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We propose a new variable selection algorithm, subsample-ordered least-angleregression (solar), and its coordinate descent generalization, solar-cd. Solarre-constructs lasso paths using the $L_0$ norm and averages the resultingsolution paths across subsamples. Path averaging retains the rankinginformation of the informative variables while averaging out sensitivity tohigh dimensionality, improving variable selection stability, efficiency, andaccuracy. We prove that: (i) with a high probability, path averaging perfectlyseparates informative variables from redundant variables on the average $L_0$path; (ii) solar variable selection is consistent and accurate; and (iii) theprobability that solar omits weak signals is controllable for finite samplesize. We also demonstrate that: (i) solar yields, with less than $1/3$ of thelasso computation load, substantial improvements over lasso in terms of thesparsity (64-84\% reduction in redundant variable selection) and accuracy ofvariable selection; (ii) compared with the lasso safe/strong rule and variablescreening, solar largely avoids selection of redundant variables and rejectionof informative variables in the presence of complicated dependence structures;(iii) the sparsity and stability of solar conserves residual degrees of freedomfor data-splitting hypothesis testing, improving the accuracy of post-selectioninference on weak signals with limited $n$; (iv) replacing lasso with solar inbootstrap selection (e.g., bolasso or stability selection) produces amulti-layer variable ranking scheme that improves selection sparsity andranking accuracy with the computation load of only one lasso realization; and(v) given the computation resources, solar bootstrap selection is substantiallyfaster (98\% lower computation time) than the theoretical maximum speedup forparallelized bootstrap lasso (confirmed by Amdahl's law).

Solar: $L_0$ solution path averaging for fast and accurate variable selection in high-dimensional data