arXiv subjects

Haizhao Yang

Publications and source records attributed to Haizhao Yang.

At least 73 records · Page 4Linked to original sources

arXiv2024

Towards Better Generalization: Weight Decay Induces Low-rank Bias for Neural Networks

We study the implicit bias towards low-rank weight matrices when training neural networks (NN) with Weight Decay (WD). We prove that when a ReLU NN is sufficiently trained with Stochastic Gradient Descent (SGD) and WD, its weight matrix is approximately a rank-two matrix. Empirically, we demonstrate that WD is a necessary condition for inducing this low-rank bias across both regression and classification tasks. Our work differs from previous studies as our theoretical analysis does not rely on common assumptions regarding the training data distribution, optimality of weight matrices, or specific training procedures. Furthermore, by leveraging the low-rank bias, we derive improved generalization error bounds and provide numerical evidence showing that better generalization can be achieved. Thus, our work offers both theoretical and empirical insights into the strong generalization performance of SGD when combined with WD.

cs.LG↗

arXiv2024

FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model

The fast simulation of dynamical systems is a key challenge in many scientific and engineering applications, such as weather forecasting, disease control, and drug discovery. With the recent success of deep learning, there is increasing interest in using neural networks to solve differential equations in a data-driven manner. However, existing methods are either limited to specific types of differential equations or require large amounts of data for training. This restricts their practicality in many real-world applications, where data is often scarce or expensive to obtain. To address this, we propose a novel multi-modal foundation model, named \textbf{FMint} (\textbf{F}oundation \textbf{M}odel based on \textbf{In}i\textbf{t}ialization), to bridge the gap between human-designed and data-driven models for the fast simulation of dynamical systems. Built on a decoder-only transformer architecture with in-context learning, FMint utilizes both numerical and textual data to learn a universal error correction scheme for dynamical systems, using prompted sequences of coarse solutions from traditional solvers. The model is pre-trained on a corpus of 40K ODEs, and we perform extensive experiments on challenging ODEs that exhibit chaotic behavior and of high dimensionality. Our results demonstrate the effectiveness of the proposed model in terms of both accuracy and efficiency compared to classical numerical solvers, highlighting FMint's potential as a general-purpose solver for dynamical systems. Our approach achieves an accuracy improvement of 1 to 2 orders of magnitude over state-of-the-art dynamical system simulators, and delivers a 5X speedup compared to traditional numerical algorithms. The code for FMint is available at \url{https://github.com/margotyjx/FMint}.

cs.LG↗

arXiv2024

PNOD: An Efficient Projected Newton Framework for Exact Optimal Experimental Designs

Computing the exact optimal experimental design has been a longstanding challenge in various scientific fields. This problem, when formulated using a specific information function, becomes a mixed-integer nonlinear programming (MINLP) problem, which is typically NP-hard, thus making the computation of a globally optimal solution extremely difficult. The branch and bound (BnB) method is a widely used approach for solving such MINLPs, but its practical efficiency heavily relies on the ability to solve continuous relaxations effectively within the BnB search tree. In this paper, we propose a novel projected Newton framework, combining with a vertex exchange method for efficiently solving the associated subproblems, designed to enhance the BnB method. This framework offers strong convergence guarantees by utilizing recent advances in solving self-concordant optimization and convex quadratic programming problems. Extensive numerical experiments on A-optimal and D-optimal design problems, two of the most commonly used models, demonstrate the framework's promising numerical performance. Specifically, our framework significantly improves the efficiency of node evaluation within the BnB search tree and enhances the accuracy of solutions compared to state-of-the-art methods. The proposed framework is implemented in an open source Julia package called \texttt{PNOD.jl}, which opens up possibilities for its application in a wide range of real-world scenarios.

stat.ME↗

arXiv2024

Accelerating Multi-Block Constrained Optimization Through Learning to Optimize

Learning to Optimize (L2O) approaches, including algorithm unrolling, plug-and-play methods, and hyperparameter learning, have garnered significant attention and have been successfully applied to the Alternating Direction Method of Multipliers (ADMM) and its variants. However, the natural extension of L2O to multi-block ADMM-type methods remains largely unexplored. Such an extension is critical, as multi-block methods leverage the separable structure of optimization problems, offering substantial reductions in per-iteration complexity. Given that classical multi-block ADMM does not guarantee convergence, the Majorized Proximal Augmented Lagrangian Method (MPALM), which shares a similar form with multi-block ADMM and ensures convergence, is more suitable in this setting. Despite its theoretical advantages, MPALM's performance is highly sensitive to the choice of penalty parameters. To address this limitation, we propose a novel L2O approach that adaptively selects this hyperparameter using supervised learning. We demonstrate the versatility and effectiveness of our method by applying it to the Lasso problem and the optimal transport problem. Our numerical results show that the proposed framework outperforms popular alternatives. Given its applicability to generic linearly constrained composite optimization problems, this work opens the door to a wide range of potential real-world applications.

math.OC↗

arXiv2024

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

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 $O(ε^{-4})$ sample complexity to an $ε$-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 $O(ε^{-4})$ to $O(ε^{-3})$ 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.

cs.LG↗

arXiv2024

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

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/.

cs.AI↗

arXiv2024

Let data talk: data-regularized operator learning theory for inverse problems

Regularization plays a pivotal role in integrating prior information into inverse problems. While many deep learning methods have been proposed to solve inverse problems, determining where to apply regularization remains a crucial consideration. Typical methods regularize neural networks via architecture, wherein neural network functions parametrize the parameter of interest or the regularization term. We introduce a novel approach, denoted as the "data-regularized operator learning" (DaROL) method, designed to address PDE inverse problems. The DaROL method trains a neural network on data, regularized through common techniques such as Tikhonov variational methods and Bayesian inference. The DaROL method offers flexibility across different frameworks, faster inverse problem-solving, and a simpler structure that separates regularization and neural network training. We demonstrate that training a neural network on the regularized data is equivalent to supervised learning for a regularized inverse map. Furthermore, we provide sufficient conditions for the smoothness of such a regularized inverse map and estimate the learning error in terms of neural network size and the number of training samples.

math.NA↗

arXiv2024

Solving PDEs on Unknown Manifolds with Machine Learning

This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a supervised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural networks (NNs). In a well-posed elliptic PDE setting, when the hypothesis space consists of neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, gradient descent can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions, ranging from simple manifolds with low and high co-dimensions, to rough surfaces with and without boundaries. We also show that the proposed NN solver can robustly generalize the PDE solution on new data points with generalization errors that are almost identical to the training errors, superseding a Nystrom-based interpolation method.

math.NA↗

arXiv2024

A Distributed Block Chebyshev-Davidson Algorithm for Parallel Spectral Clustering

We develop a distributed Block Chebyshev-Davidson algorithm to solve large-scale leading eigenvalue problems for spectral analysis in spectral clustering. First, the efficiency of the Chebyshev-Davidson algorithm relies on the prior knowledge of the eigenvalue spectrum, which could be expensive to estimate. This issue can be lessened by the analytic spectrum estimation of the Laplacian or normalized Laplacian matrices in spectral clustering, making the proposed algorithm very efficient for spectral clustering. Second, to make the proposed algorithm capable of analyzing big data, a distributed and parallel version has been developed with attractive scalability. The speedup by parallel computing is approximately equivalent to $\sqrt{p}$, where $p$ denotes the number of processes. {Numerical results will be provided to demonstrate its efficiency in spectral clustering and scalability advantage over existing eigensolvers used for spectral clustering in parallel computing environments.}

cs.LG↗

arXiv2023

Neural Network Approximation for Pessimistic Offline Reinforcement Learning

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 $\mathcal{C}$-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 $\mathcal{C}$-mixing sequences and the neural network approximation theory for the Hölder 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.

cs.LG↗

arXiv2023

On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: a Riemannian optimization interpretation

In many applications, it is desired to obtain extreme eigenvalues and eigenvectors of large Hermitian matrices by efficient and compact algorithms. In particular, orthogonalization-free methods are preferred for large-scale problems for finding eigenspaces of extreme eigenvalues without explicitly computing orthogonal vectors in each iteration. For the top $p$ eigenvalues, the simplest orthogonalization-free method is to find the best rank-$p$ approximation to a positive semi-definite Hermitian matrix by algorithms solving the unconstrained Burer-Monteiro formulation. We show that the nonlinear conjugate gradient method for the unconstrained Burer-Monteiro formulation is equivalent to a Riemannian conjugate gradient method on a quotient manifold with the Bures-Wasserstein metric, thus its global convergence to a stationary point can be proven. Numerical tests suggest that it is efficient for computing the largest $k$ eigenvalues for large-scale matrices if the largest $k$ eigenvalues are nearly distributed uniformly.

math.NA↗

arXiv2023

Deep Operator Learning Lessens the Curse of Dimensionality for PDEs

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-invariance in operator learning.

cs.LG↗

arXiv2023

On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solver

The large-scale simulation of dynamical systems is critical in numerous scientific and engineering disciplines. However, traditional numerical solvers are limited by the choice of step sizes when estimating integration, resulting in a trade-off between accuracy and computational efficiency. To address this challenge, we introduce a deep learning-based corrector called Neural Vector (NeurVec), which can compensate for integration errors and enable larger time step sizes in simulations. Our extensive experiments on a variety of complex dynamical system benchmarks demonstrate that NeurVec exhibits remarkable generalization capability on a continuous phase space, even when trained using limited and discrete data. NeurVec significantly accelerates traditional solvers, achieving speeds tens to hundreds of times faster while maintaining high levels of accuracy and stability. Moreover, NeurVec's simple-yet-effective design, combined with its ease of implementation, has the potential to establish a new paradigm for fast-solving differential equations based on deep learning.

cs.CE↗

arXiv2023

Hierarchical Interpolative Factorization for Self Green's Function in 3D Modified Poisson-Boltzmann Equations

The modified Poisson-Boltzmann (MPB) equations are often used to describe equilibrium particle distribution of ionic systems. In this paper, we propose a fast algorithm to solve MPB equations with the self Green's function as the self energy in three dimensions, where the solution of the self Green's function poses a computational bottleneck due to the need to solve a high-dimensional partial differential equation. Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green's function by extending our previous result of two dimensions. This leads to an $O(N\log N)$ algorithm through the strategical utilization of locality and low-rank characteristics of the corresponding operators. Furthermore, the estimated $O(N)$ complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction. Extensive numerical results are performed to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.

physics.comp-ph↗

arXiv2023

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

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.

cs.LG↗

arXiv2023

Deep Learning via Neural Energy Descent

This paper proposes the Nerual Energy Descent (NED) via neural network evolution equations for a wide class of deep learning problems. We show that deep learning can be reformulated as the evolution of network parameters in an evolution equation and the steady state solution of the partial differential equation (PDE) provides a solution to deep learning. This equation corresponds to a gradient descent flow of a variational problem and hence the proposed time-dependent PDE solves an energy minimization problem to obtain a global minimizer of deep learning. This gives a novel interpretation and solution to deep learning optimization. The computational complexity of the proposed energy descent method can be enhanced by randomly sampling the spatial domain of the PDE leading to an efficient NED. Numerical examples are provided to demonstrate the numerical advantage of NED over stochastic gradient descent (SGD).

math.NA↗

arXiv2023

Convergence Analysis of the Deep Galerkin Method for Weak Solutions

This paper analyzes the convergence rate of a deep Galerkin method for the weak solution (DGMW) of second-order elliptic partial differential equations on $\mathbb{R}^d$ with Dirichlet, Neumann, and Robin boundary conditions, respectively. In DGMW, a deep neural network is applied to parametrize the PDE solution, and a second neural network is adopted to parametrize the test function in the traditional Galerkin formulation. By properly choosing the depth and width of these two networks in terms of the number of training samples $n$, it is shown that the convergence rate of DGMW is $\mathcal{O}(n^{-1/d})$, which is the first convergence result for weak solutions. The main idea of the proof is to divide the error of the DGMW into an approximation error and a statistical error. We derive an upper bound on the approximation error in the $H^{1}$ norm and bound the statistical error via Rademacher complexity.

math.NA↗

arXiv2023

Neural Network Architecture Beyond Width and Depth

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.

cs.LG↗