# Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems

@article{Sattar2020NonasymptoticAA, title={Non-asymptotic and Accurate Learning of Nonlinear Dynamical Systems}, author={Yahya Sattar and Samet Oymak}, journal={ArXiv}, year={2020}, volume={abs/2002.08538} }

We consider the problem of learning stabilizable systems governed by nonlinear state equation $h_{t+1}=\phi(h_t,u_t;\theta)+w_t$. Here $\theta$ is the unknown system dynamics, $h_t $ is the state, $u_t$ is the input and $w_t$ is the additive noise vector. We study gradient based algorithms to learn the system dynamics $\theta$ from samples obtained from a single finite trajectory. If the system is run by a stabilizing input policy, we show that temporally-dependent samples can be approximated…

## 22 Citations

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## References

SHOWING 1-10 OF 84 REFERENCES

### Stochastic Gradient Descent Learns State Equations with Nonlinear Activations

- Computer ScienceCOLT
- 2019

It is proved that SGD estimate linearly converges to the ground truth weights while using near-optimal sample size and a novel SGD convergence result with nonlinear activations is published.

### Finite-time Analysis of Approximate Policy Iteration for the Linear Quadratic Regulator

- Computer ScienceNeurIPS
- 2019

A simple adaptive procedure based on $\varepsilon$-greedy exploration which relies on approximate PI as a sub-routine and obtains regret is constructed, improving upon a recent result of Abbasi-Yadkori et al.

### How fast can linear dynamical systems be learned?

- Computer Science, MathematicsArXiv
- 2018

Finite time error bounds for estimating general linear time-invariant systems from a single observed trajectory using the method of least squares are provided and it is demonstrated that the least squares solution may be statistically inconsistent under certain conditions even when the signal-to-noise ratio is high.

### Stability Bounds for Non-i.i.d. Processes

- Computer ScienceNIPS
- 2007

Novel stability-based generalization bounds that hold even with this more general setting are proved, which strictly generalize the bounds given in the i.i.d. case.

### Regret Analysis for Adaptive Linear-Quadratic Policies

- Computer Science, Mathematics
- 2017

To establish high probability regret bounds for the classical problem of Linear-Quadratic control, certain novel techniques are introduced to comprehensively address the probabilistic behavior of dependent random matrices with heavy-tailed distributions.

### Near optimal finite time identification of arbitrary linear dynamical systems

- MathematicsICML
- 2019

This work derives finite time error bounds for estimating general linear time-invariant (LTI) systems from a single observed trajectory using the method of least squares and demonstrates that the least squares solution may be statistically inconsistent under certain conditions even when the signal-to-noise ratio is high.

### Generalization bounds for non-stationary mixing processes

- Computer ScienceMachine Learning
- 2016

The first generalization bounds for time series prediction with a non-stationary mixing stochastic process is presented and it is proved that fast learning rates can be achieved by extending existing local Rademacher complexity analyses to the non-i.i.d. data.

### Derivative-Free Methods for Policy Optimization: Guarantees for Linear Quadratic Systems

- Mathematics, Computer ScienceAISTATS
- 2019

This work characterizes the convergence rate of a canonical stochastic, two-point, derivative-free method for linear-quadratic systems in which the initial state of the system is drawn at random, and shows that for problems with effective dimension $D$, such a method converges to an $\epsilon$-approximate solution within $\widetilde{\mathcal{O}}(D/\ep silon)$ steps.

### Certainty Equivalent Control of LQR is Efficient

- Computer ScienceArXiv
- 2019

The results show that certainty equivalent control with $\varepsilon$-greedy exploration achieves $\tilde{\mathcal{O}}(\sqrt{T})$ regret in the adaptive LQR setting, yielding the first guarantee of a computationally tractable algorithm that achieves nearly optimal regret for adaptive L QR.