Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions
neural stochastic differential equations;Markov process;scoring rule
Got it. I agree wo do need the separability for the last paragraph in the proof of Theorem 5. I'll add the Polish assumption for sure. 

The reviewer DrUo mentioned  $\mathcal{E}$ could be a space of graphs, so would $\mathcal{E}$ being Euclidian a bit too restrictive? 

Thank you again for the insightful and timely comments.
I misinterpreted your $X,Y$ taking value in $\mathcal E$. I thought you were saying that $\omega\rightarrow X(\omega,\cdot )$ is in $\mathcal E$. In any case, you do need the processes to be separable (continuity would suffice), i.e. the FDD is determined by a dense subset of $[0,T)$, to argue for the last part of Theorem 5. This is not explicitly stated in the assumption. I would suggest you just go with $X\in C_{\mathcal E}[0,T]$ or $D_{\mathcal E}[0,T]$ where $\mathcal E$ is a Euclidian space.

For the sensitivity, I am wondering if the scoring rule is smooth (and how smooth) in a change in the parameter of the neural SDE. This will affect the convergence rates of your algorithm. I think this is interesting especially for the volatility part, as we know that the KL is infinity if you perturb the volatility (the two laws are not absolutely continuous).
Polish and sensitivity
Hi,

Thank you for the review, and we're happy to address the raised concerns.

1. We're going to fix the template in the forthcoming revision.

2. *Continuity and Markov Assumption*

As mentioned in the response to reviewer SBGZ ([link](https://openreview.net/forum?id=d4qMoUSMLT&noteId=zKB258o4r6)), the proof can be directly extended to cadlag Markov processes defined on an open interval $[0, T)$. The only modification needed in the proof is to construct the sequences to converge from the right in the last paragraph of the proof of Theorem 5. If you and the other reviewers deem it necessary, we are happy to extend the proof to the cadlag case. This allows our theory to cover jump processes.

Regarding the Markov assumption, we believe this can be partially addressed by augmenting the paths with more information or leveraging a hidden Markov model. However, these are beyond the scope of neural SDEs and may warrant a separate paper. If you feel it is necessary, we can add additional discussion on the relaxation of these assumptions.

We thank you again for your insights, and please let us know if there is anything unclear; we're more than happy to clarify!
For the sensitive properties, could you recommend any literature? We'd deeply appreciate it.
Thank you for the quick response!

Indeed, the proof can be directly extended to cadlag Markov processes defined on an open interval $[0, T)$. The only modification needed in the proof is to construct the sequences to converge from the right in the last paragraph of the proof of Theorem 5. As the focus of the paper is on neural SDEs, we did not include the proof for the cadlag case. That being said, if you and the other reviewers feel it is necessary, we are happy to extend the proof to cover the cadlag case.
I see, I forgot about the property of the scoring rule. 

By the way, in responding to the concerns of the other reviewer about the jumps, I think this should work for (at least a large subclass of) Feller processes. Because essentially the law of the Feller process will be determined by the generator. This essentially means that you only need to match the joint distribution over an infinitesimal interval. So, the a.e. Lebesgue would suffice. 

In this case, I will raise my score to 5 for now. If you can clarify your paper as suggested by me and other reviews, I will further improve the score to 6. 

As the reviewer DrUo helpfully pointed out, this would be a better submission if you could say something about the qualities/properties of the scoring rule, especially sensitivity properties,  i.e. how perturbations in $\mu$ and $\sigma$ would translate to $S$.
Response to the clarification
Hi,

Thank you for the detailed review and we're happy to address the raise concerns.

1. For the proof, it is important that we choose $\nu$ to be an equivalent measure to the Lebesgue measure $\mu$ on $[0, T] \times [0, T]$. So there exists a $\mu$-a.e. positive function $\lambda(t_1, t_2)$ such that $d\nu = \lambda d\mu$. (in case you need a proof, see https://math.stackexchange.com/questions/1393425/equivalent-finite-measures-if-and-only-if-strictly-positive-radon-nikodym-deriv)

Then 
$$E_{(t_1, t_2) \sim \nu} S(P_{\pi_{t_1, t_2} (X)}, P_{\pi_{t_1, t_2} (Y)}) = E_{(t_1, t_2) \sim \nu} S(P_{\pi_{t_1, t_2} (Y)}, P_{\pi_{t_1, t_2} (Y)})$$ 
suggests
$$E_{(t_1, t_2) \sim \mu} \lambda(t_1, t_2) S(P_{\pi_{t_1, t_2} (X)}, P_{\pi_{t_1, t_2} (Y)}) = E_{(t_1, t_2) \sim \mu} \lambda(t_1, t_2) S(P_{\pi_{t_1, t_2} (Y)}, P_{\pi_{t_1, t_2} (Y)}),$$
which further implies 
$$E_{(t_1, t_2) \sim \mu} \lambda(t_1, t_2) [ S(P_{\pi_{t_1, t_2} (Y)}, P_{\pi_{t_1, t_2} (Y)}) - S(P_{\pi_{t_1, t_2} (X)}, P_{\pi_{t_1, t_2} (Y)})]  = 0.$$

Recall that by definition of the proper scoring rule $S(P_{\pi_{t_1, t_2} (Y)}, P_{\pi_{t_1, t_2} (Y)}) \geq S(P_{\pi_{t_1, t_2} (X)}, P_{\pi_{t_1, t_2} (Y)})$ and $\lambda(t_1, t_2) > 0$ $\mu$-a.e. due to the equivalence of $\nu$ and $\mu$. This makes the integrand to be non-negative $\mu$-a.e., forcing $\lambda(t_1, t_2) [ S(P_{\pi_{t_1, t_2} (Y)}, P_{\pi_{t_1, t_2} (Y)}) - S(P_{\pi_{t_1, t_2} (X)}, P_{\pi_{t_1, t_2} (Y)})]  = 0$ $\mu$-a.e.. 

Again, due to the fact that $\lambda(t_1, t_2) > 0$ $\mu$-a.e., $S(P_{\pi_{t_1, t_2} (Y)}, P_{\pi_{t_1, t_2} (Y)}) - S(P_{\pi_{t_1, t_2} (X)}, P_{\pi_{t_1, t_2} (Y)}) = 0$ $\mu$-a.e.. 

I think two important points here are the equivalence of between $\mu$ and $\nu$ and that $s$ is a strictly proper scoring rule.

2. I'm checking where we need $\mathcal{E}$ to be Polish. The disintegration theorem we used (Theorem 8.5 of Kallenberg (2021)) only requires the value space to be Borel. We'd sincerely appreciate it you can help us to point it out.

3. For the other minor issues, we can fix them in the upcoming revision.

We thank you again for the insights and please let us know if you feel there is anything unclear; we're more than happy to clarify!
Hi,

We deeply appreciate the detailed reviews and are happy to address the concerns.

*1. In Theorem 2, does the result hold for any scoring rule $s$, or does $s$ also need to be strictly proper?*  
Yes. $s$ has to be strictly proper. This is stated in the first paragraph of section 4.1: "*Let $s$ be any strictly proper scoring rule defined on ...*". We agree this is probably not sufficiently clear and will move this condition to the theorem statement in the forthcoming revision.

*2. The complexity reduction claim*  
We apologize for this confusion. $D$ refers to the discretization step in the numerical integration of the SDE, i.e., for the SDE
$$
dZ_t = \mu^{\theta}(t, Z_t)  dt + \sigma^{\theta}(t, Z_t)  dW_t,
$$
we evaluate the integral
$$
Z_T = Z_0 + \int_{0}^{T} \mu^{\theta}(t, Z_t)  dt + \int_{0}^{T} \sigma^{\theta}(t, Z_t)  dW_t
$$
by numerical integration with $D$ discretization steps using the Euler-Maruyama method:
$$
Z_{t_{k+1}} = Z_{t_k} + \mu^{\theta}(t_k, Z_{t_k}) \Delta t + \sigma^{\theta}(t_k, Z_{t_k}) \Delta W_k
$$
where $\Delta t = T/D$, $t_{k+1}=t_k + \Delta t$, and $\Delta W_k$ is the Wiener increment over the interval $[t_k, t_{k+1}]$.

$B$ in Algorithm 1 refers to the batch size of the SGD optimizer and is different from $D$. 

The $O(D^2)$ complexity comes from the previous state-of-the-art Neural SDE training method proposed in [Issa et al., 2023](https://arxiv.org/pdf/2305.16274), which requires solving a PDE
$$
f(s, t) = 1 + \int_{0}^{s} \int_{0}^{t} f(u, v) \langle dx_u, dy_v \rangle_1  dv du
$$
(see (2) in the linked paper) and backpropagate the gradients through the PDE solver. The double integral is typically numerically approximated using a rectangular rule with $D$ discretization steps:
$$
\int_{0}^{T} \int_{0}^{T} f(u, v) \langle dx_u, dy_v \rangle_1 dv du \approx \sum_{i=1}^{D} \sum_{j=1}^{D} f(u_i, v_j) \langle dx_{u_i}, dy_{v_j} \rangle \Delta u \Delta v,
$$
where $\Delta u = T/D$, $\Delta v = T/D$, and $u_i = i \Delta u$, $v_j = j \Delta v$ for $i, j = 1, \dots, D$. The double sum requires $O(D^2)$. Also, the method of [Issa et al., 2023](https://arxiv.org/pdf/2305.16274) does not have a better complexity on $B$ as their objective also involves a double sum over $B$ (see (4) in the linked paper, our $B$ is their $m$, the double integral occurs in their $k_{sig}$). 

So overall, our method reduces the complexity from $O(D^2)$ to $O(D)$ (or, from $O(D^2 B^2)$ to $O(D B^2)$ if you prefer to also include $B$ ) as we don't need to solve the PDE with the double integral. 

*3. Not sufficient preliminary material on scoring rules or background on how these rules are used to measure divergence between two Markov processes.*  
Thank you for pointing out the lack of clarity. We're happy to add more explanation to the background of scoring rules in the forthcoming revision. We'll move the RBF kernel example earlier.

We thank you again for the insights and please let us know if you feel there is anything unclear; we're more than happy to clarify!
Yes
No ethics review needed.
In Theorem 2, does the result hold for any scoring rule $s$, or does $s$ also need to be strictly proper? Could you clarify if there are specific conditions on $s$ required for Theorem 2 to apply?
This paper addresses an important problem: the high computational complexity (quadratic in time steps) associated with training Neural SDEs using scoring rules. The authors propose a reduced complexity method, aiming for linear complexity to enhance performance. They also provide a theoretically grounded approach in designing a new scoring rule. However, despite this strong motivation, the results are somewhat unconvincing due to certain weaknesses noted below.
This paper proposes a new method called Finite Dimensional Matching (FDM) for training Neural SDEs by identifying a class of strictly proper scoring rules for comparing continuous Markov processes. Using this scoring rule, they claim to reduce the training complexity from quadratic in discretization timesteps to linear.
This paper has several notable weaknesses. While the authors propose a reduced complexity approach for training Neural SDEs, they do not adequately explain key concepts, making the paper difficult to follow, especially for readers without a strong background in this area. For example, despite experience with SDEs in score-based generative models and deep learning theory, I found the explanations lacking in detail and context and some results are not convincing.

The authors do not provide sufficient preliminary material on scoring rules or background on how these rules are used to measure divergence between two Markov processes. A concrete example of the scoring rule $s(P, z)$ with an RBF kernel, presented early on, would have improved clarity.

Additionally, the complexity reduction claim is unconvincing. For example, at the beginning, the authors claim they reduce the complexity to linear using $D$ to denote the time steps, however, this notation $D$ is never used again in the rest of the paper. Instead, in Section 4.2 Algorithm, they compare two stochastic processes and use $B$ to denote the total number of time steps. However, the nested summations in the top equation on page 5 suggest quadratic rather than linear complexity, i.e., $B^2$.

Finally, the paper uses the **ICLR 2024 format rather than the ICLR 2025** format.
Yes
No ethics review needed.
The main concern I had was in regard to the proof. Please help me to understand or fix the issue. 

If this and the writing issues I raised above can be addressed, I will happily give you at least a 6. However, since I feel that your main result is wrong, I have to give a low score for now.

If this (or a variant of the) scoring rule for Markov processes is indeed correct, I think the authors could improve the paper by exploring the sensitivity properties of this scoring rule; for example, what kernel to use and how the score behaves when P and Q are close? Is there a simple formula to compute the gradient?
The method is very simple and elegant and the implication of this result, **if true**, could be impactful to the neural SDE and generative modeling community.
This paper introduces a very simple yet elegant way of comparing the similarity of the laws of two homogeneous Markov processes. The idea follows from the fact that the distribution of the process is completely determined by the transition kernel. Hence the ability to compare the transition distribution should give enable a comparison between the laws. 

The authors then use this finding as a scoring rule to learn to simulate from a neural SDE, given the observations from a ground truth process. Then, they test their scoring rule and their FDD matching procedure in numerical experiments, showing that their method outperforms existing law matching techniques.
I think there is a significant issue in the proof that could invalidate your main theorem 2. In the middle part of page 14, you have the integral over $t_1,t_2$ of the $S(P_{t_1,t_2}, P_{t_1,t_2}')$ scoring rules being equal, but then you conclude that $$S(P_{t_1,t_2},P_{t_1,t_2}') = S(P_{t_1,t_2},P_{t_1,t_2})$$ a.e.. This is not true or I am missing something? 

My intuition is that this is not an easy fix: You want to match the distribution over all $t_1,t_2$, so, in some sense, you want the expectation equality to hold for all test measures $\nu$ instead of just one particular $\nu$. I would be very happy to raise the score and rewrite my review if I am wrong. However, it does seem that proving a result like yours is possible, given that the generator or the resolvent will determine the law of the Markov process (see Either and Kurtz). 

I appreciate the rigor in defining the math notations and results. But the writing and explanations in the paper need improvements. For example, what do you mean by "Update the model parameters $\theta$ through backpropogation to maximize $\hat S$" (inside the algorithm)? Also, there is no explanation of what the data is. What are the "Average KS test scores"? Are you repeating the experiment across multiple batches and producing the percentage of rejection "chance of rejecting the null hypothesis (%) at 5%-significance level on marginal"? 

Other minor issues include: 
1. I think you need $\mathcal{E}$ to be Polish. 
2. The radial basis function (RBF) kernel is not defined. 
3. The $\pi$ notation seems a little distracting, why not just use $(x_t,x_s)$ for $\pi_{t,s}(x)$. 
4. It would be nice to emphasize that the Markov processes you define are homogeneous Markov processes.