A Proof of Exact Convergence Rate of Gradient Descent. Part I. Performance Criterion $\Vert \nabla f(x_N)\Vert^2/(f(x_0)-f_*)$

TL;DR

This paper establishes the precise worst-case convergence rate of gradient descent for smooth strongly convex functions using a specific performance criterion, employing the performance estimation methodology.

Contribution

It provides a novel proof of the exact convergence rate, differing from previous work, and applies the performance estimation methodology to this problem.

Findings

Exact worst-case convergence rate derived

Proof differs from previous methods by Rotaru et al.

Utilizes performance estimation methodology for analysis

Abstract

We prove the exact worst-case convergence rate of gradient descent for smooth strongly convex optimization, with respect to the performance criterion $\Vert \nabla f(x_N)\Vert^2/(f(x_0)-f_*)$. The proof differs from the previous one by Rotaru \emph{et al.} [RGP24], and is based on the performance estimation methodology [DT14].