{"id":392960,"date":"2017-04-07T00:00:32","date_gmt":"2017-04-07T07:00:32","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=392960"},"modified":"2021-07-23T10:57:57","modified_gmt":"2021-07-23T17:57:57","slug":"optimal-algorithms-non-smooth-distributed-optimization-networks","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/optimal-algorithms-non-smooth-distributed-optimization-networks\/","title":{"rendered":"Optimal Algorithms for Non-Smooth Distributed Optimization in Networks"},"content":{"rendered":"

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in\u00a0O<\/em>(1\/\u221at<\/em>), the structure of the communication network only impacts a second-order term in\u00a0O<\/em>(1\/t<\/em>), where\u00a0t<\/em>\u00a0is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a\u00a0d1\/4<\/sup>\u00a0multiplicative factor of the optimal convergence rate, where\u00a0d<\/em> is the underlying dimension.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order 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