\section{Conclusions}
We study the problem of learning Nash equilibrium of black-box games with a Bayesian approach using Gaussian processes as surrogates for the unknown utilities. We characterize the equilibrium computation problem as optimizing an unknown objective function. As a result, finding the Nash equilibrium of the game is equivalent to minimizing the unknown objective function. We also proposed a no-regret learning approach to minimize the unknown objective function with principled ROI identification and acquisition maximization. Our study shows the proposed algorithm improves upon existing methods both with novel theoretical results and strong empirical performance across various tasks. 

Our results open the possibilities for many other interesting questions. For example, our work and prior research primarily address learning NE in normal-form games, where agents act simultaneously. Another intriguing domain is Stackelberg games, where agents move sequentially (cf. Appendix \ref{sec:additional_related}). Hence, exploring Stackelberg equilibrium computation presents another interesting problem to investigate. Furthermore, we assume the GPs of distinct agents are independent. Investigating the correlation between agents' utility functions and constructing multivariate GPs presents an intriguing avenue for future exploration as well.