Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials

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).