In this paper, we follow Eftekhari's work to give a non-local convergence analysis of deep linear networks. Specifically, we consider optimizing deep linear networks which have a layer with one neuron under quadratic loss. We describe the convergent point of trajectories with arbitrary starting point under gradient flow, including the paths which converge to one of the saddle points or the original point. We also show specific convergence rates of trajectories that converge to the global minimizer by stages. To achieve these results, this paper mainly extends the machinery in Eftekhari's work to provably identify the rank-stable set and the global minimizer convergent set. We also give specific examples to show the necessity of our definitions. Crucially, as far as we know, our results appear to be the first to give a non-local global analysis of linear neural networks from arbitrary initialized points, rather than the lazy training regime which has dominated the literature of neural networks, and restricted benign initialization in Eftekhari's work. We also note that extending our results to general linear networks without one hidden neuron assumption remains a challenging open problem.
Recent studies revealed complex convergence dynamics in gradient-based methods, which has been little understood so far. Changing the step size to balance between high convergence rate and small generalization error may not be sufficient: maximizing the test accuracy usually requires a larger learning rate than minimizing the training loss. To explore the dynamic bounds of convergence rate, this study introduces \textit{differential capability} into an optimization process, which measures whether the test accuracy increases as fast as a model approaches the decision boundary in a classificatio…
Duke University（杜克大学）；Princeton University（普林斯顿大学）
While over-parameterization is widely believed to be crucial for the success of optimization for the neural networks, most existing theories on over-parameterization do not fully explain the reason -- they either work in the Neural Tangent Kernel regime where neurons don't move much, or require an enormous number of neurons. In practice, when the data is generated using a teacher neural network, even mildly over-parameterized neural networks can achieve 0 loss and recover the directions of teacher neurons. In this paper we develop a local convergence theory for mildly over-parameterized two-la…
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Princeton University（普林斯顿大学）；Center for Data Science, New York University（纽约大学数据科学中心）
Neural networks with a large number of parameters admit a mean-field description, which has recently served as a theoretical explanation for the favorable training properties of "overparameterized" models. In this regime, gradient descent obeys a deterministic partial differential equation (PDE) that converges to a globally optimal solution for networks with a single hidden layer under appropriate assumptions. In this work, we propose a non-local mass transport dynamics that leads to a modified PDE with the same minimizer. We implement this non-local dynamics as a stochastic neuronal birth-dea…
With the motive of training all the parameters of a neural network, we study why and when one can achieve this by iteratively creating, training, and combining randomly selected subnetworks. Such scenarios have either implicitly or explicitly emerged in the recent literature: see e.g., the Dropout family of regularization techniques, or some distributed ML training protocols that reduce communication/computation complexities, such as the Independent Subnet Training protocol. While these methods are studied empirically and utilized in practice, they often enjoy partial or no theoretical support…
Understanding implicit bias of gradient descent for generalization capability of ReLU networks has been an important research topic in machine learning research. Unfortunately, even for a single ReLU neuron trained with the square loss, it was recently shown impossible to characterize the implicit regularization in terms of a norm of model parameters (Vardi & Shamir, 2021). In order to close the gap toward understanding intriguing generalization behavior of ReLU networks, here we examine the gradient flow dynamics in the parameter space when training single-neuron ReLU networks. Specifically, …
To understand learning the dynamics of deep ReLU networks, we investigate the dynamic system of gradient flow $w(t)$ by decomposing it to magnitude $w(t)$ and angle $ϕ(t):= π- θ(t) $ components. In particular, for multi-layer single ReLU neurons with spherically symmetric data distribution and the square loss function, we provide upper and lower bounds for magnitude and angle components to describe the dynamics of gradient flow. Using the obtained bounds, we conclude that small scale initialization induces slow convergence speed for deep single ReLU neurons. Finally, by exploiting the relation…
University of Münster（明斯特大学）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））
In this article we study the stochastic gradient descent (SGD) optimization method in the training of fully-connected feedforward artificial neural networks with ReLU activation. The main result of this work proves that the risk of the SGD process converges to zero if the target function under consideration is constant. In the established convergence result the considered artificial neural networks consist of one input layer, one hidden layer, and one output layer (with $d \in \mathbb{N}$ neurons on the input layer, $H \in \mathbb{N}$ neurons on the hidden layer, and one neuron on the output l…
We theoretically study the fundamental problem of learning a single neuron with a bias term ($\mathbf{x} \mapsto σ(<\mathbf{w},\mathbf{x}> + b)$) in the realizable setting with the ReLU activation, using gradient descent. Perhaps surprisingly, we show that this is a significantly different and more challenging problem than the bias-less case (which was the focus of previous works on single neurons), both in terms of the optimization geometry as well as the ability of gradient methods to succeed in some scenarios. We provide a detailed study of this problem, characterizing the critical points o…
Yale University（耶鲁大学）；Georgia Institute of Technology（佐治亚理工学院）
Over-parametrization has become a popular technique in deep learning. It is observed that by over-parametrization, a larger neural network needs a fewer training iterations than a smaller one to achieve a certain level of performance -- namely, over-parametrization leads to acceleration in optimization. However, despite that over-parametrization is widely used nowadays, little theory is available to explain the acceleration due to over-parametrization. In this paper, we propose understanding it by studying a simple problem first. Specifically, we consider the setting that there is a single tea…
We consider optimizing two-layer neural networks in the mean-field regime where the learning dynamics of network weights can be approximated by the evolution in the space of probability measures over the weight parameters associated with the neurons. The mean-field regime is a theoretically attractive alternative to the NTK (lazy training) regime which is only restricted locally in the so-called neural tangent kernel space around specialized initializations. Several prior works (\cite{chizat2018global, mei2018mean}) establish the asymptotic global optimality of the mean-field regime, but it is…
Stochastic gradient descent with backpropagation is the workhorse of artificial neural networks. It has long been recognized that backpropagation fails to be a biologically plausible algorithm. Fundamentally, it is a non-local procedure -- updating one neuron's synaptic weights requires knowledge of synaptic weights or receptive fields of downstream neurons. This limits the use of artificial neural networks as a tool for understanding the biological principles of information processing in the brain. Lillicrap et al. (2016) propose a more biologically plausible "feedback alignment" algorithm th…
Graph neural networks (GNNs) are highly effective on a variety of graph-related tasks; however, they lack interpretability and transparency. Current explainability approaches are typically local and treat GNNs as black-boxes. They do not look inside the model, inhibiting human trust in the model and explanations. Motivated by the ability of neurons to detect high-level semantic concepts in vision models, we perform a novel analysis on the behaviour of individual GNN neurons to answer questions about GNN interpretability, and propose new metrics for evaluating the interpretability of GNN neuron…
Universidade Federal Rural de Pernambuco（伯南布哥联邦农村大学）；Harvard University（哈佛大学）
The activation function plays a fundamental role in the artificial neural network learning process. However, there is no obvious choice or procedure to determine the best activation function, which depends on the problem. This study proposes a new artificial neuron, named global-local neuron, with a trainable activation function composed of two components, a global and a local. The global component term used here is relative to a mathematical function to describe a general feature present in all problem domain. The local component is a function that can represent a localized behavior, like a t…
NeuronSeek: On Stability and Expressivity of Task-driven Neurons
Frontier of Artificial Networks (FAN) Lab, Department of Data Science, City University of Hong Kong（前沿人工智能网络实验室，数据科学系，香港城市大学）；Department of Industrial and Systems Engineering, The Hong Kong Polytechnic University（工业与系统工程系，香港理工大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所高级智能项目中心）；Department of Applied Mathematics, The Hong Kong Polytechnic University（应用数学系，香港理工大学）
Neural network models generally involve two important components, i.e., network architecture and neuron model. Although there are abundant studies about network architectures, only a few neuron models have been developed, such as the MP neuron model developed in 1943 and the spiking neuron model developed in the 1950s. Recently, a new bio-plausible neuron model, Flexible Transmitter (FT) model, has been proposed. It exhibits promising behaviors, particularly on temporal-spatial signals, even when simply embedded into the common feedforward network architecture. This paper attempts to understan…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
Solving for the frequency-domain scattered wavefield via physics-informed neural network (PINN) has great potential in seismic modeling and inversion. However, when dealing with high-frequency wavefields, its accuracy and training cost limits its applications. Thus, we propose a novel implementation of PINN using frequency upscaling and neuron splitting, which allows the neural network model to grow in size as we increase the frequency while leveraging the information from the pre-trained model for lower-frequency wavefields, resulting in fast convergence to high-accuracy solutions. Numerical …
University of Electronic Science and Technology of China（电子科技大学）；Peking University（北京大学）；Microsoft Research Asia（微软亚洲研究院）
Initialization plays a critical role in the training of deep neural networks (DNN). Existing initialization strategies mainly focus on stabilizing the training process to mitigate gradient vanish/explosion problems. However, these initialization methods are lacking in consideration about how to enhance generalization ability. The Information Bottleneck (IB) theory is a well-known understanding framework to provide an explanation about the generalization of DNN. Guided by the insights provided by IB theory, we design two criteria for better initializing DNN. And we further design a neuron campa…
We examine one-hidden-layer neural networks with random weights. It is well-known that in the limit of infinitely many neurons they simplify to Gaussian processes. For networks with a polynomial activation, we demonstrate that the rate of this convergence in 2-Wasserstein metric is $O(n^{-\frac{1}{2}})$, where $n$ is the number of hidden neurons. We suspect this rate is asymptotically sharp. We improve the known convergence rate for other activations, to power-law in $n$ for ReLU and inverse-square-root up to logarithmic factors for erf. We explore the interplay between spherical harmonics, St…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Southern California（南加利福尼亚大学）
One of the main difficulties in analyzing neural networks is the non-convexity of the loss function which may have many bad local minima. In this paper, we study the landscape of neural networks for binary classification tasks. Under mild assumptions, we prove that after adding one special neuron with a skip connection to the output, or one special neuron per layer, every local minimum is a global minimum.
Soft Foundry Institute, Seoul National University（首尔国立大学软铸造研究所）
The gradient information of multilayer perceptron with a linear neuron is modified with functional derivative for the global minimum search benchmarking problems. From this approach, we show that the landscape of the gradient derived from given continuous function using functional derivative can be the MLP-like form with ax+b neurons. In this extent, the suggested algorithm improves the availability of the optimization process to deal all the parameters in the problem set simultaneously. The functionality of this method could be improved through intentionally designed convex function with Kull…
Fuzhou University（福州大学）；East China Normal University（华东师范大学）
The opacity of neural networks leads their vulnerability to backdoor attacks, where hidden attention of infected neurons is triggered to override normal predictions to the attacker-chosen ones. In this paper, we propose a novel backdoor defense method to mark and purify the infected neurons in the backdoored neural networks. Specifically, we first define a new metric, called benign salience. By combining the first-order gradient to retain the connections between neurons, benign salience can identify the infected neurons with higher accuracy than the commonly used metric in backdoor defense. Th…
The University of Iowa（爱荷华大学）；University of Central Florida（中佛罗里达大学）
Dropout has been witnessed with great success in training deep neural networks by independently zeroing out the outputs of neurons at random. It has also received a surge of interest for shallow learning, e.g., logistic regression. However, the independent sampling for dropout could be suboptimal for the sake of convergence. In this paper, we propose to use multinomial sampling for dropout, i.e., sampling features or neurons according to a multinomial distribution with different probabilities for different features/neurons. To exhibit the optimal dropout probabilities, we analyze the shallow l…
Croatian Institute for Brain Research（克罗地亚脑研究所）；University of Zagreb, School of Medicine（萨格勒布大学医学院）；Ruđer Bošković Institute（鲁杰尔·博什科维奇研究所）；Faculty of Electrical Engineering and Computing（电气工程与计算学院）；University of Zagreb（萨格勒布大学）
In this paper, we present a novel method for analysis and segmentation of laminar structure of the cortex based on tissue characteristics whose change across the gray matter underlies distinctive between cortical layers. We develop and analyze features of individual neurons to investigate changes in cytoarchitectonic differentiation and present a novel high-performance, automated framework for neuron-level histological image analysis. Local tissue and cell descriptors such as density, neuron size and other measures are used for development of more complex neuron features used in machine learni…
We derive global convergence bounds for the Frank Wolfe algorithm when training one hidden layer neural networks. When using the ReLU activation function, and under tractable preconditioning assumptions on the sample data set, the linear minimization oracle used to incrementally form the solution can be solved explicitly as a second order cone program. The classical Frank Wolfe algorithm then converges with rate $O(1/T)$ where $T$ is both the number of neurons and the number of calls to the oracle.
Recent results in nonparametric regression show that deep learning, i.e., neural network estimates with many hidden layers, are able to circumvent the so-called curse of dimensionality in case that suitable restrictions on the structure of the regression function hold. One key feature of the neural networks used in these results is that their network architecture has a further constraint, namely the network sparsity. In this paper we show that we can get similar results also for least squares estimates based on simple fully connected neural networks with ReLU activation functions. Here either …
Neuroscientific theory suggests that dopaminergic neurons broadcast global reward prediction errors to large areas of the brain influencing the synaptic plasticity of the neurons in those regions. We build on this theory to propose a multi-agent learning framework with spiking neurons in the generalized linear model (GLM) formulation as agents, to solve reinforcement learning (RL) tasks. We show that a network of GLM spiking agents connected in a hierarchical fashion, where each spiking agent modulates its firing policy based on local information and a global prediction error, can learn comple…
