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

TL;DR

This paper introduces an optimal stochastic optimization algorithm that efficiently handles high-dimensional problems with strongly convex and sparse structures, improving convergence rates over previous methods.

Contribution

The authors develop a new algorithm combining $ ext{l}_1$ regularization and Nesterov's dual averaging to achieve optimal convergence rates for sparse, strongly convex problems.

Findings

Achieves $ ext{O}(rac{ ext{sp}}{ ext{T}})$ convergence rate for sparse solutions.

Extends to approximate sparsity with similar rates.

Validated through numerical experiments on least-squares regression.

Abstract

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 $\order((\spindex \log\pdim)/T)$, 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.