Without-Replacement Sampling for Stochastic Gradient Methods: Convergence Results and Application to Distributed Optimization
This addresses the gap between theoretical analysis and practical implementation of sampling methods in optimization, with incremental improvements for distributed settings.
The paper tackles the problem of analyzing stochastic gradient methods with without-replacement sampling, providing competitive convergence guarantees for algorithms like SGD and SVRG, and applies this to develop a nearly-optimal distributed algorithm for regularized least squares with communication and runtime complexities up to logarithmic factors.
Stochastic gradient methods for machine learning and optimization problems are usually analyzed assuming data points are sampled \emph{with} replacement. In practice, however, sampling \emph{without} replacement is very common, easier to implement in many cases, and often performs better. In this paper, we provide competitive convergence guarantees for without-replacement sampling, under various scenarios, for three types of algorithms: Any algorithm with online regret guarantees, stochastic gradient descent, and SVRG. A useful application of our SVRG analysis is a nearly-optimal algorithm for regularized least squares in a distributed setting, in terms of both communication complexity and runtime complexity, when the data is randomly partitioned and the condition number can be as large as the data size per machine (up to logarithmic factors). Our proof techniques combine ideas from stochastic optimization, adversarial online learning, and transductive learning theory, and can potentially be applied to other stochastic optimization and learning problems.