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For decades, best subset selection (BSS) has eluded statisticians mainly dueto its computational bottleneck. However, until recently, modern computationalbreakthroughs have rekindled theoretical interest in BSS and have led to newfindings. Recently, \cite{guo2020best} showed that the model selectionperformance of BSS is governed by a margin quantity that is robust to thedesign dependence, unlike modern methods such as LASSO, SCAD, MCP, etc.Motivated by their theoretical results, in this paper, we also study thevariable selection properties of best subset selection for high-dimensionalsparse linear regression setup. We show that apart from the identifiabilitymargin, the following two complexity measures play a fundamental role incharacterizing the margin condition for model consistency: (a) complexity of\emph{residualized features}, (b) complexity of \emph{spurious projections}. Inparticular, we establish a simple margin condition that depends only on theidentifiability margin and the dominating one of the two complexity measures.Furthermore, we show that a margin condition depending on similar marginquantity and complexity measures is also necessary for model consistency ofBSS. For a broader understanding, we also consider some simple illustrativeexamples to demonstrate the variation in the complexity measures that refinesour theoretical understanding of the model selection performance of BSS underdifferent correlation structures.

Understanding Best Subset Selection: A Tale of Two C(omplex)ities