{"ID":6138120,"CreatedAt":"2026-07-09T01:07:32.349475501Z","UpdatedAt":"2026-07-11T06:53:25.465901322Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.07078","arxiv_id":"2607.07078","title":"Forgetting-Factor Regret for Online Zero-Sum Games","abstract":"This paper studies dynamic equilibrium tracking in online two-player zero-sum games with time-varying convex-concave payoff functions. Existing regret metrics for online saddle-point problems usually aggregate historical payoffs with uniform weights, and hence may fail to characterize the real-time tracking performance with respect to the current Nash equilibrium (NE). To address this issue, we introduce a zero-sum game regret function with a forgetting factor, which assigns exponentially decaying weights to past saddle gaps and emphasizes recent performance. This metric directly links regret minimization to the tracking of time-varying NEs. Within this framework, we investigate three online algorithms under different computational and information settings. For first-order feedback, we analyze projected gradient descent-ascent and design a projection-free online Frank-Wolfe method to reduce the computational cost of projections. For zeroth-order feedback, we develop a deterministic finite-difference method that only uses function-value queries. For all three algorithms, we establish forgetting-factor regret bounds that explicitly characterize the effects of NE variation, payoff variation, and gradient-estimation error. We further provide sufficient conditions under which the proposed regret converges to zero, thereby certifying asymptotic tracking of time-varying NEs. The numerical example validates the theoretical results and illustrates the tracking advantage of the proposed regret metric.","short_abstract":"This paper studies dynamic equilibrium tracking in online two-player zero-sum games with time-varying convex-concave payoff functions. Existing regret metrics for online saddle-point problems usually aggregate historical payoffs with uniform weights, and hence may fail to characterize the real-time tracking performance...","url_abs":"https://arxiv.org/abs/2607.07078","url_pdf":"https://arxiv.org/pdf/2607.07078v1","authors":"[\"Yuhang Liu\",\"Zi'ang Yan\",\"Wenjun Mei\",\"Wenxiao Zhao\"]","published":"2026-07-08T07:08:52Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}