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Payoff-based learning with matrix multiplicative weights in quantum games

Kyriakos Lotidis, Panayotis Mertikopoulos, Nicholas Bambos, José Blanchet

Published Dec 10, 2023• 0 citations
OpenAlex

Abstract

In this paper, we study the problem of learning in quantum gamesand 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 O(1/ √ 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 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.

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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.

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In this paper, we study the problem of learning in quantum gamesand 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 O(1/ √ 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 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.
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 O(1/T)\mathcal{O}(1/\sqrt{T})O(1/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 O(T−1/4)\mathcal{O}(T^{-1/4})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.
In this paper, we study the equilibrium convergence and stability properties of the widely used matrix multiplicative weights (MMW) dynamics for learning in general quantum games. A key difficulty in this endeavor is that the induced quantum state dynamics decompose naturally 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 the classical replicator dynamics; and (ii) a non-commutative component for the system's eigenvectors. This non-commutative component has no classical counterpart and, as a result, requires the introduction of novel notions of (asymptotic) stability to account for the nonlinear geometry of the game's quantum space. In this general context, we show that (i) only pure quantum equilibria can be stable and attracting under the MMW dynamics; and (ii) as a partial converse, pure quantum states that satisfy a certain “variational stability” condition are always attracting. This allows us to fully characterize the structure of quantum Nash equilibria that are stable and attracting under the MMW dynamics, a fact with significant implications for predicting the outcome of a multi-agent quantum learning process.
Recent developments in domains such as non-local games, quantum interactive proofs, and quantum generative adversarial networks have renewed interest in quantum game theory and, specifically, quantum zero-sum games. Central to classical game theory is the efficient algorithmic computation of Nash equilibria, which represent optimal strategies for both players. In 2008, Jain and Watrous proposed the first classical algorithm for computing equilibria in quantum zero-sum games using the Matrix Multiplicative Weight Updates (MMWU) method to achieve a convergence rate of O(d/ϵ2) iterations to ϵ -Nash equilibria in the 4d -dimensional spectraplex. In this work, we propose a hierarchy of quantum optimization algorithms that generalize MMWU via an extra-gradient mechanism. Notably, within this proposed hierarchy, we introduce the Optimistic Matrix Multiplicative Weights Update (OMMWU) algorithm and establish its average-iterate convergence complexity as O(d/ϵ) iterations to ϵ -Nash equilibria. This quadratic speed-up relative to Jain and Watrous' original algorithm sets a new benchmark for computing ϵ -Nash equilibria in quantum zero-sum games.
In recent years, quantum game theory has gained significant attention as a framework for studying decision-making in multi-agent systems using quantum principles. However, computing equilibrium strategies is challenging because the dimension of the joint Hilbert space grows as the product of the players' local dimensions. In this paper, we consider an extended Gutoski-Watrous (EGW) game in which each player's quantum strategy is represented by a local density matrix. We derive tensor-contraction expressions for the payoff functions and their gradients, thereby avoiding the explicit construction of the full joint density matrix and its computationally expensive multiplication by the payoff operators. Building on the resulting effective Hamiltonians, we propose the Matrix Exponential Fixed-Point Iteration with Annealing (MEFPIA) algorithm to search for equilibrium points in EGW games. We compare MEFPIA with the Matrix Multiplicative Weights Update (MMWU) algorithm in terms of convergence. For the tested instances and parameter settings, both algorithms approach the same strategy profiles and payoffs, while MEFPIA achieves lower relative error in fewer iterations. These results indicate that MEFPIA is a promising numerical method for equilibrium search in multi-agent quantum games. Our findings provide important insights into the quantum game theory's potential for addressing complex decision-making processes, as well as opening up new paths for future research and exploration in multi-agent quantum systems.
Learning in games has emerged as a powerful tool for machine learning with numerous applications. Quantum games model interactions between strategic players who have access to quantum resources, and several recent works have studied learning in the competitive regime of quantum zero-sum games. Going beyond this setting, we introduce quantum common-interest games (CIGs) where players have density matrices as strategies and their interests are perfectly aligned. We bridge the gap between optimization and game theory by establishing the equivalence between KKT (first-order stationary) points of an instance of the Best Separable State (BSS) problem and the Nash equilibria of its corresponding quantum CIG. This allows learning dynamics for the quantum CIG to be seen as decentralized algorithms for the BSS problem. Taking the perspective of learning in games, we then introduce non-commutative extensions of the continuous-time replicator dynamics and the discrete-time best response dynamics/linear multiplicative weights update for learning in quantum CIGs. We prove analogues of classical convergence results of the dynamics and explore differences which arise in the quantum setting. Finally, we corroborate our theoretical findings through extensive experiments.
Recent advances in quantum computing and in particular, the introduction of quantum GANs, have led to increased interest in quantum zero-sum game theory, extending the scope of learning algorithms for classical games into the quantum realm. In this paper, we focus on learning in quantum zero-sum games under Matrix Multiplicative Weights Update (a generalization of the multiplicative weights update method) and its continuous analogue, Quantum Replicator Dynamics. When each player selects their state according to quantum replicator dynamics, we show that the system exhibits conservation laws in a quantum-information theoretic sense. Moreover, we show that the system exhibits Poincare recurrence, meaning that almost all orbits return arbitrarily close to their initial conditions infinitely often. Our analysis generalizes previous results in the case of classical games.