The theory of Koopman operators allows to deploy non-parametric machine learning algorithms to predict and analyze complex dynamical systems.

Cohen et al. (2021) empirically study the evolution of the largest eigenvalue of the loss Hessian, also known as sharpness, along the gradient descent (GD) trajectory and observe the Edge of Stability (EoS) phenomenon.

We explore the methodology and theory of reward-directed generation via conditional diffusion models.

We explore the methodology and theory of reward-directed generation via conditional diffusion models.

They are used to test the proposed enhancement approach iGA T. In general, ADP employs an ensemble by averaging, i.e., (C 1) ( C 1) Adversarial examples are generated to compute the losses by using the PGD attack. Our main theorem builds on a supporting Lemma 2.1. We start from the cross-entropy loss curvature measured by Eq. The above new expression of T (x) helps bound the difference between h(x) and h(x). Note that these three cases are mutually exclusive.

Guided by this theory, we propose an effective approach to improve ensemble adversarial defense, named interactive global adversarial training (iGA T).

The proof of this lemma requires Lemma A.1, which characterizes the distribution of the residual By Pinsker's inequality, this implies d By Lemma A.1, we have E[ X ( null w w The proof is inspired by Theorem 11.2 in [20], with modifications to our setting. First, we construct a "ghost" dataset The most challenging aspect of the ReLU setting is that we do not have an expression for the TV suffered by the MLE, such as Lemma 4.2 in the linear case. The proof of this Lemma, as well as other Lemmas in this section, can be found in Appendix B.1. Using Lemma B.2 and Lemma B.3, we can form a uniform bound, such that all A straight forward combination of Lemma 4.3 and Lemma B.4 gives the following Theorem. Now we can apply Bernstein's inequality (Theorem 2.10 of [8]).