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.

翻译：在分类问题中，使用替代损失函数进行惩罚经验风险最小化通常用于推导高维线性决策规则。尽管现有文献主要关注泛化误差，但缺乏有效的推断程序来识别估计决策规则的关键驱动因素，尤其是当替代损失函数不可微时。本文提出一种核平滑去相关得分方法，用于构建利用分段线性替代损失（具有不连续梯度和非正则海塞矩阵）估计的线性决策规则的假设检验与区间估计。具体而言，我们采用核逼近在不连续点附近平滑不连续梯度，并近似替代损失的非正则海塞矩阵。在涉及额外 nuisance 参数的应用场景中，我们提出一种新型交叉拟合版本，以适应灵活的 nuisance 估计与核逼近。我们在高维框架下建立了核平滑去相关得分及其交叉拟合版本的极限分布。通过模拟实验与真实数据分析验证了所提方法的有效性与优越性。