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

TL;DR

This paper introduces an angular calibration method for high-dimensional binary classifiers that is provably well-calibrated and Bregman-optimal, and shows that classical Platt scaling can also achieve these properties under certain conditions.

Contribution

It presents the first calibration strategy that guarantees both calibration and optimality in high-dimensional settings, using angular interpolation and Bregman divergence minimization.

Findings

Angular calibration is well-calibrated in high dimensions.

The estimator of the angle between weights is consistent.

Classical Platt scaling converges to the Bregman-optimal solution under certain conditions.

Abstract

We study the fundamental problem of calibrating a linear binary classifier of the form $\sigma(\hat{w}^\top x)$, where the feature vector $x$ is Gaussian, $\sigma$ is a link function, and $\hat{w}$ is an estimator of the true linear weight $w^\star$. By interpolating with a noninformative $\textit{chance classifier}$, we construct a well-calibrated predictor whose interpolation weight depends on the angle $\angle(\hat{w}, w_\star)$ between the estimator $\hat{w}$ 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 $\angle(\hat{w}, w_\star)$ can be consistently estimated. Furthermore, the resulting predictor is uniquely $\textit{Bregman-optimal}$, 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.