# Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials

arXiv: https://arxiv.org/abs/2410.02979 · HTML: https://mlsanity.com/paper/2410.02979/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Oct 2024.

## Built on by (most cited first, 1 of 1)

- [On the Robustness of Langevin Dynamics to Score Function Error](https://mlsanity.com/paper/2603.11319/) (2026)

## Abstract (arXiv)

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 mubeta = e-beta F/Z at low temperature. In particular, under mild regularity assumptions on the convergence rate of Gradient Flow, we establish that mubeta satisfies a Poincar'e Inequality with constant O(C'+1/beta) for beta >= Omega(d), where C' is the Poincar'e constant of mubeta restricted to a neighborhood of the global minimizers of F. Under an additional mild condition on F, we show that mubeta satisfies a Log-Sobolev Inequality with constant O(beta max(S, 1) max(C', 1)) where S denotes the second moment of mubeta. 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 mubeta satisfies a Weak Poincar'e Inequality with parameters (O(C'+1/beta), O(mubeta(S))) for β = 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).