Supplementary material: Weston-Watkins Hinge Loss and Ordered Partitions

Sections S3 to S5 contain the proofs for all results stated in the matching sections from the main article. References to contents in the supplementary material are prefixed with an "S", e.g., Lemma S3.20, eq. References to contents in the main article do not have any prefix, e.g., Theorem 1.3 and Section 1.4. We introduce notations in addition to those already defined in in the main article's Section 1.3. L always denotes the WW-hinge loss (Definition 1.4) and l always denotes the ordered partition loss (Definition 2.2).