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- W2912297034 abstract "This paper considers the minimization of a sum of an expectation-valued smooth nonconvex function and a nonsmooth block-separable convex regularizer. By combining a randomized block-coordinate descent method with a proximal variable sample-size stochastic gradient (VSSG) method, we propose a randomized block proximal VSSG algorithm. In each iteration, a single block is randomly chosen to updates its estimates by {a VSSG scheme} with an increasing batch of sampled gradients, while the remaining blocks are kept invariant. By appropriately chosen batch sizes, we prove that every limit point for almost every sample path is a stationary point when blocks are chosen either randomly or cyclically. We further show that the ergodic mean-squared error of the gradient mapping {diminishes at the rate of $mathcal{O}(1/K) $ where $K$denotes the iteration index} and establish that the iteration and oracle complexity to obtain an $epsilon$-stationary point are $mathcal{O}(1/epsilon )$ and $mathcal{O}(1/epsilon^2)$, respectively. Furthermore, under a $ {mu}$-proximal Polyak-{L}ojasiewicz condition with the batch size increasing at a suitable geometric rate, we prove that the suboptimality diminishes at a {em geometric} rate, the {em optimal} deterministic rate. In addition, if $L_{rm ave}$ denotes the average of block-specific Lipschitz constants, the iteration and oracle complexity to obtain an $epsilon$-optimal solution are $mathcal{O}( {(L_{rm ave}/mu)}ln(1/epsilon))$ and $mathcal{O}left( (1/epsilon)^{1+c} right)$, respectively, {matching} the deterministic result. When $n=1$, we obtainthe {em optimal} red{oracle complexity bound} $mathcal{O}(1/epsilon) $ while $c>0$ when $ngeq 2$ represents the positive cost of multiple blocks. Finally, preliminary numerical experiments support our theoretical findings." @default.
- W2912297034 created "2019-02-21" @default.
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- W2912297034 date "2018-08-07" @default.
- W2912297034 modified "2023-09-27" @default.
- W2912297034 title "A Randomized Block Proximal Variable Sample-size Stochastic Gradient Method for Composite Nonconvex Stochastic Optimization" @default.
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