We describe and analyze two stochastic methods for $\ell_1$ regularized loss minimization problems, such as the Lasso. The first method updates the weight of a single feature at each iteration while the second method updates the entire weight vector but only uses a single training example at each iteration. In both methods, the choice of feature/example is uniformly at random. Our theoretical runtime analysis suggests that the stochastic methods should outperform state-of-the-art deterministic approaches, including their deterministic counterparts, when the size of the problem is large. We demonstrate the advantage of stochastic methods by experimenting with synthetic and natural data sets.
Discussion
We have made the C++ code for both SCD and SMIDAS algorithms described in the paper available at:
http://ttic.uchicago.edu/~tewari/code/
Please try it out and let us know of your feedback!
Shai Shalev-Shwartz and Ambuj Tewari