Calibration of predicted probabilities is critical for reliable machine learning, yet it is poorly understood how standard training procedures yield well-calibrated models. This work provides the first theoretical proof that canonical $L_{2}$-regularized empirical risk minimization directly controls the smooth calibration error (smCE) without post-hoc correction or specialized calibration-promoting regularizer. We establish finite-sample generalization bounds for smCE based on optimization error, regularization strength, and the Rademacher complexity. We then instantiate this theory for models in reproducing kernel Hilbert spaces, deriving concrete guarantees for kernel ridge and logistic regression. Our experiments confirm these specific guarantees, demonstrating that $L_{2}$-regularized ERM can provide a well-calibrated model without boosting or post-hoc recalibration. The source code to reproduce all experiments is available at https://github.com/msfuji0211/erm_calibration.

翻译：预测概率的校准对于可靠的机器学习至关重要，然而标准训练过程如何产生校准良好的模型却鲜为人知。本工作首次从理论上证明，经典的$L_{2}$正则化经验风险最小化能够直接控制平滑校准误差，而无需进行事后校正或使用专门的促进校准的正则化项。我们基于优化误差、正则化强度和Rademacher复杂度，为平滑校准误差建立了有限样本泛化界。随后，我们在再生核希尔伯特空间模型中实例化了该理论，为核岭回归和逻辑回归推导出具体的保证。我们的实验证实了这些具体保证，表明$L_{2}$正则化经验风险最小化无需进行提升或事后重新校准即可提供校准良好的模型。用于复现所有实验的源代码可在 https://github.com/msfuji0211/erm_calibration 获取。