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-P{\L}$^\circ$ measures $\mu_\epsilon \propto \exp(-V/\epsilon)$, where the potential $V$ satisfies a local Polyak-{\L}ojasiewicz (P{\L}) 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 Poincar\'e constant in the Poincar\'e 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.