Forward-reflected-backward method with variance reduction

We propose a variance reduced algorithm for solving monotone variational inequalities. Without assuming strong monotonicity, cocoercivity, or boundedness of the domain, we prove almost sure convergence of the iterates generated by the algorithm to a solution. In the monotone case, the ergodic average converges with the optimal O(1/k) rate of convergence. When strong monotonicity is assumed, the algorithm converges linearly, without requiring the knowledge of strong monotonicity constant. We finalize with extensions and applications of our results to monotone inclusions, a class of non-monotone variational inequalities and Bregman projections.Funding Agencies|EPFL Lausanne; European Research Council (ERC) under the European UnionEuropean Research Council (ERC) [725594]; Swiss National Science Foundation (SNSF)Swiss National Science Foundation (SNSF) [200021_178865/1]; Department of the Navy, Office of Naval Research (ONR)Office of Naval Research [N62909-17-1-211]; Hasler Foundation Program: Cyber Human Systems [16066]; Knut and Alice Wallenberg FoundationKnut &amp; Alice Wallenberg Foundation [305286]</p

Distributed forward-backward methods for ring networks

In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion in the product space. Each iteration of the studied algorithms requires one resolvent evaluation per set-valued operator, one forward evaluation per cocoercive operator, and two forward evaluations per monotone operator. Unlike existing methods, the structure of the proposed algorithms are suitable for distributed, decentralised implementation in ring networks without needing global summation to enforce consensus between nodes.Funding Agencies|CRUE-CSIC agreement; Springer Nature; Ministry of Science, Innovation and Universities of Spain; European Regional Development Fund (ERDF) of the European Commission [PGC2018097960-B-C22]; Generalitat Valenciana [AICO/2021/165]; Wallenberg Al, Autonomous Systems and Software Program (WASP) - Knut and Alice Wallenberg Foundation [305286]; Australian Research Council [DE200100063]; MINECO; European Social Fund [PRE2019-090751]</p