Targeted Maximum Likelihood Learning: An Optimization Perspective

Targeted maximum likelihood estimation ( TMLE ) is a widely used debiasing al-gorithm for plug-in estimation. While its statistical guarantees, such as double robustness and asymptotic efficiency, are well-studied, the convergence properties of TMLE as an iterative optimization scheme have remained underexplored. To bridge this gap, we study TMLE ’s iterative updates through an optimization-theoretic lens, establishing global convergence under standard assumptions and regularity conditions. We begin by providing the first complete characterization of different stopping criteria and their relationship to convergence in TMLE . Next, we provide geometric insights . We show that each submodel induces a smooth, non-self-intersecting path (homotopy) through the probability simplex. We then analyze the solution space of the estimating equation and loss landscape. We show that all valid solutions form a submanifold of the statistical model, with the difference in dimension (i.e., codimension) exactly matching the dimension of the target parameter. Building on these geometric insights, we deliver the first strict proof of TMLE ’s convergence from an optimization viewpoint , 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 a homotopy path, highlighting a promising avenue for designing enhanced debias algorithms.