Rademacher Complexity of Neural ODEs via Chen-Fliess Series
This provides theoretical insights into the generalization of continuous-depth models, which is incremental but addresses a known bottleneck in understanding neural ODEs.
The paper tackles the problem of analyzing the generalization capabilities of neural ODEs by framing them as single-layer, infinite-width networks using the Chen-Fliess series expansion, resulting in compact expressions for their Rademacher complexity.
We show how continuous-depth neural ODE models can be framed as single-layer, infinite-width nets using the Chen--Fliess series expansion for nonlinear ODEs. In this net, the output ``weights'' are taken from the signature of the control input -- a tool used to represent infinite-dimensional paths as a sequence of tensors -- which comprises iterated integrals of the control input over a simplex. The ``features'' are taken to be iterated Lie derivatives of the output function with respect to the vector fields in the controlled ODE model. The main result of this work applies this framework to derive compact expressions for the Rademacher complexity of ODE models that map an initial condition to a scalar output at some terminal time. The result leverages the straightforward analysis afforded by single-layer architectures. We conclude with some examples instantiating the bound for some specific systems and discuss potential follow-up work.