Optimal treatment regime is the individualized treatment decision rule which yields the optimal treatment outcomes in expectation. A simple case of treatment decision rule is the linear decision rule, which is characterized by its coefficients and its threshold. As patients heterogeneity data accumulates, it is of interest to estimate the optimal treatment regime with a linear decision rule in high-dimensional settings. Single timepoint optimal treatment regime can be estimated using Concordance-assisted learning (CAL), which is based on pairwise comparison. CAL is flexible and achieves good results in low dimensions. However, with an indicator function inside it, CAL is difficult to optimize in high dimensions. Recently, researchers proposed a smoothing approach using a family of cumulative distribution functions to replace indicator functions. In this paper, we introduce smoothed concordance-assisted learning (SMCAL), which applies the smoothing method to CAL using a family of sigmoid functions. We then prove the convergence rates of the estimated coefficients by analyzing the approximation and stochastic errors for the cases when the covariates are continuous. We also consider discrete covariates cases, and establish similar results. Simulation studies are conducted, demonstrating the advantage of our method.

翻译：最优治疗方案是一种个体化治疗决策规则，可使期望治疗结果达到最优。线性决策规则是其中的简单情形，其特性由系数和阈值共同决定。随着患者异质性数据的不断积累，在高维环境中估计具有线性决策规则的最优治疗方案具有重要意义。基于成对比较的一致性辅助学习（CAL）可用于估计单时间点最优治疗方案。CAL方法灵活且在低维情境中表现良好，但其内部包含指示函数，导致在高维条件下难以优化。近期，研究者提出采用一族累积分布函数替代指示函数的平滑方法。本文引入了平滑一致性辅助学习（SMCAL），该方法通过使用一族Sigmoid函数对CAL应用平滑技术。我们通过分析协变量连续情形下的逼近误差与随机误差，证明了估计系数的收敛速率；同时考虑离散协变量情形，建立了类似结论。仿真实验表明了我们方法的优越性。