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We consider estimation of a functional parameter of a realistically modeleddata distribution based on observing independent and identically distributedobservations. We define an $m$-th order Spline Highly Adaptive Lasso MinimumLoss Estimator (Spline HAL-MLE) of a functional parameter that is defined byminimizing the empirical risk function over an $m$-th order smoothness class offunctions. We show that this $m$-th order smoothness class consists of allfunctions that can be represented as an infinitesimal linear combination oftensor products of $\leq m$-th order spline-basis functions, and involvesassuming $m$-derivatives in each coordinate. By selecting $m$ withcross-validation we obtain a Spline-HAL-MLE that is able to adapt to theunderlying unknown smoothness of the true function, while guaranteeing a rateof convergence faster than $n^{-1/4}$, as long as the true function is cadlag(right-continuous with left-hand limits) and has finite sectional variationnorm. The $m=0$-smoothness class consists of all cadlag functions with finitesectional variation norm and corresponds with the original HAL-MLE defined invan der Laan (2015).  In this article we establish that this Spline-HAL-MLE yields anasymptotically efficient estimator of any smooth feature of the functionalparameter under an easily verifiable global undersmoothing condition. Asufficient condition for the latter condition is that the minimum of theempirical mean of the selected basis functions is smaller than a constant times$n^{-1/2}$, which is not parameter specific and enforces the selection of the$L_1$-norm in the lasso to be large enough to include sparsely supported basis.We demonstrate our general result for the $m=0$-HAL-MLE of the averagetreatment effect and of the integral of the square of the data density. We alsopresent simulations for these two examples confirming the theory.

Efficient Estimation of Pathwise Differentiable Target Parameters with the Undersmoothed Highly Adaptive Lasso