On the Consistency of Max-Margin Losses

TL;DR

This paper investigates the limitations of Max-Margin losses in multi-label settings, identifies conditions for their consistency, and proposes a corrected and generalized version called Restricted-Max-Margin that works under milder assumptions.

Contribution

It reveals the restrictive conditions for Max-Margin loss consistency in multi-label problems, proves consistency for tree-structured distances, and introduces Restricted-Max-Margin as a more flexible alternative.

Findings

Max-Margin loss is consistent only under restrictive conditions.

Tree-structured distances satisfy the conditions for consistency.

Restricted-Max-Margin generalizes binary SVM and is consistent under milder conditions.

Abstract

The foundational concept of Max-Margin in machine learning is ill-posed for output spaces with more than two labels such as in structured prediction. In this paper, we show that the Max-Margin loss can only be consistent to the classification task under highly restrictive assumptions on the discrete loss measuring the error between outputs. These conditions are satisfied by distances defined in tree graphs, for which we prove consistency, thus being the first losses shown to be consistent for Max-Margin beyond the binary setting. We finally address these limitations by correcting the concept of Max-Margin and introducing the Restricted-Max-Margin, where the maximization of the loss-augmented scores is maintained, but performed over a subset of the original domain. The resulting loss is also a generalization of the binary support vector machine and it is consistent under milder conditions on the discrete loss.