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Consider the case that we observe $n$ independent and identically distributedcopies of a random variable with a probability distribution known to be anelement of a specified statistical model. We are interested in estimating aninfinite dimensional target parameter that minimizes the expectation of aspecified loss function. In \cite{generally_efficient_TMLE} we defined anestimator that minimizes the empirical risk over all multivariate real valuedcadlag functions with variation norm bounded by some constant $M$ in theparameter space, and selects $M$ with cross-validation. We referred to thisestimator as the Highly-Adaptive-Lasso estimator due to the fact that theconstrained can be formulated as a bound $M$ on the sum of the coefficients alinear combination of a very large number of basis functions. Specifically, inthe case that the target parameter is a conditional mean, then it can beimplemented with the standard LASSO regression estimator. In\cite{generally_efficient_TMLE} we proved that the HAL-estimator is consistentw.r.t. the (quadratic) loss-based dissimilarity at a rate faster than$n^{-1/2}$ (i.e., faster than $n^{-1/4}$ w.r.t. a norm), even when theparameter space is completely nonparametric. The only assumption required forthis rate is that the true parameter function has a finite variation norm. Theloss-based dissimilarity is often equivalent with the square of an$L^2(P_0)$-type norm. In this article, we establish that under some weakcontinuity condition, the HAL-estimator is also uniformly consistent.

Uniform Consistency of the Highly Adaptive Lasso Estimator of Infinite Dimensional Parameters