We study the overparametrization bounds required for the global convergence of stochastic gradient descent algorithm for a class of one hidden layer feed-forward neural networks, considering most of the activation functions used in practice, including ReLU. We improve the existing state-of-the-art results in terms of the required hidden layer width. We introduce a new proof technique combining nonlinear analysis with properties of random initializations of the network. First, we establish the global convergence of continuous solutions of the differential inclusion being a nonsmooth analogue of the gradient flow for the MSE loss. Second, we provide a technical result (working also for general approximators) relating solutions of the aforementioned differential inclusion to the (discrete) stochastic gradient descent sequences, hence establishing linear convergence towards zero loss for the stochastic gradient descent iterations.
We consider the problem of training a deep orthogonal linear network, which consists of a product of orthogonal matrices, with no non-linearity in-between. We show that training the weights with Riemannian gradient descent is equivalent to training the whole factorization by gradient descent. This means that there is no effect of overparametrization and implicit bias at all in this setting: training such a deep, overparametrized, network is perfectly equivalent to training a one-layer shallow network.
Smoothing the Edges: Smooth Optimization for Sparse Regularization using Hadamard Overparametrization
Department of Statistics, LMU Munich（慕尼黑大学统计系）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）；Institute of Computational Biology, Helmholtz Munich（亥姆霍兹慕尼黑计算生物学研究所）；Center for Computational Mathematics, Flatiron Institute（熨斗研究所计算数学中心）
A recent line of work has shown that an overparametrized neural network can perfectly fit the training data, an otherwise often intractable nonconvex optimization problem. For (fully-connected) shallow networks, in the best case scenario, the existing theory requires quadratic over-parametrization as a function of the number of training samples. This paper establishes that linear overparametrization is sufficient to fit the training data, using a simple variant of the (stochastic) gradient descent. Crucially, unlike several related works, the training considered in this paper is not limited to…
CSAIL, MIT（麻省理工学院计算机科学与人工智能实验室）；GoogleX（谷歌X实验室）；Center for Brains, Minds and Machines, MIT（麻省理工学院大脑、心智与机器中心）；IIT（意大利理工学院）；Claremont Graduate University（克莱蒙特研究生大学）；Alphabet (Google) X（Alphabet（谷歌）X实验室）
A main puzzle of deep networks revolves around the absence of overfitting despite large overparametrization and despite the large capacity demonstrated by zero training error on randomly labeled data. In this note, we show that the dynamics associated to gradient descent minimization of nonlinear networks is topologically equivalent, near the asymptotically stable minima of the empirical error, to linear gradient system in a quadratic potential with a degenerate (for square loss) or almost degenerate (for logistic or crossentropy loss) Hessian. The proposition depends on the qualitative theory…
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Recent theoretical work has guaranteed that overparameterized networks trained by gradient descent achieve arbitrarily low training error, and sometimes even low test error. The required width, however, is always polynomial in at least one of the sample size $n$, the (inverse) target error $1/ε$, and the (inverse) failure probability $1/δ$. This work shows that $\widetildeΘ(1/ε)$ iterations of gradient descent with $\widetildeΩ(1/ε^2)$ training examples on two-layer ReLU networks of any width exceeding $\mathrm{polylog}(n,1/ε,1/δ)$ suffice to achieve a test misclassification error of $ε$. We a…
Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially…
Two aspects of neural networks that have been extensively studied in the recent literature are their function approximation properties and their training by gradient descent methods. The approximation problem seeks accurate approximations with a minimal number of weights. In most of the current literature these weights are fully or partially hand-crafted, showing the capabilities of neural networks but not necessarily their practical performance. In contrast, optimization theory for neural networks heavily relies on an abundance of weights in over-parametrized regimes. This paper balances thes…
University of California, Berkeley（加州大学伯克利分校）；Nvidia Corporation（英伟达公司）
This paper defines linear range as the range of parameter perturbations which lead to approximately linear perturbations in the states of a network. We compute linear range from the difference between actual perturbations in states and the tangent solution. Linear range is a new criterion for estimating the effectivenss of gradients and thus having many possible applications. In particular, we propose that the optimal learning rate at the initial stages of training is such that parameter changes on all minibatches are within linear range. We demonstrate our algorithm on two shallow neural netw…
We revisit on-average algorithmic stability of GD for training overparameterised shallow neural networks and prove new generalisation and excess risk bounds without the NTK or PL assumptions. In particular, we show oracle type bounds which reveal that the generalisation and excess risk of GD is controlled by an interpolating network with the shortest GD path from initialisation (in a sense, an interpolating network with the smallest relative norm). While this was known for kernelised interpolants, our proof applies directly to networks trained by GD without intermediate kernelisation. At the s…
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 …
In this paper we first identify a basic limitation in gradient descent-based optimization methods when used in conjunctions with smooth kernels. An analysis based on the spectral properties of the kernel demonstrates that only a vanishingly small portion of the function space is reachable after a polynomial number of gradient descent iterations. This lack of approximating power drastically limits gradient descent for a fixed computational budget leading to serious over-regularization/underfitting. The issue is purely algorithmic, persisting even in the limit of infinite data. To address this s…
Princeton University（普林斯顿大学）；Institute for Advanced Study（高等研究院）；Google Brain（谷歌大脑）
Conventional wisdom in deep learning states that increasing depth improves expressiveness but complicates optimization. This paper suggests that, sometimes, increasing depth can speed up optimization. The effect of depth on optimization is decoupled from expressiveness by focusing on settings where additional layers amount to overparameterization - linear neural networks, a well-studied model. Theoretical analysis, as well as experiments, show that here depth acts as a preconditioner which may accelerate convergence. Even on simple convex problems such as linear regression with $\ell_p$ loss, …
University of Birmingham（伯明翰大学）；Alibaba Group（阿里巴巴集团）；State University of New York at Albany（纽约州立大学奥尔巴尼分校）
While significant theoretical progress has been achieved, unveiling the generalization mystery of overparameterized neural networks still remains largely elusive. In this paper, we study the generalization behavior of shallow neural networks (SNNs) by leveraging the concept of algorithmic stability. We consider gradient descent (GD) and stochastic gradient descent (SGD) to train SNNs, for both of which we develop consistent excess risk bounds by balancing the optimization and generalization via early-stopping. As compared to existing analysis on GD, our new analysis requires a relaxed overpara…
A method named simple2complex for modeling and training deep neural networks is proposed. Simple2complex train deep neural networks by smoothly adding more and more layers to the shallow networks, as the learning procedure going on, the network is just like growing. Compared with learning by end2end, simple2complex is with less possibility trapping into local minimal, namely, owning ability for global optimization. Cifar10 is used for verifying the superiority of simple2complex.
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Department of Chemistry, Stanford University（斯坦福大学化学系）；Center for Data Science, New York University（纽约大学数据科学中心）
Recent theoretical works have characterized the dynamics of wide shallow neural networks trained via gradient descent in an asymptotic mean-field limit when the width tends towards infinity. At initialization, the random sampling of the parameters leads to deviations from the mean-field limit dictated by the classical Central Limit Theorem (CLT). However, since gradient descent induces correlations among the parameters, it is of interest to analyze how these fluctuations evolve. Here, we use a dynamical CLT to prove that the asymptotic fluctuations around the mean limit remain bounded in mean …
Carnegie Mellon University（卡内基梅隆大学）；Massachusetts Institute of Technology（麻省理工学院）
One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies this surprising phenomenon for two-layer fully connected ReLU activated neural networks. For an $m$ hidden node shallow neural network with ReLU activation and $n$ training data, we show as long as $m$ is large enough and no two inputs are parallel, randomly initialized gradient descent converges to a globally optimal solution at a linear convergence rate fo…
University College London（伦敦大学学院）；Inria（法国国家信息与自动化研究所）
We focus on a specific class of shallow neural networks with a single hidden layer, namely those with $L_2$-normalised data and either a sigmoid-shaped Gaussian error function ("erf") activation or a Gaussian Error Linear Unit (GELU) activation. For these networks, we derive new generalisation bounds through the PAC-Bayesian theory; unlike most existing such bounds they apply to neural networks with deterministic rather than randomised parameters. Our bounds are empirically non-vacuous when the network is trained with vanilla stochastic gradient descent on MNIST and Fashion-MNIST.
The key to generalization is controlling the complexity of the network. However, there is no obvious control of complexity -- such as an explicit regularization term -- in the training of deep networks for classification. We will show that a classical form of norm control -- but kind of hidden -- is present in deep networks trained with gradient descent techniques on exponential-type losses. In particular, gradient descent induces a dynamics of the normalized weights which converge for $t \to \infty$ to an equilibrium which corresponds to a minimum norm (or maximum margin) solution. For suffic…
The evolution of a deep neural network trained by the gradient descent can be described by its neural tangent kernel (NTK) as introduced in [20], where it was proven that in the infinite width limit the NTK converges to an explicit limiting kernel and it stays constant during training. The NTK was also implicit in some other recent papers [6,13,14]. In the overparametrization regime, a fully-trained deep neural network is indeed equivalent to the kernel regression predictor using the limiting NTK. And the gradient descent achieves zero training loss for a deep overparameterized neural network.…
Johns Hopkins University（约翰斯·霍普金斯大学）；Technion, Israel Institute of Technology（以色列理工学院）；Colorado School of Mines（科罗拉多矿业学院）
We examine the squared error loss landscape of shallow linear neural networks. We show---with significantly milder assumptions than previous works---that the corresponding optimization problems have benign geometric properties: there are no spurious local minima and the Hessian at every saddle point has at least one negative eigenvalue. This means that at every saddle point there is a directional negative curvature which algorithms can utilize to further decrease the objective value. These geometric properties imply that many local search algorithms (such as the gradient descent which is widel…
Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学）；Inovalabs Digital S.L. (TECHEYE)（Inovalabs Digital有限公司（TECHEYE））
We propose to optimize neural networks with a uniformly-distributed random learning rate. The associated stochastic gradient descent algorithm can be approximated by continuous stochastic equations and analyzed within the Fokker-Planck formalism. In the small learning rate regime, the training process is characterized by an effective temperature which depends on the average learning rate, the mini-batch size and the momentum of the optimization algorithm. By comparing the random learning rate protocol with cyclic and constant protocols, we suggest that the random choice is generically the best…
University of Chicago（芝加哥大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Steklov Mathematical Institute（斯捷克洛夫数学研究所）
We continue a long line of research aimed at proving convergence of depth 2 neural networks, trained via gradient descent, to a global minimum. Like in many previous works, our model has the following features: regression with quadratic loss function, fully connected feedforward architecture, RelU activations, Gaussian data instances and network initialization, adversarial labels. It is more general in the sense that we allow both layers to be trained simultaneously and at {\em different} rates. Our results improve on state-of-the-art [Oymak Soltanolkotabi 20] (training the first layer only) a…
Université Paris-Saclay（巴黎-萨克雷大学）；CNRS（法国国家科学研究中心）；CEA（法国原子能和替代能源委员会）；Institut de physique théorique（理论物理研究所）；Courant Institute, New York University（纽约大学柯朗研究所）
We study the dynamics of optimization and the generalization properties of one-hidden layer neural networks with quadratic activation function in the over-parametrized regime where the layer width $m$ is larger than the input dimension $d$. We consider a teacher-student scenario where the teacher has the same structure as the student with a hidden layer of smaller width $m^*\le m$. We describe how the empirical loss landscape is affected by the number $n$ of data samples and the width $m^*$ of the teacher network. In particular we determine how the probability that there be no spurious minima …
We consider shallow (single hidden layer) neural networks and characterize their performance when trained with stochastic gradient descent as the number of hidden units $N$ and gradient descent steps grow to infinity. In particular, we investigate the effect of different scaling schemes, which lead to different normalizations of the neural network, on the network's statistical output, closing the gap between the $1/\sqrt{N}$ and the mean-field $1/N$ normalization. We develop an asymptotic expansion for the neural network's statistical output pointwise with respect to the scaling parameter as t…
University of California, Riverside（加州大学河滨分校）；University of Southern California（南加州大学）
Many modern neural network architectures are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Sufficiently overparameterized neural network architectures in principle have the capacity to fit any set of labels including random noise. However, given the highly nonconvex nature of the training landscape it is not clear what level and kind of overparameterization is required for first order methods to converge to a global optima that perfectly interpolate any labels. A number of recent theoretical works have shown that for very wide…
In this paper we study the problem of learning a shallow artificial neural network that best fits a training data set. We study this problem in the over-parameterized regime where the number of observations are fewer than the number of parameters in the model. We show that with quadratic activations the optimization landscape of training such shallow neural networks has certain favorable characteristics that allow globally optimal models to be found efficiently using a variety of local search heuristics. This result holds for an arbitrary training data of input/output pairs. For differentiable…
Bielefeld Universität（比勒费尔德大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Institute of Mathematics of NAS of Ukraine（乌克兰国家科学院数学研究所）
The convergence of stochastic interacting particle systems in the mean-field limit to solutions of conservative stochastic partial differential equations is established, with optimal rate of convergence. As a second main result, a quantitative central limit theorem for such SPDEs is derived, again, with optimal rate of convergence. The results apply, in particular, to the convergence in the mean-field scaling of stochastic gradient descent dynamics in overparametrized, shallow neural networks to solutions of SPDEs. It is shown that the inclusion of fluctuations in the limiting SPDE improves th…
Center for Data Science, New York University（纽约大学数据科学中心）；Inria - Département d’Informatique de l’École Normale Supérieure（法国国家信息与自动化研究所 - 巴黎高等师范学校计算机科学系）；PSL Research University（巴黎文理研究大学）
Deep networks are often considered to be more expressive than shallow ones in terms of approximation. Indeed, certain functions can be approximated by deep networks provably more efficiently than by shallow ones, however, no tractable algorithms are known for learning such deep models. Separately, a recent line of work has shown that deep networks trained with gradient descent may behave like (tractable) kernel methods in a certain over-parameterized regime, where the kernel is determined by the architecture and initialization, and this paper focuses on approximation for such kernels. We show …
In this paper we study the problem of learning Rectified Linear Units (ReLUs) which are functions of the form $max(0,<w,x>)$ with $w$ denoting the weight vector. We study this problem in the high-dimensional regime where the number of observations are fewer than the dimension of the weight vector. We assume that the weight vector belongs to some closed set (convex or nonconvex) which captures known side-information about its structure. We focus on the realizable model where the inputs are chosen i.i.d.~from a Gaussian distribution and the labels are generated according to a planted weight vect…
University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
The NP-hard problem of optimizing a shallow ReLU network can be characterized as a combinatorial search over each training example's activation pattern followed by a constrained convex problem given a fixed set of activation patterns. We explore the implications of this combinatorial aspect of ReLU optimization in this work. We show that it can be naturally modeled via a geometric and combinatoric object known as a zonotope with its vertex set isomorphic to the set of feasible activation patterns. This assists in analysis and provides a foundation for further research. We demonstrate its usefu…
University of Waterloo（滑铁卢大学）；Vector Research Institute（向量研究所）；École de technologie supérieure ÉTS（高等工程技术学院）
The simplicity of gradient descent (GD) made it the default method for training ever-deeper and complex neural networks. Both loss functions and architectures are often explicitly tuned to be amenable to this basic local optimization. In the context of weakly-supervised CNN segmentation, we demonstrate a well-motivated loss function where an alternative optimizer (ADM) achieves the state-of-the-art while GD performs poorly. Interestingly, GD obtains its best result for a "smoother" tuning of the loss function. The results are consistent across different network architectures. Our loss is motiv…
The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；Shenzhen Research Institute of Big Data（深圳大数据研究院）；University of Münster（明斯特大学）
In this article we study fully-connected feedforward deep ReLU ANNs with an arbitrarily large number of hidden layers and we prove convergence of the risk of the GD optimization method with random initializations in the training of such ANNs under the assumption that the unnormalized probability density function of the probability distribution of the input data of the considered supervised learning problem is piecewise polynomial, under the assumption that the target function (describing the relationship between input data and the output data) is piecewise polynomial, and under the assumption …
Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics
Institut de Mathematiques de Bourgogne, Universite Bourgogne Europe, CNRS（勃艮第数学研究所，勃艮第欧洲大学，法国国家科学研究中心）；Universidad Autónoma de Madrid（马德里自治大学）；Fundación Deusto（德乌斯托基金会）
To enable learning on edge devices with fast convergence and low memory, we present a novel backpropagation-free optimization algorithm dubbed Target Projection Stochastic Gradient Descent (tpSGD). tpSGD generalizes direct random target projection to work with arbitrary loss functions and extends target projection for training recurrent neural networks (RNNs) in addition to feedforward networks. tpSGD uses layer-wise stochastic gradient descent (SGD) and local targets generated via random projections of the labels to train the network layer-by-layer with only forward passes. tpSGD doesn't requ…
Iowa State University（爱荷华州立大学）；Texas A&M University（得克萨斯农工大学）
We provide a series of results for unsupervised learning with autoencoders. Specifically, we study shallow two-layer autoencoder architectures with shared weights. We focus on three generative models for data that are common in statistical machine learning: (i) the mixture-of-gaussians model, (ii) the sparse coding model, and (iii) the sparsity model with non-negative coefficients. For each of these models, we prove that under suitable choices of hyperparameters, architectures, and initialization, autoencoders learned by gradient descent can successfully recover the parameters of the correspon…
Multi-dimensional Hawkes process (MHP) is a class of self and mutually exciting point processes that find wide range of applications -- from prediction of earthquakes to modelling of order books in high frequency trading. This paper makes two major contributions, we first find an unbiased estimator for the log-likelihood estimator of the Hawkes process to enable efficient use of the stochastic gradient descent method for maximum likelihood estimation. The second contribution is, we propose a specific single hidden layered neural network for the non-parametric estimation of the underlying kerne…
University of Cambridge（剑桥大学）；Harvard University（哈佛大学）
Finding parameters that minimise a loss function is at the core of many machine learning methods. The Stochastic Gradient Descent algorithm is widely used and delivers state of the art results for many problems. Nonetheless, Stochastic Gradient Descent typically cannot find the global minimum, thus its empirical effectiveness is hitherto mysterious. We derive a correspondence between parameter inference and free energy minimisation in statistical physics. The degree of undersampling plays the role of temperature. Analogous to the energy-entropy competition in statistical physics, wide but shal…
We explore the ability of overparameterized shallow neural networks to learn Lipschitz regression functions with and without label noise when trained by Gradient Descent (GD). To avoid the problem that in the presence of noisy labels, neural networks trained to nearly zero training error are inconsistent on this class, we propose an early stopping rule that allows us to show optimal rates. This provides an alternative to the result of Hu et al. (2021) who studied the performance of $\ell 2$ -regularized GD for training shallow networks in nonparametric regression which fully relied on the infi…
