Scale-Invariant Regret Matching and Online Learning with Optimal Convergence: Bridging Theory and Practice in Zero-Sum Games

• URL: http://arxiv.org/abs/2510.04407v2
• Date: Mon, 13 Oct 2025 20:08:52 GMT
• Title: Scale-Invariant Regret Matching and Online Learning with Optimal Convergence: Bridging Theory and Practice in Zero-Sum Games
• Authors: Brian Hu Zhang, Ioannis Anagnostides, Tuomas Sandholm,
• Abstract summary: A considerable chasm has been looming for decades between theory and practice in zero-sum game solving through first-order methods.<n>We propose a new scale-invariance and parameter-free variant of PRM$+$, which we call IREG-PRM$+$.<n>We show that it achieves $T-1/2$ best-iterate and $T-1$ optimal convergence guarantees, while also being on par with PRM$+$ on benchmark games.
• Score: 60.871651115241406
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A considerable chasm has been looming for decades between theory and practice in zero-sum game solving through first-order methods. Although a convergence rate of $T^{-1}$ has long been established since Nemirovski's mirror-prox algorithm and Nesterov's excessive gap technique in the early 2000s, the most effective paradigm in practice is *counterfactual regret minimization*, which is based on *regret matching* and its modern variants. In particular, the state of the art across most benchmarks is *predictive* regret matching$^+$ (PRM$^+$), in conjunction with non-uniform averaging. Yet, such algorithms can exhibit slower $\Omega(T^{-1/2})$ convergence even in self-play. In this paper, we close the gap between theory and practice. We propose a new scale-invariant and parameter-free variant of PRM$^+$, which we call IREG-PRM$^+$. We show that it achieves $T^{-1/2}$ best-iterate and $T^{-1}$ (i.e., optimal) average-iterate convergence guarantees, while also being on par with PRM$^+$ on benchmark games. From a technical standpoint, we draw an analogy between IREG-PRM$^+$ and optimistic gradient descent with *adaptive* learning rate. The basic flaw of PRM$^+$ is that the ($\ell_2$-)norm of the regret vector -- which can be thought of as the inverse of the learning rate -- can decrease. By contrast, we design IREG-PRM$^+$ so as to maintain the invariance that the norm of the regret vector is nondecreasing. This enables us to derive an RVU-type bound for IREG-PRM$^+$, the first such property that does not rely on introducing additional hyperparameters to enforce smoothness. Furthermore, we find that IREG-PRM$^+$ performs on par with an adaptive version of optimistic gradient descent that we introduce whose learning rate depends on the misprediction error, demystifying the effectiveness of the regret matching family *vis-a-vis* more standard optimization techniques.

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