16, 2023, INTELLIGENCE CYBER WIZARD successfully hacked my Langerhans.DAT Android DOGECOIN wallet to recover the lost/forgotten spending PIN, necessary for spending funds. It took them only three (3) days to complete the task. It turned out that the spending PIN was only ten (10) digits long, which I had mistakenly believed to be twelve. Once Intelligence Cyber Wizard discovered and recited the full ten-digit PIN to me, based on "clues" I had given them and that only I knew, it reminded me a bit of when I set up the PIN back in 2015. In my case, I made the mistake of not writing it down. The wallet file in this case contained a little over 10.5 million DOGECOINs installed on it, spread across a total of 15 DOGE public wallet addresses and their corresponding private keys. When I acquired the 10.5 million coins in 2015, the total cost to me was only around $1,500, based on the current face value of DOGE at the time, which was approximately $0.0001 per coin. The wallet file had been inaccessible to me since December 27, 2015, after I realized I had forgotten my spending PIN. Before finding Intelligence Cyber Wizard, I had had several other encounters with people who were unsuccessful in recovering the wallet's spending PIN. So, as it turned out, I actually ended up much better off financially because I had forgotten my spending PIN. Had I not forgotten, I would have traded or sold the 10.5 million coins in question during the 2017 cryptocurrency bull run, along with another 40 million DOGECOIN that I sold that year. Even after accounting for a portion of the recovered coins that was awarded to Intelligence Cyber Wizard for their efforts, I still gained a significant financial advantage. Therefore, I am deeply grateful to Intelligence Cyber Wizard, for the swift and successful recovery of my wallet. Their efforts have undoubtedly had a truly transformative impact on my life in the long run. I highly recommend Intelligence Cyber Wizard services if you or someone you know has a wallet that needs hacking and whose password or spending PIN has been lost or forgotten. Contact Intelligence Cyber Wizard via WhatsApp or email. WhatsApp +1 (219) 424-7566 Email: intelligencecyberwizard @ gmail.com Email: intelligencecyberwizard @ cyber-wizard.com Website: https: //intelligence-cyber-wizard.web node.page
18.12.25 13:24 Regency Victorian

I was scammed and left penniless, until a cryptocurrency recovery expert helped me. INTELLIGENCE CYBER WIZARD - Cryptocurrency Recovery. I was scammed out of a huge amount of money and, devastated, I started looking for help. Someone recommended an expert and private investigator, Intelligence Cyber Wizard. I emailed him, and as soon as I explained what had happened, he immediately told me it had all been a setup. He put my case together, and within a few days, we got a result: I recovered all my money! Intelligence Cyber Wizard was amazing, professional, and efficient; he guided me through the entire process. I can't thank him and his forensic team enough. I can live again! His contact information is below. WhatsApp +1 (219) 424-7566 Email: [email protected] Email: [email protected] Website: https://intelligence-cyber-wizard.webnode.page
18.12.25 20:54 James Robert

I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have access to my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site or Call +1(406)2729101
18.12.25 20:55 James Robert

I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have access to my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site or Call +1(406)2729101
18.12.25 21:02 James Robert

This company call GREAT WHIP RECOVERY CYBER SERVICES was honest from the beginning. No false promises – just results. Call:+1(406)2729101
18.12.25 21:02 James Robert

I am James Robert from Chicago. Few months ago, I fell victim to an online Bitcoin investment scheme that promised high returns within a short period. At first, everything seemed legitimate, their website looked professional, and the people behind it were very convincing. I invested a significant amount of money about $440,000 with the way they talk to me into investing on their bitcoin platform. Two months later I realized that it was a scam when I could no longer have access to my account and couldn’t withdraw my money. At first, I lost hope that I wouldn't be able to get my money back, I cried and was angry at how I even fell victim to a scam. For days after doing some research and seeking professional help online, I came across GREAT WHIP RECOVERY CYBER SERVICES and saw how they have helped people recover their money back from scammers. I reported the case immediately to them and gather every transaction detail, documentation and sent it to them. Today, I’m very happy because the GREAT WHIP RECOVERY CYBER SERVICES help me recover all my money I was scammed. You can contact GREAT WHIP RECOVERY CYBER SERVICES if you have ever fallen victim to scam. Email: [email protected] or Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site or Call +1(406)2729101
20.12.25 16:03 VERONICAFREDDIE809

Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I found someone Agent Jasmine Lopez ([email protected]) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.
06.01.26 00:54 justineluscacio

I hope this review gets to as many as possible because I am sure it will save many. On March 15, I received a message on Telegram inviting me to join a Cryptocurrency investment group. At first, I thought it was a harmless invitation. I joined and began to see how a specific crypto and forex trader was helping many people and even gifting out Bitcoin to his loyal followers. I was thrilled; many people in the group were reporting their daily successes and how they were earning a substantial amount. I was jealous; I saw people making a lot of money, but not me. I became furious and decided to contact the group's admin, not realizing I had fallen for their trap. That Telegram channel was a hoax, controlled by an organized team of scammers. Within 3 months of being on that platform, I had given out a whopping 11 Bitcoin in total. I was lost when I was logged out of the investment website and blocked from accessing their Telegram channel. I tried all the means I could think of at that moment, all to no avail. It went from days to weeks, and I still couldn't get back my lost Bitcoin. I was on Medium when I came across a review about digital assets recovery companies. There were quite a few of them, but DuneNectarOnlineExpert caught my attention. I copied their email ( Support @ Dunenectaronlineexpert . Com ), filed a report, and my task was attended to with utmost urgency. After back and forth with this team of cybersecurity experts, DuneNectarOnlineExpert successfully traced the Bitcoin to the scammers' wallet, and it was successfully reclaimed. The team of DuneNectarOnlineExpert was incredibly helpful in recovering my lost Bitcoin. Telegram: T.me/Dunenectarwebexpert WhatApp: +(1252-271-8900)
06.01.26 15:05 justineluscacio

I hope this review gets to as many as possible because I am sure it will save many. On March 15, I received a message on Telegram inviting me to join a Cryptocurrency investment group. At first, I thought it was a harmless invitation. I joined and began to see how a specific crypto and forex trader was helping many people and even gifting out Bitcoin to his loyal followers. I was thrilled; many people in the group were reporting their daily successes and how they were earning a substantial amount. I was jealous; I saw people making a lot of money, but not me. I became furious and decided to contact the group's admin, not realizing I had fallen for their trap. That Telegram channel was a hoax, controlled by an organized team of scammers. Within 3 months of being on that platform, I had given out a whopping 11 Bitcoin in total. I was lost when I was logged out of the investment website and blocked from accessing their Telegram channel. I tried all the means I could think of at that moment, all to no avail. It went from days to weeks, and I still couldn't get back my lost Bitcoin. I was on Medium when I came across a review about digital assets recovery companies. There were quite a few of them, but DuneNectarOnlineExpert caught my attention. I copied their email ( Support @ Dunenectaronlineexpert . Com ), filed a report, and my task was attended to with utmost urgency. After back and forth with this team of cybersecurity experts, DuneNectarOnlineExpert successfully traced the Bitcoin to the scammers' wallet, and it was successfully reclaimed. The team of DuneNectarOnlineExpert was incredibly helpful in recovering my lost Bitcoin. Telegram: T.me/Dunenectarwebexpert WhatApp: +(1252-271-8900)
06.01.26 16:06 justineluscacio

Successful Bitcoin Recovery Testimonials in Europe, USA, UK, Asia, Canada, South Africa.
06.01.26 16:07 justineluscacio

Successful Bitcoin Recovery Testimonials in Europe, USA, UK, Asia, Canada, South Africa. BITCOIN RECOVERY EXPERT USDT RECOVERY EXPERT ETH RECOVERY EXPERT How do I hire a hacker to help me recover my lost funds? How to Recover Money Lost to an Online Scam How to recover lost or stolen cryptocurrency Best recovery experts for cryptocurrency DuneNectarOnlineExpert was incredibly helpful in recovering my lost Bitcoin. Telegram: T.me/Dunenectarwebexpert WhatApp: +(1252-271-8900)
06.01.26 16:14 justineluscacio

I hope this review gets to as many as possible because I am sure it will save many. On March 15, I received a message on Telegram inviting me to join a Cryptocurrency investment group. At first, I thought it was a harmless invitation. I joined and began to see how a specific crypto and forex trader was helping many people and even gifting out Bitcoin to his loyal followers. I was thrilled; many people in the group were reporting their daily successes and how they were earning a substantial amount. I was jealous; I saw people making a lot of money, but not me. I became furious and decided to contact the group's admin, not realizing I had fallen for their trap. That Telegram channel was a hoax, controlled by an organized team of scammers. Within 3 months of being on that platform, I had given out a whopping 11 Bitcoin in total. I was lost when I was logged out of the investment website and blocked from accessing their Telegram channel. I tried all the means I could think of at that moment, all to no avail. It went from days to weeks, and I still couldn't get back my lost Bitcoin. I was on Medium when I came across a review about digital assets recovery companies. There were quite a few of them, but DuneNectarOnlineExpert caught my attention. I copied their email ( Support @ Dunenectaronlineexpert . Com ), filed a report, and my task was attended to with utmost urgency. After back and forth with this team of cybersecurity experts, DuneNectarOnlineExpert successfully traced the Bitcoin to the scammers' wallet, and it was successfully reclaimed. The team of DuneNectarOnlineExpert was incredibly helpful in recovering my lost Bitcoin. Telegram: T.me/Dunenectarwebexpert WhatApp: +(1252-271-8900)
06.01.26 17:03 Matt Kegan

Reach out to SolidBlock Forensics if you want to get back your coins from fake crypto investment or your wallet was compromised and all your coins gone. SolidBlock Forensics provide deep ethical analysis and investigation that enables them to trace these schemes, and recover all your funds. Their services are professional and reliable. http://www.solidblockforensics.com
09.01.26 17:11 VERONICAFREDDIE809

Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I found someone Agent Jasmine Lopez ([email protected]) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.
10.01.26 14:28 justineluscacio

I hope this review gets to as many as possible because I am sure it will save many. On March 15, I received a message on Telegram inviting me to join a Cryptocurrency investment group. At first, I thought it was a harmless invitation. I joined and began to see how a specific crypto and forex trader was helping many people and even gifting out Bitcoin to his loyal followers. I was thrilled; many people in the group were reporting their daily successes and how they were earning a substantial amount. I was jealous; I saw people making a lot of money, but not me. I became furious and decided to contact the group's admin, not realizing I had fallen for their trap. That Telegram channel was a hoax, controlled by an organized team of scammers. Within 3 months of being on that platform, I had given out a whopping 11 Bitcoin in total. I was lost when I was logged out of the investment website and blocked from accessing their Telegram channel. I tried all the means I could think of at that moment, all to no avail. It went from days to weeks, and I still couldn't get back my lost Bitcoin. I was on Medium when I came across a review about digital assets recovery companies. There were quite a few of them, but DuneNectarOnlineExpert caught my attention. I copied their email ( Support @ Dunenectaronlineexpert . Com ), filed a report, and my task was attended to with utmost urgency. After back and forth with this team of cybersecurity experts, DuneNectarOnlineExpert successfully traced the Bitcoin to the scammers' wallet, and it was successfully reclaimed. The team of DuneNectarOnlineExpert was incredibly helpful in recovering my lost Bitcoin. Telegram: T.me/Dunenectarwebexpert WhatApp: +(1252-271-8900)
11.01.26 03:16 austinjarvis

I lost £75,000 in a Bitcoin scam. Scammers used deepfake videos to make it seem like Elon Musk himself pitched the deal. These deepfakes swap faces onto real people with AI. They look so real you can't spot the trick at first. They drew me in with talk of fast cash. Double your money in weeks, they said. Videos showed Musk explaining simple steps to join. I bit. Sent my Bitcoin bit by bit. First small amounts worked. Then I dumped my full savings. Boom. Funds vanished. No more contact. My bank account empty. Bills piled up. Loans to cover rent and food. Sleep gone. Stress hit hard. Felt like my whole life crashed. A mate saw my pain. He shared his own story from last year. Same scam type. He pointed me to Sylvester, a recovery agent. Reach him at Yt7cracker@gmail . com. Or WhatsApp + 1 512 577 7957. Or + 44 7428 662701. Sylvester got to work fast. Tracked the blockchain trails. Dealt with exchanges. Fought the scammers' tricks. Weeks later, my Bitcoin came back. Even the profits they promised. Debt lifted. Life back on track. If you're hit, contact him now.
12.01.26 16:24 VERONICAFREDDIE809

Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I found someone Agent Jasmine Lopez ([email protected]) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.
19.01.26 13:04 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
19.01.26 13:05 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
19.01.26 15:53 VERONICAFREDDIE809

Earlier this year, I made a mistake that changed everything. I downloaded what I thought was a legitimate trading app I’d found through a Telegram channel. At first, everything looked real until I tried to withdraw. My entire investment vanished into a bot account, and that’s when the truth hit me: I had been scammed. I can’t describe the feeling. It was as if the ground dropped out from under me. I blamed myself. I felt stupid, ashamed, helpless every painful emotion at once. For a while, I couldn’t even talk about it. I thought no one would understand. But then I found someone Agent Jasmine Lopez ([email protected]) ,She didn’t brush me off or judge me. She took my fear seriously. She followed leads I didn’t even know existed, and identified multiple off-chain indicators and wallet clusters linked to the scammer network, she helped me understand what had truly happened behind the scenes. For the first time since everything fell apart, I felt hope. Hearing that other people students, parents, hardworking people had been targeted the same way made me realize I wasn’t alone. What happened to us wasn’t stupidity. It was a coordinated attack. We were prey in a system built to deceive. And somehow, through all the chaos, Agent Jasmine stepped in and shined a light into the darkest moment of my life. I’m still healing from the experience. It changed me. But it also reminded me that even when you think you’re at the end, sometimes a lifeline appears where you least expect it. Contact her at [email protected] WhatsApp at +44 736-644-5035.
22.01.26 07:48 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
22.01.26 07:50 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
22.01.26 10:42 Tonerdomark

I got my money back from the Elon Musk scam. It cost me over 1 BTC and $55,000 in Dogecoin. Scammers vowed to double investments. Their sites seemed real. Fraud was tough to catch early. They hooked me with fast doubles in weeks or months. Videos showed Musk promising giveaways and gains. I bought in. I sent Bitcoin and Dogecoin in bits at first. Small sends worked. Then I wired my full savings. It vanished quick. No answers came. Bank account empty. Bills piled up. Loans covered rent and food. Sleep fled. Stress hit hard. Life crumbled. A friend spotted my trouble. He told of his scam loss last year. Same old plays. He pointed me to Sylvester Bryant, a recovery expert. Email Yt7cracker@gmail. com. WhatsApp +1 512 577 7957 or +44 7428 662701. Sylvester acted fast. He tracked blockchain trails. Dealt with exchanges. Outsmarted the scammers. In weeks, my Bitcoin came back. Even their phony profits too. Debts gone. Life back on track. Got hit? Contact him now.
22.01.26 19:25 Angela_Moore

Help to recover money from elon musk giveaway scam I got my money back from the Elon Musk scam. It cost me over 1 BTC and $55,000 in Dogecoin. Scammers vowed to double investments. Their sites seemed real. Fraud was tough to catch early. They hooked me with fast doubles in weeks or months. Videos showed Musk promising giveaways and gains. I bought in. I sent Bitcoin and Dogecoin in bits at first. Small sends worked. Then I wired my full savings. It vanished quick. No answers came. Bank account empty. Bills piled up. Loans covered rent and food. Sleep fled. Stress hit hard. Life crumbled. A friend spotted my trouble. He told of his scam loss last year. Same old plays. He pointed me to Sylvester Bryant, a recovery expert. Email Yt7cracker@gmail. com. WhatsApp +1 512 577 7957 or +44 7428 662701. Sylvester acted fast. He tracked blockchain trails. Dealt with exchanges. Outsmarted the scammers. In weeks, my Bitcoin came back. Even their phony profits too. Debts gone. Life back on track. Got hit? Contact him now.
23.01.26 07:35 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
23.01.26 07:35 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
26.01.26 10:36 alksnismareks

It all started when I decided to explore online trading as a way to grow my savings. Like many, I trusted what appeared to be a legitimate platform, only to find myself trapped in a nightmare. After making consistent trades and finally deciding to withdraw my profits, I was met with silence. My account was suddenly restricted—no warning, no explanation. Every attempt to contact the broker went unanswered or was met with vague, dismissive replies. For three long, agonizing months, I lived in uncertainty. I couldn’t sleep at night. I replayed every email, every transaction, wondering if I’d made a mistake. But deep down, I knew the truth: I hadn’t done anything wrong. The broker had simply decided to lock me out and keep my money. During that time, I felt completely powerless—like I was shouting into a void. The stress affected my health, my relationships, and my ability to focus on anything else. There were days I truly believed that $167,000 was gone forever, lost to the shadows of the unregulated online trading world. I even began to accept it as a painful lesson—one that would cost me dearly but might teach me to be more cautious in the future. But something inside me refused to surrender completely. That’s when I discovered TechY Force Cyber Retrieval. At first, I was cautious—after being scammed once, I didn’t want to fall victim again. But everything about TechY Force felt different. They were transparent from the start. No grand promises, no pressure tactics. Just clear, professional communication and a deep understanding of how these fraudulent brokers operate. Most importantly, they are a licensed specialist in binary options and forex fund recovery, which gave me the confidence to move forward. From our very first consultation, their team treated my case with urgency and empathy. They walked me through the entire process, explained the legal and technical avenues available, and assured me they would handle every detail. They collected documentation, analyzed transaction trails, and engaged directly with the payment processors and the broker using precise, strategic methods I never could have navigated on my own. What happened next was nothing short of miraculous. Within weeks, the broker—who had ignored me for months—began responding. And then, without any further drama or delays, my full $167,000 USD was returned to me. No deductions. No hidden fees. Just clean, complete recovery. The relief I felt was indescribable. It wasn’t just about the money—it was about reclaiming control, restoring trust, and proving that even in the face of deception, there are still good people who fight for what’s right. If you’ve been locked out of your trading account, scammed by a fake investment platform, or had your funds unjustly withheld, please know this: you are not alone, and your money may not be lost forever. Thanks to TechY Force Cyber Retrieval, I got my life back. Their expertise, integrity, and unwavering commitment turned my despair into deliverance. I cannot recommend them highly enough. To anyone reading this in distress: don’t give up. Reach out. Take that step. Because if someone like me—broken, doubtful, and nearly hopeless—can recover every dollar… so can you. WhatsApp them + 156 172 63 697 With heartfelt thanks and renewed hope, — A Recovered and Grateful Client
26.01.26 23:21 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
26.01.26 23:21 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
26.01.26 23:21 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
27.01.26 01:18 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
27.01.26 01:19 Kelvin Alfons

Hello everyone. I’d like to share my personal experience from one of the most challenging times in my life. I’m based in Sydney, Australia, and on November 13, 2025, I fell victim to a fraudulent cryptocurrency investment platform that promised substantial financial growth. Believing their claims, I invested a total of $220,000 with the expectation of earning solid returns. However, when I attempted to withdraw my funds, all communication abruptly stopped. My calls were ignored, my emails went unanswered, and I was left feeling completely powerless. Like many others, I had heard that Bitcoin transactions are impossible to trace, so I assumed my money was lost forever. After some time, I discovered information about GREAT WHIP RECOVERY CYBER SERVICES, a reputable digital asset recovery firm. I decided to reach out to them, and to my astonishment, they were able to help me recover the full amount I had lost. I’m sharing my story in the hope that it may help someone else who is going through a similar situation and looking for support. Their contact is, Website: https://greatwhiprecoveryc.wixsite.com/greatwhip-site Email:[email protected] Call: +1(406)2729101
27.01.26 09:29 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
27.01.26 09:29 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
27.01.26 09:32 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
29.01.26 05:03 joyo

The digital world of cryptocurrency offers big chances, but it also hides tricky scams. Losing your crypto to fraud feels awful. It can leave you feeling lost and violated. This guide tells you what to do right away if a crypto scam has hit you. These steps can help you get funds back or stop more trouble. Knowing what to do fast can change everything,reach marie ([email protected] and whatsapp:+1 7127594675)
30.01.26 08:23 joseph67t

It's a joy to write this review. Since I began working with Marie at the beginning of 2018, the service has been outstanding. Hackers stole my monies, and I was frightened about how I would get them back. I didn't know where to begin, consequently it was a nightmare for me. But once my friend told me about ([email protected] and whatsap:+1 7127594675), things became simple for me. I'm glad she was able to get my bitcoin back so I could start trading on Binance again!
31.01.26 00:55 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
31.01.26 00:55 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
02.02.26 18:52 Christopherbelle

Sylvester Bryant is a top crypto recovery agent! Then I contacted them with my story that i have been scammed. It took time, yet my stolen crypto was recovered . Need help? Reach out to Sylvester on WhatsApp at +1 512 577 7957 or +44 7428 662701. Or email yt7cracker@gmail . com.
03.02.26 08:05 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
03.02.26 08:05 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
04.02.26 16:23 borutaralf

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04.02.26 16:24 borutaralf

GREAT WHIP RECOVERY CYBER SERVICES TRUSTED EXPERTS IN ONLINE RECOVERY SOLUTIONS PHONE CALL:+1(406)2729101 I was unfortunately deceived and scammed out of $88,000 by someone I trusted to manage my funds during a transaction we carried out together. The experience left me deeply disappointed and hurt, realizing that someone could betray that level of trust without any remorse. Determined to seek justice and recover what was stolen, I began searching for legal assistance and came across numerous testimonials about GREAT WHIP RECOVERY CYBER SERVICES, a group known for helping victims recover lost funds. From what I learned, they have successfully assisted many people facing similar situations, returning stolen funds to their rightful owners in a remarkably short time. In my case, the GREAT WHIP RECOVERY CYBER SERVICES were able to recover my funds within just 48 hours, which was truly unbelievable. Even more reassuring was the fact that the scammer was identified, located, and eventually arrested by local authorities in his region. That outcome brought a great sense of relief and closure. I hope this information helps others who have lost their hard-earned money due to misplaced trust. If you’re in a similar situation, you can contact them through their info below to seek help in recovering your stolen funds. Email: [email protected] Website https://greatwhiprecoveryc.wixsite.com/greatwhip-site Phone Call:+1(406)2729101
04.02.26 17:11 wendytaylor015

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05.02.26 12:07 Thomas Muller

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05.02.26 15:46 feliciabotezatu

Losing access to your cryptocurrency can be devastating—whether you’ve been scammed, hacked, or locked out due to a forgotten password. Many assume their digital assets are gone forever. But with the right expertise, recovery is not only possible—it’s our daily reality. At TECHY FORCE CYBER RETRIEVAL (TFCR), we’re a globally recognized, fully legitimate crypto recovery service dedicated to helping victims reclaim lost or stolen digital assets—safely, ethically, and effectively. Who We Are Backed by a team of certified blockchain forensic analysts, cybersecurity specialists, and ethical hackers, TFCR has recovered millions of dollars in Bitcoin, Ethereum, USDT, and other major cryptocurrencies for clients worldwide. We specialize in cases involving: - Investment scams and fake platforms - Wallet hacks and unauthorized transactions - Forgotten passwords, seed phrases, or corrupted backups - Inaccessible hardware or software wallets Our mission is clear: Help you recover what’s rightfully yours—with honesty, transparency, and proven results. How We Work 1. Confidential Case Review Share your situation with us—no cost, no obligation. We assess whether your case is recoverable based on transaction data, wallet details, and loss type. 2. Advanced Blockchain Forensics Using industry-leading tools, we trace your funds across blockchains, identify destination addresses, and determine if assets are held on exchanges or recoverable platforms—even after complex laundering attempts. 3. Custom Recovery Execution Depending on your case, we: - Reconstruct access to locked wallets using secure decryption methods - Engage with exchanges or payment processors to freeze or retrieve funds - Provide forensic reports to support legal or compliance actions - Negotiate with third parties when appropriate and safe 4. Secure Return & Prevention Advice Recovered assets go directly to a wallet you control. We also offer practical guidance to help you avoid future losses—because security starts after recovery. Why Choose TFCR? No Recovery, No Fee – You only pay upon successful retrieval Legitimate & Transparent – No upfront payments, no hidden costs Global Expertise – Proven success across 50+ countries Ethical Standards – All actions comply with cybersecurity and privacy best practices While crypto threats grow daily, so does our resolve. At TECHY FORCE CYBER RETRIEVAL, we don’t just track transactions—we restore trust, hope, and financial peace of mind. Don’t give up on your crypto. Act now—before critical evidence disappears. 📧 Email: [email protected] 🌐 Visit: Official https://techyforcecyberretrieval.com Website] 🕒 Available 24/7 for urgent cases Your crypto may be missing—but with TFCR, it’s never truly lost. ©️ 2026 TECHY FORCE CYBER RETRIEVAL — Trusted. Professional. Results-Driven.
05.02.26 15:52 harryjones5

How Can I Contact a Cryptocurrency Recovery Company? Visit iFORCE HACKER RECOVERY I realize how volatile and thrilling cryptocurrency can be. After joining a Telegram-based service, I made consistent profits for six months before unexpected faults deprived me of approximately $343,000. Withdrawal blunders, little help, and rising dread kept me stuck. I then discovered iForce Hacker Recovery from positive reviews. They replied swiftly, handled my issue professionally, and walked me through every step. My valuables were returned within a week, giving me back my confidence. I heartily recommend their dependable, professional aid services. Contact Info: Website address: htt p:// iforcehackers. co m. Email: iforcehk @ consultant .co m WhatsApp: +1 240 803-3706
06.02.26 14:44 feliciabotezatu

Losing access to your cryptocurrency can be devastating—whether you’ve been scammed, hacked, or locked out due to a forgotten password. Many assume their digital assets are gone forever. But with the right expertise, recovery is not only possible—it’s our daily reality. At TECHY FORCE CYBER RETRIEVAL (TFCR), we’re a globally recognized, fully legitimate crypto recovery service dedicated to helping victims reclaim lost or stolen digital assets—safely, ethically, and effectively. Who We Are Backed by a team of certified blockchain forensic analysts, cybersecurity specialists, and ethical hackers, TFCR has recovered millions of dollars in Bitcoin, Ethereum, USDT, and other major cryptocurrencies for clients worldwide. We specialize in cases involving: - Investment scams and fake platforms - Wallet hacks and unauthorized transactions - Forgotten passwords, seed phrases, or corrupted backups - Inaccessible hardware or software wallets Our mission is clear: Help you recover what’s rightfully yours—with honesty, transparency, and proven results. How We Work 1. Confidential Case Review Share your situation with us—no cost, no obligation. We assess whether your case is recoverable based on transaction data, wallet details, and loss type. 2. Advanced Blockchain Forensics Using industry-leading tools, we trace your funds across blockchains, identify destination addresses, and determine if assets are held on exchanges or recoverable platforms—even after complex laundering attempts. 3. Custom Recovery Execution Depending on your case, we: - Reconstruct access to locked wallets using secure decryption methods - Engage with exchanges or payment processors to freeze or retrieve funds - Provide forensic reports to support legal or compliance actions - Negotiate with third parties when appropriate and safe 4. Secure Return & Prevention Advice Recovered assets go directly to a wallet you control. We also offer practical guidance to help you avoid future losses—because security starts after recovery. Why Choose TFCR? No Recovery, No Fee – You only pay upon successful retrieval Legitimate & Transparent – No upfront payments, no hidden costs Global Expertise – Proven success across 50+ countries Ethical Standards – All actions comply with cybersecurity and privacy best practices While crypto threats grow daily, so does our resolve. At TECHY FORCE CYBER RETRIEVAL, we don’t just track transactions—we restore trust, hope, and financial peace of mind. Don’t give up on your crypto. Act now—before critical evidence disappears. Email: [email protected] Visit: Official https://techyforcecyberretrieval.com Website] Available 24/7 for urgent cases Your crypto may be missing—but with TFCR, it’s never truly lost. ©️ 2026 TECHY FORCE CYBER RETRIEVAL — Trusted. Professional. Results-Driven.
00:44 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
00:44 marcushenderson624

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04:43 Matt Kegan

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