Improved Mutual Information Estimation
We propose a new variational lower bound on the KL divergence and show that the Mutual Information (MI) can be estimated by maximizing this bound using a witness function on a hypothesis function class and an auxiliary scalar variable. If the function class is in a Reproducing Kernel Hilbert Space (RKHS), this leads to a jointly convex problem. We analyze the bound by deriving its dual formulation and show its connection to a likelihood ratio estimation problem. We show that the auxiliary variable introduced in our variational form plays the role of a Lagrange multiplier that enforces a normalization constraint on the likelihood ratio. By extending the function space to neural networks, we propose an efficient neural MI estimator, and validate its performance on synthetic examples, showing advantage over the existing baselines. We then demonstrate the strength of our estimator in large-scale self-supervised representation learning through MI maximization.
The paper contains one main contribution, the unbiased version of MINE obtained using the eta-trick, however a lot of theory is presented and not always very clearly. While some results are interesting (like the constrained ratio estimation interpretation of the dual) some parts are unclear and of lesser relevance. The section on “What Do Neural Estimators of LK or MI Learn” is a collection of different remarks without much coherence, some of which are imprecisely stated. The comparison with the estimator from Pool et al. (2019) could also be much simplified, in particular I would review the list of remarks below theorem 2. Another weakness of the paper is that two important aspects in assessing the quality of an estimator are overlooked: the variance of the estimator and the performance on finite data. Bias is not the only property which matters, both the variance and the dependence on the number of samples should be assessed experimentally, especially when no discussion or theoretical results are provided. I liked to see experiments performed on different tasks and datasets but overall that section could be significantly improved. The experiment comparing different MI estimators on synthetic Gaussian data is interesting. The plots are however difficult to read, it would be good to make them larger or split the results in different panels. For the 2D and the 20D case, the results reported in the MINE paper are much closer to the true MI than what is reported here, could the authors explain this difference? It would also be good to see some experiments done with a higher ground truth MI, 3.5 is not a lot for higher dimensional cases. Concerning the Deep InfoMax experiment, we see some improvements when using the proposed MI estimator with Deep InfoMax, however I doubt that this task is an ideal test case for an MI estimator since it has been shown (On Mutual Information Maximization for Representation Learning, Tschannen et al.) that the performance on downstream supervised tasks often does not clearly depend on the quality of MI estimation. Some additional points: Assuming a finite dimensional feature map in section 3 actually is a loss of generality. The proof of lemma 3 is hard to follow with some notations used without being properly introduced or changed in unexpected ways (eta switches places in f_b(w, eta), what does q do here?, what does PSD mean?). The proof of theorem 1 is again unclear. It starts with a typo in the second line (E_y~Q should be replaced by an integral). A couple of lines below a function alpha is introduced without further explanation. Typos: There is one x missing in the proof of Lemma 1 Appendix A.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Improved Mutual Information Estimation
We propose a new variational lower bound on the KL divergence and show that the Mutual Information (MI) can be estimated by maximizing this bound using a witness function on a hypothesis function class and an auxiliary scalar variable. If the function class is in a Reproducing Kernel Hilbert Space (RKHS), this leads to a jointly convex problem. We analyze the bound by deriving its dual formulation and show its connection to a likelihood ratio estimation problem. We show that the auxiliary variable introduced in our variational form plays the role of a Lagrange multiplier that enforces a normalization constraint on the likelihood ratio. By extending the function space to neural networks, we propose an efficient neural MI estimator, and validate its performance on synthetic examples, showing advantage over the existing baselines. We then demonstrate the strength of our estimator in large-scale self-supervised representation learning through MI maximization.
user
<think>Okay, to do a decent peer review, I should start with summarizing what the work is about. This includes novelty, contributions, clarity, and impact. I might include suggestions to issues that I find critical to the usefulness of the work. Starting with novelty, there are different types of novelty. The first aspect of novelty is scope and scale. Is the work covering a broader perspective than previous works? The second aspect is methodology. Is the proposed methodology novel compared to previous methods? Do they provide any significant advantage over previous methodologies? The third aspect is results. Are the results different compared previous work? Does this lead to any novel theories or an extension of current theories? Next is clarity, I will be reading the work in depth and analyze if any sentences can be potentially ambiguous or misleading. If any sentences are ambiguous, I will state them and give a suggestion on how to rephrase such sentence or how to make more clear to the audience. Lastly, impact is important. I will attempt to predict how such work will impact the scientific community. This might include the applications of such work and how the work might transform the way we study a certain topic. Finally, I will might only mention issues that are critical. This is to keep the review as concise and helpful as possible.</think>



The paper contains one main contribution, the unbiased version of MINE obtained using the eta-trick, however a lot of theory is presented and not always very clearly. While some results are interesting (like the constrained ratio estimation interpretation of the dual) some parts are unclear and of lesser relevance. The section on “What Do Neural Estimators of LK or MI Learn” is a collection of different remarks without much coherence, some of which are imprecisely stated. The comparison with the estimator from Pool et al. (2019) could also be much simplified, in particular I would review the list of remarks below theorem 2. Another weakness of the paper is that two important aspects in assessing the quality of an estimator are overlooked: the variance of the estimator and the performance on finite data. Bias is not the only property which matters, both the variance and the dependence on the number of samples should be assessed experimentally, especially when no discussion or theoretical results are provided. I liked to see experiments performed on different tasks and datasets but overall that section could be significantly improved. The experiment comparing different MI estimators on synthetic Gaussian data is interesting. The plots are however difficult to read, it would be good to make them larger or split the results in different panels. For the 2D and the 20D case, the results reported in the MINE paper are much closer to the true MI than what is reported here, could the authors explain this difference? It would also be good to see some experiments done with a higher ground truth MI, 3.5 is not a lot for higher dimensional cases. Concerning the Deep InfoMax experiment, we see some improvements when using the proposed MI estimator with Deep InfoMax, however I doubt that this task is an ideal test case for an MI estimator since it has been shown (On Mutual Information Maximization for Representation Learning, Tschannen et al.) that the performance on downstream supervised tasks often does not clearly depend on the quality of MI estimation. Some additional points: Assuming a finite dimensional feature map in section 3 actually is a loss of generality. The proof of lemma 3 is hard to follow with some notations used without being properly introduced or changed in unexpected ways (eta switches places in f_b(w, eta), what does q do here?, what does PSD mean?). The proof of theorem 1 is again unclear. It starts with a typo in the second line (E_y~Q should be replaced by an integral). A couple of lines below a function alpha is introduced without further explanation. Typos: There is one x missing in the proof of Lemma 1 Appendix A.
assistant
The paper contains one main contribution, the unbiased version of MINE obtained using the eta-trick, however a lot of theory is presented and not always very clearly. While some results are interesting (like the constrained ratio estimation interpretation of the dual) some parts are unclear and of lesser relevance. The section on “What Do Neural Estimators of LK or MI Learn” is a collection of different remarks without much coherence, some of which are imprecisely stated. The comparison with the estimator from Pool et al. (2019) could also be much simplified, in particular I would review the list of remarks below theorem 2. Another weakness of the paper is that two important aspects in assessing the quality of an estimator are overlooked: the variance of the estimator and the performance on finite data. Bias is not the only property which matters, both the variance and the dependence on the number of samples should be assessed experimentally, especially when no discussion or theoretical results are provided. I liked to see experiments performed on different tasks and datasets but overall that section could be significantly improved. The experiment comparing different MI estimators on synthetic Gaussian data is interesting. The plots are however difficult to read, it would be good to make them larger or split the results in different panels. For the 2D and the 20D case, the results reported in the MINE paper are much closer to the true MI than what is reported here, could the authors explain this difference? It would also be good to see some experiments done with a higher ground truth MI, 3.5 is not a lot for higher dimensional cases. Concerning the Deep InfoMax experiment, we see some improvements when using the proposed MI estimator with Deep InfoMax, however I doubt that this task is an ideal test case for an MI estimator since it has been shown (On Mutual Information Maximization for Representation Learning, Tschannen et al.) that the performance on downstream supervised tasks often does not clearly depend on the quality of MI estimation. Some additional points: Assuming a finite dimensional feature map in section 3 actually is a loss of generality. The proof of lemma 3 is hard to follow with some notations used without being properly introduced or changed in unexpected ways (eta switches places in f_b(w, eta), what does q do here?, what does PSD mean?). The proof of theorem 1 is again unclear. It starts with a typo in the second line (E_y~Q should be replaced by an integral). A couple of lines below a function alpha is introduced without further explanation. Typos: There is one x missing in the proof of Lemma 1 Appendix A.