🤖 AI Summary
This paper addresses the real-time tracking of solutions to time-varying variational inequalities (VIs), relaxing the restrictive assumptions—namely strong monotonicity/strong convexity and sublinear growth of solution trajectories—prevalent in prior work. We consider two novel settings: (1) non-monotone time-varying VIs with sublinearly growing solution paths, and (2) periodically time-varying VIs whose solution paths need not be sublinear. Our method employs a unified analytical framework grounded in dynamical systems modeling, Lyapunov stability analysis, and nonlinear stability theory. Contributions include: (i) the first universal upper bound on tracking error for non-monotone time-varying VIs; (ii) rigorous proof that discrete-time dynamical systems for periodic time-varying VIs exhibit coexisting chaotic and asymptotically convergent behaviors; and (iii) numerical simulations validating both the tightness of the theoretical error bound and the accuracy of predicted dynamical behaviors.
📝 Abstract
Tracking the solution of time-varying variational inequalities is an important problem with applications in game theory, optimization, and machine learning. Existing work considers time-varying games or time-varying optimization problems. For strongly convex optimization problems or strongly monotone games, these results provide tracking guarantees under the assumption that the variation of the time-varying problem is restrained, that is, problems with a sublinear solution path. In this work we extend existing results in two ways: In our first result, we provide tracking bounds for (1) variational inequalities with a sublinear solution path but not necessarily monotone functions, and (2) for periodic time-varying variational inequalities that do not necessarily have a sublinear solution path-length. Our second main contribution is an extensive study of the convergence behavior and trajectory of discrete dynamical systems of periodic time-varying VI. We show that these systems can exhibit provably chaotic behavior or can converge to the solution. Finally, we illustrate our theoretical results with experiments.