In the context of over-parameterization, there is a line of work demonstrating that randomly initialized (stochastic) gradient descent (GD) converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. However, the learning rate of GD for training two-layer neural networks exhibits poor dependence on the sample size and the Gram matrix, leading to a slow training process. In this paper, we show that for training two-layer $\text{ReLU}^3$ Physics-Informed Neural Networks (PINNs), the learning rate can be improved from $\mathcal{O}(λ_0)$ to $\mathcal{O}(1/\|\bm{H}^{\infty}\|_2)$, implying that GD actually enjoys a faster convergence rate. Despite such improvements, the convergence rate is still tied to the least eigenvalue of the Gram matrix, leading to slow convergence. We then develop the positive definiteness of Gram matrices with general smooth activation functions and provide the convergence analysis of natural gradient descent (NGD) in training two-layer PINNs, demonstrating that the learning rate can be $\mathcal{O}(1)$ and at this rate, the convergence rate is independent of the Gram matrix. In particular, for smooth activation functions, the convergence rate of NGD is quadratic. Numerical experiments are conducted to verify our theoretical results.
We investigate the role of noise in optimization algorithms for learning over-parameterized models. Specifically, we consider the recovery of a rank one matrix $Y^*\in R^{d\times d}$ from a noisy observation $Y$ using an over-parameterization model. We parameterize the rank one matrix $Y^*$ by $XX^\top$, where $X\in R^{d\times d}$. We then show that under mild conditions, the estimator, obtained by the randomly perturbed gradient descent algorithm using the square loss function, attains a mean square error of $O(σ^2/d)$, where $σ^2$ is the variance of the observational noise. In contrast, the …
The reinforcement learning problem of finding a control policy that minimizes the minimum time objective for the Mountain Car environment is considered. Particularly, a class of parameterized nonlinear feedback policies is optimized over to reach the top of the highest mountain peak in minimum time. The optimization is carried out using quasi-Stochastic Gradient Descent (qSGD) methods. In attempting to find the optimal minimum time policy, a new parameterized policy approach is considered that seeks to learn an optimal policy parameter for different regions of the state space, rather than rely…
The University of Tokyo（东京大学）；RIKEN（理化学研究所）；ENSAE（法国国家统计与经济管理学院）；CREST（法国国家经济统计与管理研究中心）
Recently, several studies have proven the global convergence and generalization abilities of the gradient descent method for two-layer ReLU networks. Most studies especially focused on the regression problems with the squared loss function, except for a few, and the importance of the positivity of the neural tangent kernel has been pointed out. On the other hand, the performance of gradient descent on classification problems using the logistic loss function has not been well studied, and further investigation of this problem structure is possible. In this work, we demonstrate that the separabi…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the problem of training deep neural networks with Rectified Linear Unit (ReLU) activation function using gradient descent and stochastic gradient descent. In particular, we study the binary classification problem and show that for a broad family of loss functions, with proper random weight initialization, both gradient descent and stochastic gradient descent can find the global minima of the training loss for an over-parameterized deep ReLU network, under mild assumption on the training data. The key idea of our proof is that Gaussian random initialization followed by (stochastic) gra…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
Empirical studies show that gradient-based methods can learn deep neural networks (DNNs) with very good generalization performance in the over-parameterization regime, where DNNs can easily fit a random labeling of the training data. Very recently, a line of work explains in theory that with over-parameterization and proper random initialization, gradient-based methods can find the global minima of the training loss for DNNs. However, existing generalization error bounds are unable to explain the good generalization performance of over-parameterized DNNs. The major limitation of most existing …
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…
In this paper, we study the generalization performance of global minima for implementing empirical risk minimization (ERM) on over-parameterized deep ReLU nets. Using a novel deepening scheme for deep ReLU nets, we rigorously prove that there exist perfect global minima achieving almost optimal generalization error bounds for numerous types of data under mild conditions. Since over-parameterization is crucial to guarantee that the global minima of ERM on deep ReLU nets can be realized by the widely used stochastic gradient descent (SGD) algorithm, our results indeed fill a gap between optimiza…
Duke University（杜克大学）；Princeton University（普林斯顿大学）；Stanford University（斯坦福大学）
Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an $l$-th order tensor in $(R^d)^{\otimes l}$ of rank $r$ (where $r\ll d$), can variants of gradient descent find a rank $m$ decomposition where $m > r$? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least $m = Ω(d^{l-1})$, while a variant of gradient descent can fi…
University of California, Berkeley（加州大学伯克利分校）；ICSI（国际计算机科学研究所）
Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks has motivated a theoretical analysis of the double descent phenomenon in classical models such as linear regression which can also generalize well in the over-parameterized regime. We provide the first exact non-asymptotic expressions for double descent of the minimum norm linear e…
Over-parameterization and adaptive methods have played a crucial role in the success of deep learning in the last decade. The widespread use of over-parameterization has forced us to rethink generalization by bringing forth new phenomena, such as implicit regularization of optimization algorithms and double descent with training progression. A series of recent works have started to shed light on these areas in the quest to understand -- why do neural networks generalize well? The setting of over-parameterized linear regression has provided key insights into understanding this mysterious behavi…
In this study, we investigate learning rate adaption at different levels based on the hyper-gradient descent framework and propose a method that adaptively learns the optimizer parameters by combining multiple levels of learning rates with hierarchical structures. Meanwhile, we show the relationship between regularizing over-parameterized learning rates and building combinations of adaptive learning rates at different levels. The experiments on several network architectures, including feed-forward networks, LeNet-5 and ResNet-18/34, show that the proposed multi-level adaptive approach can outp…
An operator preconditioning perspective on training in physics-informed machine learning
ETH Zürich（苏黎世联邦理工学院）；Sorbonne Université（索邦大学）；Extrality（埃克斯特拉蒂（Extrality））；ETH AI Center（苏黎世联邦理工学院人工智能中心）；Seminar for Applied Mathematics（应用数学研讨班）；Institute of Intelligent Systems and Robotics（智能系统与机器人研究所）
Peking University（北京大学）；Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；University of Washington（华盛顿大学）
It is believed that Gradient Descent (GD) induces an implicit bias towards good generalization in training machine learning models. This paper provides a fine-grained analysis of the dynamics of GD for the matrix sensing problem, whose goal is to recover a low-rank ground-truth matrix from near-isotropic linear measurements. It is shown that GD with small initialization behaves similarly to the greedy low-rank learning heuristics (Li et al., 2020) and follows an incremental learning procedure (Gissin et al., 2019): GD sequentially learns solutions with increasing ranks until it recovers the gr…
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）
We study the training and generalization of deep neural networks (DNNs) in the over-parameterized regime, where the network width (i.e., number of hidden nodes per layer) is much larger than the number of training data points. We show that, the expected $0$-$1$ loss of a wide enough ReLU network trained with stochastic gradient descent (SGD) and random initialization can be bounded by the training loss of a random feature model induced by the network gradient at initialization, which we call a neural tangent random feature (NTRF) model. For data distributions that can be classified by NTRF mod…
University of British Columbia（不列颠哥伦比亚大学）；INRIA（法国国家信息与自动化研究所）；ENS（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
Modern machine learning focuses on highly expressive models that are able to fit or interpolate the data completely, resulting in zero training loss. For such models, we show that the stochastic gradients of common loss functions satisfy a strong growth condition. Under this condition, we prove that constant step-size stochastic gradient descent (SGD) with Nesterov acceleration matches the convergence rate of the deterministic accelerated method for both convex and strongly-convex functions. We also show that this condition implies that SGD can find a first-order stationary point as efficientl…
Taylor University（泰勒大学）；Oak Ridge National Laboratory（橡树岭国家实验室）
Variational training of parameterized quantum circuits (PQCs) underpins many workflows employed on near-term noisy intermediate scale quantum (NISQ) devices. It is a hybrid quantum-classical approach that minimizes an associated cost function in order to train a parameterized ansatz. In this paper we adapt the qualitative loss landscape characterization for neural networks introduced in \cite{goodfellow2014qualitatively,li2017visualizing} and tests for connectivity used in \cite{draxler2018essentially} to study the loss landscape features in PQC training. We present results for PQCs trained on…
Sharp Generalization for Nonparametric Regression in Interpolation Space by Over-Parameterized Neural Networks Trained with Preconditioned Gradient Descent and Early Stopping
School of Computing and Augmented Intelligence（计算与增强智能学院）；Arizona State University（亚利桑那州立大学）；VecML Inc.（VecML公司）
Graduate School of Informatics, Kyoto University（京都大学信息学研究生院）；School of Informatics and Mathematical Science, Faculty of Engineering, Kyoto University（京都大学工学部信息与数学科学学院）
Over-parameterized neural networks generalize well in practice without any explicit regularization. Although it has not been proven yet, empirical evidence suggests that implicit regularization plays a crucial role in deep learning and prevents the network from overfitting. In this work, we introduce the gradient gap deviation and the gradient deflection as statistical measures corresponding to the network curvature and the Hessian matrix to analyze variations of network derivatives with respect to input parameters, and investigate how implicit regularization works in ReLU neural networks from…
Efficient Over-parameterized Matrix Sensing from Noisy Measurements via Alternating Preconditioned Gradient Descent
State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences（机器人国家重点实验室，沈阳自动化研究所，中国科学院）；University of Chinese Academy of Sciences（中国科学院大学）；State University of New York at Buffalo（纽约州立大学布法罗分校）；School of Management, Xi’an Jiaotong University（西安交通大学管理学院）
University of Bath（巴斯大学）；CNRS（法国国家科学研究中心）；ENS（巴黎高等师范学院）；PSL Université（巴黎文理研究大学）
Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for the definition of robust loss functions and scale-free functionals such as square-root Lasso. Standard approaches to deal with non-smoothness leverage either proximal splitting or coordinate descent. These approaches are effective but usually require parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a dif…
University of California, Santa Barbara（加州大学圣巴巴拉分校）
Deep neural networks with remarkably strong generalization performances are usually over-parameterized. Despite explicit regularization strategies are used for practitioners to avoid over-fitting, the impacts are often small. Some theoretical studies have analyzed the implicit regularization effect of stochastic gradient descent (SGD) on simple machine learning models with certain assumptions. However, how it behaves practically in state-of-the-art models and real-world datasets is still unknown. To bridge this gap, we study the role of SGD implicit regularization in deep learning systems. We …
Shenzhen Research Institute of Big Data（深圳大数据研究院）；Princeton University（普林斯顿大学）；HKUST（香港科技大学）
Recently, over-parameterized neural networks have been extensively analyzed in the literature. However, the previous studies cannot satisfactorily explain why fully trained neural networks are successful in practice. In this paper, we present a new theoretical framework for analyzing over-parameterized neural networks which we call neural feature repopulation. Our analysis can satisfactorily explain the empirical success of two level neural networks that are trained by standard learning algorithms. Our key theoretical result is that in the limit of infinite number of hidden neurons, over-param…
The noise in stochastic gradient descent (SGD) provides a crucial implicit regularization effect for training overparameterized models. Prior theoretical work largely focuses on spherical Gaussian noise, whereas empirical studies demonstrate the phenomenon that parameter-dependent noise -- induced by mini-batches or label perturbation -- is far more effective than Gaussian noise. This paper theoretically characterizes this phenomenon on a quadratically-parameterized model introduced by Vaskevicius et el. and Woodworth et el. We show that in an over-parameterized setting, SGD with label noise r…
California Institute of Technology（加州理工学院）；Department of Computing and Mathematical Sciences（计算与数学科学系）；Department of Electrical Engineering（电气工程系）
Stochastic descent methods (of the gradient and mirror varieties) have become increasingly popular in optimization. In fact, it is now widely recognized that the success of deep learning is not only due to the special deep architecture of the models, but also due to the behavior of the stochastic descent methods used, which play a key role in reaching "good" solutions that generalize well to unseen data. In an attempt to shed some light on why this is the case, we revisit some minimax properties of stochastic gradient descent (SGD) for the square loss of linear models---originally developed in…
Oden Institute for Computational Engineering and Sciences（奥登计算工程与科学研究所）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
The curse of dimensionality is a longstanding challenge in Bayesian inference in high dimensions. In this work, we propose a projected Stein variational gradient descent (pSVGD) method to overcome this challenge by exploiting the fundamental property of intrinsic low dimensionality of the data informed subspace stemming from ill-posedness of such problems. We adaptively construct the subspace using a gradient information matrix of the log-likelihood, and apply pSVGD to the much lower-dimensional coefficients of the parameter projection. The method is demonstrated to be more accurate and effici…
We study generalization properties of random features (RF) regression in high dimensions optimized by stochastic gradient descent (SGD) in under-/over-parameterized regime. In this work, we derive precise non-asymptotic error bounds of RF regression under both constant and polynomial-decay step-size SGD setting, and observe the double descent phenomenon both theoretically and empirically. Our analysis shows how to cope with multiple randomness sources of initialization, label noise, and data sampling (as well as stochastic gradients) with no closed-form solution, and also goes beyond the commo…
Peking University（北京大学）；Huazhong University of Science and Technology（华中科技大学）；Peng Cheng Laboratory（鹏城实验室）
A deep equilibrium model (DEQ) is implicitly defined through an equilibrium point of an infinite-depth weight-tied model with an input-injection. Instead of infinite computations, it solves an equilibrium point directly with root-finding and computes gradients with implicit differentiation. The training dynamics of over-parameterized DEQs are investigated in this study. By supposing a condition on the initial equilibrium point, we show that the unique equilibrium point always exists during the training process, and the gradient descent is proved to converge to a globally optimal solution at a …
Carnegie Mellon University（卡内基梅隆大学）；University of Southern California（南加州大学）；Peking University（北京大学）；Beijing Institute of Big Data Research（北京大数据研究院）；Massachusetts Institute of Technology（麻省理工学院）
Gradient descent finds a global minimum in training deep neural networks despite the objective function being non-convex. The current paper proves gradient descent achieves zero training loss in polynomial time for a deep over-parameterized neural network with residual connections (ResNet). Our analysis relies on the particular structure of the Gram matrix induced by the neural network architecture. This structure allows us to show the Gram matrix is stable throughout the training process and this stability implies the global optimality of the gradient descent algorithm. We further extend our …
This work is substituted by the paper in arXiv:2011.14066. Stochastic gradient descent is the de facto algorithm for training deep neural networks (DNNs). Despite its popularity, it still requires fine tuning in order to achieve its best performance. This has led to the development of adaptive methods, that claim automatic hyper-parameter optimization. Recently, researchers have studied both algorithmic classes via toy examples: e.g., for over-parameterized linear regression, Wilson et. al. (2017) shows that, while SGD always converges to the minimum-norm solution, adaptive methods show no suc…
University of California, Riverside（加州大学河滨分校）；University of Southern California（南加州大学）
Modern neural networks are typically trained in an over-parameterized regime where the parameters of the model far exceed the size of the training data. Such neural networks in principle have the capacity to (over)fit any set of labels including pure noise. Despite this, somewhat paradoxically, neural network models trained via first-order methods continue to predict well on yet unseen test data. This paper takes a step towards demystifying this phenomena. Under a rich dataset model, we show that gradient descent is provably robust to noise/corruption on a constant fraction of the labels despi…
Although deep learning has shown its powerful performance in many applications, the mathematical principles behind neural networks are still mysterious. In this paper, we consider the problem of learning a one-hidden-layer neural network with quadratic activations. We focus on the under-parameterized regime where the number of hidden units is smaller than the dimension of the inputs. We shall propose to solve the problem via a provable gradient-based method with random initialization. For the non-convex neural networks training problem we reveal that the gradient descent iterates are able to e…
Accurate estimates of network parameters are essential for modeling, monitoring, and control in power distribution systems. In this paper, we develop a physics-informed graphical learning algorithm to estimate network parameters of three-phase power distribution systems. Our proposed algorithm uses only readily available smart meter data to estimate the three-phase series resistance and reactance of the primary distribution line segments. We first develop a parametric physics-based model to replace the black-box deep neural networks in the conventional graphical neural network (GNN). Then we d…
Data-induced multiscale losses and efficient multirate gradient descent schemes
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）；Oden Institute for Computational Engineering and Sciences（奥登计算工程与科学研究所）
Gauss-Newton Natural Gradient Descent for Shape Learning
Seminar for Applied Mathematics (SAM), D-Math, ETH Zürich（应用数学研讨会（SAM）、D-Math、苏黎世联邦理工学院）；Institute for Machine Learning, JKU Linz（机器学习研究所、杰克逊维尔大学）；ETH AI Center, ETH Zürich（苏黎世联邦理工学院人工智能中心）
This paper proposes a new mean-field framework for over-parameterized deep neural networks (DNNs), which can be used to analyze neural network training. In this framework, a DNN is represented by probability measures and functions over its features (that is, the function values of the hidden units over the training data) in the continuous limit, instead of the neural network parameters as most existing studies have done. This new representation overcomes the degenerate situation where all the hidden units essentially have only one meaningful hidden unit in each middle layer, and further leads …
Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
In this paper we investigate how gradient-based algorithms such as gradient descent, (multi-pass) stochastic gradient descent, its persistent variant, and the Langevin algorithm navigate non-convex loss-landscapes and which of them is able to reach the best generalization error at limited sample complexity. We consider the loss landscape of the high-dimensional phase retrieval problem as a prototypical highly non-convex example. We observe that for phase retrieval the stochastic variants of gradient descent are able to reach perfect generalization for regions of control parameters where the gr…
The University of Tokyo（东京大学）；Austrian Academy of Sciences（奥地利科学院）；Technische Universität Wien（维也纳工业大学）
Optimizing parameterized quantum circuits is a key routine in using near-term quantum devices. However, the existing algorithms for such optimization require an excessive number of quantum-measurement shots for estimating expectation values of observables and repeating many iterations, whose cost has been a critical obstacle for practical use. We develop an efficient alternative optimization algorithm, stochastic gradient line Bayesian optimization (SGLBO), to address this problem. SGLBO reduces the measurement-shot cost by estimating an appropriate direction of updating circuit parameters bas…
An increasing number of machine learning problems, such as robust or adversarial variants of existing algorithms, require minimizing a loss function that is itself defined as a maximum. Carrying a loop of stochastic gradient ascent (SGA) steps on the (inner) maximization problem, followed by an SGD step on the (outer) minimization, is known as Epoch Stochastic Gradient \textit{Descent Ascent} (ESGDA). While successful in practice, the theoretical analysis of ESGDA remains challenging, with no clear guidance on choices for the inner loop size nor on the interplay between inner/outer step sizes.…
A Validation Approach to Over-parameterized Matrix and Image Recovery
Department of Mathematics, University of California San Diego（加州大学圣地亚哥分校数学系）；Department of Computer Science and Engineering, Ohio State University（俄亥俄州立大学计算机科学与工程系）；H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology（佐治亚理工学院H. Milton Stewart工业与系统工程学院）
College of Computer Science and Technology/Artificial Intelligence, Nanjing University of Aeronautics and Astronautics（南京航空航天大学计算机科学与技术学院/人工智能学院）；MIIT Key Laboratory of Pattern Analysis and Machine Intelligence（工业和信息化部模式分析与机器智能重点实验室）
Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems, but they are still trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit stochastic gradient descent (ISGD) method to train PINNs for improving the stability of training process. We heuristically analyze how ISGD overcome stiffness in the gradient flow dynamics of PINNs, especially for problems with multi-scale solutions. We theoretically prove t…
The Ohio State University（俄亥俄州立大学）；University of California, San Diego（加州大学圣地亚哥分校）；Halicioğlu Data Science Institute, University of California, San Diego（加州大学圣地亚哥分校哈利西奥卢数据科学研究所）
The success of deep learning is due, to a large extent, to the remarkable effectiveness of gradient-based optimization methods applied to large neural networks. The purpose of this work is to propose a modern view and a general mathematical framework for loss landscapes and efficient optimization in over-parameterized machine learning models and systems of non-linear equations, a setting that includes over-parameterized deep neural networks. Our starting observation is that optimization problems corresponding to such systems are generally not convex, even locally. We argue that instead they sa…
