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Smoothed-SGDmax: A Stability-Inspired Algorithm to Improve Adversarial Generalization
Jiancong Xiao · Jiawei Zhang · Zhiquan Luo · Asuman Ozdaglar
Unlike standard training, deep neural networks can suffer from serious overfitting problems in adversarial settings. Recent research [40,39] suggests that adversarial training can have nonvanishing generalization error even if the sample size $n$ goes to infinity. A natural question arises: can we eliminate the generalization error floor in adversarial training? This paper gives an affirmative answer. First, by an adaptation of information-theoretical lower bound on the complexity of solving Lipschitz-convex problems using randomized algorithms, we establish a minimax lower bound $\Omega(s(T)/n)$ given a training loss of $1/s(T)$ for the adversarial generalization gap, where $T$ is the number of iterations, and $s(T)\rightarrow+\infty$ as $T\rightarrow+\infty$. Next, by observing that the nonvanishing generalization error of existing adversarial training algorithms comes from the non-smoothness of the adversarial loss function, we employ a smoothing technique to smooth the adversarial loss function. Based on the smoothed loss function, we design a smoothed SGDmax algorithm achieving a generalization bound $\mathcal{O}(s(T)/n)$, which eliminates the generalization error floor and matches the minimax lower bound. Experimentally, we show that our algorithm improves adversarial generalization on common datasets.

#### Author Information

##### Asuman Ozdaglar (Massachusetts Institute of Technology)

Asu Ozdaglar received the B.S. degree in electrical engineering from the Middle East Technical University, Ankara, Turkey, in 1996, and the S.M. and the Ph.D. degrees in electrical engineering and computer science from the Massachusetts Institute of Technology, Cambridge, in 1998 and 2003, respectively. She is currently a professor in the Electrical Engineering and Computer Science Department at the Massachusetts Institute of Technology. She is also the director of the Laboratory for Information and Decision Systems. Her research expertise includes optimization theory, with emphasis on nonlinear programming and convex analysis, game theory, with applications in communication, social, and economic networks, distributed optimization and control, and network analysis with special emphasis on contagious processes, systemic risk and dynamic control. Professor Ozdaglar is the recipient of a Microsoft fellowship, the MIT Graduate Student Council Teaching award, the NSF Career award, the 2008 Donald P. Eckman award of the American Automatic Control Council, the Class of 1943 Career Development Chair, the inaugural Steven and Renee Innovation Fellowship, and the 2014 Spira teaching award. She served on the Board of Governors of the Control System Society in 2010 and was an associate editor for IEEE Transactions on Automatic Control. She is currently the area co-editor for a new area for the journal Operations Research, entitled "Games, Information and Networks. She is the co-author of the book entitled âConvex Analysis and Optimizationâ (Athena Scientific, 2003).