ResearchPod Summary
Neural network verification often relies on overapproximating non-linear activation functions with linear relaxations. While effective for simple functions like ReLU, extending these methods to modern, complex activation functions (e.g., GELU, SWISH, MISH) typically requires labor-intensive, hand-crafted relaxations that are difficult to optimize. The authors introduce SLiR (Shifting-based Linear Relaxations), a framework that decouples the construction of sound linear relaxations from the optimization process. By parameterizing relaxations by their slope and using a shifting procedure to compute the corresponding offset, SLiR enables efficient gradient-based optimization for any activation function, provided one can compute its critical points or determine a Lipschitz constant.
The core innovation is a shifting formalism: for a chosen slope m, the optimal offset b is computed as the maximum of f(x) - mx over the input domain. For functions where the derivative's pre-image is computable, this is solved exactly. For more complex functions, the authors construct a piecewise linear (PWL) surrogate once, then extend it to sound upper and lower envelopes using a modified Piyavskii-based Lipschitz optimization. This allows the verifier to optimize the slope parameter m during the verification process without re-establishing soundness at every iteration, significantly improving efficiency.
SLiR demonstrates high versatility and performance. By requiring only a formula for critical points or a Lipschitz constant, the authors reduced the implementation effort for new activation functions from over 1,300 lines of code (in existing frameworks like alpha-CROWN) to approximately 154 lines. Experimental results show that SLiR produces tighter relaxations for functions like GELU and tanh compared to existing hand-crafted methods and enables the verification of up to 7.8x more properties, proving its effectiveness in practical, safety-critical applications.
As neural networks are increasingly deployed in safety-critical domains, the ability to formally verify their behavior is essential. Current verification tools are often bottlenecked by the difficulty of supporting diverse, non-linear activation functions. SLiR provides a scalable, automated path to support virtually any activation function, lowering the barrier to entry for verifying modern, high-performance neural architectures.
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