Related papers: Global Convergence of SGD For Logistic Loss on Two…
In this note, we consider appropriately regularized $\ell_2-$empirical risk of depth $2$ nets with any number of gates and show bounds on how the empirical loss evolves for SGD iterates on it -- for arbitrary data and if the activation is…
The phenomenon that stochastic gradient descent (SGD) favors flat minima has played a critical role in understanding the implicit regularization of SGD. In this paper, we provide an explanation of this striking phenomenon by relating the…
In recent years, stochastic gradient descent (SGD) based techniques has become the standard tools for training neural networks. However, formal theoretical understanding of why SGD can train neural networks in practice is largely missing.…
The remarkable generalization ability of neural networks is usually attributed to the implicit bias of SGD, which often yields models with lower complexity using simpler (e.g. linear) and low-rank features. Recent works have provided…
In many numerical simulations stochastic gradient descent (SGD) type optimization methods perform very effectively in the training of deep neural networks (DNNs) but till this day it remains an open problem of research to provide a…
Implicit deep learning has received increasing attention recently due to the fact that it generalizes the recursive prediction rules of many commonly used neural network architectures. Its prediction rule is provided implicitly based on the…
Stochastic Gradient Descent (SGD) is a central tool in machine learning. We prove that SGD converges to zero loss, even with a fixed (non-vanishing) learning rate - in the special case of homogeneous linear classifiers with smooth monotone…
In this paper, we examine the time it takes for stochastic gradient descent (SGD) to reach the global minimum of a general, non-convex loss function. We approach this question through the lens of randomly perturbed dynamical systems and…
Stochastic gradient descent (SGD) has been found to be surprisingly effective in training a variety of deep neural networks. However, there is still a lack of understanding on how and why SGD can train these complex networks towards a…
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…
Multi-layer neural networks are among the most powerful models in machine learning, yet the fundamental reasons for this success defy mathematical understanding. Learning a neural network requires to optimize a non-convex high-dimensional…
Non-convex optimization problems are ubiquitous in machine learning, especially in Deep Learning. While such complex problems can often be successfully optimized in practice by using stochastic gradient descent (SGD), theoretical analysis…
Deep neural networks (DNNs) have demonstrated dominating performance in many fields; since AlexNet, networks used in practice are going wider and deeper. On the theoretical side, a long line of works has been focusing on training neural…
The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is violated for cases where the objective…
Normalized gradient descent has shown substantial success in speeding up the convergence of exponentially-tailed loss functions (which includes exponential and logistic losses) on linear classifiers with separable data. In this paper, we go…
The convergence of GD and SGD when training mildly parameterized neural networks starting from random initialization is studied. For a broad range of models and loss functions, including the most commonly used square loss and cross entropy…
The $L_{2}$-regularized loss of Deep Linear Networks (DLNs) with more than one hidden layers has multiple local minima, corresponding to matrices with different ranks. In tasks such as matrix completion, the goal is to converge to the local…
The generalization mystery of overparametrized deep nets has motivated efforts to understand how gradient descent (GD) converges to low-loss solutions that generalize well. Real-life neural networks are initialized from small random values…
We study the gradient descent (GD) dynamics of a depth-2 linear neural network with a single input and output. We show that GD converges at an explicit linear rate to a global minimum of the training loss, even with a large stepsize --…
The multiplicative structure of parameters and input data in the first layer of neural networks is explored to build connection between the landscape of the loss function with respect to parameters and the landscape of the model function…