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

We propose a new variable selection algorithm, subsample-ordered least-angle regression (solar), and its coordinate descent generalization, solar-cd. Solar re-constructs lasso paths using the $L_0$ norm and averages the resulting solution paths across subsamples. Path averaging retains the ranking information of the informative variables while averaging out sensitivity to high dimensionality, improving variable selection stability, efficiency, and accuracy. We prove that: (i) with a high probability, path averaging perfectly separates informative variables from redundant variables on the average $L_0$ path; (ii) solar variable selection is consistent and accurate; and (iii) the probability that solar omits weak signals is controllable for finite sample size. We also demonstrate that: (i) solar yields, with less than $1/3$ of the lasso computation load, substantial improvements over lasso in terms of the sparsity (64-84\% reduction in redundant variable selection) and accuracy of variable selection; (ii) compared with the lasso safe/strong rule and variable screening, solar largely avoids selection of redundant variables and rejection of informative variables in the presence of complicated dependence structures; (iii) the sparsity and stability of solar conserves residual degrees of freedom for data-splitting hypothesis testing, improving the accuracy of post-selection inference on weak signals with limited $n$; (iv) replacing lasso with solar in bootstrap selection (e.g., bolasso or stability selection) produces a multi-layer variable ranking scheme that improves selection sparsity and ranking accuracy with the computation load of only one lasso realization; and (v) given the computation resources, solar bootstrap selection is substantially faster (98\% lower computation time) than the theoretical maximum speedup for parallelized bootstrap lasso (confirmed by Amdahl's law).