This paper investigates the partial linear model by Least Absolute Deviation (LAD) regression. We parameterize the nonparametric term using Deep Neural Networks (DNNs) and formulate a penalized LAD problem for estimation. Specifically, our model exhibits the following challenges. First, the regularization term can be nonconvex and nonsmooth, necessitating the introduction of infinite dimensional variational analysis and nonsmooth analysis into the asymptotic normality discussion. Second, our network must expand (in width, sparsity level and depth) as more samples are observed, thereby introducing additional difficulties for theoretical analysis. Third, the oracle of the proposed estimator is itself defined through a ultra high-dimensional, nonconvex, and discontinuous optimization problem, which already entails substantial computational and theoretical challenges. Under such the challenges, we establish the consistency, convergence rate, and asymptotic normality of the estimator. Furthermore, we analyze the oracle problem itself and its continuous relaxation. We study the convergence of a proximal subgradient method for both formulations, highlighting their structural differences lead to distinct computational subproblems along the iterations. In particular, the relaxed formulation admits significantly cheaper proximal updates, reflecting an inherent trade-off between statistical accuracy and computational tractability.

翻译：本文研究基于最小绝对偏差回归的部分线性模型。我们使用深度神经网络对非参数项进行参数化，并构建惩罚LAD问题进行估计。具体而言，我们的模型面临以下挑战：首先，正则化项可能为非凸且非光滑，这要求将无限维变分分析与非光滑分析引入渐近正态性讨论；其次，随着观测样本增加，网络需在宽度、稀疏性水平和深度上扩展，从而为理论分析带来额外困难；第三，所提估计量的真值本身通过一个超高维、非凸且不连续的优化问题定义，这已带来显著的计算与理论挑战。在此类挑战下，我们建立了估计量的一致性、收敛速率及渐近正态性。此外，我们分析了真值问题本身及其连续松弛形式，研究了两种形式下邻近次梯度方法的收敛性，强调其结构差异导致迭代过程中产生不同的计算子问题。特别地，松弛形式允许显著更廉价的邻近更新，这反映了统计精度与计算可处理性之间的内在权衡。