Title: Speeding Up Min-Max Optimization Through Power-Law Step Sizes

Abstract: This study re-examines the convergence properties of the Extragradient (EG) algorithm when applied to unconstrained biaffine min-max problems. While it is established that EG utilizing a constant step size yields a last-iterate convergence rate of $\Theta(T^{-1/2})$—a pace lagging behind the optimal $\mathcal{O}(T^{-1})$ achievable through techniques like anchoring—recent developments indicate that dynamic step sizes can markedly hasten gradient descent. Inspired by these findings, we investigate whether analogous acceleration can be realized for EG’s last-iterate convergence, delivering the first affirmative answer. We introduce a deterministic dynamic step size schedule that boosts EG’s convergence to $\mathcal{O}(T^{-2/3+\varepsilon})$ for any $\varepsilon > 0$. Furthermore, we demonstrate that this performance bound is tight under the constraint that the extrapolation and update phases of EG share an identical step size. However, decoupling the step sizes for these two phases enables a further enhancement, reaching a near-optimal rate of $\mathcal{O}(T^{-1+\varepsilon})$. By framing step size scheduling as an optimization challenge, our approach yields a schedule based on a discretized power-law distribution. These findings and the underlying methodology are not limited to EG; they also apply to Optimistic Gradient (OG) and suggest wider relevance across general min-max optimization contexts.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC