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

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∘ measures μϵ∝exp(−V/ϵ), 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 C2 embedding submanifold of Rd 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 ϵ-independent lower bound for the Poincaré constant in the Poincaré inequality of μϵ. As a direct consequence, Langevin dynamics with such non-convex potential V and diffusion coefficient ϵ converges to its equilibrium μϵ at a rate of Õ (1/ϵ), provided ϵ is sufficiently small. Here Õ  hides logarithmic terms.