Inference with non-differentiable surrogate loss in a general high-dimensional classification framework

• URL: http://arxiv.org/abs/2405.11723v1
• Date: Mon, 20 May 2024 01:50:35 GMT
• Title: Inference with non-differentiable surrogate loss in a general high-dimensional classification framework
• Authors: Muxuan Liang, Yang Ning, Maureen A Smith, Ying-Qi Zhao,
• Abstract summary: We propose a kernel-smoothed decorrelated score to construct hypothesis testing and interval estimations. Specifically, we adopt kernel approximations to smooth the discontinuous gradient near discontinuity points. We establish the limiting distribution of the kernel-smoothed decorrelated score and its cross-fitted version in a high-dimensional setup.
• Score: 4.792322531593389
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Penalized empirical risk minimization with a surrogate loss function is often used to derive a high-dimensional linear decision rule in classification problems. Although much of the literature focuses on the generalization error, there is a lack of valid inference procedures to identify the driving factors of the estimated decision rule, especially when the surrogate loss is non-differentiable. In this work, we propose a kernel-smoothed decorrelated score to construct hypothesis testing and interval estimations for the linear decision rule estimated using a piece-wise linear surrogate loss, which has a discontinuous gradient and non-regular Hessian. Specifically, we adopt kernel approximations to smooth the discontinuous gradient near discontinuity points and approximate the non-regular Hessian of the surrogate loss. In applications where additional nuisance parameters are involved, we propose a novel cross-fitted version to accommodate flexible nuisance estimates and kernel approximations. We establish the limiting distribution of the kernel-smoothed decorrelated score and its cross-fitted version in a high-dimensional setup. Simulation and real data analysis are conducted to demonstrate the validity and superiority of the proposed method.

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