2025/02/08 by Gong Yun, Gong, Yun, Zebang Shen +3
Economics, Econometrics and Finance · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Probability (math.PR) #Spectral Theory in Mathematical Physics #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2502.06862
openalex publication_date 2025/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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∝ 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 \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 O(1/ε), provided ε is sufficiently small. Here O hides logarithmic terms.