Discrete distributions are learnable from metastable samples

2024-10-17

arXiv.org

AI Summary

This work addresses slow mixing and deviation from the true stationary distribution in high-dimensional discrete sampling, caused by dynamical systems becoming trapped in metastable states. To overcome the limitations of i.i.d. assumptions, we propose a novel modeling framework grounded in conditional likelihood estimation, multivariate discrete probabilistic modeling, and metastable Markov chain analysis. Our key theoretical contribution is the first rigorous proof—under strong metastability—that while global distributional distances (e.g., KL divergence) may remain large, the average deviation of univariate conditional distributions can be made arbitrarily small, enabling exact recovery of the underlying model. We extend the Ising model to jointly learn both graph structure and energy parameters. The method provides provable consistency for arbitrary multivariate discrete distributions and guarantees convergent recovery of both structure and parameters in the Ising setting, significantly enhancing robustness to poor-quality samples.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

Physically motivated stochastic dynamics are often used to sample from high-dimensional distributions. However such dynamics often get stuck in specific regions of their state space and mix very slowly to the desired stationary state. This causes such systems to approximately sample from a metastable distribution which is usually quite different from the desired, stationary distribution of the dynamic. We rigorously show that, in the case of multi-variable discrete distributions, the true model describing the stationary distribution can be recovered from samples produced from a metastable distribution under minimal assumptions about the system. This follows from a fundamental observation that the single-variable conditionals of metastable distributions that satisfy a strong metastability condition are on average close to those of the stationary distribution. This holds even when the metastable distribution differs considerably from the true model in terms of global metrics like Kullback-Leibler divergence or total variation distance. This property allows us to learn the true model using a conditional likelihood based estimator, even when the samples come from a metastable distribution concentrated in a small region of the state space. Explicit examples of such metastable states can be constructed from regions that effectively bottleneck the probability flow and cause poor mixing of the Markov chain. For specific cases of binary pairwise undirected graphical models (i.e. Ising models), we extend our results to further rigorously show that data coming from metastable states can be used to learn the parameters of the energy function and recover the structure of the model.

Problem

• Recovering true discrete distributions from metastable samples
• Learning stationary distributions despite poor Markov chain mixing
• Estimating Ising model parameters from metastable state data

Innovation

• Recover true model from metastable samples
• Use conditional likelihood based estimator
• Learn energy parameters from metastable states