Tight Second-Order Certificates for Randomized Smoothing
This work addresses the need for provable adversarial robustness in machine learning models, though it is incremental as it builds on existing randomized smoothing methods with a focus on tightness and efficiency.
The paper tackles the problem of providing robustness certificates against adversarial attacks via randomized smoothing, introducing a second-order certificate that is proven tight, but shows limited gains in certified robustness, and proposes a variant with improved sample efficiency achieving marginal improvements on high-dimensional datasets like CIFAR-10 and ImageNet.
Randomized smoothing is a popular way of providing robustness guarantees against adversarial attacks: randomly-smoothed functions have a universal Lipschitz-like bound, allowing for robustness certificates to be easily computed. In this work, we show that there also exists a universal curvature-like bound for Gaussian random smoothing: given the exact value and gradient of a smoothed function, we compute a lower bound on the distance of a point to its closest adversarial example, called the Second-order Smoothing (SoS) robustness certificate. In addition to proving the correctness of this novel certificate, we show that SoS certificates are realizable and therefore tight. Interestingly, we show that the maximum achievable benefits, in terms of certified robustness, from using the additional information of the gradient norm are relatively small: because our bounds are tight, this is a fundamental negative result. The gain of SoS certificates further diminishes if we consider the estimation error of the gradient norms, for which we have developed an estimator. We therefore additionally develop a variant of Gaussian smoothing, called Gaussian dipole smoothing, which provides similar bounds to randomized smoothing with gradient information, but with much-improved sample efficiency. This allows us to achieve (marginally) improved robustness certificates on high-dimensional datasets such as CIFAR-10 and ImageNet. Code is available at https://github.com/alevine0/smoothing_second_order.