Local and Global Uniform Convexity Conditions

TL;DR

This paper reviews uniform convexity and smoothness in finite-dimensional spaces, connecting geometric properties with optimization convergence analysis, and introduces local conditions to improve complexity bounds in learning and optimization.

Contribution

It establishes local versions of uniform convexity conditions, enhancing the understanding of their role in optimization complexity and providing practical examples in machine learning.

Findings

Local convexity conditions lead to sharper complexity bounds.

Connections between geometry of Banach spaces and optimization analysis.

Practical examples demonstrate benefits in machine learning.

Abstract

We review various characterizations of uniform convexity and smoothness on norm balls in finite-dimensional spaces and connect results stemming from the geometry of Banach spaces with \textit{scaling inequalities} used in analysing the convergence of optimization methods. In particular, we establish local versions of these conditions to provide sharper insights on a recent body of complexity results in learning theory, online learning, or offline optimization, which rely on the strong convexity of the feasible set. While they have a significant impact on complexity, these strong convexity or uniform convexity properties of feasible sets are not exploited as thoroughly as their functional counterparts, and this work is an effort to correct this imbalance. We conclude with some practical examples in optimization and machine learning where leveraging these conditions and localized assumptions lead to new complexity results.