From ea523909108af4733c461f4cfa3cefa1ba00a1fe Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Wed, 4 Oct 2023 09:52:25 +0200 Subject: [PATCH] Add SMM descent methods --- src/ainotes.cls | 3 +- .../img/_gradient_contour.pdf | Bin 0 -> 467964 bytes .../main.tex | 1 + .../sections/_gradient_methods.tex | 139 ++++++++++++++++++ 4 files changed, 142 insertions(+), 1 deletion(-) create mode 100644 src/statistical-and-mathematical-methods-for-ai/img/_gradient_contour.pdf create mode 100644 src/statistical-and-mathematical-methods-for-ai/sections/_gradient_methods.tex diff --git a/src/ainotes.cls b/src/ainotes.cls index 0323a80..b41a4ae 100644 --- a/src/ainotes.cls +++ b/src/ainotes.cls @@ -39,7 +39,8 @@ showspaces = false, showstringspaces = true, showtabs = false, - tabsize = 3 + tabsize = 3, + belowskip = -0.8\baselineskip } \lstset{style=mystyle} \lstset{language=Python} diff --git 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\vec{p}) < f(\vec{x}) \] +\end{description} + +\begin{theorem} + Let $\vec{p} \in \mathbb{R}^N$, $\vec{p} \neq \nullvec$. + \[ \text{If } \vec{p}^T \nabla f(\vec{x}) < 0 \Rightarrow \vec{p} \text{ descent direction of } f \text{ in } x \] +\end{theorem} + +\begin{theorem} + For all $\vec{x}$, $\vec{p} = -\nabla f(\vec{x})$ is a descent direction of $f$ in $x$. +\end{theorem} +\begin{proof} + \[ + \begin{split} + \vec{p}^T \nabla f(\vec{x}) < 0 &\iff -(\nabla f(\vec{x}))^T \nabla f(\vec{x}) < 0 \\ + &\iff - \Vert \nabla f(\vec{x}) \Vert_2^2 < 0 + \end{split} + \] + This holds as the norm is always positive. +\end{proof} + +\begin{description} + \item[Gradient-like methods] \marginnote{Gradient-like methods} + Gradient-like methods are descent methods that use $-\nabla f$ as step. +\end{description} + + +\subsection{Choice of the step length} +\begin{description} + \item[Constant] + In machine learning, it is common to set a constant value for the step (learning rate), + but it can be proved that this does not guarantee convergence. + + \item[Backtracking procedure] \marginnote{Backtracking procedure} + $\alpha_k$ is chose such that it respects the Wolfe condition\footnote{\url{https://en.wikipedia.org/wiki/Wolfe_conditions}}: + \begin{lstlisting}[mathescape=true] + def backtracking($\tau$, $c_1$): + $\alpha_k$ = 1 # Initial guess + while $f(x_k - \alpha_k \nabla f(\vec{x}_k))$ > $f(\vec{x}_k)$ + $c_1 \alpha_k \nabla f(\vec{x}_k)^T \nabla f(\vec{x}_k)$: + $\alpha_k$ = $\alpha_k$ / $\tau$ + return $\alpha_k$ + \end{lstlisting} + It can be proved that, by using the backtracking procedure, gradient methods converge to a local minimum. +\end{description} + + +\subsection{Stopping condition} +\marginnote{Stopping condition} +We can stop iterating when $\vec{x}_k \approx \vec{x}^*$, that is, $\nabla f(\vec{x}_k) \approx \nullvec$. +We can verify this by checking the norm of the gradient against a tolerance $\tau$: +\begin{descriptionlist} + \item[Absolute condition] $\Vert \nabla f(x_k) \Vert_2 < \tau$ + \item[Relative condition] $\frac{\Vert \nabla f(x_k) \Vert_2}{\Vert \nabla f(x_0) \Vert_2} < \tau$ +\end{descriptionlist} + +A generic gradient-like method can then be defined as: +\begin{lstlisting}[mathescape=true] + def gradientMethod($f$, $\vec{x}_0$): + $k$ = 0 + while stoppingCondition($f$, $\vec{x}_k$, $\vec{x}_0$): + $p_k$ = $-\nabla f(\vec{x}_k)$ + $\alpha_k$ = backtracking($\dots$) + $\vec{x}_{k+1}$ = $\vec{x}_k$ + $\alpha_k \vec{p}_k$ + $k$ = $k$ + 1 + return $x_k$ +\end{lstlisting} \ No newline at end of file