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

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.