When and Why is Optimistic Multiplicative Weights Slow? The Geometry of Energy Dissipation

This paper studies the convergence of the Optimistic Multiplicative Weights Update algorithm (OMWU) in two-player zero-sum games. Recent works have identified instances on which the last-iterate of OMWU can converge arbitrarily slowly, but understanding when and why this slow convergence occurs has remained open. In this work, we develop a new analysis framework that gives sharp, quantitative explanations for this behavior. Our analysis is based on viewing the algorithm's dual iterates as an *optimistic skew-gradient descent* with respect to an energy function. We prove over the dual iterates that energy is dissipative, and by establishing tight bounds on the magnitude of dissipation, our analysis quantifies the geometric bottlenecks that arise when the corresponding primal iterates are close to the simplex boundary. This further translates into a new linear last-iterate convergence rate in KL divergence on games with a unique and interior Nash equilibrium. Compared to prior work, this new rate contains a much sharper dependence on game-specific constants, and we prove this dependence is optimal. Moreover, these geometric insights further translate into new separations on *uniform* convergence rates for OMWU. On the one hand, we prove *constant lower bounds* on the uniform *best-iterate* convergence rate in KL divergence and Total Variation distance from Nash. On the other hand, we establish for the $2\times 2$ setting a new $\widetilde O(T^{-1/2})$ best-iterate rate in duality gap, improving substantially over prior work. Together, this shows in general that uniform convergence rate guarantees do not transfer across different measures of distance to Nash.