New insights in smoothness and strong convexity with improved convergence of gradient descent

TL;DR

This paper revisits the fundamental properties of smoothness and strong convexity, offering new perspectives that lead to improved linear convergence rates for gradient descent methods.

Contribution

It introduces novel insights into smoothness and strong convexity, resulting in enhanced convergence guarantees for gradient descent.

Findings

Revised characterizations of smoothness and strong convexity

Improved linear convergence rates for gradient descent

Deeper understanding of convergence conditions

Abstract

The starting assumptions to study the convergence and complexity of gradient-type methods may be the smoothness (also called Lipschitz continuity of gradient) and the strong convexity. In this note, we revisit these two basic properties from a new perspective that motivates their definitions and equivalent characterizations, along with an improved linear convergence of the gradient descent method.