We study dropout in two-layer neural networks with rectified linear unit (ReLU) activations. Under mild overparametrization and assuming that the limiting kernel can separate the data distribution with a positive margin, we show that the dropout training with logistic loss achieves $\epsilon$-suboptimality in the test error in $O(1/\epsilon)$ iterations.

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function (which maps input vectors to output vectors) follows the so-called kernel gradient associated with a new object, which we call the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

Neural Information Processing Systems http://nips.cc/

We study dropout in two-layer neural networks with rectified linear unit (ReLU) activations. Under mild overparametrization and assuming that the limiting kernel can separate the data distribution with a positive margin, we show that the dropout training with logistic loss achieves \epsilon -suboptimality in the test error in O(1/\epsilon) iterations.

The authors prove that networks of infinite width trained with SGD and (infinitely) small step size evolve according to a differential equation, the solution of which depends only on the covariance kernel of the data and, in the case of L2 regression, on the eigenspectrum of the Kernel. I believe this is a breakthrough result in the field of neural network theory. It elevates the analysis of infinitely wide networks from the study of the static initial function to closely predicting the entire training path. There are a plethora of powerful consequences about infinitely wide, fully-connected networks: - They cannot learn information not contained in the covariance matrix - Change to latent representation and parameters tends to zero as width goes to infinity. Therefore choosing nonlinearities in all layers reduces to choosing a single 1d function.

For modern artificial intelligence (AI) applications such as large language models (LLMs), the training paradigm has recently shifted to pre-training followed by fine-tuning. Furthermore, owing to dwindling open repositories of data and thanks to efforts to democratize access to AI models, pre-training is expected to increasingly migrate from the current centralized deployments to federated learning (FL) implementations. Meta-learning provides a general framework in which pre-training and fine-tuning can be formalized. Meta-learning-based personalized FL (meta-pFL) moves beyond basic personalization by targeting generalization to new agents and tasks. This paper studies the generalization performance of meta-pFL for a wireless setting in which the agents participating in the pre-training phase, i.e., meta-learning, are connected via a shared wireless channel to the server. Adopting over-the-air computing, we study the trade-off between generalization to new agents and tasks, on the one hand, and convergence, on the other hand. The trade-off arises from the fact that channel impairments may enhance generalization, while degrading convergence. Extensive numerical results validate the theory.

We introduce and study methods of inserting synaptic noise into dynamically-driven recurrent neural networks and show that ap(cid:173) plying a controlled amount of noise during training may improve convergence and generalization. In addition, we analyze the effects of each noise parameter (additive vs. multiplicative, cumulative vs. non-cumulative, per time step vs. per string) and predict that best overall performance can be achieved by injecting additive noise at each time step. Extensive simulations on learning the dual parity grammar from temporal strings substantiate these predictions.

Convergence and generalization are two crucial aspects of performance in neural networks. When analyzed separately, these properties may lead to contradictory results. Optimizing a convergence rate yields fast training, but does not guarantee the best generalization error. To avoid the conflict, recent studies suggest adopting a moderately large step size for optimizers, but the added value on the performance remains unclear. We propose the LIGHT function with the four configurations which regulate explicitly an improvement in convergence and generalization on testing. This contribution allows to: 1) improve both convergence and generalization of neural networks with no need to guarantee their stability; 2) build more reliable and explainable network architectures with no need for overparameterization. We refer to it as "painless" step size adaptation.

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function (which maps input vectors to output vectors) follows the so-called kernel gradient associated with a new object, which we call the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space.

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function (which maps input vectors to output vectors) follows the so-called kernel gradient associated with a new object, which we call the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.