  This paper considers adaptive, minimax estimation of a quadratic functional
in a nonparametric instrumental variables (NPIV) model, which is an important
problem in optimal estimation of a nonlinear functional of an ill-posed inverse
regression with an unknown operator. We first show that a leave-one-out, sieve
NPIV estimator of the quadratic functional can attain a convergence rate that
coincides with the lower bound previously derived in Chen and Christensen
[2018]. The minimax rate is achieved by the optimal choice of the sieve
dimension (a key tuning parameter) that depends on the smoothness of the NPIV
function and the degree of ill-posedness, both are unknown in practice. We next
propose a Lepski-type data-driven choice of the key sieve dimension adaptive to
the unknown NPIV model features. The adaptive estimator of the quadratic
functional is shown to attain the minimax optimal rate in the severely
ill-posed case and in the regular mildly ill-posed case, but up to a
multiplicative $\sqrt{\log n}$ factor in the irregular mildly ill-posed case.
