  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.
