公共测地线无法保证结构化支持向量机的费希尔一致性：最小反例与树度量分类

中文摘要 AI 辅助

结构化支持向量机（Structured SVM）的费希尔一致性存在一个已知必要条件，要求任务损失为一种度量，其中每个输出三元组都有一个公共测地线点。我们证明，该条件对于规范逐坐标argmax解码器并非充分条件：四输出单元星型图存在一个精确最优得分向量，其所有最大化器均严格非贝叶斯，且四输出是满足该条件的最小度量。随后，我们对顶点集为输出空间的正权重树度量进行完整分类：argmax一致性成立当且仅当该树为路径。分支树的失效仅局限于边界分布，每棵树在所有满支撑分布下均保留argmax性质。在满足公共测地线条件的度量中，五输出是满支撑反例的必要且充分条件；$K_{2,3}$是无穷$K_{m,n}$族中最小成员。我们还给出三维汉明立方体的一个满支撑反例。所有最优性断言均有精确原始-对偶证书。这些反例揭示了一个具体解码器缺口：在该多面体场景中，嵌入可保证校准链接的存在，而无需在每个 surrogate风险最小化器上验证规定的argmax链接。

英文摘要

A known necessary condition for Fisher consistency of the structured support vector machine requires the task loss to be a metric for which every output triple has a common geodesic point. We show that this condition is not sufficient for the canonical coordinate-wise argmax decoder. A four-output unit star admits an exactly optimal score vector whose maximizers are all strictly non-Bayes, and four outputs are minimal among metrics satisfying the condition. We then completely classify positively weighted tree metrics whose vertex set is the output space: argmax consistency holds if and only if the tree is a path. The failure on branching trees is confined to boundary distributions; every tree retains the argmax property at every full-support distribution. Among metrics satisfying the common-geodesic condition, five outputs are necessary and sufficient for a full-support counterexample; $K_{2,3}$ is the smallest member of an infinite $K_{m,n}$ family. We additionally give a full-support counterexample for the three-dimensional Hamming cube. All optimality claims have exact primal-dual certificates. The counterexamples expose a concrete decoder gap: in this polyhedral setting, an embedding can guarantee the existence of a calibrated link without validating a prescribed argmax link on every surrogate-risk minimizer.

发表机构

• Tsinghua University（清华大学）

机构由 AI 辅助整理，请以论文原文为准。

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14 pages, 1 figure