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We develop and analyze stochastic optimization algorithms for problems inwhich the expected loss is strongly convex, and the optimum is (approximately)sparse. Previous approaches are able to exploit only one of these twostructures, yielding an $\order(\pdim/T)$ convergence rate for strongly convexobjectives in $\pdim$ dimensions, and an $\order(\sqrt{(\spindex \log\pdim)/T})$ convergence rate when the optimum is $\spindex$-sparse. Ouralgorithm is based on successively solving a series of $\ell_1$-regularizedoptimization problems using Nesterov's dual averaging algorithm. We establishthat the error of our solution after $T$ iterations is at most$\order((\spindex \log\pdim)/T)$, with natural extensions to approximatesparsity. Our results apply to locally Lipschitz losses including the logistic,exponential, hinge and least-squares losses. By recourse to statistical minimaxresults, we show that our convergence rates are optimal up to multiplicativeconstant factors. The effectiveness of our approach is also confirmed innumerical simulations, in which we compare to several baselines on aleast-squares regression problem.

Stochastic optimization and sparse statistical recovery: An optimal algorithm for high dimensions