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Haizhao Yang

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14 papers

2 author rows

JMLR Journal 2025 Journal Article

Finite Expression Method for Solving High-Dimensional Partial Differential Equations

Senwei Liang
Haizhao Yang

Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in designing numerical schemes that scale in dimension. This paper introduces a new methodology that seeks an approximate PDE solution in the space of functions with finitely many analytic expressions and, hence, this methodology is named the finite expression method (FEX). It is proved in approximation theory that FEX can avoid the curse of dimensionality. As a proof of concept, a deep reinforcement learning method is proposed to implement FEX for various high-dimensional PDEs in different dimensions, achieving high and even machine accuracy with a memory complexity polynomial in dimension and an amenable time complexity. An approximate solution with finite analytic expressions also provides interpretable insights into the ground truth PDE solution, which can further help to advance the understanding of physical systems and design postprocessing techniques for a refined solution. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2025. ( edit, beta )

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JMLR Journal 2024 Journal Article

Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces

Hao Liu
Haizhao Yang
Minshuo Chen
Tuo Zhao
Wenjing Liao

Learning operators between infinitely dimensional spaces is an important learning task arising in machine learning, imaging science, mathematical modeling and simulations, etc. This paper studies the nonparametric estimation of Lipschitz operators using deep neural networks. Non-asymptotic upper bounds are derived for the generalization error of the empirical risk minimizer over a properly chosen network class. Under the assumption that the target operator exhibits a low dimensional structure, our error bounds decay as the training sample size increases, with an attractive fast rate depending on the intrinsic dimension in our estimation. Our assumptions cover most scenarios in real applications and our results give rise to fast rates by exploiting low dimensional structures of data in operator estimation. We also investigate the influence of network structures (e.g., network width, depth, and sparsity) on the generalization error of the neural network estimator and propose a general suggestion on the choice of network structures to maximize the learning efficiency quantitatively. [abs] [ pdf ][ bib ] &copy JMLR 2024. ( edit, beta )

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AAAI Conference 2024 Conference Paper

Neural Network Approximation for Pessimistic Offline Reinforcement Learning

Di Wu
Yuling Jiao
Li Shen
Haizhao Yang
Xiliang Lu

Deep reinforcement learning (RL) has shown remarkable success in specific offline decision-making scenarios, yet its theoretical guarantees are still under development. Existing works on offline RL theory primarily emphasize a few trivial settings, such as linear MDP or general function approximation with strong assumptions and independent data, which lack guidance for practical use. The coupling of deep learning and Bellman residuals makes this problem challenging, in addition to the difficulty of data dependence. In this paper, we establish a non-asymptotic estimation error of pessimistic offline RL using general neural network approximation with C-mixing data regarding the structure of networks, the dimension of datasets, and the concentrability of data coverage, under mild assumptions. Our result shows that the estimation error consists of two parts: the first converges to zero at a desired rate on the sample size with partially controllable concentrability, and the second becomes negligible if the residual constraint is tight. This result demonstrates the explicit efficiency of deep adversarial offline RL frameworks. We utilize the empirical process tool for C-mixing sequences and the neural network approximation theory for the Holder class to achieve this. We also develop methods to bound the Bellman estimation error caused by function approximation with empirical Bellman constraint perturbations. Additionally, we present a result that lessens the curse of dimensionality using data with low intrinsic dimensionality and function classes with low complexity. Our estimation provides valuable insights into the development of deep offline RL and guidance for algorithm model design.

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TMLR Journal 2024 Journal Article

On the Stochastic (Variance-Reduced) Proximal Gradient Method for Regularized Expected Reward Optimization

Ling Liang
Haizhao Yang

We consider a regularized expected reward optimization problem in the non-oblivious setting that covers many existing problems in reinforcement learning (RL). In order to solve such an optimization problem, we apply and analyze the classical stochastic proximal gradient method. In particular, the method has shown to admit an $O(\epsilon^{-4})$ sample complexity to an $\epsilon$-stationary point, under standard conditions. Since the variance of the classical stochastic gradient estimator is typically large, which slows down the convergence, we also apply an efficient stochastic variance-reduce proximal gradient method with an importance sampling based ProbAbilistic Gradient Estimator (PAGE). Our analysis shows that the sample complexity can be improved from $O(\epsilon^{-4})$ to $O(\epsilon^{-3})$ under additional conditions. Our results on the stochastic (variance-reduced) proximal gradient method match the sample complexity of their most competitive counterparts for discounted Markov decision processes under similar settings. To the best of our knowledge, the proposed methods represent a novel approach in addressing the general regularized reward optimization problem.

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TMLR Journal 2024 Journal Article

What is the Solution for State-Adversarial Multi-Agent Reinforcement Learning?

Songyang Han
Sanbao Su
Sihong He
Shuo Han
Haizhao Yang
Shaofeng Zou
Fei Miao

Various methods for Multi-Agent Reinforcement Learning (MARL) have been developed with the assumption that agents' policies are based on accurate state information. However, policies learned through Deep Reinforcement Learning (DRL) are susceptible to adversarial state perturbation attacks. In this work, we propose a State-Adversarial Markov Game (SAMG) and make the first attempt to investigate different solution concepts of MARL under state uncertainties. Our analysis shows that the commonly used solution concepts of optimal agent policy and robust Nash equilibrium do not always exist in SAMGs. To circumvent this difficulty, we consider a new solution concept called robust agent policy, where agents aim to maximize the worst-case expected state value. We prove the existence of robust agent policy for finite state and finite action SAMGs. Additionally, we propose a Robust Multi-Agent Adversarial Actor-Critic (RMA3C) algorithm to learn robust policies for MARL agents under state uncertainties. Our experiments demonstrate that our algorithm outperforms existing methods when faced with state perturbations and greatly improves the robustness of MARL policies. Our code is public on https://songyanghan.github.io/what_is_solution/.

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TMLR Journal 2023 Journal Article

Deep Operator Learning Lessens the Curse of Dimensionality for PDEs

Ke Chen
Chunmei Wang
Haizhao Yang

Deep neural networks (DNNs) have achieved remarkable success in numerous domains, and their application to PDE-related problems has been rapidly advancing. This paper provides an estimate for the generalization error of learning Lipschitz operators over Banach spaces using DNNs with applications to various PDE solution operators. The goal is to specify DNN width, depth, and the number of training samples needed to guarantee a certain testing error. Under mild assumptions on data distributions or operator structures, our analysis shows that deep operator learning can have a relaxed dependence on the discretization resolution of PDEs and, hence, lessen the curse of dimensionality in many PDE-related problems including elliptic equations, parabolic equations, and Burgers equations. Our results are also applied to give insights about discretization-invariant in operator learning.

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NeurIPS Conference 2023 Conference Paper

Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivatives

Yahong Yang
Haizhao Yang
Yang Xiang

This paper addresses the problem of nearly optimal Vapnik--Chervonenkis dimension (VC-dimension) and pseudo-dimension estimations of the derivative functions of deep neural networks (DNNs). Two important applications of these estimations include: 1) Establishing a nearly tight approximation result of DNNs in the Sobolev space; 2) Characterizing the generalization error of machine learning methods with loss functions involving function derivatives. This theoretical investigation fills the gap of learning error estimations for a wide range of physics-informed machine learning models and applications including generative models, solving partial differential equations, operator learning, network compression, distillation, regularization, etc.

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ICML Conference 2022 Conference Paper

Deep Network Approximation in Terms of Intrinsic Parameters

Zuowei Shen
Haizhao Yang
Shijun Zhang

One of the arguments to explain the success of deep learning is the powerful approximation capacity of deep neural networks. Such capacity is generally accompanied by the explosive growth of the number of parameters, which, in turn, leads to high computational costs. It is of great interest to ask whether we can achieve successful deep learning with a small number of learnable parameters adapting to the target function. From an approximation perspective, this paper shows that the number of parameters that need to be learned can be significantly smaller than people typically expect. First, we theoretically design ReLU networks with a few learnable parameters to achieve an attractive approximation. We prove by construction that, for any Lipschitz continuous function $f$ on $[0, 1]^d$ with a Lipschitz constant $\lambda>0$, a ReLU network with $n+2$ intrinsic parameters (those depending on $f$) can approximate $f$ with an exponentially small error $5 \lambda \sqrt{d} \, 2^{-n}$. Such a result is generalized to generic continuous functions. Furthermore, we show that the idea of learning a small number of parameters to achieve a good approximation can be numerically observed. We conduct several experiments to verify that training a small part of parameters can also achieve good results for classification problems if other parameters are pre-specified or pre-trained from a related problem.

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JMLR Journal 2022 Journal Article

Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons

Shijun Zhang
Zuowei Shen
Haizhao Yang

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a simple, computable, and continuous activation function $\sigma$ leveraging a triangular-wave function and the softsign function. We first prove that $\sigma$-activated networks with width $36d(2d+1)$ and depth $11$ can approximate any continuous function on a $d$-dimensional hypercube within an arbitrarily small error. Hence, for supervised learning and its related regression problems, the hypothesis space generated by these networks with a size not smaller than $36d(2d+1)\times 11$ is dense in the continuous function space $C([a,b]^d)$ and therefore dense in the Lebesgue spaces $L^p([a,b]^d)$ for $p\in [1,\infty)$. Furthermore, we show that classification functions arising from image and signal classification are in the hypothesis space generated by $\sigma$-activated networks with width $36d(2d+1)$ and depth $12$ when there exist pairwise disjoint bounded closed subsets of $\mathbb{R}^d$ such that the samples of the same class are located in the same subset. Finally, we use numerical experimentation to show that replacing the rectified linear unit (ReLU) activation function by ours would improve the experiment results. [abs] [ pdf ][ bib ] &copy JMLR 2022. ( edit, beta )

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TMLR Journal 2022 Journal Article

From Optimization Dynamics to Generalization Bounds via Łojasiewicz Gradient Inequality

Fusheng Liu
Haizhao Yang
Soufiane Hayou
Qianxiao Li

Optimization and generalization are two essential aspects of statistical machine learning. In this paper, we propose a framework to connect optimization with generalization by analyz- ing the generalization error based on the optimization trajectory under the gradient flow algorithm. The key ingredient of this framework is the Uniform-LGI, a property that is generally satisfied when training machine learning models. Leveraging the Uniform-LGI, we first derive convergence rates for gradient flow algorithm, then we give generalization bounds for a large class of machine learning models. We further apply our framework to three distinct machine learning models: linear regression, kernel regression, and two-layer neural networks. Through our approach, we obtain generalization estimates that match or extend previous results.

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JMLR Journal 2022 Journal Article

Integral Autoencoder Network for Discretization-Invariant Learning

Yong Zheng Ong
Zuowei Shen
Haizhao Yang

Discretization invariant learning aims at learning in the infinite-dimensional function spaces with the capacity to process heterogeneous discrete representations of functions as inputs and/or outputs of a learning model. This paper proposes a novel deep learning framework based on integral autoencoders (IAE-Net) for discretization invariant learning. The basic building block of IAE-Net consists of an encoder and a decoder as integral transforms with data-driven kernels, and a fully connected neural network between the encoder and decoder. This basic building block is applied in parallel in a wide multi-channel structure, which is repeatedly composed to form a deep and densely connected neural network with skip connections as IAE-Net. IAE-Net is trained with randomized data augmentation that generates training data with heterogeneous structures to facilitate the performance of discretization invariant learning. The proposed IAE-Net is tested with various applications in predictive data science, solving forward and inverse problems in scientific computing, and signal/image processing. Compared with alternatives in the literature, IAE-Net achieves state-of-the-art performance in existing applications and creates a wide range of new applications where existing methods fail. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2022. ( edit, beta )

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NeurIPS Conference 2022 Conference Paper

Neural Network Architecture Beyond Width and Depth

Shijun Zhang
Zuowei Shen
Haizhao Yang

This paper proposes a new neural network architecture by introducing an additional dimension called height beyond width and depth. Neural network architectures with height, width, and depth as hyper-parameters are called three-dimensional architectures. It is shown that neural networks with three-dimensional architectures are significantly more expressive than the ones with two-dimensional architectures (those with only width and depth as hyper-parameters), e. g. , standard fully connected networks. The new network architecture is constructed recursively via a nested structure, and hence we call a network with the new architecture nested network (NestNet). A NestNet of height $s$ is built with each hidden neuron activated by a NestNet of height $\le s-1$. When $s=1$, a NestNet degenerates to a standard network with a two-dimensional architecture. It is proved by construction that height-$s$ ReLU NestNets with $\mathcal{O}(n)$ parameters can approximate $1$-Lipschitz continuous functions on $[0, 1]^d$ with an error $\mathcal{O}(n^{-(s+1)/d})$, while the optimal approximation error of standard ReLU networks with $\mathcal{O}(n)$ parameters is $\mathcal{O}(n^{-2/d})$. Furthermore, such a result is extended to generic continuous functions on $[0, 1]^d$ with the approximation error characterized by the modulus of continuity. Finally, we use numerical experimentation to show the advantages of the super-approximation power of ReLU NestNets.

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AAAI Conference 2020 Conference Paper

DIANet: Dense-and-Implicit Attention Network

Zhongzhan Huang
Senwei Liang
Mingfu Liang
Haizhao Yang

Attention networks have successfully boosted the performance in various vision problems. Previous works lay emphasis on designing a new attention module and individually plug them into the networks. Our paper proposes a novel-andsimple framework that shares an attention module throughout different network layers to encourage the integration of layer-wise information and this parameter-sharing module is referred to as Dense-and-Implicit-Attention (DIA) unit. Many choices of modules can be used in the DIA unit. Since Long Short Term Memory (LSTM) has a capacity of capturing long-distance dependency, we focus on the case when the DIA unit is the modiﬁed LSTM (called DIA-LSTM). Experiments on benchmark datasets show that the DIA-LSTM unit is capable of emphasizing layer-wise feature interrelation and leads to signiﬁcant improvement of image classiﬁcation accuracy. We further empirically show that the DIA-LSTM has a strong regularization ability on stabilizing the training of deep networks by the experiments with the removal of skip connections (He et al. 2016a) or Batch Normalization (Ioffe and Szegedy 2015) in the whole residual network.

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AAAI Conference 2020 Conference Paper

Instance Enhancement Batch Normalization: An Adaptive Regulator of Batch Noise

Senwei Liang
Zhongzhan Huang
Mingfu Liang
Haizhao Yang

Batch Normalization (BN) (Ioffe and Szegedy 2015) normalizes the features of an input image via statistics of a batch of images and hence BN will bring the noise to the gradient of training loss. Previous works indicate that the noise is important for the optimization and generalization of deep neural networks, but too much noise will harm the performance of networks. In our paper, we offer a new point of view that the self-attention mechanism can help to regulate the noise by enhancing instance-speciﬁc information to obtain a better regularization effect. Therefore, we propose an attention-based BN called Instance Enhancement Batch Normalization (IEBN) that recalibrates the information of each channel by a simple linear transformation. IEBN has a good capacity of regulating the batch noise and stabilizing network training to improve generalization even in the presence of two kinds of noise attacks during training. Finally, IEBN outperforms BN with only a light parameter increment in image classiﬁcation tasks under different network structures and benchmark datasets.

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