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

TL;DR

The paper addresses convergence of Langevin dynamics in nonconvex landscapes with nonisolated minima by introducing Log-PL$^\circ$ measures, where the global minimizers form a compact $C^2$ embedding submanifold $S$ without boundary. It establishes a temperature-independent lower bound on the Poincaré constant in terms of the first nonzero eigenvalue $\lambda_1(S)$ of the Laplacian-Beltrami operator on $S$, yielding a sub-exponential $\tilde{O}(1/\epsilon)$ convergence rate for small $\epsilon$. The proof advances in two steps: first reducing the PI constant to a Neumann eigenvalue problem on a tubular neighborhood $U=S^{\sqrt{C\epsilon}}$ via a Lyapunov framework, then showing spectral stability that ties $\lambda_1^{n}(U)$ to $\lambda_1(S)$. This work connects geometric spectral theory on $S$ with stochastic sampling dynamics, offering a novel route to analyze nonconvex, nonisolated minima in high-dimensional settings and informing the design of efficient sampling algorithms in the 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 $μ_ε\propto \exp(-V/ε)$, 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 ${C}^2$ 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 $ε$-independent lower bound for the Poincaré constant in the Poincaré inequality of $μ_ε$. As a direct consequence, Langevin dynamics with such non-convex potential $V$ and diffusion coefficient $ε$ converges to its equilibrium $μ_ε$ at a rate of $\tilde{O}(1/ε)$, provided $ε$ is sufficiently small. Here $\tilde{O}$ hides logarithmic terms.

Figures (1)

• Figure 1: The circle in (a) can be represented using two local charts (blue and green). Using the tubular neighborhood theorem, in a local region of $U$ (outlined with the red dashed line) we transform the uniform measure to a pair of decoupled measures on the tangent and normal directions. (a) Uniform measure $\mu_U$ (over $(x, y)$) under the Cartisian coordinate $(x, y)$; (b) Uniform measure $\mu_U$ (over $(x, y)$) under the local coordinate $(\theta, r)$; (c) Uniform measure (over $(\theta, r)$) under the local coordinate $(\theta, r)$. Importantly, when the radius of the tubular neighborhood is small, the densities in (b) and (c) point-wisely control each other.

Theorems & Definitions (28)

• Theorem 1: informal
• Definition 1: Poincaré-Wirtinger Inequality
• Proposition 1
• Proposition 2: Holley-Stroock perturbation principle
• Definition 2: Embedding submanifold in $\mathbb{R}^d$
• Remark 1
• Definition 3: Neumann eigenvalue
• Definition 4: Eigenvalue of the Laplacian-Beltrami operator
• Remark 2
• Remark 3: Dimension of ${{S}^{}}$