Constant Regret, Generalized Mixability, and Mirror Descent

Abstract

We consider the setting of prediction with expert advice; a learner makes predictions by aggregating those of a group of experts. Under this setting, and for the right choice of loss function and mixing algorithm, it is possible for the learner to achieve a constant regret regardless of the number of prediction rounds. For example, a constant regret can be achieved for mixable losses using the aggregating algorithm. The Generalized Aggregating Algorithm (GAA) is a name for a family of algorithms parameterized by convex functions on simplices (entropies), which reduce to the aggregating algorithm when using the Shannon entropy $\mathrm{S}$. For a given entropy $\Phi$, losses for which a constant regret is possible using the GAA are called $\Phi$-mixable. Which losses are $\Phi$-mixable was previously left as an open question. We fully characterize $\Phi$-mixability and answer other open questions posed by (Reid et al. 2015). We show that the Shannon entropy $\mathrm{S}$ is fundamental in nature when it comes to mixability; any $\Phi$-mixable loss is necessarily $\mathrm{S}$-mixable, and the lowest worst-case regret of the GAA is achieved using the Shannon entropy. Finally, by leveraging the connection between the mirror descent algorithm and the update step of the GAA, we suggest a new adaptive generalized aggregating algorithm and analyze its performance in terms of the regret bound.

Publication
Advances in Neural Information Processing Systems 31
Zak Mhammedi
Zak Mhammedi
PhD Student

My research interests include the theory of online learning and statistical generalization.