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Register a new userDistributed Optimization with Global Constraints Using Noisy Measurements
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We propose a new distributed optimization algorithm for solving a class of constrained optimization problems in which (a) the objective function is separable (i.e., the sum of local objective functions of agents), (b) the optimization variables of distributed agents, which are subject to nontrivial local constraints, are coupled by global constraints, and (c) only noisy observations are available to estimate (the gradients of) local objective functions. In many practical scenarios, agents may not be willing to share their optimization variables with others. For this reason, we propose a distributed algorithm that does not require the agents to share their optimization variables with each other; instead, each agent maintains a local estimate of the global constraint functions and share the estimate only with its neighbors. These local estimates of constraint functions are updated using a consensus-type algorithm, while the local optimization variables of each agent are updated using a first-order method based on noisy estimates of gradient. We prove that, when the agents adopt the proposed algorithm, their optimization variables converge with probability 1 to an optimal point of an approximated problem based on the penalty method.
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The paper proves convergence to global optima for a class of distributed algorithms for nonconvex optimization in network-based multi-agent settings. Agents are permitted to communicate over a time-varying undirected graph. Each agent is assumed to possess a local objective function (assumed to be smooth, but possibly nonconvex). The paper considers algorithms for optimizing the sum function. A distributed algorithm of the consensus+innovations type is proposed which relies on first-order information at the agent level. Under appropriate conditions on network connectivity and the cost objective, convergence to the set of global optima is achieved by an annealing-type approach, with decaying Gaussian noise independently added into each agents update step. It is shown that the proposed algorithm converges in probability to the set of global minima of the sum function.
This paper considers a distributed convex optimization problem over a time-varying multi-agent network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling equality constraints. Over directed graphs, a distributed algorithm is proposed that incorporates the push-sum protocol into dual subgradient methods. Under the convexity assumption, the optimality of primal and dual variables, and constraint violations is first established. Then the explicit convergence rates of the proposed algorithm are obtained. Finally, some numerical experiments on the economic dispatch problem are provided to demonstrate the efficacy of the proposed algorithm.
This paper first proposes an N-block PCPM algorithm to solve N-block convex optimization problems with both linear and nonlinear constraints, with global convergence established. A linear convergence rate under the strong second-order conditions for optimality is observed in the numerical experiments. Next, for a starting point, an asynchronous N-block PCPM algorithm is proposed to solve linearly constrained N-block convex optimization problems. The numerical results demonstrate the sub-linear convergence rate under the bounded delay assumption, as well as the faster convergence with more short-time iterations than a synchronous iterative scheme.
We consider continuous-time dynamics for distributed optimization with set constraints in the note. To handle the computational complexity of projection-based dynamics due to solving a general quadratic optimization subproblem with projection, we propose a distributed projection-free dynamics by employing the Frank-Wolfe method, also known as the conditional gradient algorithm. The process searches a feasible descent direction with solving an alternative linear optimization instead of a quadratic one. To make the algorithm implementable over weight-balanced digraphs, we design one dynamics for the consensus of local decision variables and another dynamics of auxiliary variables to track the global gradient. Then we prove the convergence of the dynamical systems to the optimal solution, and provide detailed numerical comparisons with both projection-based dynamics and other distributed projection-free algorithms.
One of the most widely used methods for solving large-scale stochastic optimization problems is distributed asynchronous stochastic gradient descent (DASGD), a family of algorithms that result from parallelizing stochastic gradient descent on distributed computing architectures (possibly) asychronously. However, a key obstacle in the efficient implementation of DASGD is the issue of delays: when a computing node contributes a gradient update, the global model parameter may have already been updated by other nodes several times over, thereby rendering this gradient information stale. These delays can quickly add up if the computational throughput of a node is saturated, so the convergence of DASGD may be compromised in the presence of large delays. Our first contribution is that, by carefully tuning the algorithms step-size, convergence to the critical set is still achieved in mean square, even if the delays grow unbounded at a polynomial rate. We also establish finer results in a broad class of structured optimization problems (called variationally coherent), where we show that DASGD converges to a global optimum with probability $1$ under the same delay assumptions. Together, these results contribute to the broad landscape of large-scale non-convex stochastic optimization by offering state-of-the-art theoretical guarantees and providing insights for algorithm design.
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