用凸优化方法提升神经网络的鲁棒性，训练更快更稳定。
A Scalable Approach for Safe and Robust Learning via Lipschitz-Constrained Networks
- 通过环形变换重构网络，实现可验证的凸约束训练
- 在多个数据集上达到更强的抗干扰能力与更快训练速度
- 适合对安全性要求高的工业部署场景
在安全关键应用中，神经网络的可认证鲁棒性至关重要。一种主流方法是限制网络的全局Lipschitz常数。然而，现有精确方法常因依赖全局半定规划（SDP）而面临非凸性与可扩展性差的问题。本文提出一种基于半定松弛的凸训练框架，通过环形变换重构网络，推导出可解析的凸可行性条件，实现高效且可验证的训练。为克服全局SDP规模限制，进一步设计随机子空间线性矩阵不等式（RS-LMI）方法，将全局约束分解为低维子空间上的逐层投影约束，得到平滑且内存高效的训练目标。在MNIST、CIFAR-10和ImageNet上的实验表明，该框架在保持竞争性准确率的同时，显著提升了Lipschitz界与运行效率。
原文摘要 · Abstract (English)
Certified robustness is a critical property for deploying neural networks (NN) in safety-critical applications. A principle approach to achieving such guarantees is to constrain the global Lipschitz constant of the network. However, accurate methods for Lipschitz-constrained training often suffer from non-convex formulations and poor scalability due to reliance on global semidefinite programs (SDPs). In this letter, we propose a convex training framework that enforces global Lipschitz constraints via semidefinite relaxation. By reparameterizing the NN using loop transformation, we derive a convex admissibility condition that enables tractable and certifiable training. While the resulting formulation guarantees robustness, its scalability is limited by the size of global SDP. To overcome this, we develop a randomized subspace linear matrix inequalities (RS-LMI) approach that decomposes the global constraints into sketched layerwise constraints projected onto low-dimensional subspaces, yielding a smooth and memory-efficient training objective. Empirical results on MNIST, CIFAR-10, and ImageNet demonstrate that the proposed framework achieves competitive accuracy with significantly improved Lipschitz bounds and runtime performance.
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