Symmetry-Breaking Convergence Analysis of Certain Two-layered Neural Networks with ReLU nonlinearity
In this paper, we use dynamical system to analyze the nonlinear weight dynamics of two-layered bias-free networks in the form of , where is ReLU nonlinearity. We assume that the input follow Gaussian distribution. The network is trained using gradient descent to mimic the output of a teacher network of the same size with fixed parameters using loss. We first show that when , the nonlinear dynamics can be written in close form, and converges to with at least probability, if random weight initializations of proper standard derivation ( ) is used, verifying empirical practice. For networks with many ReLU nodes ( ), we apply our close form dynamics and prove that when the teacher parameters forms orthonormal bases, (1) a symmetric weight initialization yields a convergence to a saddle point and (2) a certain symmetry-breaking weight initialization yields global convergence to without local minima. To our knowledge, this is the first proof that shows global convergence in nonlinear neural network without unrealistic assumptions on the independence of ReLU activations. In addition, we also give a concise gradient update formulation for a multilayer ReLU network when it follows a teacher of the same size with loss. Simulations verify our theoretical analysis.
The paper proposes a convergence analysis of some two-layer NNs with ReLUs. It is not the first such analysis, but maybe it is novel on the assumptions used in the analysis, and the focus on ReLU nonlinearity that is pretty popular in practice. The paper is quite hard to read, with many English mistakes and typos. Nevertheless, the analysis seems to be generally correct. The novelty and the key insights are however not always well motivated or presented. And the argument that the work uses realistic assumptions (Gaussian inputs for example) as opposed to other works, is quite debatable actually. Overall, the paper looks like a correct analysis work, but its form is really suboptimal in terms of writing/presentation, and the novelty and relevance of the results are not always very clear, unfortunately. The main results and intuition should be more clearly presented, and details could be moved to appendices for example - that could only help to improve the visibility and impact of these interesting results.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Symmetry-Breaking Convergence Analysis of Certain Two-layered Neural Networks with ReLU nonlinearity
In this paper, we use dynamical system to analyze the nonlinear weight dynamics of two-layered bias-free networks in the form of , where is ReLU nonlinearity. We assume that the input follow Gaussian distribution. The network is trained using gradient descent to mimic the output of a teacher network of the same size with fixed parameters using loss. We first show that when , the nonlinear dynamics can be written in close form, and converges to with at least probability, if random weight initializations of proper standard derivation ( ) is used, verifying empirical practice. For networks with many ReLU nodes ( ), we apply our close form dynamics and prove that when the teacher parameters forms orthonormal bases, (1) a symmetric weight initialization yields a convergence to a saddle point and (2) a certain symmetry-breaking weight initialization yields global convergence to without local minima. To our knowledge, this is the first proof that shows global convergence in nonlinear neural network without unrealistic assumptions on the independence of ReLU activations. In addition, we also give a concise gradient update formulation for a multilayer ReLU network when it follows a teacher of the same size with loss. Simulations verify our theoretical analysis.
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 proposes a convergence analysis of some two-layer NNs with ReLUs. It is not the first such analysis, but maybe it is novel on the assumptions used in the analysis, and the focus on ReLU nonlinearity that is pretty popular in practice. The paper is quite hard to read, with many English mistakes and typos. Nevertheless, the analysis seems to be generally correct. The novelty and the key insights are however not always well motivated or presented. And the argument that the work uses realistic assumptions (Gaussian inputs for example) as opposed to other works, is quite debatable actually. Overall, the paper looks like a correct analysis work, but its form is really suboptimal in terms of writing/presentation, and the novelty and relevance of the results are not always very clear, unfortunately. The main results and intuition should be more clearly presented, and details could be moved to appendices for example - that could only help to improve the visibility and impact of these interesting results.
assistant
The paper proposes a convergence analysis of some two-layer NNs with ReLUs. It is not the first such analysis, but maybe it is novel on the assumptions used in the analysis, and the focus on ReLU nonlinearity that is pretty popular in practice. The paper is quite hard to read, with many English mistakes and typos. Nevertheless, the analysis seems to be generally correct. The novelty and the key insights are however not always well motivated or presented. And the argument that the work uses realistic assumptions (Gaussian inputs for example) as opposed to other works, is quite debatable actually. Overall, the paper looks like a correct analysis work, but its form is really suboptimal in terms of writing/presentation, and the novelty and relevance of the results are not always very clear, unfortunately. The main results and intuition should be more clearly presented, and details could be moved to appendices for example - that could only help to improve the visibility and impact of these interesting results.