Computing the Variance of Shuffling Stochastic Gradient Algorithms via Power Spectral Density Analysis
This work provides incremental theoretical insights into variance reduction for finite-sum optimization, benefiting researchers in machine learning optimization.
The paper tackled the problem of analyzing the stationary variances of shuffling-based stochastic gradient algorithms (SGD-RR and SGD-SO) compared to standard SGD, finding that their leading terms decrease in that order, with simple approximations derived under a stochastic noise model. Experiments on quadratic objectives validated these findings.
When solving finite-sum minimization problems, two common alternatives to stochastic gradient descent (SGD) with theoretical benefits are random reshuffling (SGD-RR) and shuffle-once (SGD-SO), in which functions are sampled in cycles without replacement. Under a convenient stochastic noise approximation which holds experimentally, we study the stationary variances of the iterates of SGD, SGD-RR and SGD-SO, whose leading terms decrease in this order, and obtain simple approximations. To obtain our results, we study the power spectral density of the stochastic gradient noise sequences. Our analysis extends beyond SGD to SGD with momentum and to the stochastic Nesterov's accelerated gradient method. We perform experiments on quadratic objective functions to test the validity of our approximation and the correctness of our findings.