We study the distributed optimization problem over a graphon with a continuum of nodes, which is regarded as the limit of the distributed networked optimization as the number of nodes goes to infinity. Each node has a private local cost function. The global cost function, which all nodes cooperatively minimize, is the integral of the local cost functions on the node set. We propose stochastic gradient descent and gradient tracking algorithms over the graphon. The two algorithms are the special cases of a class of graphon particle systems with time-varying random coefficients. For the class of graphon particle systems, we prove the existence and uniqueness of the solutions and establish the law of large numbers, which shows that the stochastic gradient descent algorithm over the graphon can be regarded as the spatio-temporal approximation of the discrete-time distributed stochastic gradient descent algorithm over the large-scale network. We also study the convergence of the two algorithms. For both kinds of algorithms, we show that by choosing the time-varying algorithm gains properly, all nodes' states achieve \(L\infty\)-concensus for a connected graphon. Furthermore, if the local cost functions are strongly convex, then all nodes' states converge to the minimizer of the global cost function and the auxiliary states in the stochastic gradient tracking algorithm converge to the gradient value of the global cost function at the minimizer uniformly in mean square.

Biography: Yan Chen received the B.S. degree in mathematics and applied mathematics from East China Normal University, Shanghai, China, in 2019, and is now pursuing the Ph.D. degree in operations research and cybernetics in the School of Mathematical Sciences, East China Normal University. Her research interests are stochastic control, mean field games, graphon mean field theory, multi-agent reinforcement learning and distributed optimization.