\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: draft.}
\title{Smooth and strongly convex function}

\begin{document}

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

\begin{frame}
  \frametitle{Strong convexity VS smoothness}

  \begin{columns}
    \begin{column}{0.5\linewidth}
      $\mu$-strong convexity
      \[
        f(y) \geq f(x) + \dotp{\nabla f(y)}{y-x} + \frac{\mu}{2}\norm{x - y}^{2}
      \]
    \end{column}
    \begin{column}{0.5\linewidth}
      $L$-smoothness
      \[
        f(y) \leq f(x) + \langle \nabla f(x), y-x \rangle + \frac{L}{2} \| x-y \|^{2}
      \]
    \end{column}
  \end{columns}
  \begin{center}
    \begin{tikzpicture}
      % Axis
      \draw[->] (-6,0) -- (6,0);
      \draw[->] (0,-0.1) -- (0,4);
    \end{tikzpicture}
    {\tiny (live)}
  \end{center}
\end{frame}

\end{document}