Recent advances in machine learning and artificial intelligence have relied on fitting highly overparam-eterized models, notably deep neural networks, to observed data; e.g.

Neural networks with wide layers have attracted significant attention due to their equivalence to Gaussian processes, enabling perfect fitting of training data while maintaining generalization performance, known as benign overfitting. However, existing results mainly focus on shallow or finite-depth networks, necessitating a comprehensive analysis of wide neural networks with infinite-depth layers, such as neural ordinary differential equations (ODEs) and deep equilibrium models (DEQs). In this paper, we specifically investigate the deep equilibrium model (DEQ), an infinite-depth neural network with shared weight matrices across layers. Our analysis reveals that as the width of DEQ layers approaches infinity, it converges to a Gaussian process, establishing what is known as the Neural Network and Gaussian Process (NNGP) correspondence. Remarkably, this convergence holds even when the limits of depth and width are interchanged, which is not observed in typical infinite-depth Multilayer Perceptron (MLP) networks. Furthermore, we demonstrate that the associated Gaussian vector remains non-degenerate for any pairwise distinct input data, ensuring a strictly positive smallest eigenvalue of the corresponding kernel matrix using the NNGP kernel. These findings serve as fundamental elements for studying the training and generalization of DEQs, laying the groundwork for future research in this area.

Normalization methods play an important role in enhancing the performance of deep learning while their theoretical understandings have been limited. To theoretically elucidate the effectiveness of normalization, we quantify the geometry of the parameter space determined by the Fisher information matrix (FIM), which also corresponds to the local shape of the loss landscape under certain conditions. We analyze deep neural networks with random initialization, which is known to suffer from a pathologically sharp shape of the landscape when the network becomes sufficiently wide. We reveal that batch normalization in the last layer contributes to drastically decreasing such pathological sharpness if the width and sample number satisfy a specific condition. In contrast, it is hard for batch normalization in the middle hidden layers to alleviate pathological sharpness in many settings. We also found that layer normalization cannot alleviate pathological sharpness either. Thus, we can conclude that batch normalization in the last layer significantly contributes to decreasing the sharpness induced by the FIM.

A longstanding goal in deep learning research has been to precisely characterize training and generalization. However, the often complex loss landscapes of neural networks have made a theory of learning dynamics elusive. In this work, we show that for wide neural networks the learning dynamics simplify considerably and that, in the infinite width limit, they are governed by a linear model obtained from the first-order Taylor expansion of the network around its initial parameters. Furthermore, mirroring the correspondence between wide Bayesian neural networks and Gaussian processes, gradient-based training of wide neural networks with a squared loss produces test set predictions drawn from a Gaussian process with a particular compositional kernel. While these theoretical results are only exact in the infinite width limit, we nevertheless find excellent empirical agreement between the predictions of the original network and those of the linearized version even for finite practically-sized networks. This agreement is robust across different architectures, optimization methods, and loss functions.

The success of over-parameterized neural networks trained to near-zero training error has caused great interest in the phenomenon of benign overfitting, where estimators are statistically consistent even though they interpolate noisy training data. While benign overfitting in fixed dimension has been established for some learning methods, current literature suggests that for regression with typical kernel methods and wide neural networks, benign overfitting requires a high-dimensional setting, where the dimension grows with the sample size. In this paper, we show that the smoothness of the estimators, and not the dimension, is the key: benign overfitting is possible if and only if the estimator's derivatives are large enough. We generalize existing inconsistency results to non-interpolating models and more kernels to show that benign overfitting with moderate derivatives is impossible in fixed dimension. Conversely, we show that benign overfitting is possible for regression with a sequence of spiky-smooth kernels with large derivatives. Using neural tangent kernels, we translate our results to wide neural networks. We prove that while infinite-width networks do not overfit benignly with the ReLU activation, this can be fixed by adding small high-frequency fluctuations to the activation function. Our experiments verify that such neural networks, while overfitting, can indeed generalize well even on low-dimensional data sets.

Knowledge distillation is a strategy of training a student network with guide of the soft output from a teacher network. It has been a successful method of model compression and knowledge transfer. However, currently knowledge distillation lacks a convincing theoretical understanding. On the other hand, recent finding on neural tangent kernel enables us to approximate a wide neural network with a linear model of the network's random features. In this paper, we theoretically analyze the knowledge distillation of a wide neural network. First we provide a transfer risk bound for the linearized model of the network. Then we propose a metric of the task's training difficulty, called data inefficiency. Based on this metric, we show that for a perfect teacher, a high ratio of teacher's soft labels can be beneficial. Finally, for the case of imperfect teacher, we find that hard labels can correct teacher's wrong prediction, which explains the practice of mixing hard and soft labels.

Deep neural networks (DNNs) defy the classical bias-variance trade-off: adding parameters to a DNN that interpolates its training data will typically improve its generalization performance. Explaining the mechanism behind this ``benign overfitting'' in deep networks remains an outstanding challenge. Here, we study the last hidden layer representations of various state-of-the-art convolutional neural networks and find that if the last hidden representation is wide enough, its neurons tend to split into groups that carry identical information and differ from each other only by statistically independent noise. The number of such groups increases linearly with the width of the layer, but only if the width is above a critical value. We show that redundant neurons appear only when the training is regularized and the training error is zero.

We perform a careful, thorough, and large scale empirical study of the correspondence between wide neural networks and kernel methods. By doing so, we resolve a variety of open questions related to the study of infinitely wide neural networks. Our experimental results include: kernel methods outperform fully-connected finite-width networks, but underperform convolutional finite width networks; neural network Gaussian process (NNGP) kernels frequently outperform neural tangent (NT) kernels; centered and ensembled finite networks have reduced posterior variance and behave more similarly to infinite networks; weight decay and the use of a large learning rate break the correspondence between finite and infinite networks; the NTK parameterization outperforms the standard parameterization for finite width networks; diagonal regularization of kernels acts similarly to early stopping; floating point precision limits kernel performance beyond a critical dataset size; regularized ZCA whitening improves accuracy; finite network performance depends non-monotonically on width in ways not captured by double descent phenomena; equivariance of CNNs is only beneficial for narrow networks far from the kernel regime. Our experiments additionally motivate an improved layer-wise scaling for weight decay which improves generalization in finite-width networks. Finally, we develop improved best practices for using NNGP and NT kernels for prediction, including a novel ensembling technique. Using these best practices we achieve state-of-the-art results on CIFAR-10 classification for kernels corresponding to each architecture class we consider.

Natural Gradient Descent (NGD) helps to accelerate the convergence of gradient descent dynamics, but it requires approximations in large-scale deep neural networks because of its high computational cost. Empirical studies have confirmed that some NGD methods with approximate Fisher information converge sufficiently fast in practice. Nevertheless, it remains unclear from the theoretical perspective why and under what conditions such heuristic approximations work well. In this work, we reveal that, under specific conditions, NGD with approximate Fisher information achieves the same fast convergence to global minima as exact NGD. We consider deep neural networks in the infinite-width limit, and analyze the asymptotic training dynamics of NGD in function space via the neural tangent kernel.

In this paper, we investigate the limiting behavior of a continuous-time counterpart of the Stochastic Gradient Descent (SGD) algorithm applied to two-layer overparameterized neural networks, as the number or neurons (i.e., the size of the hidden layer) $N \to \plusinfty$. Following a probabilistic approach, we show `propagation of chaos' for the particle system defined by this continuous-time dynamics under different scenarios, indicating that the statistical interaction between the particles asymptotically vanishes. In particular, we establish quantitative convergence with respect to $N$ of any particle to a solution of a mean-field McKean-Vlasov equation in the metric space endowed with the Wasserstein distance. In comparison to previous works on the subject, we consider settings in which the sequence of stepsizes in SGD can potentially depend on the number of neurons and the iterations. We then identify two regimes under which different mean-field limits are obtained, one of them corresponding to an implicitly regularized version of the minimization problem at hand. We perform various experiments on real datasets to validate our theoretical results, assessing the existence of these two regimes on classification problems and illustrating our convergence results.