Sorting out Lipschitz function approximation (1811.05381v2)

Abstract: Training neural networks under a strict Lipschitz constraint is useful for provable adversarial robustness, generalization bounds, interpretable gradients, and Wasserstein distance estimation. By the composition property of Lipschitz functions, it suffices to ensure that each individual affine transformation or nonlinear activation is 1-Lipschitz. The challenge is to do this while maintaining the expressive power. We identify a necessary property for such an architecture: each of the layers must preserve the gradient norm during backpropagation. Based on this, we propose to combine a gradient norm preserving activation function, GroupSort, with norm-constrained weight matrices. We show that norm-constrained GroupSort architectures are universal Lipschitz function approximators. Empirically, we show that norm-constrained GroupSort networks achieve tighter estimates of Wasserstein distance than their ReLU counterparts and can achieve provable adversarial robustness guarantees with little cost to accuracy.

• The paper shows that combining GroupSort activation with norm constraints preserves gradient norms, enabling universal Lipschitz function approximation.
• It empirically demonstrates that norm-constrained GroupSort networks yield tighter Wasserstein distance estimates and robust performance compared to ReLU networks.
• The method offers potential for provable adversarial robustness and improved generalization in scalable neural architectures.

An Overview of "Sorting Out Lipschitz Function Approximation"

The paper "Sorting Out Lipschitz Function Approximation" addresses a significant challenge in neural network research: training networks that maintain a strict Lipschitz constraint without sacrificing expressive power, which is essential for provable adversarial robustness, sharper generalization bounds, interpretable gradients, and effective Wasserstein distance estimation. The authors identify a crucial property for achieving expressive architectures: gradient norm preservation across network layers during backpropagation. By introducing a novel combination of activation functions and norm constraints, they demonstrate that it is possible to create universal Lipschitz function approximators.

The authors propose the use of the GroupSort activation function alongside norm-constrained weight matrices to ensure that networks preserve the gradient norm. This combination is critical for maintaining the expressive power of Lipschitz-constrained networks. The GroupSort activation function is a variant that groups and sorts activations, and importantly, it is gradient norm preserving and Lipschitz. The paper employs a sophisticated variant of the Stone-Weierstrass theorem to prove that networks using norm-constrained GroupSort architectures can universally approximate Lipschitz functions.

A key empirical contribution of the paper is the demonstration that norm-constrained GroupSort networks provide tighter estimates of Wasserstein distances than ReLU networks and can offer provable adversarial robustness with minimal accuracy loss. The authors present evidence that ReLU networks underperform in approximating basic Lipschitz functions, particularly as problem dimensionality increases, because they fail to preserve gradient norms adequately. This issue is exacerbated in ReLU networks where activations tend to remain positive, thereby limiting non-linear expressivity.

The broader implications of this paper are substantial. The proposed architectures can offer improved robustness against adversarial examples and can be employed in machine learning problems requiring strong generalization guarantees. Moreover, these networks provide theoretical guarantees for adversarial training and enable a more comprehensive understanding of the trade-offs involved in designing resilient networks.

Future developments in this domain may explore extensions to convolutional neural networks and further delve into real-world applications where adversarial robustness is paramount. Additionally, while the paper focuses on 2-norm and $\infty$-norm constraints, exploring other norm constraints and their practical implications remains an open question.

Overall, the paper makes significant strides in understanding how to effectively balance expressivity and robustness in neural networks under Lipschitz constraints, presenting a solid foundation for future work in this area of machine learning.