# Polyak-Lojasiewicz inequality is essentially no more general than strong convexity for C2 functions

arXiv: https://arxiv.org/abs/2512.05285 · HTML: https://mlsanity.com/paper/2512.05285/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Dec 2025.

## Built on by (most cited first, 2 of 2)

- [CogGen: Cognitive-Load-Inspired Fully Unsupervised Deep Generative Modeling for Compressively Sampled MRI Reconstruction](https://mlsanity.com/paper/2603.04438/) (2026)
- [Optimal Asymptotic Rates for (Stochastic) Gradient Descent under the Local PL-Condition: A Geometric Approach](https://mlsanity.com/paper/2605.14663/) (2026)

## Abstract (arXiv)

The Polyak-Lojasiewicz (PL) 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 PL 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 C2 PL 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≤ 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.