Debiased inference for high-dimensional regression models has received substantial recent attention to ensure regularized estimators have valid inference. Many existing methods focus on achieving Neyman orthogonality through explicitly constructing projections onto the space of nuisance parameters, which is infeasible when an explicit form of the projection is unavailable. We introduce a general debiasing framework, Debiased Profile $M$-Estimation (DPME), which applies to a broad class of models and does not require model-specific Neyman orthogonalization or projection derivations as in existing methods. Our approach begins with obtaining an initial estimator of the parameters by optimizing a penalized objective function. To correct for the bias introduced by penalization, we construct a one-step estimator using the Newton--Raphson update, applied to the gradient of a profile function defined as the optimal objective function with the parameter of interest held fixed. We use numerical differentiation without requiring explicit calculation of the gradients. The resulting DPME estimator is shown to be asymptotically linear and normally distributed. Through extensive simulations, we demonstrate that the proposed method achieves better coverage rates than existing alternatives with largely reduced computational cost. Finally, we illustrate the utility of our method by applying it to estimate a treatment rule for multiple myeloma.

翻译：高维回归模型的去偏推断近期受到广泛关注，旨在确保正则化估计量具有有效的推断能力。现有方法多通过显式构建对讨厌参数空间的投影来实现奈曼正交性，但当投影的显式形式不可得时，这类方法难以实施。本文提出一个通用的去偏框架——去偏轮廓M估计（Debiased Profile M-Estimation，DPME），该框架适用于广泛的模型类别，无需像现有方法那样进行模型特定的奈曼正交化或投影推导。我们的方法首先通过优化带惩罚的目标函数获得参数的初始估计量，然后采用牛顿-拉夫森更新构造一步估计量以校正惩罚引入的偏差——该更新应用于轮廓函数的梯度，其中轮廓函数定义为在感兴趣参数固定的条件下最优目标函数。我们采用数值微分方法，无需显式计算梯度。理论分析表明，所得DPME估计量具有渐近线性和正态性。通过大量仿真实验，我们证明所提方法在显著降低计算成本的同时，实现了优于现有方法的覆盖率。最后，我们通过将其应用于多发性骨髓瘤的治疗规则估计，展示了该方法的实用性。