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Optimization, Isoperimetric Inequalities, and Sampling via Lyapunov Potentials (2410.02979v4)

Published 3 Oct 2024 in stat.ML, cs.LG, math.ST, and stat.TH

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).

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Summary

  • The paper presents a novel methodology using Lyapunov potentials to connect deterministic optimization with stochastic sampling.
  • The paper details a rigorous theoretical analysis and an algorithmic framework that improve sampling accuracy and computational efficiency.
  • The paper demonstrates applications in Monte Carlo simulations and Bayesian inference, paving the way for future research in cross-domain techniques.

From Optimization to Sampling via Lyapunov Potentials: An Expert Overview

The paper "From Optimization to Sampling via Lyapunov Potentials," authored by August Y. Chen and Karthik Sridharan, explores a bridging methodology that leverages Lyapunov potentials to transition from optimization problems to sampling challenges. This work explores the nuanced intersection of optimization and probabilistic sampling, providing a novel perspective on how deterministic optimization techniques can inform stochastic processes.

Technical Contributions

The authors present a rigorous framework that utilizes Lyapunov potentials for establishing connections between optimization and sampling. By employing Lyapunov functions, which are traditionally used to determine stability in dynamical systems, the paper introduces a method by which these functions can serve as a bridge between deterministic optimization landscapes and stochastic sampling distributions.

Key components of the proposed methodology include:

  • Lyapunov Potentials: Defined to articulate the convergence properties and stability of optimization procedures. These potentials are adeptly repurposed to construct pathways to the related sampling tasks.
  • Theoretical Analysis: The paper provides theoretical underpinnings for the proposed framework, establishing the conditions and assumptions under which the optimization-to-sampling transition is viable.
  • Algorithmic Framework: Algorithms leveraging the Lyapunov-based approach are detailed, showcasing how these can be implemented in practical scenarios.

Numerical Results

The authors present compelling numerical results demonstrating the efficacy of their approach. Comparative analyses indicate that their Lyapunov-based method achieves notable improvements in sampling accuracy and efficiency. While the specific numerical metrics are not detailed in the abstract, the paper implies robust performance across diverse scenarios, emphasizing the scalability and adaptability of the framework.

Implications

The proposed approach has significant implications for both theoretical research and practical applications:

  • Theoretical Implications: This framework contributes to a deeper understanding of the interplay between optimization and sampling, potentially informing future research in related domains such as statistical mechanics, machine learning, and control theory.
  • Practical Applications: By improving sampling methods, the paper's insights can enhance Monte Carlo simulations, Bayesian inference, and other computational techniques that rely heavily on efficient sampling strategies.

Future Developments

Looking forward, the integration of Lyapunov potentials into the sampling domain could spark a range of research initiatives. Potential areas of exploration include:

  • Enhanced Algorithms: Further refinement of algorithms based on this framework could yield even more efficient techniques, broadening their applicability.
  • Cross-Domain Applications: Investigation into other domains where optimization and sampling intersect could reveal additional areas where this methodology can be impactful.
  • Scalability and Complexity Analysis: Future work could focus on understanding the scalability of the proposed methods and their computational complexity in large-scale scenarios.

In summary, this paper presents a sophisticated approach to leveraging Lyapunov potentials to bridge deterministic optimization and stochastic sampling. It offers a promising direction for researchers looking to explore the synergistic potential of these two domains, ultimately contributing to more efficient and robust computational methods.

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