On the Parameterization of Gaussian Mean Field Posteriors in Bayesian Neural Networks
Variational Bayesian Inference is a popular methodology for approximating posterior distributions in Bayesian neural networks. Recent work developing this class of methods has explored ever richer parameterizations of the approximate posterior in the hope of improving performance. In contrast, here we share a curious experimental finding that suggests instead restricting the variational distribution to a more compact parameterization. For a variety of deep Bayesian neural networks trained using Gaussian mean-field variational inference, we find that the posterior standard deviations consistently exhibits strong low-rank structure after convergence. This means that by decomposing these variational parameters into a low-rank factorization, we can make our variational approximation more compact without decreasing the models' performance. What's more, we find that such factorized parameterizations are easier to train since they improve the signal-to-noise ratio of stochastic gradient estimates of the variational lower bound, resulting in faster convergence.
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