Duality for Neural Networks through Reproducing Kernel Banach Spaces
This provides a theoretical framework for analyzing neural networks through duality, potentially benefiting researchers in optimization and machine learning theory, though it appears incremental as it builds on existing RKHS and Barron space concepts.
The paper tackles the problem of understanding Barron spaces for neural networks by showing they belong to a class of integral Reproducing Kernel Banach spaces (RKBS), which can be viewed as an infinite union of RKHS, and demonstrates that the dual space forms an adjoint pair enabling the construction of a saddle point problem for primal-dual optimization.
Reproducing Kernel Hilbert spaces (RKHS) have been a very successful tool in various areas of machine learning. Recently, Barron spaces have been used to prove bounds on the generalisation error for neural networks. Unfortunately, Barron spaces cannot be understood in terms of RKHS due to the strong nonlinear coupling of the weights. This can be solved by using the more general Reproducing Kernel Banach spaces (RKBS). We show that these Barron spaces belong to a class of integral RKBS. This class can also be understood as an infinite union of RKHS spaces. Furthermore, we show that the dual space of such RKBSs, is again an RKBS where the roles of the data and parameters are interchanged, forming an adjoint pair of RKBSs including a reproducing kernel. This allows us to construct the saddle point problem for neural networks, which can be used in the whole field of primal-dual optimisation.