Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials

2024-10-03

AI Summary

This work establishes an equivalence between the global optimizability of arbitrary functions $F$ under gradient flow at low temperatures and the satisfaction of Poincaré or Log-Sobolev inequalities by the associated Gibbs measure $mu_eta$, thereby providing theoretical foundations for efficient sampling. Using Lyapunov potential analysis, gradient flow dynamics, and spectral theory of Gibbs measures, we rigorously prove that global optimizability directly implies isoperimetric inequalities—and hence Poincaré inequalities—for $mu_eta$, a novel connection not previously established. Our framework transcends classical log-concavity assumptions, enabling efficient sampling even for non-log-concave densities. Moreover, under weak regularity conditions, we derive improved discrete-time sampling guarantees for log-concave distributions, surpassing the recent result of Lehec (2023). Collectively, these contributions unify optimization and sampling perspectives, significantly extending the theoretical frontier of high-dimensional non-convex sampling.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements

Abstract

In this paper, we prove that optimizability of any F using Gradient Flow from all initializations implies a Poincar'e Inequality for Gibbs measures mu_{beta} = e^{-eta 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(S beta C') 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 (C', mu_{beta}(S)) for eta = 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).

Problem

• Establishes Poincaré and Log-Sobolev inequalities for low-temperature Gibbs measures.
• Shows optimizability via Gradient Flow implies efficient sampling from initializations.
• Demonstrates efficient sampling for non-log-concave densities under mild conditions.

Innovation

• Gradient Flow optimizability implies Poincaré Inequality.
• Log-Sobolev Inequality derived from Gradient Flow conditions.
• Efficient sampling for non-log-concave densities demonstrated.