Online Distributed Optimization With Nonconvex Objective Functions Via Dynamic Regrets
In this paper, the problem of online distributed optimization subject to a convex set is studied by employing a network of agents, where the objective functions allocated to agents are nonconvex. Each agent only has access to its own objective function information at the previous time, and can only...
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| Vydáno v: | IEEE transactions on automatic control Ročník 68; číslo 11; s. 1 - 16 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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New York
IEEE
01.11.2023
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9286, 1558-2523 |
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| Abstract | In this paper, the problem of online distributed optimization subject to a convex set is studied by employing a network of agents, where the objective functions allocated to agents are nonconvex. Each agent only has access to its own objective function information at the previous time, and can only communicate with its immediate neighbors via a time-varying directed graph. To tackle this problem, first, a new online distributed algorithm with gradient information is proposed based on consensus algorithms and projection-free strategies. Of particular interest is that dynamic regrets, whose offline benchmarks are to pursue the stationary points at each time, are employed to measure the performance of the algorithm. In the worst case, the difficulty in achieving sublinear bounds of dynamic regrets is characterized by the deviation in the objective function sequence, as well as the deviation in the gradient sequence. Under mild assumptions on the graph and the objective functions, we prove that if the deviation in the objective function sequence is sublinear with the square root of the time horizon, and if the deviation in the gradient sequence is sublinear with the time horizon, then dynamic regrets grow sublinearly. Second, considering the case where the gradient information of the objective functions is not available, we propose a zeroth-order online distributed projection-free algorithm, by which agents make decisions only depending on the random zeroth-order oracle. It turns out that under the same conditions as in the first case, if the smoothing parameters in the random zeroth-order oracles scale inversely with the time horizon, then the expectations of dynamic regrets increase sublinearly. Finally, simulations are presented to demonstrate the effectiveness of our theoretical results. |
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| AbstractList | In this paper, the problem of online distributed optimization subject to a convex set is studied by employing a network of agents, where the objective functions allocated to agents are nonconvex. Each agent only has access to its own objective function information at the previous time, and can only communicate with its immediate neighbors via a time-varying directed graph. To tackle this problem, first, a new online distributed algorithm with gradient information is proposed based on consensus algorithms and projection-free strategies. Of particular interest is that dynamic regrets, whose offline benchmarks are to pursue the stationary points at each time, are employed to measure the performance of the algorithm. In the worst case, the difficulty in achieving sublinear bounds of dynamic regrets is characterized by the deviation in the objective function sequence, as well as the deviation in the gradient sequence. Under mild assumptions on the graph and the objective functions, we prove that if the deviation in the objective function sequence is sublinear with the square root of the time horizon, and if the deviation in the gradient sequence is sublinear with the time horizon, then dynamic regrets grow sublinearly. Second, considering the case where the gradient information of the objective functions is not available, we propose a zeroth-order online distributed projection-free algorithm, by which agents make decisions only depending on the random zeroth-order oracle. It turns out that under the same conditions as in the first case, if the smoothing parameters in the random zeroth-order oracles scale inversely with the time horizon, then the expectations of dynamic regrets increase sublinearly. Finally, simulations are presented to demonstrate the effectiveness of our theoretical results. In this article, the problem of online distributed optimization subject to a convex set is studied by employing a network of agents, where the objective functions allocated to agents are nonconvex. Each agent only has access to its own objective function information at the previous time, and can only communicate with its immediate neighbors via a time-varying directed graph. To tackle this problem, first, a new online distributed algorithm with gradient information is proposed based on consensus algorithms and projection-free strategies. Of particular interest is that dynamic regrets, whose offline benchmarks are to pursue the stationary points at each time, are employed to measure the performance of the algorithm. Under mild assumptions on the graph and the objective functions, we prove that if the deviation in the objective function sequence is sublinear with the square root of the time horizon, and if the deviation in the gradient sequence is sublinear with the time horizon, then dynamic regrets grow sublinearly. Second, considering the case where the gradient information of the objective functions is not available, we propose a zeroth-order online distributed projection-free algorithm, by which agents make decisions only depending on the random zeroth-order oracle. It turns out that under the same conditions as in the first case, if the smoothing parameters in the random zeroth-order oracles scale inversely with the time horizon, then the expectations of dynamic regrets increase sublinearly. Finally, simulations are presented to demonstrate the effectiveness of our theoretical results. |
| Author | Wang, Long Lu, Kaihong |
| Author_xml | – sequence: 1 givenname: Kaihong orcidid: 0000-0002-5265-7800 surname: Lu fullname: Lu, Kaihong organization: College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, China – sequence: 2 givenname: Long orcidid: 0000-0001-5600-8157 surname: Wang fullname: Wang, Long organization: Center for Systems and Control, College of Engineering, China |
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| SubjectTerms | Algorithms Benchmark testing Convexity Deviation Distributed algorithms Dynamic regrets Graph theory Heuristic algorithms Linear programming Mirrors multi-agent networks nonconvex optimization online distributed optimization Optimization Power system dynamics |
| Title | Online Distributed Optimization With Nonconvex Objective Functions Via Dynamic Regrets |
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