Global Optimality Conditions for Deep Neural Networks
We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and sufficient conditions for a critical point of the risk function to be a global minimum. Surprisingly, our conditions provide an efficiently checkable test for global optimality, while such tests are typically intractable in nonconvex optimization. We further extend these results to deep nonlinear neural networks and prove similar sufficient conditions for global optimality, albeit in a more limited function space setting.
 -I think title is misleading, as the more concise results in this paper is about linear networks I recommend adding linear in the title i.e. changing the title to … deep LINEAR networks - Theorems 2.1, 2.2 and the observation (2) are nice! - Theorem 2.2 there is no discussion about the nature of the saddle point is it strict? Does this theorem imply that the global optima can be reached from a random initialization? Regardless of if this theorem can deal with these issues, a discussion of the computational implications of this theorem is necessary. - I’m a bit puzzled by Theorems 4.1 and 4.2 and why they are useful. Since these results do not seem to have any computational implications about training the neural nets what insights do we gain about the problem by knowing this result? Further discussion would be helpful.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Global Optimality Conditions for Deep Neural Networks
We study the error landscape of deep linear and nonlinear neural networks with the squared error loss. Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete. For deep linear networks, we present necessary and sufficient conditions for a critical point of the risk function to be a global minimum. Surprisingly, our conditions provide an efficiently checkable test for global optimality, while such tests are typically intractable in nonconvex optimization. We further extend these results to deep nonlinear neural networks and prove similar sufficient conditions for global optimality, albeit in a more limited function space setting.
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>



 -I think title is misleading, as the more concise results in this paper is about linear networks I recommend adding linear in the title i.e. changing the title to … deep LINEAR networks - Theorems 2.1, 2.2 and the observation (2) are nice! - Theorem 2.2 there is no discussion about the nature of the saddle point is it strict? Does this theorem imply that the global optima can be reached from a random initialization? Regardless of if this theorem can deal with these issues, a discussion of the computational implications of this theorem is necessary. - I’m a bit puzzled by Theorems 4.1 and 4.2 and why they are useful. Since these results do not seem to have any computational implications about training the neural nets what insights do we gain about the problem by knowing this result? Further discussion would be helpful.
assistant
 -I think title is misleading, as the more concise results in this paper is about linear networks I recommend adding linear in the title i.e. changing the title to … deep LINEAR networks - Theorems 2.1, 2.2 and the observation (2) are nice! - Theorem 2.2 there is no discussion about the nature of the saddle point is it strict? Does this theorem imply that the global optima can be reached from a random initialization? Regardless of if this theorem can deal with these issues, a discussion of the computational implications of this theorem is necessary. - I’m a bit puzzled by Theorems 4.1 and 4.2 and why they are useful. Since these results do not seem to have any computational implications about training the neural nets what insights do we gain about the problem by knowing this result? Further discussion would be helpful.