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.
Categories: math.ST cs.LG stat.ME stat.ML stat.TH