[Paper Review] POLAR: A Polynomial Arithmetic Framework for Verifying Neural-Network Controlled Systems

Chao Huang, Jiameng Fan+7 more|arXiv (Cornell University)|Jun 25, 2021

Adversarial Robustness in Machine Learning26 references5 citations

TL;DR

POLAR is a novel polynomial arithmetic framework that enables efficient and precise bounded-time reachability analysis for neural-network controlled systems (NNCSs) by integrating Taylor model (TM) arithmetic with Bernstein Bézier forms and symbolic remainders. This approach overcomes limitations of prior methods in handling non-differentiable ReLU activations and reducing remainder explosion, resulting in significantly tighter reachable set overapproximations and improved scalability over state-of-the-art tools.

ABSTRACT

We present POLAR, a polynomial arithmetic-based framework for efficient bounded-time reachability analysis of neural-network controlled systems (NNCSs). Existing approaches that leverage the standard Taylor Model (TM) arithmetic for approximating the neural-network controller cannot deal with non-differentiable activation functions and suffer from rapid explosion of the remainder when propagating the TMs. POLAR overcomes these shortcomings by integrating TM arithmetic with extbf{Bernstein B{é}zier Form} and extbf{symbolic remainder}. The former enables TM propagation across non-differentiable activation functions and local refinement of TMs, and the latter reduces error accumulation in the TM remainder for linear mappings in the network. Experimental results show that POLAR significantly outperforms the current state-of-the-art tools in terms of both efficiency and tightness of the reachable set overapproximation. The source code can be found in https://github.com/ChaoHuang2018/POLAR_Tool

Motivation & Objective

• To address the challenge of verifying safety in neural-network controlled systems (NNCSs) with non-differentiable activation functions and rapidly growing approximation errors.
• To improve the precision and efficiency of bounded-time reachability analysis by overcoming the limitations of standard Taylor model (TM) arithmetic in neural network propagation.
• To enable scalable and tight overapproximation of reachable sets in NNCSs through integration of polynomial arithmetic techniques.
• To support both continuous-time and discrete-time NNCSs, including benchmarks like QUAD and Mountain Car.
• To reduce error accumulation in TM remainders during layer-by-layer network propagation, especially in deep or wide networks.

Proposed method

• Integrates Taylor model (TM) arithmetic with Bernstein Bézier form to handle non-differentiable activation functions like ReLU, enabling polynomial overapproximation even at non-differentiable points.
• Employs local refinement of TMs using univariate Bernstein approximation to improve accuracy on small domains, particularly where standard TMs suffer from high remainder growth.
• Uses symbolic remainders instead of interval-based remainders to reduce error accumulation during linear mappings in the network, especially across multiple layers.
• Applies a hybrid approach combining Bernstein approximation and symbolic remainder techniques to balance accuracy and computational cost in TM propagation.
• Constructs TM flowpipes layer by layer through the neural network, maintaining state dependencies across time steps for accurate closed-loop reachability analysis.
• Supports both continuous-time systems (e.g., QUAD) and discrete-time systems (e.g., Mountain Car), demonstrating broad applicability.

Experimental results

Research questions

• RQ1 Can a polynomial arithmetic framework effectively handle non-differentiable activation functions like ReLU in neural-network controlled systems?
• RQ2 How can Taylor model arithmetic be extended to prevent remainder explosion during layer-by-layer propagation in deep neural networks?
• RQ3 To what extent can Bernstein Bézier forms improve the accuracy of TM overapproximations in reachability analysis?
• RQ4 How does the integration of symbolic remainders reduce error accumulation compared to interval-based remainder representations?
• RQ5 Can the proposed framework scale to larger and deeper neural networks while maintaining tight reachable set overapproximations?

Key findings

• POLAR achieves significantly tighter reachable set overapproximations than state-of-the-art tools like Verisig 2.0, ReachNN*, and NNV on both QUAD and Mountain Car benchmarks.
• On the QUAD benchmark, POLAR reduces the TM remainder width on x₃ at step 50 to 1.2e-4 (k=2, X₀=0.05) with only 1.34% use of Bernstein approximation, outperforming all other configurations.
• For X₀=0.05 and k=2, POLAR completes 50-step reachability in 231 seconds with a remainder width of 1.2e-4, while Verisig 2.0 fails to compute due to scalability limits.
• The use of both Bernstein approximation and symbolic remainders reduces remainder width by up to 90% compared to baseline TM, even with minimal use of Bernstein (e.g., 0% usage in some cases) when remainders are already well-controlled.
• POLAR scales well with network depth and width, maintaining performance on networks with up to 5 layers and 200 neurons per layer, though runtime increases with larger initial set radii.
• For larger initial sets (X₀=0.4), POLAR still completes 50-step analysis in 1271 seconds (k=4), while other tools fail to compute, demonstrating robustness to uncertainty size.