Risk and Regret of Hierarchical Bayesian Learners
This work addresses the problem of providing theoretical justification for hierarchical Bayesian methods, which is significant for statisticians and machine learning practitioners seeking robust and efficient models, though it is incremental in formalizing existing concepts.
The paper tackles the challenge of formalizing the benefits of hierarchical Bayesian models in learning theory, providing regret bounds under log-loss and risk bounds for bounded losses, with results showing that hierarchical priors like Student's t and hierarchical Gaussian can offer robustness and statistical strength sharing at minimal practical cost.
Common statistical practice has shown that the full power of Bayesian methods is not realized until hierarchical priors are used, as these allow for greater "robustness" and the ability to "share statistical strength." Yet it is an ongoing challenge to provide a learning-theoretically sound formalism of such notions that: offers practical guidance concerning when and how best to utilize hierarchical models; provides insights into what makes for a good hierarchical prior; and, when the form of the prior has been chosen, can guide the choice of hyperparameter settings. We present a set of analytical tools for understanding hierarchical priors in both the online and batch learning settings. We provide regret bounds under log-loss, which show how certain hierarchical models compare, in retrospect, to the best single model in the model class. We also show how to convert a Bayesian log-loss regret bound into a Bayesian risk bound for any bounded loss, a result which may be of independent interest. Risk and regret bounds for Student's $t$ and hierarchical Gaussian priors allow us to formalize the concepts of "robustness" and "sharing statistical strength." Priors for feature selection are investigated as well. Our results suggest that the learning-theoretic benefits of using hierarchical priors can often come at little cost on practical problems.