# Optimal and Provable Calibration in High-Dimensional Binary Classification: Angular Calibration and Platt Scaling

arXiv: https://arxiv.org/abs/2502.15131 · HTML: https://mlsanity.com/paper/2502.15131/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 95 references that are in the map.
- First posted on arXiv 21 Feb 2025.

## Builds on (most cited first, 30 of 53)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [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)
- [MiniLM: Deep Self-Attention Distillation for Task-Agnostic Compression of Pre-Trained Transformers](https://mlsanity.com/paper/2002.10957/) (2020)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [OpenML: networked science in machine learning](https://mlsanity.com/paper/1407.7722/) (2014)
- [Surprises in High-Dimensional Ridgeless Least Squares Interpolation](https://mlsanity.com/paper/1903.08560/) (2019)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [BatchEnsemble: An Alternative Approach to Efficient Ensemble and Lifelong Learning](https://mlsanity.com/paper/2002.06715/) (2020)
- [Beyond temperature scaling: Obtaining well-calibrated multiclass probabilities with Dirichlet calibration](https://mlsanity.com/paper/1910.12656/) (2019)
- [ChemBERTa: Large-Scale Self-Supervised Pretraining for Molecular Property Prediction](https://mlsanity.com/paper/2010.09885/) (2020)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networks](https://mlsanity.com/paper/2002.10118/) (2020)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [Ensemble Distribution Distillation](https://mlsanity.com/paper/1905.00076/) (2019)
- [Statistical physics of inference: Thresholds and algorithms](https://mlsanity.com/paper/1511.02476/) (2015)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization](https://mlsanity.com/paper/2008.06786/) (2020)
- [A modern maximum-likelihood theory for high-dimensional logistic regression](https://mlsanity.com/paper/1803.06964/) (2018)
- [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)
- [Universality of empirical risk minimization](https://mlsanity.com/paper/2202.08832/) (2022)
- [Moment Multicalibration for Uncertainty Estimation](https://mlsanity.com/paper/2008.08037/) (2020)
- [Evaluating Uncertainty Quantification in End-to-End Autonomous Driving Control](https://mlsanity.com/paper/1811.06817/) (2018)
- [A unifying tutorial on Approximate Message Passing](https://mlsanity.com/paper/2105.02180/) (2021)
- [The phase transition for the existence of the maximum likelihood estimate in high-dimensional logistic regression](https://mlsanity.com/paper/1804.09753/) (2018)
- [A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-ℓ1-Norm Interpolated Classifiers](https://mlsanity.com/paper/2002.01586/) (2020)
- [The Lasso with general Gaussian designs with applications to hypothesis testing](https://mlsanity.com/paper/2007.13716/) (2020)
- [Don't Just Blame Over-parametrization for Over-confidence: Theoretical Analysis of Calibration in Binary Classification](https://mlsanity.com/paper/2102.07856/) (2021)

## Built on by (most cited first, 2 of 2)

- [Too Sharp, Too Sure: When Calibration Follows Curvature](https://mlsanity.com/paper/2604.20614/) (2026)
- [Multi-Environment GLAMP: Approximate Message Passing for Transfer Learning with Applications to Lasso-based Estimators](https://mlsanity.com/paper/2505.22594/) (2025)

## Abstract (arXiv)

We study the fundamental problem of calibrating a linear binary classifier of the form σ(hatw^top x), where the feature vector x is Gaussian, σ is a link function, and hatw is an estimator of the true linear weight w^star. By interpolating with a noninformative textitchance classifier, we construct a well-calibrated predictor whose interpolation weight depends on the angle angle(hatw, w_star) between the estimator hatw and the true linear weight w_star. We establish that this angular calibration approach is provably well-calibrated in a high-dimensional regime where the number of samples and features both diverge, at a comparable rate. The angle angle(hatw, w_star) can be consistently estimated. Furthermore, the resulting predictor is uniquely textitBregman-optimal, minimizing the Bregman divergence to the true label distribution within a suitable class of calibrated predictors. Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions. Additionally, we identify conditions under which a classical Platt-scaling predictor converges to our Bregman-optimal calibrated solution. Thus, Platt-scaling also inherits these desirable properties provably in high dimensions.