Information-Theoretic Analysis of Epistemic Uncertainty in Bayesian Meta-learning
This work addresses the challenge of understanding uncertainty in meta-learning for researchers in Bayesian methods, though it is incremental as it extends prior information-theoretic frameworks to a meta-learning context.
The paper tackles the problem of quantifying epistemic uncertainty in Bayesian meta-learning by deriving exact characterizations for log-loss and bounds for general losses, showing how uncertainty depends on the number of tasks and per-task data.
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 use the proposed information-theoretic bounds, evaluated via neural mutual information estimators, to compare the performance of conventional learning and meta-learning as the number of meta-learning tasks increases.