AI 速览：作者报告交替在线镜像下降（AltOMD）在两人零和双线性博弈中可达到 O(log T) 的外部遗憾和，从而正面回答其是否具有近常数遗憾的开放问题。
Alternating online mirror descent (AltOMD) is a simple learning rule for zero-sum games, which improves upon simultaneous OMD numerically and theoretically. Previous work obtained $O(T^{1/3})$ and $O(T^{1/5})$ regret bounds for AltOMD, by leveraging modified-energy-function-based analyses. However, settling whether AltOMD achieves near-constant regret has remained an open question. In this paper we resolve the question of whether AltOMD achieves near-constant regret in the affirmative. In particular, we show an $O(log(T))$ regret bound on the sum of the players' external regrets for any two-player zero-sum bilinear game with compact convex strategy sets, under mild regularity assumptions on the regularizers. We verify that our assumptions hold for the negative entropy regularizer, thereby obtaining logarithmic regret for the alternating multiplicative weights updates algorithm. The regret bound also yields an $O(log(T)/T)$ duality gap bound for the averaged strategies.
We consider $T$ repeated second-price auctions across $n$ pacing bidders with i.i.d. item arrivals, and show that it is computationally impossible for all bidders to achieve no-regret simultaneously while satisfying their respective budget or return-on-investment constraints. Therefore, in the prominent internet auction systems where buyers simultaneously deploy pacing algorithms or automated bidding agents, we cannot hope to achieve joint no-regret guarantees in general. We establish a deterministic relationship that high variance in online bidding systems implies high regret. By showing that one can extract a pacing equilibrium from a low-variance system, we prove PPAD-hardness of attaining joint no-regret. For several widely used bidding algorithms, we also show evidence from dynamical systems and numerical simulations that they can indeed display high-variance behavior through cycling, even on simple market instances.