Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networks
This work addresses the problem of ensuring stability and invertibility in neural networks for applications like robust learning and optimization, though it appears incremental in building on existing Lipschitz and monotonic network concepts.
The paper introduces BiLipNet, a bi-Lipschitz invertible neural network for controlling output sensitivity and input distinguishability, and PLNet, a scalar-output network that satisfies the Polyak-Lojasiewicz condition to learn non-convex surrogate losses with a unique global minimum, achieving tighter bounds than spectral normalization.
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.