arXiv:2502.15131v3 Announce Type: replace-cross
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.