Title: Targeted Maximum Likelihood Learning: An Optimization Perspective
Venue: NeurIPS 2025
Accepted: yes
Keywords: Targeted MLE

Iterative Optimization

Semi-parametric Learning
Abstract: Targeted maximum likelihood estimation (TMLE) is a widely used debiasing algorithm 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-selfintersecting 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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses a significant gap in understanding TMLE's convergence properties as an iterative algorithm. The authors establish geometric insights about TMLE's homotopy paths, prove global convergence under some technical conditions, derive conditions for one-step termination, and discover an overshooting phenomenon.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\nThis seems to be the first ever work to rigorously analyze TMLE through an optimization lens. It derives comprehensive theoretical results including global convergence guarantees (Theorem 4), a geometric characterization of solution manifolds (Theorem 3). The discussion around the overshooting phenomenon and the identification of cases where TMLE can overshoot feasible solutions is genuinely novel and has practical implications.\n\nWeaknesses:\nThe paper is extremely notation-heavy and dense, although the authors provide a notaion-dictionary which I appreciate, but I think it is important to have better proof sketches/an overall better presentation. Overshooting phenomenon probability bounds though novel and very interesting, I would like to see simulations to validate these results. Could you please provide some? I totally understand the paper is theoretical and does not contain any experiment and it is not a weakness in my opinion given the strength of the theory content. However, to me, this is an obvious place where we can have an understanding of how tight your bounds are. Apart from that, I would like to understand a few steps in the proofs before recommending full acceptance.\n\nQUESTIONS:\n1) Inequality (41) (in the appendix) seems really one of the key steps in the proof. Could you please re-explain why the final result is zero (especially the last steps (43) and (44))?\n\n2) In the proof of Theorem 3, I do not understand what is happening (around step (b)) after line 691 (I am familiar with complex analysis but I fail to see the necessity to bring this up here. How is it helpful in the process?)\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work mainly focused on the theoretical investigation of the optimization aspects of the targeted maximum likelihood estimation (TMLE). It provides some interesting aspects, including some geometrical insights, on when and why the iterative algorithms of the TMLE will convergence or not.  The manuscript provides the sketches of the proof. But the detailed proofs are all in the supplementary materials.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\n1.\tThe paper is well written from a theoretical aspect with clear logic flow. It aims to address an important aspect of learning: not just the asymptotic properties, but also the numerical convergence. \n\n2.\tThe connection to the geometrical aspects provides some interesting insights. \n\n3.\tThe authors also provide good remarks based on the theoretical results.\n\nWeakness: \n1.\tThe manuscript has a lot of notations. It will be more approachable that the authors provide a through notation before the technical description. \n\n2.\tThe theoretical investigation is kind of general from the optimization aspect. It will be more appealing to illustrate the results under some specific statistical models. \n\n3.\tThe investigation of the iterative algorithm depends on the properties of the initial estimators. However, there is not sufficient discussion how the initial estimator will affect the convergence properties of the TMLE algorithm, especially on the targeted parameter \\psi.\n\n4.\tThere is no numerical study to evaluate the convergence results of the theoretical investigation.\n\n5.\tAlthough the theoretical investigation is kind of comprehensive, there is no illustration of such theoretical analysis under some specific statistical models. \n\n6.\tFor the investigation of the TMLE, it will be more appealing to connect the optimization convergence to the asymptotic convergence. For example, how the sample size $n$ would affect both convergence.\n\nQUESTIONS:\n1.\tThe linear reparameterication in Example 1 is not clear.  By additive perturbation, the resultant function may not be a proper density function. \n\n2.\tIn Definition 1, it is not clear the probability distribution can be a continuous distribution or a discrete distribution.\n\n3.\tThe main results in Theorem 1 are about the threshold equivalence. But it is not clear to the reviewer what does “thresholding” mean here? The iterative algorithm seems not involving the thresholding operator.\n\n4.\tThe global convergence in Section 4.2 is mainly about the convergence of the density estimator p_{n}. But the manuscript overlook the convergence property of the targeted parameter \\psi_{n}.\n\n5.\tThe scope of investigating TMLE in the manuscript is mainly for the unconstraint MLE problem. Many MLE problems in the statistical model can have constraints on parameter spaces. It is not clear how the theoretical investigation in this work can be applicable for the constrained MLE problems. \n\n6.\tIn Section 4.4, the authors aim to address the overshoot problem. However, it is better to mathematically define what is the “overshoot” in a rigorous manner. \n\n7.\tIt is better to have some numerical study to evaluate the convergence results of the theoretical investigation.\n\n8.\tIt is better to have some illustration of such theoretical analysis under some specific statistical models.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work addresses what appears to be a glaring gap in the literature concerning general conditions under which TMLE converges — in particular, the lack of general global convergence guarantees, which the authors provide. This is augmented by more nuanced contributions in the understanding of the procedure, such as why TMLE can work despite complex likelihoods, and the fact that convergence of TMLE is sufficient but not necessary for \"visiting\" densities during optimization that yield efficient estimators.\n\nSTRENGTHS AND WEAKNESSES:\nThe authors seem to have closed a large gap in both TMLE convergence guarantees and in current understanding of the algorithm's behavior as an optimization routine. With the caveat that I am somewhat new and non-expert in this area, their contributions seem highly impactful and long overdue. I also appreciate their careful explanation and organization in the \"narrative arc\" outlining their contributions.\n\nThe largest weakness, to my mind, is that their contributions need better contextualization in the current literature. At best, the poor contextualization \"buries the lede\" and results in missed opportunities to scaffold the significance of the contribution for readers, and at worst can misrepresent the scope of the novelty.\n\nIn particular, the discussion around theorem 1 (and its placement inside inside a gray box, which indicates a \"new result\") reads to me as if the entirety of the theorem is novel. The original TMLE paper proves, however, that if TMLE converges, it does indeed yield an efficient estimator (the E3 arrow in figures 1 and 5). And while I am not fully confident, I also believe they may prove that either convergence of empirical risks implies convergence of the epsilon iterates, or perhaps vice-versa (please correct me otherwise)? These established results are mentioned by the authors, but are scattered throughout the paper (e.g. lines 167/168 saying that convergence implies a locally efficient estimator, e.g. lemma 1, and e.g. lines 288/289 saying that E3/E4 are exactly or at least follow straightforwardly from established results).\n\nSimilarly, the section on global convergence could benefit with a discussion of the fact that global convergence results do exist for certain settings — this is alluded to, for example, in the mention of D ́iaz and Rosenblum's work (2015), which I believe establish global convergence results for exponential family submodels (by convexity) (please clarify if this is not the case). On this note, it might help readers if the authors concretize some practical applications of the kinds of model families that were previously lacking the convergence guarantees that this paper provides.\n\nLastly, I believe a number of the one-step convergence conditions in theorem 5 are already known? I would appreciate clarification on the novelty there, too.\n\nDespite the scope of novelty being somewhat murky in the main body of the paper, I do believe the authors are addressing a highly impactful gap. Especially for familiar-but-non-expert readers (e.g. TMLE users), a very precise explanation of exactly what has and has not been established in the literature should be included. That discussion should then be revisited when formally laying out each theoretical result. To the best of my understanding, this paper addresses a significant gap in a complex literature — this contribution deserves for readers to have a very clear and precise sense of the gap being closed.\n\nRelatedly, except where required for sake of mathematical precision, the jargon levels seem excessive to a point of obscuring the contributions. Consider this sentence from the abstract: \"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.\" In my admittedly subjective opinion, the most important point here is not the details of the geometric findings (however illuminating), but the \"implications for algorithmic stability\", which are not described here at all. Likewise, only after two very jargon heavy sentences in the abstract, do we reach what I see as the most significant and novel contribution: general conditions for global convergence of TMLE. That seems so central, even, that I wonder why it isn't in the title of the paper.\n\nThose broad criticisms aside, and with the caveat that I lack the expertise required to check the proofs in detail, I thank the authors for doing this work!\n\nQUESTIONS:\n(Note: I restate here some of the questions I asked in the main body of the review above)\n\n1. Is the necessity and sufficiency of the convergence of empirical risks for the convergence of epsilon iterates novel?\n2. What are some concrete applications unlocked by the sub-model families that you provide novel TMLE global convergence results for?\n3. Which one-step convergence conditions are known that are presently listed in theorem 5? Which are new and made possible by your analysis here?\n4. It is mentioned that these analyses expand our understanding of the behavior of the TMLE algorithm in the non-asymptotic regime — from my read, finite sample insights seem limited to the rate of occurrence of the overshoot phenomenon? Is that correct? If so, the introductory claims on this front may need to be tempered.\n5. Does the overshoot phenomenon lead only to a failure detect what amount to early stopping opportunities, i.e. skipping over an efficient estimator? I.e it leads to suboptimal performance only *computationally*?\n\nI am willing to increase my score pending clarifications around the scope of novelty, both in the discussion phase and in the paper.\n\nAs a minor point, the paper has a number of grammatical issues involving definite articles, spelling, etc. and should undergo additional proofreading in a potential camera ready period.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 3, "DATE": "2025-07-03"}