Stochastic Gradient Descent (SGD) has become popular for solving large scale supervised machine learning optimization problems such as SVM, due to their strong theoretical guarantees. While the closely related Dual Coordinate Ascent (DCA) method has been implemented in various software packages, it has so far lacked good convergence analysis. This paper presents a new analysis of Stochastic Dual Coordinate Ascent (SDCA) showing that this class of methods enjoy strong theoretical guarantees that are comparable or better than SGD. This analysis justifies the effectiveness of SDCA for practical applications.
In addition, @cite_0 , and later @cite_7 have analyzed randomized versions of coordinate descent for unconstrained and constrained minimization of smooth convex functions. [Theorem 4] HsiehChLiKeSu08 applied these results to the dual SVM formulation. However, the resulting convergence rate is @math which is, as mentioned before, inferior to the results we obtain here. Furthermore, neither of these analyses can be applied to logistic regression due to their reliance on the smoothness of the dual objective function which is not satisfied for the dual formulation of logistic regression. We shall also point out again that all of these bounds are for the dual sub-optimality, while as mentioned before, we are interested in the primal sub-optimality.
@cite_0 @cite_7
2154685722 2095984592
We describe and analyze two stochastic methods for l1 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 or 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. In this paper we propose new methods for solving huge-scale optimization problems. For problems of this size, even the simplest full-dimensional vector operations are very expensive. Hence, we propose to apply an optimization technique based on random partial update of decision variables. For these methods, we prove the global estimates for the rate of convergence. Surprisingly enough, for certain classes of objective functions, our results are better than the standard worst-case bounds for deterministic algorithms. We present constrained and unconstrained versions of the method, and its accelerated variant. Our numerical test confirms a high efficiency of this technique on problems of very big size.