  We study the estimation of causal parameters when not all confounders are
observed and instead negative controls are available. Recent work has shown how
these can enable identification and efficient estimation via two so-called
bridge functions. In this paper, we tackle the primary challenge to causal
inference using negative controls: the identification and estimation of these
bridge functions. Previous work has relied on completeness conditions on these
functions to identify the causal parameters and required uniqueness assumptions
in estimation, and they also focused on parametric estimation of bridge
functions. Instead, we provide a new identification strategy that avoids the
completeness condition. And, we provide new estimators for these functions
based on minimax learning formulations. These estimators accommodate general
function classes such as Reproducing Kernel Hilbert Spaces and neural networks.
We study finite-sample convergence results both for estimating bridge functions
themselves and for the final estimation of the causal parameter under a variety
of combinations of assumptions. We avoid uniqueness conditions on the bridge
functions as much as possible.
