Payoff-based learning with matrix multiplicative weights in quantum games

In this paper, we study the problem of learning in quantum games - and other classes of semidefinite games - with scalar, payoff-based feedback. For concreteness, we focus on the widely used matrix multiplicative weights (MMW) algorithm and, instead of requiring players to have full knowledge of the game (and/or each other's chosen states), we introduce a suite of minimal-information matrix multiplicative weights (3MW) methods tailored to different information frameworks. The main difficulty to attaining convergence in this setting is that, in contrast to classical finite games, quantum games have an infinite continuum of pure states (the quantum equivalent of pure strategies), so standard importance-weighting techniques for estimating payoff vectors cannot be employed. Instead, we borrow ideas from bandit convex optimization and we design a zeroth-order gradient sampler adapted to the semidefinite geometry of the problem at hand. As a first result, we show that the 3MW method with deterministic payoff feedback retains the $\mathcal{O}(1/\sqrt{T})$ convergence rate of the vanilla, full information MMW algorithm in quantum min-max games, even though the players only observe a single scalar. Subsequently, we relax the algorithm's information requirements even further and we provide a 3MW method that only requires players to observe a random realization of their payoff observable, and converges to equilibrium at an $\mathcal{O}(T^{-1/4})$ rate. Finally, going beyond zero-sum games, we show that a regularized variant of the proposed 3MW method guarantees local convergence with high probability to all equilibria that satisfy a certain first-order stability condition.

Learning in quantum games

In this paper, we introduce a class of learning dynamics for general quantum games, that we call "follow the quantum regularized leader" (FTQL), in reference to the classical "follow the regularized leader" (FTRL) template for learning in finite games. We show that the induced quantum state dynamics decompose into (i) a classical, commutative component which governs the dynamics of the system's eigenvalues in a way analogous to the evolution of mixed strategies under FTRL; and (ii) a non-commutative component for the system's eigenvectors which has no classical counterpart. Despite the complications that this non-classical component entails, we find that the FTQL dynamics incur no more than constant regret in all quantum games. Moreover, adjusting classical notions of stability to account for the nonlinear geometry of the state space of quantum games, we show that only pure quantum equilibria can be stable and attracting under FTQL while, as a partial converse, pure equilibria that satisfy a certain "variational stability" condition are always attracting. Finally, we show that the FTQL dynamics are Poincar\'e recurrent in quantum min-max games, extending in this way a very recent result for the quantum replicator dynamics.

On the convergence of policy gradient methods to Nash equilibria in general stochastic games

Learning in stochastic games is a notoriously difficult problem because, in addition to each other's strategic decisions, the players must also contend with the fact that the game itself evolves over time, possibly in a very complicated manner. Because of this, the convergence properties of popular learning algorithms - like policy gradient and its variants - are poorly understood, except in specific classes of games (such as potential or two-player, zero-sum games). In view of this, we examine the long-run behavior of policy gradient methods with respect to Nash equilibrium policies that are second-order stationary (SOS) in a sense similar to the type of sufficiency conditions used in optimization. Our first result is that SOS policies are locally attracting with high probability, and we show that policy gradient trajectories with gradient estimates provided by the REINFORCE algorithm achieve an $\mathcal{O}(1/\sqrt{n})$ distance-squared convergence rate if the method's step-size is chosen appropriately. Subsequently, specializing to the class of deterministic Nash policies, we show that this rate can be improved dramatically and, in fact, policy gradient methods converge within a finite number of iterations in that case.