\documentclass[t,aspectratio=169]{beamer}
\usetheme{metropolis}
\metroset{block=fill}

\input{../../_assets/basicslide1}
\input{../../_assets/macros}

\author{Samuel Vaiter}
\date{}
\institute{
Created: 2023-11-08.\\
Last update: 2023-11-08.\\
Status: inprogress.}
\title{Smooth function \emph{a.k.a.} function with Lipschitz gradient}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}
  \frametitle{Lipschitz continuity}

  \begin{definition}
    $\phi: \Omega \subseteq E \to F$ is $L$-Lipschitz continuous if
    \[
      \forall x, y \in \Omega, \quad
      \norm{\phi(x) - \phi(y)}_{E} \leq L \norm{x - y}_{F} .
    \]
  \end{definition}

  \begin{center}
    \begin{tikzpicture}
      % Axis
      \draw[->] (-4,0) -- (4,0);
      \draw[->] (0,-0.1) -- (0,3);

      \draw (0,0.5) -- (3,2);
      \draw (0,0.5) to [bend left] (-4,-1);
    \end{tikzpicture}
    {\tiny (live)}
  \end{center}
\end{frame}

\begin{frame}
  \frametitle{Smooth function: quadratic upper-bound}

  \faWarning\, ``smooth'' refers to a lot of different concept in mathematics.

  \begin{definition}
  A differentiable function $f: \mathbb{R}^{n} \to \mathbb{R}$ is said to be \(L\)-smooth if $\nabla f$ is \(L\)-Lipschitz
  \[
    \forall x, y \in \dom f, \quad
    \| \nabla f(x) - \nabla f(y) \| \leq L \| x - y \| .
  \]
\end{definition}
\end{frame}

\begin{frame}
  \frametitle{Smooth function: quadratic upper-bound}

  $L$-smooth: $\forall x, y \in \dom f, \| \nabla f(x) - \nabla f(y) \| \leq L \| x - y \| .$

  \begin{proposition}
    Let $f$ a \(L\)-smooth function.
    Then, for all $x, y \in \mathrm{dom}f$,
    \begin{equation}\label{eq:lsmooth-cs}
      \langle \nabla f(x) - \nabla f(y), x-y \rangle \leq L \|x-y\|^{2} .
    \end{equation}
    Moreover, if $\mathrm{dom}f$ is convex, then \eqref{eq:lsmooth-cs} is equivalent to
    \begin{equation}\label{eq:lsmooth-upper}
      f(y) \leq f(x) + \langle \nabla f(x), y-x \rangle + \frac{L}{2} \| x-y \|^{2},
      \quad
      \forall x, y \in \mathrm{dom}f .
    \end{equation}
  \end{proposition}
\end{frame}

\end{document}