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\author{Samuel Vaiter}
\date{}
\institute{
Created: 2023-11-10.\\
Last update: 2023-11-10.\\
Status: inprogress.}
\title{Constant stepsize gradient descent on smooth strongly convex function}

\begin{document}

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  \titlepage
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\begin{frame}
  \frametitle{Strongly convex $L$-smooth function}

  \begin{proposition}
    Let $f: \RR^{d} \to \RR$ be a \alert{$\mu$-strongly convex $L$-smooth} function.
    For any $x^{(0)}$, the GD iterates $x^{(t)}$ with $t^{(t)} = \eta \in (0,\frac{2}{\mu + L}]$ converge towards \alert{the optimal point} of $f$
    \[
      \norm{x^{(t)} - x^{\star}}^{2} \leq
      \left( 1 - \frac{2 \eta \mu L}{\mu + L} \right)^{t} \norm{x^{(0)} - x^{\star}}^{2} .
    \]
    Moreover, if $\eta = \frac{2}{\mu + L}$, then (with $K_{f} = L / \mu$)
    \begin{align*}
      \norm{x^{(t)} - x^{\star}}^{2} & \leq
      \left( \frac{K_{f} - 1}{K_{f} + 1} \right)^{2t} \norm{x^{(0)} - x^{\star}}^{2} \\
      f(x^{(t)}) - f^{\star} &\leq
      \frac{L}{2} \left( \frac{K_{f} - 1}{K_{f} + 1} \right)^{2t} \norm{x^{(0)} - x^{\star}}^{2} .
    \end{align*}
  \end{proposition}
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\end{document}