ACHIEVING OPTIMAL ADVERSARIAL ACCURACY FOR ADVERSARIAL DEEP LEARNING USING STACKELBERG GAMES

The purpose of adversarial deep learning is to train robust DNNs against adversarial attacks,and this is one of the major research focuses of deep learning.Game theory has been used to answer some of the basic questions about adversarial deep learning,such as those regarding the existence of a classifier with optimal robustness and the existence of optimal adversarial samples for a given class of classifiers.In most previous works,adversarial deep learning was formulated as a simultaneous game and the strategy spaces were assumed to be certain probability distributions in order for the Nash equilibrium to exist.However,this assumption is not applicable to practical situations.In this paper,we give answers to these basic questions for the practical case where the classifiers are DNNs with a given structure;we do that by formulating adversarial deep learning in the form of Stackelberg games.The existence of Stackelberg equilibria for these games is proven.Furthermore,it is shown that the equilibrium DNN has the largest adversarial accuracy among all DNNs with the same structure,when Carlini-Wagner s margin loss is used.The trade-off between robustness and accuracy in adversarial deep learning is also studied from a game theoretical perspective.

PROPER REPARAMETRIZATION FOR INHERENTLY IMPROPER UNIRATIONAL VARIETIES

In this paper, a class of lattice supports in the lattice space Zm is found to be inherently improper because any rational parametrization from Cm to Cm defined on such a support is improper. The improper index for such a lattice support is defined to be the gcd of the normalized volumes of all the simplex sub-supports. The structure of an improper support S is analyzed and shrinking transformations are constructed to transform S to a proper one. For a generic rational parametrization RP defined on an improper support S, we prove that its improper index is the improper index of S and give a proper reparametrization algorithm for RP. Finally, properties for rational parametrizations defined on an improper support and with numerical coefficients are also considered.

ON FUNCTIONAL DECOMPOSITION OF MULTIVARIATE POLYNOMIALS WITH DIFFERENTIATION AND HOMOGENIZATION

This paper gives a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which were proposed by Ye, Dai, and Lam （1999） and were developed by Faugere, Perret （2006, 2008, 2009）. The authors show that a degree proper functional decomposition for a set of randomly decomposable quartic homoge- nous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dal, and Lam （1999）. The authors also propose a conjecture which asserts that the decomposition for a set of polynomials can be computed from that of its homogenization and show that the conjecture is valid with high probability for quartic polynomials. Finally, the authors prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space.

Preface

The Silkroad Mathematics Center（SMC）was established in September 2016 by the Chinese Mathematical Society under the support of the China Association for Science and Technology.The main task of the center is to promote mathematics exchanges and cooperation among the countries along the Belt and Road.Professor Ya-xiang Yuan is the current director of SMC.The founding member societies of SMC include Chinese Mathematical Society,