In this paper, we propose a simple variant of the original SVRG, called variance reduced stochastic gradient descent (VR-SGD). Unlike the choices of snapshot and starting points in SVRG and its proximal variant, Prox-SVRG, the two vectors of VR-SGD are set to the average and last iterate of the previous epoch, respectively. The settings allow us to use much larger learning rates, and also make our convergence analysis more challenging. We also design two different update rules for smooth and non-smooth objective functions, respectively, which means that VR-SGD can tackle non-smooth and/or non-strongly convex problems directly without any reduction techniques. Moreover, we analyze the convergence properties of VR-SGD for strongly convex problems, which show that VR-SGD attains linear convergence. Different from its counterparts that have no convergence guarantees for non-strongly convex problems, we also provide the convergence guarantees of VR-SGD for this case, and empirically verify that VR-SGD with varying learning rates achieves similar performance to its momentum accelerated variant that has the optimal convergence rate $\mathcal{O}(1/T^2)$. Finally, we apply VR-SGD to solve various machine learning problems, such as convex and non-convex empirical risk minimization, leading eigenvalue computation, and neural networks. Experimental results show that VR-SGD converges significantly faster than SVRG and Prox-SVRG, and usually outperforms state-of-the-art accelerated methods, e.g., Katyusha.

In this paper, we propose a simple variant of the original stochastic variance reduction gradient (SVRG), where hereafter we refer to as the variance reduced stochastic gradient descent (VR-SGD). Different from the choices of the snapshot point and starting point in SVRG and its proximal variant, Prox-SVRG, the two vectors of each epoch in VR-SGD are set to the average and last iterate of the previous epoch, respectively. This setting allows us to use much larger learning rates or step sizes than SVRG, e.g., 3/(7L) for VR-SGD vs 1/(10L) for SVRG, and also makes our convergence analysis more challenging. In fact, a larger learning rate enjoyed by VR-SGD means that the variance of its stochastic gradient estimator asymptotically approaches zero more rapidly. Unlike common stochastic methods such as SVRG and proximal stochastic methods such as Prox-SVRG, we design two different update rules for smooth and non-smooth objective functions, respectively. In other words, VR-SGD can tackle non-smooth and/or non-strongly convex problems directly without using any reduction techniques such as quadratic regularizers. Moreover, we analyze the convergence properties of VR-SGD for strongly convex problems, which show that VR-SGD attains a linear convergence rate. We also provide the convergence guarantees of VR-SGD for non-strongly convex problems. Experimental results show that the performance of VR-SGD is significantly better than its counterparts, SVRG and Prox-SVRG, and it is also much better than the best known stochastic method, Katyusha.