An Embedding Framework for Consistent Polyhedral Surrogates
An Embedding Framework for Consistent Polyhedral Surrogates
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发表时间：
2019-07
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通讯作者：
J. Finocchiaro;Rafael M. Frongillo;Bo Waggoner
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作者：
J. Finocchiaro;Rafael M. Frongillo;Bo Waggoner
我们形式化并研究了通过嵌入来解决分类或排序等问题的凸代理损失函数的自然方法。在这种方法中，将有限多个预测（例如类别）中的每一个嵌入为 R^d 中的一个点，将原始损失值分配给这些点，并将其之间的损失凸化以获得代理。我们证明这种方法在很大程度上等同于处理多面体（分段线性凸）损失。此外，给定任何多面体损失$L$，我们给出了一个链接函数的构造，通过该函数$L$是它嵌入的损失的一致代理。我们继续通过文献中各种多面体代理的一致性或不一致的简洁证明来说明这种嵌入框架的强大功能。
We formalize and study the natural approach of designing convex surrogate loss functions via embeddings for problems such as classification or ranking. In this approach, one embeds each of the finitely many predictions (e.g. classes) as a point in R^d, assigns the original loss values to these points, and convexifies the loss in between to obtain a surrogate. We prove that this approach is equivalent, in a strong sense, to working with polyhedral (piecewise linear convex) losses. Moreover, given any polyhedral loss $L$, we give a construction of a link function through which $L$ is a consistent surrogate for the loss it embeds. We go on to illustrate the power of this embedding framework with succinct proofs of consistency or inconsistency of various polyhedral surrogates in the literature.