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a/notes/2025/mvp/chapters/7-gp.tex +++ b/notes/2025/mvp/chapters/7-gp.tex @@ -527,7 +527,7 @@ \section{Gaussian Process Regression (GPR)} \Sigma_\star \bigr), \end{equation} -with mean +with posterior mean \begin{equation} \mu_\star = k(x_\star,X)^\top (K + \sigma^2 I)^{-1} y, @@ -564,7 +564,7 @@ \section{Gaussian Process Regression (GPR)} Gram matrix $K_\ell$ converges to a diagonal matrix: \begin{equation} \lim_{\ell \to 0} K_\ell - = \sigma_f^2 I_n. + = \sigma_f^2 I. \end{equation} In the noise-free case (i.e.\ $\sigma^2 = 0$ in the observation @@ -581,7 +581,7 @@ \section{Gaussian Process Regression (GPR)} \qquad K_\ell^{-1} \;\longrightarrow\; - \frac{1}{\sigma_f^2} I_n, + \frac{1}{\sigma_f^2} I, \end{equation} where $e_j$ is the $j$-th standard basis vector in $\mathbb R^n$. Hence @@ -595,5 +595,82 @@ \section{Gaussian Process Regression (GPR)} exactly at every training input: each training point is matched perfectly by the predictive mean. \end{remark} +%=================================== +From the discussion above, we know that the \emph{posterior mean} of a Gaussian +Process provides the predictive value for a test input, while the associated +\emph{uncertainty} can be quantified through the \emph{predictive variance}. +The relative standard deviation is simply the square root of this predictive +variance. + +Gaussian Process Regression (GPR) serves as the mathematical foundation of +\textbf{Bayesian Optimization (BO)}, a framework for performing optimization +when the objective function is expensive, noisy, or lacks analytic structure. + +\begin{definition}[Black-box Optimization] +An optimization problem is called \emph{black-box optimization} if and only if +the analytical form of the objective function is unknown and no gradient +information is available. +\end{definition} +Since the gradient of the objective function is unavailable, gradient-based +methods such as gradient descent cannot be applied in black-box settings. +The only feasible operation is \emph{point-wise evaluation}: we may query the +black box at a finite number of input locations and observe the corresponding +outputs. + +Bayesian Optimization (BO) aims to construct a probabilistic surrogate model +of the black-box function and use it to locate the maximizer of $y$ with as +few evaluations as possible. This is particularly important when each +evaluation is expensive, for instance in hyperparameter tuning. + +The typical BO procedure proceeds as follows: +\begin{enumerate} + \item Randomly or uniformly select a small set of initial points + $x_1,\ldots,x_n$ and obtain their evaluations $y_1,\ldots,y_n$. + \item Fit a Gaussian Process Regression (GPR) model using the collected + data. + \item Use an acquisition function $a(x)$ to select one or a batch of new + query points; evaluate them, augment the dataset, and refit the GPR model. + \item Repeat the process until the evaluation budget is exhausted, and + return the point achieving the maximum observed value of $y$. +\end{enumerate} +Here are two commonly used acquisition functions in Bayesian Optimization: + +\begin{enumerate} + \item \textbf{Expected Improvement (EI)}: + \begin{equation} + a(x) = \mathbb{E}\big[(y(x) - y_{\max})^{+}\big], + \end{equation} + where + \begin{equation} + [z]^{+} := \max(0, z). + \end{equation} + Under the GPR model, the predictive distribution is + \begin{equation} + y(x) \sim \mathcal{N}\big(\mu(x),\, \Sigma(x)\big), + \end{equation} + so EI admits a closed-form expression obtained by integrating over the + tail above $y_{\max}$: + \begin{equation} + a(x)=\int_{y_{\max}}^{+\infty} \big(y(x)-y_{\max}\big)\, + \mathcal{N}\!\left(y(x)\,\middle|\,\mu(x),\,\Sigma(x)\right)\, + \mathrm{d}y(x). + \end{equation} + + \item \textbf{Upper Confidence Bound (UCB)}: + \begin{equation} + a(x)=\mu(x) + \kappa\,\sigma(x), + \end{equation} + where $\sigma(x)$ denotes the predictive standard deviation and + $\kappa>0$ controls the exploration–exploitation balance. + In essence, UCB prefers points where the model is either promising + (large $\mu$) or highly uncertain (large $\sigma$). +\end{enumerate} + +\begin{remark} + \textbf{Optuna} is a widely used automatic hyperparameter tuning library, + recommended in class for practical Bayesian Optimization. +\end{remark} + + \end{document} \ No newline at 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z8hi9N(_6LXSL;DE5tV5{{}W=ZsL_GG+qr zPPjoh=~7DD@rv>DHO!~&H!{wCwjy^}t>j!{eqXkOcAWj!qUjg=_@#>OuwKlkci!n8 zXrzPR@27VZ(g#bY543LmTCg5IuP9iCu{qmE8Jw`IiUUxdcJyBXEh&dyw|ol$$B6q6 zr8XZzbwY@pUm3?wplHqvC&Wqcx-E;&a>ZHFZ*fD!oGb|s*#bcR6N4cQ%X^gbe6QpR zjk0C8hz{7NIU;J@JpE0?d#P*MU-}8LUwB&8Vp9I`pMbhprQth1Eh#r)l6;nakY}*| zaIS@b%+ei60q^%E&>5e9dS(=$7UzTTnOdH$cs`SyTT;)?ElW^#N<`0Yn}WOlu=MO7 zDJr9wDz=1IajFlV^I%_nCH)Yh71`Vb<)4I2A}PA=uU+Nw9b=C4%<~etUkLU=4B8Fj&bB zXGI{#4S8m`lO{T@A>9y2eStf&OAte8JEWft{msx|V)$D{$ZIGo#Dk6>R%~x=Y7FQa zyds#22mRqIIrz1+Y&-w;;P0J5&eivYRsFDGWs5i71+OZXH{^x1ih7NsvMI$ZWj;bT zn3Q_$lSMtsXea~X+UiLjIgGD3Ih(SGucL=!y02v7m#6B4uk+eUy1g$Q0xr?9<|Vb{ zT7Of{C~;&iCL)7n+cqTbdG0FNjD;M?(a+<_XrcZrnFn?rMPF%)E#q`c_zh-hv7 zLZYFg#dsqX9)=$WRKJ6cs#Gw#lQM^O6~GE3#$U)fYDq^`Be=FC-(0r~@bHaR->kI@ ziV8X<;|Hr#_W=wBW5BvIBE0%S{f1JoA0}fn!X5VbuWv1^tIlz(2F;nwy%v>>N10$, define + \begin{equation} + a_{\mathrm{UCB}}(x) + = \mu(x) + \kappa\,\sigma(x). + \end{equation} + When $\kappa=1$, $a_{\mathrm{UCB}}(x)$ follows the upper envelope + $\mu(x)+\sigma(x)$ of the GP posterior. Larger $\kappa$ encourages more + exploration (sampling high-uncertainty regions), while smaller $\kappa$ + favors exploitation. +\end{itemize} +In practice, BO alternates between refining the surrogate in regions where data +are scarce and zooming in on promising areas, progressively improving our +estimate of both the location and value of the optimum. + +\begin{remark}[Practical Bayesian Optimization] + Many modern hyperparameter optimization libraries implement Bayesian + Optimization with GP or related surrogates. For example, \texttt{Optuna} + provides a flexible interface where the user specifies: + \begin{itemize} + \item the search space of hyperparameters; + \item an evaluation budget (maximum number of trials); + \item an objective function that trains and evaluates the model. + \end{itemize} + The library then automatically manages the BO loop, returning (approximately) + optimal hyperparameters within the given budget. +\end{remark} + +\section{Bias--Variance Decomposition} + +Consider a supervised learning setting with training dataset $D$, drawn from +some unknown data-generating distribution. Let $x$ denote a test input with +true label $y$, and let +\begin{equation} + f(x;D) +\end{equation} +denote the prediction of a learning algorithm trained on $D$ when evaluated at +$x$. Under squared loss, the prediction error at $(x,y)$ for a given training +set $D$ is +\begin{equation} + \bigl(f(x;D) - y\bigr)^2. +\end{equation} +Since $D$ is random, it is natural to measure performance by averaging over the +random draw of $D$. We therefore consider +\begin{equation} + \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr], +\end{equation} +which quantifies the expected squared error at $x$, averaged over the +randomness in the training data. + +Define the \emph{average prediction} at $x$ as +\begin{equation} + \bar{f}(x) + := \mathbb{E}_{D}\bigl[f(x;D)\bigr], +\end{equation} +that is, the expected prediction we would obtain if we could repeatedly sample +training sets $D$ from the underlying distribution and retrain the model. +Insert and subtract $\bar{f}(x)$ inside the square: +\begin{align} + \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr] + &= \mathbb{E}_{D} + \Bigl[\bigl(f(x;D) - \bar{f}(x) + \bar{f}(x) - y\bigr)^2\Bigr]\\ + &= \mathbb{E}_{D} + \Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)^2\Bigr] + + \bigl(\bar{f}(x) - y\bigr)^2\\ + &\quad + + 2\,\mathbb{E}_{D}\Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)\bigl(\bar{f}(x) - y\bigr)\Bigr]. +\end{align} +The cross term vanishes because +\begin{equation} + \mathbb{E}_{D}[f(x;D) - \bar{f}(x)] + = \mathbb{E}_{D}[f(x;D)] - \bar{f}(x) = 0, +\end{equation} +and $\bar{f}(x)-y$ does not depend on $D$. Hence +\begin{equation} + \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr] + = + \underbrace{\mathbb{E}_{D} + \Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)^2\Bigr]}_{\text{variance at }x} + + + \underbrace{\bigl(\bar{f}(x) - y\bigr)^2}_{\text{bias}^2 \text{ at }x}. +\end{equation} + +\begin{definition}[Bias and variance at a test point] + For a fixed test input $x$ with true label $y$, the + \emph{variance} and \emph{(squared) bias} of a learning algorithm are + defined as + \begin{align} + \mathrm{Var}_D\bigl[f(x;D)\bigr] + &= \mathbb{E}_{D} + \Bigl[\bigl(f(x;D)-\bar{f}(x)\bigr)^2\Bigr],\\ + \mathrm{Bias}^2(x) + &= \bigl(\bar{f}(x) - y\bigr)^2. + \end{align} +\end{definition} + +Intuitively: +\begin{itemize} + \item The variance term measures how sensitive the predictor $f(x;D)$ is to + fluctuations in the training data: for different draws of $D$, predictions + at $x$ may vary significantly around their mean $\bar{f}(x)$. + \item The bias term measures how far, on average, the prediction is from + the true label $y$. Even if we could train on infinitely many different + datasets, the average predictor $\bar{f}(x)$ might still systematically + miss the true target due to limited model capacity or misspecification. +\end{itemize} + +High variance typically arises from \emph{overfitting} particular training sets, +while high bias indicates that the model class itself is too simple to capture +the underlying relationship, even when averaged over many datasets. + +\begin{example}[Linear model on nonlinear data] + Suppose the data arise from a quadratic relationship, but we insist on + fitting a linear model. Even if we repeatedly resample training sets and + average the resulting fitted lines, the averaged predictor $\bar{f}(x)$ + remains linear and cannot match the true quadratic curve everywhere. This + manifests as a persistent bias term. +\end{example} + +Bias and variance suggest complementary strategies: +\begin{itemize} + \item To reduce \textbf{variance}, we can increase the training set size or + average multiple diverse models (as in bagging and random forests). + \item To reduce \textbf{bias}, we can increase model capacity, e.g.\ by + using more flexible hypothesis classes or boosting weak learners. +\end{itemize} + +\section{Decision Trees and Feature Selection} + +\subsection{Decision Trees and Axis-Aligned Splits} + +Previously, we used linear models (or linear models in a high-dimensional +feature space) to perform binary classification. The classifier can be +written as +\begin{equation} +f(x) = +\begin{cases} ++1, & \text{if } w^\top x + b \ge 0,\\[4pt] +-1, & \text{if } w^\top x + b < 0. +\end{cases} +\end{equation} +The decision boundary is the hyperplane +\begin{equation} +w^\top x + b = 0. +\end{equation} + +Another way to obtain more flexible, nonlinear decision boundaries is to use +\emph{tree-based models}. A simple decision tree implementing the rule above +can be drawn as follows: +\begin{figure}[h] + \centering + \includegraphics{../../tikz/8/1.pdf} +\end{figure} +\begin{definition}[Decision Tree] + A \emph{decision tree} is a tree consisting of a root node, internal nodes, + leaf nodes, and directed edges. + Each internal node partitions the data according to some feature. + For any input example, its final prediction is given by the label of the leaf + node it reaches. + \end{definition} + + \begin{example} + \newcommand{\cmark}{\ding{51}} % ✓ +\newcommand{\xmark}{\ding{55}} % ✗ + +\begin{wraptable}[20]{r}{0.4\linewidth} + \centering + % \caption{Feature table for the decision tree example} + \vspace{2mm} + \begin{tabular}{c|ccc|c} + \textbf{ID} & \textbf{A} & \textbf{B} & \textbf{C} & \textbf{y} \\ \hline + 1 & \cmark & \cmark & \cmark & \cmark \\ + 2 & \cmark & \xmark & \xmark & \cmark \\ + 3 & \cmark & \xmark & \cmark & \xmark \\ + 4 & \xmark & \cmark & \cmark & \xmark \\ + 5 & \xmark & \xmark & \xmark & \xmark \\ + 6 & \xmark & \xmark & \xmark & \xmark \\ + 7 & \xmark & \cmark & \xmark & \xmark \\ + 8 & \cmark & \xmark & \cmark & \xmark \\ + 9 & \xmark & \cmark & \cmark & \xmark \\ + \end{tabular} +\end{wraptable} + + We consider a binary classification setting where the label set is + \begin{equation} + Y \in \{-1, +1\}. + \end{equation} + Here, the interpretation is: + \begin{equation} + +1: \text{ a good researcher}, \qquad + -1: \text{ a bad researcher}. + \end{equation} + + Assume we have three binary features: + + \begin{itemize} + \item Feature \(A\): + \begin{equation} + +1: \text{ hardworking}, \qquad -1: \text{ not hardworking}. + \end{equation} + + \item Feature \(B\): + \begin{equation} + +1: \text{ has good vision}, \qquad -1: \text{ not}. + \end{equation} + + \item Feature \(C\): + \begin{equation} + +1: \text{ likes bananas}, \qquad -1: \text{ not}. + \end{equation} + \end{itemize} + + A decision tree may use these features at its internal nodes to determine + whether an example will ultimately be predicted as \(+1\) or \(-1\). + \end{example} + \begin{solution} + + \vspace{1em} + \begin{center} + \begin{minipage}{0.55\linewidth} + \centering + \includegraphics[width=\linewidth]{../../tikz/8/2.pdf} + \end{minipage} + \hfill + \begin{minipage}{0.33\linewidth} + \centering + \includegraphics[width=\linewidth]{../../tikz/8/3.pdf} + \end{minipage} + \end{center} + + \vspace{1em} + \end{solution} + +If we choose feature $C$ as the root node, the samples are not separated as cleanly as when we use feature $A$. +This leads to a natural question: \emph{given many candidate features, how can we choose the one that produces the ``purest'' split at the root?} + +From the example above, a good feature should split the training set into subsets whose labels are as pure (i.e., as close to all positive or all negative) as possible. +To quantify this notion of purity---or, conversely, of randomness---we introduce the notion of entropy. + +\subsection{Entropy, Information Gain, and Mutual Information} + +\begin{definition}[Entropy] + For a discrete random variable $X$ with distribution $P$, the (Shannon) + entropy of $P$ is defined as + \begin{equation} + H(P) + := -\sum_{x} P(x)\log P(x). + \end{equation} + Entropy measures the intrinsic uncertainty of the distribution. Equivalently, + \begin{equation} + H(P) + = \mathbb{E}_{X\sim P}\bigl[-\log P(X)\bigr], + \end{equation} + so $-\log P(X)$ can be viewed as the \emph{information content} of the outcome + $X$, and $H(P)$ is its average over many draws. +\end{definition} +Entropy has several essential properties. +\begin{enumerate} +\item +\textbf{Non-negativity.} Since $\log P(x) \le 0$ for all $x$, we have +\begin{equation} + H(P) = -\sum_x P(x)\log P(x) \ge 0, +\end{equation} +with equality if and only if the distribution is degenerate (i.e., $P(x)=1$ +for some $x$). +\item \textbf{Upper bound.} +Using Jensen's inequality: +\begin{equation} + H(P) + = -\mathbb{E}_{P}[\log P(X)] + \le -\log \mathbb{E}_{P}[P(X)] + = -\log \sum_x P(x)^2. +\end{equation} +In particular, for a distribution supported on $n$ distinct outcomes, +\begin{equation} + H(P) \le \log n, +\end{equation} +and equality holds if and only if $P$ is the uniform distribution: +\begin{equation} + P(x) = \frac{1}{n}. +\end{equation} +\end{enumerate} +\begin{remark} + Beyond measuring uncertainty, entropy also characterizes the fundamental + limit of lossless data compression. + + For a discrete source with distribution $P$, any prefix-free coding scheme + assigns a codeword of length $\ell(x)$ to each symbol $x$. Shannon's source + coding theorem states that the expected code length satisfies + \begin{equation} + \mathbb{E}[\ell(X)] + \;\ge\; + H(P) + := -\sum_x P(x)\log P(x), + \end{equation} + with equality achievable asymptotically. + + Thus, the entropy $H(P)$ represents the minimal achievable average number of + bits required to encode samples drawn from $P$. In this sense, entropy + quantifies both the intrinsic uncertainty of a distribution and the optimal + compression rate for a lossless encoding. + \end{remark} +\begin{definition}[Cross-Entropy] +Let $P$ and $Q$ be two distributions over the same support. +The cross-entropy between $P$ and $Q$ is defined as +\begin{equation} + H(P,Q) + := -\sum_x P(x)\log Q(x) + = \mathbb{E}_{P}[-\log Q(X)]. +\end{equation} +\end{definition} + +Observe: +\begin{equation} + H(P,Q) - H(P) + = -\sum_x P(x)\log Q(x) + +\sum_x P(x)\log P(x) + = \sum_x P(x)\log\frac{P(x)}{Q(x)}. +\end{equation} +\begin{definition}[Kullback--Leibler (KL) Divergence] + Let $P$ and $Q$ be two probability distributions over the same support. + The Kullback--Leibler divergence from $P$ to $Q$ is defined as + \begin{equation} + D_{\mathrm{KL}}(P \,\|\, Q) + := \sum_{x} P(x)\,\log\frac{P(x)}{Q(x)} + \end{equation} + in the discrete case, or + \begin{equation} + D_{\mathrm{KL}}(P \,\|\, Q) + := \int p(x)\,\log\frac{p(x)}{q(x)}\,\mathrm{d}x + \end{equation} + in the continuous case. + + KL divergence measures how different the distribution $P$ is from $Q$. + It satisfies + \begin{equation} + D_{\mathrm{KL}}(P\|Q) \ge 0, + \end{equation} + with equality if and only if $P = Q$ almost everywhere. + \end{definition} +Thus, +\begin{equation} + H(P,Q) - H(P) + = D_{\mathrm{KL}}(P\|Q) \ge 0, +\end{equation} +with equality if and only if $P = Q$. + +Cross-entropy decomposes into: +\begin{equation} + H(P,Q) = H(P) + D_{\mathrm{KL}}(P\|Q), +\end{equation} +meaning that cross-entropy is always at least as large as entropy and grows +with the divergence between $P$ and $Q$. +\begin{remark}[Huffman Coding] + Huffman coding is a greedy algorithm for constructing an optimal + prefix-free code when the symbol probabilities are known. + More frequent symbols receive shorter codewords and rare symbols receive + longer ones. Shannon's source coding theorem guarantees that the expected + code length of a Huffman code is no larger than $H(P)+1$ bits per symbol, + so Huffman coding is essentially optimal among all lossless prefix codes. + \begin{center} + \vspace{0.3em} + \includegraphics{../../tikz/8/4.pdf} + \end{center} + Huffman coding is tightly connected to entropy and cross-entropy. + + \textbf{1. Entropy as the optimal code length.} + For a source with distribution $P$, the entropy + \begin{equation} + H(P) = -\sum_x P(x)\log P(x) + \end{equation} + is the theoretical lower bound on the average number of bits needed to encode + symbols drawn from $P$. Huffman coding produces the optimal prefix-free code, + and its expected code length $L_{\mathrm{Huff}}$ satisfies + \begin{equation} + H(P) \le L_{\mathrm{Huff}} < H(P)+1. + \end{equation} + + \textbf{2. Cross-entropy as the cost of using a wrong code.} + Suppose we design a Huffman code based on a \emph{wrong} distribution $Q$ + but the true data come from $P$. Then the expected code length becomes + \begin{equation} + \mathbb{E}_{P}[-\log Q(X)] + = H(P,Q), + \end{equation} + which is the cross-entropy between $P$ and $Q$. + + \textbf{3. Extra penalty equals KL divergence.} + The additional number of bits required due to the mismatch between $P$ and $Q$ + is + \begin{equation} + H(P,Q) - H(P) + = D_{\mathrm{KL}}(P\|Q). + \end{equation} + Thus, the KL divergence quantifies the inefficiency of using a code optimized + for $Q$ when the true distribution is $P$. + + \textbf{Summary.} + Entropy gives the minimal achievable rate, Huffman coding attains it (up to one + bit), and cross-entropy / KL divergence measure how much extra cost arises when + the coding distribution does not match the true distribution. + \end{remark} + \begin{theorem}[Shannon's Source Coding Theorem*] + Let $X$ be a discrete memoryless source with distribution $P$. + For any prefix-free code with codeword lengths $\{\ell(x)\}$, the + expected code length satisfies + \begin{equation} + \mathbb{E}[\ell(X)] \;\ge\; H(P). + \end{equation} + Moreover, for every $\varepsilon>0$, there exists a prefix-free code such that + \begin{equation} + \mathbb{E}[\ell(X)] \;\le\; H(P)+\varepsilon, + \end{equation} + for sufficiently long block coding. + Thus $H(P)$ is the optimal achievable rate for lossless compression. + \end{theorem} +\begin{proof} + The theorem consists of two parts: a converse (no code beats entropy) and + an achievability (entropy can be approached arbitrarily closely). + + \textbf{1. Converse: No prefix-free code can beat $H(P)$.} + For any prefix-free code, Kraft's inequality gives + \begin{equation} + \sum_x 2^{-\ell(x)} \le 1. + \end{equation} + Multiplying both sides by $P(x)$ and applying Jensen's inequality to the + convex function $-\log(\cdot)$ yields + \begin{equation} + \mathbb{E}[\ell(X)] + \;=\; \sum_x P(x)\ell(x) + \;\ge\; -\sum_x P(x)\log P(x) + \;=\; H(P). + \end{equation} + Thus entropy is a lower bound on any valid coding scheme. + + \textbf{2. Achievability: Constructing codes approaching $H(P)$.} + Consider $n$ i.i.d. samples $X^n$ from the source. + The Asymptotic Equipartition Property (AEP) states that for any + $\varepsilon>0$, with probability approaching $1$, sequences in the + typical set $\mathcal{T}_\varepsilon^{(n)}$ satisfy + \begin{equation} + -\frac{1}{n}\log P(X^n) + \in [H(P)-\varepsilon,\; H(P)+\varepsilon]. + \end{equation} + The typical set contains approximately $2^{nH(P)}$ sequences. + We can therefore assign to each typical sequence a codeword of length + \[ + nH(P)+o(n), + \] + while atypical sequences receive longer codewords but contribute + negligibly to the expected length. + This yields a block code with + \begin{equation} + \mathbb{E}[\ell(X^n)] + \le n(H(P)+\varepsilon). + \end{equation} + + \textbf{3. Conclusion.} + Combining the converse and achievability gives the optimal rate: + \[ + \lim_{n\to\infty} \frac{1}{n}\mathbb{E}[\ell(X^n)] = H(P). + \] + Thus $H(P)$ is both a fundamental lower bound and an achievable rate for + lossless source coding. + \end{proof} + + To measure how ``pure'' the labels in a dataset are, we use the notion of + \emph{information gain}. + + Let $D$ be a training set. + Labels lie in a finite set $Y\in\{1,\ldots,K\}$. + For each label $k$, define the subset + \begin{equation} + C_k := \{x\in D : y(x)=k\}. + \end{equation} + + The entropy of $D$ with respect to its empirical label distribution is + \begin{equation} + H(D) + = -\sum_{k=1}^K + \frac{|C_k|}{|D|} + \log\frac{|C_k|}{|D|} + \approx + -\sum_{y=1}^K P(y)\log P(y) + = H(Y), + \end{equation} + where $P(y)$ denotes the empirical frequency of label $y$. + + Let $A$ be a feature taking values in $\{a_1,\ldots,a_m\}$, and let + \begin{equation} + D_{a_i} := \{x\in D : A(x)=a_i\} + \end{equation} + be the subset of samples with feature value $a_i$. + + The conditional entropy of labels given the feature is + \begin{equation} + H(D\mid A) + = + \sum_{i=1}^m + \frac{|D_{a_i}|}{|D|} + H(D_{a_i}) + = + \sum_{i=1}^m + \frac{|D_{a_i}|}{|D|} + \left( + -\sum_{k=1}^K + \frac{|D_{a_i}\cap C_k|}{|D_{a_i}|} + \log \frac{|D_{a_i}\cap C_k|}{|D_{a_i}|} + \right), + \end{equation} +\begin{note} + \begin{equation} + H(D\mid A)=\mathbb E_{x|A}[H(D_x)] + \end{equation} +\end{note} + The \emph{information gain} of feature $A$ is defined as + \begin{definition}[Information gain (Mutual Information)] + \begin{equation} + g(D,A) + = + H(D) - H(D\mid A). + \end{equation} + \end{definition} + + A large value of $g(D,A)$ means that feature $A$ significantly reduces label + uncertainty, producing purer label subsets and therefore acting as a good + splitting feature in decision tree construction. + + More generally, for two random variables $X$ and $Y$, their \emph{mutual + information} is + \begin{equation} + I(X;Y) + = H(X) - H(X\mid Y) + = H(Y) - H(Y\mid X) + = H(X) + H(Y) - H(X,Y). + \end{equation} + In our setting, information gain is exactly the mutual information between the + feature and the label: + \begin{equation} + g(D,A) \approx I(Y;A). + \end{equation} + We therefore choose the best feature $A$ by + \begin{equation} + A^\star=\argmax_A g(D,A). + \end{equation} + + If two random variables satisfy $I(X;Y)=0$, then they are independent: knowing + one of them does not reduce the uncertainty about the other. + + Mutual information has several important properties: + \begin{enumerate} + \item \textbf{Symmetry.} $I(X;Y) = I(Y;X)$. + \item \textbf{Non-negativity.} $I(X;Y) \ge 0$, with equality if and only if $X$ and $Y$ are independent. + \item \textbf{Alternative forms.} $I(X;Y) = H(X) + H(Y) - H(X,Y)$. + \end{enumerate} + + The larger the mutual information, the more correlated $X$ and $Y$ are. When $X$ and $Y$ are highly correlated, knowing $X$ significantly reduces the uncertainty about $Y$, and vice versa. + +\begin{figure}[h] + \centering + \includegraphics{../../tikz/8/5.pdf} +\end{figure} + +\subsection{Information Gain Ratio} + +Information gain has a significant drawback: it tends to favor features with many possible values. + +\begin{example} + Consider a dataset with two features: $A$ (taking values $a_1, a_2$) and $B$ (taking values $b_1, \ldots, b_{10}$). Suppose that: + \begin{itemize} + \item Feature $A$ splits the data into two subsets, each containing 5 classes uniformly. + \item Feature $B$ splits the data into 10 pure subsets, each containing only one class. + \end{itemize} + Then $g(D,A) = \log 10 - \log 5 = \log 2$, while $g(D,B) = \log 10 - 0 = \log 10$. + Information gain would prefer feature $B$, even though it may have poor generalization ability. + + More extremely, if we use the index (row number) as a feature, each example would fall into its own pure subset, yielding maximum information gain $g(D,\mathrm{index}) = H(D)$, but such a feature has no generalization ability for new test examples. +\end{example} + +To address this issue, we introduce the \emph{information gain ratio}, which penalizes features with many values. + +\begin{definition}[Information Gain Ratio] + The information gain ratio of feature $A$ is defined as + \begin{equation} + \mathrm{GR}(D,A) + = \frac{g(D,A)}{H(D,A)}, + \end{equation} + where $H(D,A)$ is the entropy of feature $A$ itself: + \begin{equation} + H(D,A) + = -\sum_{i=1}^m \frac{|D_{a_i}|}{|D|}\log\frac{|D_{a_i}|}{|D|}. + \end{equation} + This measures the intrinsic uncertainty of the feature $A$'s distribution. +\end{definition} + +The denominator $H(D,A)$ penalizes features with many values. When $A$ has many uniformly distributed values, $H(D,A)$ approaches $\log m$, which reduces the gain ratio. This helps prevent overfitting to features that achieve high purity simply by having many possible values. + +\subsection{Gini Index} + +Another common purity measure is the \emph{Gini index}, which serves as an alternative to entropy. + +\begin{definition}[Gini Index] + For a dataset $D$ with labels in $\{1,\ldots,K\}$, the Gini index is defined as + \begin{equation} + \mathrm{Gini}(D) + = 1 - \sum_{k=1}^K \left(\frac{|C_k|}{|D|}\right)^2 + = \sum_{k=1}^K \frac{|C_k|}{|D|}\left(1 - \frac{|C_k|}{|D|}\right), + \end{equation} + where $C_k = \{x \in D : y(x) = k\}$. +\end{definition} + +The Gini index measures the expected error rate if we randomly label examples according to the class distribution. It satisfies: +\begin{itemize} + \item $\mathrm{Gini}(D) = 0$ when $D$ is pure (all examples have the same label). + \item $\mathrm{Gini}(D)$ is maximized when the class distribution is uniform, taking the value $1 - 1/K$ for $K$ classes. + \item For binary classification ($K=2$), $\mathrm{Gini}(D) = 2p(1-p)$, where $p$ is the proportion of the positive class. +\end{itemize} + +The conditional Gini index given feature $A$ is: +\begin{equation} + \mathrm{Gini}(D \mid A) + = \sum_{i=1}^m \frac{|D_{a_i}|}{|D|} \mathrm{Gini}(D_{a_i}). +\end{equation} + +For feature selection, we choose the feature that minimizes the conditional Gini index: +\begin{equation} + A^\star = \argmin_A \mathrm{Gini}(D \mid A). +\end{equation} + +\begin{remark} + For binary classification, the Gini index and entropy (scaled by $1/2$) are very similar functions of the class probability $p$. Both achieve their minimum at $p \in \{0,1\}$ and maximum at $p = 1/2$. +\end{remark} + +\subsection{Feature Selection for Regression} + +For regression problems, where labels are continuous, we use a different criterion based on squared loss. + +\begin{definition}[L2 Loss Criterion] + For a regression problem, after splitting by feature $A$ into subsets $D_1,\ldots,D_m$, define the mean label in each subset: + \begin{equation} + \bar{y}_{D_i} = \frac{1}{|D_i|}\sum_{j \in D_i} y_j. + \end{equation} + The L2 loss criterion is: + \begin{equation} + L(D,A) + = \sum_{i=1}^m \sum_{j \in D_i} (y_j - \bar{y}_{D_i})^2. + \end{equation} + We choose the feature that minimizes this criterion: + \begin{equation} + A^\star = \argmin_A L(D,A). + \end{equation} +\end{definition} + +This criterion measures the variance within each subset after splitting. A smaller L2 loss indicates purer subsets, where examples in each subset have labels close to their subset mean. The mean $\bar{y}_{D_i}$ serves as the prediction for examples in subset $D_i$. + +\subsection{Building Decision Trees: Greedy Algorithm} + +Given multiple feature selection criteria (information gain, information gain ratio, Gini index, or L2 loss), we can build a decision tree. The naive approach of enumerating all possible trees is computationally infeasible (super-exponential complexity). Instead, we use a \emph{greedy algorithm}. + +The greedy algorithm builds the tree recursively: + + \textbf{Input:} Training set $D$, feature set $\mathcal{F}$. + + \textbf{Procedure:} + \begin{enumerate} + \item \textbf{Choose a feature} $A \in \mathcal{F}$ according to a purity metric (e.g., information gain, information gain ratio, Gini index, or L2 loss). + \item \textbf{Partition} $D$ into subsets $D_1,\ldots,D_m$ based on feature $A$'s values $a_1,\ldots,a_m$. + \item \textbf{Recursively build subtrees} for each subset $D_i$ using the remaining features $\mathcal{F} \setminus \{A\}$. + + \textbf{Why remove $A$?} Since all examples in $D_i$ share the same value for feature $A$, further splitting by $A$ would be redundant. + + \item \textbf{Termination conditions:} + \begin{itemize} + \item The feature set $\mathcal{F}$ is empty (no more features available). + \item The subset $D_i$ is pure (contains only one class for classification, or has very low variance for regression). + \item Maximum depth is reached. + \item Purity improvement is below a threshold. + \item Number of samples in a node is below a minimum threshold. + \end{itemize} + \item When termination occurs, the node becomes a \textbf{leaf node} with label equal to the majority class (for classification) or the mean value (for regression) in that subset. + \end{enumerate} + + \textbf{Prediction:} For a test example, traverse the tree according to its feature values until reaching a leaf node, then use the leaf's label as the prediction. + + +\begin{remark}[Computational Complexity] + The naive approach of enumerating all possible trees would require exploring all permutations of feature selection orders, leading to super-exponential complexity. The greedy algorithm reduces this to polynomial time by making locally optimal choices at each step, though it does not guarantee global optimality. +\end{remark} + +\begin{remark}[Regularization] + To prevent overfitting, common regularization techniques include: + \begin{itemize} + \item Limiting the maximum depth of the tree. + \item Setting a minimum number of samples required to split a node. + \item Setting a minimum purity improvement threshold. + \item Using a validation set to monitor performance and stop early. + \end{itemize} +\end{remark} + +\begin{remark} + The greedy algorithm is heuristic: it makes locally optimal choices at each step but does not guarantee a globally optimal tree. Despite this, decision trees and their ensemble variants (Random Forests, Gradient Boosting) are among the most powerful methods in practice, especially for tabular data. +\end{remark} + +\subsection{Continuous Features in Decision Trees} + +In the preceding discussion, we implicitly assumed that features are discrete, +taking values in a finite set. For such a feature $A$ with possible values +$\{a_1,\dots,a_m\}$, a split on $A$ partitions the training set into $m$ +subsets $D_{a_1},\dots,D_{a_m}$. + +In practice, many features are continuous. To handle a continuous feature $A$, +we can discretize it by introducing \emph{thresholds}. Let +\begin{equation} + \{a^{(1)},\dots,a^{(n)}\} +\end{equation} +be the distinct values of $A$ observed in the training set (for some feature +dimension). Sort them so that +\begin{equation} + a^{(1)} < a^{(2)} < \dots < a^{(n)}. +\end{equation} +Candidate thresholds can then be placed between successive values: +\begin{equation} + \tau_j = \frac{a^{(j)} + a^{(j+1)}}{2}, \qquad j=1,\dots,n-1. +\end{equation} +Each threshold $\tau_j$ defines a binary split +\begin{equation} + A \le \tau_j + \quad\text{vs.}\quad + A > \tau_j, +\end{equation} +which we can evaluate using any of our usual criteria (information gain, Gini +index, squared loss for regression, etc.). By scanning over all candidate +thresholds and choosing the best one according to the purity measure, the tree +handles continuous features while remaining axis-aligned. + +At prediction time, a test example with continuous feature value $A(x)$ is +routed through the tree by comparing $A(x)$ with the learned thresholds at each +internal node, just as if the feature had been discrete. + +\section{From Single Trees to Ensembles} + +So far, we have seen how to build and select a \emph{single} decision tree, by +choosing features that maximize information gain at each split. In practice, +however, a single tree often suffers from two issues: +\begin{itemize} + \item \textbf{High variance.} Small changes in the training data may lead to + very different trees, especially when the tree is grown deep and fits the + training set closely. + \item \textbf{Limited accuracy.} A single tree is easy to interpret, but its + predictive performance may lag behind more powerful models such as kernel + methods or neural networks. +\end{itemize} + +Ensemble learning addresses these issues by combining the predictions of many +base learners (often decision trees) to form a stronger model. + +\begin{definition}[Ensemble Learning] + Ensemble learning constructs a predictor by aggregating a collection of + base learners (also called \emph{weak learners}). Given base predictors + $f_1,\dots,f_M$, the ensemble predictor takes the form + \begin{equation} + F(x) = + \begin{cases} + \mathrm{majority\ vote}\bigl(f_1(x),\dots,f_M(x)\bigr), + & \text{(classification)},\\[4pt] + \dfrac{1}{M}\sum_{m=1}^M f_m(x), + & \text{(regression)}. + \end{cases} + \end{equation} + The key idea is that even if each $f_m$ is only moderately accurate, their + combination can be significantly more accurate and more robust. +\end{definition} + +Two classical ensemble strategies are: +\begin{itemize} + \item \emph{Bagging} (Bootstrap Aggregating): build many base learners + independently on randomized versions of the data, and average or vote. + \item \emph{Boosting}: build base learners sequentially, each one focusing + on correcting the mistakes of the previous ones. +\end{itemize} + +In modern machine learning practice, decision-tree ensembles based on bagging +and boosting are among the most powerful off-the-shelf methods, with famous +examples including Random Forests, Gradient Boosting Trees, and XGBoost. + +\subsection{Bagging: Bootstrap Aggregating} + +Bagging is designed to reduce the variance of an unstable base learner (such as +a deep decision tree) by averaging many independently trained copies. + +\begin{definition}[Bagging] + Let $D$ be a training dataset of size $n$, and let $\mathcal{A}$ be a base + learning algorithm that outputs a predictor $f = \mathcal{A}(D)$. + Bagging constructs $M$ bootstrap datasets $D^{(1)},\dots,D^{(M)}$, where + each $D^{(m)}$ is obtained by sampling $n$ points \emph{with replacement} + from $D$. + + On each bootstrap dataset $D^{(m)}$, we train a base learner + \begin{equation} + f_m = \mathcal{A}\bigl(D^{(m)}\bigr), + \qquad m=1,\dots,M. + \end{equation} + The bagged ensemble predictor is + \begin{equation} + F_{\mathrm{bag}}(x) + = + \begin{cases} + \mathrm{sign}\Bigl(\dfrac{1}{M}\sum_{m=1}^M f_m(x)\Bigr), + & \text{classification},\\[6pt] + \dfrac{1}{M}\sum_{m=1}^M f_m(x), + & \text{regression}. + \end{cases} + \end{equation} +\end{definition} + +Intuitively, each bootstrap sample $D^{(m)}$ can be viewed as a noisy version +of the original dataset. The base learners $f_m$ will differ from one another, +and averaging their predictions cancels out part of the randomness. + +\begin{remark}[Variance Reduction] + Consider a simplified setting where we average $M$ identically distributed + base predictors $f_1,\dots,f_M$ for regression, each with variance + $\sigma^2$ and pairwise correlation $\rho$. Then the variance of the + averaged predictor is + \begin{equation} + \mathrm{Var}\bigl(F_{\mathrm{bag}}(x)\bigr) + = \frac{1}{M^2}\sum_{m=1}^M\sum_{m'=1}^M + \mathrm{Cov}\bigl(f_m(x), f_{m'}(x)\bigr) + \approx \sigma^2\left(\rho + \frac{1-\rho}{M}\right). + \end{equation} + As $M$ grows, the second term $(1-\rho)/M$ vanishes, so the variance is + dominated by $\rho\sigma^2$. Thus, bagging is most effective when base + learners are accurate but have high variance and are not too strongly + correlated. +\end{remark} + +\begin{figure}[h] + \centering + \includegraphics{../../tikz/8/6.pdf} +\end{figure} + +\subsection{Random Forests} + +Random Forests specialize bagging to decision trees, and further inject +randomness at the \emph{feature} level to decorrelate the trees. + +\begin{definition}[Random Forest] + A Random Forest is an ensemble of decision trees trained with two sources + of randomness: + \begin{enumerate} + \item \textbf{Bootstrap sampling of data.} + For each tree $m$, sample a bootstrap dataset $D^{(m)}$ from $D$. + \item \textbf{Random feature selection at each split.} + When splitting a node, instead of considering all features, randomly + select a subset $\mathcal{F}$ of features (of size $d_{\mathrm{sub}}$), + and choose the best split only among features in $\mathcal{F}$. + \end{enumerate} + The final prediction aggregates all trees by majority vote or averaging, + as in bagging. +\end{definition} + +The additional randomness in feature selection has two important effects: +\begin{itemize} + \item It reduces the correlation between different trees, which + strengthens the variance reduction effect of averaging. + \item It forces each tree to explore different feature combinations, + sometimes discovering useful patterns that a single greedy tree might miss. +\end{itemize} + +In practice, Random Forests are strong general-purpose models with relatively +few hyperparameters. Common choices include: +\begin{itemize} + \item the number of trees $M$ (often in the hundreds), + \item the maximum depth of each tree, + \item the number of features $d_{\mathrm{sub}}$ considered at each split + (e.g., $\sqrt{d}$ for classification, where $d$ is the total number of + features). +\end{itemize} + +\begin{remark}[Out-of-Bag Evaluation] + In each bootstrap sample $D^{(m)}$, roughly a fraction $1-1/e\approx 0.63$ + of the original training points are included, and the remaining points are + left out. For any training example, we can average the predictions of all + trees that did \emph{not} see this example during training; this is called + the \emph{out-of-bag} prediction. Aggregating these predictions over the + training set provides an internal estimate of the generalization error, + without using a separate validation set. +\end{remark} + +\subsection{Boosting} + +While bagging focuses on variance reduction by averaging many independent +learners, boosting builds an ensemble \emph{sequentially}. Each new learner +tries to correct the mistakes of the current ensemble, effectively turning a +collection of weak learners into a strong one. + +\begin{definition}[Boosting (High-Level View)] + Given a training set $D = \{(x_i,y_i)\}_{i=1}^n$, boosting maintains an + ensemble predictor + \begin{equation} + F_0(x) \equiv 0,\qquad + F_M(x) = \sum_{m=1}^M \alpha_m f_m(x), + \end{equation} + where each $f_m$ is a base learner (often a shallow tree) and the weights + $\alpha_m$ control their influence. + + At iteration $m$, boosting chooses $f_m$ to focus on the current residual + errors or misclassified examples of $F_{m-1}$, and then updates the ensemble + to $F_m$. +\end{definition} + +One classical example is AdaBoost for binary classification. It maintains a +distribution of weights over training samples, gives higher weights to +misclassified points, and fits a new weak learner to this reweighted dataset at +each iteration. + +\begin{remark}[Boosting vs.\ Bagging] + \begin{itemize} + \item Bagging trains base learners \emph{in parallel} on resampled + datasets and primarily reduces variance. + \item Boosting trains base learners \emph{sequentially}, focusing on + difficult samples, and can reduce both bias and variance, but is also + more prone to overfitting if not regularized (e.g., via tree depth, + learning rate, or early stopping). + \end{itemize} + Modern implementations such as Gradient Boosting Trees and XGBoost can be + viewed as performing a form of functional gradient descent in function + space, where each tree fits the negative gradient of the loss with respect + to the current ensemble prediction. +\end{remark} + +\begin{figure}[h] + \centering + \includegraphics{../../tikz/8/7.pdf} +\end{figure} + +\begin{note} + From a geometric perspective, both Random Forests and boosting-based tree + ensembles can be seen as constructing a complex, highly nonlinear decision + boundary by patching together many simple, axis-aligned splits. Whereas a + single tree corresponds to a small set of such partitions, an ensemble can + carve out increasingly intricate decision regions, often achieving + state-of-the-art performance on tabular data. +\end{note} + +\end{document} diff --git a/notes/2025/mvp/main.pdf b/notes/2025/mvp/main.pdf index 8cf0e9dddab58884d2713fcda103a03655a7e8c5..465b5839018885024e6c86ec67caa0e6102a75cc 100644 GIT binary patch delta 50074 zcmZ5{19W8F5^n6IW7{?-wllG9+v-l5#i@Euv#gk3Hl7(K%*fFe zK@W{I2q8ZrWtmc8jRhpg6vZ1|@e&!`qD$@q_xJtbzBS|AT zV#=ZYgxZZ*9*@;Fi^a81IWlfZz{R|t59+qyU){_@AF!|uLbpIx%<0PuN3jz$d8+oe zoWrCzkkVMgbm^grTgTB^oKwUDu6BsCG8$$iE=h_7#sC1 z1z=aUdP=|C9Baq@HgeX?Wx#h8px3=0?BHE2PsK1(Y8TU5R6jqHZ{TOa$w{$pDPbHx zu|J;&mY1;HJ-5V8YjXNr_LML7K1mBUzNYZkm%L#is(Uqg0KL-Jd%b->yJzMc>__7U zZttg#LcwBbzhdzb_XLIc-C6pztkUsFxWOpskt2}^VSb61$8Ece4dX%&3{l(Rc-J{I z>@ZX|&2K{<6(9h;yDuRLnYl!1h1i6E2<~m@;ut32vU>P)4q(0E8h?FGiSr`SN5KGR1F2vFq*AWnI3YM$Q|RDHAvm~G6yRkcIN4G% 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\node[circle,draw,minimum width=13mm,label=right:{$w^\top x + b \ge 0$}] (root) {}; + + % leaves + \node[draw,rectangle,below left=1.4cm and 2.0cm of root] (plus) {$+1$}; + \node[draw,rectangle,below right=1.4cm and 2.0cm of root] (minus) {$-1$}; + + % edges + \draw[->] (root) -- node[above left] {yes} (plus); + \draw[->] (root) -- node[above right] {no} (minus); +\end{tikzpicture} + +\end{document} \ No newline at end of file diff --git a/notes/2025/tikz/8/2.bbl b/notes/2025/tikz/8/2.bbl new file mode 100644 index 0000000..e69de29 diff --git a/notes/2025/tikz/8/2.blg b/notes/2025/tikz/8/2.blg new file mode 100644 index 0000000..41db3d9 --- /dev/null +++ b/notes/2025/tikz/8/2.blg @@ -0,0 +1,3 @@ +[0] Config.pm:308> INFO - This is Biber 2.20 +[0] Config.pm:311> INFO - Logfile is '2.blg' +[42] biber:340> INFO - === Wed Nov 19, 2025, 15:59:11 diff --git a/notes/2025/tikz/8/2.pdf b/notes/2025/tikz/8/2.pdf new file mode 100644 index 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24mm of A, + minimum width=12mm,minimum height=9mm] (Rminus) {$-1$}; + + % left internal node (feature B) + \node[circle,draw,below left=16mm and 10mm of A, + minimum size=11mm, + label=left:{\small $B$}] (B) {}; + + % left leaf under B: +1, set {1,2} + \node[rectangle,draw,below left=16mm and 8mm of B, + minimum width=12mm,minimum height=9mm] (Lplus) {$+1$}; + + % right leaf under B: -1, set {3,8} + \node[rectangle,draw,below right=16mm and 8mm of B, + minimum width=12mm,minimum height=9mm] (Lminus) {$-1$}; + + % edges from root + \draw[->] (A) -- node[above left,pos=0.45] {\scriptsize yes} + node[left,pos=0.55] {\scriptsize $\{1,2,3,8\}$} + (B); + \draw[->] (A) -- node[above right,pos=0.45] {\scriptsize no} + node[right,pos=0.55] {\scriptsize $\{4,5,6,7,9\}$} + (Rminus); + + % edges from B + \draw[->] (B) -- node[above left,pos=0.45] {\scriptsize yes} + node[left,pos=0.65] {\scriptsize $\{1,2\}$} + (Lplus); + \draw[->] (B) -- node[above right,pos=0.45] {\scriptsize no} + node[right,pos=0.65] {\scriptsize $\{3,8\}$} + (Lminus); + +\end{tikzpicture} + +\end{document} \ No newline at end of file diff --git a/notes/2025/tikz/8/3.pdf b/notes/2025/tikz/8/3.pdf new file mode 100644 index 0000000000000000000000000000000000000000..da92626e87c14600e31a950b0cc6889737f01b5f GIT binary patch literal 26033 zcma&tQ{T+no)=2lL|4g_?fRt8SS z!p4TSM#j**ywHwL4#o!7&~6*esxq;Q^oZT3YEwu-+?V&Qu~kE=2dW#`cA87abt0Oy zU*2RbiEHK?VWQ&J$Xs-@k2{g2$#=lWF4@`~DEOp^^1`KlunB~2ysox=akwPf}`xZyd#Z#srS-8 z0SdQxoCfplW!!3%pPj<+#D6RoK~ZoU81cp)u4}c{9@`5*M}xaf0B`0|9rYD5)6JilhHp2%5RUSnY}Wm z(HtT%>Af>~^P9<(i7@0xn+`q@73*J$;L{#67hvKjo13m;`l~7RI&QDc;z`T= zD=yrOqu-A@rs;QY;YTwoZ9h*$#vl~|#)jBU$+v@e;o<>h&((>wa3TI&J(MD1B|#I} zmaHLYC0k{pvEa^CWgz!pVXx)6|4ZwhUb#(5%*vK(pC7!#IMz0=&MJ=kAxZ6AUahQH z>yj*jTRf%;;p=mpo0Z&~=nKTy1?~)OY-99)W$C}&|DHnzmj7QM83-7d>Dm5AJQ)dC z8QB^B$BBu6{(o!<*ch1^{#SDw?FylsxVgX@Kp;{Sp&Sm0!wJ>>stYElU_k&RC?SY^ zPS9QWw3Afvq+3L(!XHOOg-fXjnxqTbqdwF9y7jAf`Kh`h`7*`Lb{B{!}S{V 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diff --git a/notes/2025/tikz/8/4.tex b/notes/2025/tikz/8/4.tex new file mode 100644 index 0000000..1dd715c --- /dev/null +++ b/notes/2025/tikz/8/4.tex @@ -0,0 +1,40 @@ +\documentclass[tikz]{standalone} +\usepackage{tikz} +\usetikzlibrary{positioning} + +\begin{document} + +\begin{tikzpicture}[% + >=stealth, + level distance=18mm, + sibling distance=26mm, + every node/.style={font=\small} +] + +% root = 1.0 +\node[circle, draw, minimum size=9mm] (root) {$1.0$} + child[left] { + % apple leaf + node[align=center] (apple) {apple\\$0.2$} + edge from parent node[midway, left] {0} + } + child[right] { + % internal node for orange + banana + node[circle, draw, minimum size=9mm] (node2) {$0.8$} + edge from parent node[midway, right] {1} + }; + +% children of node2 +\node[align=center, below=12.7mm of node2, xshift=8mm] (orange) + {orange\\$0.3$}; + +\node[align=center, below=12mm of node2, xshift=-8mm] (banana) + {banana\\$0.5$}; + +% edges from node2 to leaves manually +\draw[->] (node2) -- (orange) node[midway,right ] {0}; +\draw[->] (node2) -- (banana) node[midway, left] {1}; + +\end{tikzpicture} + +\end{document} \ No newline at end of file diff --git a/notes/2025/tikz/8/5.pdf b/notes/2025/tikz/8/5.pdf new file mode 100644 index 0000000000000000000000000000000000000000..abc54303a86aa8111f5281660a29dad0e8b1740b GIT binary patch literal 28346 zcmb5VQ;;r9&~DkbvD@Bl+qP}nz1y~J+qV6-ZQHhu`A*FI6EQL8Vy3P#?^i@s<+GAh zK}3v}k&YFHbZ%*Q4Tg=7f$*P^6$}p#484r0ow>6GAuA^f;s1GH=*28;oJ}1G>BVde zolQkdjsKaL!tn9II5|6-8rs6RZ(OHI$1MuLjo!Ya@eL5^dFf>!W8epZw3&Jtfa5AC zMj?svAb9&^nUT?6i~EC`WLr3MZ%Rgnr}wh)5M@_B?6 zX$lOgCtB*)euMpP#9duanG^D3`q~j>+NsH`%Imsb#F+K;dWD2=dh}p{HZ|vbof|tT zj#d+IeHhDSDa#V#!osoGFqZVXDx>w?{iUf$Iz#Ztx>cW7(V1@h)H>=3sXzdg&IN** zG@)sH`ZFG%0o3R!G2?vWZU_^jDLr1XNI5nxn_KR3hG$w~5i~?sHQtWoggYHA@{|-$GWauf%kD){ zc61~Jxnwh(;)qG}DwkWARi)15Kf0G4@cgIHVyPO?7badu;HL#Z*G)U;v7$9LK=S@8 zt*HDabU9t_q3d}#Flq*VttX~#PoXE!5fnVk)R+~gIpt+@i^;C_~7gITz 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corners,minimum width=18mm,minimum height=7mm,align=center}, + tree/.style={draw,rectangle,minimum width=10mm,minimum height=7mm,align=center} +] + +% Original dataset +\node[box] (data) {Training\\data $D$}; + +% Bootstrap samples +\node[box,below left=10mm and 6mm of data] (b1) {$D^{(1)}$}; +\node[box,below =10mm of data] (b2) {$D^{(2)}$}; +\node[box,below right=10mm and 6mm of data] (b3) {$D^{(M)}$}; + +% Arrows from original data to bootstraps +\draw[->] (data.south west) -- (b1.north); +\draw[->] (data.south) -- (b2.north); +\draw[->] (data.south east) -- (b3.north); + +% Base learners (trees) +\node[tree,below=10mm of b1] (t1) {$f_1$}; +\node[tree,below=10mm of b2] (t2) {$f_2$}; +\node[tree,below=10mm of b3] (t3) {$f_M$}; + +\draw[->] (b1) -- (t1); +\draw[->] (b2) -- (t2); +\draw[->] (b3) -- (t3); + +% Aggregation box +\node[box,below=16mm of t2,minimum width=26mm] (agg) {Average /\\majority vote}; + +\draw[->] (t1.south) -- ([xshift=-8mm]agg.north); +\draw[->] (t2.south) -- 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+\end{document} + From c39296735ebfaab648f55b8d9db6eaabfd62b6f8 Mon Sep 17 00:00:00 2001 From: Photon Date: Sat, 29 Nov 2025 17:40:54 +0800 Subject: [PATCH 2/4] main --- notes/2025/mvp/main.pdf | Bin 1920704 -> 2774166 bytes notes/2025/mvp/main.synctex(busy) | Bin 311296 -> 0 bytes 2 files changed, 0 insertions(+), 0 deletions(-) delete mode 100644 notes/2025/mvp/main.synctex(busy) diff --git a/notes/2025/mvp/main.pdf b/notes/2025/mvp/main.pdf index 465b5839018885024e6c86ec67caa0e6102a75cc..789dc78789ffcb8a743e2b3fc4d1868eb9fced14 100644 GIT binary patch delta 1001935 zcmce6bzB^6vMwIn-Q5}7-GjTkySolSg1fuByIXLF0D<5d+$A_S-=39k_nti~XYc)^ z`q$GvJyZ2o)$=}8U47YZDdpv$Kng`M2?k~WJ1oWC<=F!)2Qd?|gRwO%A0MNtm!lb@ zrh>7xnTac-ikq?PZ=WR`obCQ_`0Xw+Goy%zgQpJDuZNhK{vdjcaz?JsR-QV{zpnkd z^4n35QQ6GJ!Ohvk%mtS7*9B!p2`gJyGiOE#TO**WnV6Z0gQ*##teL%qt0gfDtAGHk zi>tGlksYk(iVBmWqXv`WM~zveX+`R35D;uI@FZ9>d(%IDJ0ps)|#31-(Kv$v(~`Bi4kVDe^T*(O>g$U zr#J9_72A@|4sMRW(nW<))!E42#nH&w%-+O{QB;Ld%*@@xrJ$V))rcPYU9TaNuc 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b/notes/2025/mvp/chapters/8-tel.tex @@ -90,8 +90,8 @@ \subsection{Black-Box Optimization with Gaussian Processes} very costly, so we must find good $x$ using as \emph{few} evaluations as possible. -\begin{remark}[Basic Bayesian Optimization Loop] - Bayesian Optimization with a GP surrogate proceeds iteratively: +\begin{remark} + Bayesian Optimization with a GP surrogate proceeds iteratively: \begin{enumerate} \item \textbf{Initialization.} Select an initial design $\{x_i\}_{i=1}^n$ (e.g.\ random or space-filling) and evaluate @@ -164,7 +164,7 @@ \subsection{Acquisition Functions} are scarce and zooming in on promising areas, progressively improving our estimate of both the location and value of the optimum. -\begin{remark}[Practical Bayesian Optimization] +\begin{remark} Many modern hyperparameter optimization libraries implement Bayesian Optimization with GP or related surrogates. For example, \texttt{Optuna} provides a flexible interface where the user specifies: @@ -502,7 +502,7 @@ \subsection{Entropy, Information Gain, and Mutual Information} \end{equation} meaning that cross-entropy is always at least as large as entropy and grows with the divergence between $P$ and $Q$. -\begin{remark}[Huffman Coding] +\begin{remark} Huffman coding is a greedy algorithm for constructing an optimal prefix-free code when the symbol probabilities are known. More frequent symbols receive shorter codewords and rare symbols receive @@ -782,8 +782,6 @@ \subsection{Gini Index} For binary classification, the Gini index and entropy (scaled by $1/2$) are very similar functions of the class probability $p$. Both achieve their minimum at $p \in \{0,1\}$ and maximum at $p = 1/2$. \end{remark} -\subsection{Feature Selection for Regression} - For regression problems, where labels are continuous, we use a different criterion based on squared loss. \begin{definition}[L2 Loss Criterion] @@ -834,11 +832,11 @@ \subsection{Building Decision Trees: Greedy Algorithm} \textbf{Prediction:} For a test example, traverse the tree according to its feature values until reaching a leaf node, then use the leaf's label as the prediction. -\begin{remark}[Computational Complexity] +\begin{remark} The naive approach of enumerating all possible trees would require exploring all permutations of feature selection orders, leading to super-exponential complexity. 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$\mathcal{X}$ be an unlabeled dataset. + Unsupervised learning aims to learn a mapping $f:\mathcal{X}\to\mathcal{Z}$ or a generative model $p_\theta(x)$ that captures the intrinsic structure of the underlying data distribution $p(x)$. + Typical objectives include clustering, dimensionality reduction, density estimation, and representation learning. + Importantly, no labeled pairs $(x,y)$ are observed during training. +\end{definition} + +Unsupervised learning can be divided into several tasks: +\begin{enumerate} + \item Dimensionality reduction: datapoints are often lie in high-dimensional manifold, while the dimension that really matter only contribute as a small part. Dimensionality reduction is the process to train the model to find these crutial dimension from enormous amount of dimensions. + \item Clustering. Train model to group similar data togather. + \item Generate models. Various generative model, like SoRA, LLM/GPT, are trained to learn how to generate contents like text or image by learning the distribution of the target context. +\end{enumerate} +\section{Dimensionality reduction} +\subsection{Principal Component Analysis (PCA)} + +\begin{theorem}[Principle of PCA] + PCA seeks a direction $w$ (with $\|w\|=1$) along which the projected data $w^\top x$ + achieves the \emph{maximum possible variance}. + Formally, the goal is to maximize + \begin{equation} + \operatorname{Var}(w^\top X). + \end{equation} +\end{theorem} + + Let the dataset be + \begin{equation} + X = + \begin{pmatrix} + x_1^{\top} \\ + x_2^{\top} \\ + \vdots \\ + x_n^{\top} + \end{pmatrix} + \in \mathbb{R}^{n\times d}, + \end{equation} + with sample mean + \begin{equation} + \bar{x} = \frac{1}{n}\sum_{i=1}^n x_i . + \end{equation} + Each $x_i \in \mathbb{R}^d$ is an observation. + \begin{figure}[h] + \centering + \includegraphics{../../tikz/9/1.pdf} + \end{figure} + + The empirical covariance matrix is defined as + \begin{equation} + \Sigma = \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})(x_i - \bar{x})^{\top} + \in \mathbb{R}^{d\times d}. + \end{equation} + Entrywise, + \begin{equation} + \Sigma_{jk} + = \frac{1}{n}\sum_{i=1}^n + (x_i^{(j)} - \bar{x}^{(j)})(x_i^{(k)} - \bar{x}^{(k)}). + \end{equation} + \begin{remark} + The covariance matrix defined with the factor $\tfrac{1}{n}$ is technically a biased estimator of the true population covariance. + The unbiased version uses $\tfrac{1}{n-1}$ instead. + However, when $n$ is large, the difference between $\tfrac{1}{n}$ and $\tfrac{1}{n-1}$ is negligible: + \begin{equation} + \frac{1}{n} = \frac{1}{n-1} \left( 1 - \frac{1}{n} \right), + \end{equation} + and thus affects the covariance only by a vanishing scalar factor. + + Moreover, PCA depends solely on the \emph{eigenvectors} of the covariance matrix, i.e., the principal directions. + Multiplying $\Sigma$ by any positive constant does not change its eigenvectors. + Therefore, for PCA, the choice between $\tfrac{1}{n}$ and $\tfrac{1}{n-1}$ has no effect on the principal components. + \end{remark} + \begin{lemma}[Variance of a Linear Projection] + Let $w$ be any unit vector in $\mathbb{R}^d$. + The projection of a data point $x_i$ onto $w$ is the scalar + \begin{equation} + z_i = w^\top x_i . + \end{equation} + The sample mean of the projected data is + \begin{equation} + \bar{z} = \frac{1}{n}\sum_{i=1}^n z_i + = \frac{1}{n}\sum_{i=1}^n w^\top x_i + = w^\top \bar{x}. + \end{equation} + Thus + \begin{equation} + z_i - \bar{z} + = w^\top (x_i - \bar{x}). + \end{equation} + The sample variance of the projected data is + \begin{equation} + \operatorname{Var}(w^\top X) + = \frac{1}{n} \sum_{i=1}^n (z_i - \bar{z})^2 + = \frac{1}{n} \sum_{i=1}^n + \bigl( w^\top (x_i - \bar{x}) \bigr)^2 . + \end{equation} + Using the identity + \[ + (w^\top a)^2 = w^\top (a a^\top) w, + \] + we obtain + \begin{equation} + \operatorname{Var}(w^\top X) + = w^\top + \left( + \frac{1}{n}\sum_{i=1}^n (x_i - \bar{x})(x_i - \bar{x})^\top + \right) + w + = w^\top \Sigma w. + \end{equation} + \end{lemma} + Let $u_1$ be the unit projection vector on $\mathbb R^d$ to project each $x_i$ onto a 1-D space, then PCA is equivalent to solve the below problem + \begin{equation} + \argmax_{\|u_1\|=1}\; u_1^\top \Sigma u_1 . + \end{equation} + By introducing a Lagrange multiplier $\lambda$ for the constraint $u_1^\top u_1 = 1$, we consider the Lagrangian + \begin{equation} + \mathcal{L}(w,\lambda) + = u_1^\top \Sigma u_1 - \lambda \bigl( u_1^\top u_1 - 1 \bigr). + \end{equation} + Taking the derivative with respect to $u_1$ and setting it to zero gives + \begin{equation} + \frac{\partial \mathcal{L}}{\partial u_1} + = 2 \Sigma u_1 - 2 \lambda u_1 = 0, + \end{equation} + hence + \begin{equation} + \Sigma u_1 = \lambda u_1. + \end{equation} + The derivative with respect to $\lambda$ enforces the constraint + \begin{equation} + \frac{\partial \mathcal{L}}{\partial \lambda} + = -\bigl(u_1^\top u_1 - 1\bigr) = 0 + \quad\Longrightarrow\quad + u_1^\top u_1 = 1. + \end{equation} + Therefore, any maximizer \textbf{$u_1$ must be an eigenvector of $\Sigma$}, with $\lambda$ equal to the corresponding eigenvalue. + Since + \begin{equation} + u_1^\top \Sigma u_1 = \lambda + \end{equation} + for any unit eigenvector $u_1$, the optimization problem is solved by choosing $u_1$ as the eigenvector associated with the \textbf{largest eigenvalue of $\Sigma$}. + This $u_1$ is the first principal component. + + After obtaining the first principal component $u_1$, the second principal component $u_2$ is defined as the direction that maximizes the projected variance subject to two constraints: +\begin{equation} + \max_{u_2}\; u_2^\top \Sigma u_2 +\end{equation} +subject to +\begin{equation} + u_2^\top u_2 = 1, + \qquad + u_2^\top u_1 = 0. +\end{equation} +The first condition enforces unit length; the second ensures orthogonality to $u_1$. + +Introduce Lagrange multipliers $\lambda_2$ and $\alpha$. +Consider the Lagrangian +\begin{equation} +\mathcal{L}(u_2,\lambda_2,\alpha) + = u_2^\top \Sigma u_2 + + \lambda_2 (1 - u_2^\top u_2) + + \alpha\, u_2^\top u_1. +\end{equation} +Taking the derivative with respect to $u_2$ and setting it to zero yields +\begin{equation} + \frac{\partial \mathcal{L}}{\partial u_2} + = 2 \Sigma u_2 - 2 \lambda_2 u_2 + \alpha u_1 = 0. +\end{equation} + +Rearranging the stationarity condition gives +\begin{equation} + \Sigma u_2 = \lambda_2 u_2 - \frac{\alpha}{2} u_1. +\end{equation} +Using the orthogonality constraint $u_1^\top u_2 = 0$ and the fact that +\begin{equation} + \Sigma u_1 = \lambda_1 u_1, +\end{equation} +we obtain +\begin{equation} + u_1^\top \Sigma u_2 = \lambda_2\, u_1^\top u_2 - \frac{\alpha}{2} u_1^\top u_1 + = -\frac{\alpha}{2}. +\end{equation} +On the other hand, +\begin{equation} + u_1^\top \Sigma u_2 = (\Sigma u_1)^\top u_2 = \lambda_1 u_1^\top u_2 = 0. +\end{equation} +Thus $\alpha = 0$, and the stationarity condition reduces to the eigenvalue equation +\begin{equation} + \Sigma u_2 = \lambda_2 u_2. +\end{equation} +Therefore $u_2$ must be an eigenvector of $\Sigma$ associated with the second largest eigenvalue. + +By induction: +\begin{theorem} + The first $K$ principal components are exactly the eigenvectors of the covariance matrix $\Sigma$ corresponding to its $K$ largest eigenvalues. +\end{theorem} +Denote these eigenvectors by +\begin{equation} + U_{1:K} + = +( + u_1,u_2 ,\cdots , u_K +) + \in \mathbb{R}^{d \times K}. +\end{equation} +Each $u_i$ satisfies$ + \Sigma u_i = \lambda_i u_i, + \, + \lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_K.$ + + Given the matrix of the top $K$ principal directions $U_{1:K}$, the projection of a data point $x \in \mathbb{R}^d$ onto the $K$-dimensional PCA subspace is $ + z = U_{1:K}^{\top} x + \in \mathbb{R}^{n\times K}. +$ + For a dataset $X \in \mathbb{R}^{n \times d}$, the projected data matrix is + \begin{equation} + X U_{1:K} + \in \mathbb{R}^{n \times K}. + \end{equation} + This reduces the dimensionality from $d$ to $K$ while retaining the directions of maximal variance. + + \begin{definition}[Centered Data Matrix] + Let the centered data matrix be + \begin{equation} + \hat{X} + = + \begin{pmatrix} + x_1^{\top} - \bar{x}^{\top} \\ + x_2^{\top} - \bar{x}^{\top} \\ + \vdots \\ + x_n^{\top} - \bar{x}^{\top} + \end{pmatrix} + \in \mathbb{R}^{n \times d}. + \end{equation} + Each row of $\hat{X}$ is a mean-subtracted data point. + \end{definition} + + The empirical covariance matrix can be written compactly as + \begin{equation} + \Sigma = \frac{1}{n}\, \hat{X}^{\top}\hat{X}. + \end{equation} + + \begin{theorem}[Eigen-Decomposition of the Covariance] + Because $\Sigma$ is real symmetric, it admits an orthogonal eigendecomposition: + \begin{equation} + \Sigma + = U \Lambda U^{\top}, + \end{equation} + where $U$ is an orthogonal matrix whose columns are eigenvectors + and $\Lambda$ is a diagonal matrix containing the eigenvalues of $\Sigma$. + \end{theorem} + + In practice, numerical libraries such as \texttt{numpy.linalg.eig} or + \texttt{numpy.linalg.eigh} compute the eigen-decomposition of $\Sigma$ directly. + The eigenvectors $\{u_1,\ldots,u_d\}$ form the principal directions of the data. +\subsection{Singular Value Decomposition (SVD)} + Consider the singular value decomposition of the centered data matrix: + \begin{equation} + \hat{X} = U_X\, S\, V^{\top}. + \end{equation} + \newcommand{\mblock}[2]{% + \begingroup + \setlength{\fboxsep}{1pt}% + \colorbox{#1!20}{$\displaystyle #2$}% + \endgroup +} + +\begin{equation} + \hat{X} \in \mathbb{R}^{n\times d} + = + \mblock{blue}{ U_{X\,\mathbb{R}^{n\times r}}}\, + \mblock{green}{\begin{pmatrix} + \sigma_1&\cdots &0&0&\cdots&0\\ + 0&\sigma_2&\cdots&0&\cdots&0\\ + \vdots&\ddots &&&&\vdots\\ + 0&\cdots&0&\sigma_n&\cdots&0 + \end{pmatrix} _{ \mathbb{R}^{r\times r}}}\, + \mblock{red}{V^{\top}_ {\mathbb{R}^{r\times d}} }. +\end{equation} + Then + \begin{equation} + \hat{X}^{\top}\hat{X} + = V S^{2} V^{\top}, + \end{equation} + which implies that + \begin{equation} + \Sigma = \frac{1}{n} V S^{2} V^{\top}. + \end{equation} + Thus \textbf{the right singular vectors of $\hat{X}$ are exactly the eigenvectors of $\Sigma$}, + and the eigenvalues of $\Sigma$ are the squared singular values of $\hat{X}$ scaled by $\tfrac{1}{n}$. + + + \begin{remark} + This relationship provides an alternative way to compute principal components: + the top $K$ principal directions are simply the first $K$ right singular vectors of $\hat{X}$. + This method is numerically more stable, especially when $n \neq d$ or when $d$ is large. + \end{remark} + \begin{note} + In practice, computing a full SVD of a \(d\times d\) matrix costs \(O(d^3)\) time, + which is prohibitive for high-dimensional data. + Modern numerical linear algebra therefore relies on \emph{approximate} SVD methods + that dramatically reduce the computational burden while preserving the leading singular components. + + A common approach is the \emph{truncated SVD}: + instead of computing all singular values, we only approximate the top \(K\) components. + Techniques such as randomized sketching, subspace iteration, and the Nyström method + project the data onto a low-dimensional subspace in which SVD becomes cheap. + These methods achieve a cost of roughly + \begin{equation*} + O(ndK) \quad\text{or}\quad O(d^2K), + \end{equation*} + depending on the algorithm, reducing the complexity significantly when \(K \ll d\). + + This is why real-world PCA implementations (e.g., \emph{\texttt{sklearn}, \texttt{numpy.linalg.svd}}, + and randomized PCA algorithms) compute only the leading singular vectors rather than performing a full decomposition. + \end{note} +\subsection{t-distributed Stochastic Neighbor Embedding (t-SNE)*} + +t-SNE is a nonlinear dimensionality reduction method designed for visualization +of high-dimensional datasets. +It constructs two probability distributions: +\begin{itemize} + \item one on the pairwise similarities in the high-dimensional space, + \item one on the pairwise similarities in the low-dimensional embedding, +\end{itemize} +and finds an embedding that makes these two distributions as close as possible. + + +\begin{definition}[High-Dimensional Similarity] +For datapoints $x_i, x_j$ in the original space, define a conditional probability +\begin{equation} + p_{j|i} + = \frac{\exp\!\left(-\|x_i - x_j\|^2 / 2\sigma_i^2\right)} + {\sum_{k \neq i} \exp\!\left(-\|x_i - x_k\|^2 / 2\sigma_i^2\right)}. +\end{equation} +The bandwidth $\sigma_i$ is chosen such that the \emph{perplexity} +of the distribution matches a user-specified value. +The symmetric joint probability is +\begin{equation} + p_{ij} = \frac{p_{j|i} + p_{i|j}}{2n}. +\end{equation} +\end{definition} +\begin{note} + It simply applies an RBF kernel to these data points. +\end{note} +\begin{definition}[Low-Dimensional Similarity] +For embedding points $y_i, y_j \in \mathbb{R}^2$, +t-SNE uses a Student-\(t\) distribution with one degree of freedom: +\begin{equation} + q_{ij} + = + \frac{\left(1 + \|y_i - y_j\|^2\right)^{-1}} + {\sum_{k \neq \ell} + \left(1 + \|y_k - y_\ell\|^2\right)^{-1}}. +\end{equation} +The heavy-tailed distribution alleviates the “crowding problem.” +\end{definition} + +\begin{theorem}[t-SNE Objective] +t-SNE seeks an embedding that minimizes the Kullback–Leibler divergence +between the high-dimensional and low-dimensional similarity distributions: +\begin{equation} + \mathcal{L}(Y) + = \mathrm{KL}(P \parallel Q) + = \sum_{i \neq j} p_{ij} \log \frac{p_{ij}}{q_{ij}}. +\end{equation} +Gradient descent on $\mathcal{L}$ yields the final visualization. +\end{theorem} + +t-SNE preserves local neighborhood structure. +Pairs with large $p_{ij}$ are forced to satisfy large $q_{ij}$, keeping +nearby points close. +The heavy-tailed $t$-distribution allows distant points to be modeled far apart, +preventing clusters from collapsing together. + +t-SNE is excellent for visualization but not for general-purpose embedding: +its geometry is not globally meaningful, and repeated runs +may differ due to initialization and stochasticity. +\begin{figure}[h] + \centering + \includegraphics{../../tikz/9/2.pdf} +\end{figure} +\section{Clustering} +\subsection{K-means Clustering Algorithm} +Clustering is a fundamental task in unsupervised learning. +Given a dataset +$ + D = \{x_1, x_2, \ldots, x_n\} \subset \mathbb{R}^d, +$ +our goal is to divide these points into \(K\) meaningful groups. +Here \(K\) is a user-chosen hyperparameter, reflecting how many clusters we expect to find. + +To formalize this idea, assume that each cluster \(k \in [K]\) is represented by a \textbf{center} +$ + m_k \in \mathbb{R}^d, +$ +and each data point chooses exactly one cluster. +We denote the assignment using an indicator variable \(r_{ik} \in \{0,1\}\): +\begin{equation} + r_{ik} = 1 + \quad\Longleftrightarrow\quad + x_i \text{ belongs to cluster } k. +\end{equation} +Because every point must join one and only one cluster, we enforce +\begin{equation} + \sum_{k=1}^{K} r_{ik} = 1, + \qquad \forall i \in [n]. +\end{equation} + +The intuitive goal of K-means is simple: +each point should be close to the center of the cluster it joins. +This leads to the objective function +\begin{theorem} + \begin{equation} + L = + \sum_{i=1}^{n} + \sum_{k=1}^{K} + r_{ik} \, \|x_i - m_k\|^2, + \end{equation} +\end{theorem} +which measures the total within-cluster variance. + +At first glance, minimizing \(L\) over both the assignments \(r\) and the centers \(m\) seems challenging, +because the problem is non-convex and the discrete variables \(r_{ik}\) make the search space combinatorial. +However, K-means becomes surprisingly tractable once we notice a key structural property of its objective. +Although optimizing both the assignments \(r\) and the centers \(m\) jointly is difficult, +each subproblem becomes very simple when the other variable is held fixed: + +\begin{itemize} + \item \textbf{Fixing the centers \(\{m_k\}\):} + assigning each point \(x_i\) reduces to choosing the closest center. + + \item \textbf{Fixing the assignments \(\{r_{ik}\}\):} + updating each center becomes a simple averaging step, since the optimal \(m_k\) is just the mean of the points assigned to cluster \(k\). +\end{itemize} + +This observation directly motivates the alternating minimization strategy used by the K-means algorithm: +we first update the assignments, then update the centers, and repeat the process until convergence. +The objective value decreases at every iteration, and although the algorithm is not guaranteed to reach the global optimum, +it converges to a stable \textbf{local minimum} in practice. + +\textbf{Assignment step (fix the centers).} +For each datapoint \(x_i\), we assign it to the nearest cluster center: +\begin{equation} + r_{ik} = + \begin{cases} + 1, + & k = \displaystyle\argmin_{j\in[K]} \|x_i - m_j\|^2, \\[6pt] + 0, + & \text{otherwise}. + \end{cases} +\end{equation} + +\textbf{Update step (fix the assignments).} +To update \(m_k\), we minimize the partial objective +\begin{equation} + \sum_{i=1}^{n} r_{ik} \|x_i - m_k\|^2 +\end{equation} +with respect to \(m_k\). +Taking the derivative and setting it to zero, +\begin{equation} + \frac{\partial}{\partial m_k} + \sum_{i=1}^{n} r_{ik} \|x_i - m_k\|^2 + = + -2 \sum_{i=1}^{n} r_{ik} (x_i - m_k) + = 0, +\end{equation} +which gives the closed-form mean update: +\begin{equation} + m_k + = + \frac{\sum_{i=1}^{n} r_{ik} x_i} + {\sum_{i=1}^{n} r_{ik}}. +\end{equation} + +\begin{remark} + At each iteration, K-means performs a greedy improvement step: + the assignment update minimizes the objective given the centers, + and the center update minimizes the objective given the assignments. + Hence the objective value is non-increasing and bounded below, + which guarantees convergence. +\end{remark} + +\begin{remark} + However, precisely because each step is greedy and only optimizes a partial variable, + K-means may converge to a \emph{local} minimum rather than the global optimum. + Different initializations can therefore lead to different final solutions. +\end{remark} + +Together, these two steps define the classical K-means iteration, +which alternates between assigning points to their closest centers +and recomputing each center as the average of its assigned datapoints. +\begin{figure}[h] + \centering + \includegraphics[width=\textwidth]{../../tikz/9/3.pdf} +\end{figure} +\begin{note} +At each iteration, the K-means algorithm computes the \emph{centroids} of the current approximate clusters and then uses these centroids as the new cluster centers. + In other words, K-means repeatedly replaces each center with the center of mass of the points currently assigned to it. +\end{note} +\subsection{Mixture of Gaussians (MoG)} +K-means assumes that points belonging to each cluster are concentrated around a center, +but it does not model the \emph{shape} or \emph{spread} of each cluster. +A more expressive model is the \emph{Mixture of Gaussians}, +which assumes that points assigned to the same cluster are generated from a Gaussian distribution with its own mean and covariance. + +Formally, for cluster \(k \in [K]\), assume +$ + (x \mid z_k = 1) \sim \mathcal{N}(\mu_k,\, \Sigma_k). +$ +Here \(z = (z_1,\ldots,z_K)\) is a latent indicator vector with +\begin{equation} + z_k \in \{0,1\}, + \qquad + \sum_{k=1}^{K} z_k = 1. +\end{equation} +Thus \(z_k = 1\) indicates that \(x\) is generated from cluster \(k\). + +The prior distribution over latent clusters is modeled using a categorical distribution: +\begin{equation} + P(z_k = 1) = \pi_k, + \qquad + 0 \le \pi_k \le 1, + \qquad + \sum_{k=1}^{K} \pi_k = 1. +\end{equation} + +Given the latent variable \(z\), the conditional likelihood is +\begin{equation} + P(x \mid z_k = 1) + = + \mathcal{N}(x \mid \mu_k, \Sigma_k). +\end{equation} + +Since MoG is a \emph{generative model}, the marginal density of \(x\) is obtained by summing over all latent clusters: +\begin{equation} + P(x) + = + \sum_{k=1}^{K} + P(z_k = 1)\, P(x \mid z_k = 1) + = + \sum_{k=1}^{K} + \pi_k \, \mathcal{N}(x \mid \mu_k, \Sigma_k). +\end{equation} + +% In practice, the parameters +% \(\{\pi_k, \mu_k, \Sigma_k\}_{k=1}^{K}\) +% are estimated by maximizing the likelihood of the observed data, +% typically using the EM algorithm (Expectation–Maximization), +% which alternates between inferring the latent cluster probabilities and updating the Gaussian parameters. +\begin{figure}[h] + \centering + \includegraphics{../../tikz/9/4.pdf} +\end{figure} +\begin{note} + MoG can be viewed as a soft, probabilistic extension of K-means: +instead of forcing each point into exactly one cluster, +it allows fractional responsibilities and models not only the center but also the spread and orientation of each cluster. +\end{note} +We introduce the latent variable +$ + Z := (Z_1, Z_2, \ldots, Z_K)^\top, +$ +where each \(Z_k \in \{0,1\}\) and +\begin{equation} + \sum_{k=1}^{K} Z_k = 1. +\end{equation} +Thus \(Z\) is a \textbf{one-hot encoding }indicating which Gaussian component generates the observation \(X\). +\begin{definition}[One–Hot Encoding] + For a categorical variable taking one of \(K\) possible states, its \emph{one–hot encoding} is the vector + \begin{equation} + Z = (Z_1, Z_2, \ldots, Z_K)^\top \in \{0,1\}^K, + \end{equation} + where exactly one entry equals \(1\) and all others equal \(0\): + \begin{equation} + \sum_{k=1}^{K} Z_k = 1. + \end{equation} + If the variable takes state \(k\), then \(Z_k = 1\) and \(Z_j = 0\) for all \(j \neq k\). + \end{definition} +Using this notation, the mixture density becomes +\begin{equation} + P(x) + = + \sum_{z} P(x \mid z)\, P(z) + = + \sum_{k=1}^{K} + \pi_k\, \mathcal{N}(x \mid \mu_k, \Sigma_k). +\end{equation} + +We aim to estimate the parameters +\begin{equation} + \Theta = \{\pi_k, \mu_k, \Sigma_k\}_{k=1}^{K} +\end{equation} +from data \(x_1,\ldots,x_n\) via maximum likelihood estimation. +The log-likelihood is +\begin{equation} + \argmax_{\Theta} + \sum_{i=1}^{n} + \log P(x_i) + = + \argmax_{\Theta} + \sum_{i=1}^{n} + \log\!\bigg( + {\color{red}\sum_{k=1}^{K}} + \pi_k\, \mathcal{N}(x_i \mid \mu_k, \Sigma_k) + \bigg). +\end{equation} + +Direct optimization is difficult because of the logarithm of a sum. +In contrast, if the latent assignment were known, that is, if for each \(x_i\) we knew which \(Z_{ik}=1\), then the complete-data likelihood would factorize: +\begin{equation} + P(x_i, Z_{ik}=1) + = + \pi_k\, \mathcal{N}(x_i \mid \mu_k, \Sigma_k), +\end{equation} +and maximizing with respect to \(\pi_k, \mu_k, \Sigma_k\) reduces to fitting separated Gaussian clusters, which is straightforward. This gap motivates the \textbf{Expectation–Maximization (EM) algorithm}. + +EM initializes with soft assignments +\(\gamma_{ik} \in [0,1]\) satisfying +\begin{equation} + \sum_{k=1}^{K} \gamma_{ik} = 1, \qquad \forall\, i\in[n], +\end{equation} +where \(\gamma_{ik}\) serves as the initial responsibility of cluster \(k\) for data point \(x_i\). + +EM alternates between: + +\textbf{E-step:} compute posterior responsibilities +\begin{equation} + \gamma_{ik}:=P(Z_{ik}=1 \mid x_i) + = + \frac{\pi_k\, \mathcal{N}(x_i \mid \mu_k, \Sigma_k)} + {\sum_{j\in[K]} \pi_j\, \mathcal{N}(x_i \mid \mu_j, \Sigma_j)}, +\end{equation} + +\textbf{M-step:} maximize the expected complete-data log-likelihood to update \(\pi_k\), \(\mu_k\), and \(\Sigma_k\): +\begin{gather} + \mu_k=\frac{\sum_{i\in[n]} \gamma_{ij}x_i}{\sum_{i\in[n]} \gamma_{ik}}\\ + \Sigma_k=\frac{\sum_{i\in[n]} \gamma_{ik}(x_i-\mu_k)(x_i-\mu_k)^{\top}} + {\sum_{i\in[n]} \gamma_{ik}}\\ + \pi_k=\frac{\sum_{i\in[n]} \gamma_{ik}}{n} +\end{gather} 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0.3/0.7, 0.5/0.8, + 0.7/1.0, 0.9/1.2, 1.0/1.4, + % lower-left small cloud + -2.0/-0.7, -1.8/-0.6, -1.7/-0.4, -1.5/-0.5, -1.3/-0.3, + -1.2/-0.2, -1.0/-0.1, -0.9/0.0, -0.8/0.1 + }{ + \fill (\x,\y) circle[radius=1.6pt]; + } + + % PCA axes + % PC1: along the main diagonal direction (roughly slope ~1) + \draw[ thick,blue,->] (-3,-1.8) -- (3,3.2) node[anchor=south west] {$z_1$}; + + % PC2: perpendicular to PC1 + \draw[ thick,red,<-] (-2.5,3) -- (2.5,-3) node[anchor=north west] {$z_2$}; + +\end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/notes/2025/tikz/9/2.pdf b/notes/2025/tikz/9/2.pdf new file mode 100644 index 0000000000000000000000000000000000000000..935a0edda928e033d352e54597567bbfaacbc1d4 GIT binary patch literal 35986 zcma&NV~j3L6rkBQ-nMPqHf|fYZQHtS+qP}nwr$(?ezUtX$!6!r)UT>ks!};QsXUeE z6q$mkI2{u`8w}ap((oD#J0T;XouMTRFE0#(jESw8vpFF%D+@E>|J`61#4W6yO&kds z#H|gSO+-zM?2Jud`1xU+oE=RJY+&3snl;;FtXq+$KWQ*Eg_O5a=a{P?IdMFhHx+RT 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circle (1.7pt); + } + \node[red!70] at (2.2,1.6) {\small $\mu_B$}; + + % Cluster 3 (green) + \foreach \x/\y in { + -0.4/-1.2, -0.2/-1.0, 0.0/-1.3, 0.2/-1.1, + -0.1/-1.5, 0.3/-1.4, -0.3/-1.0, 0.1/-1.6 + }{ + \fill[green!70!black] (\x,\y) circle (1.7pt); + } + \node[green!70!black] at (0.4,-0.6) {\small $\mu_C$}; + + % A few “outliers” + \fill[gray!70] (2.4,-1.5) circle (1.7pt); + \fill[gray!70] (-2.3,-1.8) circle (1.7pt); + +\end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/notes/2025/tikz/9/3.pdf b/notes/2025/tikz/9/3.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1f1ecefe8973876e7bc241de3a17cc32b7389e64 GIT binary patch literal 14674 zcma*ObC4%Nv?biOZF?HOwr$(CZQGo-?Vh%6+jdXecJI7>?|rfH?MCcY)E{{>Z&lvN zs*`ym^Bi&o5ivSOdR7?nxuxMX7&amRk-d=>3=ak*RPfVTqtt6O?Qtd;C!PhVyPDw&p| z6-M4brhUN#CBnVSf>2YJwO+F;%jVY&ImRvbpPUY8DlzEw{+TNUdBndS#b_CLqNUsV zr=Faj!4KmK7-{w4go4qL!^__~utE}*Bbs@JCXW>TnqX&g=rQ&wLP2BTi#`vEii}LH 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However, this situation is not desirable in practice, since it implies poor robustness. Hence a natural question arises: under the condition of linear separability, how can we find a well-chosen separating hyperplane that maximizes robustness? The answer will be presented in the next chapter, where we introduce the Support Vector Machine (SVM). The SVM optimizes $\hat w, \hat b$ by maximizing the margin (the distance between data points and the separating hyperplane), instead of simply minimizing the cross-entropy loss. +Figure~\ref{fig:2-hyperplane} illustrates a linearly separable setting, and the final state leads to $\|W\| \to \infty, \quad \|b\| \to \infty$. However, this situation is not desirable in practice, since it implies poor robustness. Hence a natural question arises: under the condition of linear separability, how can we find a well-chosen separating hyperplane that maximizes robustness? The answer will be presented in the next chapter, where we introduce the Support Vector Machine (SVM). The SVM optimizes $\hat w, \hat b$ by maximizing the margin (the distance between data points and the separating hyperplane), instead of simply minimizing the cross-entropy loss. Although logistic regression may suffer from divergence of parameters under separable data, it often achieves better performance than SVM in practice, due to the following reasons: \begin{enumerate} @@ -261,7 +259,7 @@ \section{Rethink of Linear Regression} \centering \includegraphics{../../tikz/2/3.pdf} \caption{Normal distribution (95\%).} - \label{2-lr} + \label{fig:2-normal} \end{figure} While the Central Limit Theorem (CLT) does not imply that most datasets are normally distributed, it motivates modeling additive noise as Gaussian. We assume: diff --git a/notes/2025/mvp/chapters/3-svm.tex b/notes/2025/mvp/chapters/3-svm.tex index d1e9a01..8f5c036 100644 --- a/notes/2025/mvp/chapters/3-svm.tex +++ b/notes/2025/mvp/chapters/3-svm.tex @@ -237,7 +237,8 @@ \subsection{Hard Margin} \begin{figure}[H] \centering \includegraphics{../../tikz/3/1.pdf} - \label{2-lr} + \caption{Distance from a point to a hyperplane.} + \label{fig:3-hyperplane-distance} \end{figure} \begin{proof} \begin{enumerate} @@ -962,7 +963,7 @@ \section{Kernel} \begin{equation} K(x, z) = x^\top z; \end{equation} - \item \textbf{Polynomial kernel}: maps $\mathbb{R}^n \to \mathbb{R}^{\scriptsize\begin{pmatrix}n+p\\p\end{pmatrix}}$, + \item \textbf{Polynomial kernel}: can be understood as mapping $\mathbb{R}^n$ into a feature space of dimension $\binom{n+p}{p}$, \begin{equation} K(x, z) = (x^\top z + 1)^p; \end{equation} diff --git a/notes/2025/mvp/chapters/8-tel.pdf b/notes/2025/mvp/chapters/8-tel.pdf index 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Given training inputs @@ -58,7 +58,7 @@ \section{Review: Gaussian Processes and Bayesian Optimization} $\mu_\star \pm \sigma_\star$, we obtain a smooth predictive curve together with credible intervals that quantify uncertainty. -\subsection{Black-Box Optimization with Gaussian Processes} +\subsection*{Black-Box Optimization with Gaussian Processes} An important application of GP regression is \emph{Bayesian Optimization} (BO), which addresses \emph{black-box optimization} problems: @@ -117,7 +117,7 @@ \subsection{Black-Box Optimization with Gaussian Processes} $\mu(x)$ is large). \end{remark} -\subsection{Acquisition Functions} +\subsection*{Acquisition Functions} Let $y_{\max}$ denote the best function value observed so far, and suppose the GP posterior at $x$ is Gaussian: @@ -177,105 +177,6 @@ \subsection{Acquisition Functions} optimal hyperparameters within the given budget. \end{remark} -\section{Bias--Variance Decomposition} - -Consider a supervised learning setting with training dataset $D$, drawn from -some unknown data-generating distribution. Let $x$ denote a test input with -true label $y$, and let -\begin{equation} - f(x;D) -\end{equation} -denote the prediction of a learning algorithm trained on $D$ when evaluated at -$x$. Under squared loss, the prediction error at $(x,y)$ for a given training -set $D$ is -\begin{equation} - \bigl(f(x;D) - y\bigr)^2. -\end{equation} -Since $D$ is random, it is natural to measure performance by averaging over the -random draw of $D$. We therefore consider -\begin{equation} - \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr], -\end{equation} -which quantifies the expected squared error at $x$, averaged over the -randomness in the training data. - -Define the \emph{average prediction} at $x$ as -\begin{equation} - \bar{f}(x) - := \mathbb{E}_{D}\bigl[f(x;D)\bigr], -\end{equation} -that is, the expected prediction we would obtain if we could repeatedly sample -training sets $D$ from the underlying distribution and retrain the model. -Insert and subtract $\bar{f}(x)$ inside the square: -\begin{align} - \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr] - &= \mathbb{E}_{D} - \Bigl[\bigl(f(x;D) - \bar{f}(x) + \bar{f}(x) - y\bigr)^2\Bigr]\\ - &= \mathbb{E}_{D} - \Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)^2\Bigr] - + \bigl(\bar{f}(x) - y\bigr)^2\\ - &\quad - + 2\,\mathbb{E}_{D}\Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)\bigl(\bar{f}(x) - y\bigr)\Bigr]. -\end{align} -The cross term vanishes because -\begin{equation} - \mathbb{E}_{D}[f(x;D) - \bar{f}(x)] - = \mathbb{E}_{D}[f(x;D)] - \bar{f}(x) = 0, -\end{equation} -and $\bar{f}(x)-y$ does not depend on $D$. Hence -\begin{equation} - \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr] - = - \underbrace{\mathbb{E}_{D} - \Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)^2\Bigr]}_{\text{variance at }x} - + - \underbrace{\bigl(\bar{f}(x) - y\bigr)^2}_{\text{bias}^2 \text{ at }x}. -\end{equation} - -\begin{definition}[Bias and variance at a test point] - For a fixed test input $x$ with true label $y$, the - \emph{variance} and \emph{(squared) bias} of a learning algorithm are - defined as - \begin{align} - \mathrm{Var}_D\bigl[f(x;D)\bigr] - &= \mathbb{E}_{D} - \Bigl[\bigl(f(x;D)-\bar{f}(x)\bigr)^2\Bigr],\\ - \mathrm{Bias}^2(x) - &= \bigl(\bar{f}(x) - y\bigr)^2. - \end{align} -\end{definition} - -Intuitively: -\begin{itemize} - \item The variance term measures how sensitive the predictor $f(x;D)$ is to - fluctuations in the training data: for different draws of $D$, predictions - at $x$ may vary significantly around their mean $\bar{f}(x)$. - \item The bias term measures how far, on average, the prediction is from - the true label $y$. Even if we could train on infinitely many different - datasets, the average predictor $\bar{f}(x)$ might still systematically - miss the true target due to limited model capacity or misspecification. -\end{itemize} - -High variance typically arises from \emph{overfitting} particular training sets, -while high bias indicates that the model class itself is too simple to capture -the underlying relationship, even when averaged over many datasets. - -\begin{example}[Linear model on nonlinear data] - Suppose the data arise from a quadratic relationship, but we insist on - fitting a linear model. Even if we repeatedly resample training sets and - average the resulting fitted lines, the averaged predictor $\bar{f}(x)$ - remains linear and cannot match the true quadratic curve everywhere. This - manifests as a persistent bias term. -\end{example} - -Bias and variance suggest complementary strategies: -\begin{itemize} - \item To reduce \textbf{variance}, we can increase the training set size or - average multiple diverse models (as in bagging and random forests). - \item To reduce \textbf{bias}, we can increase model capacity, e.g.\ by - using more flexible hypothesis classes or boosting weak learners. -\end{itemize} - \section{Decision Trees and Feature Selection} \subsection{Decision Trees and Axis-Aligned Splits} @@ -920,6 +821,104 @@ \section{From Single Trees to Ensembles} combination can be significantly more accurate and more robust. \end{definition} +\subsection{Bias--Variance Decomposition} +Consider a supervised learning setting with training dataset $D$, drawn from +some unknown data-generating distribution. Let $x$ denote a test input with +true label $y$, and let +\begin{equation} + f(x;D) +\end{equation} +denote the prediction of a learning algorithm trained on $D$ when evaluated at +$x$. Under squared loss, the prediction error at $(x,y)$ for a given training +set $D$ is +\begin{equation} + \bigl(f(x;D) - y\bigr)^2. +\end{equation} +Since $D$ is random, it is natural to measure performance by averaging over the +random draw of $D$. We therefore consider +\begin{equation} + \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr], +\end{equation} +which quantifies the expected squared error at $x$, averaged over the +randomness in the training data. + +Define the \emph{average prediction} at $x$ as +\begin{equation} + \bar{f}(x) + := \mathbb{E}_{D}\bigl[f(x;D)\bigr], +\end{equation} +that is, the expected prediction we would obtain if we could repeatedly sample +training sets $D$ from the underlying distribution and retrain the model. +Insert and subtract $\bar{f}(x)$ inside the square: +\begin{align} + \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr] + &= \mathbb{E}_{D} + \Bigl[\bigl(f(x;D) - \bar{f}(x) + \bar{f}(x) - y\bigr)^2\Bigr]\\ + &= \mathbb{E}_{D} + \Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)^2\Bigr] + + \bigl(\bar{f}(x) - y\bigr)^2\\ + &\quad + + 2\,\mathbb{E}_{D}\Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)\bigl(\bar{f}(x) - y\bigr)\Bigr]. +\end{align} +The cross term vanishes because +\begin{equation} + \mathbb{E}_{D}[f(x;D) - \bar{f}(x)] + = \mathbb{E}_{D}[f(x;D)] - \bar{f}(x) = 0, +\end{equation} +and $\bar{f}(x)-y$ does not depend on $D$. Hence +\begin{equation} + \mathbb{E}_{D}\Bigl[\bigl(f(x;D) - y\bigr)^2\Bigr] + = + \underbrace{\mathbb{E}_{D} + \Bigl[\bigl(f(x;D) - \bar{f}(x)\bigr)^2\Bigr]}_{\text{variance at }x} + + + \underbrace{\bigl(\bar{f}(x) - y\bigr)^2}_{\text{bias}^2 \text{ at }x}. +\end{equation} + +\begin{definition}[Bias and variance at a test point] + For a fixed test input $x$ with true label $y$, the + \emph{variance} and \emph{(squared) bias} of a learning algorithm are + defined as + \begin{align} + \mathrm{Var}_D\bigl[f(x;D)\bigr] + &= \mathbb{E}_{D} + \Bigl[\bigl(f(x;D)-\bar{f}(x)\bigr)^2\Bigr],\\ + \mathrm{Bias}^2(x) + &= \bigl(\bar{f}(x) - y\bigr)^2. + \end{align} +\end{definition} + +Intuitively: +\begin{itemize} + \item The variance term measures how sensitive the predictor $f(x;D)$ is to + fluctuations in the training data: for different draws of $D$, predictions + at $x$ may vary significantly around their mean $\bar{f}(x)$. + \item The bias term measures how far, on average, the prediction is from + the true label $y$. Even if we could train on infinitely many different + datasets, the average predictor $\bar{f}(x)$ might still systematically + miss the true target due to limited model capacity or misspecification. +\end{itemize} + +High variance typically arises from \emph{overfitting} particular training sets, +while high bias indicates that the model class itself is too simple to capture +the underlying relationship, even when averaged over many datasets. + +\begin{example}[Linear model on nonlinear data] + Suppose the data arise from a quadratic relationship, but we insist on + fitting a linear model. Even if we repeatedly resample training sets and + average the resulting fitted lines, the averaged predictor $\bar{f}(x)$ + remains linear and cannot match the true quadratic curve everywhere. This + manifests as a persistent bias term. +\end{example} + +Bias and variance suggest complementary strategies: +\begin{itemize} + \item To reduce \textbf{variance}, we can increase the training set size or + average multiple diverse models (as in bagging and random forests). + \item To reduce \textbf{bias}, we can increase model capacity, e.g.\ by + using more flexible hypothesis classes or boosting weak learners. +\end{itemize} + Two classical ensemble strategies are: \begin{itemize} \item \emph{Bagging} (Bootstrap Aggregating): build many base learners diff --git a/notes/2025/mvp/chapters/9-em-mog.pdf b/notes/2025/mvp/chapters/9-em-mog.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9929aa68cd02c21827b37ba8c3b74749f8ee1754 GIT binary patch literal 87085 zcmd42V~{29w(eWiUAAo-vuxY8ZQHhO+qP|^%j{B@(dD|mPMmx8I(MzJ*IhgIKjM7I zj5!lC#xtLH{4(E>BT40jMQIu7SfEIcZZ4mpSP2*i>KUV*Io1&z8Tfb?6mM zoa|g2jZB=N*#2x#q!+cYb~bUO7qvEUHW4;4vNJZJmo~9Ab2cXcF!Az2IXOF;7}!9$ zZzwUy+p96ii>uAU&&pHG0s&!s`<@JCVr%?AkNETHZ^3f>`O&{JXvV)u`EQS!@o#_5 zUm7;xFA1CJuMV5}FA1CFF9}=B(ay#GPqHY{D?1w4I@udIn%Ek7&4)v$4d?_D?f_LRp>x1OPiAEtDS*oGd-7GVeCUB5yyT1_2@nS`G#z zT?jq?kD0;vw|x8WA7lT%YWDv|)yjXTx(RyXpQ!$8fbegt2K*~-{5xJN{+;S!=*fRn z4ftz#^6#tuSJ?S?RIC1->J{jzf1>)Y#u4Bz!4cs95*z{k8*$X~cQ}%vf}Z~;X8kod z`j6WSz`x?mf9d`L@P7$UnEoYC*#G}uJDC5MTW~UMEN_n{uL_zYi|8DNci{N`d8%mx7kz3840YRW| zTo1AwkPaQ&-liO00AGLkTjAZg@%o^T9%TP+Vu48h>A|)Y5LAjNapA35gp72s5NeOe z)_{mF$X;3^3+qcI`QD(Bx)J?;4^4hr zLMkwoze3BVE^qsM^POJkx#N3p;1^5bKtVCJn+G!jnf$JJ9&_xx!ijoO(5go7^Y5^s zyenH!qT?YyB*(!yxii|GDf5uYlfy=TWOJBiH|!z%`_8M?+C%!a0Zz)5T{t4YvU=S{8$vCPe$+HW~L z2EO<(DKlwRBJc}`g69gMr4};C*FtS3;j>Cts%q z8$IHv!g}Z`w4vYzvyk05;=19S9uY!_Y_g0+FnVh4q|+=yk2>G(9SE~SOY`*aCKXzp zxTI9F28Hu1($jgf;gI*zS4)rvKI$WOXIbrd=Vf=Zm%z4;&E6CRUCTzp)?-{?4uNl; z)Sn0*yz*%2g49Vsqc>bKao8tO_y+?VWT9bHWw%(rtGow5At2Bu4)TQiJ0DDQ>B2XT z3yQ(6?%CWx+Zp>my$UT`d}W9SfNA+S*>cuj!7GlqM1Y6XwT)%fl=sppUKqY~z6$8ic|fcr)o zEyJ=9V_>qa%g@j+vS*qZxs(8>_cnH?%RbT=QI@bj10h7u!^SF3=64$rvJKc^{D}TyJZMKv$xHx{a-c21%It$}*8nsEve%RK-ly&2m}2 zwE&Vd$IpUT{>)!IISham4_SM<{oZ4#UA}qVLWfDMyazEcf{MYecM}kagjwB4Mt<#X3O*92pYOU#(ueH zZkc_u=}v3p9tk{Vp-s2W(}auTIap97cj%P=9i{{Ld96eo5US1npsFiqD9>{&@+_g3 zpbUo6dYHQDveNIN6t8%>aVN0Yr?_1b;CWxgFr6Se1m`*^kQ@uTmpoweIp*ii;OApP zsiO~+J%u3zEF$nA25Bw6`WuHLdH*#l-Yb=yNiA#TiX`2*sfY1mVr*2#m6G`Ds3d2+ znM71B?xh@IE}HbUPPR`f)z`JruRB?O9&Pr~80@)JE$f__UlsN5;n$zcLv*3hR*!&;dH_aF;ovO3N*OTl$@Yk}bc zJyywa=X)zF1GDa2%$^!7RK>$QTD3FdgEq6o?M5Rme&)@;3rf6X{kF*zq+B&XVK#)N zA_)OAWlzg_X4I?65F6V*WX1K<2~WGF!lMr>Zp;D)oE9ap$nriN@}-O%=}O3s z7NZVJIvK)G&Fih18G2^N*qpgmoX&?N-#Fr#y>bbxT1@F?cMoD)g(YR)KO1A-)#$t?rIK_kLfVzHo zgd?204*vZxE5Kw&Hixma>39@&mH?&ib2-}_`V)jARy?^@Hfq<@D7v>=13PGE)smh2MAr}q(;Tx0j*s0Ft#oCls;N6*7%Zm zb!62eS_Q{LZCrbBq?023pX~d-F;vT_KU!(_k=YZy9z%I|YT6h6o~`JQ z&}TE(}`9NP8DVkTO}^<@;J3< z2kBnbl1Lt{c3D#w!g}$buEhR%;>~QQyJFsAe#R$>&EU*L+J(DV6685Y%-2kfZl%Jr z2vAz00mLijw_4y3MQywr5aUy{r^$+&^K9WTw<0c1Wsf1c4lYe2?O|N=^H|nzXWbCaLc;l&PbVs^J!XTl@ z;BtM=^?{5rlqINJ#C&B@|Ef*a_|gQp4dw@j0$B54-LQSwb@~MuPX%B>v4uhv5pwNO zTnkg5pw}KKP%X%LQ8U$dJY~mS1Yr3m=9T_w_Ma^^Cn{(ssZj?jA$EH z1e?zFX|sW3et18}N#shOpf6f~J8%1FOkV4d-kq9k!XpmQ}pr`Zubc_qAznp zL|W!13arRN@l%Elg6#3#;?O!XuVsUIW)^e_;(1nlnXdz+FU7PiPOb^wHi#axoljZ^ zUT3lUy7Ww|E;o_r8n!PoM--h}y5jy^@u!CiWgK!==Q2}e`Oh{AF0lvT6TQrfEa<6^EgVjZUSt`J;hIT#nzU$TOd4n` zrP#vVMTDWD)Cvwhd8Drg6j0pQYk4Q3^gNpCraZFxvVdJ#NggwEWyg)Iwf+U%4f&zx z;z##Ig0D<)PB>Ud%buGs(n{N5^O<3DL9-3sZK|xI8BC<>bj<^R;zKh%ya6h(SYk^`Xa6K8qb9fFcJl4?4uYM#y>@*|k|?>=0}A=l+%M(h z!DsgM;`(?!e)y%tB3)P4$6=r&KKmZ~UAYT0cmU0UV-4!- z23Y&rw1T5`5cShQFJ?yn=9FM#wZVq|9GHOn$#hD7)?WKP&jk# zgp5zkLUY7TulL>i_2^iOU_UE@^7`Rnn;E>mBPUp2v*TSQ&|6+tOb1Ezb227C^3Yf}`Z(RrGAo43;b!xaeMSzTY*T{AO) zm}XKT>$+Y-h>S>?8vSn_&VaU$45%xD#&JD|p+Vc*Tj>sS^i**4#D2{X&BxC%qeCoK zi?eZeY>`ef7tfFj#tqkAbgv5y$?})Z0PJ`geDDWvblGN%9*(~hwm?Q>*5qKIY0cJBd zFlH0nc;=^%Yx_jgmi@Z!jP?&7v8wd3twB40+M$no*-=@&uM1cHGk1pzndf8wP!=a~ z?6VhYixGXOkoW#A7mcISM&~yb1LB%F`p`ASw2-@nhb+*aDdt%zunk97V!h3R))#vj zd+}T#o(sF!o`_|Qfy8H^9!!K7M`%bLnFFyFd`sq=u@b-dh~1=UdLGZ^f!1Yk0)qOj zA)$D&3iuhXFDukWU5kMi6;QNeS&K42r7B*EfJw2)n#q>m!^$X#j#ps?vje@Wu$Sv7 zgT5)gz5CQBnu?<0eAvL`2rc6Buk8xmpmeIauNHNM8aD5!&CZzfbjJd@86Y1hb?SC) z)?SQ-lor3-$0)%Sc0CSWUIf@X!*MA7^br7f7S0!7jr|t6W438J6=*_;^&nwY=IqNT zJV0i_U(RV9Mak3~_FMYOzTL{~O9^wz+($annK`p><48+^9}a5Lec#R3`AegbtE`#WhV0cuu-M2 zAU*f#<8@y()x#gX^+|33ra_4(z(92%7o*vNz`c%g)+X>cL=}QGj}^eCT_T=+hwFFZL<&5%kNnjOf#@d?8zhO%$DpExnm2tJRWi zjBvxW3ghk;RBI#M&(mAB+wfNFMExunORPU&aP`&`z01AR zWc+(k3jS(bZRW)`{|fpJ(|0@-K3Iva&qMi@6NL$}BSzAh)|PNAhof3Jyw!Hl=KaW- zo6QExt5}7Zm_)bApP77?Or3fos9AmNG14T#2R+`6oSk%7TM2TBc+7iy^+E z8R1`FOnw|r&w|2gVXFWMTFg3T+c2j6V?dw+uL1Px8%=Dy(_yMr6~echi502XKE%lD z)8O1JG%GrVtB=wwj)sz4|3$Otp;MF&iMzyL{n}KFA5eA>R_P!*`Afp-rTjAnkU2vH zCT&}5E*Y`SD7QJiu|&#Xg(pr8F+UgDkzuS9;M=AkpMAgkS)AmXgcfUa>39)HD|?JL zjn~4f*jvu$t`kn35|J+A;acN55o=~Z3$L&})#3c1<=S{xW zBr3#?(;B|kwC0rFke};1el_

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+\section{Mixture of Gaussians (MoG): a generative view of clustering} +We revisit clustering from a probabilistic and generative perspective. +Instead of assigning each datapoint to a cluster deterministically, we assume +each datapoint is generated by \emph{first} choosing a latent cluster index and +\emph{then} sampling from a Gaussian distribution attached to that cluster. + +\subsection{Latent variable and generative process} +Let $K\in\mathbb N$ be the number of clusters (components). For each datapoint +$x\in\mathbb R^d$, introduce a latent discrete variable +\begin{equation} + G \in \{1,2,\dots,K\}, +\end{equation} +where $G=k$ indicates that $x$ is generated from the $k$-th Gaussian component. +We model +\begin{equation} + p(G=k)=\pi_k,\qquad \pi_k\ge 0,\qquad \sum_{k=1}^K \pi_k = 1, +\end{equation} +and the conditional likelihood +\begin{equation} + p(x\mid G=k)=\mathcal N(x\mid \mu_k,\Sigma_k), +\end{equation} +where $\mu_k\in\mathbb R^d$ and $\Sigma_k\in\mathbb R^{d\times d}$ is symmetric +positive definite. + +\begin{note} + Earlier chapters may encode the latent cluster by a one-hot vector + $z\in\{0,1\}^K$ with $\sum_k z_k=1$. The integer notation here is + equivalent but notationally lighter: + $G=k \Longleftrightarrow z_k=1$. +\end{note} + +\subsection{Marginal likelihood and the MLE objective} +By marginalizing $G$, the density of $x$ becomes a mixture: +\begin{equation}\label{eq:mog-marginal} + p(x;\theta) + = + \sum_{k=1}^K p(G=k)\,p(x\mid G=k) + = + \sum_{k=1}^K \pi_k\,\mathcal N(x\mid \mu_k,\Sigma_k), +\end{equation} +where we collect all parameters as +\begin{equation} + \theta := \Bigl\{\pi_k,\mu_k,\Sigma_k\Bigr\}_{k=1}^K. +\end{equation} + +Given i.i.d.\ data $\{x_i\}_{i=1}^n$, maximum likelihood estimation solves +\begin{equation}\label{eq:mog-mle} + \argmax_{\theta}\; + \sum_{i=1}^n \log p(x_i;\theta) + = + \argmax_{\theta}\; + \sum_{i=1}^n + \log\!\left( + \sum_{k=1}^K \pi_k\,\mathcal N(x_i\mid \mu_k,\Sigma_k) + \right). +\end{equation} + +\begin{remark} + Unlike supervised learning objectives of the form $\sum_i \log p(y_i\mid + x_i;\theta)$, unsupervised density estimation maximizes $\sum_i \log + p(x_i;\theta)$ directly because no labels are observed. +\end{remark} + +\subsection{Why direct optimization is non-trivial} +The objective \eqref{eq:mog-mle} is difficult for at least two reasons. +\begin{itemize} + \item \textbf{Log-sum structure:} $\log\big(\sum_k \pi_k \mathcal N(\cdot)\big)$ + couples all components inside a logarithm, preventing simple closed-form + derivatives from decoupling. + \item \textbf{Constraints:} $\{\pi_k\}$ must lie on the simplex and each + $\Sigma_k$ must be positive definite. One can use constrained optimization, + penalty methods, or reparameterization, but a naive unconstrained gradient + method is not directly applicable. +\end{itemize} +\begin{remark} + Gradient descent can be made to work with non-trivial adaptations (e.g., + softmax parameters for $\pi$, Cholesky factors for $\Sigma$), but EM + typically provides a faster and more elegant framework for this class of + latent-variable MLE problems. +\end{remark} + +\section{From latent-variable MLE to ELBO} +We now derive the Expectation--Maximization (EM) algorithm in a general setting. + +\subsection{General latent-variable likelihood} +Let $x$ be observed and $z$ be a latent variable (discrete or continuous). +Assume a joint model $p(x,z;\theta)$. The marginal likelihood is +\begin{equation} + p(x;\theta) = \sum_z p(x,z;\theta)\qquad + (\text{or } \int p(x,z;\theta)\,\mathrm dz). +\end{equation} +In MLE, the objective is +\begin{equation} + \argmax_{\theta}\; \sum_{i=1}^n \log p(x_i;\theta). +\end{equation} +The marginalization over $z$ often makes $\log p(x;\theta)$ hard to optimize. + +\subsection{KL divergence: definition and key properties} +\begin{definition}[Kullback--Leibler divergence] + For distributions $Q$ and $P$ on the same variable $Z$, the KL divergence + is + \begin{equation} + \mathrm{KL}(Q\|P) + := \mathbb E_{Z\sim Q}\!\left[\log\frac{Q(Z)}{P(Z)}\right]. + \end{equation} +\end{definition} + +\begin{proposition}[Basic properties of KL]\label{prop:kl-basic} + For any $Q,P$, + \begin{enumerate} + \item $\mathrm{KL}(Q\|P)\ge 0$. + \item $\mathrm{KL}(Q\|P)=0$ if and only if $Q=P$ almost surely. + \item In general, $\mathrm{KL}(Q\|P)\neq \mathrm{KL}(P\|Q)$ (not symmetric). + \end{enumerate} +\end{proposition} +\begin{proof} + By Jensen's inequality applied to the convex function $-\log(\cdot)$, + \begin{align*} + \mathrm{KL}(Q\|P) + &= -\mathbb E_{Q}\!\left[\log \frac{P(Z)}{Q(Z)}\right] + \ge -\log \mathbb E_{Q}\!\left[\frac{P(Z)}{Q(Z)}\right] + = -\log\!\left(\sum_z P(z)\right) + = 0, + \end{align*} + and equality holds iff $\frac{P(Z)}{Q(Z)}$ is constant $Q$-a.s., which gives + $Q=P$ a.s. Non-symmetry follows from counterexamples. +\end{proof} + +\begin{remark} + KL can be written as \textbf{cross-entropy minus entropy}: + \begin{equation} + \mathrm{KL}(Q\|P) + = + \underbrace{\mathbb E_Q[-\log P(Z)]}_{\text{cross-entropy}} + - + \underbrace{\mathbb E_Q[-\log Q(Z)]}_{\text{entropy}}. + \end{equation} + This explains why using the ``wrong code'' $P$ to encode samples from $Q$ + necessarily incurs extra expected description length. +\end{remark} + +\subsection{ELBO decomposition} +Introduce an arbitrary distribution $q(z)$ (a ``variational distribution''). +Then the following identity holds: +\begin{theorem}[ELBO decomposition]\label{thm:elbo} + For any $q(z)$ and any $\theta$, + \begin{equation}\label{eq:elbo-decomp} + \log p(x;\theta) + = + \underbrace{ + \mathbb E_{z\sim q}\!\left[\log \frac{p(x,z;\theta)}{q(z)}\right] + }_{\mathcal L(q,\theta)\;\;(\text{ELBO})} + + + \underbrace{ + \mathrm{KL}\!\left(q(z)\,\middle\|\,p(z\mid x;\theta)\right) + }_{\ge 0}. + \end{equation} + Consequently, $\mathcal L(q,\theta)\le \log p(x;\theta)$ for all $q$. +\end{theorem} +\begin{proof} + Starting from $\log p(x;\theta)$ and taking expectation over $q$, + \begin{align*} + \log p(x;\theta) + &= \sum_z q(z)\,\log p(x;\theta) \\ + &= \sum_z q(z)\,\log \frac{p(x,z;\theta)}{p(z\mid x;\theta)} \\ + &= \sum_z q(z)\,\log \frac{p(x,z;\theta)}{q(z)} + + \sum_z q(z)\,\log \frac{q(z)}{p(z\mid x;\theta)}, + \end{align*} + where the last term is exactly the KL divergence. +\end{proof} + +\begin{definition}[Evidence lower bound (ELBO)] + The functional + \begin{equation} + \mathcal L(q,\theta) + := + \mathbb E_{z\sim q}\!\left[\log p(x,z;\theta)\right] + - + \mathbb E_{z\sim q}\!\left[\log q(z)\right] + = + \mathbb E_{z\sim q}\!\left[\log \frac{p(x,z;\theta)}{q(z)}\right] + \end{equation} + is called the \emph{evidence lower bound}. +\end{definition} + +\begin{remark} + The abbreviation \textbf{ELBO} is commonly read as ``elbow'' in talks. + The bound gap is precisely + $\mathrm{KL}\big(q(z)\|p(z\mid x;\theta)\big)$. + When $q(z)=p(z\mid x;\theta)$, the bound is tight. +\end{remark} + +\begin{note} + When the posterior $p(z\mid x;\theta)$ is intractable, one typically + restricts $q$ to a tractable family and maximizes ELBO approximately (this + is the core idea of variational inference). EM can be viewed as the special + case where the E-step posterior is tractable and makes the bound tight at + the current iterate. +\end{note} + +\section{The EM algorithm} +EM is an iterative algorithm that alternates between optimizing $q$ (E-step) +and optimizing $\theta$ (M-step), using ELBO as a surrogate objective. + +\subsection{E-step and M-step from ELBO} +Suppose we are at $\theta^{(t)}$. +\begin{itemize} + \item \textbf{E-step (fix $\theta$):} + choose $q^{(t+1)}$ to maximize $\mathcal L(q,\theta^{(t)})$. + Since $\log p(x;\theta^{(t)})$ is a constant w.r.t.\ $q$, + Theorem~\ref{thm:elbo} implies + \begin{equation}\label{eq:e-step} + q^{(t+1)}(z) + = + p\bigl(z\mid x;\theta^{(t)}\bigr), + \end{equation} + which minimizes the KL gap to $0$. + + \item \textbf{M-step (fix $q$):} + update $\theta$ by maximizing the ELBO + \begin{equation}\label{eq:m-step} + \theta^{(t+1)} + \in + \argmax_{\theta}\; \mathcal L\!\left(q^{(t+1)},\theta\right). + \end{equation} + Because the term $-\mathbb E_{q^{(t+1)}}[\log q^{(t+1)}(z)]$ does not depend + on $\theta$, the M-step equivalently maximizes + \begin{equation}\label{eq:q-function} + Q(\theta\mid \theta^{(t)}) + := + \mathbb E_{z\sim p(z\mid x;\theta^{(t)})}\!\left[\log p(x,z;\theta)\right], + \end{equation} + the expected complete-data log-likelihood. +\end{itemize} + +\begin{remark} + The ``expectation'' in \textbf{E-step} refers to taking expectation with + respect to the posterior $p(z\mid x;\theta^{(t)})$, which is computed in the + E-step and then used to form the expected complete log-likelihood in the + M-step. +\end{remark} + +\begin{note} + The overall algorithm repeats \textbf{E-step $\rightarrow$ M-step} until + convergence, e.g.\ until $\theta^{(t+1)}$ is sufficiently close to + $\theta^{(t)}$ or the log-likelihood improvement becomes negligible. +\end{note} + +\subsection{Monotonicity and local optima} +\begin{proposition}[Monotonic improvement] + Each EM iteration does not decrease the data log-likelihood: + \[ + \log p(x;\theta^{(t+1)}) \ge \log p(x;\theta^{(t)}). + \] +\end{proposition} +\begin{proof} + In the E-step, the ELBO is made tight at $\theta^{(t)}$ by choosing + $q^{(t+1)}(z)=p(z\mid x;\theta^{(t)})$, hence + $\mathcal L(q^{(t+1)},\theta^{(t)})=\log p(x;\theta^{(t)})$. + In the M-step, we maximize the ELBO in $\theta$, so + \[ + \mathcal L(q^{(t+1)},\theta^{(t+1)}) + \ge + \mathcal L(q^{(t+1)},\theta^{(t)}) + = + \log p(x;\theta^{(t)}). + \] + Since ELBO is always a lower bound, $\log p(x;\theta^{(t+1)})\ge + \mathcal L(q^{(t+1)},\theta^{(t+1)})$, proving the claim. +\end{proof} + +\begin{remark} + EM typically converges to a \emph{stationary point}, which may be a local + optimum. Different initializations can lead to different solutions. +\end{remark} + +% \begin{figure}[h] +% \centering +% \includegraphics[width=0.9\textwidth]{../../mvp/tikz/9/5.pdf} +% \caption{Geometric intuition: each E-step picks an ELBO that ``touches'' the log-likelihood at the current $\theta^{(t)}$, and the M-step maximizes that ELBO to obtain a new parameter $\theta^{(t+1)}$ with increased likelihood.} +% \label{fig:em-elbo-geometry} +% \end{figure} + +\section{EM for Mixture of Gaussians} +We now apply the general EM framework to MoG. + +\subsection{Complete-data likelihood} +For a single datapoint $(x_i,G_i)$, +\begin{equation} + p(x_i,G_i=k;\theta) + = + p(G_i=k)\,p(x_i\mid G_i=k) + = + \pi_k\,\mathcal N(x_i\mid \mu_k,\Sigma_k). +\end{equation} +For the dataset, the complete-data log-likelihood is +\begin{equation} + \log p(\{x_i\},\{G_i\};\theta) + = + \sum_{i=1}^n \sum_{k=1}^K \mathbb I\{G_i=k\} + \Bigl( + \log \pi_k + \log \mathcal N(x_i\mid \mu_k,\Sigma_k) + \Bigr). +\end{equation} + +\subsection{E-step: responsibilities} +Define the \emph{responsibility} (posterior assignment probability) +\begin{equation}\label{eq:responsibility} + \gamma_{ik} + := + p(G_i=k \mid x_i;\theta^{(t)}). +\end{equation} +By Bayes' rule, +\begin{equation}\label{eq:mog-e-step} + \gamma_{ik} + = + \frac{ + \pi_k^{(t)}\,\mathcal N(x_i\mid \mu_k^{(t)},\Sigma_k^{(t)}) + }{ + \sum_{j=1}^K \pi_j^{(t)}\,\mathcal N(x_i\mid \mu_j^{(t)},\Sigma_j^{(t)}) + }. +\end{equation} +Intuitively, $\gamma_{ik}$ measures the fraction of ``credit'' component $k$ +takes for explaining datapoint $x_i$. + +\subsection{M-step: closed-form updates} +Given $\{\gamma_{ik}\}$, the EM objective for MoG becomes +\begin{equation}\label{eq:mog-Q} + Q(\theta\mid \theta^{(t)}) + = + \sum_{i=1}^n \sum_{k=1}^K \gamma_{ik} + \Bigl( + \log \pi_k + \log \mathcal N(x_i\mid \mu_k,\Sigma_k) + \Bigr). +\end{equation} +Let +\begin{equation} + N_k := \sum_{i=1}^n \gamma_{ik}. +\end{equation} + +\subsubsection{Update of $\pi_k$ (simplex constraint)} +We maximize \eqref{eq:mog-Q} w.r.t.\ $\pi$ subject to $\sum_k \pi_k=1$. +Introduce a Lagrange multiplier $\lambda$: +\begin{equation} + \mathcal J(\pi,\lambda) + = + \sum_{i=1}^n \sum_{k=1}^K \gamma_{ik}\log \pi_k + + \lambda\left(1-\sum_{k=1}^K \pi_k\right). +\end{equation} +Setting $\partial \mathcal J/\partial \pi_k=0$ gives +\begin{equation} + \pi_k + = + \frac{1}{n}\sum_{i=1}^n \gamma_{ik} + = + \frac{N_k}{n}. +\end{equation} + +\subsubsection{Update of $\mu_k$} +Using the Gaussian log-density, the terms depending on $\mu_k$ yield the +weighted least-squares problem whose solution is +\begin{equation}\label{eq:mog-mu-update} + \mu_k + = + \frac{\sum_{i=1}^n \gamma_{ik} x_i}{\sum_{i=1}^n \gamma_{ik}} + = + \frac{1}{N_k}\sum_{i=1}^n \gamma_{ik} x_i. +\end{equation} + +\subsubsection{Update of $\Sigma_k$} +Similarly, maximizing w.r.t.\ $\Sigma_k$ gives +\begin{equation}\label{eq:mog-sigma-update} + \Sigma_k + = + \frac{\sum_{i=1}^n \gamma_{ik} (x_i-\mu_k)(x_i-\mu_k)^\top}{\sum_{i=1}^n \gamma_{ik}} + = + \frac{1}{N_k}\sum_{i=1}^n \gamma_{ik} (x_i-\mu_k)(x_i-\mu_k)^\top. +\end{equation} + +\begin{remark} + The updates \eqref{eq:mog-mu-update}--\eqref{eq:mog-sigma-update} are + \textbf{soft-assignment weighted} empirical mean and covariance. If + $\gamma_{ik}\in\{0,1\}$ becomes hard assignment, they reduce to the usual + sample mean/covariance of points in cluster $k$. +\end{remark} + +\subsection{Relationship to K-means as a limiting case} +Assume an isotropic shared covariance $\Sigma_k=\sigma^2 I$ for all $k$. +Then +\begin{equation} + \gamma_{ik} + \propto + \pi_k \exp\!\left(-\frac{\|x_i-\mu_k\|^2}{2\sigma^2}\right). +\end{equation} +As $\sigma\to 0$, the softmax distribution concentrates on the closest mean: +\begin{equation} + \gamma_{ik} + \to + \begin{cases} + 1, & k=\argmin_j \|x_i-\mu_j\|^2,\\ + 0, & \text{otherwise}, + \end{cases} +\end{equation} +which recovers the hard assignment step of K-means. With these hard assignments, +the M-step mean update reduces to the K-means centroid update. + +\begin{remark} + This connection explains a common interpretation: \textbf{K-means is a + special/limiting case of MoG} (and can be seen as a degenerate EM). +\end{remark} + +\section{A clarification: PCA vs regression (from Q\&A)} +\begin{note} + PCA is an \emph{unsupervised} problem: all coordinates are treated + symmetrically and the objective is to find a low-dimensional subspace that + best explains the variance/geometry of $x$. + In contrast, regression is \emph{supervised}: $y$ plays a special role and + the objective is to minimize prediction error of $y$ given $x$. + Therefore, even if one augments $x$ with $y$, the resulting optimization is + not equivalent to PCA because the symmetry between coordinates is broken by + the learning objective. +\end{note} + +\end{document} + + diff --git a/notes/2025/mvp/chapters/9-ul.pdf b/notes/2025/mvp/chapters/9-ul.pdf index 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