Title: VATT-EG: Vanishingly-Anchored Two-Timescale Extragradient
Primary area: optimization
Keywords: min-max, optimization, extragradient, theory

Abstract: We resolve the open problem of whether a first-order algorithm can converge only to local
minimax equilibria in smooth nonconvex-nonconcave zero-sum games. Building on the two-
timescale extragradient method, we develop VATT-EG (Vanishingly-Anchored Two-Timescale
Extragradient), which introduces vanishing Tikhonov-style anchors to eliminate neutral modes
without biasing the stationary set. We show that under the standard calmness (bounded-
ratio) assumption: (i) a point is asymptotically stable for VATT-EG if and only if it is a calm
local minimax equilibrium; (ii) every non-minimax stationary point is a hyperbolic repeller
with a quantitative spectral margin; and lastly (iii) from almost every initialization, iterates
converge with probability one to a calm local minimax point. The proposed analysis depends
on a discrete-time extragradient expansion with constants; a Lyapunov metric built from the
restricted Schur complement, and a Robbins-Siegmund argument combined with a measure-zero
stable-manifold theorem. Experiments on canonical two-dimensional games confirm the neutral-
mode damping of vanishing anchors, while a delayed-corrector stress test shows that VATT-EG
stabilizes regimes where GDA and standard extragradient diverge or spiral out. Finally, in
a practical adversarial debiasing example (namely Colored-MNIST), VATT-EG achieves digit
accuracy comparable to baselines and consistently reducing adversary accuracy, yielding more
invariant features. Together, these results close the first-order selection gap and establish VATT-
EG as both a theoretically exact and practically robust algorithm for nonconvex-nonconcave
minimax optimization.