Beyond Time-Average Convergence: Near-Optimal Uncoupled Online Learning via Clairvoyant Multiplicative Weights Update
This work addresses the challenge of efficient and uncoupled online learning in multi-agent systems, offering a novel algorithm with improved convergence rates for game theory applications.
The paper tackles the problem of regret minimization in general games by introducing the Clairvoyant Multiplicative Weights Update (CMWU) algorithm, which achieves constant regret of ln(m)/η in normal-form games and converges to a Coarse Correlated Equilibrium with a rate of O(nV log m log T / T), improving on prior state-of-the-art uncoupled online learning dynamics.
In this paper, we provide a novel and simple algorithm, Clairvoyant Multiplicative Weights Updates (CMWU) for regret minimization in general games. CMWU effectively corresponds to the standard MWU algorithm but where all agents, when updating their mixed strategies, use the payoff profiles based on tomorrow's behavior, i.e. the agents are clairvoyant. CMWU achieves constant regret of $\ln(m)/η$ in all normal-form games with m actions and fixed step-sizes $η$. Although CMWU encodes in its definition a fixed point computation, which in principle could result in dynamics that are neither computationally efficient nor uncoupled, we show that both of these issues can be largely circumvented. Specifically, as long as the step-size $η$ is upper bounded by $\frac{1}{(n-1)V}$, where $n$ is the number of agents and $[0,V]$ is the payoff range, then the CMWU updates can be computed linearly fast via a contraction map. This implementation results in an uncoupled online learning dynamic that admits a $O (\log T)$-sparse sub-sequence where each agent experiences at most $O(nV\log m)$ regret. This implies that the CMWU dynamics converge with rate $O(nV \log m \log T / T)$ to a \textit{Coarse Correlated Equilibrium}. The latter improves on the current state-of-the-art convergence rate of \textit{uncoupled online learning dynamics} \cite{daskalakis2021near,anagnostides2021near}.