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

Smoothed Concordance-Assisted Learning for Optimal Treatment Decision in High Dimensional Data