Stochastic Optimization and Sparse Statistical Recovery: An Optimal Algorithm in High Dimensions

Alekh Agarwal
Microsoft Research

We develop and analyze stochastic optimization algorithms for problems in which the expected loss is strongly convex, and the optimum is (approximately) sparse. Previous approaches are able to exploit only one of these two structures, yielding an $order(pdim/T)$ convergence rate for strongly convex objectives in $pdim$ dimensions, and an $order(sqrt{(spindex log pdim)/T})$ convergence rate when the optimum is $spindex$-sparse. Our algorithm is based on successively solving a series of $ell_1$-regularized optimization problems using Nesterov's dual averaging algorithm. We establish that the error of our solution after $T$ iterations is at most $order((spindex logpdim)/T)$, with natural extensions to approximate sparsity. Our results apply to locally Lipschitz losses including the logistic, exponential, hinge and least-squares losses. By recourse to statistical minimax results, we show that our convergence rates are optimal up to multiplicative constant factors. The effectiveness of our approach is also confirmed in numerical simulations, in which we compare to several baselines on a least-squares regression problem. [Based on joint work with Sahand Negahban and Martin Wainwright]

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