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This paper studies the distributed optimization problem over directed networks with noisy information-sharing. To resolve the imperfect communication issue over directed networks, a series of noise-robust variants of Push-Pull/AB method have been developed. These methods improve the robustness of Push-Pull method against the information-sharing noise through adding small factors on weight matrices and replacing the global gradient tracking with the cumulative gradient tracking. Based on the two techniques, we propose a new variant of the Push-Pull method by presenting a novel mechanism of inter-agent information aggregation, named variance-reduced aggregation (VRA). VRA helps us to release some conditions on the objective function and networks. When the objective function is convex and the sharing-information noise is variance-unbounded, it can be shown that the proposed method converges to the optimal solution almost surely. When the objective function is strongly convex and the sharing-information noise is variance-bounded, the proposed method achieves the convergence rate of $\mathcal{O}\left(k^{-(1-\epsilon)}\right)$ in the mean square sense, where $\epsilon$ could be close to 0 infinitely. Simulated experiments on ridge regression problems verify the effectiveness of the proposed method.

The paper studies the distributed stochastic compositional optimization problems over networks, where all the agents' inner-level function is the sum of each agent's private expectation function. Focusing on the aggregative structure of the inner-level function, we employ the hybrid variance reduction method to obtain the information on each agent's private expectation function, and apply the dynamic consensus mechanism to track the information on each agent's inner-level function. Then by combining with the standard distributed stochastic gradient descent method, we propose a distributed aggregative stochastic compositional gradient descent method. When the objective function is smooth, the proposed method achieves the optimal convergence rate $\mathcal{O}\left(K^{-1/2}\right)$. We further combine the proposed method with the communication compression and propose the communication compressed variant distributed aggregative stochastic compositional gradient descent method. The compressed variant of the proposed method maintains the optimal convergence rate $\mathcal{O}\left(K^{-1/2}\right)$. Simulated experiments on decentralized reinforcement learning verify the effectiveness of the proposed methods.

We study the distributed stochastic compositional optimization problems over directed communication networks in which agents privately own a stochastic compositional objective function and collaborate to minimize the sum of all objective functions. We propose a distributed stochastic compositional gradient descent method, where the gradient tracking and the stochastic correction techniques are employed to adapt to the networks' directed structure and increase the accuracy of inner function estimation. When the objective function is smooth, the proposed method achieves the convergence rate $\mathcal{O}\left(k^{-1/2}\right)$ and sample complexity $\mathcal{O}\left(\frac{1}{\epsilon^2}\right)$ for finding the ($\epsilon$)-stationary point. When the objective function is strongly convex, the convergence rate is improved to $\mathcal{O}\left(k^{-1}\right)$. Moreover, the asymptotic normality of Polyak-Ruppert averaged iterates of the proposed method is also presented. We demonstrate the empirical performance of the proposed method on model-agnostic meta-learning problem and logistic regression problem.

The popular $\mathcal{AB}$/push-pull method for distributed optimization problem may unify much of the existing decentralized first-order methods based on gradient tracking technique. More recently, the stochastic gradient variant of $\mathcal{AB}$/Push-Pull method ($\mathcal{S}$-$\mathcal{AB}$) has been proposed, which achieves the linear rate of converging to a neighborhood of the global minimizer when the step-size is constant. This paper is devoted to the asymptotic properties of $\mathcal{S}$-$\mathcal{AB}$ with diminishing stepsize. Specifically, under the condition that each local objective is smooth and the global objective is strongly-convex, we first present the boundedness of the iterates of $\mathcal{S}$-$\mathcal{AB}$ and then show that the iterates converge to the global minimizer with the rate $\mathcal{O}\left(1/\sqrt{k}\right)$. Furthermore, the asymptotic normality of Polyak-Ruppert averaged $\mathcal{S}$-$\mathcal{AB}$ is obtained and applications on statistical inference are discussed. Finally, numerical tests are conducted to demonstrate the theoretic results.

Considering the constrained stochastic optimization problem over a time-varying random network, where the agents are to collectively minimize a sum of objective functions subject to a common constraint set, we investigate asymptotic properties of a distributed algorithm based on dual averaging of gradients. Different from most existing works on distributed dual averaging algorithms that mainly concentrating on their non-asymptotic properties, we not only prove almost sure convergence and the rate of almost sure convergence, but also asymptotic normality and asymptotic efficiency of the algorithm. Firstly, for general constrained convex optimization problem distributed over a random network, we prove that almost sure consensus can be archived and the estimates of agents converge to the same optimal point. For the case of linear constrained convex optimization, we show that the mirror map of the averaged dual sequence identifies the active constraints of the optimal solution with probability 1, which helps us to prove the almost sure convergence rate and then establish asymptotic normality of the algorithm. Furthermore, we also verify that the algorithm is asymptotically optimal. To the best of our knowledge, it seems to be the first asymptotic normality result for constrained distributed optimization algorithms. Finally, a numerical example is provided to justify the theoretical analysis.