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Considering the constrained stochastic optimization problem over atime-varying random network, where the agents are to collectively minimize asum of objective functions subject to a common constraint set, we investigateasymptotic properties of a distributed algorithm based on dual averaging ofgradients. Different from most existing works on distributed dual averagingalgorithms that mainly concentrating on their non-asymptotic properties, we notonly prove almost sure convergence and the rate of almost sure convergence, butalso asymptotic normality and asymptotic efficiency of the algorithm. Firstly,for general constrained convex optimization problem distributed over a randomnetwork, we prove that almost sure consensus can be archived and the estimatesof agents converge to the same optimal point. For the case of linearconstrained convex optimization, we show that the mirror map of the averageddual sequence identifies the active constraints of the optimal solution withprobability 1, which helps us to prove the almost sure convergence rate andthen establish asymptotic normality of the algorithm. Furthermore, we alsoverify that the algorithm is asymptotically optimal. To the best of ourknowledge, it seems to be the first asymptotic normality result for constraineddistributed optimization algorithms. Finally, a numerical example is providedto justify the theoretical analysis.

Asymptotic properties of dual averaging algorithm for constrained distributed stochastic optimization