Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization

Authors: Mikhail Persiianov, Arip Asadulaev, Nikita Andreev, Nikita Starodubcev, Dmitry Baranchuk, Anastasis Kratsios, Evgeny Burnaev, Alexander Korotin
Primary area: generative models
Keywords: semi-supervised domain translation, likelihood maximization, inverse entropic optimal transport

Abstract:
Learning conditional distributions $\pi^*(\cdot|x)$ is a central problem in machine learning, which is typically approached via supervised methods with paired data $(x,y) \sim \pi^*$. However, acquiring paired data samples is often challenging, especially in problems such as domain translation. This necessitates the development of *semi-supervised* models that utilize both limited paired data and additional unpaired i.i.d. samples $x \sim \pi^*_x$ and $y \sim \pi^*_y$ from the marginal distributions. The usage of such combined data is complex and often relies on heuristic approaches. To tackle this issue, we propose a new learning paradigm that integrates both paired and unpaired data **seamlessly** through the data likelihood maximization techniques. We demonstrate that our approach also connects intriguingly with inverse entropic optimal transport (OT). This finding allows us to apply recent advances in computational OT to establish a **light** learning algorithm to get $\pi^*(\cdot|x)$. In addition, we derive the universal approximation property demonstrating that our approach can theoretically recover true conditional distributions with arbitrarily small error. Furthermore, we demonstrate through empirical tests that our method effectively learns conditional distributions using paired and unpaired data simultaneously.

Decision: reject
Reviewer scores: 3, 5, 6

Reviews and discussion:
<thread>
creation date:
1730490911894

thread_type:
official_review

<thread_object>
creator: reviewer_nafh

<content>
summary:
The paper proposes a semi-supervised objective for the translation task that can leverage both paired and unpaired samples from the source and target domains.
The paper decomposes the standard MLE objective into three parts corresponding to a joint expectation and two marginal expectations over the unpaired source and target samples.
The key challenge is that parts of the objective require integration.
To avoid this computational challenge, the paper makes several critical assumptions about the joint cost function and the target-specific function that makes the necessary functions known in closed-form.
In particular, a Gaussian mixture model assumption and a simple log-sum-exp form of the cost function.
With these closed-form solutions, the paper then demonstrates the method on a 2D synthetic dataset and a weather prediction task.

strengths:
- Proposes a promising MLE approach to incorporating both paired and unpaired data for the translation task.

- The proposed algorithm does not seem to have any complex hyperparameters.

- Shows a nice connection between using an energy-based conditional generative model and inverse entropic optimal transport. This opens up using insights and methods from computational OT.

- The paper was easy to read.

weaknesses:
- Intuition and theoretic understanding of objective function is not clear. This makes it difficult to understand and extend. What is the purpose of the objectives corresponding to the Y and X marginals? Why does this make sense? What if these are misestimated?

- The assumptions are not clearly discussed and their corresponding limitations. What is the limitations of assuming the particular form of the needed parametric functions? It seems restrictive. Also, why is it required to have an energy-based model? For better or worse, score-based/diffusion-based generative models are far more common now. Is using an energy-based model significantly more restrictive?

- The practical motivation of the scenario is lacking. When would you expect this situation to hold in the real-world? Which real-world scenarios naturally have both paired and unpaired data?

- The experiments are relatively weak with only one 2D synthetic experiment and 1 weather experiment, but the weather experiment only includes a simple MLP regression baseline (i.e., none of the other baseline methods).

questions:
- What is the intuitive interpretation of the new objective? This seems a bit ad-hoc/heuristic to introduce the $f(y)$ term for the sole purpose of using marginal samples from $p(y)$.

- Why use the parametrization in 16 and 17? What is the intuition of this parametrization? Is it a universal type of parametrization (i.e., can it represent all interesting cost functions)? Can you give examples of cost functions that do not have this form?

- How does this compare to the most natural likelihood objective that is $L(\theta) = E_{x,y}[-\log \pi^\theta(y|x)] + E_y[-\log E_x[\pi^\theta(y|x)]]$, which is basically the expected conditional KL plus a regularization term on the conditional that ensures it matches the marginal of p(y), i.e., integrating the conditional over the marginal of x should yield the marginal of y.  This seems to be a more straightforward (albeit hard-to-compute) semi-supervised objective.

- What does it mean that the x and y terms are asymmetric? It would seem that the marginal optimization terms should be symmetric as the unpaired data has no directionality per se.

- What if the paired data is not iid from the joint distribution? For example, if there are certain regions of the space that are more likely to have paired samples? This seems quite natural and reasonable in real-world scenarios. How would your method fair in these scenarios?

- How do you ensure $c^\theta(x,y)$ is a valid cost function?

*Summary of Review*
Overall, I found the paper to be intriguing, particularly in the connection between inverse optimal transport and the proposed objective. However, the derivations and development lacked strong intuition and explanation---rather seeming like mostly algebraic manipulation without a clear reason. The paper could be improved significantly with a much deeper understanding and interpretation of the objective. Furthermore, the assumptions lacked significant discussion while being relatively restrictive. Finally, the empirical results were relatively weak only having one 2D synthetic experiment and 1 weather experiment on a very small dataset with less than 1000 samples. Thus, the scalability and applicability of the method are questionable.
</content>
</thread_object>

</thread>
<thread>
creation date:
1730705919745

thread_type:
official_review

<thread_object>
creator: reviewer_vgg8

<content>
summary:
This paper presents a novel approach to semi-supervised domain translation by introducing a likelihood-based loss function that naturally incorporates both paired and unpaired samples. The key theoretical contribution is establishing a connection between their loss and inverse entropic optimal transport - a connection that hasn't been previously explored in the literature. The authors propose using a Gaussian mixture parameterization that allows closed-form expressions for their loss terms. They empirically validate their method against standard supervised/unsupervised approaches on synthetic 2D data and a weather prediction task.

While the paper presents an interesting theoretical contribution connecting inverse entropic optimal transport with likelihood estimation, the lack of comparisons against competing semi-supervised methods and limited experimental validation makes it difficult to assess the practical value of the proposed approach. Thus, currently the paper does not reach the bar for acceptance.

strengths:
- Novel likelihood maximization loss for semi-supervised domain translation that elegantly handles both paired and unpaired data within a single objective.
- Novel theoretical connection established between inverse entropic optimal transport and likelihood estimation.
- Empirical validation demonstrating how performance scales with both paired and unpaired sample sizes.

weaknesses:
- Experimental validation:
	- While the related work section discusses multiple approaches to semi-supervised domain translation, the paper lacks empirical comparisons against any of these existing methods. This makes it difficult to assess the relative advantages of the proposed approach compared to established methods that also leverage both paired and unpaired data.
	- The paper would be substantially strengthened by: 1) clearly articulating the advantages of the proposed method over existing approaches, and 2) providing direct empirical comparisons against state-of-the-art semi-supervised methods. Currently, comparisons are limited to baseline methods that were not specifically designed for semi-supervised domain translation.
	- The experimental validation would benefit from including tests on higher-dimensional problems, such as the image translation tasks explored in [1], to demonstrate the method's scalability beyond simple domains.
- Reliance on Gaussian mixture parameterization:
	- The authors acknowledge that their choice of Gaussian mixture parameterization inherently limits the method's scalability to more complex domains.
	- The discussion of alternative parameterizations seems incomplete, leaving questions about potential extensions to more expressive models.
	- While the authors claim their parameterization is "light-speed," this assertion lacks proper justification or comparative analysis.

[1] Xiaole Tang, Xin Hu, Xiang Gu, and Jian Sun. "Residual-conditioned optimal transport: Towards structure-preserving unpaired and paired image restoration". ICML 2024.

questions:
- The authors describe their parameterization as "light-speed" - what are the specific computational advantages that justify this claim?
- What makes Gaussian mixture parameterization particularly suitable for the proposed loss compared to alternative parameterizations?
- How does the proposed method compare to existing semi-supervised domain translation techniques in terms of both theoretical properties and empirical performance?
</content>
</thread_object>

</thread>
<thread>
creation date:
1731200940754

thread_type:
official_review

<thread_object>
creator: reviewer_obqy

<content>
summary:
This paper introduces a novel semi-supervised learning approach that leverages both paired and unpaired data to learn conditional distributions through a data likelihood maximization framework. The proposed method constructs a new loss function that integrates paired and unpaired data directly into the objective, with a connection to inverse entropic optimal transport (OT). By framing the problem as an inverse entropic OT, the authors develop an algorithm that employs Gaussian mixture parameterization to approximate the conditional distribution in a computationally efficient manner. The empirical evaluation demonstrates that this method can achieve effective learning outcomes even with limited paired data, as long as sufficient unpaired data is available, showing promising results on synthetic and real-world datasets.

strengths:
Clarity: The paper is reasonably structured and provides some intuition on connecting OT with likelihood maximization.

Empirical Results: The empirical tests show potential for the method to handle limited paired data, a typical challenge in semi-supervised setups.

weaknesses:
Justification of Loss Function: The paper lacks sufficient theoretical or empirical justification for the specific loss function design. The decomposition into paired and unpaired data components seems plausible, but more discussion on its necessity or optimality would strengthen the work. Additionally, while the paper claims the loss solves inverse entropic OT, the practical implications of this for general semi-supervised tasks are underexplored.

Limited Comparisons: The experimental section lacks comparisons with several prominent semi-supervised learning methods that do not rely on OT or Gaussian mixtures. Without comparisons to these methods, it is challenging to gauge whether the proposed approach offers an advantage over non-OT-based techniques.

questions:
See weakness
</content>
</thread_object>

</thread>
<thread>
creation date:
1732361979606

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
General response and revision

comment:
Dear Reviewers,

We sincerely appreciate your thoughtful comments and constructive feedback on our work. We are grateful for the time and effort each of you (oBqy, vGg8, Nafh) dedicated to reviewing our manuscript. We are particularly pleased that reviewers (vGg8, Nafh) acknowledged the significance of our novel maximum-likelihood framework, which combines paired and unpaired data within a theoretically grounded approach that requires minimal hyperparameters (as specifically noted by reviewer vGg8). Additionally, we are encouraged by the positive feedback from all reviewers (oBqy, vGg8, Nafh) regarding the connection between our proposed loss function and inverse entropic optimal transport, as well as the clarity of our paper’s exposition. For ease of review, all changes made in the revised manuscript are highlighted in **blue**.

We have thoroughly considered your suggestions and implemented substantial revisions to the manuscript, which address the following key points:

1. **Justification of Parametrization (oBqy, vGg8, Nafh):**

   To address concerns regarding our choice of Gaussian mixture parametrization, we have added **Theorem 3.1**, which demonstrates that this parametrization, under mild assumptions, can approximate a broad class of conditional plans with arbitrary accuracy. Detailed proofs supporting this result are now provided in **Appendix C.2**.

2. **Exploration of Advanced Parametrizations (oBqy, vGg8, Nafh):**

   To further emphasize the flexibility of our framework, we have included a new section in **Appendix A** that explores an implementation using an energy-based model. This model leverages a neural network to parametrize both the potential and cost functions, illustrating the applicability of our framework beyond Gaussian mixtures. We provided a proof-of-concept experiment in the Appendix A.

3. **Comparison with Alternative Loss Functions (Reviewer Nafh) and Baseline Models (oBqy, vGg8, Nafh):**

   To thoroughly evaluate our proposed loss function, we conducted additional experiments comparing it with an alternative loss suggested by Reviewer Nafh. These new results, presented in the revised manuscript, underscore the strengths of our approach. We have also expanded baseline comparisons from Section 5.1 (2D illustrative example) to include results in Section 5.2 (Weather experiments), providing a more comprehensive empirical evaluation.

We believe these revisions have substantially strengthened the manuscript and have addressed your concerns. **We have also updated our codebase to include all relevant baselines, ensuring reproducibility.**

Thank you once again for your valuable feedback and guidance. Detailed responses to your individual comments will be provided below in responses to each of your.
</content>
</thread_object>

</thread>


Metareview:
metareview:
This paper proposes a semi-supervised approach for domain translation that combines paired and unpaired data within a likelihood maximization framework. The authors establish a connection between their proposed loss function and inverse entropic optimal transport, introducing a Gaussian mixture parameterization for computation. The method is evaluated on synthetic 2D data and a weather prediction task.

The paper's main strength lies in its theoretical approach to semi-supervised domain translation, integrating paired and unpaired data in a single objective. The connection to inverse entropic optimal transport is noteworthy.

However, the paper suffers from several significant weaknesses. The experimental validation is limited, especially for higher-dimensional problems, and lacks comparisons with state-of-the-art semi-supervised domain translation methods. The justification for the Gaussian mixture parameterization is insufficient, raising concerns about its scalability. The intuition and explanation for some aspects of the objective function are incomplete, and the practical motivation for scenarios with both paired and unpaired data is unclear.

The primary reasons for rejecting this paper are the limited experimental validation and the lack of comparisons with existing state-of-the-art methods. The practical impact and scalability of the proposed approach remain unclear, and the authors' responses did not sufficiently address these concerns. The reliance on Gaussian mixture parameterization raises questions about the method's applicability to more complex, high-dimensional problems. Furthermore, the insufficient explanation of the objective function's intuition makes it challenging to assess the method's full potential and limitations.