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\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}

\begin{center}
{\Large CS 6220: Machine Learning Theory} \\
Problem Set 4 \\
Your Name \\
\today
\end{center}

\hrule
\bigskip

\section*{Problem 1}
\textit{Show that the softmax function is invariant to adding a constant to all its inputs, and use this fact to derive a numerically stable implementation.}

\textbf{Solution.} Let $z \in \mathbb{R}^n$ and $c \in \mathbb{R}$. By definition,
\[
\text{softmax}(z + c)_i = \frac{e^{z_i + c}}{\sum_j e^{z_j + c}} = \frac{e^c e^{z_i}}{e^c \sum_j e^{z_j}} = \text{softmax}(z)_i.
\]

\begin{enumerate}
  \item[(a)] Since the identity holds for any $c$, choose $c = -\max_i z_i$ so that every exponent $z_i + c \leq 0$, avoiding overflow.
  \item[(b)] The stabilized computation is
  \[
  \text{softmax}(z)_i = \frac{e^{z_i - \max_j z_j}}{\sum_k e^{z_k - \max_j z_j}}.
  \]
\end{enumerate}

\section*{Problem 2}
\textit{Prove that the KL divergence $D_{KL}(P \| Q) \geq 0$ for probability distributions $P, Q$ over a finite set, with equality iff $P = Q$.}

\textbf{Proof.} By definition,
\[
D_{KL}(P \| Q) = \sum_i P(i) \log \frac{P(i)}{Q(i)}.
\]
By Jensen's inequality applied to the concave function $\log$,
\[
-D_{KL}(P \| Q) = \sum_i P(i) \log \frac{Q(i)}{P(i)} \leq \log \sum_i P(i) \frac{Q(i)}{P(i)} = \log \sum_i Q(i) = \log 1 = 0.
\]
Hence $D_{KL}(P \| Q) \geq 0$, with equality iff $Q(i)/P(i)$ is constant over the support, i.e. $P = Q$. $\blacksquare$

\section*{Problem 3}
\textit{Compute the gradient of the cross-entropy loss with respect to the logits when combined with a softmax output layer.}

\textbf{Solution.} For logits $z$, softmax output $\hat{y} = \text{softmax}(z)$, and one-hot target $y$, the loss is $L = -\sum_i y_i \log \hat{y}_i$. Differentiating gives the well-known simplification
\[
\frac{\partial L}{\partial z_i} = \hat{y}_i - y_i.
\]

\bigskip
\hrule
Collaborators: discussed Problem 2 with Mara Okafor during office hours.

\end{document}

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