Deep Stochastic Processes via Functional Markov Transition Operators

Summary

The paper introduces a new class of Bayesian nonparametric models by extending neural processes. The fundamental idea is to create a Markov chain of stochastic processes, culminating in a flexible enough stochastic process.

Strengths

The paper introduces a simple but a clever modification of vanilla neural processes. The idea allows for more expressive representation as demonstrated by the experiments. I believe the idea is significant enough for it to be of interest to the community.

Weaknesses

The paper uses an unclear notation for probabilistic notions. It’s not just that this choice is unconventional, forcing the reader to keep a mental map of how things are redefined, but also that it needlessly complicates exposition. For example, what is $p$? It is ubiquitous in the paper but is never clearly defined. This makes the discussion too hand-wavy — what is $p_{x_{1:n}}(y_{1:n})$? Is it the distribution function of the random vector $(F(x_1), \ldots, F(x_n))$? If so, what even is $y_{1:n}$? Is the density function of the random vector $(F(x_1), \ldots, F(x_n))$? What is $f$? Since $\mathcal X$ is simply the indexing set, and for real-valued random variables we should have $\mathcal Y = \mathbb R$, is $f$ simply a real-valued function defined on the indexing set? The notation for transition operators is also very troublesome. My point in raising such questions is that I would have preferred if the authors used standard measure-theoretic notation (it is standard for a good reason) to write much of the paper (I really like the treatment in Ghosal and van der Vaart’s book).

Questions

None

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

1 poor

Contribution

3 good

Limitations

Not applicable.

Summary

This paper proposes Markov neural process, a new class of stochastic processes. The work is based on prior art on neural process, which was published in 2018. The paper is reasonably written. The effectiveness of the new method is very clear from the experimentations.

Strengths

1. The proposed markov neural process adds more expressiveness to the traditional neural process, which potentially can open the door to new applications; 2. The theoretical analysis appears sounds 2. The proposed method appears to be competitive when compared to the baselines.

Weaknesses

1. The explanation of the overall MNP is hard to follow. Fig 2 is crucial to explain the overall structure, plus the training and inference flow. But it's unclear. For example, the three type of the arrows in Fig 2(b) aren't explained well. 2. The experiments are impressive in the tables. But the manuscript didn't provide any intuitive explanation why MNP works better. In the contextual bandit example, MNP shows significant improvement over NP when $\delta = 0.99$. Some explanation the reason or an ablation study with respect to the number of markov stages is needed for the readers to get a better feeling.

Questions

1. please elaborate the statement "the exchangeability and consistency in MNPs could improve the robustness and reliability of neural SP model".

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

2 fair

Contribution

3 good

Limitations

No potential negative societal impact.

Summary

The paper attempts to improve the flexibility of Neural Processes without sacrificing exchangeability and consistency properties of classical stochastic processes. The paper proposes a diffusion-like iterative refinement of the predicted values, however, ensuring exchangeability and consistency.

Strengths

- Theoretically ensures exchangeability and consistency. - Performs better than the main baseline in this category i.e., NPs.

Weaknesses

I discuss this in the Questions section.

Questions

1. I feel Eq. (11) is somehow incomplete considering that $y_i^{(t)}$ (for $t=0, \ldots, T-1$) are also random variables that do not appear here. Are they absorbed in the reconstruction term (i.e., the first term)? 2. Is there an empirical way to test how big of a problem marginal consistency poses, for instance, by measuring KL between LHS and RHS of Eq. (2). What I am getting at is: Is marginal consistency really a big issue empirically speaking, especially considering the flexibility of the state of the art models like TNPs or NDPs. 3. The paper does not cite or discuss Transformer Neural Processes (TNPs). Perhaps one may also compare with Neural Diffusion Processes (NDPs). It may help position the performance of MNPs even if it cannot outperform these considering that these models relax the consistency requirement. 4. One aspect I think is that MNPs do not require pairwise interaction between the query points required by the transformer-based NP architectures. I wonder if one might expect some computational benefits of MNPs with respect to this class of models. 5. It would be nice to provide a derivation of the second line of equation (10).

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

Yes, the paper discusses the limitations.

Summary

The paper presents Markov Neural Processes (MNPs), a novel class of Stochastic Processes (SPs) that leverage neural networks for data modeling. MNPs enhance the flexibility and expressivity of the Neural Processes (NPs) framework without sacrificing consistency or imposing restrictions. The proposed iterative construction is fully generative and maintains consistency under conditioning. MNPs demonstrate superior performance over baseline models across various tasks. The primary contribution is the MNP model, which employs a neural process as the transition operator, thereby adding flexibility to the stochastic process. This method increases expressivity while preserving the fully generative nature and consistency of NPs. The conditional consistency defined in this paper plays a crucial role in preventing uncertainty mismatches.

Strengths

he model is theoretically proven to satisfy the valid conditions. The experiments are well-designed to demonstrate the model's better expressivity, advantage over traditional models on non-Gaussian data, and better performance on scientific problems. The method has high theoretical novelty, especially in the use of Markov operators that lifts the Gaussian restrictions of the previous models.

Weaknesses

The experiments are run on a few small-scale synthetic datasets. The model's performance would be better evaluated if there are experiments on a larger variety of, especially large real-world, datasets. Besides, experiments that evaluates the time consumption among the models would be helpful.

Questions

How is the scalability of this method? Were any tricks used to make the method more scalable? Can you provide an analysis on the time/space complexity?

Rating

8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

Yes.

Summary

The paper introduces Markov Neural Processes (MNPs), a new variant of Stochastic Processes (SP). The paper points out that predictive SP models that construct context-to-target mappings no longer satisfy conditional consistency property and also are not fully generative. An expressive NP variant called MNP is proposed, the basis of which is stacking sequences of Markov transition operators in the function space. The paper adapts the definitions of consistency and exchangeability to MNPs, and show that they are consistent and exchangeable by construction. The paper shows the potential of MNPs in a variety of benchmarks from 1D function regression to geological inference.

Strengths

The paper introduces the framework from a very generic point of view, and also provides context in the specific choices made for the experiments. For instance, the usage of instance-wise conditional normalizing flows to parameterize the MNP is very interesting; as flows are universal approximators, this gives the MNPs high flexibility. Similarly, the usage of permutation-invariant neural networks is also interesting. The idea is simple but powerful, and the exposition of the idea is very neat.

Weaknesses

The empirical evaluation of the paper is somewhat limited. More empirical results may attract readers to this paper - for instance, few-shot image classification results would be interesting to see. Results in more large-scale datasets will also be useful.

Questions

Figure 2: I could not understand clearly why the auxiliary latent variables should be inferred in the reverse order. Can you expand further? The paper uses a specific variant of normalizing flows in its experiments called spline flows. It would be useful to justify the usage of the specific variant of flows. More flexible variants such as deep sigmoidal flows can also be tried out.

Rating

8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

4 excellent

Presentation

4 excellent

Contribution

4 excellent

Limitations

The training of MNPs is computationally intensive as the authors state; this limitation can be addressed in future works.

I've read the rebuttal

Thanks for the updated figure. It's much clearer than the previous version. > As the results show, in this setting the performance improves as the number of steps increases. We will run a more extensive ablation study and include the detailed results in the next version of the paper. I would strongly encourage the authors to complete a more comprehensive ablation study in the revision.

Thank You for the Response

I read the rebuttal and I thank the authors for their response. Thanks for the updated figure and clarification about the deterministic dependencies. It helps a lot. I appreciate the clarification that certain applications may require consistency as a guarantee rather than as a may-be. It makes sense. I have revised my rating.