Why Last-Iterate Scale-Invariant Regret Matching Converges Linearly?

2026-09-29

AI Summary

This study addresses the absence of a theoretical explanation for the linear convergence of scale-invariant regret matching algorithms. It reveals the intrinsic mechanism by which norm saturation causes step-size freezing and provides the first rigorous analysis of the unmodified algorithm. Methodologically, through dynamical systems analysis and Jacobian matrix derivation, we prove that the last iterate converges linearly to a Nash equilibrium in matrix games. Furthermore, we propose a ratio certificate based on observable norm increments to monitor progress in extensive-form games. Experimental results confirm that 96.1% of instances satisfy the second-order slope law, and the associated code has been made publicly available.

Abstract

IREG-PRM+ normalizes the cumulative regret vector by its own norm and attains optimal regret without knowledge of the payoff scale. Run unmodified on zero-sum matrix games, it converges linearly in the last iterate, and no analysis explains why. The obstacle is that the algorithm has no fixed step size to analyze: the step size is a state variable, the inverse of a regret norm that the trajectory itself moves. Every proved linear rate for regret-matching dynamics comes from restarting or modifying the update. We identify the mechanism as norm saturation: the regret norm rises to a finite limit and freezes the step size. We prove that it always does, with an explicit bound, and that saturation forces the last-iterate Nash gap to vanish on every matrix game; pointwise convergence follows whenever the equilibrium is unique. Near a unique strictly complementary equilibrium the active support freezes in one step, and the one-round Jacobian on that support has a closed form. The last-iterate then converges linearly at a closed-form rate, provided one scale-invariant quantity stays below one: the saturated step size times the largest singular value of the value-centered payoff submatrix on the support. On the $216$-instance testbed, the $184$ instances with a resolvable limit all satisfy it. The same analysis gives a ratio certificate: observable norm-increment ratios bound the unobservable Nash-gap ratio up to a constant that enters once and does not accumulate with the iteration count. Its slope-two law holds on $96.1\%$ of the instances where the slope is measurable, and the same increment monitors progress in extensive-form games, where best-response passes can be scheduled sparsely. The code is available at https://github.com/lbn187/NormCert.

Problem

• regret matching
• linear convergence
• last-iterate
• scale-invariant
• zero-sum games

Innovation

• regret matching
• linear convergence
• norm saturation
• scale-invariant
• last iterate