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Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of $Φ$-Mutual Information (2402.17067v3)

Published 26 Feb 2024 in math.ST, cs.IT, math.IT, stat.ML, and stat.TH

Abstract: The mixing time of a Markov chain determines how fast the iterates of the Markov chain converge to the stationary distribution; however, it does not control the dependencies between samples along the Markov chain. In this paper, we study the question of how fast the samples become approximately independent along popular Markov chains for continuous-space sampling: the Langevin dynamics in continuous time, and the Unadjusted Langevin Algorithm and the Proximal Sampler in discrete time. We measure the dependence between samples via $\Phi$-mutual information, which is a broad generalization of the standard mutual information, and which is equal to $0$ if and only if the the samples are independent. We show that along these Markov chains, the $\Phi$-mutual information between the first and the $k$-th iterate decreases to $0$ exponentially fast in $k$ when the target distribution is strongly log-concave. Our proof technique is based on showing the Strong Data Processing Inequalities (SDPIs) hold along the Markov chains. To prove fast mixing of the Markov chains, we only need to show the SDPIs hold for the stationary distribution. In contrast, to prove the contraction of $\Phi$-mutual information, we need to show the SDPIs hold along the entire trajectories of the Markov chains; we prove this when the iterates along the Markov chains satisfy the corresponding $\Phi$-Sobolev inequality, which is implied by the strong log-concavity of the target distribution.

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Summary

  • The paper introduces a novel framework that quantifies independence by examining mutual information decay in both continuous (Langevin diffusion) and discrete (ULA) settings.
  • It demonstrates exponential decay under strong log-concavity and polynomial convergence under weak log-concavity, linking mixing time with sample independence.
  • Rigorous bounds on the required sampling time are established, providing actionable insights for effective Bayesian inference and machine learning applications.

Exploring Independence Along the Langevin Diffusion and the Unadjusted Langevin Algorithm

Introduction

Sampling from complex distributions is central to numerous applications in statistics, machine learning, and computational sciences. Markov chain Monte Carlo (MCMC) methods, particularly those based on Langevin dynamics, play a pivotal role in this endeavor. Among these, the Langevin diffusion in continuous time and its discretized counterpart, the Unadjusted Langevin Algorithm (ULA), are of substantial interest due to their simplicity and grounding in stochastic calculus. This paper explores the independence time of these chains, specifically, the rate at which samples drawn become independent, quantified through the lens of mutual information decay.

Key Findings

The paper presents rigorous analyses and results concerning the exponential and polynomial convergence rates of mutual information to zero for the Langevin diffusion and ULA under varying concavity conditions of the target distribution. The primary contributions can be distilled into the following points:

  • For the Langevin diffusion in a strongly log-concave setting, mutual information is shown to converge exponentially fast to zero, echoing the analogous mixing time behavior. In contrast, under weak log-concavity, the convergence emerges at a polynomial rate.
  • Transitioning to the Unadjusted Langevin Algorithm in discrete time, the paper proves exponential decay in mutual information for strongly log-concave and smooth targets. Here, smoothness is a critical added assumption aligning with discrete-time analysis necessities.
  • A novel methodological framework is developed, integrating the mutual version of functional analysis, strong data processing inequalities (SDPIs), and the exploration of regularity properties associated with these stochastic processes.
  • Resulting from the methodological advancements, bounds on the independence time are established, offering insights into the operational time frame required before one can draw an approximately independent sample from the target distribution.

Implications and Significance

The theoretical implications stretch across multiple domains, reinforcing the utility and efficiency of Langevin-based sampling methods. Practically, the findings enhance our understanding of sampling dynamics, guiding the optimal setup of Langevin dynamics and ULA for effective sampling tasks.

Moreover, the introduction of mutual information as a metric for independence time opens novel avenues for assessing the quality and independence of samples in complex high-dimensional spaces. This is particularly relevant in machine learning applications, like Bayesian inference, where the quality of samples directly impacts model performance.

Future Directions

Looking ahead, several questions beckon further investigation:

  • Extension of mutual information convergence results under broader conditions, such as isoperimetry, presents a natural next step.
  • The exploration of mutual information dynamics in other Markov chains, including the underdamped Langevin dynamics, could yield additional insights into sampling methodologies.
  • Investigating convergence rates in alternative divergences, like Rényi or χ², could offer a more nuanced understanding of sample independence.
  • Finally, conceptualizing and realizing the gradient flow for mutual information presents an intriguing challenge with potential algorithmic breakthroughs in sampling methods.

Closing Remarks

This paper underscores the continued relevance and potency of Langevin dynamics in the computational toolbox for sampling. By rigorously characterizing the rate of independence via mutual information, the work paves the way for more informed and efficacious use of these methods in a broad spectrum of applications, from statistical physics to artificial intelligence.

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