An Embedding Framework for the Design and Analysis of Consistent Polyhedral Surrogates
This work addresses the challenge of ensuring consistency in machine learning models using surrogate losses, offering a theoretical framework that is incremental but clarifies and extends existing methods for researchers in optimization and learning theory.
The paper tackles the problem of designing convex surrogate loss functions for tasks like classification and ranking by formalizing an embedding approach, establishing that every discrete loss can be embedded by a polyhedral loss and vice versa, and providing constructive results for consistency proofs and regret bounds.
We formalize and study the natural approach of designing convex surrogate loss functions via embeddings, for problems such as classification, ranking, or structured prediction. In this approach, one embeds each of the finitely many predictions (e.g. rankings) as a point in $R^d$, assigns the original loss values to these points, and "convexifies" the loss in some way to obtain a surrogate. We establish a strong connection between this approach and polyhedral (piecewise-linear convex) surrogate losses: every discrete loss is embedded by some polyhedral loss, and every polyhedral loss embeds some discrete loss. Moreover, an embedding gives rise to a consistent link function as well as linear surrogate regret bounds. Our results are constructive, as we illustrate with several examples. In particular, our framework gives succinct proofs of consistency or inconsistency for various polyhedral surrogates in the literature, and for inconsistent surrogates, it further reveals the discrete losses for which these surrogates are consistent. We go on to show additional structure of embeddings, such as the equivalence of embedding and matching Bayes risks, and the equivalence of various notions of non-redudancy. Using these results, we establish that indirect elicitation, a necessary condition for consistency, is also sufficient when working with polyhedral surrogates.