Chao Zhou

Chao Zhou, Yuan-Gen Wang, Zi-jia Wang et al.

Adversarial examples have shown a powerful ability to make a well-trained model misclassified. Current mainstream adversarial attack methods only consider one of the distortions among $L_0$-norm, $L_2$-norm, and $L_\infty$-norm. $L_0$-norm based methods cause large modification on a single pixel, resulting in naked-eye visible detection, while $L_2$-norm and $L_\infty$-norm based methods suffer from weak robustness against adversarial defense since they always diffuse tiny perturbations to all pixels. A more realistic adversarial perturbation should be sparse and imperceptible. In this paper, we propose a novel $L_p$-norm distortion-efficient adversarial attack, which not only owns the least $L_2$-norm loss but also significantly reduces the $L_0$-norm distortion. To this aim, we design a new optimization scheme, which first optimizes an initial adversarial perturbation under $L_2$-norm constraint, and then constructs a dimension unimportance matrix for the initial perturbation. Such a dimension unimportance matrix can indicate the adversarial unimportance of each dimension of the initial perturbation. Furthermore, we introduce a new concept of adversarial threshold for the dimension unimportance matrix. The dimensions of the initial perturbation whose unimportance is higher than the threshold will be all set to zero, greatly decreasing the $L_0$-norm distortion. Experimental results on three benchmark datasets show that under the same query budget, the adversarial examples generated by our method have lower $L_0$-norm and $L_2$-norm distortion than the state-of-the-art. Especially for the MNIST dataset, our attack reduces 8.1$\%$ $L_2$-norm distortion meanwhile remaining 47$\%$ pixels unattacked. This demonstrates the superiority of the proposed method over its competitors in terms of adversarial robustness and visual imperceptibility.

Huafu Liao, Alpár R. Mészáros, Chenchen Mou et al.

This paper deals with a class of neural SDEs and studies the limiting behavior of the associated sampled optimal control problems as the sample size grows to infinity. The neural SDEs with $N$ samples can be linked to the $N$-particle systems with centralized control. We analyze the Hamilton-Jacobi-Bellman equation corresponding to the $N$-particle system and establish regularity results which are uniform in $N$. The uniform regularity estimates are obtained by the stochastic maximum principle and the analysis of a backward stochastic Riccati equation. Using these uniform regularity results, we show the convergence of the minima of the objective functionals and optimal parameters of the neural SDEs as the sample size $N$ tends to infinity. The limiting objects can be identified with suitable functions defined on the Wasserstein space of Borel probability measures. Furthermore, quantitative convergence rates are also obtained.

Qiang Du, Yiqi Gu, Haizhao Yang et al.

Identifying hidden dynamics from observed data is a significant and challenging task in a wide range of applications. Recently, the combination of linear multistep methods (LMMs) and deep learning has been successfully employed to discover dynamics, whereas a complete convergence analysis of this approach is still under development. In this work, we consider the deep network-based LMMs for the discovery of dynamics. We put forward error estimates for these methods using the approximation property of deep networks. It indicates, for certain families of LMMs, that the $\ell^2$ grid error is bounded by the sum of $O(h^p)$ and the network approximation error, where $h$ is the time step size and $p$ is the local truncation error order. Numerical results of several physically relevant examples are provided to demonstrate our theory.