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

Abstract

We study the fundamental problem of calibrating a linear binary classifier of the form $\ddot{I}ƒ({\hat{w}}^{\top }x)$, where the feature vector $x$ is Gaussian, $\ddot{I}ƒ$ is a link function, and $\hat{w}$ is an estimator of the true linear weight $w^{\star }$. By interpolating with a noninformative $\text{chance classifier}$, 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 $\text{Bregman-optimal}$, 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.

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