Discretely Relaxing Continuous Variables for tractable Variational\n Inference

We explore a new research direction in Bayesian variational inference with\ndiscrete latent variable priors where we exploit Kronecker matrix algebra for\nefficient and exact computations of the evidence lower bound (ELBO). The\nproposed "DIRECT" approach has several advantages over its predecessors; (i) it\ncan exactly compute ELBO gradients (i.e. unbiased, zero-variance gradient\nestimates), eliminating the need for high-variance stochastic gradient\nestimators and enabling the use of quasi-Newton optimization methods; (ii) its\ntraining complexity is independent of the number of training points, permitting\ninference on large datasets; and (iii) its posterior samples consist of sparse\nand low-precision quantized integers which permit fast inference on hardware\nlimited devices. In addition, our DIRECT models can exactly compute statistical\nmoments of the parameterized predictive posterior without relying on Monte\nCarlo sampling. The DIRECT approach is not practical for all likelihoods,\nhowever, we identify a popular model structure which is practical, and\ndemonstrate accurate inference using latent variables discretized as extremely\nlow-precision 4-bit quantized integers. While the ELBO computations considered\nin the numerical studies require over $10^{2352}$ log-likelihood evaluations,\nwe train on datasets with over two-million points in just seconds.\n

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