# L2-Regularized Empirical Risk Minimization Guarantees Small Smooth Calibration Error

arXiv: https://arxiv.org/abs/2510.13450 · HTML: https://mlsanity.com/paper/2510.13450/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 36 references that are in the map.
- First posted on arXiv 15 Oct 2025.

## Builds on (most cited first, 22 of 22)

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Rotation Equivariant CNNs for Digital Pathology](https://mlsanity.com/paper/1806.03962/) (2018)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [Distribution-free binary classification: prediction sets, confidence intervals and calibration](https://mlsanity.com/paper/2006.10564/) (2020)
- [Non-Parametric Calibration for Classification](https://mlsanity.com/paper/1906.04933/) (2019)
- [A Unifying Theory of Distance from Calibration](https://mlsanity.com/paper/2211.16886/) (2022)
- [A Consistent and Differentiable Lp Canonical Calibration Error Estimator](https://mlsanity.com/paper/2210.07810/) (2022)
- [Distribution-free calibration guarantees for histogram binning without sample splitting](https://mlsanity.com/paper/2105.04656/) (2021)
- [Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenon](https://mlsanity.com/paper/1812.11167/) (2018)
- [When Does Optimizing a Proper Loss Yield Calibration?](https://mlsanity.com/paper/2305.18764/) (2023)
- [Calibrated and Sharp Uncertainties in Deep Learning via Density Estimation](https://mlsanity.com/paper/2112.07184/) (2021)
- [Multicalibration as Boosting for Regression](https://mlsanity.com/paper/2301.13767/) (2023)
- [A Large-Scale Study of Probabilistic Calibration in Neural Network Regression](https://mlsanity.com/paper/2306.02738/) (2023)
- [Calibration tests beyond classification](https://mlsanity.com/paper/2210.13355/) (2022)
- [Loss Minimization Yields Multicalibration for Large Neural Networks](https://mlsanity.com/paper/2304.09424/) (2023)
- [When is Multicalibration Post-Processing Necessary?](https://mlsanity.com/paper/2406.06487/) (2024)
- [On Computationally Efficient Multi-Class Calibration](https://mlsanity.com/paper/2402.07821/) (2024)
- [Testing Calibration in Nearly-Linear Time](https://mlsanity.com/paper/2402.13187/) (2024)
- [Information-theoretic Generalization Analysis for Expected Calibration Error](https://mlsanity.com/paper/2405.15709/) (2024)
- [PAC-Bayes Analysis for Recalibration in Classification](https://mlsanity.com/paper/2406.06227/) (2024)

## Abstract (arXiv)

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 L2-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 L2-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/ermcalibration.