Information-Theoretic Analysis of Epistemic Uncertainty in Bayesian Meta-learning
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The overall predictive uncertainty of a trained predictor can be decomposed into separate contributions due to epistemic and aleatoric uncertainty. Under a Bayesian formulation, assuming a well-specified model, the two contributions can be exactly expressed (for the log-loss) or bounded (for more general losses) in terms of information-theoretic quantities (Xu and Raginsky, 2020). This paper addresses the study of epistemic uncertainty within an information-theoretic framework in the broader setting of Bayesian meta-learning. A general hierarchical Bayesian model is assumed in which hyperparameters determine the per-task priors of the model parameters. Exact characterizations (for the log-loss) and bounds (for more general losses) are derived for the epistemic uncertainty - quantified by the minimum excess meta-risk (MEMR)- of optimal meta-learning rules. This characterization is leveraged to bring insights into the dependence of the epistemic uncertainty on the number of tasks and on the amount of per-task training data. Experiments are presented that compare the proposed information-theoretic bounds, evaluated via neural mutual information estimators, with the performance of a novel approximate fully Bayesian meta-learning strategy termed Langevin-Stein Bayesian Meta-Learning (LS-BML).
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