# Payoff-based learning with matrix multiplicative weights in quantum games

arXiv: https://arxiv.org/abs/2311.02423 · HTML: https://mlsanity.com/paper/2311.02423/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 61 references that are in the map.
- First posted on arXiv 4 Nov 2023.

## Builds on (most cited first, 16 of 16)

- [Quantum generative adversarial networks](https://mlsanity.com/paper/1804.08641/) (2018)
- [Tsallis-INF: An Optimal Algorithm for Stochastic and Adversarial Bandits](https://mlsanity.com/paper/1807.07623/) (2018)
- [An Optimal Algorithm for Bandit and Zero-Order Convex Optimization with Two-Point Feedback](https://mlsanity.com/paper/1507.08752/) (2015)
- [Learning in games with continuous action sets and unknown payoff functions](https://mlsanity.com/paper/1608.07310/) (2016)
- [Kernel-based methods for bandit convex optimization](https://mlsanity.com/paper/1607.03084/) (2016)
- [Online Learning of Quantum States](https://mlsanity.com/paper/1802.09025/) (2018)
- [Bandit learning in concave N-person games](https://mlsanity.com/paper/1810.01925/) (2018)
- [Quantum Wasserstein Generative Adversarial Networks](https://mlsanity.com/paper/1911.00111/) (2019)
- [A unified stochastic approximation framework for learning in games](https://mlsanity.com/paper/2206.03922/) (2022)
- [Sublinear quantum algorithms for training linear and kernel-based classifiers](https://mlsanity.com/paper/1904.02276/) (2019)
- [Learning with minimal information in continuous games](https://mlsanity.com/paper/1806.11506/) (2018)
- [Distributed stochastic optimization via matrix exponential learning](https://mlsanity.com/paper/1606.01190/) (2016)
- [Quantum Speed-ups for Semidefinite Programming](https://mlsanity.com/paper/1609.05537/) (2016)
- [Logarithmic-Regret Quantum Learning Algorithms for Zero-Sum Games](https://mlsanity.com/paper/2304.14197/) (2023)
- [Sublinear classical and quantum algorithms for general matrix games](https://mlsanity.com/paper/2012.06519/) (2020)
- [Semidefinite games](https://mlsanity.com/paper/2202.12035/) (2022)

## Built on by (most cited first, 2 of 2)

- [Coherent Swap Regret and Channel-Proof Learning](https://mlsanity.com/paper/2606.02655/) (2026)
- [Guiding Agents of Quantum Games to Equilibrium using Matrix Exponential Fixed-Point Iteration](https://mlsanity.com/paper/2609.21944/) (2026)

## Abstract (arXiv)

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) 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.