单调、双李普希茨与 Polyak-Łojasiewicz 网络
Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networks
AI总结：
本文提出双李普希茨可逆网络 BiLipNet 和满足 Polyak-Łojasiewicz 条件的标量输出网络 PLNet，通过基于增量二次约束的可逆残差层实现更紧的性质认证，并将逆计算转化为三算子分裂问题以高效求解全局最小值。
摘要：
本文提出了一种新的双李普希茨可逆神经网络 BiLipNet，它能够平滑地同时控制其李普希茨性（输出对输入扰动的敏感性）和逆李普希茨性（从不同输出区分输入的能力）。第二个主要贡献是一种新的标量输出网络 PLNet，它由 BiLipNet 与一个二次势函数复合而成。我们证明 PLNet 满足 Polyak-Łojasiewicz 条件，可用于学习具有唯一且可高效计算的全局最小值的非凸代理损失。这些网络的核心技术要素是一种新颖的可逆残差层，该层具有经过认证的强单调性和李普希茨性，我们将其与正交层组合以构建 BiLipNet。这些性质的认证基于增量二次约束，从而获得比谱归一化所能实现的更紧的界。此外，我们将 BiLipNet 的逆计算——以及因此 PLNet 的最小值计算——表述为一系列三算子分裂问题，对此可以应用快速算法。
英文摘要：
This paper presents a new bi-Lipschitz invertible neural network, the BiLipNet, which has the ability to smoothly control both its Lipschitzness (output sensitivity to input perturbations) and inverse Lipschitzness (input distinguishability from different outputs). The second main contribution is a new scalar-output network, the PLNet, which is a composition of a BiLipNet and a quadratic potential. We show that PLNet satisfies the Polyak-Lojasiewicz condition and can be applied to learn non-convex surrogate losses with a unique and efficiently-computable global minimum. The central technical element in these networks is a novel invertible residual layer with certified strong monotonicity and Lipschitzness, which we compose with orthogonal layers to build the BiLipNet. The certification of these properties is based on incremental quadratic constraints, resulting in much tighter bounds than can be achieved with spectral normalization. Moreover, we formulate the calculation of the inverse of a BiLipNet -- and hence the minimum of a PLNet -- as a series of three-operator splitting problems, for which fast algorithms can be applied.
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