Efficient Stochastic Gradient Hard Thresholding
Stochastic gradient hard thresholding methods have recently been shown to work favorably in solving large-scale empirical risk minimization problems under sparsity or rank constraint. Despite the improved iteration complexity over full gradient methods, the gradient evaluation and hard thresholding complexity of the existing stochastic algorithms usually scales linearly with data size, which could still be expensive when data is huge and the hard thresholding step could be as expensive as singular value decomposition in rank-constrained problems. To address these deficiencies, we propose an efficient hybrid stochastic gradient hard thresholding (HSG-HT) method that can be provably shown to have sample-size-independent gradient evaluation and hard thresholding complexity bounds. Specifically, we prove that the stochastic gradient evaluation complexity of HSG-HT scales linearly with inverse of sub-optimality and its hard thresholding complexity scales logarithmically. By applying the heavy ball acceleration technique, we further propose an accelerated variant of HSG-HT which can be shown to have improved factor dependence on restricted condition number in the quadratic case. Numerical results confirm our theoretical affirmation and demonstrate the computational efficiency of the proposed methods.
The article analyses convergence in hard-thresholding algorithms and proposes an accelerated stochastic hybrid hard thresholding method that displays better convergence with respect to the compared methods.  The article is dense but relatively fine to follow. Theoretical development seems to be complete and accurate, though I admit I have not throughly followed the full derivation. Experimental section is in accordance with the theoretical claims and is more than sufficient.  Just for the sake of reproducibility of the results an exhaustive pseudocode or repository should be made available as a companion to the article to further strength the autor's points.