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1. Wide and deep neural networks achieve consistency for classification

   https://doi.org/10.1073/pnas.2208779120  

   Radhakrishnan, Adityanarayanan; Belkin, Mikhail; Uhler, Caroline (April 2023, Proceedings of the National Academy of Sciences)

   While neural networks are used for classification tasks across domains, a long-standing open problem in machine learning is determining whether neural networks trained using standard procedures are consistent for classification, i.e., whether such models minimize the probability of misclassification for arbitrary data distributions. In this work, we identify and construct an explicit set of neural network classifiers that are consistent. Since effective neural networks in practice are typically both wide and deep, we analyze infinitely wide networks that are also infinitely deep. In particular, using the recent connection between infinitely wide neural networks and neural tangent kernels, we provide explicit activation functions that can be used to construct networks that achieve consistency. Interestingly, these activation functions are simple and easy to implement, yet differ from commonly used activations such as ReLU or sigmoid. More generally, we create a taxonomy of infinitely wide and deep networks and show that these models implement one of three well-known classifiers depending on the activation function used: 1) 1-nearest neighbor (model predictions are given by the label of the nearest training example); 2) majority vote (model predictions are given by the label of the class with the greatest representation in the training set); or 3) singular kernel classifiers (a set of classifiers containing those that achieve consistency). Our results highlight the benefit of using deep networks for classification tasks, in contrast to regression tasks, where excessive depth is harmful. 

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2. Dynamics of finite width Kernel and prediction fluctuations in mean field neural networks

   https://doi.org/10.1088/1742-5468/ad642b  

   Bordelon, Blake; Pehlevan, Cengiz (October 2024, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Starting from a dynamical mean field theory description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the $O\left(1/\sqrt{\text{width}}\right)$fluctuations of the dynamical mean field theory order parameters over random initializations of the network weights. Our results, while perturbative in width, unlike prior analyses, are non-perturbative in the strength of feature learning. We find that once the mean field/µP parameterization is adopted, the leading finite size effect on the dynamics is to introduce initialization variance in the predictions and feature kernels of the networks. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with a variance that can be computed self-consistently. In two layer networks, we show how feature learning can dynamically reduce the variance of the final tangent kernel and final network predictions. We also show how initialization variance can slow down online learning in wide but finite networks. In deeper networks, kernel variance can dramatically accumulate through subsequent layers at large feature learning strengths, but feature learning continues to improve the signal-to-noise ratio of the feature kernels. In discrete time, we demonstrate that large learning rate phenomena such as edge of stability effects can be well captured by infinite width dynamics and that initialization variance can decrease dynamically. For convolutional neural networks trained on CIFAR-10, we empirically find significant corrections to both the bias and variance of network dynamics due to finite width. 

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3. The Recurrent Neural Tangent Kernel

   Alemohammad, Sina; Wang, Zichao; Balestriero, Randall; Baraniuk, Richard (May 2021, The International Conference on Learning Representations)

   The study of deep neural networks (DNNs) in the infinite-width limit, via the so-called neural tangent kernel (NTK) approach, has provided new insights into the dynamics of learning, generalization, and the impact of initialization. One key DNN architecture remains to be kernelized, namely, the recurrent neural network (RNN). In this paper we introduce and study the Recurrent Neural Tangent Kernel (RNTK), which provides new insights into the behavior of overparametrized RNNs. A key property of the RNTK should greatly benefit practitioners is its ability to compare inputs of different length. To this end, we characterize how the RNTK weights different time steps to form its output under different initialization parameters and nonlinearity choices. A synthetic and 56 real-world data experiments demonstrate that the RNTK offers significant performance gains over other kernels, including standard NTKs, across a wide array of data sets. 

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4. Self-consistent dynamical field theory of kernel evolution in wide neural networks ^*

   https://doi.org/10.1088/1742-5468/ad01b0  

   Bordelon, Blake; Pehlevan, Cengiz (November 2023, Journal of Statistical Mechanics: Theory and Experiment)

   Abstract We analyze feature learning in infinite-width neural networks trained with gradient flow through a self-consistent dynamical field theory. We construct a collection of deterministic dynamical order parameters which are inner-product kernels for hidden unit activations and gradients in each layer at pairs of time points, providing a reduced description of network activity through training. These kernel order parameters collectively define the hidden layer activation distribution, the evolution of the neural tangent kernel (NTK), and consequently, output predictions. We show that the field theory derivation recovers the recursive stochastic process of infinite-width feature learning networks obtained by Yang and Hu with tensor programs. For deep linear networks, these kernels satisfy a set of algebraic matrix equations. For nonlinear networks, we provide an alternating sampling procedure to self-consistently solve for the kernel order parameters. We provide comparisons of the self-consistent solution to various approximation schemes including the static NTK approximation, gradient independence assumption, and leading order perturbation theory, showing that each of these approximations can break down in regimes where general self-consistent solutions still provide an accurate description. Lastly, we provide experiments in more realistic settings which demonstrate that the loss and kernel dynamics of convolutional neural networks at fixed feature learning strength are preserved across different widths on a image classification task. 

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5. Analyzing convergence in quantum neural networks: deviations from neural tangent kernels

   You, Xuchen; Chakrabarti, Shouvanik; Chen, Boyang; Wu, Xiaodi (July 2023, ICML'23: Proceedings of the 40th International Conference on Machine Learning)

   A quantum neural network (QNN) is a parameterized mapping efficiently implementable on near-term Noisy Intermediate-Scale Quantum (NISQ) computers. It can be used for supervised learning when combined with classical gradient-based optimizers. Despite the existing empirical and theoretical investigations, the convergence of QNN training is not fully understood. Inspired by the success of the neural tangent kernels (NTKs) in probing into the dynamics of classical neural networks, a recent line of works proposes to study over-parameterized QNNs by examining a quantum version of tangent kernels. In this work, we study the dynamics of QNNs and show that contrary to popular belief it is qualitatively different from that of any kernel regression: due to the unitarity of quantum operations, there is a nonnegligible deviation from the tangent kernel regression derived at the random initialization. As a result of the deviation, we prove the at-most sublinear convergence for QNNs with Pauli measurements, which is beyond the explanatory power of any kernel regression dynamics. We then present the actual dynamics of QNNs in the limit of over-parameterization. The new dynamics capture the change of convergence rate during training, and implies that the range of measurements is crucial to the fast QNN convergence. 

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