On the Consistency of Top-k Surrogate Losses
This work addresses a theoretical gap in top-k classification for computer vision, providing foundational insights that could improve performance in ambiguous label scenarios, though it is incremental in advancing existing calibration theory.
The paper tackles the problem of ensuring that top-k classification methods are theoretically consistent, meaning they align with Bayes optimality, by analyzing top-k calibration and proposing new surrogate losses. It shows that previously proposed hinge-like losses are not calibrated, introduces a new consistent hinge loss, and demonstrates its advantages over cross entropy in linear settings.
The top-$k$ error is often employed to evaluate performance for challenging classification tasks in computer vision as it is designed to compensate for ambiguity in ground truth labels. This practical success motivates our theoretical analysis of consistent top-$k$ classification. Surprisingly, it is not rigorously understood when taking the $k$-argmax of a vector is guaranteed to return the $k$-argmax of another vector, though doing so is crucial to describe Bayes optimality; we do both tasks. Then, we define top-$k$ calibration and show it is necessary and sufficient for consistency. Based on the top-$k$ calibration analysis, we propose a class of top-$k$ calibrated Bregman divergence surrogates. Our analysis continues by showing previously proposed hinge-like top-$k$ surrogate losses are not top-$k$ calibrated and suggests no convex hinge loss is top-$k$ calibrated. On the other hand, we propose a new hinge loss which is consistent. We explore further, showing our hinge loss remains consistent under a restriction to linear functions, while cross entropy does not. Finally, we exhibit a differentiable, convex loss function which is top-$k$ calibrated for specific $k$.