基于Chen-Fliess级数的神经ODE的Rademacher复杂度
Rademacher Complexity of Neural ODEs via Chen-Fliess Series
AI总结：
本研究利用Chen–Fliess级数将连续深度神经ODE构建为单层无限宽网络，推导了这类ODE模型的Rademacher复杂度简洁表达式，给出特定系统的界实例并展望后续工作。
摘要：
我们展示了如何利用非线性常微分方程（ODE）的Chen–Fliess级数展开，将连续深度神经ODE模型构建为单层无限宽度网络。在该网络中，输出“权重”取自控制输入的signature（签名）——一种将无限维路径表示为张量序列的工具，由控制输入在单纯形上的迭代积分构成。“特征”则取为输出函数关于受控ODE模型中向量场的迭代李导数。本研究的主要成果是应用该框架，推导出将初始条件映射到某一终端时刻标量输出的ODE模型的Rademacher复杂度的简洁表达式。该结果利用了单层架构所带来的简便分析优势。我们最后给出了一些为特定系统实例化该界的示例，并讨论了潜在的后续工作方向。
英文摘要：
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.
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