In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Input: Consider Input: Deep neural networks are known to be vulnerable to adversarial perturbations. In this paper, we bridge adversarial robustness of neural nets with Lyapunov stability of dynamical systems. From this viewpoint, training neural nets is equivalent to finding an optimal control of the discrete dynamical system, which allows one to utilize methods of successive approximations, an optimal control algorithm based on Pontryagin's maximum principle, to train neural nets. This decoupled training method allows us to add constraints to the optimization, which makes the deep model more robust. The constrained optimization problem can be formulated as a semi-definite programming problem and hence can be solved efficiently. Experiments show that our method effectively improves deep model's adversarial robustness.

Output: An adversarial defense method bridging robustness of deep neural nets with Lyapunov stability


Input: Consider Input: We extend the recent results of (Arora et al., 2019) by a spectral analysis of representations corresponding to kernel and neural embeddings. They showed that in a simple single layer network, the alignment of the labels to the eigenvectors of the corresponding Gram matrix determines both the convergence of the optimization during training as well as the generalization properties. We generalize their result to kernel and neural representations and show that these extensions improve both optimization and generalization of the basic setup studied in (Arora et al., 2019).

Output: Spectral analysis for understanding how different representations can improve optimization and generalization.


Input: Consider Input: This paper is focused on investigating and demystifying an intriguing robustness phenomena in over-parameterized neural network training. In particular we provide empirical and theoretical evidence that first order methods such as gradient descent are provably robust to noise/corruption on a constant fraction of the labels despite over-parameterization under a rich dataset model. In particular: i) First, we show that in the first few iterations where the updates are still in the vicinity of the initialization these algorithms only fit to the correct labels essentially ignoring the noisy labels. ii) Secondly, we prove that to start to overfit to the noisy labels these algorithms must stray rather far from from the initial model which can only occur after many more iterations. Together, these show that gradient descent with early stopping is provably robust to label noise and shed light on empirical robustness of deep networks as well as commonly adopted early-stopping heuristics.


Output: We prove that gradient descent is robust to label corruption despite over-parameterization under a rich dataset model.