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.

We would be especially grateful if you would consider further acknowledging the contributions of our theoretical work.

【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.

We 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.

【Comment 2】I think it is important to have better proof sketches/an overall better presentation

Thank 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).

【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.

We 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.

When 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:

• 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.

• 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.

• 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.

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.

【Comment 4】However, to me, this is an obvious place where we can have an understanding of how tight your bounds are.

Many 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.

One 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).

As the community's theoretical understanding of TMLE continues to advance, we would be delighted to further investigate this attractive phenomenon in the future.

【Question 1】(part I) Inequality (41) seems really one of the key steps in the proof

Your observation is absolutely correct, as it provides the simultaneous lower and upper bounds on the fluctuation $\epsilon\_n^k$.

【Question 1】(part II) why the final result is zero (especially the last steps (43) and (44))?

Thank 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.

The 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.

Importantly, motivated by your question, we will add more technical comment to this derivation.

【Question 2】(part I) I do not understand what is happening (around step (b)) after line 691

After 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).

• How does (a) become (b)?

We 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.

• How does (b) become (c)?

This part is complex-algebraic and exploits that $|\zeta|=\rho>1$, $\frac{1}{\zeta-1}=\sum_{m=0}^{\infty} \frac{1}{\zeta^{m+1}}$. Pull 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

Similar 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.

We also found that the $\le$ at step (c) technically should be $=$. We appreciate your constructive question again.

【Question 2】(part II) I fail to see the necessity to bring this up here. How is it helpful in the process?

Thank you for the thoughtful question. Below we explain why the complex‑analytic approach is used for our step (b).

• 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.

• 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.

• 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.

In 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.

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.

I have updated my ratings accordingly. Thank you.

We sincerely thank the Reviewer Mj6b for your constructive and detailed feedback. We appreciate your effort in thoroughly evaluating our submission.

We believe we have addressed each point raised and remain eager to engage in any further discussion.

Our Clarification

We 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.

Please kindly allow us to make a quick clarification at the outset:

• 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.

• Consequently, our results naturally inherit all original strengths of TMLE, both from a statistical analysis perspective and in terms of empirical applicability.

【Weakness 3】However, there is not sufficient discussion how the initial estimator will affect ... especially on the targeted parameter \psi.

【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}.

We believe that the questions you raised refer primarily to plug-in estimation and have already been addressed in depth in foundational TMLE literature.

For 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.

【Weakness 6】... connect the optimization convergence to the asymptotic convergence. For example, how the sample size $n$ would affect both convergence.

Thank you very much for your illuminating comment.

In 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.

As 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.

【Question 1】The linear reparameterication in Example 1 is not clear. By additive perturbation, the resultant function may not be a proper density function.

In 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.

【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.

We fully agree with your perspective, though addressing this point is beyond the scope of our current paper.

• 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.

• 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).

Your Concern

【Weakness 2】The theoretical investigation is kind of general ... .. illustrate the results under some specific statistical models.

【Weakness 5】Although ... comprehensive, there is no illustration of such theoretical analysis under some specific statistical models.

【Question 8】... illustration of such theoretical analysis under some specific statistical models.

We highly appreciate your stimulating question. We address your comments as follows:

• 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.

• 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.

• 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.

We hope this response addresses your concerns to some extent.

【Weakness 4】There is no numerical study to evaluate the convergence results of the theoretical investigation.

【Question 7】... some numerical study to evaluate the convergence results of the theoretical investigation.

Thank you very much for raising your concern. We have indeed considered incorporating convergence experiments, and would like to respectfully clarify our perspective as follows:

• 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.

• 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 ($k <\infty$).

• 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.

• 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.

【Question 6】it is better to mathematically define what is the “overshoot” in a rigorous manner.

We 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.

Remaining Questions

【Weakness 1】The manuscript has a lot of notations. ... provide a through notation before the technical description.

Thanks 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.

【Weakness 3】there is not sufficient discussion how the initial estimator will affect the convergence properties of the TMLE algorithm

In 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.

Motivated 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.

【Question 2】In Definition 1, it is not clear the probability distribution can be a continuous distribution or a discrete distribution.

Our analysis does not impose any specific requirement on this matter, i.e., the setting can be either continuous or discrete.

【Question 3】what does “thresholding” mean here? The iterative algorithm seems not involving the thresholding operator.

Thank 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.

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.

Simulations on (i), (ii) of Thm 5

Our experimental setup largely mirrors the previous one.

1. We first verify the condition (i) within Thm 5. For simplicity, we choose $p$ to be a degenerate distribution. We then follow the TMLE algorithm and execute one step of the pseudocode, obtaining the resulting $\epsilon$ from the solver (from SciPy) of the subproblem. Note that for each coordinate

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.

In the following, we address each of the technical points appeared in your review.

Novelty of the Theorem 1

We sincerely apologize if the phrasing in our manuscript led to any confusion or misunderstanding.

(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, ...

Strictly 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.

For 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

... These established results are mentioned by the authors, but are scattered throughout the paper ...

Thank 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.

Although $(\text{E3})$ 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.

either convergence of empirical risks implies convergence of the epsilon iterates, or perhaps vice-versa

As 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.

This also answers your【Question 1】. Please keep us updated if you would like to further discuss any specific conditions or applications.

Novelty of the Theorem 5

We appreciate your careful attention to our Theorem 5.

Specifically, 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).

Note 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.

Following 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.

This also answers your【Question 3】. Please inform us if you would like to further discuss any specific conditions or applications.

Title and Abstract

Consider this sentence from the abstract: ... but the "implications for algorithmic stability", which are not described here at all.

We wish to once again thank you for your careful observations and insightful comment.

The 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.

During 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.

In 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.

the jargon levels seem excessive to a point of obscuring the contributions.

We 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.

That seems so central, even, that I wonder why it isn't in the title of the paper.

It 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.

We 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.

However, if Reviewer bWRS believes it would be beneficial, we would certainly be willing to explicitly include 'convergence'. We welcome your thoughts on this stuff.

Remaining Questions

【Comment 1】global convergence results ... if the authors concretize some practical applications of the kinds of model families

Your 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:

• The mean of an outcome missing at random

• Median regression

• The causal effect of a continuous exposure

We 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.

We warmly invite further discussion about specific applications of interest to you.

This also answers your【Question 2】. Please tell us if you would like to further discuss any specific instantiation.

【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

We 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.

【Question 4】finite sample insights seem limited to the rate of occurrence of the overshoot phenomenon? Is that correct?

Thank you for posing such a thought-provoking question.

In 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.

We 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.

【Question 5】skipping over an efficient estimator? I.e it leads to suboptimal performance only computationally?

Based 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.

【Question 6】the paper has a number of grammatical issues involving definite articles, spelling, etc.

We do apologize for the typos. In response, we have conducted two more rounds of proofreading and fixed all typos identified, including

• Page 2, we changed "behavior of algorithm" to "behavior of the algorithm";

• Page 3, we changed consist to consists in "the dataset … consist of $n$";

• Page 3, we added the before "optimization community";

• Page 5, we changed the to that in "requires the every parametric submodel";

• Remark 3, we changed dose to does;

• Theorem 5, we changed satisfied to satisfies in "the loss satisfied line integral".

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.

I appreciate the clarifications around novelty — I hope the mentioned changes will help highlight the importance of the work for readers by emphasizing the gaps being addressed.

Regarding the point about "convergence" typically referring to asymptotic statistics here: I agree with the authors' assessment. Perhaps emphasizing the (optimization) global convergence result earlier in the abstract might be a good middle ground? That way context can be provided clarifying that the result does not relate to statistical convergence.

Re. comment 1: it might be helpful here (perhaps in the appendix) to provide some examples of impactful statistical tasks and sub-model families for which global convergence was not guaranteed before your work (you may have done this already, if so, please kindly point me to the content!). And to clarify the "by convexity" statement, I was referring to assumptions that (I believe) D'iaz and Rosenblum make. On further read of their work, it seems they may not provide global convergence result for their more restricted setting — would you mind clarifying whether you're aware of global convergence results that exist already for more restricted settings and assumptions that you work with in this paper?

Regarding other comments and questions, thank you for the clarifications! Overall, I maintain my assessment of the importance of this paper.

Thanks for these additional clarifications.

I wonder if some version of the following statement can be incorporated into the introduction somewhere:

existing convergence results in the literature have focused only on scenarios achievable in one- or two-step iterations. Unfortunately, we have not found prior work addressing convergence from a computational perspective in more general settings.

I think connecting this to convexity properties could help as well, in addition to explicitly juxtaposing convexity with the non-self-intersecting result as being a much more general means of establishing convergence.

I'm sure I am missing some nuance here, so take those suggestions with "a grain of salt", of course.