Targeted Maximum Likelihood Learning: An Optimization Perspective
Authors: Diyang Li
Kyra Gan
Venue: NeurIPS 2025 poster ()
Keywords: Targeted MLE
Iterative Optimization
Semi-parametric Learning
Primary area: optimization

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.

Targeted Maximum Likelihood Learning: An Optimization Perspective
Diyang Li
Kyra Gan Cornell University kyragan@cornell. edu
Cornell University diyang01@cs cornell.ccccclcooco edu
Abstract
Targeted maximum likelihood estimation (TMLE) is widely used debiasing al- gorithm for plug-in estimation. While its statistical guarantees, such as double robustness and asymptotic efficiency, well-studied, the convergence properties TMLE as an iterative optimization scheme have remained underexplored. To bridge this gap, study TMLE's iterative updates through aptimization--hhoree 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 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 submanifold of the statistical model, with the difference in dimension (i.e., codimension) exactly matching the dimension of the target parame- ter. 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 single update. As by-product, we discover an unidentified overshooting phenomenon wherein the algorithm can surpass feasible roots to the estimating equation along homotopy path, highlighting promising avenue for designing enhanced debias algorithms.
Introduction
Plug-in estimation, the approach of first estimating the data- eataa-gnneratig distribution and then eval- uating the target parameter on this estimate, a natural strategy for estimating quantities such as quantiles, variance, average treatment effects (ATEs), and feature importance measures [1, 2]. Despite widespread use, these estimators often suffer from biases arising from the nonlinearity parameter mappings, the finite-sample variability inherent to empirical distributions, model misspecification. which can lead to unreliable inference high-dimensional or complex settings [3, 4J.
To address these limitations, Targeted Maximum Likelihood Estimation (TMLE) [5] offers principled framework for constructing data-adapa aata-adaptive estimators by combining muchine-learning (ML)-based initial estimates with targeted bias correction. At each step. TMLE updates the current distribution estimate along a fluctuation direction to reduce bias for the target parameter, achieving the desired double robustness and asymptotic efficiency under mild regularity conditions [6, 7]. These strengths have driven its adoption across areas including semi-supervised learning [8, 9], personalized medicine [10, IJ. algorithmic fairness [12, 13], and off-policy evaluation [14], equipping practitioners with estimators that are both flexible and theoretically grounded.
While TMLE's asymptotic properties (e.g., double robustness and semi-parametric efficiency) are well-established [16], its plored. Current theory often treats iterative updates as implementation details for solving estimating
behavior as an optimization procedure remains underex-
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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equations [16, 17]. While asymptotic guarantees depend on algorithmic convergence, this con- vergence is typically assumed, except in few cases with known one- or two-step convergence (e.g., [42, 44]). However, most target parameters lack such one-step guarantees, making algorithmic convergence not just an implementation concern but fundamental determinant of TMLE's debiasing performance. This gap is especially critical in finite-sample settings like healthcare [18, 19], where asymptotic guarantees offer no guidance for choosing convergence criteria, iteration limits, and tuning parameters. Without ceiiizaaionnthhoreee foundations, implementations remain ad-hoc [20, 21], risking reproducibility and silently degrading estimator quality.
We address this by analyzing TMLE's iterative trajectory and convergence through an optimization- theoretic lens. Unlike asymptotic efficiency results, our framework provides actionable insights for practical implementation. Conceptually, the convergence analysis of TMLE shares high-level similarities with that of schemes designed to optimize complex objective functions via tractable subproblems using altemnt- ing optimization frameworks. However, TMLE's sofluence-function (IF)-driven fluctuations within probability simplex-embedder homotopies pose unique challenges absent in classical EM literature [23, 24]. These technical differences necessitate tailored convergence analyses that differ significantly from those in EM. While recent works have explored optimization aspects for specific problems, e.g., causal effect estimation in exponential families [25] and off- off-policy evaluation with regularization [14], their scope remains limited to these special settings and does not apply to the general template.
(EM) algorithms [22], both involving iterative
Our contributions In this paper, we address a longstanding gap in the theoretical understanding of TMLE by recasting it as an explicit iterative optimization scheme. Our contributions are fourfold:
Geometric Insights. We establish formal connections between different stopping criteria and the resulting algorithm convergence behavior (Theorem 1). Next, we show that each submodel induces a smooth homotopy mapping (or path), embedding each fluctuation into the probability simplex without self-intersection (Theorem 2). precluding cyclic or oscillatory behavior in TMLE. We demonstrate that the set of distributions satisfying the estimating-equation forms a smooth submanifold of the statistical model, whose codimension coincides exactly with the dimensionality of the target parameter (Theorem 3). This structural perspective reveals that TMLE iterates traverse a now-dimensional manifold within the ambient probability space, explaining their effectiveness in navigating a complex landscape with irregular likelihood surfaces (Theorem 3).
Convergence Guarantee. We provide the first rigorous proof of TMLE's global convergence under mild regularity conditions (Theorem 4). Although this result asymptotic, requiring an infinite number of iterations, it confirms long-standing empirical observations of TMLE's convergence behavior. Further, we derive explicit sufficient conditions under which TMLE terminates after a single update (Theorem 5). These conditions, based on the initial estimator and IF structure, offer verifiable criteria for simplified TMLE implementations.
Our analysis reveals an unrecognized overshooting phenomenon, where TMLE may bypass feasible roots along the homotopy path (Theorem 6), potentially affecting finite- sample performance. This insight motivates safeguard strategies such step-size control or root tracking that retain asymptotic guarantees while improving runtime and stability.
Overshoot Behavior.
Interdisciplinary Impact. By casting TMLE estimation with modern optimization theory. Conventional analyses of parametric optimizer using convexity typically require the submodel objective to be strictly (or strongly) convex. In contrast, our analyses based on non-intersection do not require such assumptions and thus hold under broader settings. This perspective enables reinterpretations of influence functions via tools like mirror descent, and suggests integrating adaptive acceleration or regularization paths into TMLE, potentially leading to new, theoretically grounded algorithms.
optimization procedure, we bridge statistical
2 Preliminaries on Plug-in Estimation
This section introduces notation, reviews plug-in estimation, plug-in bias, and the influence function. Readers familiar with these concepts may skip ahead. Given the dataset (O...... On consists of n independent and identically distributed (i.i.d.) observations of a random variable o (e.g., an experimental unit) that fits an unknown frue distribution Po with sample space O. Our goal is to efficiently estimate a d-dimensional target parameter representing some statistical feature of interest (c.g., population mean and average treatment effect).

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Notation Let P be probability distribution with density p. Abusing the notation, we use P and p interchangeably to denote the probability measure. We let P, denote the empirical measure (Definition A.5), and J == E ʃ(O.). Let (P) denote the space of mean-zero, finite- variance functions with respect to the distribution P. i... Lÿ(P) :- {h- 0 - R Eph(o)² o0, ...(p))() = 0}. where o is a generic random variable drawn from distribution P. We use a S to denote that there exists a constant C' such that S Cb. Let Dj be the Fréchet derivative, and we abbreviate Pr, (o) as pn, when no ambiguity arises. We use C to denote the class of mappings that are j-times continuously differentiable. complete notation table included Appendix A.
Plug-in estimation We consider nonparametric estimation," where the model class M contains all candidate distributions for Po on o-finite measurable space (S2, F..F),,, v), where each element P M admits a Radon-Nikodym density
dP' /v with respect to a dominating measure v, satisfying p 2 0 v-a.e. and Jn dvv To ease the derivation, we work with the density p of P with respect to fixed dominating measure. The equivalent definitions in this section can be stated directly in terms of P. which more common the literature. We assume p is uniformly bounded:
Assumption (Uniform Boundedness). There exists Coo 00 s.I. I|l| Cocs Vp E M.
The target parameter functional V : M + Fd then maps each candidate distribution correspond- ing feature of interest (e.g., for d = 1, + (l)))) The plug-in estimator &.. v/ (fn) is obtained by applying Vr to an empirical estimate pa. E .M of the unknown po. Let 450 v/ (po) be the true value of our target parameter.
Plug-in Bias and Influence Function While plug-in estimators are often consistent under regularity conditions, they typically fail to achieve asymptotic linearity (Definition A.6)² due to bias inherited from the initial estimate P.." and the potential nonlinearity of Vo. Even if Pm. is consistent at the parametric rate, when V is nonlinear in p. estimation errors in P.. can propagate nonlinearly through V, introducing bias that invalidates a / Vn-linear expansion. As a result, plug-in estimators typically require bias correction to attain asymptotic linearity and efficiency.
key tool for formalizing this limitation is the influence function, which quantifies how sensitive the target parameter V is to small perturbations of the underlying distribution P. When M is nonparametric, the IF of the target parameter V at a distribution P, Di (p)(-) O - R E (P) is unique. We formally introduce the influence function in Appendix A.4, Definition A.9.
To establish the asymptotic behavior of oh can then be analyzed through von Mises expansion. Let V be pathwise differentiable (Definition A.8), and consider the perturbation path defined in Eq. (A.14) with P = P, and Q = Po. Let Po be absolutely continuous with respect to P.. and dPo/dP. E ( The estimation error in ti can be decomposed into the following [26, 27]:
$. 1/0 = -G - Pú (P, Po) Di (no))
Ra Ra(pn, Po)
plug-in bins
empirical process term
second-order remainder
where R2(pm; Po) is the second-order remainder in the difference between Pn and pn- The expansion in Eq. (1) resembles Definition A.6. While standard regularity conditions (e.g., Donsker class requirements and rate constraints on ||,,, Po l) suffice to ensure the empirical process term and the second-order remainder Opo (1/vn), the first-order plug-in bias term typically remains non-negligible without correction.
We analyze TMLE convergence in nonparametric models, revealing how target smoothness interacts with estimator complexity. Results extend to semi-paramest cases via subspace projections.
2 This commonly assumed in prior works obtain statistical guarantecs [5].
Asymptotic linearity ensures /ni-rate convergence to a normal distribution. with an asymptotic variance
determined by the influence function, thereby facilitating efficient estimation and valid statistical inference.
4 For example, fn may not be consistent at the parametric rate, especially when estimated via neural networks.
We refer readers Van Der Laan and Rubin [5] and Cho al. [27] for detailed assumptions.
3

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3
Warm-up: TMLE Template
To correct the plug-in bias term, TMLE finds the solution pz, that solves the score equation
1 Σ (e - / D& (i = 0,
P. P.D
(2)
by iterative refining the initial estimate P... We express this via sample space integration connect with optimization- thhoorrtccccc analysis.
Abusing the notation, let ph be the k-th iteration of THLE. At high-level, after obtaining "sufficiently
good" initial P.. (e.g., via ML), TMLE selects a parametric submodel {p(C)]eER C M (Definition 2) guided by the IF maximize sensitivity the target parameter Vi at p.. It then updates f.. by fitting c along this submodel (typically via minimizing a loss function) to obtain an updated estimate p.z
This procedure is repeated until no further improvement (i.e., nonzero e) can be found (if achieved). A pseudo-code of TMLE is provided in Algorithm 1. If TMLE terminates after k iterations, the final estimate Pk achieves asymptotic efficiency under standard regularity conditions. However, the actual convergence behavior of this iterative procedure remains unestablished.
Definition (TMLE Loss). Define the TELE loss function L .M- + (0 + R20) such that
Po arg min / L(p)po(o)dr(().
(3)
Similar to classical maximum likelihood estimation, we use loss function L(-) such that the mapping (3) is minimized at the true density E.g., one may use the negative log- fog-likelihood, L(p) == log
Definition 2 (Fluctuation Submodel). Let R denote open subset of R2 We define a family of fluctuation submodel {p(c): c R} (a.k.a. parametric working model) that follows (i) {p(c) : et C M; (ii) the submodel through dL (p(e)))) de at e recovers the IF (cf. Definition 3).
0; (iii) linear combination of the components of "score"
R denotes the set of values for which (e) is a proper density. Common parametric submodels include linear (Example 1) and exponential (Example 2).
Example 1 (Linear Reparameterization)), For E R. the instances of additive perturbation include
(x(()) (1 +e Dig(+t)) ph:
(4)
Example 2 (Exponential Family), For e E R, the instances of exponential tilting include
pt((e) & ((ppp)))xpp(' (T.)):)) pi:
(5)
phi(e) A (e,,, {1 exp 22 TO DV (pt))} pn
(6)
for C(e, pt) be normalizing constants (defined in Definition B.10).
Definition 3 (Relaxed Score Condition). Let A be constant matrix with ||A||| < co. TMLE requires that every parametric submodel has a sufficient statistic equal to the IF, i.e.,
(T(p(e))oo)
AA- (2) for all possible values O E O.
Definition 3 imposes the statistical connections between the loss and the IF in TMLE. Additionally, make the following mild regularity assumption, ensuring that the solution of Algorithm is achieved in the interior of M:
Assumption 2 (van der Laan et al. [6]). Let M be a sufficiently rich model class such that the iterated model locally saturated. Under this condition, the minimization invoked in line 3 of Algorithm I admits its solution strictly within the interior of M.
'In Eq. (2), we adopt an equivalent integral representation facilitate the convergence analysis. We follow the common TMLE convention of using for both submodels and probabilities, where no ambiguity arises.

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Algorithm Targeted Maximum Likelihood Estimator (TMLE)
Input: Data (0:) canonical gradient Diy of the interested functional V, initial estimator p
1: & +- O, initialize en +
2: while cx, + 0 do
3: arg min (esp)()))) / L(phe))o))pdddl))
4: p + ph (ch).
++1
|| literative debiasing
5: end while
6: P 6 Pm - Un f ( 11 plug-in estimation
Output: Targeted estimator tin-
4
Main Results
This section presents main theoretical results, framing TMLE through optimization lens for pathwise differentiable target parameter (Definition A.8). Throughout, we assume that the standard regularity conditions (Assumptions I, 2) as well as our mild assumptions (Assumptions 3, 4, 5) hold globally unless explicitly stated otherwise.
Assumption 3 (Smooth Link). Let the link function f: Ln(v) - Ln(v) be C2 smooth. We assume that (c) admits the representation (c((e) & f(E" (i(p))) where f(-) is injec- tive on its effective domain, namely the linear span of the components of Diy
Assumption 3 ensures that the perturbation parameter and the IF Di (-) always appear jointly in the form _ D& (-). Meanwhile, the injectivity imposed on the mapping f is typically mild in practice and readily satisfied by virtually all mainstream submodels such as those defined in Examples and 2. A formal justification of this injectivity condition is provided in Appendix D.1.
Assumption (Differentiabiliiyy and Lipschitz-in-pee innpit... The mappings + p(e) and L(e) are of class C² and C in respectively. The mappings p + L(p) and - Di (p) are twice continuously Fréchet-differentabll And for l E Z+, e1 E R, and p1.p2 M we have
[P* L(p (c2) S ||<r c2ll2
(L)) L (p2)llea)) $ }lp p2 p2lll
(8)
(pi(()))) - D $ l| - <2ll2-
Assumption 5 (Metric-subregularic of Gradient). There exists neighborhood of origin s.t. for k E Zt and all c,, in that neighborhood, with V L.) evaluated at + O, we have
||q 12112 S (.L(pt()))- L.L(((( E ||| -2ll2-
(9)
Assumption is a standard and widely adopted assumption within the optimization literature. Like- wise, Assumption 5 is notably mild within the context of TMLE optimization. E.g., one can trivially verify that the log-likes likeliihoood under an exponential tilt naturally satisfies (9). Further, the second inequality in (9) essentially inherited from the vanilla TMLE literature (albeit expressed differently).
4.1 What do we really mean when we talk about 'convergence"?
In TMLE practice, we observe that iterative stopping ("convergence") criteria are often applied incon- sistently. To maintain theoretical rigor, we explicitly define TMLE convergence as the convergence in probability of the iterates to a deterministic limiting distribution in M. While previous studies like [5, 6] have shown that the convergence implies solving (2), we generalize this result Theorem 1.
Theorem 1 (Stopping Condition Equivalence). The condition limg-pa c 0 is both necessary and sufficient for the convergence of Algorithm 1. If Algorithm I does converge, then the iterates { 122 admit a limit limg- p us P", where P", lies on the solution manifold of (2).
5

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Theorem rigorously characterizes and clarifies the underlying relationships among several commonly em- ployed stopping conditions, visually illustrating these connections Figure Theorem serves as an essential foundation for our subsequent optimization studies. It is also worth noting that convergence TMLE is a sufficient but not necessary condition for solving the Estimating Equation (2), which implies the potential occurrence overshooting phenomena during algorithmic iterations (as discussed later in Section 4.4).
Convergence
lim cx k-0o
Di dulp,
Figure 1: Illustration of Thm. 1.
4.2 Global convergence
The state-of-the-arr convergence result regarding Algorithm I is currently represented by Lemma 1.
Lemma 1. In Algorithm 1, the sequence of empirical risks {So L(ph pr, converges as k w co.
Proof. See Section 3 in Van Der Laan and Rubin [5] for a detailed proof.
However, it is a common consensus within the optimization community that convergence of the empirical loss alone does necessarily imply convergence of the iterates themselves. This issue becomes even more challenging in a semi-parametre context." & We provide motivating examples of TMLE divergence without standard assumptions Appendix I. Nonetheless, our analysis will build upon Lemma 1, and we first present several preparatory results essential to our analysis.
Theorem 2 (Non-self -ittteerseecciin).. The homotopy path (p(c) : c} in the probability simplex never self- inff-interseett for d = 1. Further assume the non- deneeeeenrrcyy condition
vB E (O)),, 3: (p) + 0} > 0.
(10)
Then the p(c) defines a one-to-one C¹ embedding of R TR. into the probability simplex, and its image is free of self self-intersecctins/ for every d E zt.
Remark 1. The assumption for d 2 2 is to require that the d components of Di (p)(o) are linearly independent in (v). Equivalently, L- = J D Di (p)(o) p(o)du(o) is a full-rank d x d
matrix. 9
Theorem 3 (Solution Submanifold), Assume n. < 3o and the Fréchet derivative Dy is surjective for every p M, then both the (i) IF Estimating Equation (2), and (ii) nonparametric loss landscape (in So L(p)pndv admit infinitely many solutions which form a smooth submanifold (or continuum) of whole equivalence class in M.
We know that e 0 implies p(c) = p. while Theorem 2 characterizes the converse direction, i.c., if p(e) = then O. also reveals a profound geometric structure underlying TMLE's iterative process. Specifically, the ho- motopy mapping induced by each fluctuation submodel defines a smooth embedding into the probability simplex. This embedding represents continuous deformation of the statistical model that preserves its topological struc- ture without self-intersections (cf. Figure 2), ensuring that each TMLE fluctuation traverses well-defined homotopy path that cannot revisit the same density twice, preventing potential cycling behavior. Theorem 3 implies that the solution submanifold is inherently infiiie-dimenniioall consisting of uncountably many solutions. However, it forms well-defined, smooth "sheet" of equivalent solutions that simultaneously solve the estimating equation and the (approximate) empirical-risk minimization problem. Inspired by Theorem we geometrically interpret the iterative optimization procedure of TMLE the
pí(e
Pile)
Figure An illustration of self- intersection in M. The path pi (c) (left) is self-intersect while the other p2(e) (right) is not.
'In this context, Sbemi-parameett refers settings where the target parameter defined parametrically, but the plug-in distribution is estimated using nonparametric methods.
Since 2T p(3 DE (p))² dv. where by assumption the integrand is strictly positive in part of domain.

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Fluctuation Fréchet
6.
Pole)
Tangent line
E.R.
333
2.4
P.le)
-13
M
0.6
plee)
M
(a) Understanding of itcrative process.
(b) Characterization of limiting distribution.
Figure 3:
The conceptual diagram TMLE procedure. 3a, the gray region represents model M and E.R. denotes the empirical risk. At k-th iteration, we span submodel (red curve) around př in M, then project the path onto the loss landscape (blue curve). We select the point corresponding with the minimal loss p++1 3b is the top-down perspective of 3a, which indicates the Fréchet gradient of loss evaluated at the p. is orthogonal (in the I sense) to the tangent line of the submodel at c 0.
empirical loss landscape (cf. Figure 3a), as well as characterize the geometric structure of the limiting distribution (ef. Figure 3b).
Grounded the insights from Theorems and 3, our asymptotic convergence guarantees are now provided in Theorem 4, demonstrating the infinite-step convergence behavior of TMLE even when the initial estimator heavily misspecified in M.
Theorem 4 (Iterative Convergence). Let (1r be the sequence of density estimates produced by the TMLE Algorithm I. There exists a limiting density P" E M such that limy- P P
Remark Theorem covers many well-known THLE instances like D liz and Rosenblum [25]. Note that even if Assumption 5 may not hold for some TMLE practices, we can still establish asymptotic regularity (or pseudo-comvergence) of the iterates as lim 11 pfill (a.ka., quasi- Cauchy sequence).
We emphasize that the proof of Theorem does depend the quality of the initial estimate ph meaning that Algorithm guarantees asymptotic convergence to a solution the estimating equation from any starting point (Theorem 1). However, poor initial estimate can slow the decay of the empirical process term and the second-order remainder in Eq. (1), preventing the estimator from achieving parametric-rate efficiency asymptotically [6].
4.3 One-step property
Theorem 5 (One-step Property). The semi-parametric TMLE procedure performs exactly one update when one of the following conditions is met:
(i) initial density ph. already on the solution manifold of (2) and fo L is strictly corvex in e;
(ii for Vo E O, the mapping c r> D (o) is a conservative (i.e., curl-free) vector field in and the loss satisfies line integral ((p(e)) L(p(0)) + A (o(p(u))()dduu (iii) in some of the practical problems on outcome regression (e.g., ATE, propensity-seue intervention) with the existence of "clever covariate" " and a proper fluctuation submodel; (iv) hit the user-set convergence criteria (e.g., machine precision, number of iterations).
In the (i) (ii) Theorem 5, we firstly establish a set of sufficient conditions under which the TMLE can be terminated after one single iteration. Putting together with existing conditions (ii) and (iv), the whole theorem significantly extends various scattered conditions previously dispersed throughout existing literature, e.g., [42, 43, 25, 44]. It is important to clarify that this one-step property does
7

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nor imply convergence in the mathematical sense; instead, it indicates the algorithm has achieved predefined stopping criterion (e.g., satisfying Estimating Equation (2)) and obtained desirable properties for the targeted estimator of interest.
4.4 Potential overshooting
As a by-product of our analysis into the optimization, we uncovered an interesting phenomenon wherein TMLE update may overshoot feasible solution to the estimating equation along the homotopy path induced by the fluctuation submodel (cf. Figure 4), We establish theoretical existence of this overshooting phenomenon through concrete illustrative example presented in Example 3. Example 3 (Degenerate Hyperplanc). Consider Submodel (4) with log- oggliikeihood loss. Solving
/ D (o)) (1 (1+ T (b()))))) -1
Pn Prdd((o) =
(11)
with of 0 gives the next movement of TMLE. If all of the Dy (O:) happen lie in single affine hyperplane {a: e x} where X R is a constant, the resulting distribution would exactly solve the Estimating Equation (2) while Algorithm keeps looping.
Motivated by Example 3, we present more formal definition of this phenomenon.
Definition 4 (Overshooting of TMLE). At the k-th iteration of TMLE, we say that the Algo- rithm I overshoots feasible solution if there exists et * &h such that pr(ctt E M and So Di (pt (e)) =.
Theorem 6 (Overshoot Control). If we further assume that
(i) the population Jacobian J , D (c))))) du has positive minimal eigenvalue Ao du -e ||e||2.
(ii) (.. V.L (P) SoP
Then, the probability that TMLE overshoots (o.s.) the nearest root of the estimating equation is
PP [o.s.] S 2d exp
22 + 2d expe
n2pi
111 Do ( D (12)
2B²
2B3
where Bo, Co are constants specified in Appendix H.
While a systematic characterization of the exact con- ditions leading overshooting remains open chal- lenge, we derive preliminary probabilistic guarantees under stronger assumptions in Theorem 6, particularly the uniform boundedness of second-order remainders, specified in Condition (ii) above. Our findings in- dicate that the exponent in (12) scales linearly with sample size leading to exponential reduction in the overshooting probability bound as n increases. Additionally, higher-dimensional lihood of overshooting and complicates the conver- gence. Intuitively, when d > 2, the solution set forms a complex manifold, where the risk function may ex- hibit heterogeneous curvature, flat some directions while sharply curved others. Consequently, optimizers following steepest descent directions may escape the manifold before reaching a risk minimum, overshooting potential solutions. Furthermore, common subproblem solvers employing backtracking line searches or momentum acceleration are particularly prone to this phenomenon. The combination of adaptive step sizing and inertia can cause the algorithm to leap over viable roots before the termination criteria are triggered.
overshoot
pile)
M
0
manifold
feasible solution
Figure 4: Illustration an overshooting where the TMLE update passes through an feasible ElF-root * but continues to the next iterate *
increases the like-
Proof Overview
This section sketches the high-level ideas behind our proofs.

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Proof Sketch of Theorem 1. This theorem naturally decomposes into 4 independent sub-results, which we denote by E1, E2, E3, and E4 (see Figure 5). For E1, we begin by showing that the sequence (P%) 222 enjoys tail-sum additivity property in the underlying metric space. By carefully bounding the increments and summing over the fluctuation steps, we conclude that {p"] is Cauchy sequence. In proving E2, we derive both compact upper and lower bounds on the fluctuation parameters [1 Applying the squeeze theorem we deduce that the norm of e,, must converge to the origin. Sub-result E3 follows directly from the classical analysis in Van Der Laan and Rubin [5], and it is worth noting that the proof remains straightforward under our framework. Lastly, sub-result E4 emerges as a trivial generalization of sub-result E3.
Proof Sketch of Theorem 2. Since the fluctuation submodel is defined via an injective link on its effective domain, in the scalar case (d 1) we show that the Hellinger distance between p(ez) and p(c2) never vanishes. Hence e + p(c) is strictly monotonic in the simplex. For general d, the non non-degenyracy condition (10) guarantees that D ()p(e)[h] # 0 whenever h o 0. One shows that
||e - e2ll Z l[p(e)) - ple2)ll 2 ||e - e2|1. ll. thus p a bi-Lipschitz embedding of Rd into the simplex and cannot self-intersect by Hadamard's ' global inverse function theorem.
Proof Sketch of Theorem 3. We first show that the functional + Jo Di + (o) is locally Lipschitz-type continuous in an L2(v) (v))-neihborrhoo of the reference solution po. Using a second-order Fréchet expansion we derive [l (Dg (po + h1) Diy (Po h2) brn)llddlll 2 1|4 h2 [[]))))). Given that the Fréchet derivative D (po) is assumed surjective, the rank-nullity theorem for Banach spaces ensures that the null space must be infiniteddimensionall The infinite-dimensional implicit function theorem then applies due to continuity and surjectivity conditions, which guaranteess that the solution locus is a C2 manifold. The proof of (ii) technically follows a similar process.
Proof Sketch of Theorem 4. Along the real-analytic fluctuation path, we first show that |e-l| X fo -L( (olp:
using Lipschitz-in-paan Based on Lemma 1, telescoping this inequality shows that the squared step sizes form a summable series Σx ||e 12. Therefore, we get <h ~ O, combining these facts with the equivalence statement in Theorem 1, the vanishing of e is both necessary and sufficient for convergence of the TMLE iterates.
Proof Sketch of Theorem 5. For case (i), the result follows directly by substituting into the algorith- mic framework and exploiting convexity arguments. In the analysis of (ii), we leverage fundamental properties of line integrals along with optimality conditions to demonstrate that the updated density after one iteration lies within component of the solution manifold. For case (iv), we rigorously establish both the existence and appropriateness of stopping conditions introduced therein. We omit the proof for (iii), as it is highly problem-specine and can be found in the corresponding paper. O
Proof Sketch of Theorem 6. We first linearize the empirical score map at the origin, writing it as a fixed Jacobian term plus a Lipschitz remainder. The minimal-norm root therefore lies within factor of the score norm, and a single coss-based update stays in the same radius. Overshoot can happen if the score at the origin is already large compared with that radius. Each coordinate of that score is bounded average, by applying Hoeffding's 's inequality twice we obtain the desired bound.
Remark 3. We also provide several toy examples to validate partial results in Appendix B.
6 Discussions
In this work, we have laid a rigorous optimization opiimizatoon-theoretii foundation for TMLE. Our analysis assumes exact solution of each subproblem, whereas in numerical practice one always computes an approximate et satisfying ||eh - ex l2ll2 I a. and setting p + ph (e%). How such gap affects convergence guarantees remains unclear. Meanwhile, the non-self-interssction property of submodel paths plays a subtle role and may carry implications for algorithmic stability. Further, our framework addresses only canonical first-order TMLE while extending to higher-order variants [28] is also an important directions for future research.

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Reviews:
[{"review_id": "0SPh34Mv94", "reviewer": "Reviewer_bWRS", "summary": "This 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.", "questions": "(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.", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "The 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!", "was_revised": true, "final_justification": "As far as I am aware, the authors are correct that their work establishes the first general convergence result for TMLE. Given the promised clarifications around the scope of novelty and some changes to presentation, the importance of their key result is sufficient for me to recommend acceptance.", "rebuttal": "### We sincerely appreciate Reviewer bWRS's exceptionally detailed and constructive feedback (among the most thorough reviews we’ve received at NeurIPS). We are particularly grateful for your considerable effort invested in providing an extensive review, which will definitely enhance the quality of our manuscript.\n\n### In the following, we address each of the technical points appeared in your review.\n\n--- \n\n## *Novelty of the Theorem 1*\n\nWe sincerely apologize if the phrasing in our manuscript led to any confusion or misunderstanding.\n\n> (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. ... however, ...\n\nStrictly speaking, aside from a minor part of Theorem 1 $(\\text{E3})$ and one item of Theorem 5 (explained further below), all results presented in the gray boxes are entirely novel findings.\n\nFor elegance of presentation and completeness of theoretical results, we originally integrated existing results with our new contributions in the manuscript. Although mentioned elsewhere, to better differentiate our contributions, we promise to remove this statement: `(line 126) Our new results along with their assumptions are in grey boxes`\n\n> ... These established results are mentioned by the authors, but are scattered throughout the paper ...\n\nThank you for drawing attention to Figure 5. Formally, the statements $(\\text{E1})$, $(\\text{E2})$, and $(\\text{E4})$ represent entirely new conclusions. We have provided full rigorous proofs of $(\\text{E1})$ and $(\\text{E2})$ in the appendix, and shown that $(\\text{E4})$ follows directly from $(\\text{E1})$ together with $(\\text{E3})$, thereby establishing $(\\text{E4})$ as a novel result as well.\n\nAlthough $(\\text{E3})$ was implicitly referenced earlier in Section 3 (\"Warm-up\") preceding the Main Results, it was not explicitly stated. Following your valuable feedback, we will **clarify** and better emphasize the novelty of Theorem 1 in Section 4.1 in our revised manuscript.\n\n> either convergence of empirical risks implies convergence of the epsilon iterates, or perhaps vice-versa\n\nAs stated in Section 4.2, the convergence of the empirical loss function does not imply the convergence of fluctuation $\\epsilon\\_n\\^k$. To clearly illustrate this point, we have also provided two counterexamples in Appendix H, in which the $\\epsilon\\_n\\^k$ fail to converge despite the corresponding loss risks converging. Convergence of empirical risk $\\int_{\\mathcal{O}} \\mathbf{L}(p\\_n\\^k) p\\_n d \\nu$ is only a necessary condition to the convergence of $p\\_n\\^k$, but *not* a sufficient condition.\n\nThis also answers your【Question 1】. Please keep us updated if you would like to further discuss any specific conditions or applications.\n\n--- \n\n## *Novelty of the Theorem 5*\n\nWe appreciate your careful attention to our Theorem 5.\n\nSpecifically, condition *(iii)* is based on observations already present in the existing literature, while conditions *(i)* and *(ii)* are original to our work. Condition *(iv)*, though not entirely new, had previously only been suggested implicitly or used informally in prior experiments. We have provided a comprehensive, rigorous analytical justification for this condition (line 784-799). \n\nNote that for these novel conditions, a **complete** theoretical proof is available in Appendix F (pages 40-43). For conditions already present in existing literature, relevant discussions and references are included at the corresponding parts in Appendix F.\n\nFollowing your valuable feedback, we commit to **relocating** part of the explanations currently in the appendix to Section 4.3 in the revised manuscript. We will also clearly highlight the novelty of our Theorem 5 to improve the clarity and positioning of this paper.\n\nThis also answers your【Question 3】. Please inform us if you would like to further discuss any specific conditions or applications.\n\n--- \n\n## *Title and Abstract*\n\n> Consider this sentence from the abstract: ... but the \"implications for algorithmic stability\", which are not described here at all.\n\nWe wish to once again thank you for your careful observations and insightful comment.\n\nThe non-self-intersection property of submodel paths indeed plays a subtle yet important role in establishing global convergence, therefore, we should not ignore them exactly in the abstract.\n\nDuring the working stage, we intended to also derive explicit stability analysis bounds for the TMLE algorithm (motivated by analogies from parametric optimization), but ultimately gave up this aspect due to significant technical challenges. Meanwhile, people may agree that from an intuitive perspective, the non-self-intersection property of homotopy path implies that the direction of each iterative step does not exhibit cyclic or oscillatory behavior.\n\nIn response to your valuable feedback, we have **removed** this redundant phrasing and clearly indicated this stability analysis as a direction for future research in the conclusion section.\n\n> the jargon levels seem excessive to a point of obscuring the contributions.\n\nWe agree with your comment that overly specialized and obscure terminology in the abstract may not be accessible to a broad audience. Accordingly, we will **revise** the abstract in our next version to better align with the writing style of the main body.\n\n\n> That seems so central, even, that I wonder why it isn't in the title of the paper.\n\nIt is worth noting that in prior studies of TMLE focusing predominantly on statistical properties, discussions of \"convergence\" typically refer to the *asymptotic behavior* of estimators as $n \\rightsquigarrow \\infty$. Alternatively, other works address \"convergence\" in the context of solving subproblems with specific numerical solvers. This distinction caused some initial difficulty in our literature review.\n\nWe believe that the convergence claim here might indeed be easily confused with the usage of convergence in *asymptotic statistics*. Therefore, we deliberately emphasized \"optimization perspective\" rather than \"convergence\" in our title.\n\nHowever, if Reviewer bWRS believes it would be beneficial, we would certainly be willing to explicitly include 'convergence'. We welcome your thoughts on this stuff.\n\n--- \n\n##  *Remaining Questions*\n\n> 【**Comment 1**】global convergence results ... if the authors concretize some practical applications of the kinds of model families \n\nYour interpretation aligns precisely with the intended meaning of our manuscript. While space constraints prevented us from listing exhaustive applications explicitly, in principle, any practical estimator demonstrated in that paper‘s studies can be analyzed within our theoretical framework. This may include:\n\n- The mean of an outcome missing at random\n\n- Median regression\n\n- The causal effect of a continuous exposure\n\nWe would also like to politely clarify one point regarding your mention of \"by convexity\"; in fact, convexity was not employed in our major analysis. Indeed, convexity constitutes a relatively stronger condition. While it could simplify our analysis, it might somehow limit the scope and applicability of our results compared to the current theoretical analysis.\n\nWe warmly invite further discussion about specific applications of interest to you.\n\nThis also answers your【Question 2】. Please tell us if you would like to further discuss any specific instantiation.\n\n> 【**Comment 2**】... explanation of exactly what has and has not been established in the literature should be included. ... revisited when formally laying out each theoretical result\n\nWe greatly appreciate your critical and insightful feedback, and we commit to revising the manuscript accordingly, as detailed in our responses above. We also strongly agree that clearly distinguishing technical contributions is fundamental to scientific writing.\n\n> 【**Question 4**】finite sample insights seem limited to the rate of occurrence of the overshoot phenomenon? Is that correct?\n\nThank you for posing such a thought-provoking question.\n\nIn fact, our entire analysis throughout the manuscript is conducted under the assumption of a finite sample size $n < \\infty$. Thus, the insight regarding finite $n$ should be consistently recognized across all sections, rather than limited only to Section 4.4.\n\nWe also fully agree that exploring whether our current conclusions might change in the limit $n \\rightsquigarrow \\infty$ is highly interesting. However, such an analysis involves substantial complexity and is best left for future research.\n\n> 【**Question 5**】skipping over an efficient estimator? I.e it leads to suboptimal performance only *computationally*?\n\nBased on the current state-of-the-art understanding, we would respond **Yes**. However, we acknowledge the possibility that future theoretical advancements may provide more sophisticated criteria for distinguishing among these 'efficient' estimators.\n\n> 【**Question 6**】the paper has a number of grammatical issues involving definite articles, spelling, etc.\n\nWe do apologize for the typos. In response, we have conducted two more rounds of proofreading and fixed all typos identified, including\n\n- Page 2, we changed \"behavior of algorithm\" to \"behavior of the algorithm\";\n\n- Page 3, we changed `consist` to `consists` in \"the dataset … consist of $n$\";\n\n- Page 3, we added `the` before \"optimization community\";\n\n- Page 5, we changed `the` to `that` in \"requires the every parametric submodel\";\n\n- Remark 3, we changed `dose` to `does`;\n\n- Theorem 5, we changed `satisfied` to `satisfies` in \"the loss satisfied line integral\".\n\n---\n\n### Thank you again for your insightful feedback. Due to the word limit, if you would like any further clarification on the TMLE optimization perspective, please do let us know and we will respond promptly."}, {"review_id": "T3ybyqv0hX", "reviewer": "Reviewer_Mj6b", "summary": "This 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.", "questions": "1.\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.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strength:\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.", "was_revised": true, "final_justification": "The authors have clarified my concerns about the numerical experiments to verify some of the theoretical results to some extents.", "rebuttal": "### We sincerely thank the Reviewer Mj6b for your constructive and detailed feedback. We appreciate your effort in thoroughly evaluating our submission. \n\n### We believe we have addressed each point raised and remain eager to engage in any further discussion. \n\n---\n\n## Our Clarification\n\nWe apologize if the motivation and positioning of our work within the literature were not fully conveyed, which may have led to some unintended misunderstandings about the scope and objectives of our study.\n\nPlease kindly allow us to make a quick clarification at the outset:\n\n- In our research, we do **not** propose any modifications to the original algorithmic procedure; all theoretical analyses presented are grounded entirely on the classical *vanilla TMLE algorithm* as established in the literature.\n\n- Consequently, our results naturally inherit **all** original strengths of TMLE, both from a statistical analysis perspective and in terms of empirical applicability.\n\n> 【**Weakness 3**】However, there is not sufficient discussion how the initial estimator will affect ... especially on the targeted parameter \\psi.\n\n> 【**Question 4**】The 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\nWe believe that the questions you raised refer primarily to **plug-in estimation** and have already been addressed in depth in foundational TMLE literature.\n\nFor a detailed statistical analysis connecting TMLE’s convergence point to the estimation of $\\Psi(\\cdot)$, please refer to the equations and corresponding references provided in our Section 2.2. Please feel free to let us know if you would like further clarification on any specific mathematical details.\n\n> 【**Weakness 6**】... connect the optimization convergence to the asymptotic convergence. For example, how the sample size $n$\n would affect both convergence.\n\nThank you very much for your illuminating comment.\n\nIn asymptotic statistics, the primary objective is to characterize limiting behaviors as the sample size $n$ tends towards infinity. Thus, $n$ itself functions as a variable rather than an object of study. However, we fully agree that investigating the influence of $n$ on optimization properties is both interesting and technically challenging, and is respectfully deferred to future exploration.\n\nAs mentioned in Section 1, prior works on TMLE typically presume convergence of the iterative algorithm and subsequently focus on asymptotic properties such as asymptotic linearity. Our work *naturally* bridges these two aspects, as explicitly discussed in Section 4.1, where we establish that algorithmic convergence is **a sufficient condition** for these asymptotic statistical guarantees.\n\n> 【**Question 1**】The linear reparameterication in Example 1 is not clear. By additive perturbation, the resultant function may not be a proper density function.\n\nIn Example 1, the choice of $\\epsilon$ is not arbitrary in $\\mathbb{R}\\^d$, rather, it must lie within $\\left\\\\{\\epsilon: p\\_n\\^k(\\epsilon) \\in \\mathcal{M}\\right\\\\}$ to ensure the resulting probability density remains within a valid space (line 3, Algorithm 1). If you have any technical question, please feel free to tell us or consult the original TMLE literature.\n\n> 【**Question 5**】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\nWe fully agree with your perspective, though addressing this point is beyond the scope of our current paper.\n\n- TMLE utilizes existing parametric optimization algorithms from machine learning to solve its subproblems. In cases where parameter domains are constrained, well-known algorithms such as interior-point methods or ADMM could be applied. However, these serve merely as *computational tools* within the TMLE procedure and are not inherent components of TMLE itself.\n\n- Throughout all prior analyses, it has consistently been assumed that optimal solutions to subproblems are attained within the interior of $\\mathcal{M}$. This assumption is not new to our work, but is a standard practice adopted universally across analytical studies (please refer to citation in Assumption 3).\n\n---\n\n## Your Concern\n\n> 【**Weakness 2**】The theoretical investigation is kind of general ... .. illustrate the results under some specific statistical models.\n\n> 【**Weakness 5**】Although ... comprehensive, there is no illustration of such theoretical analysis under some specific statistical models.\n\n> 【**Question 8**】... illustration of such theoretical analysis under some specific statistical models.\n\nWe highly appreciate your stimulating question. We address your comments as follows:\n\n- We have already analyzed several common statistical submodels in the manuscript (line 193-195 + Appendix C.1). While this coverage is not exhaustive, the spectrum of potential statistical models is exceedingly broad, including a wide variety of loss functions, influence functions, and statistical submodel constructions, etc. Given the theoretical depth and space of a typical conference paper, we respectfully leave further detailed analyses to future work.\n\n- Our theoretical framework is designed to provide a broad (macro-level) perspective. This *generality* is technically more challenging than analyzing specific statistical models and sets the stage for numerous potential directions for upcoming investigation.\n\n- Finally, motivated by your comments and bWRS, we will include additional discussions in Sections 4 & 6 to better guide practitioners in connecting our theoretical framework to practical models. We kindly refer you to our response to Reviewer bWRS.\n\nWe hope this response addresses your concerns to some extent.\n\n> 【**Weakness 4**】There is no numerical study to evaluate the convergence results of the theoretical investigation.\n\n> 【**Question 7**】... some numerical study to evaluate the convergence results of the theoretical investigation.\n\nThank you very much for raising your concern. We have indeed considered incorporating convergence experiments, and would like to respectfully clarify our perspective as follows:\n\n- First, the practical success of TMLE has been widely demonstrated and recognized across various research communities (Section 1), with numerous studies already providing extensive numerical results. Given the scope and theoretical focus of our paper, additional numerical experiments would contribute very limited new insights.\n\n- We politely ask the reviewer to notice the theoretical-practical gap inherent in this context. Our manuscript investigates theoretical guarantees in the limit of infinite iterations ($k \\rightsquigarrow \\infty$), which is inherently unattainable in practical computations ($k <\\infty$).\n\n- Unlike typical parametric optimization (e.g. convex optimization), the TMLE template studied here involves *semiparametric* updates, a feature not directly computable with current computational configurations. Consequently, numerical experiments would necessarily rely on specific *parametric instantiation* of the TMLE framework, which cannot fully represent the broader class of semiparametric TMLE algorithms addressed by our theoretical results.\n\n- While our work examines the optimization properties of TMLE, it does not address convergence rates or iteration error, making numerical validation of our theoretical conclusions inherently inappropriate.\n\n> 【**Question 6**】it is better to mathematically define what is the “overshoot” in a rigorous manner.\n\nWe agree with your critique. Although an informal, intuitive explanation has been provided in section 4.4 (line 257-258 + Fig. 4), we promise to **add a rigorous** mathematical definition explicitly in the main body or appendix. Currently, such a definition appears only implicitly within our proof.\n\n---\n\n##  Remaining Questions\n\n> 【**Weakness 1**】The manuscript has a lot of notations. ... provide a through notation before the technical description.\n\nThanks for your comment. We indeed provided a comprehensive introduction to notations in Supplementary Material (due to the limited page), which was also acknowledged by wW82.\n\n> 【**Weakness 3**】there is not sufficient discussion how the initial estimator will affect the convergence properties of the TMLE algorithm\n\nIn Section 4.2, we investigate global convergence rather than local characteristics. Consequently, the convergence results we established do **not** depend on the initialization, provided it is a valid initial estimate.\n\nMotivated by your insightful comment, we believe that considering the impact of initialization on convergence rate or efficiency is indeed an interesting direction, and we respectfully leave this exploration to future work.\n\n> 【**Question 2**】In Definition 1, it is not clear the probability distribution can be a continuous distribution or a discrete distribution.\n\nOur analysis does not impose any specific requirement on this matter, i.e., the setting can be either continuous or discrete.\n\n> 【**Question 3**】what does “thresholding” mean here? The iterative algorithm seems not involving the thresholding operator.\n\nThank you for your constructive question. We acknowledge that the current naming may risk confusion with the \"thresholding operator\" from sparse optimization. We commit to renaming Theorem 1 to \"Condition Equivalence\" or \"Stopping Conditions\" in the next version.\n\n--- \n\n### We respectfully invite you to reevaluate our work in light of these clarifications and to consider the potential downstream value of our theoretical results. Our sincere thanks once again."}, {"review_id": "0dGHYgKxiG", "reviewer": "Reviewer_wW82", "summary": "This 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.", "questions": "1) 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?)", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 2, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strength:\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.", "was_revised": true, "final_justification": "The authors have clarified my doubts from some parts in the appendix.", "rebuttal": "### We sincerely thank Reviewer wW82 for taking the time to review our submission and for recognizing the quality of our contribution (\"4: excellent\") and the originality of our work (\"4: excellent\"). Below, we provide a detailed response to each of the comments you raised, following the order presented in review.\n\n### We would be especially grateful if you would consider further acknowledging the contributions of our theoretical work.\n\n--- \n\n> 【**Comment 1**】This seems to be the first ever work to rigorously analyze TMLE through an optimization lens... .... is genuinely novel and has practical implications.\n\nWe greatly appreciate your recognition of our contributions. To the best of our knowledge, this manuscript represents **exactly the first work** to analyze the optimization (convergence) behavior of TMLE.\n\n> 【**Comment 2**】I think it is important to have better proof sketches/an overall better presentation\n\nThank you for your kind feedback. We are committed to continually improving the overall presentation quality and readability of the manuscript (*please see response to Reviewer bWRS*).\n\n> 【**Comment 3**】Overshooting phenomenon probability bounds though novel and very interesting, I would like to see simulations to validate these results. Could you please provide some? ... does not contain any experiment and it is not a weakness in my opinion given the strength of the theory content. \n\nWe are very pleased to hear that you also find the overshooting phenomenon interesting and potentially of independent interest. Indeed, this was one of our primary motivations for sharing these findings with the community.\n\nWhen conducting this study, we also considered how to incorporate (illustrative) experiments, and thus we fully appreciate your perspective regarding numerical validation. Accordingly, we respectfully offer the following clarifications and supplemental points:\n\n- **Minimal illustration already included.** To ground the theory and identify the existing of overshooting, we do provide a self‑contained toy example (Example 3, page 8) that can be executed in just a few lines via any numerical libraries like Matlab or SciPy. It shows how a degenerate hyper‑plane forces the TMLE to leap over the EIF root, exactly as predicted. \n\n- **The large‑scale simulations are challenging.** The probability decays exponentially with $n$. To obtain a stable Monte‑Carlo estimate when the event probability is, say, $10\\^{-4}$, one needs on the order of $10\\^6$ independent TMLE runs per $n$. At present, the general mechanism underlying overshooting (and any reliable indicators) remains unclear. Under the definition, identifying overshooting at each iteration would require traversing the entire non-monotonic loss landscape and computing the empirical EIF, which is highly expensive.\n\n- As discussed in Section 6, our analysis assumes an idealized TMLE optimization in which each subproblem is solved exactly at every $k$. In practice, however, **small discrepancies can arise** (e.g. finite machine precision), and the impact of this gap on overshooting has not yet been thoroughly investigated. \n\n**We sincerely thank you for your high regard of our novel theoretical contributions.** A deeper investigation into the underlying conditions for overshooting is highly non-trivial and would likely require a full‐scale research effort, which regrettably beyond the current scope.\n\n> 【**Comment 4**】However, to me, this is an obvious place where we can have an understanding of how tight your bounds are.\n\nMany thanks for sharing this helpful comment; we agree the tightness is a promising direction for research. We believe numerical experiments would be valuable, but as discussed above, experiment itself have inherent challenges.\n\nOne point to note is that in Theorem 6 we use $\\mathbb{P}\\lesssim$ rather than a strict $<$, as deriving sharp constant is highly non-trivial. Doing so would likely require additional assumptions and statistical tools, and it’s difficult to accommodate within the current length (Appendix, p 43-45).\n\nAs the community's theoretical understanding of TMLE continues to advance, we would be delighted to further investigate this attractive phenomenon in the future.\n\n> 【**Question 1**】(part I) Inequality (41) seems really one of the key steps in the proof\n\nYour observation is absolutely correct, as it provides the simultaneous lower and upper bounds on the fluctuation $\\epsilon\\_n^k$.\n\n> 【**Question 1**】(part II)  why the final result is zero (especially the last steps (43) and (44))?\n\nThank you for carefully reading our proof and for raising an insightful question. Now we provide a *step-by-step* explanation to clarify the last equality.\n\n$$\n\\begin{aligned}\n& \\sup\\_{k\\^{\\prime} \\geq 0} \n\\frac{\\sup\\limits\\_{\\left\\\\{\\epsilon: p\\_n\\^{k\\^{\\prime}}(\\epsilon) \\in \\mathcal{M}\\right\\\\}} \\left(\\displaystyle\\sqrt{\\int\\_{\\mathcal{O}\\^{\\prime}} \\int\\_{\\mathcal{O}\\^{\\prime}}\n\\left\\\\{\\left\\\\|\\nabla\\_\\epsilon\\^2 \\mathbf{L}\\left(p\\_n^{k\\^{\\prime}}(\\epsilon)\\right)(o)\\right\\\\| \n\\left\\\\|\\nabla\\_\\epsilon^2 \\mathbf{L}\\left(p\\_n\\^{k^{\\prime}}(\\epsilon)\\right) \\left(o^{\\prime}\\right)\\right\\\\|\\right\\\\} d \\nu(o) d \\nu(o^{\\prime})} \n-\\int\\_{\\mathcal{O}^{\\prime}}\\left\\\\|\\nabla\\_\\epsilon\\^2 \\mathbf{L}(p\\_n\\^{k\\^{\\prime}}(\\epsilon))(o)\\right\\\\| d \\nu(o)\\right)}{2 \\clubsuit\\_L \\log (n+1)} \n\\\\\\\\\n= & \\sup\\_{k\\^{\\prime} \\geq 0} \n\\frac{\\sup\\limits\\_{\\left\\\\{\\epsilon: p\\_n\\^{k\\^{\\prime}}(\\epsilon) \\in \\mathcal{M}\\right\\\\}} \\left(\\displaystyle\\sqrt{\\iint\\_{\\mathcal{O}\\^{\\prime}\\times\\mathcal{O}\\^{\\prime}} \n\\left\\\\{\\left\\\\|\\nabla\\_\\epsilon\\^2 \\mathbf{L}\\left(p\\_n^{k\\^{\\prime}}(\\epsilon)\\right)(o)\\right\\\\|  d \\nu(o)\\right\\\\} \\cdot\n\\left\\\\|\\nabla\\_\\epsilon^2 \\mathbf{L}\\left(p\\_n\\^{k^{\\prime}}(\\epsilon)\\right) \\left(o^{\\prime}\\right)\\right\\\\|  d \\nu(o^{\\prime})} \n-\\int\\_{\\mathcal{O}^{\\prime}}\\left\\\\|\\nabla\\_\\epsilon\\^2 \\mathbf{L}(p\\_n\\^{k\\^{\\prime}}(\\epsilon))(o)\\right\\\\| d \\nu(o)\\right)}{2 \\clubsuit\\_L \\log (n+1)} \n\\\\\\\\\n = & \\sup\\_{k\\^{\\prime} \\geq 0} \n\\frac{\\sup\\limits\\_{\\left\\\\{\\epsilon: p\\_n\\^{k\\^{\\prime}}(\\epsilon) \\in \\mathcal{M}\\right\\\\}} \\left(\\displaystyle\n\\int\\_{\\mathcal{O}^{\\prime}}\\left\\\\|\\nabla\\_\\epsilon\\^2 \\mathbf{L}(p\\_n\\^{k\\^{\\prime}}(\\epsilon))(o)\\right\\\\| d \\nu(o)\n-\\int\\_{\\mathcal{O}^{\\prime}}\\left\\\\|\\nabla\\_\\epsilon\\^2 \\mathbf{L}(p\\_n\\^{k\\^{\\prime}}(\\epsilon))(o)\\right\\\\| d \\nu(o)\\right)\n}{2 \\clubsuit\\_L \\log (n+1)} \n\\\\\\\\\n= & \\sup\\_{k\\^{\\prime} \\geq 0} \n\\frac{0}{2 \\clubsuit\\_L \\log (n+1)} \n\\\\\\\\\n= & ~0\n\\end{aligned}\n$$\n\nThe derivation of equation (44) follows exactly the same principle as that of (43) (plus the property of matrix trace). Due to space constraints, we have omitted the details here.\n\nImportantly, motivated by your question, **we will add more technical comment to this derivation**.\n\n> 【**Question 2**】(part I) I do not understand what is happening (around step (b)) after line 691\n\nAfter line 691, we present our equation (82), which serves as a preparatory step for establishing the required Lipschitz control (84). To clarify the derivation, we would like to explain in detail the reasoning around step (b).\n\n- *How does (a) become (b)?* \n\nWe retain the first term as is, and distribute the integral across each term inside the square brackets for the second term. This facilitates the representation used in the curly braces below. In the third term within the next line, we replace the remainder $\\mathcal{R}\\left(h\\_1, o ; D\\_{\\Psi}^\\*\\right)$ by its Cauchy formula and keep $\\mathcal{R}\\left(h\\_2, o ; D\\_{\\Psi}^\\* \\right)$. For further details, please refer to the statement provided around line 693.\n\n- *How does (b) become (c)?*\n\nThis part is complex-algebraic and exploits that $|\\zeta|=\\rho>1$, $\\frac{1}{\\zeta-1}=\\sum_{m=0}^{\\infty} \\frac{1}{\\zeta^{m+1}}$.\nPull out the first three terms of that series and notice that they are exactly the polynomial $\\frac{1}{\\zeta}+\\frac{1}{\\zeta^2}+\\frac{1}{\\zeta^3}$ that was subtracted in line (b). Omitting‌ the coefficient $d^2$, what survives is\n\n$$\n\\frac{1}{\\zeta-1}-\\left(\\frac{1}{\\zeta}+\\frac{1}{\\zeta^2}+\\frac{1}{\\zeta^3}\\right)=\\sum_{j=1}^{\\infty} \\frac{1}{\\zeta^{j+3}}=\\frac{1}{\\zeta^3} \\sum_{j=1}^{\\infty} \\frac{1}{\\zeta^j}.\n$$\n\nSimilar operations apply for the another remainder $\\mathcal{R}\\left(h\\_2, o ; D\\_{\\Psi}^\\* \\right)$ via $\\alpha$-conversion. Because the circle integral has finite length $2 \\pi \\rho$ and the integrand decays like $|\\zeta|^{-4}$ or faster, dominated convergence justifies moving the infinite sum outside the path integral.\n\nWe also found that the $\\le$ at step (c) technically should be $=$. **We appreciate your constructive question again.**\n\n> 【**Question 2**】(part II)  I fail to see the necessity to bring this up here. How is it helpful in the process?\n\nThank you for the thoughtful question. Below we explain why the complex‑analytic approach is used for our step (b).\n\n- The Banach-space Cauchy integral formula is a *well-established and mature tool* in the field of analysis, and it is particularly convenient to apply in our setting.\n\n- Because the Cauchy representation works component-wise and only the radius $\\rho$ enters the denominator, we may obtain the  *dimension-independent constant* $\\frac{2\\pi}{\\rho(\\rho-1)}$ that appears in the bound (83), which feeds directly into (84) and other proofs.\n\n- Technically, the integrand's norm is bounded on the circle, so the operator norm of each derivative is $\\leq \\frac{k!}{\\rho^k} \\sup \\_{|\\zeta|=\\rho}\\left\\\\|D\\_{\\Psi}^\\*\\left(\\hat{p}\\_0+\\zeta h\\right)\\right\\\\|$. No keeping of individual third or higher‑order terms is needed.\n\nIn all, we believe this is *one possible route* to complete the rigorous proof under our conditions. If you have more elegant approaches in mind, we would be very glad to hear them and make potential advancements.\n\n--- \n\n### Thank you again for your thoughtful and constructive feedback. Should you have any further questions or require clarification on any point, please do not hesitate to let us know."}]

Author rebuttal:

Meta-review: The submission investigates the convergence properties of targeted maximum likelihood estimation (TMLE) by providing global convergence guarantees, developing geometric insights into TMLE’s iterative paths. The novelty of the offered theoretical insights are a strenght of the submission. Possible weaknesses, regarding the clarity of some proof steps (also with respect to previous contribution), heavy notation and the lack of numerical examples, were discussed and amended during the interaction period between authors and reviewers, enhancing the submission and resolving the main concerns of the latter. As observed by multiple reviewers, the paper closes a gap in the literature about an important debiasing algorithm, possibly being the first in the theoretical analysis of its convergence: the originality of the analysis and its comprehensiveness has been appreciated by all reviewers, and I recommend therefore the acceptance of the submission.
Decision: Accept (poster)