Semi-Stochastic Gradient Descent Methods

In this paper we study the problem of minimizing the average of a large number (n) of smooth convex loss functions. We propose a new method, S2GD (Semi-Stochastic Gradient Descent), which runs for one or several epochs in each of which a single full gradient and a random number of stochastic gradients is computed, following a geometric law. The total work needed for the method to output an varepsilon-accurate solution in expectation, measured in the number of passes over data, or equivalently, in units equivalent to the computation of a single gradient of the loss, is O((kappa/n)log(1/varepsilon)), where kappa is the condition number. This is achieved by running the method for O(log(1/varepsilon)) epochs, with a single gradient evaluation and O(kappa) stochastic gradient evaluations in each. The SVRG method of Johnson and Zhang arises as a special case. If our method is limited to a single epoch only, it needs to evaluate at most O((kappa/varepsilon)log(1/varepsilon)) stochastic gradients. In contrast, SVRG requires O(kappa/varepsilon2) stochastic gradients. To illustrate our theoretical results, S2GD only needs the workload equivalent to about 2.1 full gradient evaluations to find an 10-6-accurate solution for a problem with n=109 and kappa=103.