Generalization Guarantees for Representation Learning via Data-Dependent Gaussian Mixture Priors

Authors: Milad Sefidgaran, Abdellatif Zaidi, Piotr Krasnowski
Primary area: learning theory
Keywords: Representation learning algorithm, Gaussian-Mixture, regularizer, rate-disotortion

Abstract:
We establish in-expectation and tail bounds on the generalization error of representation learning type algorithms. The bounds are in terms of the relative entropy between the distribution of the representations extracted from the training and "test'' datasets and a data-dependent symmetric prior, i.e., the Minimum Description Length (MDL) of the latent variables for the training and test datasets. Our bounds are shown to reflect the "structure" and "simplicity'' of the encoder and significantly improve upon the few existing ones for the studied model. We then use our in-expectation bound to devise a suitable data-dependent regularizer; and we investigate thoroughly the important question of the selection of the prior. We propose a systematic approach to simultaneously learning a data-dependent Gaussian mixture prior and using it as a regularizer. Interestingly, we show that a weighted attention mechanism emerges naturally in this procedure. Our experiments show that our approach outperforms the now popular Variational Information Bottleneck (VIB) method as well as the recent Category-Dependent VIB (CDVIB).

Decision: accept (spotlight)
Reviewer scores: 8, 6, 8, 6, 8

Reviews and discussion:
<thread>
creation date:
1729461174666

thread_type:
official_review

<thread_object>
creator: reviewer_9bkp

<content>
summary:
This paper provides a new generalization bound for representation learning in multi-class classification tasks. The error bound incorporates the recent notion of minimum description length (MDL), which empirically has proven to be more useful than mutual information-based bounds. In particular, the in-expectation bound and tail bound only depend on the encoder part. This has implications that in order to achieve better generalization bound, one can propose a regularizer based on the MDL. This is further studied by approximations using a Gaussian mixture whose mean and variance are data-dependent, and an update scheme for the prior is provided. Several numerical experiments verify the theory.

strengths:
1. The paper is clearly written and well-organized.

2. The derived bound is tighter than the best bound with MDL.

3. This work provides a quantitative explanation that the encoder plays the role of generalization, as reflected in the bounds in Theorem 1 and 2.

4. Based on the regularization using MDL, a practical and explicit optimization scheme for the prior is provided using Gaussian mixture models.

5. The theory is verified on a few datasets.

weaknesses:
See questions.

questions:
1. Typo in line 22 of the abstract, "date-dependent" should be "data-dependent".

2. In line 49, the author should make it clear what "MI" stands for before writing "MI-based".

3. In line 86, the author should cite earlier references about the universal approximation with Gaussian mixture, for example, some articles in JRSSB.

4. How do you determine the number of modes in the Gaussian mixture prior?
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</thread_object>

</thread>
<thread>
creation date:
1730461935451

thread_type:
official_review

<thread_object>
creator: reviewer_w8tc

<content>
summary:
In this paper, the generalization performance of representation learning algorithms (where the models consist of an encoder and a decoder) is considered. Specifically, bounds based on the compressibility of the latent variables of the models are derived. On the basis of these bounds, regularizers using priors with a Gaussian mixture model are devised. These regularizers are numerically shown to provide improved performance compared to prior work for some tasks.

strengths:
The general problem under consideration is interesting, and the twin goals of explaining generalization and finding ways to improve it are valuable. Interesting connections are drawn between, e.g., the presented bounds and classical compressibility, and between the form of the prior updates and attention. The way in which the regularizer is constructed is, as far as I am aware, new and yields promising results. The numerical experiments compare to several prior alternatives from the literature.

weaknesses:
The presentation is not always clear. For instance, one of the key concerns about the validity of the data-dependent prior, as far as I understood, is to guarantee that it is symmetric with respect to the training data and a ghost sample. However, the procedure seems to only depend on the training data, and it was unclear to me whether symmetry was preserved. Noise was added at some stages to “partially” preserve symmetry, and allusions were made to prior work which allows for conditions to be only partially fulfilled (e.g. differential privacy), but the results were discussed for exact symmetry.

The relation between prior literature and the presented generalization bounds and data-dependent priors could be discussed to a greater extent.

Minor:

— “date-dependent” in abstract

— Mix of numbering and words in some lists (“(1) First” in line 218, Lines 350-356)

questions:
1. Can you clarify the data-dependence of the prior, and how this relates to the assumptions under which Theorem 1 is derived?

2. For Theorem 1, the prior to be symmetric (equivalent to the exchangeable priors of Audibert (“A better variance control for PAC-Bayesian classification”, 2004) and Catoni (“PAC-Bayesian Supervised Classification: The Thermodynamics of Statistical Learning”, 2007). In their work, it is often sufficient to consider “almost exchangeable priors”, where permutations are restricted to only swap the $i$th element of the training sample with the $i$th sample in the ghost sample. Would a similar weaker requirement work for your results?

3. Line 266: “when the empirical risk is set to 0.05” What does this mean exactly? Does it mean that you train until the empirical risk reaches 0.05 or below and then stops?

4. In Theorem 1, the bound is provided in terms of a scaled Jensen-Shannon divergence between two Bernoullis. This is reminiscent of the Maurer-Langford-Seeger (MLS) bound in PAC-Bayes, where the corresponding KL divergence is used in the LHS. There are results (Foong et al, “How Tight Can PAC-Bayes be in the Small Data Regime?”, Thm. 4, 2021) indicating that the MLS bound is the tightest possible bound (in some sense) up to the log term. What is the relation between Thm. 1 in the present paper and such preceding bounds? Is it possible to instead use the binary KL in the LHS? (Such bounds also have a fast-rate behavior for zero training loss). I understand that the present paper considers a slightly different setup.
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</thread>
<thread>
creation date:
1730553240360

thread_type:
official_review

<thread_object>
creator: reviewer_ce5p

<content>
summary:
The paper studies the generalization error of representation learning, in which we have the encoder-decoder model. The authors establish a new generalization bound based on the minimum description length (MDL) of a symmetric prior and the induced representation distribution of the encoder. The paper explains how to choose the prior using a mixture of Gaussians, and connects the generalization bound to the regularizer of the optimization problem, in which the prior and the representation distribution are learned jointly. The authors examine their method in simple datasets and model architectures, and in both lossy and lossless scenarios.

strengths:
The paper is well-written and the motivation is clear. The authors improved the generalization bound of the encoder-decoder representation learning from $\sqrt{\text{MDL}(Q)/n}$ to $\text{MDL}(Q)/n$. The proposed symmetric prior using mixtures of Gaussians is relatively practical and easy to implement. The idea of adding the generalization bound as a regularizer and learning the prior and the induced distribution of the encoder together is interesting.

weaknesses:
While the authors have improved on the previous generalization bound, the technical work and the idea of regularization are heavily based on [1]. However, I think the mixture of Gussians is a nice addition.

The lossy generalization bound in Section 3.3 seems a bit incomplete to me. While the authors explain what lossy means, I could not follow how it results in Eq 14 and 15. I also checked the appendix, but could not find anything related to this section. I would appreciate it if the authors could elaborate on this section more, or add a theorem and the proof either in the main text or the appendix. 

I also think there are some parts that the author can improve the writing or the intuition of their work (see questions).

[1] Minimum Description Length and Generalization Guarantees for Representation Learning, Sefidgaran et al. 2023

questions:
1. In section 3.3, the bound in Eq 14 seems to be similar to [1] with $\sqrt{\text{MDL}(Q)/n}$ rather than $\text{MDL}(Q)/n$. Do the authors have any intuition/proof on why this change happens in the lossy setting?

2. Is there any intuition behind $h_C$ in Eq 6?

3. In section 4.1, M denotes the number of mixtures. Do the authors suggest any systematic way of choosing M? Or is it a hyperparameter that needs to be tuned?

Some typos also can be addressed:
- Line 342, an addition | in KL
- Eq. 17, the index of X should be batch size and not beta
</content>
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</thread>
<thread>
creation date:
1730679582455

thread_type:
official_review

<thread_object>
creator: reviewer_6gsr

<content>
summary:
This paper provides generalization guarantees for representation learning algorithms by deriving in-expectation and tail bounds on generalization error. The authors develop these bounds based on the MDL principle, using a data-dependent Gaussian mixture prior as a regularizer. By establishing bounds related to the relative entropy between representations from training and test datasets, the method aims to leverage the simplicity and structure of the encoder for improved generalization.

strengths:
1. The paper provides non-vacuous generalization bounds for representation learning.

2. The proposed regularizer has been validated in many image datasets.

3. Theoretical results have been rigorously presented and supported.

weaknesses:
1. The Gaussian mixture prior introduces additional computational overhead, especially during training, which may make the approach less practical for very large datasets.

2. Although the emergence of a weighted attention mechanism is highlighted, the paper could benefit from a more detailed analysis of this component and its role in generalization.

3. The empirical validation is limited to standard classification datasets. Testing on real-world tasks beyond classification (e.g., transfer learning, semi-supervised learning) would strengthen the claims of generalization benefits.

questions:
1. What might be the potential limitations of the current method? What could be the future work?

2. Are there any constraints on the applicability in real-world scenarios introduced by relying on a Gaussian mixture prior?
</content>
</thread_object>

</thread>
<thread>
creation date:
1730846777641

thread_type:
official_review

<thread_object>
creator: reviewer_c35e

<content>
summary:
This paper studies the setting of representation learning, where the task is to jointly learn a good representation and a good predictor. PAC-like generalization bounds are provided, which crucially depend on the complexity of the latent representation. Based on these theoretical insights, the authors then suggest a new regularizer term, which shows promising results in some applications.

strengths:
- Novel generalization bounds for representation learning: The paper derives generalization guarantees for representation learning. To my knowledge, this setting has not been studied in depth previously. I found the remarks about these bounds interesting, particularly the fact that they do not depend on the encoder component.
- Novel regularization term based on the theoretical analysis: As is often the case with generalization bounds, they can directly inspire regularization terms or other modifications to the learning pipeline. The authors propose a new regularization term for representation learning. In experiments with image classification datasets, the introduced method is shown to compete favorably with other recently introduced methods for this task.

weaknesses:
In general, I found the presentation of some results in the paper to be a bit lacking, while the discussion of related work seems somewhat insufficient. The authors discuss prior work in the introduction, but the discussion is high-level and mainly directed at 'experts' in the field. An additional (perhaps small) section covering related work would have served this paper well. Additionally, the main results of the paper (Theorem 1 and Theorem 2) are presented without much (if any) discussion on the techniques used in their proofs. Another example is in line 305, where the authors mention and emphasize the 'geometrical compression phenomenon' but do not describe it further. In various places, the manuscript would benefit from additional polishing.

questions:
I have some concrete questions:
- What is the "joint" (line 1188, Appendix C.3) learning procedure that you followed in the experiments? Can you provide more details on the loss used, etc?
- How do the proof techniques compare to the ones used by SZK23?
</content>
</thread_object>

</thread>
<thread>
creation date:
1732031024637

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Regarding the revised version

comment:
We thank all reviewers for their valuable comments and are pleased to see that overall the reviewers found our work and approach interesting and promising. We have addressed all of the reviewers' comments separately and kindly ask them to let us know if any further clarification is needed.

We have also uploaded a revised version of the paper based on the reviewers' comments. To summarize, the main changes, which are highlighted in blue in the paper, include

- As suggested by Reviewer c35E and Reviewer w8tc, we now elaborate more on the relation with some related prior-art works. 

- As suggested by Reviewer c35E, we have added, in the text, right after Theorem 1, some intuitions, insights, and elements of its proof, which we hope will help the reader better understand the result.

- We have enriched the numerical evaluation of our result of Theorem 1  reported in Figure 3 by providing figures for various values of the empirical risk (instead of one value, 0.05, in the original version) - This was also suggested by Reviewer w8tc. Interestingly, the new plot shows that smaller empirical risks result in a greater improvement of our results with respect to the previous bounds.

- We now present, in Appendix A as suggested by Reviewer CE5P, the lossy version of Theorem 1.

- In Appendix B, we have detailed the approach (previously briefly outlined in the footnote on page 7) to extend our results to the cases where the priors are "partially" symmetric, as requested by Reviewer w8tc.

- In Appendix D, we discuss the limitations of the current work and how the current work opens up several interesting future research directions, as suggested by Reviewer 6GSR. In this section, we have also discussed some potentially efficient practical approaches to choosing the number of components $M$, as suggested by Reviewer 9BKP and Reviewer CE5P, using dimensionality reduction and clustering approaches.
</content>
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</thread>


Metareview:
metareview:
a) Scientific Claims and Findings

This paper establishes novel generalization bounds for representation learning algorithms by leveraging the Minimum Description Length (MDL) principle and a data-dependent Gaussian mixture prior. Specifically, it provides in-expectation and tail bounds that depend on the relative entropy between the learned representations of training and test datasets and a data-dependent prior. Inspired by these bounds, the authors propose a regularizer that incorporates a learned Gaussian mixture prior, showing promising results in outperforming existing methods such as Variational Information Bottleneck (VIB) and Category-Dependent VIB (CDVIB). The proposed method also demonstrates connections to weighted attention mechanisms.

(b) Strengths

Rigorous derivation of generalization bounds that improve upon existing approaches in terms of encoder complexity.
Novel and practical use of data-dependent Gaussian mixture priors to devise a regularization strategy.
Comprehensive theoretical analysis coupled with numerical experiments showcasing superior performance over competing methods.
The paper addresses critical concerns raised by reviewers, enhancing its clarity and robustness.

(c) Weaknesses

The computational cost introduced by the Gaussian mixture prior during training may hinder scalability to large datasets.
Empirical validation is limited to standard classification tasks, leaving the potential for broader applications unexplored.
Some results, such as the geometrical compression phenomenon and the role of attention mechanisms, require further elaboration.

(d) Decision Rationale

I recommend acceptance of this paper. It makes a significant theoretical contribution by tightening generalization bounds and providing practical insights for representation learning. The experimental results substantiate the theoretical claims, demonstrating utility and innovation. While some aspects could benefit from additional experiments or clarity, the paper’s contributions are sufficiently impactful to merit inclusion in the conference.