Can Synthetic Data Reduce Conservatism of Distributionally Robust Adversarial Training?

TL;DR — We employ distributionally robust optimization to prevent overfitting in adversarial training and use synthetic data to reduce its conservatism.

Abstract

When the inputs of a machine learning model are subject to adversarial attacks, standard stationarity assumptions on the training and test sets are violated, typically making empirical risk minimization (ERM) ineffective. Adversarial training, which imitates the adversary during the training stage, has thus emerged as the *de facto* standard for hedging against adversarial attacks. Although adversarial training provides some robustness over ERM, it can still be subject to overfitting, which explains why recent work mixing the training set with synthetic data obtains improved out-of-sample performances. Inspired by these observations, we develop a Wasserstein distributionally robust (DR) counterpart of adversarial training for improved generalization and provide a recipe for further reducing the conservatism of this approach by adjusting its ambiguity set with respect to synthetic data. The underlying optimization problem, DR adversarial training with synthetic data, is nonconvex and comprises infinitely many constraints. To this end, by using results from robust optimization and convex analysis, we develop tractable relaxations. We focus our analyses on the logistic loss function and provide discussions for adapting this framework to several other loss functions. We demonstrate the superiority of this approach on artificial as well as standard benchmark problems.