  Unbiased and consistent variance estimators generally do not exist for
design-based treatment effect estimators because experimenters never observe
more than one potential outcome for any unit. The problem is exacerbated by
interference and complex experimental designs. Experimenters must accept
conservative variance estimators in these settings, but they can strive to
minimize conservativeness. In this paper, we show that the task of constructing
a minimally conservative variance estimator can be interpreted as an
optimization problem that aims to find the lowest estimable upper bound of the
true variance given the experimenter's risk preference and knowledge of the
potential outcomes. We characterize the set of admissible bounds in the class
of quadratic forms, and we demonstrate that the optimization problem is a
convex program for many natural objectives. The resulting variance estimators
are guaranteed to be conservative regardless of whether the background
knowledge used to construct the bound is correct, but the estimators are less
conservative if the provided information is reasonably accurate. Numerical
results show that the resulting variance estimators can be considerably less
conservative than existing estimators, allowing experimenters to draw more
informative inferences about treatment effects.
