Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling (2407.16936v1)

Abstract: We address the outstanding problem of sampling from an unnormalized density that may be non-log-concave and multimodal. To enhance the performance of simple Markov chain Monte Carlo (MCMC) methods, techniques of annealing type have been widely used. However, quantitative theoretical guarantees of these techniques are under-explored. This study takes a first step toward providing a non-asymptotic analysis of annealed MCMC. Specifically, we establish, for the first time, an oracle complexity of $\widetilde{O}\left(\frac{d\beta^2{\cal A}^2}{\varepsilon^6}\right)$ for simple annealed Langevin Monte Carlo algorithm to achieve $\varepsilon^2$ accuracy in Kullback-Leibler divergence to the target distribution $\pi\propto{\rm e}^{-V}$ on $\mathbb{R}^d$ with $\beta$-smooth potential $V$. Here, ${\cal A}$ represents the action of a curve of probability measures interpolating the target distribution $\pi$ and a readily sampleable distribution.

• The paper presents a novel annealed Langevin MC algorithm that uses intermediate distributions to overcome challenges in sampling from non-log-concave, multimodal targets.
• It establishes a non-asymptotic complexity bound of Õ(dβ²A²/ε⁶) for achieving KL-divergence accuracy without relying on log-concavity or isoperimetric assumptions.
• Rigorous analysis via the Girsanov theorem and optimal transport theory demonstrates practical improvements and paves the way for future research in complex sampling.

Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling

Introduction

The paper "Provable Benefit of Annealed Langevin Monte Carlo for Non-log-concave Sampling" by Wei Guo, Molei Tao, and Yongxin Chen addresses the computational challenge of sampling from probability distributions that are non-log-concave and potentially multimodal. Common methods such as Langevin Monte Carlo (LMC) demonstrate efficiency under strong log-concavity assumptions but falter when facing complex, multimodal distributions, making improvements essential for practical applications in fields like Bayesian inference and computational physics.

Contributions and Summary

The authors present several key contributions:

1. Non-Asymptotic Complexity Bound: For the first time, an oracle complexity of $$\widetilde{O}\left(\frac{d\beta^2{\cal A}^2}{\varepsilon^6}\right)$$ is established for achieving $\varepsilon^2$ accuracy in terms of KL-divergence between the sample distribution and the target.
2. Annealed Langevin Monte Carlo (ALMC): The paper introduces a novel annealed LMC algorithm incorporating intermediate distributions to traverse from an easy-to-sample distribution to the desired complex target distribution.
3. Theoretical Guarantees: By leveraging the Girsanov theorem and optimal transport theory, the paper provides rigorous non-asymptotic analysis, bypassing the need for log-concavity or isoperimetric assumptions.

Methodological Insights

The paper defines the ALMC algorithm through an annealed framework that uses a series of intermediate distributions $\{\pi_i\}$ between an initial distribution $\pi_0$ and the target $\pi_M$. This involves:

• Annealed Langevin Diffusion (ALD): A continuous process where the diffusion dynamics are modified progressively to follow the intermediate distributions.
• Annealed LMC: A discretized algorithm derived from ALD, which employs an exponential-integrator scheme to reduce discretization error.

Theoretical Analysis

Key theoretical contributions include:

• Action Functional and Wasserstein Geometry: The analysis includes bounding the KL divergence between the path measures of ALD and a reference process through a novel application of Wasserstein geometry principles.
• Girsanov Theorem Application: The Girsanov theorem is used to relate the path measures, offering a non-asymptotic upper bound on the sampling error by summing the contributions of the continuous dynamics and discretization errors.

Numerical Results

The paper provides a comparative analysis of the oracle complexities for various sampling algorithms, highlighting that their proposed ALMC method operates under the least stringent assumptions while presenting the most favorable dependence on $\varepsilon$ among isoperimetry-free methods.

Practical and Theoretical Implications

The findings have significant implications for both theoretical and practical domains:

• Practical Improvements: These results suggest that ALMC can efficiently sample from complex, non-log-concave distributions without requiring strong log-concavity or isoperimetric inequalities, which broadens its applicability to a variety of real-world problems.
• Theoretical Advancements: The approach provides a new paradigm for analyzing and understanding annealed methods in stochastic processes and their convergence properties.

Future Directions

The paper opens several avenues for future research:

• Optimal Annealing Schedules: Identifying schedules that minimize the action functional to further reduce complexity.
• Extensions to Other Distributions: Adapting the analysis for distributions with non-smooth potentials or heavy tails.
• Tight Complexity Bounds: Refining the current bounds to achieve closer alignment with empirical performance and possibly proving lower bounds for non-log-concave sampling problems.

Conclusion

In summary, this paper makes a substantial contribution to the field of non-log-concave sampling by providing a robust theoretical framework and practical algorithm that enhances the efficiency and reliability of sampling methods under challenging conditions. The novel use of the Girsanov theorem and optimal transport properties for bounding the KL divergence in a non-asymptotic manner represents a significant theoretical advancement, with promising implications for both practical applications and future research directions in the domain of stochastic sampling.