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Bayesian deep learning plays an important role especially for its abilityevaluating epistemic uncertainty (EU). Due to computational complexity issues,approximation methods such as variational inference (VI) have been used inpractice to obtain posterior distributions and their generalization abilitieshave been analyzed extensively, for example, by PAC-Bayesian theory; however,little analysis exists on EU, although many numerical experiments have beenconducted on it. In this study, we analyze the EU of supervised learning inapproximate Bayesian inference by focusing on its excess risk. First, wetheoretically show the novel relations between generalization error and thewidely used EU measurements, such as the variance and mutual information ofpredictive distribution, and derive their convergence behaviors. Next, weclarify how the objective function of VI regularizes the EU. With thisanalysis, we propose a new objective function for VI that directly controls theprediction performance and the EU based on the PAC-Bayesian theory. Numericalexperiments show that our algorithm significantly improves the EU evaluationover the existing VI methods.

Excess risk analysis for epistemic uncertainty with application to variational inference