Non Asymptotic Bounds for Optimization via Online Multiplicative Stochastic Gradient Descent

The gradient noise of Stochastic Gradient Descent (SGD) is considered to play a key role in its properties (e.g. escaping low potential points and regularization). Past research has indicated that the covariance of the SGD error done via minibatching plays a critical role in determining its regularization and escape from low potential points. It is however not much explored how much the distribution of the error influences the behavior of the algorithm. Motivated by some new research in this area, we prove universality results by showing that noise classes that have the same mean and covariance structure of SGD via minibatching have similar properties. We mainly consider the Multiplicative Stochastic Gradient Descent (M-SGD) algorithm as introduced by Wu et al., which has a much more general noise class than the SGD algorithm done via minibatching. We establish nonasymptotic bounds for the M-SGD algorithm mainly with respect to the Stochastic Differential Equation corresponding to SGD via minibatching. We also show that the M-SGD error is approximately a scaled Gaussian distribution with mean $0$ at any fixed point of the M-SGD algorithm. We also establish bounds for the convergence of the M-SGD algorithm in the strongly convex regime.