Primary Submission Category: Heterogeneous Treatment Effects
Hierarchical Approximations to the Universal Path for Efficient Targeted Maximum Likelihood Estimation
Authors: Kaiwen HOU, Mark van der Laan,
Presenting Author: Kaiwen HOU*
TMLE promises efficiency by constructing paths in the statistical model space, solving the efficient score equation in as few updates as possible. In principle, the universal least favorable path (UFLP) achieves single-step convergence by exactly matching the canonical gradient at every measure along the path, but for many important causal parameters—such as the variance of CATE—this path is intractable to construct.
We propose a hierarchy of approximate paths derived from standard perturbation expansions of the ULFP’s defining PDE. The local path in standard TMLE enforces a first-order condition only at the path’s initial measure and solves the efficient score equation up to O(initial rate^2). Moving to second order dramatically reduces the remainder: we show that imposing an expectation-based second-order condition yields a path whose explicit construction minimizes a KL divergence with a natural interpretation of maximizing the Cramér–Rao lower bound. Moreover, this path is highly computable via off-the-shelf convex-optimization routines. An alternative second-order path enforces pointwise second-order conditions, achieving O(initial rate^3) with little additional computational cost.
Extending to higher-order local paths provides an increasingly refined approximation of the ULFP, theoretically approaching single-step TMLE. Simulations show that these refined paths are computationally feasible and yield better finite-sample performance than the basic first-order path.