A Proof of the Exact Convergence Rate of Gradient Descent

Jungbin Kim

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arXiv · 2412.04427

A Proof of the Exact Convergence Rate of Gradient Descent

Jungbin Kim

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Abstract

We prove the exact worst-case convergence rate of gradient descent for smooth strongly convex optimization on $\mathbb{R}^d$. Concretely, assuming that the objective function $f$ is $\mu$-strongly convex and $L$-smooth, we identify the smallest possible value of $\tau$ for which the inequality $f(x_{N})-f_{*}\leq\tau\|x_{0}-x_{*}\|^{2}$ always holds. The result was previously conjectured by Drori and Teboulle for the case $\mu=0$, and by Taylor, Hendrickx, and Glineur for the case $\mu>0$.

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Jungbin Kim. 2024-12-05. A Proof of the Exact Convergence Rate of Gradient Descent. https://arxiv.org/abs/2412.04427

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