From a96ddfc7b6fdf77b084ddf92ac5133217514a269 Mon Sep 17 00:00:00 2001 From: Egor Tkachenko <68852747+HalcyonForest@users.noreply.github.com> Date: Mon, 27 Jun 2022 14:42:48 +0300 Subject: [PATCH] =?UTF-8?q?=D0=9A=D0=BE=D0=BD=D1=81=D0=BF=D0=B5=D0=BA?= =?UTF-8?q?=D1=82=20=D0=BF=D0=BE=20=D0=BB=D0=B5=D0=BA=D1=86=D0=B8=D0=B8=20?= =?UTF-8?q?"=D0=93=D0=BB=D0=B8=D0=BD=D1=8F=D0=BD=D1=8B=D0=B5=20=D0=BC?= =?UTF-8?q?=D0=B5=D1=82=D0=BE=D0=B4=D1=8B=20=D1=80=D0=B0=D0=BD=D0=B6=D0=B8?= =?UTF-8?q?=D1=80=D0=BE=D0=B2=D0=B0=D0=BD=D0=B8=D1=8F"?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit пдф + исходник --- .../lecture_23_Advanced_Ranking.pdf | Bin 0 -> 221095 bytes .../lecture_23_Advanced_Ranking.tex | 133 ++++++++++++++++++ 2 files changed, 133 insertions(+) create mode 100644 2021-spring/lecture-notes/lecture_23_Advanced_Ranking.pdf create mode 100644 2021-spring/lecture-notes/lecture_23_Advanced_Ranking.tex diff --git 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+\usepackage{amsmath} +\usepackage{bigints} +\usepackage[russian]{babel} +\usepackage{array} +\usepackage{theorem} +\usepackage{ifthen} +\usepackage[ruled,section]{algorithm} +\usepackage[noend]{algorithmic} +\usepackage[all]{xy} +\usepackage{graphicx} +\usepackage{floatflt} +%%\usepackage{pb-diagram} +\usepackage{multicol} +\usepackage[footnotesize]{caption2} +\usepackage{mathrsfs} +\usepackage{color} +\usepackage{verbatim} + +\textheight=240mm +\textwidth=160mm +\topmargin=-14mm +\headsep=7mm +%\oddsidemargin=7mm\evensidemargin=-3mm +\oddsidemargin=0mm\evensidemargin=0mm +\marginparwidth=36pt +\tolerance=3000 +\hbadness=2000 +%\flushbottom +\raggedbottom +% подавить эффект "висячих стpок" +\clubpenalty=10000 +\widowpenalty=10000 +\renewcommand{\baselinestretch}{1} +%\renewcommand{\baselinestretch}{1.1} %для печати с большим интервалом + +\title{"Глиняные методы ранжирования"} +\author{Автор: Егор Ткаченко} +\date{June 2022} + +\begin{document} + +\maketitle + +\section{Pairwise ranking} +В прошлой лекции мы остановились на поточечном (Pointwise) методе ранжирования. \\ Вида $X = (q_i, d_i, y_i)_i^l$. - запрос, документ и релевантность. Все просто - предсказываем $y_i=a(q_i, d_i)$. Как обсуждалось ранее, этот метод имеет ощутимый минус - мы предсказываем меру релевантности, в то время, как нам нужен всего лишь порядок. Возникает потребность в более специфичных методах, которые будут более точными и конкретными.\\ \\ + +\section{RankNet} +Pairwise метод: рассмотрим множество объектов $R$ = $(i_k, j_k)_k^l$. Имеем некое количество таких пар. Если такая пара входит в $R$, то для $(i,j) \in R$, справедливо $a(x_i) < a(x_j)$. Т.е j-ый объект имеет более высокую позицию в нашем ранжировании относительно i-го объекта. \\ +Запишем следующий функционал: $$\mathbb{L} = \sum\limits_{(i,j) \in R}{[a(x_j)-a(x-i) < 0]} $$. Тоесть мы штрафуем, если j-ый объект получил скор меньше, чем i-ый объект. \\ +Мы не можем минимизировать это явным образом, т.к индикатор, но мы умеем работать с такими вещами. Мы заменим сумму индикаторов на верхнюю оценку, как и раньше. \\ +$$\sum\limits_{(i,j) \in R}{[a(x_j)-a(x-i) < 0]} \leqslant \sum\limits_{(i,j) \in R}{\tilde{\mathcal{L}}(a(x_j)-a(x_i))} \rightarrow \min_{a} $$. \\Где оценка сверху - +$\tilde{\matchcal{L}}(z) = \log(1+e^{-\sigma z}), \sigma \in \mathbb{R}_{+}$ \\\\ В этом методе мы берем $a(x) = \langle w,x \rangle, \tilde{\mathcal{L}}$, обучаем нашу модель с помощью SGD. В итоге получаем следующую формулу обновления весов: \\ +\[w := w + \eta \frac{\sigma(x_j-x_i) }{1 + exp(\sigma \langle x_j-x_i,w\rangle)} \] Дальше была придумана одна очень хорошая эмпирическая штука. Она позволила перейти от решения задачи про долю дефектных парк, к задаче оптимизации некоторой специфичной метрики качества ранжирования. \\ \\ +\section{LambdaRank} +Что если хотим оптимизировать DCG ? \\Отличие DCG от нашей попарной оптимизации из начала конспекта в том, что в DCG нам важны только пары, которые должны стоять высоко. Нижние объекты нас слабо интересуют. Если для двух объектов мы знаем, что они обы должны стоять где-то нгде-то внизу - нам не важен их порядок относительно друг друга с точки зрения DCG. Чтобы это учесть предлагается добавить в нашу формулу обновления весов множитель $|\vartriangle F_{ij}|$. \\Этот множитель показывает насколько изменится целевая метрика, если мы поменяем $x_i$ и $x_j$ местами. Если нам попадается эта пара, то от того, что мы поменяем их местами в ранжировании наша метрика не изменится. Поэтому мы просто домножим шаг на 0, и скажем модели не подгонять модель под эту пару $(x_i, x_j)$. \\\\ + +\section{DSSM} + +Пусть на входе имеется запрос q и документ d. Далее поверх них мы наворачиваем какую-то нейросеть. Например Embedding-слой, Encoder, RNN и так далее. Что угодно, что на выходе дает нам вектор, который является представлением запроса или документа. \\ На выходе имеем два вектора: $v_q$ и $v_d$. Мы считаем между ними косинусное расстояние: \\ $$\frac{\langle v_q, v_d\rangle}{\|v_q \| \|v_d \|} += a(q,d)$$ Функция потерь для этого метода: \\ Обозначим вероятность того, что документ d релевантен для запроса q: $$p(d|q)=\frac{\exp(\sigma \cdot a(q,d))}{\sum\limits_{d' \in \mathcal{D}}{\exp(\sigma \cdot a(q, d'))}}, \sigma \in \mathbb{R}_+$$ \\ $$L: -\log\prod\limits_{(q,d) \in R}{p(d|q)} \rightarrow min \text{, где R - множество кликов.}$$ + +Заметим, что как правило у нас очень большое множество документов, так что напрямую посчитать $\sum\limits_{d' \in \mathcal{D}}{\exp(\sigma \cdot a(q, d'))}$ не выйдет. Как нам с этим справиться? Изменим формулу: \\ +$$p(d|q)=\frac{ +\exp(\sigma \cdot a(q,d)) +}{ +\exp(\sigma \cdot a(q, d)) + \sum\limits_{q, d_-}{\exp(\sigma \cdot a(q, d_-))} +}$$, где $d_-$ - подмножество документов, из множества некликнутых документов. Чтобы не насемплировать ну совсем нерелевантные документы в $D_-$, мы берем документ с вероятностью, пропорциональной a(q, d). + +\section{ListWise ranking} +\textbf{5.1 ListNet}\\ + +Допустим у нас есть конкретный запрос q и в ответ на этот запрос мы должны отранжировать какое-то количество документов: $\{d_1, d_2, ... , d_{n_{q}}\}$. Также мы имеем истинные релевантности для этих документов относительно запроса q: $\{ y_1, y_2, ... y_{n_{q}}\}$. Допустим мы получили оценки релевантности от нашей модели: $\{z_1, z_2, ... z_{n_{q}} \}$. Нам нужно задать функцию потерь, чтобы обучать параметры такой модели. \\ + +Предлагается следющая концепция: когда модель выдает скоры, она выдает конкретную сортировку этих документов. Предположим, что у нас умная модель и вместо одной сортировки она выдает распределение вероятностей на всех перестановках наших документов. \\ + +Зададим распределение: $$P_{z}(\pi) = \prod\limits_{j=1}^n{\left( +\frac{ + \phi(z_{\pi_{j}}) + }{ + \sum\limits_{k=j}^{n_q}{\phi(z_{\pi_{k}})} + } +\right)} +\text{, где $\pi_j$ - на какую позицию в $\pi$ встает j-ый документ.}$$ \\ А $\phi(z)$ - неубывающая, строго положительная функция. ($\exp(z)$). Это распределение имеет следующие свойства: +\begin{enumerate} +\item Это распределение (Т.е сумма вероятностей по нему = 1). +\item $P_z(\pi)$ - действительно отражает выходы нашей модели. Просто она их сглаживает. \\Допустим перестановка $\pi$ ставит $x_i$, выше $x_j$, при этом согласно нашей модели: $z_i > z_j$.\\ Тогда если мы поменяем местами i-ый и j-ый объекты, то вероятность такой конкретной перестановки уменьшится. +\item Максимальная $P_z(\pi)$ у перестановки, сортирующей документы в точности по убыванию $z_i$ +\end{enumerate} + +Рассмотрим $P_z(\cdot)$ - сглаженный выход модели. Мы можем посчитать такое же распределение для правильных релевантностей - $P_y(\cdot)$. \\ Очевидно, что мы хотим, чтобы эти распределения были максимально близки друг к другу. Что мы делаем, когда хотим измерить расстояние между распределениями? Правильно - считаем KL дивегренцию. \\ +Формула KL дивергенции: $KL(p || q) = \sum\limits_{c=1}^{M}p_c \log{\frac{p_c}{q_c}}$ \\ + +Получим функционал для модели: $KL(P_y || P_z) \rightarrow $ min. В чем его проблема? Для его подсчета мы вычисляем сумму по всем перестановкам. Получается, что при $n_q$ документах в запросе у нас получается $n_q! $ слагаемых, что очень много. \\ + +Предлагается вместо распределения на всех перестановках рассмотреть вероятность документа попасть на первое место. И ранжировать уже по нему.\\ +$$P_z(j) = \frac{\phi(z_j)}{\sum\limits_{k}^{n_q}{\phi(z_k)}}$$ +\\ +Получим итоговый функционал: $$-\sum\limits_{j}^{n_q}{P_y(j) \cdot \log P_z(j)} \rightarrow min.$$\\\\ +\textbf{5.2 SoftRank}\\ + +В этом методе будем считать что у нас есть какая-то метрика качества ранжирования. Мы хотим опять ввести какую-то вероятность. \\ +Допустим мы имеем скор модели $a(q_i,d_j)$. Давайте скажем что это не число, а нормальное распределение вокруг того, что выдала модель $\tilda \mathcal{N}(a(q, d_j),\sigma^2)$. Дальше скажем, что мы, пользуясь распределениями, можем посчитать вероятность того, что i-ый документ получит скор больше \\, чем j-ый. $\pi_{ij} = \matchbb{P}(s_i > s_j) = \mathbb{P}(s_i - s_j > 0)$, где $(s_i - s_j)$ - случайная величина, распределенная, как $\mathcal{N}(a(q, d_i) - a(q, d_j), 2\sigma_s^2)$ \\ + +$\mathbb{P}(s_i - s_j > 0) = \bigints\limits_{ \text{ 0}}^{\infty}{\mathcal{N}(a(q, d_i) - a(q, d_j), 2\sigma_s^2)ds}$ - вероятность, что $d_i$ окажется выше $d_j$. \\ + +Зная это мы хотим посчитать распределение для $r_j$ - позиция по которому встает $d_j$ в ранжировании по модели. На это можно смотреть так: есть j-ый документ. Он соревнуется с каждым другим документом. И то, сколько раз он проиграл - на такую позицию он и встает. \\ +Это называется: Rank-Binomial distribution - количество успехов в $n-1$ соревновании, у каждого из которых своя вероятность. Вероятности вычисляются итерационно: \\\\ +$\mathbb{P}_{j}^{(1)}(r) = [r = 0] $ - вероятность для $d_j$ стоять на первом месте \\ +$\mathbb{P}_{j}^{(i)}(r) = \mathbb{P}_{j}^{(i-1)}(r-1)\pi_{ij} + \mathbb{P}_{j}^{(i-1)}(r)(1-\pi_{ij})$ - вероятность для $d_j$ стоять на i-ом месте\\\\ +Рассмотрим эти формулы подробнее:\\ + +В каком случае j-ый документ попадет на позицию $r$? Если у нас j-ый документ стоял на одну позицию выше $(r-1)$, затем появляется i-ый документ и побеждает j-ый документ. Это приводит к тому, что j-ый документ опускается на позицию $r$ : $\mathbb{P}_{j}^{(i-1)}(r-1)\pi_{ij}$ \\ +Либо же он может и до этого стоять на позиции $r$ и победить i-ый документ. В этом случае его позиция не меняется: $\mathbb{P}_{j}^{(i-1)}(r)(1-\pi_{ij})$ \\\\ + +Запишем метрику, которую хотим оптимизировать: DCG\\ +$$DCG@k(q): \sum\limits_{i=1}^k{g(y_{(i)})d(i)} \text{, где g(y) = } 2^{y} \text{, } d(i) = \frac{1}{\log(i+1)}$$ +Напомним что эта метрика требует, чтобы у нас на высоких позициях были релевантые документы. Если релевантный документ с большим y оказывается на высокой позиции, то мы его почти не штрафуем. В случае, если он оказывается низко, то штраф получается большой. Значение функции получается меньше, а нам нужно ее максимизировать. +\\ $$DCG@k(q): \sum\limits_{i=1}^k{g(y_{i})d(r_i)}$$ Давайте теперь согласно тому распределению, что у нас есть мы вместо обычного DCG посчитаем его матожидание. \\ +$$\mathbb{E}[DCG@k(q)] = \sum\limits_{i=1}^k{}g(y_i)\mathbb{E}[d(r_i)]$$ А $d_{\any i}(r_i)$ - распределение рангом мы знаем. Мы его посчитали. Получаем: $$\sum\limits_{i=1}^k{}g(y_i) \sum\limits_{r=0}^{n_q}{d(r)\cdot p_i(r)} \text{, где $p_i(r)$ - вероятность, что i-ый документ встанет на позицию r.}$$ + +\section{YetAnotherMethod (Pi-Rank)} + +$$DCG@k(q): \sum\limits_{i=1}^k{\frac{g(y_{(i)})}{\log(1+i)}} = +\sum\limits_{i=1}^k{\frac{[P_z (g) ]_i}{\log(1+i)}} $$ $P_z$ - матрица перестановки соответсвующей сортировки объектов по $z_i$ = $a(q, d_i)$ \\ +Идея: сгладить перестановочную матрицу, так как изначально она разреженная и бинаризированная.\\ + +Заведем унимодальную матрицу A$ = (a_{i,j})_{i,j=1}^n$ такая что: $ \sum\limits_{j=1}^n{a_{i,j}}=1$. И мы потребуем, чтобы максимальный элемент в каждой строке находился в уникальном столбце. Наша идея в том, что выбирая построчные максимумы в такой матрице мы получаем такую же перестановку, как и в оригинальной матрице $P_z$. Далее на лекции говорилось о существовании формул, которые позволяют выписать такую матрицу А, но они настолько глиномесные, что даже в саму лекцию по глиномесным методам не попали. Поверим наслово. Далее просто берем эту параметризацию и оптимизируем. +\end{document}