Abstract: We study Bayesian persuasion under approximate best response, where the receiver may choose any action that is not too much suboptimal, given their posterior belief upon receiving the signal. We focus on the computational aspects of the problem, aiming to design algorithms that efficiently compute (almost) optimal strategies for the sender. Despite the absence of the revelation principle --- which has been one of the most powerful tools in Bayesian persuasion --- we design polynomial-time exact algorithms for the problem when either the state space or the action space is small, as well as a quasi-polynomial-time approximation scheme (QPTAS) for the general problem. On the negative side, we show there is no polynomial-time exact algorithm for the general problem unless $\mathsf{P} = \mathsf{NP}$. Our results build on several new algorithmic ideas, which might be useful in other principal-agent problems where robustness is desired.