Accelerated Stochastic Min-Max Optimization Based on Bias-corrected Momentum

This paper revisits momentum in the context of min-max optimization. Momentum is a celebrated mechanism for accelerating gradient dynamics in settings like convex minimization, but its direct use in min-max optimization makes gradient dynamics diverge. Surprisingly, Gidel et al. 2019 showed that negative momentum can help fix convergence. However, despite these promising initial results and progress since, the power of momentum remains unclear for min-max optimization in two key ways. (1) Generality: is global convergence possible for the foundational setting of convex-concave optimization? This is the direct analog of convex minimization and is a standard testing ground for min-max algorithms. (2) Fast convergence: is accelerated convergence possible for strongly-convex-strong-concave optimization (the only non-linear setting where global convergence is known)? Recent work has even argued that this is impossible. We answer both these questions in the affirmative. Together, these results put negative momentum on more equal footing with competitor algorithms, and show that negative momentum enables convergence significantly faster and more generally than was known possible.

We introduce a new single-loop algorithmic framework for smooth nonconvex--concave minimax optimization. The resulting projected damped extragradient method combines projected extragradient updates, dual momentum, and a moving proximal center. Under both the optimization-stationarity and game-stationarity criteria, our method achieves the best-known complexity among single-loop first-order methods. For optimization stationarity, our method achieves a gradient complexity of $O(L^2D_Y\barΔ_0\varepsilon^{-3})$, where $L$ is the gradient Lipschitz constant, $D_Y$ bounds the diameter of the dual feasible set, and $\barΔ_0$ is an initialization quantity involving the value-function gap and the initial gradients. Moreover, by incorporating a fixed-center warm-up phase, the complexity can be improved to $O(L^2D_YΔ_φ\varepsilon^{-3})$, up to an additive lower-order cost, where $Δ_φ:=φ(x_0)-\inf_xφ(x)$. We further establish a lower bound of $Ω(L^2D_YΔ_φ\varepsilon^{-3})$ for optimization stationarity over projected zero-respecting first-order methods. This lower bound proves that the warm-started version of our algorithm is optimal up to a constant factor for optimization stationarity within this oracle class. For game stationarity, our method achieves $\mathcal{O}\!(L^{3/2}D_Y^{1/2}Δ_φ\varepsilon^{-5/2})$ gradient complexity. This matches the best-known complexity of multi-loop first-order methods, thereby establishing the same complexity with a single-loop algorithmic structure. Under dual strong concavity, the proposed framework achieves $O\!(\sqrtκ\,LΔ_φ\varepsilon^{-2})$ leading complexity for both stationarity criteria, where $κ=L/μ$ is the dual condition number, up to an additive initialization cost. The $\varepsilon^{-2}$ accuracy dependence is optimal under fixed regularity and initialization bounds.

Stochastic compositional optimization minimizes objectives of the form $\min_{\bm{x} \in \mathcal{X}} F(\bm{f}(\bm{x}), \bm{x})$, where $\bm{f}$ is accessible only through noisy stochastic queries. Existing methods for this problem assume that the outer function $F$ is continuously differentiable, which excludes many practically important applications such as robust max-of-losses, Conditional Value-at-Risk, and norm regularizers. We propose the Hybrid Momentum Stochastic Frank--Wolfe algorithm, which drops the smoothness assumption on $F$. By combining a momentum-based Jacobian tracker with a Taylor-corrected function tracker, the algorithm feeds an entire stochastic linearization -- rather than a single gradient -- into a generalized linear minimization oracle. We establish an $\mathcal{O}(K^{-1/4})$ convergence rate in the generalized Frank--Wolfe gap for non-convex objectives with $L_F$-Lipschitz outer functions, matching the optimal complexity for projection-free single-sample stochastic methods under expected smoothness. The analysis extends to heavy-tailed noise oracles with bounded $r$-th moments for $r \in (1, 2]$ and recovers the deterministic rates of Vladarean et al (2023) as the noise vanishes.