Randomised Splitting Methods and Stochastic Gradient Descent

We explore an explicit link between stochastic gradient descent using common batching strategies and splitting methods for ordinary differential equations. From this perspective, we introduce a new minibatching strategy (called Symmetric Minibatching Strategy) for stochastic gradient optimisation which shows greatly reduced stochastic gradient bias (from $\mathcal{O}(h^2)$ to $\mathcal{O}(h^4)$ in the optimiser stepsize $h$), when combined with momentum-based optimisers. We justify why momentum is needed to obtain the improved performance using the theory of backward analysis for splitting integrators and provide a detailed analytic computation of the stochastic gradient bias on a simple example. Further, we provide improved convergence guarantees for this new minibatching strategy using Lyapunov techniques that show reduced stochastic gradient bias for a fixed stepsize (or learning rate) over the class of strongly-convex and smooth objective functions. Via the same techniques we also improve the known results for the Random Reshuffling strategy for stochastic gradient descent methods with momentum. We argue that this also leads to a faster convergence rate when considering a decreasing stepsize schedule. Both the reduced bias and efficacy of decreasing stepsizes are demonstrated numerically on several motivating examples.

Random Reshuffling for Stochastic Gradient Langevin Dynamics

We examine the use of different randomisation policies for stochastic gradient algorithms used in sampling, based on first-order (or overdamped) Langevin dynamics, the most popular of which is known as Stochastic Gradient Langevin Dynamics. Conventionally, this algorithm is combined with a specific stochastic gradient strategy, called Robbins-Monro. In this work, we study an alternative strategy, Random Reshuffling, and show convincingly that it leads to improved performance via: a) a proof of reduced bias in the Wasserstein metric for strongly convex, gradient Lipschitz potentials; b) an analytical demonstration of reduced bias for a Gaussian model problem; and c) an empirical demonstration of reduced bias in numerical experiments for some logistic regression problems. This is especially important since Random Reshuffling is typically more efficient due to memory access and cache reasons. Such acceleration for the Random Reshuffling policy is familiar from the optimisation literature on stochastic gradient descent.