STABILITY AND CONVERGENCE THEORY FOR LEARNING RESNET: A FULL CHARACTERIZATION
ResNet structure has achieved great success since its debut. In this paper, we study the stability of learning ResNet. Specifically, we consider the ResNet block where is ReLU activation and is a scalar. We show that for standard initialization used in practice, is a sharp value in characterizing the stability of forward/backward process of ResNet, where is the number of residual blocks. Specifically, stability is guaranteed for while conversely forward process explodes when for a positive constant . Moreover, if ResNet is properly over-parameterized, we show for gradient descent is guaranteed to find the global minima footnote{We use to hide logarithmic factor.}, which significantly enlarges the range of that admits global convergence in previous work. We also demonstrate that the over-parameterization requirement of ResNet only weakly depends on the depth, which corroborates the advantage of ResNet over vanilla feedforward network. Empirically, with , deep ResNet can be easily trained even without normalization layer. Moreover, adding can also improve the performance of ResNet with normalization layer.
The paper presents a non-asymptotic analysis of ResNet stability which determines a *cutoff* value for the residual block scale factor characterizing stability vs output explosion, and improves upon prior results by finding a larger range for the scale factor that lead to global convergence in non-convex optimization. Theoretical findings are corroborated via experiments confirming the validity of the *cutoff* value. Additional experiments are conducted indicating that using the cutoff value, ResNet can be trained even without normalization layers, and that the cutoff value is also beneficial with normalization as it allows effective training of very deep ResNet. This is an interesting submission which presents important theoretical results while also showing their practical pertinence via experimental validation. The paper is well written and the presentation is clear. Prior work is reviewed appropriately. The reviewer found it particularly informative to provide intuition and present a proof sketch next to each theoretical results. The convergence analysis leads to a depth dependence of ResNet and the authors claim that this is not a *real* dependence and attributes it to bounding techniques handling non smooth activation. The authors further indicate in their experiments that for a given width the convergence of ResNet does not depends *much* on depth, compared to feedforward networks. So there might still be a dependence though much milder than that found by the current theory. It would be interesting to more precisely characterize the dependence observed in experiments. The authors claim that their analysis leading to the non-asymptotic bound on the spectral norm of the feedforward process via martingale theory might be relevant for other problems. It would be useful if the authors could elaborate and in particular indicate if their technique could carry to analyse other structures beyond ResNet. Minor comments: *What else values* ---> Are there other values *our first claim is a new spectral norm* --> is a new bound of the spectral norm *we does no make such* --> we do not make
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
STABILITY AND CONVERGENCE THEORY FOR LEARNING RESNET: A FULL CHARACTERIZATION
ResNet structure has achieved great success since its debut. In this paper, we study the stability of learning ResNet. Specifically, we consider the ResNet block where is ReLU activation and is a scalar. We show that for standard initialization used in practice, is a sharp value in characterizing the stability of forward/backward process of ResNet, where is the number of residual blocks. Specifically, stability is guaranteed for while conversely forward process explodes when for a positive constant . Moreover, if ResNet is properly over-parameterized, we show for gradient descent is guaranteed to find the global minima footnote{We use to hide logarithmic factor.}, which significantly enlarges the range of that admits global convergence in previous work. We also demonstrate that the over-parameterization requirement of ResNet only weakly depends on the depth, which corroborates the advantage of ResNet over vanilla feedforward network. Empirically, with , deep ResNet can be easily trained even without normalization layer. Moreover, adding can also improve the performance of ResNet with normalization layer.
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 presents a non-asymptotic analysis of ResNet stability which determines a *cutoff* value for the residual block scale factor characterizing stability vs output explosion, and improves upon prior results by finding a larger range for the scale factor that lead to global convergence in non-convex optimization. Theoretical findings are corroborated via experiments confirming the validity of the *cutoff* value. Additional experiments are conducted indicating that using the cutoff value, ResNet can be trained even without normalization layers, and that the cutoff value is also beneficial with normalization as it allows effective training of very deep ResNet. This is an interesting submission which presents important theoretical results while also showing their practical pertinence via experimental validation. The paper is well written and the presentation is clear. Prior work is reviewed appropriately. The reviewer found it particularly informative to provide intuition and present a proof sketch next to each theoretical results. The convergence analysis leads to a depth dependence of ResNet and the authors claim that this is not a *real* dependence and attributes it to bounding techniques handling non smooth activation. The authors further indicate in their experiments that for a given width the convergence of ResNet does not depends *much* on depth, compared to feedforward networks. So there might still be a dependence though much milder than that found by the current theory. It would be interesting to more precisely characterize the dependence observed in experiments. The authors claim that their analysis leading to the non-asymptotic bound on the spectral norm of the feedforward process via martingale theory might be relevant for other problems. It would be useful if the authors could elaborate and in particular indicate if their technique could carry to analyse other structures beyond ResNet. Minor comments: *What else values* ---> Are there other values *our first claim is a new spectral norm* --> is a new bound of the spectral norm *we does no make such* --> we do not make
assistant
The paper presents a non-asymptotic analysis of ResNet stability which determines a *cutoff* value for the residual block scale factor characterizing stability vs output explosion, and improves upon prior results by finding a larger range for the scale factor that lead to global convergence in non-convex optimization. Theoretical findings are corroborated via experiments confirming the validity of the *cutoff* value. Additional experiments are conducted indicating that using the cutoff value, ResNet can be trained even without normalization layers, and that the cutoff value is also beneficial with normalization as it allows effective training of very deep ResNet. This is an interesting submission which presents important theoretical results while also showing their practical pertinence via experimental validation. The paper is well written and the presentation is clear. Prior work is reviewed appropriately. The reviewer found it particularly informative to provide intuition and present a proof sketch next to each theoretical results. The convergence analysis leads to a depth dependence of ResNet and the authors claim that this is not a *real* dependence and attributes it to bounding techniques handling non smooth activation. The authors further indicate in their experiments that for a given width the convergence of ResNet does not depends *much* on depth, compared to feedforward networks. So there might still be a dependence though much milder than that found by the current theory. It would be interesting to more precisely characterize the dependence observed in experiments. The authors claim that their analysis leading to the non-asymptotic bound on the spectral norm of the feedforward process via martingale theory might be relevant for other problems. It would be useful if the authors could elaborate and in particular indicate if their technique could carry to analyse other structures beyond ResNet. Minor comments: *What else values* ---> Are there other values *our first claim is a new spectral norm* --> is a new bound of the spectral norm *we does no make such* --> we do not make