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We explore a new research direction in Bayesian variational inference withdiscrete latent variable priors where we exploit Kronecker matrix algebra forefficient and exact computations of the evidence lower bound (ELBO). Theproposed "DIRECT" approach has several advantages over its predecessors; (i) itcan exactly compute ELBO gradients (i.e. unbiased, zero-variance gradientestimates), eliminating the need for high-variance stochastic gradientestimators and enabling the use of quasi-Newton optimization methods; (ii) itstraining complexity is independent of the number of training points, permittinginference on large datasets; and (iii) its posterior samples consist of sparseand low-precision quantized integers which permit fast inference on hardwarelimited devices. In addition, our DIRECT models can exactly compute statisticalmoments of the parameterized predictive posterior without relying on MonteCarlo sampling. The DIRECT approach is not practical for all likelihoods,however, we identify a popular model structure which is practical, anddemonstrate accurate inference using latent variables discretized as extremelylow-precision 4-bit quantized integers. While the ELBO computations consideredin the numerical studies require over $10^{2352}$ log-likelihood evaluations,we train on datasets with over two-million points in just seconds.

Discretely Relaxing Continuous Variables for tractable Variational Inference