Entropy-SGD optimizes the prior of a PAC-Bayes bound: Data-dependent PAC-Bayes priors via differential privacy
We show that Entropy-SGD (Chaudhari et al., 2017), when viewed as a learning algorithm, optimizes a PAC-Bayes bound on the risk of a Gibbs (posterior) classifier, i.e., a randomized classifier obtained by a risk-sensitive perturbation of the weights of a learned classifier. Entropy-SGD works by optimizing the bound’s prior, violating the hypothesis of the PAC-Bayes theorem that the prior is chosen independently of the data. Indeed, available implementations of Entropy-SGD rapidly obtain zero training error on random labels and the same holds of the Gibbs posterior. In order to obtain a valid generalization bound, we show that an ?-differentially private prior yields a valid PAC-Bayes bound, a straightforward consequence of results connecting generalization with differential privacy. Using stochastic gradient Langevin dynamics (SGLD) to approximate the well-known exponential release mechanism, we observe that generalization error on MNIST (measured on held out data) falls within the (empirically nonvacuous) bounds computed under the assumption that SGLD produces perfect samples. In particular, Entropy-SGLD can be configured to yield relatively tight generalization bounds and still fit real labels, although these same settings do not obtain state-of-the-art performance.
 1) I would like to ask for the clarification regarding the generalization guarantees. The original Entropy-SGD paper shows improved generalization over SGD using uniform stability, however the analysis of the authors rely on an unrealistic assumption regarding the eigenvalues of the Hessian (they are assumed to be away from zero, which is not true at least at local minima of interest). What is the enabling technique in this submission that avoids taking this assumption? (to clarify: the analysis is all-together different in both papers, however this aspect of the analysis is not fully clear to me). 2) It is unclear to me what are the unrealistic assumptions made in the paper. Please, list them all in one place in the paper and discuss in details.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Entropy-SGD optimizes the prior of a PAC-Bayes bound: Data-dependent PAC-Bayes priors via differential privacy
We show that Entropy-SGD (Chaudhari et al., 2017), when viewed as a learning algorithm, optimizes a PAC-Bayes bound on the risk of a Gibbs (posterior) classifier, i.e., a randomized classifier obtained by a risk-sensitive perturbation of the weights of a learned classifier. Entropy-SGD works by optimizing the bound’s prior, violating the hypothesis of the PAC-Bayes theorem that the prior is chosen independently of the data. Indeed, available implementations of Entropy-SGD rapidly obtain zero training error on random labels and the same holds of the Gibbs posterior. In order to obtain a valid generalization bound, we show that an ?-differentially private prior yields a valid PAC-Bayes bound, a straightforward consequence of results connecting generalization with differential privacy. Using stochastic gradient Langevin dynamics (SGLD) to approximate the well-known exponential release mechanism, we observe that generalization error on MNIST (measured on held out data) falls within the (empirically nonvacuous) bounds computed under the assumption that SGLD produces perfect samples. In particular, Entropy-SGLD can be configured to yield relatively tight generalization bounds and still fit real labels, although these same settings do not obtain state-of-the-art performance.
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>



 1) I would like to ask for the clarification regarding the generalization guarantees. The original Entropy-SGD paper shows improved generalization over SGD using uniform stability, however the analysis of the authors rely on an unrealistic assumption regarding the eigenvalues of the Hessian (they are assumed to be away from zero, which is not true at least at local minima of interest). What is the enabling technique in this submission that avoids taking this assumption? (to clarify: the analysis is all-together different in both papers, however this aspect of the analysis is not fully clear to me). 2) It is unclear to me what are the unrealistic assumptions made in the paper. Please, list them all in one place in the paper and discuss in details.
assistant
 1) I would like to ask for the clarification regarding the generalization guarantees. The original Entropy-SGD paper shows improved generalization over SGD using uniform stability, however the analysis of the authors rely on an unrealistic assumption regarding the eigenvalues of the Hessian (they are assumed to be away from zero, which is not true at least at local minima of interest). What is the enabling technique in this submission that avoids taking this assumption? (to clarify: the analysis is all-together different in both papers, however this aspect of the analysis is not fully clear to me). 2) It is unclear to me what are the unrealistic assumptions made in the paper. Please, list them all in one place in the paper and discuss in details.