The mean field theory of multilayer neural networks centers around a particular infinite-width scaling, in which the learning dynamics is shown to be closely tracked by the mean field limit. A random fluctuation around this infinite-width limit is expected from a large-width expansion to the next order. This fluctuation has been studied only in the case of shallow networks, where previous works employ heavily technical notions or additional formulation ideas amenable only to that case. Treatment of the multilayer case has been missing, with the chief difficulty in finding a formulation that must capture the stochastic dependency across not only time but also depth. In this work, we initiate the study of the fluctuation in the case of multilayer networks, at any network depth.

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

Multilayer Neural Networks (MNNs) are commonly trained using gradient descent-based methods, such as BackPropagation (BP). Inference in probabilistic graphical models is often done using variational Bayes methods, such as Expectation Propagation (EP). We show how an EP based approach can also be used to train deterministic MNNs. Specifically, we approximate the posterior of the weights given the data using a "mean-field" factorized distribution, in an online setting. Using online EP and the central limit theorem we find an analytical approximation to the Bayes update of this posterior, as well as the resulting Bayes estimates of the weights and outputs. Despite a different origin, the resulting algorithm, Expectation BackPropagation (EBP), is very similar to BP in form and efficiency. However, it has several additional advantages: (1) Training is parameter-free, given initial conditions (prior) and the MNN architecture. This is useful for large-scale problems, where parameter tuning is a major challenge.

We propose a first near complete (that will make explicit sense in the main text) nonasymptotic generalization theory for multilayer neural networks with arbitrary Lipschitz activations and general Lipschitz loss functions (with some very mild conditions). In particular, it doens't require the boundness of loss function, as commonly assumed in the literature. Our theory goes beyond the bias-variance tradeoff, aligned with phenomenon typically encountered in deep learning. It is therefore sharp different with other existing nonasymptotic generalization error bounds for neural networks. More explicitly, we propose an explicit generalization error upper bound for multilayer neural networks with arbitrary Lipschitz activations $\sigma$ with $\sigma(0)=0$ and broad enough Lipschitz loss functions, without requiring either the width, depth or other hyperparameters of the neural network approaching infinity, a specific neural network architect (e.g. sparsity, boundness of some norms), a particular activation function, a particular optimization algorithm or boundness of the loss function, and with taking the approximation error into consideration. General Lipschitz activation can also be accommodated into our framework. A feature of our theory is that it also considers approximation errors. Furthermore, we show the near minimax optimality of our theory for multilayer ReLU networks for regression problems. Notably, our upper bound exhibits the famous double descent phenomenon for such networks, which is the most distinguished characteristic compared with other existing results. This work emphasizes a view that many classical results should be improved to embrace the unintuitive characteristics of deep learning to get a better understanding of it.

Multilayer Neural Networks (MNNs) are commonly trained using gradient descent-based methods, such as BackPropagation (BP). Inference in probabilistic graphical models is often done using variational Bayes methods, such as Expectation Propagation (EP). We show how an EP based approach can also be used to train deterministic MNNs. Specifically, we approximate the posterior of the weights given the data using a "mean-field" factorized distribution, in an online setting. Using online EP and the central limit theorem we find an analytical approximation to the Bayes update of this posterior, as well as the resulting Bayes estimates of the weights and outputs. Despite a different origin, the resulting algorithm, Expectation BackPropagation (EBP), is very similar to BP in form and efficiency. However, it has several additional advantages: (1) Training is parameter-free, given initial conditions (prior) and the MNN architecture. This is useful for large-scale problems, where parameter tuning is a major challenge.

The mean field theory of multilayer neural networks centers around a particular infinite-width scaling, in which the learning dynamics is shown to be closely tracked by the mean field limit. A random fluctuation around this infinite-width limit is expected from a large-width expansion to the next order. This fluctuation has been studied only in the case of shallow networks, where previous works employ heavily technical notions or additional formulation ideas amenable only to that case. Treatment of the multilayer case has been missing, with the chief difficulty in finding a formulation that must capture the stochastic dependency across not only time but also depth.In this work, we initiate the study of the fluctuation in the case of multilayer networks, at any network depth. Leveraging on the neuronal embedding framework recently introduced by Nguyen and Pham, we systematically derive a system of dynamical equations, called the second-order mean field limit, that captures the limiting fluctuation distribution.

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

The accurate prediction of molecular energetics in chemical compound space is a crucial ingredient for rational compound design. The inherently graph-like, non-vectorial nature of molecular data gives rise to a unique and difficult machine learning problem. In this paper, we adopt a learning-from-scratch approach where quantum-mechanical molecular energies are predicted directly from the raw molecular geometry. The study suggests a benefit from setting flexible priors and enforcing invariance stochastically rather than structurally. Our results improve the state-of-the-art by a factor of almost three, bringing statistical methods one step closer to chemical accuracy.

Imitation learning enables robots to learn and replicate human behavior from training data. Recent advances in machine learning enable end-to-end learning approaches that directly process high-dimensional observation data, such as images. However, these approaches face a critical challenge when processing data from multiple modalities, inadvertently ignoring data with a lower correlation to the desired output, especially when using short sampling periods. This paper presents a useful method to address this challenge, which amplifies the influence of data with a relatively low correlation to the output by inputting the data into each neural network layer. The proposed approach effectively incorporates diverse data sources into the learning process. Through experiments using a simple pick-and-place operation with raw images and joint information as input, significant improvements in success rates are demonstrated even when dealing with data from short sampling periods.

We consider the problem of on-line gradient descent learning for general two-layer neural networks. An analytic solution is pre(cid:173) sented and used to investigate the role of the learning rate in con(cid:173) trolling the evolution and convergence of the learning process. Learning in layered neural networks refers to the modification of internal parameters {J} which specify the strength of the interneuron couplings, so as to bring the map fJ implemented by the network as close as possible to a desired map 1. The degree of success is monitored through the generalization error, a measure of the dissimilarity between fJ and 1. Consider maps from an N-dimensional input space e onto a scalar (, as arise in the formulation of classification and regression tasks.