DeepBern-Nets: Taming the Complexity of Certifying Neural Networks using Bernstein Polynomial Activations and Precise Bound Propagation

Formal certification of Neural Networks (NNs) is crucial for ensuring their safety, fairness, and robustness. Unfortunately, on the one hand, sound and complete certification algorithms of ReLU-based NNs do not scale to large-scale NNs. On the other hand, incomplete certification algorithms are easier to compute, but they result in loose bounds that deteriorate with the depth of NN, which diminishes their effectiveness. In this paper, we ask the following question; can we replace the ReLU activation function with one that opens the door to incomplete certification algorithms that are easy to compute but can produce tight bounds on the NN's outputs? We introduce DeepBern-Nets, a class of NNs with activation functions based on Bernstein polynomials instead of the commonly used ReLU activation. Bernstein polynomials are smooth and differentiable functions with desirable properties such as the so-called range enclosure and subdivision properties. We design a novel algorithm, called Bern-IBP, to efficiently compute tight bounds on DeepBern-Nets outputs. Our approach leverages the properties of Bernstein polynomials to improve the tractability of neural network certification tasks while maintaining the accuracy of the trained networks. We conduct comprehensive experiments in adversarial robustness and reachability analysis settings to assess the effectiveness of the proposed Bernstein polynomial activation in enhancing the certification process. Our proposed framework achieves high certified accuracy for adversarially-trained NNs, which is often a challenging task for certifiers of ReLU-based NNs. Moreover, using Bern-IBP bounds for certified training results in NNs with state-of-the-art certified accuracy compared to ReLU networks. This work establishes Bernstein polynomial activation as a promising alternative for improving NN certification tasks across various applications

Polynomial-Time Reachability for LTI Systems with Two-Level Lattice Neural Network Controllers

In this paper, we consider the computational complexity of bounding the reachable set of a Linear Time-Invariant (LTI) system controlled by a Rectified Linear Unit (ReLU) Two-Level Lattice (TLL) Neural Network (NN) controller. In particular, we show that for such a system and controller, it is possible to compute the exact one-step reachable set in polynomial time in the size of the size of the TLL NN controller (number of neurons). Additionally, we show that it is possible to obtain a tight bounding box of the reachable set via two polynomial-time methods: one with polynomial complexity in the size of the TLL and the other with polynomial complexity in the Lipschitz constant of the controller and other problem parameters. Crucially, the smaller of the two can be decided in polynomial time for non-degenerate TLL NNs. Finally, we propose a pragmatic algorithm that adaptively combines the benefits of (semi-)exact reachability and approximate reachability, which we call L-TLLBox. We evaluate L-TLLBox with an empirical comparison to a state-of-the-art NN controller reachability tool. In these experiments, L-TLLBox was able to complete reachability analysis as much as 5000x faster than this tool on the same network/system, while producing reach boxes that were from 0.08 to 1.42 times the area

PolyARBerNN: A Neural Network Guided Solver and Optimizer for Bounded Polynomial Inequalities

Constraints solvers play a significant role in the analysis, synthesis, and formal verification of complex embedded and cyber-physical systems. In this paper, we study the problem of designing a scalable constraints solver for an important class of constraints named polynomial constraint inequalities (also known as non-linear real arithmetic theory). In this paper, we introduce a solver named PolyARBerNN that uses convex polynomials as abstractions for highly nonlinear polynomials. Such abstractions were previously shown to be powerful to prune the search space and restrict the usage of sound and complete solvers to small search spaces. Compared with the previous efforts on using convex abstractions, PolyARBerNN provides three main contributions namely (i) a neural network guided abstraction refinement procedure that helps selecting the right abstraction out of a set of pre-defined abstractions, (ii) a Bernstein polynomial-based search space pruning mechanism that can be used to compute tight estimates of the polynomial maximum and minimum values which can be used as an additional abstraction of the polynomials, and (iii) an optimizer that transforms polynomial objective functions into polynomial constraints (on the gradient of the objective function) whose solutions are guaranteed to be close to the global optima. These enhancements together allowed the PolyARBerNN solver to solve complex instances and scales more favorably compared to the state-of-art non-linear real arithmetic solvers while maintaining the soundness and completeness of the resulting solver. In particular, our test benches show that PolyARBerNN achieved 100X speedup compared with Z3 8.9, Yices 2.6, and NASALib (a solver that uses Bernstein expansion to solve multivariate polynomial constraints) on a variety of standard test benches

Attack Resilience and Recovery using Physical Challenge Response Authentication for Active Sensors Under Integrity Attacks

Embedded sensing systems are pervasively used in life- and security-critical systems such as those found in airplanes, automobiles, and healthcare. Traditional security mechanisms for these sensors focus on data encryption and other post-processing techniques, but the sensors themselves often remain vulnerable to attacks in the physical/analog domain. If an adversary manipulates a physical/analog signal prior to digitization, no amount of digital security mechanisms after the fact can help. Fortunately, nature imposes fundamental constraints on how these analog signals can behave. This work presents PyCRA, a physical challenge-response authentication scheme designed to protect active sensing systems against physical attacks occurring in the analog domain. PyCRA provides security for active sensors by continually challenging the surrounding environment via random but deliberate physical probes. By analyzing the responses to these probes, and by using the fact that the adversary cannot change the underlying laws of physics, we provide an authentication mechanism that not only detects malicious attacks but provides resilience against them. We demonstrate the effectiveness of PyCRA through several case studies using two sensing systems: (1) magnetic sensors like those found wheel speed sensors in robotics and automotive, and (2) commercial RFID tags used in many security-critical applications. Finally, we outline methods and theoretical proofs for further enhancing the resilience of PyCRA to active attacks by means of a confusion phase---a period of low signal to noise ratio that makes it more difficult for an attacker to correctly identify and respond to PyCRA's physical challenges. In doing so, we evaluate both the robustness and the limitations of PyCRA, concluding by outlining practical considerations as well as further applications for the proposed authentication mechanism.Comment: Shorter version appeared in ACM ACM Conference on Computer and Communications (CCS) 201