We study high-dimensional rank regression when data are distributed across multiple machines and the loss is a non-additive U-statistic, as in convoluted rank regression (CRR). Classical communication-efficient surrogate likelihood (CSL) methods crucially rely on the additivity of the empirical loss and therefore break down for CRR, whose global loss couples all sample pairs across machines. We propose a distributed convoluted rank regression (DCRR) framework that constructs a similar surrogate loss and demonstrate its validity under the non-additive losses. We show that this surrogate shares the same population minimizer as the full-data CRR loss and yields estimators that are statistically equivalent to centralized CRR. Building on this, we develop a two-stage sparse DCRR procedure -- an iterative $\ell_1$-penalized stage followed by a folded-concave refinement -- and establish non-asymptotic error bounds, a distributed strong oracle property, and a DHBIC-type criterion for consistent model selection. A scaling result shows that the number of machines may diverge as $M = o({N/(s^2\log p)})$ while achieving centralized oracle rates with only $O(\log N)$ communication rounds. Simulations and a large-scale real data example demonstrate substantial gains over naive divide-and-conquer, particularly under heavy-tailed errors.

翻译：本文研究高维秩回归问题，其中数据分布在多台机器上且损失函数为非可加U统计量，如卷积秩回归（CRR）所示。经典的通信高效代理似然（CSL）方法严重依赖于经验损失的加性特性，因此无法适用于CRR——其全局损失耦合了所有机器间的样本对。我们提出分布式卷积秩回归（DCRR）框架，该框架构建了类似的代理损失，并证明了其在非可加损失下的有效性。我们证明该代理损失与全数据CRR损失具有相同总体极小值点，且能产生与集中式CRR统计等价的估计量。在此基础上，我们开发了两阶段稀疏DCRR方法——迭代$\ell_1$惩罚阶段后接折叠凹修正——并建立了非渐近误差界、分布式强预言机性质以及用于一致模型选择的DHBIC型准则。尺度分析表明，在仅需$O(\log N)$轮通信的情况下，机器数量可发散至$M = o({N/(s^2\log p)})$同时达到集中式预言机速率。仿真实验与大规模实际数据案例表明，该方法相较于朴素分治策略具有显著优势，尤其在重尾误差条件下表现突出。