Linear Convergence in Games with Delayed Feedback via Extra Prediction

Feedback delays are inevitable in real-world multi-agent learning. They are known to severely degrade performance, and the convergence rate under delayed feedback is still unclear, even for bilinear games. This paper derives the rate of linear convergence of Weighted Optimistic Gradient Descent-Ascent (WOGDA), which predicts future rewards with extra optimism, in unconstrained bilinear games. To analyze the algorithm, we interpret it as an approximation of the Extra Proximal Point (EPP), which is updated based on farther future rewards than the classical Proximal Point (PP). Our theorems show that standard optimism (predicting the next-step reward) achieves linear convergence to the equilibrium at a rate $\exp(-Θ(t/m^{5}))$ after $t$ iterations for delay $m$. Moreover, employing extra optimism (predicting farther future reward) tolerates a larger step size and significantly accelerates the rate to $\exp(-Θ(t/(m^{2}\log m)))$. Our experiments also show accelerated convergence driven by the extra optimism and are qualitatively consistent with our theorems. In summary, this paper validates that extra optimism is a promising countermeasure against performance degradation caused by feedback delays.

Learning from Delayed Feedback in Games via Extra Prediction

This study raises and addresses the problem of time-delayed feedback in learning in games. Because learning in games assumes that multiple agents independently learn their strategies, a discrepancy in optimization often emerges among the agents. To overcome this discrepancy, the prediction of the future reward is incorporated into algorithms, typically known as Optimistic Follow-the-Regularized-Leader (OFTRL). However, the time delay in observing the past rewards hinders the prediction. Indeed, this study firstly proves that even a single-step delay worsens the performance of OFTRL from the aspects of regret and convergence. This study proposes the weighted OFTRL (WOFTRL), where the prediction vector of the next reward in OFTRL is weighted $n$ times. We further capture an intuition that the optimistic weight cancels out this time delay. We prove that when the optimistic weight exceeds the time delay, our WOFTRL recovers the good performances that the regret is constant ($O(1)$-regret) in general-sum normal-form games, and the strategies converge to the Nash equilibrium as a subsequence (best-iterate convergence) in poly-matrix zero-sum games. The theoretical results are supported and strengthened by our experiments.

The Hamiltonian of Poly-matrix Zero-sum Games

Understanding a dynamical system fundamentally relies on establishing an appropriate Hamiltonian function and elucidating its symmetries. By formulating agents' strategies and cumulative payoffs as canonically conjugate variables, we identify the Hamiltonian function that generates the dynamics of poly-matrix zero-sum games. We reveal the symmetries of our Hamiltonian and derive the associated conserved quantities, showing how the conservation of probability and the invariance of the Fenchel coupling are intrinsically encoded within the system. Furthermore, we propose the dissipation FTRL (DFTRL) dynamics by introducing a perturbation that dissipates the Fenchel coupling, proving convergence to the Nash equilibrium and linking DFTRL to last-iterate convergent algorithms. Our results highlight the potential of Hamiltonian dynamics in uncovering the structural properties of learning dynamics in games, and pave the way for broader applications of Hamiltonian dynamics in game theory and machine learning.