  We develop a unified theory of designs for controlled experiments that
balance baseline covariates a priori (before treatment and before
randomization) using the framework of minimax variance and a new method called
kernel allocation. We show that any notion of a priori balance must go hand in
hand with a notion of structure, since with no structure on the dependence of
outcomes on baseline covariates complete randomization (no special covariate
balance) is always minimax optimal. Restricting the structure of dependence,
either parametrically or non-parametrically, gives rise to certain covariate
imbalance metrics and optimal designs. This recovers many popular imbalance
metrics and designs previously developed ad hoc, including randomized block
designs, pairwise-matched allocation and rerandomization. We develop a new
design method called kernel allocation based on the optimal design when
structure is expressed by using kernels, which can be parametric or
non-parametric. Relying on modern optimization methods, kernel allocation,
which ensures nearly perfect covariate balance without biasing estimates under
model misspecification, offers sizable advantages in precision and power as
demonstrated in a range of real and synthetic examples. We provide strong
theoretical guarantees on variance, consistency and rates of convergence and
develop special algorithms for design and hypothesis testing.
