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Developing Hessian-free second-order adversarial examples for adversarial training

Qian Yaguan, Zhang Liangjian, Wang Yuqi, Ji Boyuan, Yao Tengteng, Wang Bin

International Journal of Applied Mathematics and Computer Science

2024

Vol. 34, no. 3

425--438

Recent studies show that deep neural networks (DNNs) are extremely vulnerable to elaborately designed adversarial examples. Adversarial training, which uses adversarial examples as training data, has been proven to be one of the most effective methods of defense against adversarial attacks. However, most existing adversarial training methods use adversarial examples relying on first-order gradients, which perform poorly against second-order adversarial attacks and make it difficult to further improve the robustness of the model. In contrast to first-order gradients, second-order gradients provide a more accurate approximation of the loss landscape relative to natural examples. Therefore, our work focuses on constructing second-order adversarial examples and utilizing them for training DNNs. However, second-order optimization involves computing the Hessian inverse, which typically consumes considerable time. To address this issue, we propose an approximation method that transforms the problem into optimization within the Krylov subspace. Compared with the Euclidean space, the Krylov subspace method typically does not require storing the entire matrix. It only needs to store vectors and intermediate results, avoiding explicitly calculating the complete Hessian matrix. We approximate the adversarial direction by a linear combination of Hessian-vector products in the Krylov subspace to reduce the computation cost. Because of the non-symmetrical Hessian matrix, we use the generalized minimum residual to search for an approximate polynomial solution of the matrix. Our method efficiently reduces computational complexity and accelerates the training process. Extensive experiments conducted on the MNIST, CIFAR-10, and ImageNet-100 datasets demonstrate that our adversarial learning using second-order adversarial samples outperforms other first-order methods, leading to improved model robustness against various attacks.

Equilibrium, kinetic and thermodynamic studies on methylene blue adsorption by Trichosanthes kirilowii Maxim shell activated carbon

Wang Yuqi, Li Yanhui, Zheng Heng

Polish Journal of Chemical Technology

2019

Vol. 21, nr 4

89--97

New kind of adsorbent was produced from Trichosanthes kirilowii Maxim shell. The KOH activation technology for preparation of Trichosanthes kirilowii Maxim shell activated carbon (TKMCK) was optimized. Using methylene blue as the sample adsorbate, the adsorption behavior was systematically investigated in terms of the activation agent and temperature, the adsorption temperature and time, the initial adsorbate pH and concentration, as well as the dosage of adsorbent. Surface physical morphology of the TKMCK prepared was observed by scanning electron microscopy (SEM), while the functional groups were determined with Fourier transform infrared (FTIR) spectra. Kinetic studies indicated that the adsorption process was more consistent with the pseudo-second-order kinetic. Both Langmuir and Freundlich isotherms were employed to fit the adsorption data at equilibrium, with the former giving a maximum adsorption capacity of 793.65 mg/g at 323 K. BET surface area of as-prepared TKMCK was 657.78 m2/g.