Abstract: Top-k classification is a generalization of multiclass classification used widely in information retrieval, image classification, and other extreme classification settings. Several hinge-like (piecewise-linear) surrogates have been proposed for the problem, yet all are either non-convex or inconsistent. For the proposed hinge-like surrogates that are convex (i.e., polyhedral), we apply the embedding framework Finocchiaro et al. (2019) to determine the prediction problem for which the surrogate is consistent. These problems can all be interpreted as variants of top-k classification, which may be better aligned with some applications. We leverage this analysis to derive constraints on the conditional label distributions under which these proposed surrogates become consistent for top-k. It has been further suggested that every convex hinge-like surrogate must be inconsistent for top-k. Yet, we use the same embedding framework to give the first consistent polyhedral surrogate for this problem.