Stein variational gradient descent (SVGD) (Liu & Wang, 2016) performs approximate Bayesian inference by representing the posterior with a set of particles. However, SVGD suffers from variance collapse, i.e. poor predictions due to underestimating uncertainty (Ba et al., 2021), even for moderately-dimensional models such as small Bayesian neural networks (BNNs). To address this issue, we generalize SVGD by letting each particle parameterize a component distribution in a mixture model. Our method, Stein Mixture Inference (SMI), optimizes a lower bound to the evidence (ELBO) and introduces user-specified guides parameterized by particles. SMI extends the Nonlinear SVGD framework (Wang & Liu, 2019) to the case of variational Bayes. SMI effectively avoids variance collapse, judging by a previously described test developed for this purpose, and performs well on standard data sets. In addition, SMI requires considerably fewer particles than SVGD to accurately estimate uncertainty for small BNNs. The synergistic combination of NSVGD, ELBO optimization and user-specified guides establishes a promising approach towards variational Bayesian inference in the case of tall and wide data.

We present a technique for applying reverse mode automatic differentiation (AD) on a non-recursive second-order functional array language that supports nested parallelism and is primarily aimed at efficient GPU execution. The key idea is to eliminate the need for a tape by relying on redundant execution to bring into each new scope all program variables that may be needed by the differentiated code. Efficient execution is enabled by the observation that perfectly nested scopes do not introduce re-execution and that such perfect nests can be readily produced by application of known compiler transformations. Our technique differentiates loops and bulk-parallel operators-e.g., map, reduce(-by-index), scan, and scatter-by specific rewrite rules and aggressively optimizes the resulting nested-parallel code. We report an evaluation that compares with established AD solutions and demonstrates competitive performance on ten common benchmarks from recent applied AD literature.