Local Signal Adaptivity: Provable Feature Learning in Neural Networks Beyond Kernels

1 Introduction . Recently , deep learning ( using multi-layer , non-linear neural networks ) has demonstrated superior performance over traditional linear learners in many machine learning tasks . These achievements have bred much theoretical investigation into whether neural networks are , in fact , superior - and why . On the one hand , the Neural Tangent Kernel ( NTK ) and derivative works show that , under certain ( limiting ) conditions , a gradient-descent-trained neural network reduces to a kernel method with a specific architecture- and initialization-determined kernel [ Jacot et al. , 2018 , Du et al. , 2019 ] . However , this does not seem to be the full story , as it fails to capture the feature learning aspect of neural network training . This distinction between a fixed feature representation and a data-adaptive feature representation has been studied from a variety of perspectives , including the lazy vs. active regime framework [ Chizat et al. , 2019 , Woodworth et al. , 2020 , Moroshko et al. , 2020 , Geiger et al. , 2020 , Wang et al. , 2020 ] . Building on these insights , there has been increasing interest in now showing a provable gap between the performance of neural networks and kernel methods in various settings [ Ghorbani et al. , 2019 , 2020 , Allen-Zhu and Li , 2019 , 2020 , Li et al. , 2020b , Malach et al. , 2021 , Kamath et al. , 2020 , Refinetti et al. , 2021 , Daniely and Malach , 2020 , Chen et al. , 2020 , Domingo-Enrich et al. , 2021 ] . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . In this work , we extend this theoretical investigation into the superiority of neural networks over linear learners and , inspired by practical settings , propose a new line of reasoning that we refer to as “ Local Signal Adaptivity ” . In particular , we explore the power of convolutional neural networks ( CNNs ) in image classification , compared to linear functions over prescribed feature mappings . Our setting : We study a simple data distribution that captures one key property of natural image classification tasks : a small set of localized “ label-determining ” features embedded within a “ noisy ” background . We formally prove that even when such background occupies a rather large fraction of an image , a CNN can be quite effective at locating the label-determining signal and thus ignoring background information that is mostly irrelevant to the true label , to achieve high accuracy . Our result : In the formal setting of our simple data distribution ( presented in Section 3 ) , we ask whether a particular two-layer CNN trained via stochastic gradient descent can provably acquire this “ signal-finding ” ability and how this compares to its associated finite-width convolutional neural tangent kernel ( CNTK ) . We answer with a separation result between our CNN and its CNTK : We formally prove that a small CNN , trained using standard SGD from random initialization , can efficiently learn to find the “ signal ” and threshold out the noise , whereas the corresponding CNTK requires a comparatively larger model ( i.e. , with more features ) in order to accomplish this . Empirical justification : While we pick a simple data distribution in our work to illustrate the main idea and obtain a formal proof , we point out that our setting is very natural in real images : in many image classification tasks , the label-determining feature only occupies a small fraction of the image , and most other parts are background noise ( Figure 1 ) . Furthermore , in neural networks trained on natural images , it is generally accepted that activation patterns become increasingly sparse throughout training , effectively zeroing out the activations of low-magnitude noise and locating the true signal ( suggestive of the denoising/LSA phenomenon studied in this work ) . For completeness , we have included such an experiment in Figure 2 above , illustrating how the average percentage of active neurons per instance decreases throughout training ( details are provided in Appendix C ) . Finally , to empirically study our theoretical results , we create new datasets by embedding CIFAR-10 and MNIST images within either random Gaussian or IMAGENET backgrounds . Our experiments show that , as the intensity of the background noise grows and thus the “ denoising task ” becomes harder , the performance of the neural network stays relatively stable , while the performance of the corresponding NTK does , in fact , degrade significantly ( Section 6 , Figure 4 ) . Based on our theorem and experiments , we therefore believe that this per-instance “ signal finding within noisy backgrounds ” ability of convolutional neural networks , which we dub “ Local Signal Adaptivity ” ( LSA ) , is one key component of the superiority of SGD-trained convolutional neural networks over fixed feature mappings . 2 Related work . There is a long line of work proving that various concept classes can be learned efficiently by SGDtrained neural networks . Many of the theoretical works taking a “ beyond-NTK ” perspective and emphasizing the non-linearity of neural networks are only proved under Gaussian inputs [ Kawaguchi , 2016 , Soudry and Carmon , 2016 , Xie et al. , 2016 , Ge et al. , 2017 , Soltanolkotabi et al. , 2017 , Tian , 2017 , Brutzkus and Globerson , 2017 , Zhong et al. , 2017 , Li and Yuan , 2017 , Boob and Lan , 2017 , Li et al. , 2018 , Vempala and Wilmes , 2018 , Ge et al. , 2018 , Bakshi et al. , 2018 , Oymak and Soltanolkotabi , 2019 , Yehudai and Shamir , 2019 , Li and Yuan , 2017 , Li and Liang , 2017 , Li et al. , 2016 , Li and Dou , 2020 , Li et al. , 2020a ] . In our work , we consider a more realistic setting in which there is a highly-structured signal hidden within random background noise . Along these lines , we recognize various aesthetic similarities to sparse blind deconvolution and related areas of the signal processing literature [ Zhang et al. , 2019 , Kuo et al. , 2019 , Sun and Donoho , 2021 , Qu et al. , 2020 ] . However , there are key structural differences as well - e.g. , rather than recovering the unknown sparse signal , we simply wish to extract the information relevant to the classification task ; thus , further exploring possible connections between LSA and the signal processing literature presents an interesting avenue for future work . We now give a more detailed comparison of our work with prior theoretical results on the separation between neural networks and kernels . Representation power of neural networks . Many existing works focus on separating the representation power of neural networks from that of other models [ Ghorbani et al. , 2020 , Refinetti et al. , 2021 , Gribonval et al. , 2020 , Suzuki , 2019 , Suzuki and Nitanda , 2019 ] . However , the fact that a function can be represented efficiently by a neural network does not mean that it can be efficiently learned . Only a subset of such works prove efficient learnability , including Suzuki and Akiyama [ 2020 ] and Daniely and Malach [ 2020 ] . In our work , we focus on the set of functions that can be efficiently learned by neural networks , in particular learned via stochastic gradient descent over the standard training objective starting from random initialization . Classification vs. regression . Many existing works only focus on separating the power of neural networks from that of other learners in a regression setting [ Ghorbani et al. , 2020 , Suzuki and Akiyama , 2020 , Allen-Zhu and Li , 2020 , 2019 , Li et al. , 2020a , Wei et al. , 2018 , Yehudai and Shamir , 2019 ] . In this case , both the neural network and the other learning methods are required to fit the exact label . Although Daniely and Malach [ 2020 ] presents one of the few results in a classification setting , their final separation result is still in terms of hinge loss instead of 0-1 loss . In our result , we focus on the binary classification setting , where the neural network and ( significantly ) the linear learner are only required to fit the sign of the label , instead of the exact labeling function . Our final result is therefore more natural in image classification settings than prior works . Learning a hidden subspace . Many of the existing works showing how neural networks are better than other learners , in particular kernel methods , focus on the case where the concept class is of the form f ( x ) = φ ( Wx ) , where W is a rank-deficient matrix [ Ghorbani et al. , 2020 , Wei et al. , 2018 ] . Thus , the learning process can be divided into two phases : ( 1 ) Identifying the hidden subspace of W ; ( 2 ) Learning the function φ over this hidden subspace . We point out that to the best of our knowledge , all the cited works only shows the superior power of neural networks in ( 1 ) . This means that , if W were known , then a neural network would have the same sample/time complexity as kernels when performing ( 2 ) . As we will show , in our work , the underlying concept class does not have a fixed subspace , rather the signal can drift between different patches , and the neural network needs to identify the signal patch while ignoring the noise patches . Our explanation of how neural networks outperform kernels is therefore fundamentally different from the cited works . Kernel lower bounds . Most of the existing theoretical works demonstrating the power of neural networks focus on showing how neural networks are better than kernel methods or linear functions over prescribed feature mappings . There are two related lines of work : ( 1 ) Showing that neural networks are better than any kernel method or any linear learner [ Allen-Zhu and Li , 2020 , Daniely and Malach , 2020 ] ; ( 2 ) Showing that neural networks are better than a particular set of linear learners , most notably , the NTKs or finite-width NTKs of the corresponding neural network [ Yehudai and Shamir , 2019 , Refinetti et al. , 2021 , Wei et al. , 2018 ] . Our work belongs to the second line . Although a result along the first line gives a much stronger separation , we point out that in the classification setting , such a separation is known to be extremely challenging in computational complexity theory [ O ’ Donnell and Servedio , 2003 , Sherstov and Wu , 2019 ] . Finite- vs. infinite-width NTK . Empirical results in the literature suggest that the infinite-width NTK generally reaches slightly higher final test accuracy than the finite-width NTK [ Allen-Zhu and Li , 2020 , Lee et al. , 2020 ] . In fact , in certain limited settings , such as low-data regimes , the infinite-width NTK has actually demonstrated performance competitive with that of the original neural network [ Arora et al. , 2020 ] . However , under certain conditions , the computational cost of the infinite-width NTK can be significant compared to that of the original neural network and its corresponding finitewidth NTK [ Allen-Zhu and Li , 2020 , Lee et al. , 2020 ] . Therefore , due to these computational and accuracy distinctions between the finite- and infinite-width NTKs , in our experiments we examine both types of NTK . Our theoretical lower bound , however , is a representational lower bound for the finite-width NTK . This is primarily due to theoretical challenges such as those discussed above . Theoretical results such as Allen-Zhu and Li [ 2020 ] suggest that it may be possible , in follow-up work , to convert our finite-width NTK representational lower bound into an infinite-width NTK sample complexity lower bound . However , proving a sample complexity gap would likely require tightening our sample complexity upper bound as well . We believe , conceptually ( and supported by our experiments ) , that the LSA phenomenon is at play for infinite-width NTKs as well , and it would thus be interesting to extend our theoretical gap to the infinite-width NTK setting as well .

Summary: The authors study a toy binary classification problem where a certain CNN architecture is trained to detect one of two templates embedded in gaussian noise. The motivation is twofold: the problem is argued to capture relevant structures present in natural image classification tasks where CNNs succeed in practice (i.e., the detection of an object/motif in the image in the presence of a nuisance background), and the theoretical analysis establishes an architectural separation via the rates one gets by analyzing the problem with the neural tangent kernel approach and via the authors' analysis. Concretely, the data model consists of a fixed unknown template embedded in one of $k+1$ consecutive disjoint length-$d$ blocks of a length $d(k+1)$ 1D signal, with the other $k$ blocks containing i.i.d. gaussian noise of a certain variance (the blocks are imagined to be disjoint "image patches", and the template a motif of interest) -- samples from this data model have the patch where the template is contained uniformly randomized and contain independent noise, and the template is given either the positive or negative sign (representing the two classes in the problem). The noise variance is set to be large enough that it is not possible to perform naive matched filtering to recover the template. Accordingly, the CNN that is trained to solve this task contains a custom denoising nonlinearity, the standard soft thresholding function -- this corresponds to a one-layer, two-neuron ReLU network that computes two $d$-strided convolutions with specific weight sharing and output averaging (the shared biases end up corresponding to the soft thresholding function's parameter) -- which is trained using finite samples from the data generating distribution by mini-batch SGD on the logistic loss with gaussian random initialization. The authors show that this procedure yields a classifier for the data whp given polynomial samples and suitable step sizes; in contrast, they show that the network's convolutional NTK is unable to solve the task whp as long as the number of filters satisfies $o(d)$ (the neural network studied corresponds to a two filter network). Experiments involving embedding CIFAR-10 images into random backgrounds (either from ImageNet or having gaussian noise) with varying intensities and comparing neural network performance to their NTK's performance computationally on the classification tasks thus generated verifies the predicted performance gap.