Stochastic Gradient Made Stable: A Manifold Propagation Approach for Large-Scale Optimization

Abstract

Stochastic gradient descent (SGD) holds as a classical method to build large scale machine learning models over big data. A stochastic gradient is typically calculated from a limited number of samples (known as mini-batch), which potentially incurs a high variance and causes the estimated parameters to bounce around the optimal solution. To improve the stability of stochastic gradient, recent years have witnessed the proposal of several semi-stochastic gradient descent algorithms, which distinguish themselves from standard SGD by incorporating global information into gradient computation. In this paper, we contribute a novel stratified semi-stochastic gradient descent (S3GD) algorithm to this nascent research area, accelerating the optimization of a large family of composite convex functions. Though theoretically converging faster, prior semi-stochastic algorithms are found to suffer from high iteration complexity, which makes them even slower than SGD in practice on many datasets. In our proposed S3GD, the semi-stochastic gradient is calculated based on efficient manifold propagation, which can be numerically accomplished by sparse matrix multiplications. This way S3GD is able to generate a highly-accurate estimate of the exact gradient from each mini-batch with largely-reduced computational complexity. Theoretic analysis reveals that the proposed S3GD elegantly balances the geometric algorithmic convergence rate against the space and time complexities during the optimization. The efficacy of S3GD is also experimentally corroborated on several large-scale benchmark datasets.