Resampled Confidence Regions with Exponential Shrinkage for the Regression Function of Binary Classification

Ambrus Tamás; Balázs Csanád Csáji

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arXiv · 2308.01835

Resampled Confidence Regions with Exponential Shrinkage for the Regression Function of Binary Classification

Ambrus Tamás·Balázs Csanád Csáji

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Abstract

The regression function is one of the key objects of binary classification, since it not only determines a Bayes optimal classifier, hence, defines an optimal decision boundary, but also encodes the conditional distribution of the output given the input. In this paper we build distribution-free confidence regions for the regression function for any user-chosen confidence level and any finite sample size based on a resampling test. These regions are abstract, as the model class can be almost arbitrary, e.g., it does not have to be finitely parameterized. We prove the strong uniform consistency of a new empirical risk minimization based approach for model classes with finite pseudo-dimensions and inverse Lipschitz parameterizations. We provide exponential probably approximately correct bounds on the $L_2$ sizes of these regions, and demonstrate the ideas on specific models. Additionally, we also consider a k-nearest neighbors based method, for which we prove strong pointwise bounds on the probability of exclusion. Finally, the constructions are illustrated on a logistic model class and compared to the asymptotic ellipsoids of the maximum likelihood estimator.

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Ambrus Tamás, Balázs Csanád Csáji. 2023-08-03. Resampled Confidence Regions with Exponential Shrinkage for the Regression Function of Binary Classification. https://arxiv.org/abs/2308.01835

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