On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length
The training of deep neural networks with Stochastic Gradient Descent (SGD) with a large learning rate or a small batch-size typically ends in flat regions of the weight space, as indicated by small eigenvalues of the Hessian of the training loss. This was found to correlate with a good final generalization performance. In this paper we extend previous work by investigating the curvature of the loss surface along the whole training trajectory, rather than only at the endpoint. We find that initially SGD visits increasingly sharp regions, reaching a maximum sharpness determined by both the learning rate and the batch-size of SGD. At this peak value SGD starts to fail to minimize the loss along directions in the loss surface corresponding to the largest curvature (sharpest directions). To further investigate the effect of these dynamics in the training process, we study a variant of SGD using a reduced learning rate along the sharpest directions which we show can improve training speed while finding both sharper and better generalizing solution, compared to vanilla SGD. Overall, our results show that the SGD dynamics in the subspace of the sharpest directions influence the regions that SGD steers to (where larger learning rate or smaller batch size result in wider regions visited), the overall training speed, and the generalization ability of the final model.
 The paper discusses connections between the properties of DNN loss surfaces and the step length SGD algorithms take, a timely topic. On the whole, reasonably well done, with some interesting observations. It makes several claims, most notably that there is an initial regime where SGD visits increasingly sharp regions of the loss surface, followed by a regime where the loss surface gets smoother. Useful to know, and characterized moderately well. A weakness is that the generality of that claim is not made clear. Like many papers in the area, it is an observation, the realm of which is not clarified. E.g., what properties of the neural network or data does it depend on. Also not clarified is how this depends on initialization, etc. The evaluation should be more systematic, as it is hard to tell how general is the claims of the paper as well as how they depend on implementation details. The discussion of Hessian directions ignores very relevant work by Yao et al (https://arxiv.org/abs/1802.08241 and follow up). The first figure in Fig 1 is probably misleading, and probably not worth having, the latter two are what is measured and thus more interesting. The obvious conclusion from the poor conditioning is that methods designed to addressed poor conditioning, i.e., second order methods, should be considered. Those should have a complementary dynamics to what is discussed. This is what is the elephant in the room when you talk about steering towards or away from regions whose curvature matches the SGD step. I don*t know what it means to say "Where applicable, the Hessian is estimated with regularization applied" Is this to speed up computation, why doesn*t this change the loss surface, etc. If you are not measuring Hessian information precisely, then all the claims of the paper fall apart. Several times claims like "SGD reaches a region in which the SGD step matches ..." Of course, the energy surface changes with training time, so it is a little unclear what is being said. The main method Nudged-SGD sounds like a poor-mans second order method. Why not describe it as such (in more than a footnote and appendix), rather than introducing a new acronym. I don*t know that I believe the "key design principle" in the appendix for second order methods. Second order methods rotate and stretch to take a locally-correct step length, and this method sounds like it is doing a poor mans version of that. There is a good question as to whether the "thresholding" into large and small that NSGD is doing causes it to do something very different, but that isn*t really evaluated. Averaging over two random seeds is not a lot.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
On the Relation Between the Sharpest Directions of DNN Loss and the SGD Step Length
The training of deep neural networks with Stochastic Gradient Descent (SGD) with a large learning rate or a small batch-size typically ends in flat regions of the weight space, as indicated by small eigenvalues of the Hessian of the training loss. This was found to correlate with a good final generalization performance. In this paper we extend previous work by investigating the curvature of the loss surface along the whole training trajectory, rather than only at the endpoint. We find that initially SGD visits increasingly sharp regions, reaching a maximum sharpness determined by both the learning rate and the batch-size of SGD. At this peak value SGD starts to fail to minimize the loss along directions in the loss surface corresponding to the largest curvature (sharpest directions). To further investigate the effect of these dynamics in the training process, we study a variant of SGD using a reduced learning rate along the sharpest directions which we show can improve training speed while finding both sharper and better generalizing solution, compared to vanilla SGD. Overall, our results show that the SGD dynamics in the subspace of the sharpest directions influence the regions that SGD steers to (where larger learning rate or smaller batch size result in wider regions visited), the overall training speed, and the generalization ability of the final model.
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 discusses connections between the properties of DNN loss surfaces and the step length SGD algorithms take, a timely topic. On the whole, reasonably well done, with some interesting observations. It makes several claims, most notably that there is an initial regime where SGD visits increasingly sharp regions of the loss surface, followed by a regime where the loss surface gets smoother. Useful to know, and characterized moderately well. A weakness is that the generality of that claim is not made clear. Like many papers in the area, it is an observation, the realm of which is not clarified. E.g., what properties of the neural network or data does it depend on. Also not clarified is how this depends on initialization, etc. The evaluation should be more systematic, as it is hard to tell how general is the claims of the paper as well as how they depend on implementation details. The discussion of Hessian directions ignores very relevant work by Yao et al (https://arxiv.org/abs/1802.08241 and follow up). The first figure in Fig 1 is probably misleading, and probably not worth having, the latter two are what is measured and thus more interesting. The obvious conclusion from the poor conditioning is that methods designed to addressed poor conditioning, i.e., second order methods, should be considered. Those should have a complementary dynamics to what is discussed. This is what is the elephant in the room when you talk about steering towards or away from regions whose curvature matches the SGD step. I don*t know what it means to say "Where applicable, the Hessian is estimated with regularization applied" Is this to speed up computation, why doesn*t this change the loss surface, etc. If you are not measuring Hessian information precisely, then all the claims of the paper fall apart. Several times claims like "SGD reaches a region in which the SGD step matches ..." Of course, the energy surface changes with training time, so it is a little unclear what is being said. The main method Nudged-SGD sounds like a poor-mans second order method. Why not describe it as such (in more than a footnote and appendix), rather than introducing a new acronym. I don*t know that I believe the "key design principle" in the appendix for second order methods. Second order methods rotate and stretch to take a locally-correct step length, and this method sounds like it is doing a poor mans version of that. There is a good question as to whether the "thresholding" into large and small that NSGD is doing causes it to do something very different, but that isn*t really evaluated. Averaging over two random seeds is not a lot.
assistant
 The paper discusses connections between the properties of DNN loss surfaces and the step length SGD algorithms take, a timely topic. On the whole, reasonably well done, with some interesting observations. It makes several claims, most notably that there is an initial regime where SGD visits increasingly sharp regions of the loss surface, followed by a regime where the loss surface gets smoother. Useful to know, and characterized moderately well. A weakness is that the generality of that claim is not made clear. Like many papers in the area, it is an observation, the realm of which is not clarified. E.g., what properties of the neural network or data does it depend on. Also not clarified is how this depends on initialization, etc. The evaluation should be more systematic, as it is hard to tell how general is the claims of the paper as well as how they depend on implementation details. The discussion of Hessian directions ignores very relevant work by Yao et al (https://arxiv.org/abs/1802.08241 and follow up). The first figure in Fig 1 is probably misleading, and probably not worth having, the latter two are what is measured and thus more interesting. The obvious conclusion from the poor conditioning is that methods designed to addressed poor conditioning, i.e., second order methods, should be considered. Those should have a complementary dynamics to what is discussed. This is what is the elephant in the room when you talk about steering towards or away from regions whose curvature matches the SGD step. I don*t know what it means to say "Where applicable, the Hessian is estimated with regularization applied" Is this to speed up computation, why doesn*t this change the loss surface, etc. If you are not measuring Hessian information precisely, then all the claims of the paper fall apart. Several times claims like "SGD reaches a region in which the SGD step matches ..." Of course, the energy surface changes with training time, so it is a little unclear what is being said. The main method Nudged-SGD sounds like a poor-mans second order method. Why not describe it as such (in more than a footnote and appendix), rather than introducing a new acronym. I don*t know that I believe the "key design principle" in the appendix for second order methods. Second order methods rotate and stretch to take a locally-correct step length, and this method sounds like it is doing a poor mans version of that. There is a good question as to whether the "thresholding" into large and small that NSGD is doing causes it to do something very different, but that isn*t really evaluated. Averaging over two random seeds is not a lot.