Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems
Yahya Sattar Department of Electrical and Computer Engineering, University of California, Riverside, CA 92521, USA. Email: Samet Oymak
Abstract
We consider the problem of learning nonlinear dynamical systems governed by nonlinear state equation $\bm{h}_{t+1}=\phi(\bm{h}_{t},{\bm{u}}_{t};\bm{\theta})+\bm{w}_{t}$ . Here $\bm{\theta}$ is the unknown system dynamics, $\bm{h}_{t}$ is the state, ${\bm{u}}_{t}$ is the input and $\bm{w}_{t}$ is the additive noise vector. We study gradient based algorithms to learn the system dynamics $\bm{\theta}$ from samples obtained from a single finite trajectory. If the system is run by a stabilizing input policy, then using a mixing-time argument we show that temporally-dependent samples can be approximated by i.i.d. samples. We then develop new guarantees for the uniform convergence of the gradients of the empirical loss induced by these i.i.d. samples. Unlike existing works, our bounds are noise sensitive which allows for learning ground-truth dynamics with high accuracy and small sample complexity. Together, our results facilitate efficient learning of a broader class of nonlinear dynamical systems as compared to the prior works. We specialize our guarantees to entrywise nonlinear activations and verify our theory in various numerical experiments.
中文速览
如何从一段有限时间序列中高效学习非线性动力系统（nonlinear dynamical system）的参数，是系统辨识领域的核心难题，因为轨迹数据在时间上相互依赖，传统的独立同分布（i.i.d.）统计理论难以直接适用。本文的核心思路是：对于满足稳定性条件的非线性系统，利用混合时间（mixing time）论证将时序相关样本近似为独立样本，从而把系统辨识问题归约为经典的统计学习框架。在此基础上，作者建立了经验损失梯度一致收敛的新理论界，关键改进在于界对噪声水平的刻画更精确，使得在低噪声环境下可以以更少样本学到更准确的真实动力学参数。最终，结合梯度下降（gradient descent）的收敛分析，论文给出了一类更广泛非线性系统的样本复杂度保证，并通过带逐元素非线性激活函数的具体模型和数值实验验证了理论结果。这项工作的意义在于为循环神经网络等非线性序列模型的系统辨识提供了有理论保障的学习算法，同时所发展的统计工具对独立感兴趣的机器学习问题也具有参考价值。
原文 arXiv:2002.08538;中英对照 + 大白话阅读 https://aha.fim.ai/paper/2002.08538v2