We appreciate the reviewer's insightful comments and supportive review.

On Gaussian Mixtures. As highlighted in Remark 4.2, the generalization of Proposition 4.1 to Gaussian mixtures with non-isotropic components is indeed straightforward. It involves considering bounds based on the minimum or maximum eigenvalues of the covariance matrices. This adjustment allows one to control the local Lipschitz constants of the score function accordingly.

Nevertheless, we would be very grateful if the reviewer could point us to the specific references they had in mind regarding recent results on the log-Lipschitz regularity of the score for general Gaussian mixtures. We would be happy to acknowledge them and expand our discussion in the revised version, possibly providing further insights into the verification of our assumptions in these more general settings.

AC assumption. Since $\pi_{\infty}$ is a Gaussian distribution, absolute continuity (AC) of $\pi_{\mathrm{data}}$ w.r.t. $\pi_{\infty}$ is equivalent to AC w.r.t. the Lebesgue measure. This condition is directly implied by our Assumption H1. This requirement is still quite mild. For instance, if $\pi_{\mathrm{data}}$ were supported on a lower-dimensional manifold, the smoothing effect of the forward process ensures that $p_t$ becomes absolutely continuous w.r.t.\ $\pi_{\infty}$ for any $t > 0$, due to convolution with Gaussian noise. This makes the assumption broadly applicable and not restrictive in practice. A comment clarifying this point has been added in the revised version of the manuscript.

Link with the HJB equation. We appreciate this insightful remark and have added a short subsection in the revised manuscript to elaborate on the connection between HJB equations, control theory, and SGMs. This addition draws on recent works (e.g., Berner et al., 2022; Zhang and Katsoulakis, 2023; Zhang et al., 2024a; Conforti et al., 2025), and aims to better contextualize our analysis by highlighting this rich and increasingly active intersection.

Typo. We are grateful to the reviewer, as this helped us to spot a typo that has now been corrected. The $\sqrt{hd}T$-dependence remains valid.

On weakly log-concave distributions. A notable class of weakly log-concave distributions arises from the convolution of distributions supported on lower-dimensional manifolds with a Gaussian kernel. This construction effectively smooths out the singularities, yielding a weakly log-concave distribution. This observation is also emphasized in Saremi et al. (2023). This insight may also help explain why early stopping techniques often perform better than directly modeling the data distribution: the intermediate distributions encountered during training are closer to weakly log-concave regimes, for which theoretical guarantees and sampling stability are more readily attainable. We have added a remark on this point in the revised version of the manuscript to clarify the broader applicability of our assumptions.

We thank the reviewer for the thoughtful and positive feedback.

Assumption H2. We emphasize that Assumption H2 is fully comparable to the standard estimation error assumptions widely used in the literature (see, e.g., Conforti et al., 2025; Benton et al., 2024; Chen et al., 2022a). In particular, our main result remains valid even if H2 is replaced by a more classical $L^2$-type control on the score approximation error of the forward process, provided that $\tilde s_{\theta^\star}(T - t_k, \cdot)$ is Lipschitz, uniformly in time. Importantly, Proposition B.2 ensures that this additional condition is not restrictive. Assumption H2 is therefore entirely consistent with standard theoretical frameworks, and its practical relevance is supported by well-established results. In particular, we refer to Appendix A of Chen et al. (2023), where this type of assumption is shown to hold in simple yet illustrative settings. We agree that a more detailed discussion of H2 would have added clarity. For this reason, we included a comment in the revised version of the manuscript to clarify this point.

Simulations. We thank the reviewer for this valuable suggestion. We refer to Appendix E.3 of Strasman et al. (2024) for an implementation of a similar bound. Our findings provide the theoretical foundation underlying these simulation results, and we believe a numerical study could be conducted by closely following the same methodological framework.

Finally, we would like to thank the reviewer for the list of misprints and other small comments, which have been happily incorporated in the revised manuscript.

We appreciate the detailed comments and constructive feedback. We highlight that, as stated in Theorem 3.4 and its accompanying discussion, this result indeed provides a bound up to a multiplicative constant depending on $\alpha, M, L_U$, and the second-order moment $m_2$ of $\pi_{\rm data}$, via the constant $B$ defined in Equation (31). Such a multiplicative constant includes all the terms listed by the reviewer. We thank them for the careful reading and address their concern in the revised manuscript as follows.

• Theorem 3.4 now displays the dependence on $m_2$. This quantity no longer appears in the multiplicative constant; instead, its influence on the bound is now explicitly reflected. We have also emphasized that the multiplicative constant depends on the parameters $\alpha$, $M$, and $L_U$, with references to the explicit expression of the constant provided in the appendix.
• We added a discussion on the leading term $e^{L_U \eta}$, highlighting that it reflects the intrinsic complexity of transforming a potentially complex, multimodal distribution into a unimodal Gaussian through an OU process. In particular, this term captures the challenge posed by large-scale structure or mode separation in the target distribution, as also highlighted in Saremi et al., 2023.
• We particularly appreciated the observation regarding the $\sqrt{h} L_U d$ term. This helped us identify and correct a typo in the proof. The correct expression is actually $\sqrt{h d} L_U$, and this has now been fixed in the revised version of the manuscript.

Comparison with $W_2$-bounds. We acknowledge that the approach in Benton et al. (2024) relies on different assumptions and methodologies, including exponentially decreasing step sizes and early stopping, while avoiding regularity assumptions on the data distribution. Given these differences, a direct side-by-side comparison is non-trivial.

Our bound is directly comparable to those of Chen et al. (2023) and Conforti et al. (2025), as it exhibits the same dependencies on $h$, $d$, $\varepsilon$, $m_2$, and $T$. A key distinction, however, lies in the fact that our result is expressed in terms of the Wasserstein distance $W_2(\pi_{\text{data}}, \pi_\infty)$ rather than the KL divergence, which makes it more practical to estimate from samples—see, e.g., Strasman et al. (2024).

Regularity assumptions. We very much agree that early stopping strategies could relax or remove the Lipschitz assumption. However, we believe that the explicit form of contraction property of the data distribution remains crucial to ensure the stability and convergence guarantees of our analysis.

We thank the reviewer for the time and effort taken to provide such detailed and insightful feedback. These comments have been valuable in the clarification of key aspects of our work and strengthen its presentation.

Assumption H2. We have clarified in the main text the comparability of Assumption H2 with the standard estimation error assumption commonly used in the literature (see, e.g., Conforti et al., 2025). Notably, our main result remains valid if H2 is replaced by an $L^2$-type condition on the score function of the continuous-time forward process, combined with the requirement that $\tilde s_{\theta^\star}(T - t_k, \cdot)$ is uniformly Lipschitz in space. Importantly, Proposition B.2 ensures that this additional condition is not restrictive. Under these assumptions, we can apply the triangle inequality together with Proposition B.2 to obtain: $||\nabla \log \tilde p_{T-t_k}( X^\star_{t_k}) - \tilde s_{\theta^\star}( T-t_k, X^\star_{t_k}) || \leq || \nabla\log \tilde p_{T-t_k}(\overleftarrow X_{t_k} ) - \tilde s_{\theta^\star}( T-t_k,\overleftarrow X_{t_k}) || +(L+L')|| X^\star_{t_k}-\overleftarrow X_{t_k}||$, with $L'$ the Lipschitz constant of the score estimator. Replacing H2 with the standard assumption, the Lipschitz regularity of the score and its estimator combined with a straightforward generalisation of Proposition C.6 in (Strasman et al., 2024) would lead to only a minor adjustment in the definition of $\delta_k$ in Section D.3. As a result, aside from slight modifications to Lemma E.2, this substitution does not materially affect the proof of the main theorem or alter the key features of the final convergence bound. Hence, Assumption H2 is fully compatible with the classical framework, and its validity directly follows from the well-established correctness of the standard score estimation assumption. We would also like to refer to Appendix A in Chen et al. (2023), where it is demonstrated that the standard estimation error assumption holds in simple yet practically relevant scenarios. We opted for the formulation in Assumption H2 as it offers the most direct and tractable path within our proof technique. We hope that highlighting the connection between these formulations helps clarify the transparency of our bound and bridges the gap between different perspectives in the literature.

Link with the HJB equation. We fully acknowledge the importance of the link with the Hamilton-Jacobi-Bellman (HJB) equation. This connection is rooted in a long history on the analysis of SGMs (see, e.g., Berner et al., 2022; Zhang and Katsoulakis, 2023; Zhang et al., 2024a; Conforti et al., 2025). Following the suggestion of the reviewer, we dedicated a discussion to clarify this connection explicitly in the revised version of the manuscript, showing the main works where this has been used.

Sentence constructions. We acknowledge the ambiguity in the phrasing “not (necessarily) contractive” and thank the reviewer for pointing this out. In the revised version of the manuscript, we better specify this making reference to the Appendix where we define $T^\star$, the actual switching point in contractivity, and introduce $T(\alpha,M)$ as a lower bound on $T^\star$. Ultimately, this is not a limitation of our proof technique but rather a direct consequence of the fact that $T^\star$ is not explicitly computable.

Finally, we have softened the original phrasing regarding the dependence on the dimension $d$. In the revised version of the paper, we now emphasize that our bound is in line with previously established results.

Additionally, we would like to thank a reviewer for the misprint suggestion. We have happily introduced the required changes in the revised manuscript.