  The theory of integral quadratic constraints (IQCs) allows the certification
of exponential convergence of interconnected systems containing nonlinear or
uncertain elements. In this work, we adapt the IQC theory to study first-order
methods for smooth and strongly-monotone games and show how to design tailored
quadratic constraints to get tight upper bounds of convergence rates. Using
this framework, we recover the existing bound for the gradient method~(GD),
derive sharper bounds for the proximal point method~(PPM) and optimistic
gradient method~(OG), and provide \emph{for the first time} a global
convergence rate for the negative momentum method~(NM) with an iteration
complexity $\mathcal{O}(\kappa^{1.5})$, which matches its known lower bound. In
addition, for time-varying systems, we prove that the gradient method with
optimal step size achieves the fastest provable worst-case convergence rate
with quadratic Lyapunov functions. Finally, we further extend our analysis to
stochastic games and study the impact of multiplicative noise on different
algorithms. We show that it is impossible for an algorithm with one step of
memory to achieve acceleration if it only queries the gradient once per batch
(in contrast with the stochastic strongly-convex optimization setting, where
such acceleration has been demonstrated). However, we exhibit an algorithm
which achieves acceleration with two gradient queries per batch.

Optimization and Control;Machine Learning;Machine Learning