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

• URL: http://arxiv.org/abs/2502.15131v1
• Date: Fri, 21 Feb 2025 01:24:27 GMT
• Title: Optimal and Provable Calibration in High-Dimensional Binary Classification: Angular Calibration and Platt Scaling
• Authors: Yufan Li, Pragya Sur,
• Abstract summary: We develop a well-calibrated predictor whosetext weight depends on the angle $angle(hatw, w_star)$ between the estimator $hatw$ and the true linear weight $w_star$.<n>Our work is the first to provide a calibration strategy that satisfies both calibration and optimality properties provably in high dimensions.
• Score: 1.342834401139078
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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