Distributed optimization over time-varying directed graphs

We consider distributed optimization by a collection of nodes, each having access to its own convex function, whose collective goal is to minimize the sum of the functions. The communications between nodes are described by a time-varying sequence of directed graphs, which is uniformly strongly connected. For such communications, assuming that every node knows its out-degree, we develop a broadcast-based algorithm, termed the subgradient-push, which steers every node to an optimal value under a standard assumption of subgradient boundedness. The subgradient-push requires no knowledge of either the number of agents or the graph sequence to implement. Our analysis shows that the subgradient-push algorithm converges at a rate of $O(\ln(t)/\sqrt{t})$, where the constant depends on the initial values at the nodes, the subgradient norms, and, more interestingly, on both the consensus speed and the imbalances of influence among the nodes

Cloud-Based Centralized/Decentralized Multi-Agent Optimization with Communication Delays

We present and analyze a computational hybrid architecture for performing multi-agent optimization. The optimization problems under consideration have convex objective and constraint functions with mild smoothness conditions imposed on them. For such problems, we provide a primal-dual algorithm implemented in the hybrid architecture, which consists of a decentralized network of agents into which centralized information is occasionally injected, and we establish its convergence properties. To accomplish this, a central cloud computer aggregates global information, carries out computations of the dual variables based on this information, and then distributes the updated dual variables to the agents. The agents update their (primal) state variables and also communicate among themselves with each agent sharing and receiving state information with some number of its neighbors. Throughout, communications with the cloud are not assumed to be synchronous or instantaneous, and communication delays are explicitly accounted for in the modeling and analysis of the system. Experimental results are presented to support the theoretical developments made.Comment: 8 pages, 4 figure

Tailoring Gradient Methods for Differentially-Private Distributed Optimization

Decentralized optimization is gaining increased traction due to its widespread applications in large-scale machine learning and multi-agent systems. The same mechanism that enables its success, i.e., information sharing among participating agents, however, also leads to the disclosure of individual agents' private information, which is unacceptable when sensitive data are involved. As differential privacy is becoming a de facto standard for privacy preservation, recently results have emerged integrating differential privacy with distributed optimization. Although such differential-privacy based privacy approaches for distributed optimization are efficient in both computation and communication, directly incorporating differential privacy design in existing distributed optimization approaches significantly compromises optimization accuracy. In this paper, we propose to redesign and tailor gradient methods for differentially-private distributed optimization, and propose two differential-privacy oriented gradient methods that can ensure both privacy and optimality. We prove that the proposed distributed algorithms can ensure almost sure convergence to an optimal solution under any persistent and variance-bounded differential-privacy noise, which, to the best of our knowledge, has not been reported before. The first algorithm is based on static-consensus based gradient methods and only shares one variable in each iteration. The second algorithm is based on dynamic-consensus (gradient-tracking) based distributed optimization methods and, hence, it is applicable to general directed interaction graph topologies. Numerical comparisons with existing counterparts confirm the effectiveness of the proposed approaches

On Stochastic Subgradient Mirror-Descent Algorithm with Weighted Averaging

This paper considers stochastic subgradient mirror-descent method for solving constrained convex minimization problems. In particular, a stochastic subgradient mirror-descent method with weighted iterate-averaging is investigated and its per-iterate convergence rate is analyzed. The novel part of the approach is in the choice of weights that are used to construct the averages. Through the use of these weighted averages, we show that the known optimal rates can be obtained with simpler algorithms than those currently existing in the literature. Specifically, by suitably choosing the stepsize values, one can obtain the rate of the order $1/k$ for strongly convex functions, and the rate $1/\sqrt{k}$ for general convex functions (not necessarily differentiable). Furthermore, for the latter case, it is shown that a stochastic subgradient mirror-descent with iterate averaging converges (along a subsequence) to an optimal solution, almost surely, even with the stepsize of the form $1/\sqrt{1+k}$, which was not previously known. The stepsize choices that achieve the best rates are those proposed by Paul Tseng for acceleration of proximal gradient methods