Formal Security Analysis of Neural Networks using Symbolic Intervals (1804.10829v3)

Abstract: Due to the increasing deployment of Deep Neural Networks (DNNs) in real-world security-critical domains including autonomous vehicles and collision avoidance systems, formally checking security properties of DNNs, especially under different attacker capabilities, is becoming crucial. Most existing security testing techniques for DNNs try to find adversarial examples without providing any formal security guarantees about the non-existence of such adversarial examples. Recently, several projects have used different types of Satisfiability Modulo Theory (SMT) solvers to formally check security properties of DNNs. However, all of these approaches are limited by the high overhead caused by the solver. In this paper, we present a new direction for formally checking security properties of DNNs without using SMT solvers. Instead, we leverage interval arithmetic to compute rigorous bounds on the DNN outputs. Our approach, unlike existing solver-based approaches, is easily parallelizable. We further present symbolic interval analysis along with several other optimizations to minimize overestimations of output bounds. We design, implement, and evaluate our approach as part of ReluVal, a system for formally checking security properties of Relu-based DNNs. Our extensive empirical results show that ReluVal outperforms Reluplex, a state-of-the-art solver-based system, by 200 times on average. On a single 8-core machine without GPUs, within 4 hours, ReluVal is able to verify a security property that Reluplex deemed inconclusive due to timeout after running for more than 5 days. Our experiments demonstrate that symbolic interval analysis is a promising new direction towards rigorously analyzing different security properties of DNNs.

• The paper introduces ReluVal, which employs symbolic interval analysis to certify DNN security properties without the computational overhead of SMT solvers.
• It achieves up to 200x faster performance than traditional SMT-based verifiers, providing scalable and precise output bounds.
• The methodology’s inherent parallelism and iterative refinement highlight its potential for deployment in safety-critical AI applications.

Formal Security Analysis of Neural Networks using Symbolic Intervals

The paper "Formal Security Analysis of Neural Networks using Symbolic Intervals" addresses a critical concern in the deployment of Deep Neural Networks (DNNs) in security-sensitive applications like autonomous vehicles and collision avoidance systems. The reliability and robustness of DNNs in these domains are paramount, amidst known susceptibility to adversarial examples—inputs that can mislead DNNs after negligible modifications.

Traditional security checks for DNNs have relied heavily on Satisfiability Modulo Theory (SMT) solvers intended to detect violations in security properties. However, these methods suffer from significant computational overhead, making scalability an issue. The authors propose a novel approach in the form of ReluVal, which utilizes interval arithmetic to calculate tight bounds on DNN outputs without the use of SMT solvers. This approach is not only efficient computationally but also inherently parallelizable, which marks a departure from the intrinsic limitations faced by solver-based techniques.

Key Methodologies

ReluVal's methodology revolves around symbolic interval analysis, which aims to minimize the overestimation of output ranges by tracking the symbolic lower and upper bounds for each neuron. This method endeavors to tackle the dependency problem endemic in interval arithmetic—a situation where the interdependencies between inputs lead to overly conservative output predictions. The symbolic analysis retains specific functional relationships between inputs across layers, thus improving precision.

Additionally, when the output range computed by symbolic intervals is overly broad and non-conclusive, ReluVal implements an iterative refinement process. This involves bisecting the input range to produce finer partitions, thereby achieving a tighter approximation of the output range. The method rests on the assurance that iterative bisecting will converge efficiently, especially under the condition of Lipschitz continuity, which is inherently satisfied by extensively utilized architectures, including those deploying the ReLU activation function.

Empirical Findings

ReluVal was benchmarked against Reluplex—a state-of-the-art SMT-based verifier—demonstrating an efficacy enhancement by approximately 200 times faster across numerous testing scenarios. It succeeded in certifying properties for which Reluplex was unable to conclude due to performance bottlenecks. These findings underline the scaling efficiency and reliability of ReluVal in offering formal security guarantees for DNNs.

Implications and Future Directions

The implications of these findings are notable in areas demanding fail-safe AI, such as aviation safety and autonomous robotics. By proving the non-existence of adversarial examples within certain constraints, ReluVal can contribute towards creating a resilient framework for DNN deployment in high-stakes environments. Moreover, the modular nature of interval arithmetic presented in this approach opens future avenues in optimizing and adapting this methodology to diverse neural architectures and other domain-specific requirements.

The trajectory of future research could explore further optimization of symbolic interval representation for non-linear functions beyond ReLU, broader generalization to support constraints expressed in various norms, notably $L_p$ norms beyond $L_\infty$, and integration with training protocols to enhance model robustness iteratively. The potential cross-pollination of symbolic interval arithmetic with other formal methods can further extend its application range and solidify its role in AI safety and verification disciplines.

The paper provides a compelling case for the continued exploration and adoption of interval arithmetic in neural model verification, emphasizing scalability, precision, and adaptability as cornerstones for robust neural network deployment in critical applications.