Smooth, globally Polyak-{\L}ojasiewicz functions are nonlinear least-squares

TL;DR

The paper proves that smooth, globally Polyak-Łojasiewicz functions on contractible manifolds are essentially sums of squares, revealing a rigid structure and dichotomy in their geometry and convexity properties.

Contribution

It establishes that such functions must be of a specific sum-of-squares form and characterizes the geometric nature of their minimizer sets, highlighting a fundamental rigidity.

Findings

Functions are sums of squares of submersions.

Minimizer sets are either Euclidean or exotic manifolds.

There exists a Riemannian metric making the function geodesically convex.

Abstract

The Polyak-{\L}ojasiewicz (P{\L}) condition is often invoked in nonconvex optimization because it allows fast convergence of algorithms beyond strong convexity. A function $f \colon \mathcal{M} \to \mathbb{R}$ on a Riemannian manifold $\mathcal{M}$ is globally P{\L} if $\|\nabla f(x)\|^2 \geq 2\mu(f(x) - f^*)$ for all $x$, where $f^* = \inf f$ and $\mu > 0$. How much does this pointwise, first-order inequality constrain $f$ and its set of minimizers $S$? We show that if $f$ is also smooth ($C^\infty$) and $\mathcal{M}$ is contractible (e.g., if $\mathcal{M} = \mathbb{R}^n$), then the P{\L} condition imposes a firm global structure: such a function is necessarily of the form $f(x) = f^* + \|\varphi(x)\|^2$ (a nonlinear sum of squares) where $\varphi \colon \mathcal{M} \to \mathbb{R}^k$ is a submersion, and $k$ is the codimension of $S$ in $\mathcal{M}$. The proof hinges on showing that the end-point map of negative gradient flow on $f$ is a trivial smooth fiber bundle over $S$. This rigidity leads to a striking dichotomy. Either $S$ is diffeomorphic to a Euclidean space, in which case $f$ can be transformed into a convex quadratic by a smooth change of coordinates. Or $S$ must display genuinely exotic geometry; for example, it can be diffeomorphic to the Whitehead manifold. As a further consequence, we show that there exists a complete Riemannian metric on $\mathcal{M}$ under which $f$ remains P{\L} and becomes geodesically convex.