We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not deterministic if depth and width simultaneously tend to infinity. Moreover, we prove that for such deep and wide networks, the NTK has a non-trivial evolution during training by showing that the mean of its first SGD update is also exponential in the ratio of network depth to width. This is sharp contrast to the regime where depth is fixed and network width is very large. Our results suggest that, unlike relatively shallow and wide networks, deep and wide ReLU networks are capable of learning data-dependent features even in the so-called lazy training regime.
This paper studies the finite depth and width corrections to the neural tangent kernel (NTK) in fully-connected ReLU networks. It gives sharp upper and lower bounds on the variance of NTK(x, x), which reveals an exponential dependence on a quantity beta=d/n, where d is depth, and n is hidden width. This implies that when beta is bounded away from 0, NTK(x, x) is not deterministic at initialization. The paper further analyzes the change of NTK(x, x) after one step of SGD on a single datapoint x, and shows that the change also depends exponentially on beta.
NTK has been a popular subject of theoretical study in deep learning, and it's an important question to understand when and to what extent NTK can capture the behavior of real neural networks. This paper makes partial progress by analyzing the diagonal entries of the NTK in fully-connected ReLU networks. It concludes that NTK(x, x) at initialization is not deterministic, and can change significantly after doing one SGD step on x, when beta is bounded away from 0. While it's nice that the authors obtained precise answers to these two questions, there are some drawbacks that limit the significance of this paper:
1. The entire paper only considers one single datapoint x, so it doesn't apply to the non-diagonal elements NTK(x, x') or the usual SGD with mini-batches containing multiple datapoints. Of course, it's already implied by the current paper that the NTK is not deterministic and can move a lot when beta is large, but it's unclear whether the reverse is true, i.e., what is the regime when the NTK becomes deterministic and frozen when an entire dataset is involved. An answer (even empirically) to this question would make the paper much more complete.
2. The result is also not so surprising given [Hanin, 2018] (and possibly some other papers by the same author) which also obtained a similar exp(beta) dependence using a similar combinatorial approach.
3. Minor issues regarding incorrect or missing references:
-- "Further, kernel method-based theorems show that even in this infinitely overparameterized regime neural networks will have non-vacuous guarantees on generalization (Wei et al., 2018)." The result of Wei et al. is not about kernel method and in particular not NTK.
-- "In fact, empirically, networks with finite but large width trained with initially large learning rates often outperform NTK predictions at infinite width." The authors should refer to the work of [Arora et al., 2019] which AFAIK is the first paper that provides empirical study of the (convolutional) NTK predictor at infinite width on relatively large datasets like CIFAR-10.
[Hanin, 2018] Which Neural Net Architectures Give Rise to Exploding and Vanishing Gradients?
[Arora et al., 2019] On Exact Computation with an Infinitely Wide Neural Net
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update: Thanks to the authors for the detailed reply, which answers most of my questions. I am updating my rating to weak accept in light of this.
Regarding Wei et al.: Their result on NTK is negative, i.e., they aimed to show that NTK is inferior to regularized NN. I find the way you cited this paper a bit weird.