Variance Reduced Stochastic Gradient Descent with Neighbors

Stochastic Gradient Descent (SGD) is a workhorse in machine learning, yet its slow convergence can be a computational bottleneck. Variance reduction techniques such as SAG, SVRG and SAGA have been proposed to overcome this weakness, achieving linear convergence. However, these methods are either based on computations of full gradients at pivot points, or on keeping per data point corrections in memory. Therefore speed-ups relative to SGD may need a minimal number of epochs in order to materialize. This paper investigates algorithms that can exploit neighborhood structure in the training data to share and re-use information about past stochastic gradients across data points, which offers advantages in the transient optimization phase. As a side-product we provide a unified convergence analysis for a family of variance reduction algorithms, which we call memorization algorithms. We provide experimental results supporting our theory.Comment: Appears in: Advances in Neural Information Processing Systems 28 (NIPS 2015). 13 page

SVAG: Stochastic Variance Adjusted Gradient Descent and Biased Stochastic Gradients

We examine biased gradient updates in variance reduced stochastic gradient methods. For this purpose we introduce SVAG, a SAG/SAGA-like method with adjustable bias. SVAG is analyzed under smoothness assumptions and we provide step-size conditions for convergence that match or improve on previously known conditions for SAG and SAGA. The analysis highlights a step-size requirement difference between when SVAG is applied to cocoercive operators and when applied to gradients of smooth functions, a difference not present in ordinary gradient descent. This difference is verified with numerical experiments. A variant of SVAG that adaptively selects the bias is presented and compared numerically to SVAG on a set of classification problems. The adaptive SVAG frequently performs among the best and always improves on the worst-case performance of the non-adaptive variant

Stochastic Optimization with Variance Reduction for Infinite Datasets with Finite-Sum Structure

Stochastic optimization algorithms with variance reduction have proven successful for minimizing large finite sums of functions. Unfortunately, these techniques are unable to deal with stochastic perturbations of input data, induced for example by data augmentation. In such cases, the objective is no longer a finite sum, and the main candidate for optimization is the stochastic gradient descent method (SGD). In this paper, we introduce a variance reduction approach for these settings when the objective is composite and strongly convex. The convergence rate outperforms SGD with a typically much smaller constant factor, which depends on the variance of gradient estimates only due to perturbations on a single example.Comment: Advances in Neural Information Processing Systems (NIPS), Dec 2017, Long Beach, CA, United State