Convergence Analysis of Accelerated Stochastic Gradient Descent under the Growth Condition

We study the convergence of accelerated stochastic gradient descent for strongly convex objectives under the growth condition, which states that the variance of stochastic gradient is bounded by a multiplicative part that grows with the full gradient, and a constant additive part. Through the lens of the growth condition, we investigate four widely used accelerated methods: Nesterov's accelerated method (NAM), robust momentum method (RMM), accelerated dual averaging method (ADAM), and implicit ADAM (iADAM). While these methods are known to improve the convergence rate of SGD under the condition that the stochastic gradient has bounded variance, it is not well understood how their convergence rates are affected by the multiplicative noise. In this paper, we show that these methods all converge to a neighborhood of the optimum with accelerated convergence rates (compared to SGD) even under the growth condition. In particular, NAM, RMM, iADAM enjoy acceleration only with a mild multiplicative noise, while ADAM enjoys acceleration even with a large multiplicative noise. Furthermore, we propose a generic tail-averaged scheme that allows the accelerated rates of ADAM and iADAM to nearly attain the theoretical lower bound (up to a logarithmic factor in the variance term)

Second-Order Stochastic Optimization for Machine Learning in Linear Time

First-order stochastic methods are the state-of-the-art in large-scale machine learning optimization owing to efficient per-iteration complexity. Second-order methods, while able to provide faster convergence, have been much less explored due to the high cost of computing the second-order information. In this paper we develop second-order stochastic methods for optimization problems in machine learning that match the per-iteration cost of gradient based methods, and in certain settings improve upon the overall running time over popular first-order methods. Furthermore, our algorithm has the desirable property of being implementable in time linear in the sparsity of the input data