  e consider the experimental design problem in an online environment, an
important practical task for reducing the variance of estimates in randomized
experiments which allows for greater precision, and in turn, improved decision
making. In this work, we present algorithms that build on recent advances in
online discrepancy minimization which accommodate both arbitrary treatment
probabilities and multiple treatments. The proposed algorithms are
computational efficient, minimize covariate imbalance, and include
randomization which enables robustness to misspecification. We provide worst
case bounds on the expected mean squared error of the causal estimate and show
that the proposed estimator is no worse than an implicit ridge regression,
which are within a logarithmic factor of the best known results for offline
experimental design. We conclude with a detailed simulation study showing
favorable results relative to complete randomization as well as to offline
methods for experimental design with time complexities exceeding our algorithm.
