Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials

Abstract

In this paper, we prove that optimizability of any function F using Gradient Flow from all initializations implies a Poincaré Inequality for Gibbs measures μ_β∝ e^-β F at low temperature. In particular, under mild regularity assumptions on the convergence rate of Gradient Flow, we establish that μ_β satisfies a Poincaré Inequality with constant O(C’) for β≥Ω(d), where C’ is the Poincaré constant of μ_β restricted to a neighborhood of the global minimizers of F. Under an additional mild condition on F, we show that μ_β satisfies a Log-Sobolev Inequality with constant O(S β C’) where S denotes the second moment of μ_β. Here asymptotic notation hides F-dependent parameters. At a high level, this establishes that optimizability via Gradient Flow from every initialization implies a Poincaré and Log-Sobolev Inequality for the low-temperature Gibbs measure, which in turn imply sampling from all initializations. Analogously, we establish that under the same assumptions, if F can be initialized from everywhere except some set 𝒮, then μ_β satisfies a Weak Poincaré Inequality with parameters (O(C’), O(μ_β(𝒮))) for β≥Ω(d). At a high level, this shows while optimizability from ‘most’ initializations implies a Weak Poincaré Inequality, which in turn implies sampling from suitable warm starts. Our regularity assumptions are mild and as a consequence, we show we can efficiently sample from several new natural and interesting classes of non-log-concave densities, an important setting with relatively few examples. As another corollary, we obtain efficient discrete-time sampling results for log-concave measures satisfying milder regularity conditions than smoothness, similar to Lehec (2023).