arXiv subjects

Zhi-Qin John Xu

Publications and source records attributed to Zhi-Qin John Xu.

85 records · Page 5Linked to original sources

arXiv2020

On the exact computation of linear frequency principle dynamics and its generalization

Recent works show an intriguing phenomenon of Frequency Principle (F-Principle) that deep neural networks (DNNs) fit the target function from low to high frequency during the training, which provides insight into the training and generalization behavior of DNNs in complex tasks. In this paper, through analysis of an infinite-width two-layer NN in the neural tangent kernel (NTK) regime, we derive the exact differential equation, namely Linear Frequency-Principle (LFP) model, governing the evolution of NN output function in the frequency domain during the training. Our exact computation applies for general activation functions with no assumption on size and distribution of training data. This LFP model unravels that higher frequencies evolve polynomially or exponentially slower than lower frequencies depending on the smoothness/regularity of the activation function. We further bridge the gap between training dynamics and generalization by proving that LFP model implicitly minimizes a Frequency-Principle norm (FP-norm) of the learned function, by which higher frequencies are more severely penalized depending on the inverse of their evolution rate. Finally, we derive an \textit{a priori} generalization error bound controlled by the FP-norm of the target function, which provides a theoretical justification for the empirical results that DNNs often generalize well for low frequency functions.

cs.LG↗

arXiv2020

Phase diagram for two-layer ReLU neural networks at infinite-width limit

How neural network behaves during the training over different choices of hyperparameters is an important question in the study of neural networks. In this work, inspired by the phase diagram in statistical mechanics, we draw the phase diagram for the two-layer ReLU neural network at the infinite-width limit for a complete characterization of its dynamical regimes and their dependence on hyperparameters related to initialization. Through both experimental and theoretical approaches, we identify three regimes in the phase diagram, i.e., linear regime, critical regime and condensed regime, based on the relative change of input weights as the width approaches infinity, which tends to $0$, $O(1)$ and $+\infty$, respectively. In the linear regime, NN training dynamics is approximately linear similar to a random feature model with an exponential loss decay. In the condensed regime, we demonstrate through experiments that active neurons are condensed at several discrete orientations. The critical regime serves as the boundary between above two regimes, which exhibits an intermediate nonlinear behavior with the mean-field model as a typical example. Overall, our phase diagram for the two-layer ReLU NN serves as a map for the future studies and is a first step towards a more systematical investigation of the training behavior and the implicit regularization of NNs of different structures.

cs.LG↗

arXiv2020

Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains

In this paper, we propose multi-scale deep neural networks (MscaleDNNs) using the idea of radial scaling in frequency domain and activation functions with compact support. The radial scaling converts the problem of approximation of high frequency contents of PDEs' solutions to a problem of learning about lower frequency functions, and the compact support activation functions facilitate the separation of frequency contents of the target function to be approximated by corresponding DNNs. As a result, the MscaleDNNs achieve fast uniform convergence over multiple scales. The proposed MscaleDNNs are shown to be superior to traditional fully connected DNNs and be an effective mesh-less numerical method for Poisson-Boltzmann equations with ample frequency contents over complex and singular domains.

physics.comp-ph↗

arXiv2020

A regularized deep matrix factorized model of matrix completion for image restoration

It has been an important approach of using matrix completion to perform image restoration. Most previous works on matrix completion focus on the low-rank property by imposing explicit constraints on the recovered matrix, such as the constraint of the nuclear norm or limiting the dimension of the matrix factorization component. Recently, theoretical works suggest that deep linear neural network has an implicit bias towards low rank on matrix completion. However, low rank is not adequate to reflect the intrinsic characteristics of a natural image. Thus, algorithms with only the constraint of low rank are insufficient to perform image restoration well. In this work, we propose a Regularized Deep Matrix Factorized (RDMF) model for image restoration, which utilizes the implicit bias of the low rank of deep neural networks and the explicit bias of total variation. We demonstrate the effectiveness of our RDMF model with extensive experiments, in which our method surpasses the state of art models in common examples, especially for the restoration from very few observations. Our work sheds light on a more general framework for solving other inverse problems by combining the implicit bias of deep learning with explicit regularization.

cs.LG↗

arXiv2020

A type of generalization error induced by initialization in deep neural networks

How initialization and loss function affect the learning of a deep neural network (DNN), specifically its generalization error, is an important problem in practice. In this work, by exploiting the linearity of DNN training dynamics in the NTK regime \citep{jacot2018neural,lee2019wide}, we provide an explicit and quantitative answer to this problem. Focusing on regression problem, we prove that, in the NTK regime, for any loss in a general class of functions, the DNN finds the same \emph{global} minima---the one that is nearest to the initial value in the parameter space, or equivalently, the one that is closest to the initial DNN output in the corresponding reproducing kernel Hilbert space. Using these optimization problems, we quantify the impact of initial output and prove that a random non-zero one increases the generalization error. We further propose an antisymmetrical initialization (ASI) trick that eliminates this type of error and accelerates the training. To understand whether the above results hold in general, we also perform experiments for DNNs in the non-NTK regime, which demonstrate the effectiveness of our theoretical results and the ASI trick in a qualitative sense. Overall, our work serves as a baseline for the further investigation of the impact of initialization and loss function on the generalization of DNNs, which can potentially guide and improve the training of DNNs in practice.

cs.LG↗

arXiv2020

A priori generalization error for two-layer ReLU neural network through minimum norm solution

We focus on estimating \emph{a priori} generalization error of two-layer ReLU neural networks (NNs) trained by mean squared error, which only depends on initial parameters and the target function, through the following research line. We first estimate \emph{a priori} generalization error of finite-width two-layer ReLU NN with constraint of minimal norm solution, which is proved by \cite{zhang2019type} to be an equivalent solution of a linearized (w.r.t. parameter) finite-width two-layer NN. As the width goes to infinity, the linearized NN converges to the NN in Neural Tangent Kernel (NTK) regime \citep{jacot2018neural}. Thus, we can derive the \emph{a priori} generalization error of two-layer ReLU NN in NTK regime. The distance between NN in a NTK regime and a finite-width NN with gradient training is estimated by \cite{arora2019exact}. Based on the results in \cite{arora2019exact}, our work proves an \emph{a priori} generalization error bound of two-layer ReLU NNs. This estimate uses the intrinsic implicit bias of the minimum norm solution without requiring extra regularity in the loss function. This \emph{a priori} estimate also implies that NN does not suffer from curse of dimensionality, and a small generalization error can be achieved without requiring exponentially large number of neurons. In addition the research line proposed in this paper can also be used to study other properties of the finite-width network, such as the posterior generalization error.

cs.LG↗

arXiv2019

Training behavior of deep neural network in frequency domain

Why deep neural networks (DNNs) capable of overfitting often generalize well in practice is a mystery [#zhang2016understanding]. To find a potential mechanism, we focus on the study of implicit biases underlying the training process of DNNs. In this work, for both real and synthetic datasets, we empirically find that a DNN with common settings first quickly captures the dominant low-frequency components, and then relatively slowly captures the high-frequency ones. We call this phenomenon Frequency Principle (F-Principle). The F-Principle can be observed over DNNs of various structures, activation functions, and training algorithms in our experiments. We also illustrate how the F-Principle help understand the effect of early-stopping as well as the generalization of DNNs. This F-Principle potentially provides insights into a general principle underlying DNN optimization and generalization.

cs.LG↗

arXiv2019

Multi-scale Deep Neural Networks for Solving High Dimensional PDEs

In this paper, we propose the idea of radial scaling in frequency domain and activation functions with compact support to produce a multi-scale DNN (MscaleDNN), which will have the multi-scale capability in approximating high frequency and high dimensional functions and speeding up the solution of high dimensional PDEs. Numerical results on high dimensional function fitting and solutions of high dimensional PDEs, using loss functions with either Ritz energy or least squared PDE residuals, have validated the increased power of multi-scale resolution and high frequency capturing of the proposed MscaleDNN.

cs.LG↗

arXiv2019

Explicitizing an Implicit Bias of the Frequency Principle in Two-layer Neural Networks

It remains a puzzle that why deep neural networks (DNNs), with more parameters than samples, often generalize well. An attempt of understanding this puzzle is to discover implicit biases underlying the training process of DNNs, such as the Frequency Principle (F-Principle), i.e., DNNs often fit target functions from low to high frequencies. Inspired by the F-Principle, we propose an effective model of linear F-Principle (LFP) dynamics which accurately predicts the learning results of two-layer ReLU neural networks (NNs) of large widths. This LFP dynamics is rationalized by a linearized mean field residual dynamics of NNs. Importantly, the long-time limit solution of this LFP dynamics is equivalent to the solution of a constrained optimization problem explicitly minimizing an FP-norm, in which higher frequencies of feasible solutions are more heavily penalized. Using this optimization formulation, an a priori estimate of the generalization error bound is provided, revealing that a higher FP-norm of the target function increases the generalization error. Overall, by explicitizing the implicit bias of the F-Principle as an explicit penalty for two-layer NNs, our work makes a step towards a quantitative understanding of the learning and generalization of general DNNs.

cs.LG↗

arXiv2018

Frequency Principle in Deep Learning with General Loss Functions and Its Potential Application

Previous studies have shown that deep neural networks (DNNs) with common settings often capture target functions from low to high frequency, which is called Frequency Principle (F-Principle). It has also been shown that F-Principle can provide an understanding to the often observed good generalization ability of DNNs. However, previous studies focused on the loss function of mean square error, while various loss functions are used in practice. In this work, we show that the F-Principle holds for a general loss function (e.g., mean square error, cross entropy, etc.). In addition, DNN's F-Principle may be applied to develop numerical schemes for solving various problems which would benefit from a fast converging of low frequency. As an example of the potential usage of F-Principle, we apply DNN in solving differential equations, in which conventional methods (e.g., Jacobi method) is usually slow in solving problems due to the convergence from high to low frequency.

cs.LG↗

arXiv2018

Swift Two-sample Test on High-dimensional Neural Spiking Data

To understand how neural networks process information, it is important to investigate how neural network dynamics varies with respect to different stimuli. One challenging task is to design efficient statistical approaches to analyze multiple spike train data obtained from a short recording time. Based on the development of high-dimensional statistical methods, it is able to deal with data whose dimension is much larger than the sample size. However, these methods often require statistically independent samples to start with, while neural data are correlated over consecutive sampling time bins. We develop an approach to pretreat neural data to become independent samples over time by transferring the correlation of dynamics for each neuron in different sampling time bins into the correlation of dynamics among different dimensions within each sampling time bin. We verify the method using simulation data generated from Integrate-and-fire neuron network models and a large-scale network model of primary visual cortex within a short time, e.g., a few seconds. Our method may offer experimenters to use the advantage of the development of statistical methods to analyze high-dimensional neural data.

q-bio.NC↗

arXiv2018

Maximum Entropy Principle Analysis in Network Systems with Short-time Recordings

In many realistic systems, maximum entropy principle (MEP) analysis provides an effective characterization of the probability distribution of network states. However, to implement the MEP analysis, a sufficiently long-time data recording in general is often required, e.g., hours of spiking recordings of neurons in neuronal networks. The issue of whether the MEP analysis can be successfully applied to network systems with data from short recordings has yet to be fully addressed. In this work, we investigate relationships underlying the probability distributions, moments, and effective interactions in the MEP analysis and then show that, with short recordings of network dynamics, the MEP analysis can be applied to reconstructing probability distributions of network states under the condition of asynchronous activity of nodes in the network. Using spike trains obtained from both Hodgkin-Huxley neuronal networks and electrophysiological experiments, we verify our results and demonstrate that MEP analysis provides a tool to investigate the neuronal population coding properties, even for short recordings.

physics.bio-ph↗

arXiv2018

Dynamical and Coupling Structure of Pulse-Coupled Networks in Maximum Entropy Analysis

Maximum entropy principle (MEP) analysis with few non-zero effective interactions successfully characterizes the distribution of dynamical states of pulse-coupled networks in many experiments, e.g., in neuroscience. To better understand the underlying mechanism, we found a relation between the dynamical structure, i.e., effective interactions in MEP analysis, and the coupling structure of pulse-coupled network to understand how a sparse coupling structure could lead to a sparse coding by effective interactions. This relation quantitatively displays how the dynamical structure is closely related to the coupling structure.

physics.bio-ph↗