Variational Inference based on Robust Divergences

TL;DR

This paper introduces a robust variational inference method that replaces the KL divergence with robust divergences, enabling deep models to be more resistant to outliers, with theoretical and empirical validation.

Contribution

It proposes a novel outlier-robust variational inference framework using robust divergences, applicable to complex models like deep networks, with proven bounded influence functions.

Findings

The method is robust to input and output outliers.

It outperforms ordinary variational inference in deep network regression.

The influence function is bounded for deep ReLU networks.

Abstract

Robustness to outliers is a central issue in real-world machine learning applications. While replacing a model to a heavy-tailed one (e.g., from Gaussian to Student-t) is a standard approach for robustification, it can only be applied to simple models. In this paper, based on Zellner's optimization and variational formulation of Bayesian inference, we propose an outlier-robust pseudo-Bayesian variational method by replacing the Kullback-Leibler divergence used for data fitting to a robust divergence such as the beta- and gamma-divergences. An advantage of our approach is that superior but complex models such as deep networks can also be handled. We theoretically prove that, for deep networks with ReLU activation functions, the \emph{influence function} in our proposed method is bounded, while it is unbounded in the ordinary variational inference. This implies that our proposed method is robust to both of input and output outliers, while the ordinary variational method is not. We experimentally demonstrate that our robust variational method outperforms ordinary variational inference in regression and classification with deep networks.