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Annealed importance sampling (AIS) and related algorithms are highlyeffective tools for marginal likelihood estimation, but are not fullydifferentiable due to the use of Metropolis-Hastings correction steps.Differentiability is a desirable property as it would admit the possibility ofoptimizing marginal likelihood as an objective using gradient-based methods. Tothis end, we propose Differentiable AIS (DAIS), a variant of AIS which ensuresdifferentiability by abandoning the Metropolis-Hastings corrections. As afurther advantage, DAIS allows for mini-batch gradients. We provide a detailedconvergence analysis for Bayesian linear regression which goes beyond previousanalyses by explicitly accounting for the sampler not having reachedequilibrium. Using this analysis, we prove that DAIS is consistent in thefull-batch setting and provide a sublinear convergence rate. Furthermore,motivated by the problem of learning from large-scale datasets, we study astochastic variant of DAIS that uses mini-batch gradients. Surprisingly,stochastic DAIS can be arbitrarily bad due to a fundamental incompatibilitybetween the goals of last-iterate convergence to the posterior and eliminationof the accumulated stochastic error. This is in stark contrast with othersettings such as gradient-based optimization and Langevin dynamics, where theeffect of gradient noise can be washed out by taking smaller steps. Thisindicates that annealing-based marginal likelihood estimation with stochasticgradients may require new ideas.

Differentiable Annealed Importance Sampling and the Perils of Gradient Noise