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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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Ethereum

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Дата

7 февр. 2026 г.

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05.07.26 14:44 lydiassmith567

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05.07.26 14:44 lydiassmith567

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06.07.26 16:20 Olga Ognjanović

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06.07.26 16:31 Joseph Weigl

Invest wisely and stay cautious. Don't be influenced by promises of unusually high returns or convincing sales pitches from brokers. I learned this the hard way after falling victim to an investment scam that promised huge profits. Fortunately, I acted quickly and reported the incident to a recovery firm for assistance. Contact: Email: [email protected] Telegram: @Resqprofirm WhatsApp: +1 985 296 9146
06.07.26 16:33 Jaran Løvlien

A heartfelt thank you to RESQPRO FIRM for their commitment and professionalism throughout the investigation of my case. Their team worked diligently and helped recover assets valued at $88,000, which were returned to my wallet. I truly appreciate their support, clear communication, and dedication, and I'm grateful for the assistance I received. Contact: Email: [email protected] Telegram: @Resqprofirm WhatsApp: +1 985 296 9146
07.07.26 18:00 robertalfred175

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07.07.26 18:01 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
09.07.26 19:06 Toivo Walli

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09.07.26 19:10 Misty Alexander

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12.07.26 03:33 Pahal Mathew

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13.07.26 23:49 [email protected]

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15.07.26 11:53 Sarah Green

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16.07.26 21:38 patricialovick86

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16.07.26 21:38 patricialovick86

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17.07.26 19:24 laimqq90

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17.07.26 20:12 martinsjude080

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18.07.26 17:12 Malthe Larsen

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18.07.26 17:42 پریا رضایی

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18.07.26 17:46 Adem Akışık

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18.07.26 23:51 bernalzenaida

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19.07.26 04:10 Fraddy Pual

I can't thank Fundsretriever enough for everything they did for me. My name is Vanessa Conway, and I'm here to tell you my story of how I recovered money I never thought I'd see again. A few months ago, I put a significant amount of money into what looked like a genuine online investment company. At first, it felt real—they showed me fake profits and convinced me to invest even more. But when I tried to cash out, they went quiet and started asking for extra fees. That's when it hit me—I had been scammed. I was heartbroken, frustrated, and didn't know where to turn. That money was my savings—months of hard work gone. Then I found Fundsretriever online. I reached out, hoping for a miracle. Right away, their team made me feel heard. They were responsive, knowledgeable, and walked me through everything. They didn't just take my case—they took it seriously and kept me in the loop every step of the way. I finally felt like I had real experts fighting for me. And guess what? They actually got my money back. It wasn't instant, and it took teamwork, but it happened. When I saw those funds returned, I cried with joy. I'm sharing this so that anyone else out there who's been scammed knows—don't lose hope. Do your research before investing, and stay far away from platforms that promise too much too fast. And if you've already been stung, don't wait—get professional help immediately. Thank you, Fundsretriever, from the bottom of my heart. You didn't just recover my money—you restored my faith. — Vanessa Conway 📧 [email protected] 📱 WhatsApp: +16035121448 💬 Telegram: @Fundsretriever
19.07.26 19:53 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text Number: +1 (336) 390-6684
19.07.26 19:53 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text Number: +1 (336) 390-6684
19.07.26 19:54 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text Number: +1 (336) 390-6684
30.07.26 00:04 Ahmed

A really cool analysis, thank you. I was especially struck by how strictly the order is defined: data processing and formatting come first, with color only at the very end. This really saves you from the typical mistake of "first a pretty palette, then we figure out what the chart is for." And the palette validator with OKLCH + colorblindness check is absolutely fantastic; you almost never see that in regular tools. By the way, while reading about rank-trajectory and the logarithmic scale, I immediately remembered how convenient it is to analyze dynamics on charts in ExpertOption—I've been trading there for quite some time now. When the data is well visualized, decisions are noticeably easier and more relaxed. I also liked the point about "no more than eight colors" and the dual-axis ban. Strict restrictions sometimes actually produce better results than complete freedom. I'll try this approach myself.
30.07.26 17:27 wendytaylor015

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30.07.26 17:27 wendytaylor015

My name is Wendy Taylor, I'm from Los Angeles, i want to announce to you Viewer how Capital Crypto Recover help me to restore my Lost Bitcoin, I invested with a Crypto broker without proper research to know what I was hoarding my hard-earned money into scammers, i lost access to my crypto wallet or had your funds stolen? Don’t worry Capital Crypto Recover is here to help you recover your cryptocurrency with cutting-edge technical expertise, With years of experience in the crypto world, Capital Crypto Recover employs the best latest tools and ethical hacking techniques to help you recover lost assets, unlock hacked accounts, Whether it’s a forgotten password, Capital Crypto Recover has the expertise to help you get your crypto back. a security company service that has a 100% success rate in the recovery of crypto assets, i lost wallet and hacked accounts. I provided them the information they requested and they began their investigation. To my surprise, Capital Crypto Recover was able to trace and recover my crypto assets successfully within 24hours. Thank you for your service in helping me recover my $647,734 worth of crypto funds and I highly recommend their recovery services, they are reliable and a trusted company to any individuals looking to recover lost money. Contact email [email protected] OR Telegram @Capitalcryptorecover Call/Text Number +1 (336)390-6684 his contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1
31.07.26 16:21 rssllhrnsb

I learned an important lesson after investing in what appeared to be a genuine opportunity. Unfortunately, I was unable to access my funds, which reinforced the importance of carrying out thorough due diligence, checking whether an investment is appropriately regulated, and seeking qualified professional advice rather than relying solely on online reviews or testimonials. During the process of addressing my case, I worked with Mrs. Doris Ashley, who communicated clearly, provided regular updates, and handled the matter in a professional manner. According to my experience, I have recovered $50,000 so far, while efforts to resolve the remaining balance are still in progress. If you wish to contact her, the details I used are: Mrs. Tatiana Sorina TEXT : (tatianasorina06 at G.Ma IL ..c 0 m ) Before committing money to any investment, take time to verify the legitimacy of the platform, confirm any relevant regulatory authorisations, and avoid investing more than you can comfortably afford to lose.
01.08.26 15:05 keithwilson9899

ETHEREUM RECOVERY ASSISTANCE: CAPITAL CRYPTO RECOVER HELPED ME RECOVER $98,000 WORTH OF LOST ETH In cases of cryptocurrency scams, having accurate information and trusted support is essential. I would like to recommend Capital Crypto Recover Service, a professional team that specializes in assisting individuals with the recovery of lost or stolen Bitcoin and Ethereum (ETH). Their experienced experts are dedicated to helping victims of digital asset fraud by carefully analyzing each case, developing strategic recovery plans, Capital Crypto Recover Service knowledgeable team's primary goals are to satisfy clients and offer significant support and working diligently toward fund retrieval. The team is committed to providing reliable assistance and maintaining a high level of client satisfaction. Based on my assessment, their reputation professionalism and a strong commitment to their clients. If you have experienced a cryptocurrency loss, you can contacting them for further assistance Phone (Call/Text): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
01.08.26 15:05 keithwilson9899

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03.08.26 20:05 Philip

I was a victim of a crypto theft involving a Pink Drainer, which resulted in the theft of my Wrapped Bitcoin (WBTC) from my Polygon network. The experience was incredibly frustrating and distressing, as I had no idea how to recover my funds. Unfortunately, by the time I noticed the theft, the funds had already been drained to an address that I had no control over, making it seem like an irreversible situation.The attack happened when I clicked on what appeared to be a legitimate link. I didn’t realize at the time that it was a phishing attempt designed to siphon off my private keys and access my wallet. The Pink Drainer, a type of malicious script used by attackers, is specifically designed to exploit such vulnerabilities in crypto wallets. The moment I realized that my WBTC had been drained from my Polygon network. I felt completely helpless, as I didn’t have direct access to the thief's address, and there was no way to reverse the transaction on my own at that point, I started searching for ways to recover my funds, but most resources only offered generic advice that wasn’t practical in this particular case. I quickly realized that if I wanted to have any hope of getting my assets back, I would need professional assistance. After some research, I came across a reliable and trusted recovery team called Aspen Recovery Experts. Their expertise in cryptocurrency recovery, especially in cases like mine, seemed promising.I decided to reach out to Aspen Recovery Experts, and I’m incredibly grateful that I did. Their team of crypto recovery experts was able to help me trace the stolen funds and identify the path the funds took after they left my wallet. Using advanced tools and techniques, they were able to track the transactions on the blockchain, helping me understand where my WBTC had been sent. More importantly, they worked tirelessly to assist me in contacting the necessary parties and even interfaced with blockchain analysts to help facilitate the recovery process.Thanks to their efforts, I was able to successfully recover my stolen funds. The entire process took some time, but the Aspen Recovery Experts dedicated team provided regular updates and kept me informed throughout the process, which gave me a sense of hope and relief during an otherwise stressful time. If you’re ever in a similar situation, I highly recommend reaching out to a trusted recovery team like Aspen Recovery Expertshhh . Their professionalism, knowledge, and expertise were critical in helping me recover my funds and regain control of my crypto assets. Whatsapp : +1 747 231 9036 Telegram : @ prohackerspy Email : [email protected]
03.08.26 20:06 Philip

I was a victim of a crypto theft involving a Pink Drainer, which resulted in the theft of my Wrapped Bitcoin (WBTC) from my Polygon network. The experience was incredibly frustrating and distressing, as I had no idea how to recover my funds. Unfortunately, by the time I noticed the theft, the funds had already been drained to an address that I had no control over, making it seem like an irreversible situation.The attack happened when I clicked on what appeared to be a legitimate link. I didn’t realize at the time that it was a phishing attempt designed to siphon off my private keys and access my wallet. The Pink Drainer, a type of malicious script used by attackers, is specifically designed to exploit such vulnerabilities in crypto wallets. The moment I realized that my WBTC had been drained from my Polygon network. I felt completely helpless, as I didn’t have direct access to the thief's address, and there was no way to reverse the transaction on my own at that point, I started searching for ways to recover my funds, but most resources only offered generic advice that wasn’t practical in this particular case. I quickly realized that if I wanted to have any hope of getting my assets back, I would need professional assistance. After some research, I came across a reliable and trusted recovery team called Aspen Recovery Experts. Their expertise in cryptocurrency recovery, especially in cases like mine, seemed promising.I decided to reach out to Aspen Recovery Experts, and I’m incredibly grateful that I did. Their team of crypto recovery experts was able to help me trace the stolen funds and identify the path the funds took after they left my wallet. Using advanced tools and techniques, they were able to track the transactions on the blockchain, helping me understand where my WBTC had been sent. More importantly, they worked tirelessly to assist me in contacting the necessary parties and even interfaced with blockchain analysts to help facilitate the recovery process.Thanks to their efforts, I was able to successfully recover my stolen funds. The entire process took some time, but the Aspen Recovery Experts dedicated team provided regular updates and kept me informed throughout the process, which gave me a sense of hope and relief during an otherwise stressful time. If you’re ever in a similar situation, I highly recommend reaching out to a trusted recovery team like Aspen Recovery Expertshhh . Their professionalism, knowledge, and expertise were critical in helping me recover my funds and regain control of my crypto assets. Whatsapp : +1 747 231 9036 Telegram : @ prohackerspy Email : [email protected]
04.08.26 11:05 Kisnoles

Excellent analysis, thank you. I was particularly struck by how rigidly the skill sets the order: data processing and form come first, with color coming last. This really cures the habit of "first a pretty palette, then we'll figure it out." A palette validator using OKLCH + color blindness + WCAG is something most chart generators lack. And regarding the boundaries of competence, it's very clear. Where there's a self-checking loop (color), you delegate freely. Where there's a heuristic (form, data interpretation), you remain the final filter. This is a universal principle, not just for /dataviz. By the way, when you look at trading dashboards (including those of decent platforms like ExpertOption), it's immediately clear who thought about readability and who just threw in rainbow lines. Tools like this skill could greatly improve the quality of analytics. Thanks again for the detailed analysis – saved.
04.08.26 12:37 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text Number: +1 (336) 390-6684
04.08.26 12:37 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] Call/Text Number: +1 (336) 390-6684
06.08.26 08:11 ROMMYHENDERSON344

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06.08.26 13:56 lydiassmith567

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06.08.26 13:56 lydiassmith567

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11.08.26 03:43 raymont0714

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12.08.26 16:37 rssllhrnsb

I learned an important lesson after investing in what appeared to be a genuine opportunity. Unfortunately, I was unable to access my funds, which reinforced the importance of carrying out thorough due diligence, checking whether an investment is appropriately regulated, and seeking qualified professional advice rather than relying solely on online reviews or testimonials. During the process of addressing my case, I worked with Mrs. Tatiana Sorina , who communicated clearly, provided regular updates, and handled the matter in a professional manner. According to my experience, I have recovered $50,000 so far, while efforts to resolve the remaining balance are still in progress. If you wish to contact her, the details I used are: Mrs. Tatiana Sorina TEXT : (tatianasorina06 at G_Ma IL dot ..c 0 m)…. Before committing money to any investment, take time to verify the legitimacy of the platform, confirm any relevant regulatory authorisations, and avoid investing more than you can comfortably afford to lose.
16.08.26 01:44 Matt Kegan

CapitalNode Analytics. They help to investigate and recover stolen digital assets from fake trading platforms. Great firm i must say.
18.08.26 04:59 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 WhatsApp/Text Number: +1 (336) 390-6684 Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
18.08.26 04:59 marcushenderson624

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19.08.26 14:50 BAYER7043

My account was locked, and I couldn’t access my own information until [email protected] +447476618364 whats?up? helped me recover it. This lady hacker was kind and professional, but this experience showed me how risky account lockouts can be. If you’re facing the same problem, do not panic consult her A. S. A. P. Be careful with recovery agents outchea, and research their name, business, and reviews before sharing any details. Never send passwords, security codes, banking information, or copies of your ID unless you’ve confirmed who you’re dealing with. Be wary of promises that sound too good to be true, especially claims that require little information or guarantee instant access with outrageous fees.
20.08.26 11:32 michaeldavenport218

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20.08.26 11:32 michaeldavenport218

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23.08.26 20:02 leslieyee

Excellent analysis by /dataviz! I was particularly struck by how strictly the order is defined: "data meaning first, color last" and how the validator actually adjusts the palette according to OKLCH, color blindness, and WCAG standards. These rules greatly improve the quality of charts. By the way, a similar approach to clean and legible charts is highly valued in trading—for example, on the ExpertOption platform, the interface and price visualization are designed to ensure information is read instantly and without unnecessary noise.
25.08.26 13:44 lydiassmith567

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25.08.26 13:44 lydiassmith567

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30.08.26 15:37 [email protected]

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30.08.26 15:37 [email protected]

Discovering I had been defrauded, I searched online for services claiming to recover stolen ETH. I contacted several companies, but none succeeded. Although some claimed they could trace assets or recover funds, I could not verify their success or recover my money. I later found SYLVESTER BRYANT Recovery through Google. After contacting him, he recovered 450,000,00. He can be reached at [email protected] or WhatsApp at +1 512 577 7957.
01.09.26 11:16 lisawerth897

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01.09.26 11:16 lisawerth897

My Experience With Cryptocurrency — CAPITAL CRYPTO RECOVER I lost over $82,000 in Bitcoin after falling victim to a fraudulent online investment scheme. After realizing I had been scammed, I spent considerable time researching possible ways to recover my funds. During my research, I came across CAPITAL CRYPTO RECOVER and decided to contact them after reading their reported success record and encouraging client positive reviews and testimonials. Their team guided me through the recovery process, and I was grateful to successfully recover my lost cryptocurrency. The experience was challenging, but I appreciated the assistance and support I received throughout the process. I’m sharing my experience in the hope that it may encourage other victims to carefully research their options and verify any recovery service before proceeding. I will forever be thankful to you CAPITAL CRYPTO RECOVER 📧 [email protected] 🌐 Website: recovercapital.wixsite.com/capital-crypto-rec-1 📧 [email protected] 📞 Call/WhatsApp: +1 (336) 390-6684
01.09.26 17:34 Garry42

Really crazy world. These fraudsters go at any length to steal your hard earned funds. I have been a victim of a bitcoin scam about 7 months back. A Con artist gained access to my cashapp account through a phishing scam. They stole $409,000. I was really devastated. I did everything to get back my funds by contacting the FBI but they claimed there was nothing they could do. A friend told me about a recovery expert. He helps fight against various phishing and investment scams and they were able to help trace and recover my funds even though it took over 2 days. you can reach out to him through recoverydarek@gmail. com . I can guarantee his services are still active.
01.09.26 17:34 Garry42

Really crazy world. These fraudsters go at any length to steal your hard earned funds. I have been a victim of a bitcoin scam about 7 months back. A Con artist gained access to my cashapp account through a phishing scam. They stole $409,000. I was really devastated. I did everything to get back my funds by contacting the FBI but they claimed there was nothing they could do. A friend told me about a recovery expert. He helps fight against various phishing and investment scams and they were able to help trace and recover my funds even though it took over 2 days. you can reach out to him through recoverydarek@gmail. com . I can guarantee his services are still active.
02.09.26 03:43 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684
02.09.26 03:43 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684
04.09.26 21:51 Kovengray

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05.09.26 21:38 [email protected]

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09.09.26 21:31 lisawerth897

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09.09.26 21:31 lisawerth897

My Experience With Cryptocurrency — CAPITAL CRYPTO RECOVER I lost over $82,000 in Bitcoin after falling victim to a fraudulent online investment scheme. After realizing I had been scammed, I spent considerable time researching possible ways to recover my funds. During my research, I came across CAPITAL CRYPTO RECOVER and decided to contact them after reading their reported success record and encouraging client positive reviews and testimonials. Their team guided me through the recovery process, and I was grateful to successfully recover my lost cryptocurrency. The experience was challenging, but I appreciated the assistance and support I received throughout the process. I’m sharing my experience in the hope that it may encourage other victims to carefully research their options and verify any recovery service before proceeding. I will forever be thankful to you CAPITAL CRYPTO RECOVER 📧 [email protected] 🌐 Website: recovercapital.wixsite.com/capital-crypto-rec-1 📧 [email protected] 📞 Call/WhatsApp: +1 (336) 390-6684
09.09.26 23:22 Fraddy Pual

Hearing that these individuals are focusing on other people makes me very sad. I had a similar situation with them and lost a lot of money, but after learning about (Cruxcipherteam @ proton DoT me), whataqq:+168160-15021, telegram: @Cruxcipherteam I was able to get my money back. It's critical that we all be watchful and keep reporting these occurrences.
11.09.26 03:23 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684
11.09.26 03:23 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684
15.09.26 03:35 elioduncan

I fell victim to a freelance web development contract scam that ultimately cost me CAD 4,000. It all started when I came across a job posting on Indeed Canada, a popular online job portal. The position seemed perfect: a freelance web developer role for an established company looking for someone to design and build a fully functional e-commerce website. The job promised a high payment of CAD 20,000 upon successful completion, which sounded like a great opportunity for me to gain experience and earn decent pay.The employer, who introduced himself as a project manager from a "well-known" tech company, was very persuasive and professional in our initial communication. He explained that the project involved developing a user-friendly, responsive online store for their client and that they had a strict timeline to meet. He assured me that the payment would be made promptly after completing the tasks. However, before starting the work, he told me that I would need to pay an upfront fee of CAD 4,000 to cover certain software tools and licensing fees required for the project. This was supposedly a part of their company policy for freelance contractors, ensuring access to their premium resources. The idea of working on a professional project, coupled with the promise of a substantial payout, convinced me to pay the upfront fee.As soon as I made the payment, the project manager became increasingly difficult to reach. He initially responded to my emails and provided some vague instructions on what the project would entail, but as time went on, communication slowed to a complete halt. I never received the required tools or any proper project details. My emails went unanswered, and any attempts to contact the company were met with silence. After waiting for weeks, I realized that I had been scammed.Desperate to recover my money, I turned to TechY Force Cyber Retrieval, a service that specializes in helping victims of online scams. They helped me track the payment and took legal steps to pursue the fraudsters. Through their guidance, I was able to recover the full CAD 4,000. TechY Force Cyber Retrieval worked with my bank to reverse the transaction and liaised with the authorities to trace the scammer's details.This was a hard lesson, and I now know to be highly cautious about online job offers that require upfront payments. It is crucial to research companies thoroughly and avoid any job that seems too good to be true, especially when it involves paying money upfront. WhatsApp https://wa.link/2x6ktp Mail. [email protected]
15.09.26 15:45 lydiassmith567

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15.09.26 15:45 lydiassmith567

HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1