An Embedding Framework for the Design and Analysis of Consistent Polyhedral Surrogates

We formalize and study the natural approach of designing convex surrogate loss functions via embeddings, for discrete 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 $\mathbb{R}^d$, 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 …

analysis and analysis classification consistent design designing embedding embeddings framework functions loss natural prediction predictions ranking rankings study values via