Finite-Time Analysis of Discrete-Time Stochastic Interpolants

Thank you for your time and effort in reviewing our paper! We are grateful for your constructive suggestions, which have significantly guided our improvements. Please find our responses to your comments below.

Other Comments Or Suggestions:

Right Column Line 170: “initial distribution mismatch (i.e., $\text{KL}( \rho(t_0)\Vert\hat{\rho}(t_N))$)” it should be $\text{KL}( \rho(t_0)\Vert\hat{\rho}(t_0))$?
Right Column Lines 361-365: There seems to be a grammatical error here, part of the sentence is repeated.
Right Column Line 373: aspeccts - > aspects

A: Thanks for your suggestion! We will update them in the final version.

Questions For Authors:

Is there some way to selected or design $\gamma$ to yield efficient sampling?

A: Thanks for your insightful question! Our current analysis regarding schedule design and its associated sample complexity is predicated on a fixed latent scale, $\gamma$. In Section 5, we adopt $\gamma(t)=\sqrt{at(1-t)}$ due to its prevalence and natural appeal, stemming from its connection to the Brownian bridge process. However, the systematic design of $\gamma$ to optimize sampling efficiency represents a compelling avenue for future research, potentially necessitating a synergistic approach encompassing both theoretical analysis and practical experimentation.

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Thank you for your time and effort in reviewing our paper! We are grateful for your constructive suggestions, which have significantly guided our improvements. Please find our responses to your comments below.

Methods And Evaluation Criteria

I believe the authors should explain the novelty of the Exponentially Decaying Time Schedule. Given that flow models and diffusion models are not substantially different, it appears that the authors may have merely transferred a time schedule from diffusion to flow models.

A: We first would like to emphasize that our work introduces the first discrete-time convergence bound for SDE-based generative models within the stochastic interpolants framework. Theorem 4.3 provides an explicit upper bound on the estimation error, expressed in terms of step sizes $h_k$, dimension $d$, latent scale $\gamma$, and the distance between two distributions measured by $\mathbb{E}\Vert x_0-x_1\Vert^p$. This theorem establishes an error bound applicable to arbitrary time schedules, thereby offering insights into the design of schedules that minimize sample complexity. The schedule presented in Section 5 is designed based on this theorem, wherein we stipulate that $h_k$ should be proportional to $\bar{\gamma}_k^2$ to ensure a well-balanced discretization error.

Specifically, for the case where $\gamma(t)=\sqrt{at(1-t)}$, the derived schedule manifests as an exponentially decaying schedule, characterized by a reduction in step size on both ends of the interval $[0,1]$. While this schedule shares similarities with the exponentially decaying schedule employed in diffusion models, a key distinction lies in the fact that diffusion models typically decay the step size on only one side of the interval.

Furthermore, it is noteworthy that for alternative choices of $\gamma$, the optimal schedule may deviate from the above schedule.

Experimental Designs Or Analyses

Although the authors provide theoretical justification for their method’s superiority, the experiments themselves appear overly simple. I recommend conducting toy experiments on low-resolution datasets such as CIFAR-10 or ImageNet32 since training flow models on these datasets are relatively fast and computationally manageable.

A: Primarily, this work constitutes a theoretical investigation, wherein we establish the first discrete-time complexity analysis for the stochastic interpolants framework. Our contributions include the derivation of an explicit error bound for the discrete-time sampler, as presented in Theorem 4.3, and the subsequent exploration of schedule design strategies aimed at minimizing sample complexity. The objective of our numerical experiments is to validate our theoretical findings.

While we acknowledge the value of demonstrating our method's efficacy on more complex datasets, such as CIFAR-10 or ImageNet32, we must emphasize that the application of our framework to these datasets introduces a multitude of practical challenges. These challenges encompass the design of effective network architectures, the meticulous tuning of hyperparameters, and the optimization of learning algorithms. Addressing these practical considerations falls outside the primary scope of this theoretical study. However, we recognize the importance of empirical validation on standard datasets and consider it an interesting direction for future research.

Given that the authors’ theory largely transfers discrete-time analysis to flow models, it would be more convincing to present additional experiments that demonstrate the claimed ability to “design efficient schedules for convergence acceleration.”

A: A core contribution of our work lies in the derivation of the inaugural theoretical error bound for distribution estimation within the stochastic interpolants framework, explicitly expressed in terms of the step sizes. Leveraging this result, we theoretically demonstrate the feasibility of designing schedules that accelerate convergence by optimizing the balance between error terms.

To empirically validate the efficacy of our schedule design methodology, we have conducted numerical experiments for both $\gamma=\sqrt{(1-t)t}$ (Section 6) and $\gamma=\sqrt{(1-t)^2t}$ (Appendix D.1), comparing the performance of our specifically designed schedules against that of the standard uniform schedule. While we acknowledge the potential for further experimentation to strengthen our claims, we consider the current numerical results as a solid validation. Furthermore, the exploration of optimal choices for $\gamma$ in practical applications represents an intriguing avenue for future research.

Essential References Not Discussed

A: Thank you for pointing them out! We will discuss them in the revision.

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Thank you for your time and effort in reviewing our paper! We are grateful for your constructive suggestions, which have significantly guided our improvements. Please find our responses to your comments below.

Experimental Designs Or Analyses

A: Thanks for the suggestion. We tested the samplers on $d$-dimensional Gaussian mixtures ($\rho_0$ and $\rho_1$) and compared the distribution estimation error for different $d$. The real drift terms used in the sampler are analytically computed [1]. You can check link1 or link2 for results. The first figure compares the error for different $d$ with fixed iterations, and the second shows convergence for higher dimensions. Both results align with our theory and validate the $d$ dependence.

Other Comments Or Suggestions

A: Thank you for your suggestions! We will update them in the revision.

Questions For Authors

A1: We first emphasize that our work primarily focuses on the theoretical analysis of the general stochastic interpolant method. We provide the first discrete-time analysis and propose an upper bound for the distribution estimation error. Additionally, we develop a discretization time schedule that optimizes this bound for a specified $\gamma$.

Theorem 4.3 offers a general upper bound, controlling the error using $\varepsilon_{b_F}^2$, the distance between distributions, data dimension $d$, and step sizes $h_k$. Assumption 4.2 quantifies how close $\hat b_F(t,x)$ is to $b_F(t,x)$ and holds for some $\varepsilon_{b_F}^2$ if $\hat b(t,x_t)$ has finite second moments. Theorem 4.3 illustrates how this impacts the final distribution error.

Our numerical experiments validate these findings. We use the loss function $\mathcal{L}[\hat{v}]=\frac{1}{2}\int_{t_0}^{t_N}\mathbb{E}[|\hat{v}|^2-2\partial_tI\cdot\hat{v}]\text{d}t$ to train the estimator $\hat{v}(t,x)$ for $v(t,x)$. This choice proved effective and is commonly used (e.g., [1], [2], [3]).

However, creating robust estimators for general applications is challenging, requiring careful designs for network architectures, learning algorithms, and learning rate schedules, which is a direction for future research.

A2: Regarding dimension dependence in Theorem 4.3, the key terms contributing to the discretization error are: $\varepsilon_{\text{dis}}\lesssim\sum_{k=0}^{N-1}h_k^3(d^3\bar \gamma_k^{-6}+d\sqrt{\mathbb{E}|x_0-x_1|^8}\bar\gamma_k^{-2})+\sum_{k=0}^{N-1}h_k^2(d^2\bar\gamma_k^{-4}+d\sqrt{\mathbb{E}|x_0-x_1|^4}\bar\gamma_k^{-2}).$ For a multivariate Gaussian $z\sim\mathcal{N}(0,I_d)$, $\mathbb{E}|z|^{2p}\le C(p)d^p$. If $x_0$ and $x_1$ are multivariate Gaussians, $\mathbb{E}|x_0-x_1|^8=O(d^4)$. Substituting this shows the dependence on $d$ is $O(d^3)$ in the $h_k^3$ term and $O(d^2)$ in the $h_k^2$ term. The dimensional dependence is also consistent in Corollary 5.2.

A3:Our work is focused on discrete-time analysis, which is new for the stochastic interpolant framework. Unlike continuous-time error bounds that use the 4th moment assumption (see [1]), our discrete-time analysis requires the 8th moment assumption. In Theorem 4.2, we control $\mathbb{E}|\nabla b_F(t,x_t)\cdot b_F(t,x_t)|^2$, requiring $\mathbb{E}|x_0-x_1|^8<\infty$. This term appears as $h_k^3\bar{\gamma}_k^{-2}d\sqrt{\mathbb{E}|x_0-x_1|^8}$ in Theorem 4.3's error bound.

However, this term doesn't dominate when the step size is small, with the error bound being dominated by $h_k^2\bar{\gamma}_k^{-2}d\sqrt{\mathbb{E}|x_0-x_1|^4}$. Relaxing the 8th moment assumption is a future research direction.

A4: It is true that the drift term $b_F(t,x)$ has larger variation near boundaries, motivating our exponentially decaying schedule in both theory and experiments. Our estimator approximates $b_F(t,x)$ well for $t_0=0.001$, so this choice does not pose a problem. We will add more details and visualizations in the revision.

A5: While analyzing the estimation errors of score and velocity terms separately is suggested, we find it unnecessary for our analysis. Partitioning $b_F(t,x)=v(t,x)+(\epsilon-\dot{\gamma}\gamma)s(t,x)$ results in $\varepsilon_{b_F}^2\lesssim\varepsilon_v^2+\sup(\epsilon-\dot{\gamma}\gamma)^2\cdot\varepsilon_s^2$, which is similar to [1] for continuous-time stochastic interpolants.

References

[1] Albergo, M. S., Boffi, N. M., and Vanden-Eijnden, E. Stochastic interpolants: A unifying framework for flows and diffusions, 2023.

[2] Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling, 2023.

[3] Albergo, M. S. and Vanden-Eijnden, E. Building normalizing flows with stochastic interpolants, 2023.

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We hope our response addresses your concerns. If so, we wonder if you could kindly consider raising your score? We will also be happy to answer any further questions you may have. Thank you very much!

Thank you for your time and effort in reviewing our paper! We are grateful for your constructive suggestions, which have significantly guided our improvements. Please find our responses to your comments below.

Claims And Evidence / Weakness

The differences introduced by the slightly more general interpolant $I(x_0, x_1, t)$ are dealt with in Lemmas B.3 and C.1, which are the main technical contribution of the work (Could the authors confirm?). A minor weakness: The technical extension over prior work might be slightly limited but it can still be worthwhile to have the bound in the literature

A: The main contributions of our work include the following. First, we propose the first discrete-time analysis for the stochastic interpolant framework, where we propose a new discrete-time sampler and provide the rigorous distribution estimation error bound for the sampler (Theorem 4.3). Second, based on the error bound, we further develop a time schedule for the discretization to achieve lower sample complexity (Corollary 5.2), when the latent scale is defined by $\gamma(t)=\sqrt{at(1-t)}$. Lastly, we validate our theoretical results with numerical experiments.

Among our technical results, Lemma B.3, Lemma C.1 and Appendix B.2 address the discretization error of the drift term $b_F(t,x)$. In this part, the main difference between our result and previous results is that we utilize a more general interpolant $I(t,x_0,x_1)+\gamma(t)z$ between two general distributions. This introduces a novel velocity term $v(t,x)$ in the process, and demands new analysis for the estimation error compared to existing results on diffusion models.

Questions

Could an additional assumption on the target distribution remove the unfortunate initial KL term in the bound?

A: Firstly, the initial KL term quantifies the discrepancy between the true initial distribution $\rho(t_0)$ and the estimated distribution $\hat{\rho}(t_0)$, where $\hat X_{t_0}$ is sampled from. We incorporate this term into our theorem to comprehensively cover the scenario where $\hat{X}_{t_0}$ differs from $\hat{\rho}(t_0)$, and provide a corresponding error bound in Theorem 4.3

Next, we look at two examples on how this term affects our error bound. If we define $I$ such that $I(t,x_0,x_1)=x_0$ within the interval $t\in[0,t_0]$, the initial error becomes exactly $0$ because we can directly sample from $\rho(t_0)$ (note that during generation, data from $\rho_0$ is accessible). Furthermore, if $\gamma^2(t)=\Theta(t)$ near $t=0$, we can achieve an $O(t_0)$ initial KL error if we choose $\hat{\rho}(t_0)$ such that $\hat{X}_{t_0}=x_0+\gamma(t)z$ for $x_0\sim\rho_0$ and $z\sim\mathcal{N}(0,I_d)$. In this case, the initial KL error is the same to the initial KL error of diffusion models with $T=\Theta(\log(1/t_0))$ [1].

References

[1] Benton, J., Bortoli, V. D., Doucet, A., and Deligiannidis, G. Nearly d-linear convergence bounds for diffusion models via stochastic localization

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