Surrogate Regret Bounds for Polyhedral Losses
This work provides theoretical insights into surrogate risk minimization, potentially guiding loss function selection in supervised learning, though it appears incremental as it builds on existing regret bound frameworks.
The paper tackles the problem of deriving surrogate regret bounds for polyhedral loss functions, showing that polyhedral surrogates yield linear regret bounds that directly translate target generalization rates, while non-polyhedral ones lead to slower square-root bounds.
Surrogate risk minimization is an ubiquitous paradigm in supervised machine learning, wherein a target problem is solved by minimizing a surrogate loss on a dataset. Surrogate regret bounds, also called excess risk bounds, are a common tool to prove generalization rates for surrogate risk minimization. While surrogate regret bounds have been developed for certain classes of loss functions, such as proper losses, general results are relatively sparse. We provide two general results. The first gives a linear surrogate regret bound for any polyhedral (piecewise-linear and convex) surrogate, meaning that surrogate generalization rates translate directly to target rates. The second shows that for sufficiently non-polyhedral surrogates, the regret bound is a square root, meaning fast surrogate generalization rates translate to slow rates for the target. Together, these results suggest polyhedral surrogates are optimal in many cases.