NeurIPS 2020

Conditioning and Processing: Techniques to Improve Information-Theoretic Generalization Bounds

Review 1

Summary and Contributions: This paper presents a probabilistic graphical representation approach to obtaining information-theoretic generalization bounds and proposes two general techniques of "conditioning" and "processing." Some recent results in the literature are explained with this proposed technique. The paper also proposes and studies the "chaining conditional mutual information" bound and shows an advantage of that bound over the recent conditional mutual information bound.

Strengths: - The paper studies information-theoretic generalization bounds and presents new techniques. The topic of generalization bounds in statistical learning is of major interest to the NeurIPS community. - Using probabilistic graphical representations for interpreting generalization bounds and as a potential framework for obtaining new bounds is new and interesting and the techniques of processing and conditioning build upon recent results in the literature. These types of visualizations may lead to new and useful generalization bounds in follow-up works. Also, combining "chaining" with "conditional mutual information" bound is new and expands the venue of combining information-theoretic techniques with probabilistic techniques for obtaining new tools in statistical learning. - The claims in the paper seem to be sound, except for one result (Theorem 6) that I did not completely understand and I mention below.

Weaknesses: 1. In Section 4, conditioning and processing are proposed as a general framework for giving a "unified view" on information-theoretic generalization bounds, however, in Section 5 to illustrate an application, suddenly the method of chaining is brought into picture which is outside that framework. In other words, it seems that the two parts of the paper are fairly disconnected. The only examples given from purely using the processing and conditioning tools correspond to known results in the literature such as [5, 15] as mentioned in the paper. It can make the paper stronger if further examples of new generalization bounds can be provided purely based on the proposed techniques of conditioning and processing. 2. What is stated as Theorem 4 and proved in the appendix - without citing a reference - is actually an old result in information theory (which should not be surprising given its simple form). For example, see Corollary of Theorem 2.8.1 in the book by Cover-Thomas (p. 35). A reference should be cited for this result. 3. The chaining conditinal mutual information bound seems to be interesting, though the only benefit of this new bound that the paper demonstrates, if I understood correctly, is that it can deduce the uniform bound of VC-dimension, while conditional mutual information will be "log n" factor bigger. This is then seems similar to the discussion in Section 28.1 of Shalev-Schwartz & Ben-David book about dropping the log factor by using the method of chaining. However, this does not look like a satisfying application since if we wanted to obtain uniform bounds such as VC-dimension then why should we use mutual information to begin with? It would make the paper stronger if any advantage of that bound can be shown over conditional mutual information in some examples, apart from uniform bounding.

Correctness: The results appear correct to me, except for something in the proof of Theorem 6 that I did not completely understand and would like the author(s) to explain in their response: How is inequality (F.3) deduced from Theorem 4? The inequality involves X_{W} while the Markov chain includes W. ----- Post Author Response ----- I decided to keep my score at 6 since I did not find the author response to the mistake in the proof of Theorem 6 (inequality F.3) adequate. The statement of Theorem 6 is likely false, as one should take into account the interaction between the scales of the chaining bound either by assuming hierarchical coverings such as Theorem 3 of or showing such interaction explicitly in the bound as in Theorem 11 of the aforementioned paper, a result which this paper can cite and use instead of Theorem 6. Assuming a deterministic algorithm makes the result too weak and restricted; in fact there is no need to use mutual information I(W_k;S) in this case as they will all equal the entropy of the marginal of the algorithm output H(W_k).

Clarity: The paper is well written except for some typos and minor comments (see below).

Relation to Prior Work: The paper has mentioned previous contributions, though it needs to be improved by elaborating more on the relation between its contributions and prior work, specifically the discussion in the paragraph 159-169. For example, in line 163 it is written that "it is not exactly what we looked for" which needs further explanation. Similarly in line 168 about the difference in constant factors (which I did not find in the appendix) and in line 179 about the difference on the learning algorithm assumptions. Also, in Section 5, the author(s) should compare the chaining conditional mutual information bound with the work by Audibert-Bousquet '07 ("Combining PAC-Bayesian and Generic Chaining Bounds") which also uses double sample sets and chaining.

Reproducibility: Yes

Additional Feedback: Minor comments: - Notations --> Notation - Let's --> Let us (please avoid contractions) - 68: Is there a typo in the equation having P^{XY}_G? - Is the Slepian-Wolf work [14] used anywhere in the paper? - 331: chain rule of mutual information.

Review 2

Summary and Contributions: 1. This paper provides a unifying framework to describe recent work on information-theoretic generalization bounds. In particular, it unifies conditional/disintegrated mutual information bounds, chaining mutual information and leave-one-out style bounds using two key steps of conditioning and preprocessing, and repeated nested applications of these procedures. 2. The paper (partially) resolves (a variant of) the conjecture of Steinke and Zakynthinou of whether information theoretic generalization bounds based on their CMI can recover tight generalization bounds based on VC-theory in the non-realizable case.

Strengths: 1. The results appear to be sound. 2. The graphical model representation is a convenient, intuitive and useful representation of the various types of techniques for information theoretic generalization bounds which have emerged in recent years. This unification will have a significant impact in the communities ability to deploy modular combinations of those results to derive tight bounds for various learning problems and algorithms. 3. The positive (partial) resolution of (a version of) the conjecture of Steinke and Zakynthinou is an important contribution which essentially validates that an information theoretic approach to generalization is at least as expressive as an approach based on uniform uniform convergence (a.k.a agnostic PAC/ Uniform Gliovenko-Cantelli). This will likely have a long term impact on the prevailing methods researchers will use for bounding generalization error in future work.

Weaknesses: 1. I did not find any particular weaknesses of the work that need to be addressed.

Correctness: The claims appear to be correct. POST REBUTTAL COMMENTS: Regarding R1's review correctly pointing out the error in (F3), I feel the author response fell a bit short. Rather than assuming that the learning algorithm is deterministic, which would be a problem for the main algorithms of interest (variants of sgd). The authors could have instead used nested coverings, as in Theorem~3 of, or used something like Theorem~11 of the same work. The result could have just been cited instead of being reproved. The determinism assumption they mentioned in rebuttal is much stronger than necessary and would adversely impact the relevance of the work. Either way, due to the error, and the weak author response, it leads me to lower my score to an 8 instead of a 9. The error must be corrected in the final version.

Clarity: 1. The paper is very easy to read and does not require specialist knowledge to understand.

Relation to Prior Work: 1.The relation to the most related prior work is thorough enough, 2. though the article would benefit from a more broad literature review if space permits in the final version. 3. It would be helpful if the exact relationship between Thm.~6,7 and the conjecture of Steinke and Zakynthinou (both from their original paper, and from the more recent preprint about various forms of the conjecture) was more clearly and explicitly expressed. This work resolves their conjecture in spirit, but not according to its precise statement, as far as I can tell, because it does not bound the CMI of their learning algorithm directly, but instead bound other information theoretic quantities that are shown to be more immediately relevant.

Reproducibility: Yes

Additional Feedback: 1. It would be helpful Table 1 included the relevant papers as a fourth column. Can Negrea et al. [10] be included in the table as well? 2. Typo "=\le" in ldisplay equation above line 171 3. In remark 1, it says that (b) is stronger than (a). But (b) does not necessarily imply (a) since if $X$ is cauchy and $P^x_{\bar G}$ is $N(x,1)$ then $P_{\bar G}$ is not subG. 4. The impact statement is underwhelming. This can easily be improved. 5. My review score (9) is based on the current state of the manuscript and my interpretation of it. Since my only feedback/suggestions are very minor and my current impression of the work is already very high, I don't foresee myself increasing my score unless the authors demonstrate in the rebuttal that I've missed or underestimated an important contribution of the work. POST REBUTTAL COMMENTS: I have reduced my score to an 8 instead of a 9, based on the error pointed out by R1. See my comments in the "Correctness" field for further details.

Review 3

Summary and Contributions: This paper introduces two techniques that can improve existing information-theoretical generalization bound. The probabilistic graphs shown in this work illustrate how the techniques can be applied. Other results include an extension to chaining mutual information and recovering the VC bound via information-theoretical tools.

Strengths: This paper deals with the high-level ideas of how to improve existing information-theoretical generalization bound. The conditioning and processing techniques could be novel to some in the machine learning community and potentially lead to concrete improvement on existing bounds.

Weaknesses: Most of the contents in this paper could be too high level to claim any concrete contributions. While it is claimed the techniques can help improving bounds, it is not clear how exactly it would improve existing bounds. The conditioning technique can provide an unconditioned bound, given a conditioned variant. It is not very clear how much the mutual information will decrease. In other words, this paper could be very inspiring, but not quite ready to appear as a concrete conference paper. More and stronger theoretical results that show improvement are needed.

Correctness: The derivations are correct.

Clarity: While the presentation is easy to follow, it is a bit unclear to identify the important results and its contribution.

Relation to Prior Work: Yes.

Reproducibility: Yes

Additional Feedback: Post-rebuttal: I have read the response. Given the authors' response I will raise my score to 6. ================================== I find it hard to imagine how the bound is improved via conditioning and processing. Is there any concrete examples that shows the bound is indeed improved?

Review 4

Summary and Contributions: This paper studied the improvement of information-theoretic generalization bounds. Additional on traditional information-theoretic generalization bounds which only considered a pair of random variables X and Y, this work aims to improve the bound to more random variables. Two techniques --- conditioning and processing --- are introduced. Conditioning upper bound is given by conditional MI I(X:Y|Z) for conditioning random variable Z. Processing upper bound is given by I(T;V) where T and V are processed random variables of X and Y respectively. These techniques can be used to provide a general framework for understanding information-theoretic generalization bounds.

Strengths: Information-theoretic generalization bounds have been actively studied recently. The idea of improving these bounds to graphical models are novel and important. This paper is the first step toward a general improvement --- information-theoretical generalization bound for complex graphical model or complex neural network. This is relevant to NeurIPS community.

Weaknesses: The mathematical content of this paper is not very novel. It is highly dependent on traditional information-theoretic generalization bound by Xu and Raginsky, plus information-processing inequality and conditioning decreases mutual information, which are standard results in information theory. It will require more work if we want to extend the information-theoretical generalization bounds to more complex graphical models

Correctness: I don't find any error in the proofs in this paper.

Clarity: Yes, this paper is written clearly.

Relation to Prior Work: This paper have cite most of the recent works about information-theoretic generalization bounds. It is better if the authors could introduce some of the recent application of information-theoretic generalization bounds to emphasize the practical usage of these bounds and the importance of the improvement in this paper. Recommended reference: Information-Theoretic Understanding of Population Risk Improvement with Model Compression. Y Bu, W Gao, S Zou, VV Veeravalli Generalization bounds of sgld for non-convex learning: Two theoretical viewpoints W Mou, L Wang, X Zhai Understanding autoencoders with information theoretic concepts S Yu, JC Principe

Reproducibility: Yes

Additional Feedback: