Distributed Delayed Stochastic Optimization

We analyze the convergence of gradient-based optimization algorithms that base their updates on delayed stochastic gradient information. The main application of our results is to the development of gradient-based distributed optimization algorithms where a master node performs parameter updates while worker nodes compute stochastic gradients based on local information in parallel, which may give rise to delays due to asynchrony. We take motivation from statistical problems where the size of the data is so large that it cannot fit on one computer; with the advent of huge datasets in biology, astronomy, and the internet, such problems are now common. Our main contribution is to show that for smooth stochastic problems, the delays are asymptotically negligible and we can achieve order-optimal convergence results. In application to distributed optimization, we develop procedures that overcome communication bottlenecks and synchronization requirements. We show $n$-node architectures whose optimization error in stochastic problems---in spite of asynchronous delays---scales asymptotically as \order(1 / \sqrt{nT}) after $T$ iterations. This rate is known to be optimal for a distributed system with $n$ nodes even in the absence of delays. We additionally complement our theoretical results with numerical experiments on a statistical machine learning task.Comment: 27 pages, 4 figure

Delayed Stochastic Algorithms for Distributed Weakly Convex Optimization

This paper studies delayed stochastic algorithms for weakly convex optimization in a distributed network with workers connected to a master node. More specifically, we consider a structured stochastic weakly convex objective function which is the composition of a convex function and a smooth nonconvex function. Recently, Xu et al. 2022 showed that an inertial stochastic subgradient method converges at a rate of $\mathcal{O}(\tau/\sqrt{K})$, which suffers a significant penalty from the maximum information delay $\tau$. To alleviate this issue, we propose a new delayed stochastic prox-linear ($\texttt{DSPL}$) method in which the master performs the proximal update of the parameters and the workers only need to linearly approximate the inner smooth function. Somewhat surprisingly, we show that the delays only affect the high order term in the complexity rate and hence, are negligible after a certain number of $\texttt{DSPL}$ iterations. Moreover, to further improve the empirical performance, we propose a delayed extrapolated prox-linear ($\texttt{DSEPL}$) method which employs Polyak-type momentum to speed up the algorithm convergence. Building on the tools for analyzing $\texttt{DSPL}$, we also develop improved analysis of delayed stochastic subgradient method ($\texttt{DSGD}$). In particular, for general weakly convex problems, we show that convergence of $\texttt{DSGD}$ only depends on the expected delay

Asynchronous Distributed Semi-Stochastic Gradient Optimization

With the recent proliferation of large-scale learning problems,there have been a lot of interest on distributed machine learning algorithms, particularly those that are based on stochastic gradient descent (SGD) and its variants. However, existing algorithms either suffer from slow convergence due to the inherent variance of stochastic gradients, or have a fast linear convergence rate but at the expense of poorer solution quality. In this paper, we combine their merits by proposing a fast distributed asynchronous SGD-based algorithm with variance reduction. A constant learning rate can be used, and it is also guaranteed to converge linearly to the optimal solution. Experiments on the Google Cloud Computing Platform demonstrate that the proposed algorithm outperforms state-of-the-art distributed asynchronous algorithms in terms of both wall clock time and solution quality

Stochastic stability of uncertain Hopfield neural networks with discrete and distributed delays

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2006 Elsevier Ltd.This Letter is concerned with the global asymptotic stability analysis problem for a class of uncertain stochastic Hopfield neural networks with discrete and distributed time-delays. By utilizing a Lyapunov–Krasovskii functional, using the well-known S-procedure and conducting stochastic analysis, we show that the addressed neural networks are robustly, globally, asymptotically stable if a convex optimization problem is feasible. Then, the stability criteria are derived in terms of linear matrix inequalities (LMIs), which can be effectively solved by some standard numerical packages. The main results are also extended to the multiple time-delay case. Two numerical examples are given to demonstrate the usefulness of the proposed global stability condition.This work was supported in part by the Engineering and Physical Sciences Research Council (EPSRC) of the UK under Grant GR/S27658/01, the Nuffield Foundation of the UK under Grant NAL/00630/G, and the Alexander von Humboldt Foundation of Germany