Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials

TL;DR

This paper establishes a deep connection between the optimization landscape of functions via Gradient Flow and the sampling properties of associated Gibbs measures, leading to new sampling algorithms for complex distributions.

Contribution

It proves that optimizability from all initializations implies Poincaré and Log-Sobolev inequalities for Gibbs measures, enabling efficient sampling of non-log-concave densities.

Findings

Gradient Flow optimizability implies Poincaré inequalities for Gibbs measures.

Under mild conditions, Gibbs measures satisfy Log-Sobolev inequalities at low temperature.

New efficient sampling methods for non-log-concave densities are derived.

Abstract

In this paper, we prove that optimizability of any function F using Gradient Flow from all initializations implies a Poincar\'e Inequality for Gibbs measures mu_{beta} = e^{-beta F}/Z at low temperature. In particular, under mild regularity assumptions on the convergence rate of Gradient Flow, we establish that mu_{beta} satisfies a Poincar\'e Inequality with constant O(C'+1/beta) for beta >= Omega(d), where C' is the Poincar\'e constant of mu_{beta} restricted to a neighborhood of the global minimizers of F. Under an additional mild condition on F, we show that mu_{beta} satisfies a Log-Sobolev Inequality with constant O(beta max(S, 1) max(C', 1)) where S denotes the second moment of mu_{beta}. Here asymptotic notation hides F-dependent parameters. At a high level, this establishes that optimizability via Gradient Flow from every initialization implies a Poincar\'e 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 mu_{beta} satisfies a Weak Poincar\'e Inequality with parameters (O(C'+1/beta), O(mu_{beta}(S))) for \beta = Omega(d). At a high level, this shows while optimizability from 'most' initializations implies a Weak Poincar\'e 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).