Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness

Jul 12, 2020

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Randomized smoothing, using just a simple isotropic Gaussian distribution, has been shown to produce good robustness guarantees against ℓ_2-norm bounded adversaries. In this work, we show that extending the smoothing technique to defend against other attack models can be challenging, especially in the high-dimensional regime. In particular, for a vast class of i.i.d. smoothing distributions, we prove that the largest ℓ_p-radius that can be certified decreases as O(1/d^1/2 - 1/p) with dimension d for p > 2. Notably, for p ≥ 2, this dependence on d is no better than that of the ℓ_p-radius that can be certified using isotropic Gaussian smoothing, essentially putting a matching lower bound on the robustness radius. When restricted to generalized Gaussian smoothing, these two bounds can be shown to be within a constant factor of each other in an asymptotic sense, establishing that Gaussian smoothing provides the best possible results, up to a constant factor, when p ≥ 2. We present experimental results on CIFAR to validate our theory. For other smoothing distributions, such as, a uniform distribution within an ℓ_1 or an ℓ_∞-norm ball, we show upper bounds of the form O(1 / d) and O(1 / d^1 - 1/p) respectively, which have an even worse dependence on d.

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The International Conference on Machine Learning (ICML) is the premier gathering of professionals dedicated to the advancement of the branch of artificial intelligence known as machine learning. ICML is globally renowned for presenting and publishing cutting-edge research on all aspects of machine learning used in closely related areas like artificial intelligence, statistics and data science, as well as important application areas such as machine vision, computational biology, speech recognition, and robotics. ICML is one of the fastest growing artificial intelligence conferences in the world. Participants at ICML span a wide range of backgrounds, from academic and industrial researchers, to entrepreneurs and engineers, to graduate students and postdocs.

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