Tracking solutions of time-varying variational inequalities

TL;DR

This work studies online tracking of solutions to time-varying variational inequalities (VIs). It first shows sublinear tracking for contractive schemes under tameness, and then develops aggregation-based methods to exploit periodic structure, achieving logarithmic or constant tracking bounds under strong monotonicity and Lipschitzness. A second major thrust analyzes the dynamics of periodic VI-induced systems and reveals a rich spectrum from convergence to chaos (Li-Yorke) when step sizes are large, highlighting the nuanced behavior of simple gradient-type methods in periodic settings. The paper also provides concrete application examples (time-varying convex optimization, saddle-point problems, and generalized linear models) and discusses limitations and directions for future work, including stochastic feedback and stochastic games. Overall, it advances both the guarantees for tracking time-varying VIs and the understanding of dynamical phenomena arising in periodic configurations, with implications for online learning and game dynamics.

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.

Figures (4)

• Figure 1: Bifurcation diagram for the dynamical system generated by gradient descent on the periodic problem \ref{['eq:exp']}, with the learning rate as the varying parameter, and with initial point $x=-0.1$. The diagram represents the accumulation points of the sequence of iterates of gradient descent. Grey areas correspond to values of the learning rate for which the sequence diverged, axis were broken to hide large areas of divergence. Details on the simulation used to create this figure can be found in Appendix \ref{['app:bifurcation_diagram']}.
• Figure 2: Evidence that $\overline \Phi_\eta(x) > 2 |x|$ for all $x \in \mathbb R$, when $\eta=2$.
• Figure 3: Evidence of the existence of periodic points for $\overline \Phi_\eta$. Each intersection of the plotted curves with the $x$-axis corresponds to a fixed point of the iterated function, and thus of a periodic orbit for $\overline \Phi_\eta$. A fixed point of $\Phi_\eta^{(3)}$ that is not a fixed point of $\Phi_\eta$ has period $3$; the existence of such points implies chaos.
• Figure 4: Illustration of Example \ref{['example:starAttractor']}. The initial $x_0$ are sampled uniformly at random from the set $[-500,500]^2$. For Figure \ref{['star']}, we sample $n = 1000$ initial $x^{(i)}_0, i \in [n]$. The plot shows $x^{(i)}_{n/2}, \dots, x^{(i)}_{n}$ for each sample $i \in [n]$. Figure \ref{['convergence']} is based on the $n = 100$ samples for the initial $x^{(i)}_0, i \in [n]$. The plot shows the average convergence over these $n$ samples $\frac{1}{n}\sum_{i=1}^n \| x_t^{(i)} \|$.

Theorems & Definitions (46)

• Definition 1.1: Tame Time-Varying VI problem
• Definition 1.2: Time-varying VI problem with periodic solutions
• Remark 1.1: Ignoring Time-Variations and Practical Relevance
• Definition 2.1: Contractive Algorithms
• Theorem 2.1
• proof
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4