提出平滑校准误差的统一收敛理论，揭示梯度可控制校准性能。
Uniform convergence of the smooth calibration error and its relationship with functional gradient
- 构建平滑校准误差的统一收敛界，连接训练误差与泛化差距。
- 证明损失函数的泛函梯度能有效抑制训练校准误差。
- 为提升模型校准性提供理论依据，适合关注可靠性建模的研究者。
校准是高风险应用中可靠概率预测的关键要求。然而，现有理论对如何同时实现高精度与良好校准的理解仍有限，多数研究仅在受限条件下提供经验验证或理论保证。本文聚焦平滑校准误差（smooth calibration error, CE），建立其统一收敛界，表明平滑CE由训练集上的平滑CE与泛化间隙之和界定。进一步证明损失函数的泛函梯度可有效控制训练阶段的平滑CE。基于此框架，分析了三种代表性算法：梯度提升树、核提升与两层神经网络，分别推导出分类与校准性能同时保障的条件。结果为设计具有可证明校准性的可靠概率模型提供了新理论视角与实践指导。
原文摘要 · Abstract (English)
Calibration is a critical requirement for reliable probabilistic prediction, especially in high-risk applications. However, the theoretical understanding of which learning algorithms can simultaneously achieve high accuracy and good calibration remains limited, and many existing studies provide empirical validation or a theoretical guarantee in restrictive settings. To address this issue, in this work, we focus on the smooth calibration error (CE) and provide a uniform convergence bound, showing that the smooth CE is bounded by the sum of the smooth CE over the training dataset and a generalization gap. We further prove that the functional gradient of the loss function can effectively control the training smooth CE. Based on this framework, we analyze three representative algorithms: gradient boosting trees, kernel boosting, and two-layer neural networks. For each, we derive conditions under which both classification and calibration performances are simultaneously guaranteed. Our results offer new theoretical insights and practical guidance for designing reliable probabilistic models with provable calibration guarantees.
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