相关论文: Optimization and generalization analysis for two-l…

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,…

While stochastic gradient descent (SGD) and variants have been surprisingly successful for training deep nets, several aspects of the optimization dynamics and generalization are still not well understood. In this paper, we present new…

Overparametrized neural networks trained by gradient descent (GD) can provably overfit any training data. However, the generalization guarantee may not hold for noisy data. From a nonparametric perspective, this paper studies how well…

The generalized Gauss-Newton (GGN) optimization method incorporates curvature estimates into its solution steps, and provides a good approximation to the Newton method for large-scale optimization problems. GGN has been found particularly…

Physics-Informed Neural Networks (PINNs) often suffer from slow convergence, training instability, and reduced accuracy on challenging partial differential equations due to the anisotropic and rapidly varying geometry of their loss…

Several works have aimed to explain why overparameterized neural networks generalize well when trained by Stochastic Gradient Descent (SGD). The consensus explanation that has emerged credits the randomized nature of SGD for the bias of the…

Many supervised learning tasks have intrinsic symmetries, such as translational and rotational symmetry in image classifications. These symmetries can be exploited to enhance performance. We formulate the symmetry constraints into a concise…

This study investigates the potential accuracy boundaries of physics-informed neural networks, contrasting their approach with previous similar works and traditional numerical methods. We find that selecting improved optimization algorithms…

We present novel empirical observations regarding how stochastic gradient descent (SGD) navigates the loss landscape of over-parametrized deep neural networks (DNNs). These observations expose the qualitatively different roles of learning…

In this paper we analyze the behaviour of the stochastic gradient descent (SGD), a widely used method in supervised learning for optimizing neural network weights via a minimization of non-convex loss functions. Since the pioneering work of…

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…

Stochastic gradient descent (SGD) is widely believed to perform implicit regularization when used to train deep neural networks, but the precise manner in which this occurs has thus far been elusive. We prove that SGD minimizes an average…

First-order methods such as stochastic gradient descent (SGD) are currently the standard algorithm for training deep neural networks. Second-order methods, despite their better convergence rate, are rarely used in practice due to the…

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,…

Most modern learning problems are highly overparameterized, meaning that there are many more parameters than the number of training data points, and as a result, the training loss may have infinitely many global minima (parameter vectors…

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional PDEs, as Richard E. Bellman first pointed…

Despite an extensive body of literature on deep learning optimization, our current understanding of what makes an optimization algorithm effective is fragmented. In particular, we do not understand well whether enhanced optimization…

In this work, we propose a new training method for finding minimum weight norm solutions in over-parameterized neural networks (NNs). This method seeks to improve training speed and generalization performance by framing NN training as a…

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…

We study the convergence of gradient descent (GD) and stochastic gradient descent (SGD) for training $L$-hidden-layer linear residual networks (ResNets). We prove that for training deep residual networks with certain linear transformations…