Tracking the Best Strategy in an Extensive-Form Game

We consider the extensive-form bandit problem, where on each trial the learner (a user coordinated by a server) plays an extensive-form game against an oblivious adversary, observing the information sets it finds itself in as well as the resulting payoff/loss. We give an algorithm for this problem that satisfies $ε$-local differential privacy and attains a regret of $\tilde{O}(\sqrt{A\ln(S)T}/ε)$, where $A$ is the total number of actions that the learner can possibly take, $S$ is the number of the learner's possible reduced strategies, and $T$ is the number of trials. On each trial, the time complexity of our algorithm is, up to a factor logarithmic in the maximum number of actions at an infoset, equal to the time required for the server to transmit the reduced strategy to the user. We note that local differential privacy is the strongest version of differential privacy and, to the best of our knowledge, this is the first work to study differential privacy of any form in the extensive-form bandit problem.

We study the two-action apple-tasting problem with switching costs against an oblivious adversary. In an equivalent normalized formulation, at each round the learner chooses between a revealing action and a blind action: the revealing action gives reward $0$ and reveals the hidden value $x_t\in[-1,1]$ of the blind action; the blind action gives reward $x_t$ but reveals nothing. The learner pays one unit whenever they switches actions, and regret is measured against the best fixed action in hindsight. General feedback-graph algorithms with switching costs give $\widetilde O(T^{2/3})$ regret guarantees for this problem. The two-action apple-tasting graph was the natural candidate for the missing $Ω(T^{2/3})$ obstruction in the switching-cost classification: such a lower bound would have transferred to a large family of still-unclassified feedback graphs. We prove that this obstruction is not there: the oblivious minimax expected regret for this problem satisfies $$\frac{1}{2\sqrt3}\cdot\sqrt T \le R_T^\star \le 2\sqrt{3}\cdot \sqrt{T}.$$

We give deterministic and uncoupled learning dynamics for finite multiplayer general-sum games under full-information feedback that achieve constant individual swap regret, independent of the horizon $T$. With $n$ players and at most $m$ actions each, the individual swap regret of every player is $O(\sqrt{n} m \log m \log^{5/2}(nm))$ at every finite horizon. Each player predicts the deviation gains, then uses these predictions to update a row-stochastic transition matrix, and plays its stationary distribution. The proof combines a potential argument exploiting stationarity with a two-scale higher-order prediction analysis, using rooted-tree representations to handle the nonlinear dependence of deviation gains on the stationary distributions. An adversarially robust variant, obtained through a generic common-prefix switching wrapper, preserves the self-play bound up to a universal constant and guarantees individual swap regret at most $7\sqrt{m T \log m}$ in the adversarial setting.