Balancing Rates and Variance via Adaptive Batch-Sizes in First-Order Stochastic Optimization

Stochastic gradient descent is a canonical tool for addressing stochastic optimization problems, and forms the bedrock of modern machine learning and statistics. In this work, we seek to balance the fact that attenuating step-sizes is required for exact asymptotic convergence with the fact that larger constant step-sizes learn faster in finite time up to an error. To do so, rather than fixing the mini-batch and step-size at the outset, we propose a strategy to allow parameters to evolve adaptively. Specifically, the batch-size is set to be a piecewise-constant increasing sequence where the increase occurs when a suitable error criterion is satisfied. Moreover, the step-size is selected as that which yields the fastest convergence. The overall algorithm, two scale adaptive (TSA) scheme, is shown to inherit the exact asymptotic convergence of stochastic gradient method. More importantly, the optimal error decreasing rate is achieved theoretically, as well as an overall reduction in sample computational cost. Experimentally, we observe a favorable tradeoff relative to standard SGD schemes absorbing their advantages, which illustrates the significant performance of proposed TSA scheme.