Polyak-{\L}ojasiewicz inequality is essentially no more general than strong convexity for $C^2$ functions

TL;DR

This paper demonstrates that for twice-differentiable functions, the Polyak-Łojasiewicz inequality essentially coincides with strong convexity, limiting its broader applicability to nonsmooth functions.

Contribution

It proves that smooth P{ }L functions with bounded minimizers are essentially strongly convex, clarifying the relationship between P{ }L inequality and strong convexity for $C^2$ functions.

Findings

P{ }L functions with bounded minimizers are strongly convex on certain sublevel sets.

The result aligns P{ }L inequality with strong convexity in the smooth case.

Implications for the assumptions used in optimization literature.

Abstract

The Polyak-{\L}ojasiewicz (P{\L}) inequality extends the favorable optimization properties of strongly convex functions to a broader class of functions. In this paper, we prove a theorem (also obtained by Criscitiello, Rebjock and Boumal in an earlier blog post) showing that the richness of the class of P{\L} 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 $C^2$ P{\L} function having a bounded set of minimizers, then it has a unique minimizer and is strongly convex on a sublevel set of the form $\{f\leq a\}$. We show that this implies a result of Asplund on properties of the squared distance function, and discuss some consequences on smoothness assumptions in results in the literature.