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

Futoshi Futami, Atsushi Nitanda

TL;DR

The paper addresses the gap between calibration and accuracy by studying the smooth calibration error (smCE) in a finite-sample setting. It develops a uniform convergence bound for smCE and connects training smCE to the norm of the empirical functional gradient, providing a principled route to improve calibration without sacrificing accuracy. Through three case studies—gradient boosting trees, kernel boosting, and two-layer neural networks—the authors derive concrete conditions under which both calibration and misclassification performance can be simultaneously guaranteed, highlighting trade-offs between iterations and model complexity. These results offer theoretical guidance for designing probabilistic predictors with provable calibration guarantees in common learning paradigms and illustrate how calibration improvements can be achieved via controlling the gradient dynamics. The framework also clarifies the relationship between smCE and traditional calibration metrics, underscoring the practical relevance of uniform convergence results for real-world probabilistic prediction tasks.

Abstract

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.

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

TL;DR

The paper addresses the gap between calibration and accuracy by studying the smooth calibration error (smCE) in a finite-sample setting. It develops a uniform convergence bound for smCE and connects training smCE to the norm of the empirical functional gradient, providing a principled route to improve calibration without sacrificing accuracy. Through three case studies—gradient boosting trees, kernel boosting, and two-layer neural networks—the authors derive concrete conditions under which both calibration and misclassification performance can be simultaneously guaranteed, highlighting trade-offs between iterations and model complexity. These results offer theoretical guidance for designing probabilistic predictors with provable calibration guarantees in common learning paradigms and illustrate how calibration improvements can be achieved via controlling the gradient dynamics. The framework also clarifies the relationship between smCE and traditional calibration metrics, underscoring the practical relevance of uniform convergence results for real-world probabilistic prediction tasks.

Abstract

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.

Paper Structure

This paper contains 50 sections, 21 theorems, 234 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem setting of binary classification
  4. Calibration metrics and loss functions
  5. Uniform convergence of the smooth CE
  6. Controlling the smooth CE via functional gradient
  7. Case study I: Gradient Boosting Tree
  8. Case study II: Kernel Boosting
  9. Case study III: Two-layer neural network
  10. Related work
  11. Conclusion and limitation
  12. Additional facts about proper losses
  13. Discussion about Calibration metrics
  14. True distance calibration and smooth CE
  15. (i) Prediction-only access
  16. ...and 35 more sections

Key Result

Theorem 1

Let $S_\mathrm{te} \sim \mathcal{D}^n$ and $S_\mathrm{tr} \sim \mathcal{D}^n$ be independent test and training datasets. Then, for any $\delta>0$, with probability at least $1-\delta$ over the draw of $S_\mathrm{te}$ and $S_\mathrm{tr}$, we have:

Figures (6)

  • Figure 1: GBT experiments on the toy dataset defined in Eq. equation \ref{['toydata_separable']} with $m \geq d$
  • Figure 2: GBT experiments on the toy dataset defined in Eq. equation \ref{['toydata_separable']} with $m < d$
  • Figure 3: GBT experiments for UCI breast cancer dataset $d=30$
  • Figure 4: Two-layer neural network using toydata defined in Eq. equation \ref{['toydata_separable']}
  • Figure 5: Two-layer neural network using toy dataset defined in Eq. equation \ref{['toy_eq_unseparable']}
  • ...and 1 more figures

Theorems & Definitions (41)

  • Definition 1: Smooth CE blasiok2023unify
  • Definition 2: Dual smooth CE blasiok2023unify
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • proof : Proof outline
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • ...and 31 more