Inverse Time Dependency in Convex Regularized Learning

In the conventional regularized learning, training time increases as the training set expands. Recent work on L2 linear SVM challenges this common sense by proposing the inverse time dependency on the training set size. In this paper, we first put forward a Primal Gradient Solver (PGS) to effectively solve the convex regularized learning problem. This solver is based on the stochastic gradient descent method and the Fenchel conjugate adjustment, employing the well-known online strongly convex optimization algorithm with logarithmic regret. We then theoretically prove the inverse dependency property of our PGS, embracing the previous work of the L2 linear SVM as a special case and enable the l_p-norm optimization to run within a bounded sphere, which qualifies more convex loss functions in PGS. We further illustrate this solver in three examples: SVM, logistic regression and regularized least square. Experimental results substantiate the property of the inverse dependency on training data size.