In this paper, we analyze the number of neurons and training parameters that a neural networks needs to approximate multivariate functions of bounded second mixed derivatives -- Korobov functions. We prove upper bounds on these quantities for shallow and deep neural networks, breaking the curse of dimensionality. Our bounds hold for general activation functions, including ReLU. We further prove that these bounds nearly match the minimal number of parameters any continuous function approximator needs to approximate Korobov functions, showing that neural networks are near-optimal function approximators.
In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $ω_f(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $\mathcal{O}(ω_f(\sqrt{d})\cdot2^{-M}+ω_{f}\left(\frac{\sqrt{d}}{N}\right))$, where $M,N\in \mathbb{N}^{+}$ denote the hyperparameters related to widths of the networks. As a consequence, we can construct ReLU-sine-$2^x$ network with the depth $5$ and width $\max\left\{\left\lceil2d^{3/2}\left(\frac{3μ}ε\right)^{1/α}\right\rceil,2\left\lceil\log_…
We consider neural networks with a single hidden layer and non-decreasing homogeneous activa-tion functions like the rectified linear units. By letting the number of hidden units grow unbounded and using classical non-Euclidean regularization tools on the output weights, we provide a detailed theoretical analysis of their generalization performance, with a study of both the approximation and the estimation errors. We show in particular that they are adaptive to unknown underlying linear structures, such as the dependence on the projection of the input variables onto a low-dimensional subspace.…
Transformers Can Overcome the Curse of Dimensionality: A Theoretical Study from an Approximation Perspective
School of Artificial Intelligence, Wuhan University（武汉大学人工智能学院）；Hubei Key Laboratory of Computational Science, Wuhan University（湖北省计算科学重点实验室）；Department of Mathematics, The Hong Kong University of Science and Technology（香港科技大学数学系）
Hong Kong University of Science and Technology（香港科技大学）；HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute（香港科技大学深港协同创新研究院）
In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of networks, which overcomes the curse of dimensionality. The key idea of the approximation is to define a Barron spectral space of functionals.
The curse of dimensionality has been studied in different aspects. However, breaking the curse has been elusive. We show for the first time that it is possible to break the curse using the recently introduced Isolation Kernel. We show that only Isolation Kernel performs consistently well in indexed search, spectral & density peaks clustering, SVM classification and t-SNE visualization in both low and high dimensions, compared with distance, Gaussian and linear kernels. This is also supported by our theoretical analyses that Isolation Kernel is the only kernel that has the provable ability to b…
In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This …
University of Vienna（维也纳大学）；ETH Zürich（苏黎世联邦理工学院）；University of Münster（明斯特大学）
The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine learning, and data analysis. In particular, these methods have been applied to the numerical solution of high-dimensional partial differential equations with great success. Recent simulations indicate that deep learning-based algorithms are capable of overcoming the curse of dimensionality for the numerical solution of Kolmogorov equations, which are widely used i…
Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are also a few results in the literature which prove that DNNs can approximate solutions of certain PDEs without the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially both in the PDE dimension and the reciprocal of the prescribed approximation accuracy. One key argument in most of these results is, first, to use a Monte Carlo approximation scheme …
Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense
Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore（数学科学系，物理与数学科学学院，南洋理工大学，新加坡）；Faculty of Mathematics, Bielefeld University, Germany（数学学院，比勒菲尔德大学，德国）
Center for Brains, Minds, and Machines（大脑、心智与机器研究中心）；McGovern Institute for Brain Research（麦戈文脑研究所）；Massachusetts Institute of Technology（麻省理工学院）；California Institute of Technology（加州理工学院）；Institute of Mathematical Sciences, Claremont Graduate University（克莱蒙特研究大学数学科学研究所）
The paper characterizes classes of functions for which deep learning can be exponentially better than shallow learning. Deep convolutional networks are a special case of these conditions, though weight sharing is not the main reason for their exponential advantage.
Cornell University（康奈尔大学）；IBM Watson Research Center（IBM沃森研究中心）
Integration is affected by the curse of dimensionality and quickly becomes intractable as the dimensionality of the problem grows. We propose a randomized algorithm that, with high probability, gives a constant-factor approximation of a general discrete integral defined over an exponentially large set. This algorithm relies on solving only a small number of instances of a discrete combinatorial optimization problem subject to randomly generated parity constraints used as a hash function. As an application, we demonstrate that with a small number of MAP queries we can efficiently approximate th…
We prove that the gradient descent training of a two-layer neural network on empirical or population risk may not decrease population risk at an order faster than $t^{-4/(d-2)}$ under mean field scaling. Thus gradient descent training for fitting reasonably smooth, but truly high-dimensional data may be subject to the curse of dimensionality. We present numerical evidence that gradient descent training with general Lipschitz target functions becomes slower and slower as the dimension increases, but converges at approximately the same rate in all dimensions when the target function lies in the …
University of Leicester（莱斯特大学）；Lobachevsky State University（罗巴切夫斯基国立大学）；Taibah University（塔伊巴大学）
The curse of dimensionality causes the well-known and widely discussed problems for machine learning methods. There is a hypothesis that using of the Manhattan distance and even fractional quasinorms lp (for p less than 1) can help to overcome the curse of dimensionality in classification problems. In this study, we systematically test this hypothesis. We confirm that fractional quasinorms have a greater relative contrast or coefficient of variation than the Euclidean norm l2, but we also demonstrate that the distance concentration shows qualitatively the same behaviour for all tested norms an…
In this note, we study how neural networks with a single hidden layer and ReLU activation interpolate data drawn from a radially symmetric distribution with target labels 1 at the origin and 0 outside the unit ball, if no labels are known inside the unit ball. With weight decay regularization and in the infinite neuron, infinite data limit, we prove that a unique radially symmetric minimizer exists, whose weight decay regularizer and Lipschitz constant grow as $d$ and $\sqrt{d}$ respectively. We furthermore show that the weight decay regularizer grows exponentially in $d$ if the label $1$ is i…
University of California at Berkeley（加州大学伯克利分校）；University of California at Davis（加州大学戴维斯分校）
Predictive rate-distortion analysis suffers from the curse of dimensionality: clustering arbitrarily long pasts to retain information about arbitrarily long futures requires resources that typically grow exponentially with length. The challenge is compounded for infinite-order Markov processes, since conditioning on finite sequences cannot capture all of their past dependencies. Spectral arguments show that algorithms which cluster finite-length sequences fail dramatically when the underlying process has long-range temporal correlations and can fail even for processes generated by finite-memor…
This article investigates the use of random feature neural networks for learning Kolmogorov partial (integro-)differential equations associated to Black-Scholes and more general exponential Lévy models. Random feature neural networks are single-hidden-layer feedforward neural networks in which only the output weights are trainable. This makes training particularly simple, but (a priori) reduces expressivity. Interestingly, this is not the case for Black-Scholes type PDEs, as we show here. We derive bounds for the prediction error of random neural networks for learning sufficiently non-degenera…
As annotations of data can be scarce in large-scale practical problems, leveraging unlabelled examples is one of the most important aspects of machine learning. This is the aim of semi-supervised learning. To benefit from the access to unlabelled data, it is natural to diffuse smoothly knowledge of labelled data to unlabelled one. This induces to the use of Laplacian regularization. Yet, current implementations of Laplacian regularization suffer from several drawbacks, notably the well-known curse of dimensionality. In this paper, we provide a statistical analysis to overcome those issues, and…
Overcoming the Curse of Dimensionality in Reinforcement Learning Through Approximate Factorization
Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Edwardson School of Industrial Engineering, Purdue University（普渡大学爱德华森工业工程学院）；School of Science and Engineering, CUHK Shenzhen（香港中文大学（深圳）理工学院）；Computing & Mathematical Sciences Department, Caltech（加州理工学院计算与数学科学系）
In recent years deep artificial neural networks (DNNs) have been successfully employed in numerical simulations for a multitude of computational problems including, for example, object and face recognition, natural language processing, fraud detection, computational advertisement, and numerical approximations of partial differential equations (PDEs). These numerical simulations indicate that DNNs seem to possess the fundamental flexibility to overcome the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially in both the reci…
Breaking the curse of dimensionality in structured density estimation
Berlin Institute for the Foundations of Learning and Data (BIFOLD)（柏林学习与数据基础研究所（BIFOLD））；Technische Universität Berlin（柏林工业大学）；Nanyang Technological University（南洋理工大学）
Georgia Institute of Technology（佐治亚理工学院）；Télécom Paris（巴黎高等电信学院）
We present a novel and practical deep learning pipeline termed RandomForestMLP. This core trainable classification engine consists of a convolutional neural network backbone followed by an ensemble-based multi-layer perceptrons core for the classification task. It is designed in the context of self and semi-supervised learning tasks to avoid overfitting while training on very small datasets. The paper details the architecture of the RandomForestMLP and present different strategies for neural network decision aggregation. Then, it assesses its robustness to overfitting when trained on realistic…
Finding global optima in high-dimensional optimization problems is extremely challenging since the number of function evaluations required to sufficiently explore the search space increases exponentially with its dimensionality. Furthermore, multimodal cost functions render local gradient-based search techniques ineffective. To overcome these difficulties, we propose to trim uninteresting regions of the search space where global optima are unlikely to be found by means of autoencoders, exploiting the lower intrinsic dimensionality of certain cost functions; optima are then searched over lower-…
Wasserstein distributionally robust optimization (DRO) aims to find robust and generalizable solutions by hedging against data perturbations in Wasserstein distance. Despite its recent empirical success in operations research and machine learning, existing performance guarantees for generic loss functions are either overly conservative due to the curse of dimensionality, or plausible only in large sample asymptotics. In this paper, we develop a non-asymptotic framework for analyzing the out-of-sample performance for Wasserstein robust learning and the generalization bound for its related Lipsc…
INESC-ID（INESC-ID（葡萄牙系统与计算机工程研究所））；Instituto Superior Técnico（里斯本高等理工学院）；University of Lisbon（里斯本大学）
It is generally assumed that the brain uses something akin to sparse distributed representations. These representations, however, are high-dimensional and consequently they affect classification performance of traditional Machine Learning models due to "the curse of dimensionality". In tasks for which there is a vast amount of labeled data, Deep Networks seem to solve this issue with many layers and a non-Hebbian backpropagation algorithm. The brain, however, seems to be able to solve the problem with few layers. In this work, we hypothesize that this happens by using Hebbian learning. Actuall…
Huazhong University of Science and Technology（华中科技大学）
Takagi-Sugeno-Kang (TSK) fuzzy system with Gaussian membership functions (MFs) is one of the most widely used fuzzy systems in machine learning. However, it usually has difficulty handling high-dimensional datasets. This paper explores why TSK fuzzy systems with Gaussian MFs may fail on high-dimensional inputs. After transforming defuzzification to an equivalent form of softmax function, we find that the poor performance is due to the saturation of softmax. We show that two defuzzification operations, LogTSK and HTSK, the latter of which is first proposed in this paper, can avoid the saturatio…
The Johns Hopkins University Applied Physics Laboratory（约翰斯·霍普金斯大学应用物理实验室）
The failure of the Euclidean norm to reliably distinguish between nearby and distant points in high dimensional space is well-known. This phenomenon of distance concentration manifests in a variety of data distributions, with iid or correlated features, including centrally-distributed and clustered data. Unsupervised learning based on Euclidean nearest-neighbors and more general proximity-oriented data mining tasks like clustering, might therefore be adversely affected by distance concentration for high-dimensional applications. While considerable work has been done developing clustering algor…
SCORE: A 1D Reparameterization Technique to Break Bayesian Optimization's Curse of Dimensionality
Laboratoire de Physique des Interfaces et des Couches Minces (LPICM)（界面与薄膜物理实验室（LPICM））；Ecole Polytechnique, Institut Polytechnique de Paris (IP Paris)（巴黎综合理工学院，巴黎理工学院（IP Paris））；Institut Photovoltaïque d’Île-de-France (IPVF)（法兰西岛光伏研究所（IPVF））
A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations
University of Vienna（维也纳大学）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；ETH Zurich（苏黎世联邦理工学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；University of Münster（明斯特大学）
The University of Tokyo（东京大学）；RIKEN（理化学研究所）；Japan Digital Design（日本数字设计公司）
Deep learning has shown high performances in various types of tasks from visual recognition to natural language processing, which indicates superior flexibility and adaptivity of deep learning. To understand this phenomenon theoretically, we develop a new approximation and estimation error analysis of deep learning with the ReLU activation for functions in a Besov space and its variant with mixed smoothness. The Besov space is a considerably general function space including the Holder space and Sobolev space, and especially can capture spatial inhomogeneity of smoothness. Through the analysis …
Google Inc.（谷歌公司）；Information Sciences Institute, University of Southern California（南加州大学信息科学研究所）；INRIA（法国国家信息与自动化研究所）
Similarity and metric learning provides a principled approach to construct a task-specific similarity from weakly supervised data. However, these methods are subject to the curse of dimensionality: as the number of features grows large, poor generalization is to be expected and training becomes intractable due to high computational and memory costs. In this paper, we propose a similarity learning method that can efficiently deal with high-dimensional sparse data. This is achieved through a parameterization of similarity functions by convex combinations of sparse rank-one matrices, together wit…
Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks
National University of Singapore（新加坡国立大学）；Worcester Polytechnic Institute（伍斯特理工学院）；Brown University（布朗大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
本文将 MC-fPINN 扩展为 MC-tfPINN，并用一维 Gaussian 求积替代 Monte Carlo 采样，在高达 100,000 维的分数阶与缓增分数阶 PDE 正、反问题中提升了精度和收敛速度。
Space-time deep neural network approximations for high-dimensional partial differential equations
Faculty of Mathematics, Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院数学学院）；Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich（苏黎世联邦理工学院数学系应用数学研讨班）；Department for Applied Mathematics, University of Freiburg（弗莱堡大学应用数学系）
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）
Adversarial training is a popular method to give neural nets robustness against adversarial perturbations. In practice adversarial training leads to low robust training loss. However, a rigorous explanation for why this happens under natural conditions is still missing. Recently a convergence theory for standard (non-adversarial) supervised training was developed by various groups for {\em very overparametrized} nets. It is unclear how to extend these results to adversarial training because of the min-max objective. Recently, a first step towards this direction was made by Gao et al. using too…
College of Intelligence and Computing, Tianjin University（天津大学智能与计算学部）；Noah’s Ark Lab, Huawei（华为诺亚方舟实验室）；The Chinese University of Hong Kong（香港中文大学）；Northwestern Polytechnical University（西北工业大学）
The state space in Multiagent Reinforcement Learning (MARL) grows exponentially with the agent number. Such a curse of dimensionality results in poor scalability and low sample efficiency, inhibiting MARL for decades. To break this curse, we propose a unified agent permutation framework that exploits the permutation invariance (PI) and permutation equivariance (PE) inductive biases to reduce the multiagent state space. Our insight is that permuting the order of entities in the factored multiagent state space does not change the information. Specifically, we propose two novel implementations: a…
Machine Learning Department（机器学习系）；Department of Statistics and Data Science（统计与数据科学系）；Auton Lab, The Robotics Institute（奥顿实验室（机器人研究所））；Carnegie Mellon University（卡内基梅隆大学）
In supervised learning, we typically leverage a fully labeled dataset to design methods for function estimation or prediction. In many practical situations, we are able to obtain alternative feedback, possibly at a low cost. A broad goal is to understand the usefulness of, and to design algorithms to exploit, this alternative feedback. In this paper, we consider a semi-supervised regression setting, where we obtain additional ordinal (or comparison) information for the unlabeled samples. We consider ordinal feedback of varying qualities where we have either a perfect ordering of the samples, a…
