Local Lipschitz Bounds of Deep Neural Networks
This work addresses a fundamental challenge in non-convex optimization for machine learning practitioners, offering theoretical insights into step size schedulers and convergence analysis, though it appears incremental as it builds on prior work by focusing on parameter-based bounds rather than input-based ones.
This paper tackles the problem of estimating Lipschitz constants for deep neural networks, which are crucial for analyzing gradient-based optimization convergence, by providing local upper and lower bounds for the Lipschitz constants of multi-layer feed-forward networks and their gradients, and extending these results to continuously deep neural networks described by controlled ODEs.
The Lipschitz constant is an important quantity that arises in analysing the convergence of gradient-based optimization methods. It is generally unclear how to estimate the Lipschitz constant of a complex model. Thus, this paper studies an important problem that may be useful to the broader area of non-convex optimization. The main result provides a local upper bound on the Lipschitz constants of a multi-layer feed-forward neural network and its gradient. Moreover, lower bounds are established as well, which are used to show that it is impossible to derive global upper bounds for the Lipschitz constants. In contrast to previous works, we compute the Lipschitz constants with respect to the network parameters and not with respect to the inputs. These constants are needed for the theoretical description of many step size schedulers of gradient based optimization schemes and their convergence analysis. The idea is both simple and effective. The results are extended to a generalization of neural networks, continuously deep neural networks, which are described by controlled ODEs.