Online Learning with Primary and Secondary Losses

TL;DR

This paper investigates online learning with primary and secondary losses, showing limitations without variance bounds and proposing algorithms that manage regret and secondary loss constraints under certain conditions.

Contribution

It introduces methods to balance primary regret minimization with secondary loss control, including algorithms that adaptively deactivate experts based on secondary loss thresholds.

Findings

Achieving low primary regret and bounded secondary loss is impossible without variance assumptions.

Switching-limited algorithms can achieve the goal if experts' secondary losses are sublinear in time.

External oracles enable algorithms to deactivate/reactivate experts to satisfy secondary loss constraints.

Abstract

We study the problem of online learning with primary and secondary losses. For example, a recruiter making decisions of which job applicants to hire might weigh false positives and false negatives equally (the primary loss) but the applicants might weigh false negatives much higher (the secondary loss). We consider the following question: Can we combine "expert advice" to achieve low regret with respect to the primary loss, while at the same time performing {\em not much worse than the worst expert} with respect to the secondary loss? Unfortunately, we show that this goal is unachievable without any bounded variance assumption on the secondary loss. More generally, we consider the goal of minimizing the regret with respect to the primary loss and bounding the secondary loss by a linear threshold. On the positive side, we show that running any switching-limited algorithm can achieve this goal if all experts satisfy the assumption that the secondary loss does not exceed the linear threshold by $o(T)$ for any time interval. If not all experts satisfy this assumption, our algorithms can achieve this goal given access to some external oracles which determine when to deactivate and reactivate experts.