Exponentially vanishing sub-optimal local minima in multilayer neural networks
Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska et al. (2015)) suggest that local minima with high error are exponentially rare in high dimensions. However, to prove low error guarantees for Multilayer Neural Networks (MNNs), previous works so far required either a heavily modified MNN model or training method, strong assumptions on the labels (e.g., “near” linear separability), or an unrealistically wide hidden layer with Omega(N) units. Results: We examine a MNN with one hidden layer of piecewise linear units, a single output, and a quadratic loss. We prove that, with high probability in the limit of N
ightarrowinfty datapoints, the volume of differentiable regions of the empiric loss containing sub-optimal differentiable local minima is exponentially vanishing in comparison with the same volume of global minima, given standard normal input of dimension d_0=	ilde{Omega}(sqrt{N}), and a more realistic number of d_1=	ilde{Omega}(N/d_0) hidden units. We demonstrate our results numerically: for example, 0% binary classification training error on CIFAR with only N/d_0 = 16 hidden neurons.
 This paper studies the question: Why does SGD on deep network is often successful, despite the fact that the objective induces bad local minima? The approach in this paper is to study a standard MNN with one hidden layer. They show that in an overparametrized regime, where the number of parameters is logarithmically larger than the number of parameters in the input, the ratio between the number of (bad) local minima to the number of global minima decays exponentially. They show this for a piecewise linear activation function, and input drawn from a standard Normal distribution. Their improvement over previous work is that the required overparameterization is fairly moderate, and that the network that they considered is similar to ones used in practice. This result seems interesting, although it is clearly not sufficient to explain even the success on the setting studied in this paper, since the number of minima of a certain type does not correspond to the probability of the SGD ending in one: to estimate the latter, the size of each basin of attraction should be taken into account. The authors are aware of this point and mention it as a disadvantage. However, since this question in general is a difficult one, any progress might be considered interesting. Hopefully, in future work it would be possible to also bound the probability of starting in one of the basins of attraction of bad local minima. The paper is well written and well presented, and the limitations of the approach, as well as its advantages over previous work, are clearly explained. As I am not an expert on the previous works in this field, my judgment relies mostly on this work and its representation of previous work. I did not verify the proofs in the appendix.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
Exponentially vanishing sub-optimal local minima in multilayer neural networks
Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska et al. (2015)) suggest that local minima with high error are exponentially rare in high dimensions. However, to prove low error guarantees for Multilayer Neural Networks (MNNs), previous works so far required either a heavily modified MNN model or training method, strong assumptions on the labels (e.g., “near” linear separability), or an unrealistically wide hidden layer with Omega(N) units. Results: We examine a MNN with one hidden layer of piecewise linear units, a single output, and a quadratic loss. We prove that, with high probability in the limit of N
ightarrowinfty datapoints, the volume of differentiable regions of the empiric loss containing sub-optimal differentiable local minima is exponentially vanishing in comparison with the same volume of global minima, given standard normal input of dimension d_0=	ilde{Omega}(sqrt{N}), and a more realistic number of d_1=	ilde{Omega}(N/d_0) hidden units. We demonstrate our results numerically: for example, 0% binary classification training error on CIFAR with only N/d_0 = 16 hidden neurons.
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>



 This paper studies the question: Why does SGD on deep network is often successful, despite the fact that the objective induces bad local minima? The approach in this paper is to study a standard MNN with one hidden layer. They show that in an overparametrized regime, where the number of parameters is logarithmically larger than the number of parameters in the input, the ratio between the number of (bad) local minima to the number of global minima decays exponentially. They show this for a piecewise linear activation function, and input drawn from a standard Normal distribution. Their improvement over previous work is that the required overparameterization is fairly moderate, and that the network that they considered is similar to ones used in practice. This result seems interesting, although it is clearly not sufficient to explain even the success on the setting studied in this paper, since the number of minima of a certain type does not correspond to the probability of the SGD ending in one: to estimate the latter, the size of each basin of attraction should be taken into account. The authors are aware of this point and mention it as a disadvantage. However, since this question in general is a difficult one, any progress might be considered interesting. Hopefully, in future work it would be possible to also bound the probability of starting in one of the basins of attraction of bad local minima. The paper is well written and well presented, and the limitations of the approach, as well as its advantages over previous work, are clearly explained. As I am not an expert on the previous works in this field, my judgment relies mostly on this work and its representation of previous work. I did not verify the proofs in the appendix.
assistant
 This paper studies the question: Why does SGD on deep network is often successful, despite the fact that the objective induces bad local minima? The approach in this paper is to study a standard MNN with one hidden layer. They show that in an overparametrized regime, where the number of parameters is logarithmically larger than the number of parameters in the input, the ratio between the number of (bad) local minima to the number of global minima decays exponentially. They show this for a piecewise linear activation function, and input drawn from a standard Normal distribution. Their improvement over previous work is that the required overparameterization is fairly moderate, and that the network that they considered is similar to ones used in practice. This result seems interesting, although it is clearly not sufficient to explain even the success on the setting studied in this paper, since the number of minima of a certain type does not correspond to the probability of the SGD ending in one: to estimate the latter, the size of each basin of attraction should be taken into account. The authors are aware of this point and mention it as a disadvantage. However, since this question in general is a difficult one, any progress might be considered interesting. Hopefully, in future work it would be possible to also bound the probability of starting in one of the basins of attraction of bad local minima. The paper is well written and well presented, and the limitations of the approach, as well as its advantages over previous work, are clearly explained. As I am not an expert on the previous works in this field, my judgment relies mostly on this work and its representation of previous work. I did not verify the proofs in the appendix.