Variational Bayesian dropout: pitfalls and fixes

TL;DR

This paper critically examines the Bayesian dropout framework, identifies fundamental issues with improper priors and singularities, and proposes Quasi-KL divergence as a novel solution to improve variational inference in neural networks.

Contribution

It highlights the pitfalls of existing Bayesian dropout methods and introduces Quasi-KL divergence to address the singularity problem in variational inference.

Findings

The variational Gaussian dropout framework suffers from irredeemable pathologies due to improper priors.

QKL divergence provides a well-defined objective for high-dimensional distribution approximation.

QKL converges to variational Bernoulli dropout under certain limits, connecting discretisation and noise motivations.

Abstract

Dropout, a stochastic regularisation technique for training of neural networks, has recently been reinterpreted as a specific type of approximate inference algorithm for Bayesian neural networks. The main contribution of the reinterpretation is in providing a theoretical framework useful for analysing and extending the algorithm. We show that the proposed framework suffers from several issues; from undefined or pathological behaviour of the true posterior related to use of improper priors, to an ill-defined variational objective due to singularity of the approximating distribution relative to the true posterior. Our analysis of the improper log uniform prior used in variational Gaussian dropout suggests the pathologies are generally irredeemable, and that the algorithm still works only because the variational formulation annuls some of the pathologies. To address the singularity issue, we proffer Quasi-KL (QKL) divergence, a new approximate inference objective for approximation of high-dimensional distributions. We show that motivations for variational Bernoulli dropout based on discretisation and noise have QKL as a limit. Properties of QKL are studied both theoretically and on a simple practical example which shows that the QKL-optimal approximation of a full rank Gaussian with a degenerate one naturally leads to the Principal Component Analysis solution.