Polyak-Łojasiewicz inequality is essentially no more general than strong convexity for $C^2$ functions

TL;DR

This work shows that for $f\in C^2(\mathbb{R}^n)$ satisfying a global Polyak-Łojasiewicz inequality with a bounded minimizer set, the minimizer is unique and the function is strongly convex on a neighborhood of that minimizer. The authors achieve this by proving the minimizer set is contractible and a closed submanifold, then using Hessian-rank arguments to force a zero-dimensional minimizer set. They also extend gradient-flow analysis to general PŁ functions without assuming compact minimizers and connect distance-function regularity to convex-analytic structure, via Asplund-type results. Overall, the paper delineates a sharp boundary: in the smooth regime, PŁ does not yield broader minimizer geometry than strong convexity, tying optimization dynamics to topological and geometric constraints.

Abstract

The Polyak-Łojasiewicz (PŁ) inequality extends the favorable optimization properties of strongly convex functions to a broader class of functions. In this paper, we show that the richness of the class of PŁ functions is rooted in the nonsmooth case since sufficient regularity forces them to be essentially strongly convex. More precisely, we prove that if $f$ is a $C^2$ PŁ function having a bounded set of minimizers, then it has a unique minimizer and is strongly convex on a sublevel set of the form $\{f\leq a\}$.

Theorems & Definitions (22)

• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Theorem 2.8
• Lemma 3.1
• Lemma 3.2