We propose dynamic sampled stochastic approximation (SA) methods for stochastic optimization with a heavy-tailed distribution (with finite 2nd moment). The objective is the sum of a smooth convex function with a convex regularizer. Typically, it is assumed an oracle with an upper bound @math on its variance (OUBV). Differently, we assume an oracle with . This rarely addressed setup is more aggressive but realistic, where the variance may not be bounded. Our methods achieve optimal iteration complexity and (near) optimal oracle complexity. For the smooth convex class, we use an accelerated SA method a la FISTA which achieves, given tolerance @math , the optimal iteration complexity of @math with a near-optimal oracle complexity of @math . This improves upon Ghadimi and Lan [, 156:59-99, 2016] where it is assumed an OUBV. For the strongly convex class, our method achieves optimal iteration complexity of @math and optimal oracle complexity of @math . This improves upon [, 134:127-155, 2012] where it is assumed an OUBV. In terms of variance, our bounds are local: they depend on variances @math at solutions @math and the per unit distance multiplicative variance @math . For the smooth convex class, there exist policies such that our bounds resemble those obtained if it was assumed an OUBV with @math . For the strongly convex class such property is obtained exactly if the condition number is estimated or in the limit for better conditioned problems or for larger initial batch sizes. In any case, if it is assumed an OUBV, our bounds are thus much sharper since typically @math .
Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardn4516420ess of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is important. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We introduce a new notion of discrepancy between functions, and use it to reduce problems of stochastic convex optimization to statistical parameter estimation, which can be lower bounded using information-theoretic methods. Using this approach, we improve upon known results and obtain tight minimax complexity estimates for various function classes.