Sampling with flows, diffusion and autoregressive neural networks: A spin-glass perspective

TL;DR

This paper analyzes the efficiency of flow, diffusion, and autoregressive neural network sampling methods on spin-glass-like problems, revealing their limitations and advantages compared to traditional sampling techniques through a theoretical physics perspective.

Contribution

It provides a theoretical analysis of these neural sampling methods on spin-glass problems, highlighting their limitations due to phase transitions and identifying parameter regions where they outperform or underperform traditional methods.

Findings

Neural sampling methods face difficulties due to phase transitions.

Traditional methods outperform neural methods in certain parameter regimes.

Neural methods excel where traditional approaches are inefficient.

Abstract

Recent years witnessed the development of powerful generative models based on flows, diffusion or autoregressive neural networks, achieving remarkable success in generating data from examples with applications in a broad range of areas. A theoretical analysis of the performance and understanding of the limitations of these methods remain, however, challenging. In this paper, we undertake a step in this direction by analysing the efficiency of sampling by these methods on a class of problems with a known probability distribution and comparing it with the sampling performance of more traditional methods such as the Monte Carlo Markov chain and Langevin dynamics. We focus on a class of probability distribution widely studied in the statistical physics of disordered systems that relate to spin glasses, statistical inference and constraint satisfaction problems. We leverage the fact that sampling via flow-based, diffusion-based or autoregressive networks methods can be equivalently mapped to the analysis of a Bayes optimal denoising of a modified probability measure. Our findings demonstrate that these methods encounter difficulties in sampling stemming from the presence of a first-order phase transition along the algorithm's denoising path. Our conclusions go both ways: we identify regions of parameters where these methods are unable to sample efficiently, while that is possible using standard Monte Carlo or Langevin approaches. We also identify regions where the opposite happens: standard approaches are inefficient while the discussed generative methods work well.