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We consider the phenomenon of adversarial examples in ReLU networks with independent gaussian parameters. For networks of constant depth and with a large range of widths (for instance, it suffices if the width of each layer is polynomial in…

Recent works show that adversarial examples exist for random neural networks [Daniely and Schacham, 2020] and that these examples can be found using a single step of gradient ascent [Bubeck et al., 2021]. In this work, we extend this line…

We prove that two-layer (Leaky)ReLU networks initialized by e.g. the widely used method proposed by He et al. (2015) and trained using gradient descent on a least-squares loss are not universally consistent. Specifically, we describe a…

We revisit the problem of learning a single neuron with ReLU activation under Gaussian input with square loss. We particularly focus on the over-parameterization setting where the student network has $n\ge 2$ neurons. We prove the global…

The implicit bias towards solutions with favorable properties is believed to be a key reason why neural networks trained by gradient-based optimization can generalize well. While the implicit bias of gradient flow has been widely studied…

Deep learning models are often successfully trained using gradient descent, despite the worst case hardness of the underlying non-convex optimization problem. The key question is then under what conditions can one prove that optimization…

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…

Despite a great deal of research, it is still not well-understood why trained neural networks are highly vulnerable to adversarial examples. In this work we focus on two-layer neural networks trained using data which lie on a low…

In this paper, we provide sufficient conditions of benign overfitting of fixed width leaky ReLU two-layer neural network classifiers trained on mixture data via gradient descent. Our results are derived by establishing directional…

A substantial body of empirical work documents the lack of robustness in deep learning models to adversarial examples. Recent theoretical work proved that adversarial examples are ubiquitous in two-layers networks with sub-exponential width…

The implicit biases of gradient-based optimization algorithms are conjectured to be a major factor in the success of modern deep learning. In this work, we investigate the implicit bias of gradient flow and gradient descent in two-layer…

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,…

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…

We consider ReLU networks with random weights, in which the dimension decreases at each layer. We show that for most such networks, most examples $x$ admit an adversarial perturbation at an Euclidean distance of…

Machine Unlearning aims to remove specific data from trained models, addressing growing privacy and ethical concerns. We provide a theoretical analysis of a simple and widely used method - gradient ascent - used to reverse the influence of…

While deep learning has outperformed other methods for various tasks, theoretical frameworks that explain its reason have not been fully established. To address this issue, we investigate the excess risk of two-layer ReLU neural networks in…

We give a simple proof for the global convergence of gradient descent in training deep ReLU networks with the standard square loss, and show some of its improvements over the state-of-the-art. In particular, while prior works require all…

Neural networks have many successful applications, while much less theoretical understanding has been gained. Towards bridging this gap, we study the problem of learning a two-layer overparameterized ReLU neural network for multi-class…

A key challenge in modern deep learning theory is to explain the remarkable success of gradient-based optimization methods when training large-scale, complex deep neural networks. Though linear convergence of such methods has been proved…

We develop Banach spaces for ReLU neural networks of finite depth $L$ and infinite width. The spaces contain all finite fully connected $L$-layer networks and their $L^2$-limiting objects under bounds on the natural path-norm. Under this…