Universality in halting time
The authors present empirical distributions for the halting time (measured by the number of iterations to reach a given accuracy) of optimization algorithms applied to two random systems: spin glasses and deep learning. Given an algorithm, which we take to be both the optimization routine and the form of the random landscape, the fluctuations of the halting time follow a distribution that remains unchanged even when the input is changed drastically. We observe two main classes, a Gumbel-like distribution that appears in Google searches, human decision times, QR factorization and spin glasses, and a Gaussian-like distribution that appears in conjugate gradient method, deep network with MNIST input data and deep network with random input data. This empirical evidence suggests presence of a class of distributions for which the halting time is independent of the underlying distribution under some conditions.
The authors explore whether the halting time distributions for various algorithms in various settings exhibit *universality*, i.e. after rescaling to zero mean and unit variance, the distribution does not depend on stopping parameter, dimensionality and ensemble. The idea of the described universality is very interesting. However I see several shortcomings in the paper: In order to be of practical relevance, the actual stopping time might be more relevant than the scaled one. The discussion of exponential tailed halting time distributions is a good start, but I am not sure how often this might be actually helpful. Still, the findings in the paper might be interesting from a theoretical point of view. Especially for ICLR, I think it would have been more interesting to look into comparisons between stochastic gradient descent, momentum, ADAM etc on different deep learning architectures. Over which of those parameters does universality hold?. How can different initializations influence the halting time distribution? I would expect a sensible initialization to cut of part of the right tail of the distribution. Additionally, I found the paper quite hard to read. Here are some clarity issues: - abstract: *even when the input is changed drastically*: From the abstract I*m not sure what *input* refers to, here - I. Introduction: *where the stopping condition is, essentially, the time to find the minimum*: this doesn*t seem to make sense, a condition is not a time. I guess the authors wanted to say that the stopping condition is that the minimum has been reached? - I.1 the notions of dimension N, epsilon and ensemble E are introduced without any clarification what they are. From the later parts of the paper I got some ideas and examples, but here it is very hard to understand what these parameters should be (just some examples would be already helpful) - I.3 *We use x^ell for ell in Z={1, dots, S} where Z is a random sample from of training samples* This formulation doesn*t make sense. Either Z is a random sample, or Z={1, ..., S}. - II.1 it took me a long time to find the meaning of M. As this parameter seems to be crucial for universality in this case, it would be very helpful to point out more explicitly what it refers to.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Universality in halting time
The authors present empirical distributions for the halting time (measured by the number of iterations to reach a given accuracy) of optimization algorithms applied to two random systems: spin glasses and deep learning. Given an algorithm, which we take to be both the optimization routine and the form of the random landscape, the fluctuations of the halting time follow a distribution that remains unchanged even when the input is changed drastically. We observe two main classes, a Gumbel-like distribution that appears in Google searches, human decision times, QR factorization and spin glasses, and a Gaussian-like distribution that appears in conjugate gradient method, deep network with MNIST input data and deep network with random input data. This empirical evidence suggests presence of a class of distributions for which the halting time is independent of the underlying distribution under some conditions.
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 authors explore whether the halting time distributions for various algorithms in various settings exhibit *universality*, i.e. after rescaling to zero mean and unit variance, the distribution does not depend on stopping parameter, dimensionality and ensemble. The idea of the described universality is very interesting. However I see several shortcomings in the paper: In order to be of practical relevance, the actual stopping time might be more relevant than the scaled one. The discussion of exponential tailed halting time distributions is a good start, but I am not sure how often this might be actually helpful. Still, the findings in the paper might be interesting from a theoretical point of view. Especially for ICLR, I think it would have been more interesting to look into comparisons between stochastic gradient descent, momentum, ADAM etc on different deep learning architectures. Over which of those parameters does universality hold?. How can different initializations influence the halting time distribution? I would expect a sensible initialization to cut of part of the right tail of the distribution. Additionally, I found the paper quite hard to read. Here are some clarity issues: - abstract: *even when the input is changed drastically*: From the abstract I*m not sure what *input* refers to, here - I. Introduction: *where the stopping condition is, essentially, the time to find the minimum*: this doesn*t seem to make sense, a condition is not a time. I guess the authors wanted to say that the stopping condition is that the minimum has been reached? - I.1 the notions of dimension N, epsilon and ensemble E are introduced without any clarification what they are. From the later parts of the paper I got some ideas and examples, but here it is very hard to understand what these parameters should be (just some examples would be already helpful) - I.3 *We use x^ell for ell in Z={1, dots, S} where Z is a random sample from of training samples* This formulation doesn*t make sense. Either Z is a random sample, or Z={1, ..., S}. - II.1 it took me a long time to find the meaning of M. As this parameter seems to be crucial for universality in this case, it would be very helpful to point out more explicitly what it refers to.
assistant
The authors explore whether the halting time distributions for various algorithms in various settings exhibit *universality*, i.e. after rescaling to zero mean and unit variance, the distribution does not depend on stopping parameter, dimensionality and ensemble. The idea of the described universality is very interesting. However I see several shortcomings in the paper: In order to be of practical relevance, the actual stopping time might be more relevant than the scaled one. The discussion of exponential tailed halting time distributions is a good start, but I am not sure how often this might be actually helpful. Still, the findings in the paper might be interesting from a theoretical point of view. Especially for ICLR, I think it would have been more interesting to look into comparisons between stochastic gradient descent, momentum, ADAM etc on different deep learning architectures. Over which of those parameters does universality hold?. How can different initializations influence the halting time distribution? I would expect a sensible initialization to cut of part of the right tail of the distribution. Additionally, I found the paper quite hard to read. Here are some clarity issues: - abstract: *even when the input is changed drastically*: From the abstract I*m not sure what *input* refers to, here - I. Introduction: *where the stopping condition is, essentially, the time to find the minimum*: this doesn*t seem to make sense, a condition is not a time. I guess the authors wanted to say that the stopping condition is that the minimum has been reached? - I.1 the notions of dimension N, epsilon and ensemble E are introduced without any clarification what they are. From the later parts of the paper I got some ideas and examples, but here it is very hard to understand what these parameters should be (just some examples would be already helpful) - I.3 *We use x^ell for ell in Z={1, dots, S} where Z is a random sample from of training samples* This formulation doesn*t make sense. Either Z is a random sample, or Z={1, ..., S}. - II.1 it took me a long time to find the meaning of M. As this parameter seems to be crucial for universality in this case, it would be very helpful to point out more explicitly what it refers to.