On sure early selection of the best subset

The early solution path, which tracks the first few variables that enter the model of a selection procedure, is of profound importance to scientific discoveries. In practice, it is often statistically hopeless to identify all the important features with no false discovery, let alone the intimidating expense of experiments to test their significance. Such realistic limitation calls for statistical guarantee for the early discoveries of a model selector. In this paper, we focus on the early solution path of best subset selection (BSS), where the sparsity constraint is set to be lower than {the true sparsity}. Under a sparse high-dimensional linear model, we establish the sufficient and (near) necessary condition for BSS to achieve sure early selection, or equivalently, zero false discovery throughout its early path. Essentially, this condition boils down to a lower bound of the minimum projected signal margin that characterizes the gap of the captured signal strength between sure selection models and those with spurious discoveries. Defined through projection operators, this margin is independent of the restricted eigenvalues of the design, suggesting the robustness of BSS against collinearity. Moreover, our model selection guarantee tolerates reasonable optimization error and thus applies to near best subsets. Finally, to overcome the computational hurdle of BSS under high dimension, we propose the "screen then select" (STS) strategy to reduce dimension for BSS. Our numerical experiments show that the resulting early path exhibits much lower false discovery rate (FDR) than LASSO, MCP and SCAD, especially in the presence of highly correlated design. We also investigate the early paths of the iterative hard thresholding algorithms, which are greedy computational surrogates for BSS, and which yield comparable FDR as our STS procedure.