# Episodic Gaussian Process-Based Learning Control with Vanishing Tracking Errors

arXiv: https://arxiv.org/abs/2307.04415 · HTML: https://mlsanity.com/paper/2307.04415/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 65 references that are in the map.
- First posted on arXiv 10 Jul 2023.

## Builds on (most cited first, 24 of 24)

- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Safe Learning in Robotics: From Learning-Based Control to Safe Reinforcement Learning](https://mlsanity.com/paper/2108.06266/) (2021)
- [On Kernelized Multi-armed Bandits](https://mlsanity.com/paper/1704.00445/) (2017)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Gaussian Processes for Data-Efficient Learning in Robotics and Control](https://mlsanity.com/paper/1502.02860/) (2015)
- [Cautious Model Predictive Control using Gaussian Process Regression](https://mlsanity.com/paper/1705.10702/) (2017)
- [Efficient Model-Based Reinforcement Learning through Optimistic Policy Search and Planning](https://mlsanity.com/paper/2006.08684/) (2020)
- [Safe Controller Optimization for Quadrotors with Gaussian Processes](https://mlsanity.com/paper/1509.01066/) (2015)
- [Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control](https://mlsanity.com/paper/1906.01376/) (2019)
- [Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression](https://mlsanity.com/paper/2105.02796/) (2021)
- [Virtual vs. Real: Trading Off Simulations and Physical Experiments in Reinforcement Learning with Bayesian Optimization](https://mlsanity.com/paper/1703.01250/) (2017)
- [Actively Learning Gaussian Process Dynamics](https://mlsanity.com/paper/1911.09946/) (2019)
- [Feedback Linearization based on Gaussian Processes with event-triggered Online Learning](https://mlsanity.com/paper/1911.06565/) (2019)
- [Deterministic error bounds for kernel-based learning techniques under bounded noise](https://mlsanity.com/paper/2008.04005/) (2020)
- [Gaussian Process bandits with adaptive discretization](https://mlsanity.com/paper/1712.01447/) (2017)
- [Control Barriers in Bayesian Learning of System Dynamics](https://mlsanity.com/paper/2012.14964/) (2020)
- [Gaussian Process Uniform Error Bounds with Unknown Hyperparameters for Safety-Critical Applications](https://mlsanity.com/paper/2109.02606/) (2021)
- [Combining Pessimism with Optimism for Robust and Efficient Model-Based Deep Reinforcement Learning](https://mlsanity.com/paper/2103.10369/) (2021)
- [Robust Uncertainty Bounds in Reproducing Kernel Hilbert Spaces: A Convex Optimization Approach](https://mlsanity.com/paper/2104.09582/) (2021)
- [Provably Robust Learning-Based Approach for High-Accuracy Tracking Control of Lagrangian Systems](https://mlsanity.com/paper/1804.01031/) (2018)
- [Localized active learning of Gaussian process state space models](https://mlsanity.com/paper/2005.02191/) (2020)
- [KPC: Learning-Based Model Predictive Control with Deterministic Guarantees](https://mlsanity.com/paper/2011.11303/) (2020)
- [Guarantees for data-driven control of nonlinear systems using semidefinite programming: A survey](https://mlsanity.com/paper/2306.16042/) (2023)
- [Learning-enhanced robust controller synthesis with rigorous statistical and control-theoretic guarantees](https://mlsanity.com/paper/2105.03397/) (2021)

## Abstract (arXiv)

Due to the increasing complexity of technical systems, accurate first principle models can often not be obtained. Supervised machine learning can mitigate this issue by inferring models from measurement data. Gaussian process regression is particularly well suited for this purpose due to its high data-efficiency and its explicit uncertainty representation, which allows the derivation of prediction error bounds. These error bounds have been exploited to show tracking accuracy guarantees for a variety of control approaches, but their direct dependency on the training data is generally unclear. We address this issue by deriving a Bayesian prediction error bound for GP regression, which we show to decay with the growth of a novel, kernel-based measure of data density. Based on the prediction error bound, we prove time-varying tracking accuracy guarantees for learned GP models used as feedback compensation of unknown nonlinearities, and show to achieve vanishing tracking error with increasing data density. This enables us to develop an episodic approach for learning Gaussian process models, such that an arbitrary tracking accuracy can be guaranteed. The effectiveness of the derived theory is demonstrated in several simulations.