Gautam Prakriya

Zi Wang, Gautam Prakriya, Somesh Jha

Fast and precise Lipschitz constant estimation of neural networks is an important task for deep learning. Researchers have recently found an intrinsic trade-off between the accuracy and smoothness of neural networks, so training a network with a loose Lipschitz constant estimation imposes a strong regularization and can hurt the model accuracy significantly. In this work, we provide a unified theoretical framework, a quantitative geometric approach, to address the Lipschitz constant estimation. By adopting this framework, we can immediately obtain several theoretical results, including the computational hardness of Lipschitz constant estimation and its approximability. Furthermore, the quantitative geometric perspective can also provide some insights into recent empirical observations that techniques for one norm do not usually transfer to another one. We also implement the algorithms induced from this quantitative geometric approach in a tool GeoLIP. These algorithms are based on semidefinite programming (SDP). Our empirical evaluation demonstrates that GeoLIP is more scalable and precise than existing tools on Lipschitz constant estimation for $\ell_\infty$-perturbations. Furthermore, we also show its intricate relations with other recent SDP-based techniques, both theoretically and empirically. We believe that this unified quantitative geometric perspective can bring new insights and theoretical tools to the investigation of neural-network smoothness and robustness.

Zi Wang, Aws Albarghouthi, Gautam Prakriya et al.

To verify safety and robustness of neural networks, researchers have successfully applied abstract interpretation, primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain for neural-network verification. First, we introduce the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function $f$ (universal approximation) as we have known for decades, but we can find a neural network, using any well-behaved activation function, whose interval bounds are an arbitrarily close approximation of the set semantics of $f$ (the result of applying $f$ to a set of inputs). We call this notion of approximation interval approximation. Our theorem generalizes the recent result of Baader et al. (2020) from ReLUs to a rich class of activation functions that we call squashable functions. Additionally, the IUA theorem implies that we can always construct provably robust neural networks under $\ell_\infty$-norm using almost any practical activation function. Second, we study the computational complexity of constructing neural networks that are amenable to precise interval analysis. This is a crucial question, as our constructive proof of IUA is exponential in the size of the approximation domain. We boil this question down to the problem of approximating the range of a neural network with squashable activation functions. We show that the range approximation problem (RA) is a $Δ_2$-intermediate problem, which is strictly harder than $\mathsf{NP}$-complete problems, assuming $\mathsf{coNP}\not\subset \mathsf{NP}$. As a result, IUA is an inherently hard problem: No matter what abstract domain or computational tools we consider to achieve interval approximation, there is no efficient construction of such a universal approximator.