Jinchao Xu
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Jinchao Xu has a total of 10 co-authors. Here are the top co-authors by number of shared publications:
- Jonathan W. Siegel
Department of Mathematics Texas A&M University College 77843 Station TX
9 shared publications Jianhong Chen
2 shared publicationsHuang Huang
School of Mathematical Sciences Peking University Beijing China
1 shared publicationJuncai He
Department of Mathematics Pennsylvania State University 16802 University Park PA USA
1 shared publicationLian Zhang
Department of Mathematics Pennsylvania State University 16802 University Park PA USA
1 shared publicationLiang Zhao
Academy of Mathematics and Systems Science State Key Laboratory of Scientific and Engineering Computing Chinese Academy of Sciences University of Chinese Academy of Sciences 100190 Beijing China
1 shared publication
Latest Publications
- Top Cited Papers
- 22 February 2023
- Published by Springer Nature in Constructive Approximation
- Vol. 57 (3) , 1109-1132
Characterization of the Variation Spaces Corresponding to Shallow Neural Networks
- 22 February 2023
- Published by Springer Nature in Constructive Approximation
- Vol. 57 (3) , 1109-1132
Top Cited PapersCharacterization of the Variation Spaces Corresponding to Shallow Neural Networks
- Top Cited Papers
- 9 November 2022
- Published by Springer Nature in Foundations of Computational Mathematics
- Vol. 24 (2) , 481-537
Sharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks
- 9 November 2022
- Published by Springer Nature in Foundations of Computational Mathematics
- Vol. 24 (2) , 481-537
Top Cited PapersSharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks
- 19 July 2022
- Published by Springer Nature in Research in the Mathematical Sciences
- Vol. 9 (3) , 1-21
Uniform approximation rates and metric entropy of shallow neural networks
- 19 July 2022
- Published by Springer Nature in Research in the Mathematical Sciences
- Vol. 9 (3) , 1-21
Uniform approximation rates and metric entropy of shallow neural networks
- Open Access
Extended Regularized Dual Averaging Methods for Stochastic Optimization
We introduce a new algorithm, extended regularized dual averaging (XRDA), for solving regularized stochastic optimization problems, which generalizes the regularized dual averaging (RDA) method. The main novelty of the method is that it allows a flexible control of the backward step size. For instance, the backward step size used in RDA grows without bound, while for XRDA the backward step size can be kept bounded. We demonstrate experimentally that additional control over the backward step size can significantly improve the convergence rate of the algorithm while preserving desired properties of the iterates, such as sparsity. Theoretically, we show that the XRDA method achieves the same convergence rate as RDA for general convex objectives.
Open AccessExtended Regularized Dual Averaging Methods for Stochastic Optimization
We introduce a new algorithm, extended regularized dual averaging (XRDA), for solving regularized stochastic optimization problems, which generalizes the regularized dual averaging (RDA) method. The main novelty of the method is that it allows a flexible control of the backward step size. For instance, the backward step size used in RDA grows without bound, while for XRDA the backward step size can be kept bounded. We demonstrate experimentally that additional control over the backward step size can significantly improve the convergence rate of the algorithm while preserving desired properties of the iterates, such as sparsity. Theoretically, we show that the XRDA method achieves the same convergence rate as RDA for general convex objectives.
- Open Access
Characterization of the Variation Spaces Corresponding to Shallow Neural Networks
We study the variation space corresponding to a dictionary of functions in $L^2(\Omega)$ for a bounded domain $\Omega\subset \mathbb{R}^d$. Specifically, we compare the variation space, which is defined in terms of a convex hull with related notions based on integral representations. This allows us to show that three important notions relating to the approximation theory of shallow neural networks, the Barron space, the spectral Barron space, and the Radon BV space, are actually variation spaces with respect to certain natural dictionaries.Open AccessCharacterization of the Variation Spaces Corresponding to Shallow Neural Networks
We study the variation space corresponding to a dictionary of functions in $L^2(\Omega)$ for a bounded domain $\Omega\subset \mathbb{R}^d$. Specifically, we compare the variation space, which is defined in terms of a convex hull with related notions based on integral representations. This allows us to show that three important notions relating to the approximation theory of shallow neural networks, the Barron space, the spectral Barron space, and the Radon BV space, are actually variation spaces with respect to certain natural dictionaries. - Open Access
Optimal Convergence Rates for the Orthogonal Greedy Algorithm
We analyze the orthogonal greedy algorithm when applied to dictionaries $\mathbb{D}$ whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $\mathbb{D}$ decays at a rate of $O(n^{-\frac{1}{2}-\alpha})$ for $\alpha > 0$, then the orthogonal greedy algorithm converges at the same rate on the variation space of $\mathbb{D}$. This improves upon the well-known $O(n^{-\frac{1}{2}})$ convergence rate of the orthogonal greedy algorithm in many cases, most notably for dictionaries corresponding to shallow neural networks. These results hold under no additional assumptions on the dictionary beyond the decay rate of the entropy of its convex hull. In addition, they are robust to noise in the target function and can be extended to convergence rates on the interpolation spaces of the variation norm. We show empirically that the predicted rates are obtained for the dictionary corresponding to shallow neural networks with Heaviside activation function in two dimensions. Finally, we show that these improved rates are sharp and prove a negative result showing that the iterates generated by the orthogonal greedy algorithm cannot in general be bounded in the variation norm of $\mathbb{D}$.Open AccessOptimal Convergence Rates for the Orthogonal Greedy Algorithm
We analyze the orthogonal greedy algorithm when applied to dictionaries $\mathbb{D}$ whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $\mathbb{D}$ decays at a rate of $O(n^{-\frac{1}{2}-\alpha})$ for $\alpha > 0$, then the orthogonal greedy algorithm converges at the same rate on the variation space of $\mathbb{D}$. This improves upon the well-known $O(n^{-\frac{1}{2}})$ convergence rate of the orthogonal greedy algorithm in many cases, most notably for dictionaries corresponding to shallow neural networks. These results hold under no additional assumptions on the dictionary beyond the decay rate of the entropy of its convex hull. In addition, they are robust to noise in the target function and can be extended to convergence rates on the interpolation spaces of the variation norm. We show empirically that the predicted rates are obtained for the dictionary corresponding to shallow neural networks with Heaviside activation function in two dimensions. Finally, we show that these improved rates are sharp and prove a negative result showing that the iterates generated by the orthogonal greedy algorithm cannot in general be bounded in the variation norm of $\mathbb{D}$. - Open Access
High-Order Approximation Rates for Shallow Neural Networks with Cosine and ReLU$^k$ Activation Functions
We study the approximation properties of shallow neural networks with an activation function which is a power of the rectified linear unit. Specifically, we consider the dependence of the approximation rate on the dimension and the smoothness in the spectral Barron space of the underlying function $f$ to be approximated. We show that as the smoothness index $s$ of $f$ increases, shallow neural networks with ReLU$^k$ activation function obtain an improved approximation rate up to a best possible rate of $O(n^{-(k+1)}\log(n))$ in $L^2$, independent of the dimension $d$. The significance of this result is that the activation function ReLU$^k$ is fixed independent of the dimension, while for classical methods the degree of polynomial approximation or the smoothness of the wavelets used would have to increase in order to take advantage of the dimension dependent smoothness of $f$. In addition, we derive improved approximation rates for shallow neural networks with cosine activation function on the spectral Barron space. Finally, we prove lower bounds showing that the approximation rates attained are optimal under the given assumptions.Open AccessHigh-Order Approximation Rates for Shallow Neural Networks with Cosine and ReLU$^k$ Activation Functions
We study the approximation properties of shallow neural networks with an activation function which is a power of the rectified linear unit. Specifically, we consider the dependence of the approximation rate on the dimension and the smoothness in the spectral Barron space of the underlying function $f$ to be approximated. We show that as the smoothness index $s$ of $f$ increases, shallow neural networks with ReLU$^k$ activation function obtain an improved approximation rate up to a best possible rate of $O(n^{-(k+1)}\log(n))$ in $L^2$, independent of the dimension $d$. The significance of this result is that the activation function ReLU$^k$ is fixed independent of the dimension, while for classical methods the degree of polynomial approximation or the smoothness of the wavelets used would have to increase in order to take advantage of the dimension dependent smoothness of $f$. In addition, we derive improved approximation rates for shallow neural networks with cosine activation function on the spectral Barron space. Finally, we prove lower bounds showing that the approximation rates attained are optimal under the given assumptions. - Open Access
Training Sparse Neural Networks using Compressed Sensing
Pruning the weights of neural networks is an effective and widely-used technique for reducing model size and inference complexity. We develop and test a novel method based on compressed sensing which combines the pruning and training into a single step. Specifically, we utilize an adaptively weighted $\ell^1$ penalty on the weights during training, which we combine with a generalization of the regularized dual averaging (RDA) algorithm in order to train sparse neural networks. The adaptive weighting we introduce corresponds to a novel regularizer based on the logarithm of the absolute value of the weights. We perform a series of ablation studies demonstrating the improvement provided by the adaptive weighting and generalized RDA algorithm. Furthermore, numerical experiments on the CIFAR-10, CIFAR-100, and ImageNet datasets demonstrate that our method 1) trains sparser, more accurate networks than existing state-of-the-art methods; 2) can be used to train sparse networks from scratch, i.e. from a random initialization, as opposed to initializing with a well-trained base model; 3) acts as an effective regularizer, improving generalization accuracy.Open AccessTraining Sparse Neural Networks using Compressed Sensing
Pruning the weights of neural networks is an effective and widely-used technique for reducing model size and inference complexity. We develop and test a novel method based on compressed sensing which combines the pruning and training into a single step. Specifically, we utilize an adaptively weighted $\ell^1$ penalty on the weights during training, which we combine with a generalization of the regularized dual averaging (RDA) algorithm in order to train sparse neural networks. The adaptive weighting we introduce corresponds to a novel regularizer based on the logarithm of the absolute value of the weights. We perform a series of ablation studies demonstrating the improvement provided by the adaptive weighting and generalized RDA algorithm. Furthermore, numerical experiments on the CIFAR-10, CIFAR-100, and ImageNet datasets demonstrate that our method 1) trains sparser, more accurate networks than existing state-of-the-art methods; 2) can be used to train sparse networks from scratch, i.e. from a random initialization, as opposed to initializing with a well-trained base model; 3) acts as an effective regularizer, improving generalization accuracy. - Open Access
Approximation Rates for Neural Networks with General Activation Functions
We prove some new results concerning the approximation rate of neural networks with general activation functions. Our first result concerns the rate of approximation of a two layer neural network with a polynomially-decaying non-sigmoidal activation function. We extend the dimension independent approximation rates previously obtained to this new class of activation functions. Our second result gives a weaker, but still dimension independent, approximation rate for a larger class of activation functions, removing the polynomial decay assumption. This result applies to any bounded, integrable activation function. Finally, we show that a stratified sampling approach can be used to improve the approximation rate for polynomially decaying activation functions under mild additional assumptions.
Open AccessApproximation Rates for Neural Networks with General Activation Functions
We prove some new results concerning the approximation rate of neural networks with general activation functions. Our first result concerns the rate of approximation of a two layer neural network with a polynomially-decaying non-sigmoidal activation function. We extend the dimension independent approximation rates previously obtained to this new class of activation functions. Our second result gives a weaker, but still dimension independent, approximation rate for a larger class of activation functions, removing the polynomial decay assumption. This result applies to any bounded, integrable activation function. Finally, we show that a stratified sampling approach can be used to improve the approximation rate for polynomially decaying activation functions under mild additional assumptions.
- Open AccessMake $\ell_1$ Regularization Effective in Training Sparse CNNCompressed Sensing using $\ell_1$ regularization is among the most powerful and popular sparsification technique in many applications, but why has it not been used to obtain sparse deep learning model such as convolutional neural network (CNN)? This paper is aimed to provide an answer to this question and to show how to make it work. We first demonstrate that the commonly used stochastic gradient decent (SGD) and variants training algorithm is not an appropriate match with $\ell_1$ regularization and then replace it with a different training algorithm based on a regularized dual averaging (RDA) method. RDA was originally designed specifically for convex problem, but with new theoretical insight and algorithmic modifications (using proper initialization and adaptivity), we have made it an effective match with $\ell_1$ regularization to achieve a state-of-the-art sparsity for CNN compared to other weight pruning methods without compromising accuracy (achieving 95\% sparsity for ResNet18 on CIFAR-10, for example).Open AccessMake $\ell_1$ Regularization Effective in Training Sparse CNNCompressed Sensing using $\ell_1$ regularization is among the most powerful and popular sparsification technique in many applications, but why has it not been used to obtain sparse deep learning model such as convolutional neural network (CNN)? This paper is aimed to provide an answer to this question and to show how to make it work. We first demonstrate that the commonly used stochastic gradient decent (SGD) and variants training algorithm is not an appropriate match with $\ell_1$ regularization and then replace it with a different training algorithm based on a regularized dual averaging (RDA) method. RDA was originally designed specifically for convex problem, but with new theoretical insight and algorithmic modifications (using proper initialization and adaptivity), we have made it an effective match with $\ell_1$ regularization to achieve a state-of-the-art sparsity for CNN compared to other weight pruning methods without compromising accuracy (achieving 95\% sparsity for ResNet18 on CIFAR-10, for example).