Tracking solutions of time-varying variational inequalities

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.

💡 Research Summary

The paper addresses the problem of tracking the solutions of time‑varying variational inequalities (VIs) in an online setting, a topic that unifies many problems in optimization, game theory, and statistical estimation. A variational inequality is defined by an operator F and a closed convex set Z; a point z* ∈ Z solves the VI if ⟨F(z*), z − z*⟩ ≥ 0 for all z ∈ Z. When the operator changes over time, denoted F₁, F₂,…, the learner must produce a sequence of iterates z₁, z₂,… without knowledge of future operators, and the quality of a method is measured by the cumulative tracking error τ_T = ∑_{t=1}^T‖z_t − z*_t‖², where z*_t is the (unique) solution of VIP(F_t, Z).

The authors distinguish two important families of time‑varying VIs.

1. Tame time‑varying VIs – The quadratic path length of the solution sequence, P*T = ∑{t=2}^T‖z*t − z*{t‑1}‖², grows sub‑linearly, i.e., P*_T ≤ c T^α for some α ∈