Neural networks have impressive performance on a variety of datasets (LeCun et al., 1998; He et al., 2015; Krizhevsky et al., 2017; Everingham et al., 2010) but can be fooled by imperceptible

While highly expressive parametric models including deep neural networks have an advantage to model complicated concepts, training such highly non-linear models is known to yield a high risk of notorious overfitting. To address this issue, this study considers a $(k,q)$th order variation regularization ($(k,q)$-VR), which is defined as the $q$th-powered integral of the absolute $k$th order derivative of the parametric models to be trained; penalizing the $(k,q)$-VR is expected to yield a smoother function, which is expected to avoid overfitting. Particularly, $(k,q)$-VR encompasses the conventional (general-order) total variation with $q=1$. While the $(k,q)$-VR terms applied to general parametric models are computationally intractable due to the integration, this study provides a stochastic optimization algorithm, that can efficiently train general models with the $(k,q)$-VR without conducting explicit numerical integration. The proposed approach can be applied to the training of even deep neural networks whose structure is arbitrary, as it can be implemented by only a simple stochastic gradient descent algorithm and automatic differentiation. Our numerical experiments demonstrate that the neural networks trained with the $(k,q)$-VR terms are more ``resilient'' than those with the conventional parameter regularization. The proposed algorithm also can be extended to the physics-informed training of neural networks (PINNs).

Existing defenses against adversarial examples such as adversarial training typically assume that the adversary will conform to a specific or known threat model, such as $\ell_p$ perturbations within a fixed budget. In this paper, we focus on the scenario where there is a mismatch in the threat model assumed by the defense during training, and the actual capabilities of the adversary at test time. We ask the question: if the learner trains against a specific "source" threat model, when can we expect robustness to generalize to a stronger unknown "target" threat model during test-time? Our key contribution is to formally define the problem of learning and generalization with an unforeseen adversary, which helps us reason about the increase in adversarial risk from the conventional perspective of a known adversary. Applying our framework, we derive a generalization bound which relates the generalization gap between source and target threat models to variation of the feature extractor, which measures the expected maximum difference between extracted features across a given threat model. Based on our generalization bound, we propose variation regularization (VR) which reduces variation of the feature extractor across the source threat model during training. We empirically demonstrate that using VR can lead to improved generalization to unforeseen attacks during test-time, and combining VR with perceptual adversarial training (Laidlaw et al., 2021) achieves state-of-the-art robustness on unforeseen attacks. Our code is publicly available at https://github.com/inspire-group/variation-regularization.

Traditional methods to infer compartmental epidemic models with time-varying dynamics can only capture continuous changes in the dynamic. However, many changes are discontinuous due to sudden interventions, such as city lockdown and opening of field hospitals. To model the discontinuities, this study introduces the tool of total variation regularization, which regulates the temporal changes of the dynamic parameters, such as the transmission rate. To recover the ground truth dynamic, this study designs a novel yet straightforward optimization algorithm, dubbed iterative Nelder-Mead, which repeatedly applies the Nelder-Mead algorithm. Experiments on the simulated data show that the proposed approach can qualitatively reproduce the discontinuities of the underlying dynamics. To extend this research to real data as well as to help researchers worldwide to fight against COVID-19, the author releases his research platform as an open-source package.