  From clinical development of cancer therapies to investigations into partisan
bias, adaptive sequential designs have become increasingly popular method for
causal inference, as they offer the possibility of improved precision over
their non-adaptive counterparts. However, even in simple settings (e.g. two
treatments) the extent to which adaptive designs can improve precision is not
sufficiently well understood. In this work, we study the problem of Adaptive
Neyman Allocation in a design-based potential outcomes framework, where the
experimenter seeks to construct an adaptive design which is nearly as efficient
as the optimal (but infeasible) non-adaptive Neyman design, which has access to
all potential outcomes. Motivated by connections to online optimization, we
propose Neyman Ratio and Neyman Regret as two (equivalent) performance measures
of adaptive designs for this problem. We present Clip-OGD, an adaptive design
which achieves $\widetilde{O}(\sqrt{T})$ expected Neyman regret and thereby
recovers the optimal Neyman variance in large samples. Finally, we construct a
conservative variance estimator which facilitates the development of
asymptotically valid confidence intervals. To complement our theoretical
results, we conduct simulations using data from a microeconomic experiment.
