# Uniform convergence of the smooth calibration error and its relationship with functional gradient

arXiv: https://arxiv.org/abs/2505.19396 · HTML: https://mlsanity.com/paper/2505.19396/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 31 of its 62 references that are in the map.
- First posted on arXiv 26 May 2025.

## Builds on (most cited first, 30 of 31)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Measuring Calibration in Deep Learning](https://mlsanity.com/paper/1904.01685/) (2019)
- [Revisiting the Calibration of Modern Neural Networks](https://mlsanity.com/paper/2106.07998/) (2021)
- [Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration](https://mlsanity.com/paper/1910.12656/) (2019)
- [Uncertainty Quantification and Deep Ensembles](https://mlsanity.com/paper/2007.08792/) (2020)
- [Soft Calibration Objectives for Neural Networks](https://mlsanity.com/paper/2108.00106/) (2021)
- [Distribution-free binary classification: prediction sets, confidence intervals and calibration](https://mlsanity.com/paper/2006.10564/) (2020)
- [Posterior calibration and exploratory analysis for natural language processing models](https://mlsanity.com/paper/1508.05154/) (2015)
- [Better Uncertainty Calibration via Proper Scores for Classification and Beyond](https://mlsanity.com/paper/2203.07835/) (2022)
- [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)
- [Low-Degree Multicalibration](https://mlsanity.com/paper/2203.01255/) (2022)
- [When Does Optimizing a Proper Loss Yield Calibration?](https://mlsanity.com/paper/2305.18764/) (2023)
- [Margins, Shrinkage, and Boosting](https://mlsanity.com/paper/1303.4172/) (2013)
- [Multicalibration as Boosting for Regression](https://mlsanity.com/paper/2301.13767/) (2023)
- [Metrics of calibration for probabilistic predictions](https://mlsanity.com/paper/2205.09680/) (2022)
- [Large Scale Prediction with Decision Trees](https://mlsanity.com/paper/2104.13881/) (2021)
- [Early stopping for kernel boosting algorithms: A general analysis with localized complexities](https://mlsanity.com/paper/1707.01543/) (2017)
- [Gradient Descent can Learn Less Over-parameterized Two-layer Neural Networks on Classification Problems](https://mlsanity.com/paper/1905.09870/) (2019)
- [T-Cal: An optimal test for the calibration of predictive models](https://mlsanity.com/paper/2203.01850/) (2022)
- [Smooth Calibration, Leaky Forecasts, Finite Recall, and Nash Dynamics](https://mlsanity.com/paper/2210.07152/) (2022)
- [Calibration by Distribution Matching: Trainable Kernel Calibration Metrics](https://mlsanity.com/paper/2310.20211/) (2023)
- [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)
- [Functional Gradient Boosting based on Residual Network Perception](https://mlsanity.com/paper/1802.09031/) (2018)
- [Minimum-Risk Recalibration of Classifiers](https://mlsanity.com/paper/2305.10886/) (2023)
- [Testing Calibration in Nearly-Linear Time](https://mlsanity.com/paper/2402.13187/) (2024)

## Abstract (arXiv)

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.