Add SMM gradient descent

src/ainotes.cls

@@ -17,6 +17,7 @@
 \usepackage{scrhack, algorithm, listings}
 \usepackage{array, makecell}
 \usepackage{acro}
+\usepackage{subcaption}
 
 \geometry{ margin=3cm, lmargin=1.5cm, rmargin=4.5cm, marginparwidth=3cm }
 \hypersetup{ colorlinks, citecolor=black, filecolor=black, linkcolor=black, urlcolor=black, linktoc=all }
@@ -55,6 +56,7 @@
 
 \theoremstyle{definition}
 \newtheorem{theorem}{Theorem}[section]
+\newtheorem{corollary}{Corollary}[theorem]
 \newtheorem*{example}{Example}
 \theoremstyle{definition}
 \newtheorem*{definition}{Def}

src/statistical-and-mathematical-methods-for-ai/sections/_gradient_methods.tex

@@ -26,6 +26,8 @@ Let $f: \mathbb{R}^N \rightarrow \mathbb{R}$ be continuous and differentiable in
         \[ f(\vec{x}^*) < f(\vec{x}) \text{ } \forall \vec{x} \in \mathbb{R}^N \]
 \end{descriptionlist}
 
+Note that $\max f(x) = \min -f(x)$.
+
 
 \subsection{Optimality conditions}
 
@@ -109,7 +111,7 @@ Note: descent methods usually converge to a local minimum.
         \begin{lstlisting}[mathescape=true, belowskip = -0.8\baselineskip]
             def backtracking($\tau$, $c_1$):
                 $\alpha_k$ = 1 # Initial guess
-                while $f(x_k - \alpha_k \nabla f(\vec{x}_k))$ > $f(\vec{x}_k)$ + $c_1 \alpha_k \nabla f(\vec{x}_k)^T \nabla f(\vec{x}_k)$:
+                while $f(x_k + \alpha_k \nabla f(\vec{x}_k))$ > $f(\vec{x}_k)$ + $c_1 \alpha_k \nabla f(\vec{x}_k)^T \nabla f(\vec{x}_k)$:
                     $\alpha_k$ = $\alpha_k$ / $\tau$
                 return $\alpha_k$
         \end{lstlisting}
@@ -137,3 +139,202 @@ A generic gradient-like method can then be defined as:
             $k$ = $k$ + 1
         return $x_k$
 \end{lstlisting}
+
+
+\subsection{Problems}
+
+\begin{description}
+    \item[Choice of the initialization point] \marginnote{Initialization point}
+        The starting point of an iterative method is a user defined parameter.
+        For simple problems, it is usually chosen randomly in $[-1, +1]$.
+        
+        For complex problems, the choice of the initialization point is critical as 
+        it may cause numerical instabilities or bad results.
+        Heuristics can be used to select an adequate starting point.
+
+    \item[Flag regions and local optima] \marginnote{Flag regions and local optima}
+        Flat regions slow down the learning speed,
+        while a local optima causes the method to converge at a poor solution.
+        \begin{figure}[ht]
+            \centering
+            \includegraphics[width=0.9\textwidth]{img/_descent_local_flat.pdf}
+            \caption{Flat regions and local minima}
+        \end{figure}
+    
+    \item[Differential curvature]
+        Different magnitudes of the partial derivatives may cause the problem of
+        vanishing and exploding gradient. \marginnote{Vanishing gradient\\Exploding gradient}
+        This causes the learning process to require more iterations to correct the direction.
+
+        In practice, as the gradient of complex functions is only an instantaneous direction of best decrease and
+        does not represent the direction to the minimum in the long term, 
+        many updates are required for a gradient method to converge.
+
+        A method to mitigate this issue is to use feature normalization techniques.
+
+    \item[Non-differentiable objective function]
+        If the objective function has a small number of non-differentiable points,
+        the gradient descent method can be applied with minor modifications.
+        
+        If lots of points are non-differentiable, the gradients will not be informative enough 
+        to determine a decrease direction.
+
+    \item[Difficult topologies]
+        \marginnote{Cliff}
+        A cliff in the objective function causes problems when evaluating the gradient at the edge.
+        With a small step size, there is a slow down in convergence. 
+        With a large step size, there is an overshoot that may cause the algorithm to diverge.
+        % a slow down when evaluating 
+        % the gradient at the edge using a small step size and 
+        % an overshoot when the step is too large.
+
+        \marginnote{Valley}
+        A valley in the objective function causes a gradient method to bounce between the sides
+        to a point where no significant progress can be made.
+
+        \begin{figure}[ht]
+            \begin{subfigure}{.5\textwidth}
+                \centering
+                \includegraphics[width=.30\linewidth]{img/cliff.png}
+                \caption{Cliff region}
+            \end{subfigure}%
+            \begin{subfigure}{.5\textwidth}
+                \centering
+                \includegraphics[width=.30\linewidth]{img/valley.png}
+                \caption{Ping pong tournament in a valley}
+            \end{subfigure}
+        \end{figure}
+\end{description}
+
+
+
+\section{Convex functions}
+
+\begin{description}
+    \item[Convex set] \marginnote{Convex set}
+        Informally, a set is convex if, for any two points of the set,
+        the points laying on the segment connecting them are also part of the set.
+
+        \begin{figure}[ht]
+            \begin{subfigure}{.5\textwidth}
+                \centering
+                \includegraphics[width=.25\linewidth]{img/convex_set.png}
+                \caption{Convex set}
+            \end{subfigure}%
+            \begin{subfigure}{.5\textwidth}
+                \centering
+                \includegraphics[width=.25\linewidth]{img/non_convex_set.png}
+                \caption{Non-convex set}
+            \end{subfigure}
+        \end{figure}
+
+    \item[Convex function] \marginnote{Convex function}
+        Let $\Omega \subseteq \mathbb{R}^n$ be a convex set and $f: \Omega \rightarrow \mathbb{R}$.
+        $f$ is convex if:
+        \[ 
+            \forall \vec{x}_1, \vec{x}_2 \in \Omega, \forall t \in [0, 1]: 
+                f(t\vec{x}_1 + (1-t)\vec{x}_2) \leq t f(\vec{x}_1) + (1-t) f(\vec{x}_2)
+        \]
+
+        In other words, the segment connecting two points of the function lays above the graph.
+        \begin{figure}[ht]
+            \centering
+            \includegraphics[width=0.55\textwidth]{img/convex_function.png}
+            \caption{Convex function}
+        \end{figure}
+
+    \item[Strictly convex function] \marginnote{Strictly convex function}
+        Let $\Omega \subseteq \mathbb{R}^n$ be a convex set and $f: \Omega \rightarrow \mathbb{R}$.
+        $f$ is strictly convex if:
+        \[ 
+            \forall \vec{x}_1, \vec{x}_2 \in \Omega, \forall t \in [0, 1]: 
+                f(t\vec{x}_1 + (1-t)\vec{x}_2) < t f(\vec{x}_1) + (1-t) f(\vec{x}_2)
+        \]
+\end{description}
+
+
+\subsection{Properties}
+\marginnote{Convex properties}
+\begin{itemize}
+    \item $\text{if } f \text{ convex} \Rightarrow \text{ any local minimum of } f \text{ is also global}$
+    \item $\text{if } f \text{ strictly convex} \Rightarrow \text{ the global minimum of } f \text{ is unique}$
+    \item $\text{if } f \text{ convex and differentiable} \Rightarrow \text{ any stationary point of } f \text{ is a global minimum}$
+\end{itemize}
+
+
+\subsection{Quadratic functions}
+\marginnote{Quadratic function}
+A quadratic function has form:
+\[ f(\vec{x}) = \frac{1}{2}\vec{x}^T\matr{A}\vec{x} - \vec{x}^T\vec{b} + c \]
+where $\matr{A} \in \mathbb{R}^{n \times n}$, $\vec{b} \in \mathbb{R}^n$ and $c \in \mathbb{R}$.
+
+\begin{theorem}
+    If $f$ is a quadratic form with $\matr{A} \in \mathbb{R}^{n \times n}$ symmetric positive semidefinite,
+    then $f$ is convex.
+\end{theorem}
+
+\begin{theorem}
+    If $f$ is a quadratic form with $\matr{A} \in \mathbb{R}^{n \times n}$ symmetric positive definite,
+    then $f$ is strictly convex.
+\end{theorem}
+
+\begin{theorem}
+    \marginnote{Least squares quadratic function}
+    The least squares problem $\Vert \matr{A}\vec{x} - \vec{b} \Vert_2^2$ is a quadratic function.
+\end{theorem}
+\begin{proof}
+    \[
+        \begin{split}
+            (\matr{A}\vec{x} - \vec{b})^T(\matr{A}\vec{x} - \vec{b}) &= (\vec{x}^T\matr{A}^T - \vec{b}^T)(\matr{A}\vec{x} - \vec{b}) \\
+                &= \vec{x}^T\matr{A}^T\matr{A}\vec{x} - \vec{b}^T\matr{A}\vec{x} - \vec{x}^T\matr{A}^T\vec{b} + \vec{b}^T\vec{b} \\
+        \end{split}
+    \]
+    As $\vec{b}^T\matr{A}\vec{x} = \vec{x}^T\matr{A}^T\vec{b}$, we have:
+    \[ \vec{x}^T\matr{A}^T\matr{A}\vec{x} - 2\vec{x}^T\matr{A}^T\vec{b} + \vec{b}^T\vec{b} \]
+
+    Let $\matr{B} = \matr{A}^T\matr{A}$, $\vec{q} = \matr{A}^T\vec{b}$ and $c = \vec{b}^T\vec{b}$, 
+    we have the quadratic form:
+    \[ \vec{x}^T\matr{B}\vec{x} - 2\vec{x}^T\vec{q} + c \]
+
+    $\matr{B}$ is symmetric positive semidefinite (i.e. $\Vert \matr{A}\vec{x} - \vec{b} \Vert_2^2$ is convex).
+    Moreover, when $\matr{A}$ is full-rank, $\matr{B}$ is symmetric positive definite (i.e. strictly convex).
+\end{proof}
+
+
+
+\section{Gradient descent with momentum}
+\marginnote{Momentum}
+The momentum is an additional term to keep track of previous iterations:
+\[
+    \Delta \vec{x}_k = \vec{x}_k - \vec{x}_{k-1} = \gamma \Delta \vec{x}_{k-1} - \alpha_{k-1}\nabla f(\vec{x}_{k-1})
+\]
+where $\gamma \in [0, 1]$. An iteration is therefore defined as:
+\[
+    \vec{x}_k = \vec{x}_{k-1} - \alpha_{k-1}\nabla f(\vec{x}_{k-1}) + \gamma \Delta\vec{x}_{k-1}
+\]
+
+
+
+\section{Stochastic gradient descent (SGD)}
+\marginnote{Stochastic gradient descent}
+SGD is a stochastic approximation of gradient descent that uses an approximation of the gradient.
+Given $N$ data points, the loss can be defined as the sum of the individual losses:
+\[ L(\vec{x}) = \sum_{n=1}^{N} L_n(\vec{x}) \]
+where $\vec{x}$ is the vector of parameters.
+The corresponding gradient can be computed as:
+\[ \nabla L(\vec{x}) = \sum_{n=1}^{N} \nabla L_n(\vec{x}) \]
+
+\marginnote{Mini-batch}
+SGD reduces the amount of computation by approximating the gradient with a subset (mini-batch) $B$ of $\nabla L_n$:
+\[ \nabla L(\vec{x}) = \sum_{i \in B} \nabla L_i(\vec{x}) \]
+
+\begin{theorem}
+    Under some assumptions and with an appropriate decrease in learning rate, 
+    SGD is guaranteed to converge to a local minimum.
+\end{theorem}
+
+Different sizes of the mini-batch result in different behavior:
+\begin{descriptionlist}
+    \item[Large mini-batches] accurate estimates of the gradient.
+    \item[Small mini-batches] faster computation.
+\end{descriptionlist}