A Rate Spectrum for Moving Proximal Centers in Sequence-Form Games

Modern solvers for two-player zero-sum extensive-form games deploy the current iterate as the policy, so the last-iterate question is when it converges. In the regularized counterfactual regret family, convergence comes from a reference strategy. The iterate is regularized toward a reference, which removes the cycling of unregularized dynamics at the price of pulling the solution away from the Nash set. On the Euclidean carrier a fixed magnet converges linearly to a regularized equilibrium at distance $O(\tau )$ from the Nash set, for proximal weight $\tau$. Periodic re-centering and hard resets reach exact equilibria at asymptotic or sublinear rates. Between these ends lies the interior speed ${\beta }_t\in (0,1)$, for which no finite-time rate is known for this update class. All three move one reference at different speeds, so their common-carrier proxies are instances of one schedule coordinate, the prox-center update spectrum, and one analysis covers the carrier-matched family. A constant speed inside an explicit window gives linear last-iterate convergence to an exact Nash equilibrium of the original game; the bound is sharp at the endpoint. Summable schedules freeze at the computable resolvent of the center's limit, with a closed-form positive floor at interior equilibria. At interior equilibria the asymptotic slope is a closed form in the speed, the proximal weight, and the game's skew spectrum, and its mode eigenvalues match measured slopes to three significant figures. On Leduc, whose equilibrium set is a manifold, every practical speed lies outside the estimated window and the finite-horizon decay at the faster speeds is fitted by a power law. Linear decay reaches a saddle gap of $10^{-10}$ on Kuhn, and the Leduc power-law exponent is $1.8$–$2.2$. Across twenty-eight carrier-matched entries the moving center reaches a saddle gap below $10^{-6}$ wherever any other method does. On Leduc it leads periodic re-centering by $2$–$3\times$ at $500$k iterations.