Targeted Maximum Likelihood Learning: An Optimization Perspective
Diyang Li, Kyra Gan
Targeted Maximum Likelihood Estimation (TMLE) is a powerful debiasing algorithm for plug-in estimation, widely employed across research communities due to its double robustness and asymptotic efficiency.While extensive literature has concentrated on its statistical guarantees, the inherent convergence properties of TMLE as an iterative optimization scheme have remained inadequately explored.To bridge this critical gap, we rigorously investigate the optimization dynamics of TMLE iterations under standard assumptions and regularity conditions.We begin with studying key thresholding rules in TMLE framework and establish their fundamental connections to convergence.Our analysis reveals that the homotopy mapping induced by each fluctuation submodel defines a smooth embedding into the probability simplex, which constitutes a topological invariance that is free of self-intersection and carries implications for algorithmic stability.We then derive a structural characterization for the solution locus of targeted estimating‐equation and loss landscape, showing that it forms a submanifold of the statistical model with codimension equal to the dimension of target parameter.Building on these geometric insights, we deliver the first strict proof of TMLE's global convergence, as well as explicit sufficient criteria under which TMLE terminates in a single update.As a by-product, we discover an unidentified overshooting phenomenon wherein the algorithm can surpass feasible roots to the estimating-equation along homotopy path, highlighting a promising avenue for designing enhanced debias algorithms.