MCMC-driven learning (2402.09598v1)

Abstract: This paper is intended to appear as a chapter for the Handbook of Markov Chain Monte Carlo. The goal of this chapter is to unify various problems at the intersection of Markov chain Monte Carlo (MCMC) and machine learning$\unicode{x2014}$which includes black-box variational inference, adaptive MCMC, normalizing flow construction and transport-assisted MCMC, surrogate-likelihood MCMC, coreset construction for MCMC with big data, Markov chain gradient descent, Markovian score climbing, and more$\unicode{x2014}$within one common framework. By doing so, the theory and methods developed for each may be translated and generalized.

• The paper introduces a novel approach that extends classical MCMC methods to solve dual optimization-integration (MOI) challenges in computational statistics.
• It details an iterative framework employing Markov chain sampling and convergence theory to optimize parameters effectively using adaptive learning rates.
• The study demonstrates practical algorithms like independence Metropolis-Hastings and transport map techniques to improve sampling efficiency in high-dimensional spaces.

MCMC-Driven Learning: A Unified Framework for Modern Computational Statistics

Overview

Markov chain Monte Carlo (MCMC) methods serve as a crucial foundation in the toolkit of modern computational statisticians and machine learners. Historically utilized for Bayesian posterior inference, MCMC methodologies have evolved, integrating with ML to address a broader array of optimization-integration (MOI) problems. MOI problems encapsulate tasks where both optimization and integration are performed simultaneously, often with the objective of minimizing some cost associated with parameter tuning based on Markov chain-generated sequences. This chapter introduces a comprehensive framework for understanding and addressing these MOI problems, leveraging MCMC alongside advanced ML techniques.

MOI Problems and Their Significance

MOI problems are characterized by their dual focus on optimization, typically of parameter selection within models or algorithms, and on integration, usually representing the inferential or predictive aspects associated with these parameters. This dual focus is reflective of the blending of inferential statistics with predictive machine learning, manifesting in a variety of applications including but not limited to adaptive MCMC, variational inference, and normalizing flow construction.

Techniques for Solving MOI Problems

Solving MOI problems effectively requires a robust set of tools and methodologies, detailed as follows:

1. Framework and Problem Formulation: The framework for MOI leverages Markovian dynamics to navigate the parameter space. This involves iterative optimization steps informed by samples drawn according to Markov processes, where the target distribution itself may adapt based on these samples.
2. Convergence Theory: A critical aspect of MOI problem-solving is ensuring convergence of the parameters to optimal points. This requires careful consideration of conditions such as the compactness and smoothness of the objective function, stability of the Markov kernels, and appropriate scheduling of the learning rates.
3. Practical Algorithms: Practical approaches to MOI problems vary based on the specific application and desired outcomes. Examples include the use of independence Metropolis-Hastings (IMH) kernels for distribution approximation, and the incorporation of transport maps to improve sampling efficiency. Algorithms are often designed to balance exploration of the parameter space with exploitation of known gradients or sufficient statistics.

Case Study: MCMC-Driven Distribution Approximation

An instructive case paper within the MOI framework is the optimization of IMH proposals via minimization of the forward KL divergence. This involves identifying a parametric family of proposal distributions that approximate the target distribution closely enough to facilitate efficient sampling. Techniques such as parallel tempering can be employed to mitigate the curse of dimensionality in high-dimensional spaces, illustrating how MOI methods adapt classical MCMC strategies for modern computational challenges.

Advancements and Extensions

The MOI framework is not static; it continually evolves to incorporate new findings and methodologies from various disciplines. Recent advances include exploration of unadjusted Langevin dynamics for quicker convergence during the burn-in phase and the use of machine learning models to construct more expressive transport maps. Furthermore, methods to stabilize learning, such as the introduction of multiple references or "legs" in tempering paths, highlight the ongoing refinement of MOI strategies.

Conclusion

The MOI framework offers a powerful lens through which to view a wide range of problems at the intersection of MCMC methods and machine learning. By formalizing these problems within a unified theoretical and practical framework, researchers and practitioners can leverage the strengths of both inferential statistics and predictive modeling. This blended approach holds promise for tackling the increasingly complex computational challenges encountered in modern data analysis.