Differentiable Annealed Importance Sampling and the Perils of Gradient Noise

TL;DR

This paper introduces Differentiable AIS (DAIS), a variant of annealed importance sampling that is fully differentiable and compatible with mini-batch gradients, enabling gradient-based optimization of marginal likelihoods.

Contribution

The paper proposes DAIS, which removes Metropolis-Hastings corrections for differentiability, provides convergence analysis for Bayesian linear regression, and reveals limitations of stochastic DAIS with mini-batch gradients.

Findings

DAIS is consistent in the full-batch setting with a sublinear convergence rate.

Stochastic DAIS can perform arbitrarily poorly due to gradient noise.

Gradient noise in stochastic DAIS cannot be mitigated by smaller steps, unlike in other methods.

Abstract

Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation, but are not fully differentiable due to the use of Metropolis-Hastings correction steps. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective using gradient-based methods. To this end, we propose Differentiable AIS (DAIS), a variant of AIS which ensures differentiability by abandoning the Metropolis-Hastings corrections. As a further advantage, DAIS allows for mini-batch gradients. We provide a detailed convergence analysis for Bayesian linear regression which goes beyond previous analyses by explicitly accounting for the sampler not having reached equilibrium. Using this analysis, we prove that DAIS is consistent in the full-batch setting and provide a sublinear convergence rate. Furthermore, motivated by the problem of learning from large-scale datasets, we study a stochastic variant of DAIS that uses mini-batch gradients. Surprisingly, stochastic DAIS can be arbitrarily bad due to a fundamental incompatibility between the goals of last-iterate convergence to the posterior and elimination of the accumulated stochastic error. This is in stark contrast with other settings such as gradient-based optimization and Langevin dynamics, where the effect of gradient noise can be washed out by taking smaller steps. This indicates that annealing-based marginal likelihood estimation with stochastic gradients may require new ideas.