Stochastic optimization and sparse statistical recovery: An optimal algorithm for high dimensions
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 leastsquares 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 leastsquares regression problem.
 Publication:

arXiv eprints
 Pub Date:
 July 2012
 arXiv:
 arXiv:1207.4421
 Bibcode:
 2012arXiv1207.4421A
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Machine Learning;
 Mathematics  Optimization and Control
 EPrint:
 2 figures