  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.
