We study the multi-agent Bayesian optimization (BO) problem, where multiple agents maximize a black-box function via iterative queries. We focus on Entropy Search (ES), a sample-efficient BO algorithm that selects queries to maximize the mutual information about the maximum of the black-box function. One of the main challenges of ES is that calculating the mutual information requires computationally-costly approximation techniques. For multi-agent BO problems, the computational cost of ES is exponential in the number of agents. To address this challenge, we propose the Gaussian Max-value Entropy Search, a multi-agent BO algorithm with favorable sample and computational efficiency. The key to our idea is to use a normal distribution to approximate the function maximum and calculate its mutual information accordingly. The resulting approximation allows queries to be cast as the solution of a closed-form optimization problem which, in turn, can be solved via a modified gradient ascent algorithm and scaled to a large number of agents. We demonstrate the effectiveness of Gaussian max-value Entropy Search through numerical experiments on standard test functions and real-robot experiments on the source-seeking problem. Results show that the proposed algorithm outperforms the multi-agent BO baselines in the numerical experiments and can stably seek the source with a limited number of noisy observations on real robots.

翻译：我们研究多智能体贝叶斯优化问题，其中多个智能体通过迭代查询最大化黑箱函数。我们聚焦于熵搜索（ES）——一种样本高效的贝叶斯优化算法，通过选择查询点最大化关于黑箱函数最大值的互信息。ES的主要挑战之一在于，计算互信息需要计算成本高昂的近似技术。对于多智能体贝叶斯优化问题，ES的计算成本随智能体数量呈指数增长。为应对这一挑战，我们提出高斯最大值熵搜索——一种兼具样本效率与计算效率的多智能体贝叶斯优化算法。其核心思想在于使用正态分布近似函数最大值，并据此计算互信息。该近似方法使得查询问题可转化为封闭形式的优化问题，进而可通过改进的梯度上升算法求解，并扩展至大规模智能体场景。我们通过标准测试函数的数值实验与真实机器人源定位实验验证了高斯最大值熵搜索的有效性。结果表明，所提算法在数值实验中优于多智能体贝叶斯优化基线方法，且能在真实机器人上以有限噪声观测实现稳定源定位。