A Modern Take on the Bias-Variance Tradeoff in Neural Networks
We revisit the bias-variance tradeoff for neural networks in light of modern empirical findings. The traditional bias-variance tradeoff in machine learning suggests that as model complexity grows, variance increases. Classical bounds in statistical learning theory point to the number of parameters in a model as a measure of model complexity, which means the tradeoff would indicate that variance increases with the size of neural networks. However, we empirically find that variance due to training set sampling is roughly constant (with both width and depth) in practice. Variance caused by the non-convexity of the loss landscape is different. We find that it decreases with width and increases with depth, in our setting. We provide theoretical analysis, in a simplified setting inspired by linear models, that is consistent with our empirical findings for width. We view bias-variance as a useful lens to study generalization through and encourage further theoretical explanation from this perspective.
 This paper studies variance-bias tradeoff as a function of depth and width of a neural network. Experiments suggest that variance may decrease as a function width and increase as a function of depth. Some analytical results are presented why this may the case for width and why the necessary assumptions for the depth are violated. Main comment on experiments: if I am correct the step size for optimization is chosen in a data-dependent way for each size of the network. This is a subtle point since it leads to a data-dependent hypothesis set. In other words, in this experiments for each width we study variance of neural nets that can be found in fixed number of iterations by a step size that is chosen in data-dependent way. It may be the case that as width grows the step size decreases faster and hence hypothesis set shrinks and we observe decreasing variance. This makes the results of experiments with width not so surprising or interesting. Further comments on experiments: it probably worth pointing out that results for depth are what we would expect from theory in general. More on experiments: it would be also interesting to see how variance behaves as a function of width for depth other than 1. On assumptions: it is not really clear why assumptions in 5.2 hold for wide shallow networks at least in some cases. Paper provides some references to prior work but it would be great to give more details. Furthermore, some statements seems to be contradicting: sentence before 5.2.1 seems to say that assumption (a) should hold for deep nets while sentence at the end of page 8 seems to say the opposite. Overall: I think this paper presents an interesting avenue of research but due to aforementioned points is not ready for publication.
You are a member of the scientific community tasked with peer review. Review the following paper title and abstract.
### Title + Abstract
A Modern Take on the Bias-Variance Tradeoff in Neural Networks
We revisit the bias-variance tradeoff for neural networks in light of modern empirical findings. The traditional bias-variance tradeoff in machine learning suggests that as model complexity grows, variance increases. Classical bounds in statistical learning theory point to the number of parameters in a model as a measure of model complexity, which means the tradeoff would indicate that variance increases with the size of neural networks. However, we empirically find that variance due to training set sampling is roughly constant (with both width and depth) in practice. Variance caused by the non-convexity of the loss landscape is different. We find that it decreases with width and increases with depth, in our setting. We provide theoretical analysis, in a simplified setting inspired by linear models, that is consistent with our empirical findings for width. We view bias-variance as a useful lens to study generalization through and encourage further theoretical explanation from this perspective.
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 variance-bias tradeoff as a function of depth and width of a neural network. Experiments suggest that variance may decrease as a function width and increase as a function of depth. Some analytical results are presented why this may the case for width and why the necessary assumptions for the depth are violated. Main comment on experiments: if I am correct the step size for optimization is chosen in a data-dependent way for each size of the network. This is a subtle point since it leads to a data-dependent hypothesis set. In other words, in this experiments for each width we study variance of neural nets that can be found in fixed number of iterations by a step size that is chosen in data-dependent way. It may be the case that as width grows the step size decreases faster and hence hypothesis set shrinks and we observe decreasing variance. This makes the results of experiments with width not so surprising or interesting. Further comments on experiments: it probably worth pointing out that results for depth are what we would expect from theory in general. More on experiments: it would be also interesting to see how variance behaves as a function of width for depth other than 1. On assumptions: it is not really clear why assumptions in 5.2 hold for wide shallow networks at least in some cases. Paper provides some references to prior work but it would be great to give more details. Furthermore, some statements seems to be contradicting: sentence before 5.2.1 seems to say that assumption (a) should hold for deep nets while sentence at the end of page 8 seems to say the opposite. Overall: I think this paper presents an interesting avenue of research but due to aforementioned points is not ready for publication.
assistant
 This paper studies variance-bias tradeoff as a function of depth and width of a neural network. Experiments suggest that variance may decrease as a function width and increase as a function of depth. Some analytical results are presented why this may the case for width and why the necessary assumptions for the depth are violated. Main comment on experiments: if I am correct the step size for optimization is chosen in a data-dependent way for each size of the network. This is a subtle point since it leads to a data-dependent hypothesis set. In other words, in this experiments for each width we study variance of neural nets that can be found in fixed number of iterations by a step size that is chosen in data-dependent way. It may be the case that as width grows the step size decreases faster and hence hypothesis set shrinks and we observe decreasing variance. This makes the results of experiments with width not so surprising or interesting. Further comments on experiments: it probably worth pointing out that results for depth are what we would expect from theory in general. More on experiments: it would be also interesting to see how variance behaves as a function of width for depth other than 1. On assumptions: it is not really clear why assumptions in 5.2 hold for wide shallow networks at least in some cases. Paper provides some references to prior work but it would be great to give more details. Furthermore, some statements seems to be contradicting: sentence before 5.2.1 seems to say that assumption (a) should hold for deep nets while sentence at the end of page 8 seems to say the opposite. Overall: I think this paper presents an interesting avenue of research but due to aforementioned points is not ready for publication.