Poincar'e Inequality for Local Log-Polyak-Lojasiewicz 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 logPLmeasure measures με propto exp(-V/ε), where the potential V satisfies a local Polyak-Lojasiewicz (PL) 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 Poincare constant in the Poincare 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 tildeO(1/ε), provided ε is sufficiently small. Here tildeO hides logarithmic terms.