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производной - + это единственная точка минимума функционала. + + \item Покажем, что минимум квантильной регрессии достигается при $ C = \tau $-квантиль выборки $ y $, обозначим его за $ q_{\tau} $. + По аналогии задачи с минимумом MAE с семинара рассмотрим три случая (для случая $ C < q_{\tau} $, обратный рассматривается аналогично): + + \small + \begin{equation*} + {\rho}_{\tau}(y_i - C) - {\rho}_{\tau}(y_i - q_{\tau}) = + \begin{cases} + (\tau - 1)(y_i - C) - (\tau - 1)(y_i - q_{\tau}), \quad y_i < C \\ + \tau(y_i - C) - (\tau - 1)(y_i - q_{\tau}), \quad C \le y_i \le q_{\tau} \\ + \tau(y_i - C) - \tau(y_i - q_{\tau}), \quad q_{\tau} < y_i + \end{cases} + \end{equation*} + + \begin{equation*} + {\rho}_{\tau}(y_i - C) - {\rho}_{\tau}(y_i - q_{\tau}) = + \begin{cases} + \tau(q_{\tau} - C) - (q_{\tau} - C), \quad y_i < C \\ + \tau(q_{\tau} - C) - (q_{\tau} - y_i), \quad C \le y_i \le q_{\tau} \\ + \tau(q_{\tau} - C), \quad q_{\tau} < y_i + \end{cases} + \end{equation*} + + \begin{equation*} + {\rho}_{\tau}(y_i - C) - {\rho}_{\tau}(y_i - q_{\tau}) \ge + \tau(q_{\tau} - C) - (q_{\tau} - C)\left[ y_i \le q_{\tau} \right] + \end{equation*} + \normalsize + + Просуммируем по всей выборке $ y $ (здесь $ Q_{\tau} $ - это средняя ошибка квантильной регрессии): + + \small + \begin{gather*} + l Q_{\tau} (C) - l Q_{\tau} (q_{\tau}) \ge + l \tau(q_{\tau} - C) - (q_{\tau} - C) \sum_{i=1}^{l} \left[ y_i \le q_{\tau} \right] \ge \\ + \ge l \tau(q_{\tau} - C) - (q_{\tau} - C) l \tau = 0 + \end{gather*} + \normalsize + + Таким образом, $ q_{\tau} $ - оптимальная константа функционала $ Q_{\tau} $ + + \item Распишем ошибку алгоритма на одном объекте: + + \small + \begin{gather*} + L(a(x_i), y_i) = {\left(y_i - \frac{\sum_{j \in B_{r(x_i)} \backslash \{ i \}} y_j}{n_{r(x_i)} - 1} \right)}^2 + \left[ n_{r(x_i)} > 1 \right] + y_i^2 \left[ n_{r(x_i)} = 1 \right] = \\ + = {\left( \frac{y_i (n_{r(x_i)} - 1) - ((\sum_{j \in B_{r(x_i)}} y_j) - y_i )}{n_{r(x_i)} - 1} \right)}^2 + \left[ n_{r(x_i)} > 1 \right] + y_i^2 \left[ n_{r(x_i)} = 1 \right] = \\ + = {\left( \frac{n_{r(x_i)} \left( y_i - \overline{y}_{r(x_i)} \right) }{n_{r(x_i)} - 1}\right)}^2 + \left[ n_{r(x_i)} > 1 \right] + y_i^2 \left[ n_{r(x_i)} = 1 \right] + \end{gather*}{} + \normalsize + + Здесь под $ n_{r(x_i)} $ имеется в виду общее число объектов выборки в ячейке $ r(x_i) $ (включая $ x_i $), + а под $ \overline{y}_{r(x_i)} $ - средний таргет по соответсвующей ячейке. Очевидным образом, обе эти величины + могут быть подсчитаны по всем ячейкам за $ O(l) $. Эти значения фигурируют в финальной формуле, которая получается + усреднением ошибки на одном объекте по всем объектам% + + \small + \begin{equation*} + L = \frac{1}{l} \sum_{i=1}^{l} \left( + {\left( \frac{n_{r(x_i)} \left( y_i - \overline{y}_{r(x_i)} \right) }{n_{r(x_i)} - 1}\right)}^2 + \left[ n_{r(x_i)} > 1 \right] + y_i^2 \left[ n_{r(x_i)} = 1 \right] \right) + \end{equation*} + \normalsize + + Таким образом, вся формула вычисляется за $ O(l) $. +\end{enumerate} + +\section{Linear Classification} + +\begin{enumerate} + \item + \item + \item + \item Для начала представим ранги элементов в аналитическом виде. Ранг элемента - это число элементов, не больших данного (включая его самого). + Для удобства обозначений будем считать, что индексация ведется по возрастанию ответов классификатора. Тогда можно записать следующее: + + \small + \begin{gather*} + r_{(i)} = \sum_{j \le i} 1 = \sum_{j \le i} \left[ y_{(j)} = -1 \right] + \sum_{j \le i} \left[ y_{(j)} = +1 \right] + \end{gather*} + \normalsize + + Теперь посчитаем $ U_+ $ статистику: + + \small + \begin{gather*} + U_+ = \sum_{i : y_{(i)} = +1} r_{(i)} - \frac{l_+(l_+ + 1)}{2} = + \sum_{i=1}^l \left[ y_{(i)} = +1 \right] r_{(i)} - \frac{l_+(l_+ + 1)}{2} = \\ + = \sum_{i=1}^l \left( \left[ y_{(i)} = +1 \right] \sum_{j < i} \left[ y_{(j)} = -1 \right] \right) + \\ + + \sum_{i=1}^l \left( \left[ y_{(i)} = +1 \right] \sum_{j \le i} \left[ y_{(j)} = +1 \right] \right) - \frac{l_+(l_+ + 1)}{2} = \\ + = \sum_{j < i} \left[ y_{(j)} < y_{(i)} \right] + \sum_{i = 1}^{l_+} i - \frac{l_+(l_+ + 1)}{2} = \\ + = \sum_{j < i} \left[ y_{(j)} < y_{(i)} \right] + \end{gather*} + \normalsize + + Таким образом, $ \frac{U_+}{l_+ l_- } $ = $ \frac{\sum_{j < i} \left[ y_{(j)} < y_{(i)} \right]}{l_+ l_-} $, а это и есть не что иное, как AUC-ROC, + то есть число пар объектов из разных классов, верно разделенных классификатором, к общему числу пар. + + \item Посчитаем $ \arg \min_{b \in \mathbb{R}} \mathbb{E} \left[ L(y, b) | x \right]$: + + \small + \begin{gather*} + \mathbb{E} \left[ L(y, b) | x \right] = p(y = +1 | x) e^{-b} + (1 - p(y = +1 | x)) e^{b} \\ + \frac{\partial \mathbb{E} \left[ L(y, b) | x \right]}{\partial b} = - p(y = +1 | x) e^{-b} + (1 - p(y = +1 | x)) e^{b} = 0 \\ + (1 - p(y = +1 | x)) e^{2b} = p(y = +1 | x) \\ + b = \frac{1}{2} \log {\left( \frac{p(y = +1 | x)}{1 - p(y = +1 | x)} \right)} + \end{gather*} + \normalsize + + Очевидно, что это минимум функционала, так как: + + \small + \begin{equation*} + \lim_{|b| \rightarrow \infty} \mathbb{E} \left[ L(y, b) | x \right] = + \infty, + \end{equation*} + \normalsize + + а полученный экстремум единственный. Мы получили функцию, которая не равна тождественно $ p(y = +1 | x) $. + Таким образом, экспоненциальная функция потерь не способна предсказывать истинные вероятности. + + \item Пусть $ w_1, b_1 $ - оптимальное решение первой оптимизационной задачи (из лекции известно, что оно существует и единственно). + Покажем, что $ t w_1, t b_1 $ - это оптимальное решение второй задачи. Для начала проверим, что эти значения подходят под ограничения задачи: + + \small + \begin{gather*} + y_i ( + b_1) \ge 1 \Rightarrow \\ + \Rightarrow y_i ( + t b_1) = t y_i ( + b_1) \ge t + \end{gather*} + \normalsize + + Это действительно так. Предположим теперь, что это решение не является оптимальным, то есть: + + \small + \begin{equation*} + \exists w_2 \in \mathbb{R}^d, b_2 \in \mathbb{R}: + \begin{cases} + {|| w_2 ||}^2 < {|| t w_1 ||}^2 \\ + y_i ( + b_2) \ge t + \end{cases} + \end{equation*} + \normalsize + + Но тогда $ \frac{w_2}{t}, \frac{b_2}{t} $ - это допустимое решение первой задачи, лучшее, чем оптимальное $ w_1, b_1 $: + + \small + \begin{equation*} + \begin{cases} + {|| \frac{w_2}{t} ||}^2 < {|| w_1 ||}^2 \\ + y_i (<\frac{w_2}{t}, x_i> + \frac{b_2}{t}) \ge 1 + \end{cases} + \end{equation*} + \normalsize + + Получили противоречие с оптимальностью решения $ w_1, b_1 $. Таким образом, $ t w_1, t b_1 $ - это оптимальное решение второй задачи, + а поскольку коэффициенты разделяющих гиперплоскостей пропорциональны, то сами гиперплоскости совпадают. + + \item Посчитаем производную сигмоиды: + + \small + \begin{gather*} + \sigma'(z) = -\frac{1}{{(1 + e^{-z})}^2} \left( -e^{-z} \right) = \frac{e^{-z}}{{(1 + e^{-z})}^2} = \\ + = \frac{1}{1 + e^{-z}} \left( 1 - \frac{1}{1 + e^{-z}} \right) = \sigma(z) (1 - \sigma(z)) + \end{gather*} + \normalsize + + Немного преобразуем функцию потерь и посчитаем градиент: + + \small + \begin{gather*} + L(x, y, w) = \log{\left(1 + \exp{(-y )}\right)} = -\log{\left(\sigma(-y )\right)} + \end{gather*} + + \begin{gather*} + \frac{\partial L}{\partial w} = + -\frac{1}{\sigma(-y )} \sigma(-y )) \left( 1 - \sigma(-y ) \right) (-y x) = \\ + = y x \left( 1 - \sigma(-y ) \right) + \end{gather*} + \normalsize + +\end{enumerate} + +\section{Trees} + +\begin{enumerate} + \item + \begin{enumerate} + \item Для случая $ L(y, c) = {(y - c)}^2 $ в качестве критерия информативности мы получаем MSE, а как известно, среднее - это оптимальная константа + для MSE. Таким образом: + + \small + \begin{equation*} + c = \frac{1}{|R|} \sum_{(x_i, y_i) \in R} y_i + \end{equation*} + + \begin{equation*} + H(R) = \frac{1}{|R|} \sum_{(x_i, y_i) \in R} \left( y_i - \frac{\sum_{(x_i, y_i) \in R} y_i}{|R|}\right) = \mathbb{D} \left[ y \right] + \end{equation*} + \normalsize + + \item Далее, считая, что $ c \in \mathbb{R}^K $, найдем оптимальный вектор для критерия информативности: + + \small + \begin{gather*} + H(R, c) = \frac{1}{|R|} \sum_{(x_i, y_i) \in R} \sum_{k=1}^K {(c_k - \left[ y_i = k \right])}^2 = \\ + = \frac{1}{|R|} \sum_{(x_i, y_i) \in R} \sum_{k=1}^K (c_k^2 - 2 c_k \left[ y_i = k \right] + \left[ y_i = k \right]) = \\ + = \frac{1}{|R|} \sum_{k=1}^K (|R| c_k^2 - 2 c_k n_k + n_k) = \sum_{k=1}^K (c_k^2 - 2 c_k p_k + p_k) + \end{gather*} + \normalsize + + Здесь $ n_k $ - число объектов класса $ k $, $ p_k $ - соответственно, их доля. Поскольку получилась сумма по всем классам, + то прооптимизируем функционал отдельно по каждому $ c_k $: + + \small + \begin{equation*} + \frac{\partial H}{\partial c_k} = 2 c_k - 2 p_k = 0 \Rightarrow c_k = p_k + \end{equation*} + \normalsize + + Очевидно, при $ c = (p_1, ..., p_k) $ достигается минимум функционала. Подставим это значение и посчитаем критерий информативности: + + \small + \begin{equation*} + H(R) = \sum_{k=1}^K (p_k^2 - 2 p_k^2 + p_k) = \sum_{k=1}^K (p_k - p_k^2) = 1 - \sum_{k=1}^K p_k^2 + \end{equation*} + \normalsize + + Это и есть не что иное, как критерий Джини. + + \item В условии забыто очень важное ограничение на $ c $: $ \sum_{k=1}^{K} c_k = 1 $ (иначе можно взять $ c_k $ сколь угодно + большими и получим $ L(y, c) $ сколь угодно маленьким, даже отрицательным). Но для начала подставим функцию потерь в критерий + информативности и запишем оптимизационную задачу. Первый шаг аналогичен первому пункту: + + \small + \begin{gather*} + H(R, c) = - \frac{1}{|R|} \sum_{(x_i, y_i) \in R} \sum_{k=1}^K \left[ y_i = k \right] \log{c_k} = - \sum_{k=1}^K p_k \log{c_k} + \end{gather*} + \normalsize + + Задача оптимизации: + + \small + \begin{gather*} + \begin{cases} + - \sum_{k=1}^K p_k \log{c_k} \rightarrow \min_{c_k} \\ + \sum_{k=1}^{K} c_k = 1 \\ + c_k > 0, k = 1 ... K + \end{cases} + \end{gather*} + \normalsize + + Решим ее с помощью метода Лагранжа для условного экстремума: + + \small + \begin{align*} + L(c) = - \sum_{k=1}^K p_k \log{c_k} + \lambda \left( \sum_{k=1}^{K} c_k - 1 \right) + \end{align*} + + \begin{align*} + \begin{cases} + \frac{\partial L}{\partial c_k} = -\frac{p_k}{c_k} + \lambda = 0 \\ + \phi(c) = \sum_{k=1}^{K} c_k - 1 = 0 + \end{cases} + \end{align*} + + \begin{align*} + \begin{cases} + c_k = \frac{p_k}{\lambda} \\ + \sum_{k=1}^{K} \frac{p_k}{\lambda} = 1 + \end{cases} + \end{align*} + + \begin{align*} + \begin{cases} + \lambda = \sum_{k=1}^{K} p_k = 1 \\ + c_k = p_k + \end{cases} + \end{align*} + \normalsize + + Таким образом: + + \begin{equation*} + H(R) = - \sum_{k=1}^K p_k \log{p_k} + \end{equation*} + + А это и есть энтропийный критерий. + \end{enumerate} + + \item При построении дерева глубины $ D $ нам необходимо выбрать предикат для $ \le 2^D - 1 $ узла. Для выбора предиката в узле дерева нужно + перебрать все $ d $ признаков, для каждого признака есть $ \le l - 1 $ возможных пороговых значений, для всех порогов критерий информативности + считается за константное время. Таким образом, общее время построения дерева: $ O((2^D - 1)d(l - 1)) = O(2^Dld) $. +\end{enumerate} + +\section{BVD} + +\begin{enumerate} + \item Найдем смещение напрямую через формулу: + + \small + \begin{gather*} + bias = \mathbb{E}_x \left[ {\left( \mathbb{E}_X \left[ \mu (X)(x) \right] - \mathbb{E}\left[ y | x \right] \right) }^2 \right] = + \mathbb{E}_x \left[ {\left( C - x^T x \right) }^2 \right] = \\ + = C^2 - 2 C \mathbb{E}_x \left[ x^T x \right] + \mathbb{E}_x \left[ {(x^T x)}^2 \right] + \end{gather*} + \normalsize + + Поскольку известно распределение $ x $, найдем матожидание напрямую: + + \small + \begin{gather*} + \mathbb{E}_x \left[ x^T x \right] = \int_{{\left[ 0, 1 \right]}^d} (x_1^2 + ... + x_d^2) d x_1 ... d x_d = d \int_0^1 x_1^2 d x_1 = + \frac{d}{3} + \end{gather*} + + \begin{gather*} + \mathbb{E}_x \left[ {(x^T x)}^2 \right] = \int_{{\left[ 0, 1 \right]}^d} {(x_1^2 + ... + x_d^2)}^2 d x_1 ... d x_d = \\ + \int_{{\left[ 0, 1 \right]}^d} \left( \sum_{i} x_i^4 + \sum_{i \ne j} x_i^2 x_j^2 \right) d x_1 ... d x_d = \\ + = d \int_0^1 x_1^4 d x_1 + d(d - 1) \int_0^1 x_1^2 d x_1 \int_0^1 x_2^2 d x_2 = \frac{d}{5} + \frac{d(d - 1)}{9} + \end{gather*} + \normalsize + + Получаем ответ: + + \begin{equation*} + bias = C^2 - \frac{2Cd}{3} + \frac{d}{5} + \frac{d(d - 1)}{9} + \end{equation*} + + \item Начнем с $ \mathbb{E}\left[ y | x \right] $, тут все просто: + + \begin{equation*} + \mathbb{E}\left[ y | x \right] = \mathbb{E}[ f_x + \varepsilon | x ] = f_x + \end{equation*} + + Теперь посчитаем $ \mathbb{E}_X \left[ \mu (X)(x) \right] $ (обратите внимание, что здесь $ f_x $ - это константа, не зависящая + от обучающей выборки, а потому она выносится за матожидание, ошибка также входит в выборку $X$, поэтому по ней берется ожидание, и она независима с распределением $ X $): + + \begin{gather*} + \mathbb{E}_X \left[ \mu (X)(x) \right] = \mathbb{E}_X \left[ \hat{f}_x \right] = + \mathbb{E}_X \left[ \frac{1}{l} \sum_{i=1}^l \left[ X_i = x \right] (f_x + \varepsilon) \right] = \\ + = \frac{f_x}{l} \mathbb{E}_X \left[ \sum_{i=1}^l \left[ X_i = x \right] \right] + + \frac{1}{l} \mathbb{E}_X \left[ \sum_{i=1}^l \left[ X_i = x \right] \right] \mathbb{E}_X \left[ \varepsilon \right] = \\ + = \frac{f_x}{l} \sum_{i=1}^l \mathbb{P}_X ( X_i = x ) = \frac{f_x}{l} \sum_{i=1}^l \frac{1}{K} = \frac{f_x}{K} + \end{gather*} + + Отсюда находим смещение: + + \begin{gather*} + bias = \mathbb{E}_x \left[ {\left( \mathbb{E}_X \left[ \mu (X)(x) \right] - \mathbb{E}\left[ y | x \right] \right) }^2 \right] = + \mathbb{E}_x \left[ {\left( \frac{f_x}{K} - f_x \right)}^2 \right] = \\ + = {\left( \frac{K - 1}{K} \right)}^2 \mathbb{E}_x \left[ f_x^2 \right] = + {\left( \frac{K - 1}{K} \right)}^2 \sum_{x = 1}^K \frac{f_x^2}{K} = \frac{{(K - 1)}^2}{K^3} \sum_{x = 1}^K f_x^2 + \end{gather*} + + \item + \begin{enumerate} + \item Заметим, что какие бы два веса мы не взяли, их сумма будет больше третьего, таким образом, для верной классификации + необходимо и достаточно, чтобы хотя бы какие-то два алгоритма выдали правильный результат. Пусть $ \xi \sim Bin(3, 1 - p) $ - число + алгоритмов, получивших верный ответ. Тогда вероятность ошибки композиции: + + \begin{gather*} + p_0 = \mathbb{P} (\xi \le 1) = p^3 + 3 p^2 (1 - p) = p^2 (3 - 2p) + \end{gather*} + + \item Тут все куда проще: поскольку $ w_2 > w_1 + w_3 $, то ответ композиции тождественно равен ответу второго алгоритма. Поэтому + вероятность ошибки равна вероятности ошибки второго алгоритма: + + \begin{equation*} + p_0 = p + \end{equation*} + \end{enumerate} +\end{enumerate} + +\end{document}