提出新条件统一验证可微代理损失的一致性
Consistency Conditions for Differentiable Surrogate Losses
- 引入强间接诱导（strong IE）作为更易验证的等价条件
- 证明在高维下普通IE失效，但强IE仍保证一致性
- 适用于可微凸损失，对模型设计有指导意义
离散预测任务中代理损失的统计一致性通常通过校准性检验。然而直接验证校准性较为困难。已有研究发现，对于多面体型代理损失，间接诱导（IE）与校准性等价且更易验证。本文首次针对非多面体型代理损失，特别是凸可微损失类，给出相应结果。我们证明，在弱条件下，一维情形下IE与校准性等价；并构造反例表明该等价性在高维不成立。为此提出强IE，其验证难度相当，且能保证校准性。进一步证明：强IE对可微代理损失是校准性的充分必要条件，尤其对强凸可微代理损失成立。最后，将这些理论应用于多种问题，展示IE与强IE在设计和分析一致可微代理损失中的强大能力。
原文摘要 · Abstract (English)
The statistical consistency of surrogate losses for discrete prediction tasks is often checked via the condition of calibration. However, directly verifying calibration can be arduous. Recent work shows that for polyhedral surrogates, a less arduous condition, indirect elicitation (IE), is still equivalent to calibration. We give the first results of this type for non-polyhedral surrogates, specifically the class of convex differentiable losses. We first prove that under mild conditions, IE and calibration are equivalent for one-dimensional losses in this class. We construct a counter-example that shows that this equivalence fails in higher dimensions. This motivates the introduction of strong IE, a strengthened form of IE that is equally easy to verify. We establish that strong IE implies calibration for differentiable surrogates and is both necessary and sufficient for strongly convex, differentiable surrogates. Finally, we apply these results to a range of problems to demonstrate the power of IE and strong IE for designing and analyzing consistent differentiable surrogates.
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