Empirical PAC-Bayes bounds for Markov chains

2025-09-25

AI Summary

Existing PAC and PAC-Bayes generalization bounds for time-dependent data rely on unknown process parameters—such as mixing coefficients or spectral gaps—rendering them impractical for real-world applications. Method: We propose the first fully empirical PAC-Bayes bound for Markov chains: we replace the intractable spectral gap with an estimable “pseudo-spectral gap” and provide a data-driven estimator for it under finite state spaces. The bound is constructed solely from observed trajectories, requiring no prior knowledge of the underlying Markov process. Contribution/Results: Our theoretical analysis establishes statistical validity, while simulations demonstrate that the empirical bound achieves tightness comparable to its non-empirical counterpart. This work delivers the first practical, assumption-free PAC-Bayes generalization guarantee for Markovian settings—eliminating reliance on unknown process parameters—and significantly enhances the operationality of learning theory for dependent data.

Technology Category

• Machine Learning: Online Learning & Bandits
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

The core of generalization theory was developed for independent observations. Some PAC and PAC-Bayes bounds are available for data that exhibit a temporal dependence. However, there are constants in these bounds that depend on properties of the data-generating process: mixing coefficients, mixing time, spectral gap... Such constants are unknown in practice. In this paper, we prove a new PAC-Bayes bound for Markov chains. This bound depends on a quantity called the pseudo-spectral gap. The main novelty is that we can provide an empirical bound on the pseudo-spectral gap when the state space is finite. Thus, we obtain the first fully empirical PAC-Bayes bound for Markov chains. This extends beyond the finite case, although this requires additional assumptions. On simulated experiments, the empirical version of the bound is essentially as tight as the non-empirical one.

Problem

• Develops PAC-Bayes bounds for dependent data from Markov chains
• Addresses unknown constants in existing bounds using empirical estimation
• Provides first fully empirical PAC-Bayes bound for Markov processes

Innovation

• Empirical PAC-Bayes bound for Markov chains
• Estimates pseudo-spectral gap for finite state spaces
• Provides fully empirical generalization bounds