  Kernel balancing weights provide confidence intervals for average treatment
effects, based on the idea of balancing covariates for the treated group and
untreated group in feature space, often with ridge regularization. Previous
works on the classical kernel ridge balancing weights have certain limitations:
(i) not articulating generalization error for the balancing weights, (ii)
typically requiring correct specification of features, and (iii) justifying
Gaussian approximation for only average effects.
  I interpret kernel balancing weights as kernel ridge Riesz representers
(KRRR) and address these limitations via a new characterization of the
counterfactual effective dimension. KRRR is an exact generalization of kernel
ridge regression and kernel ridge balancing weights. I prove strong properties
similar to kernel ridge regression: population $L_2$ rates controlling
generalization error, and a standalone closed form solution that can
interpolate. The framework relaxes the stringent assumption that the underlying
regression model is correctly specified by the features. It extends Gaussian
approximation beyond average effects to heterogeneous effects, justifying
confidence sets for causal functions. I use KRRR to quantify uncertainty for
heterogeneous treatment effects, by age, of 401(k) eligibility on assets.
