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 paper provides a theoretical analysis comparing the training dynamics and performance of a specific 1 hidden-layer convolutional neural network with those of its corresponding finite width CNTK on a simple data distribution. This analysis is used to support their claim that the gap between neural networks and kernel methods may lie in the ````''Local Signal Adaptivity'', in the sense that neural networks outperform kernel methods in distinguishing localized influential features from noisy background. Furthermore, they empirically identify a performance gap between the CNN and finite width CNTK on noise extended variants of the CIFAR-10 dataset, which emphasizes the influence of the ``''Local Signal Adaptivity''. The authors conclude that this setting degrades the performance of kernels in a more pronounced way than it affects neural networks.