Poincare Inequality for Local Log-Polyak-Łojasiewicz Measures: Non-asymptotic Analysis in Low-temperature Regime

Abstract

Potential functions in highly pertinent applications, such as deep learning in over-parameterized regime, are empirically observed to admit non-isolated minima. To understand the convergence behavior of stochastic dynamics in such landscapes, we propose to study the class of log-PŁ${}^{\circ }$ measures ${\mu }_{\epsilon }\propto \exp (-V/\epsilon )$, where the potential $V$ satisfies a local Polyak-Łojasiewicz (PŁ) inequality, and its set of local minima is provably connected. Notably, potentials in this class can exhibit local maxima and we characterize its optimal set $S$ to be a compact $C^2$ embedding submanifold of $R^d$ without boundary. The non-contractibility of $S$ distinguishes our function class from the classical convex setting topologically. Moreover, the embedding structure induces a naturally defined Laplacian-Beltrami operator on $S$, and we show that its first non-trivial eigenvalue provides an $\epsilon$-independent lower bound for the Poincaré constant in the Poincaré inequality of ${\mu }_{\epsilon }$. As a direct consequence, Langevin dynamics with such non-convex potential $V$ and diffusion coefficient $\epsilon$ converges to its equilibrium ${\mu }_{\epsilon }$ at a rate of $\tilde{O}(1/\epsilon )$, provided $\epsilon$ is sufficiently small. Here $\tilde{O}$ hides logarithmic terms.

29 pages; This is a shorter version that contains only the 2-local-PŁ case. For results in the more general $\alpha$-PL case, $\alpha <=2$, please refer to the previous version; Previously this version appeared as arXiv:2502.06862 which was submitted as a new work by accident;