This tutorial provides a systematic introduction to Gaussian process
learning-based model predictive control (GP-MPC), an advanced approach
integrating Gaussian process (GP) with model predictive control (MPC) for
enhanced control in complex systems. It begins with GP regression fundamentals,
illustrating how it enriches MPC with enhanced predictive accuracy and robust
handling of uncertainties. A central contribution of this tutorial is the first
detailed, systematic mathematical formulation of GP-MPC in literature, focusing
on deriving the approximation of means and variances propagation for GP
multi-step predictions. Practical applications in robotics control, such as
path-following for mobile robots in challenging terrains and mixed-vehicle
platooning, are discussed to demonstrate the real-world effectiveness and
adaptability of GP-MPC. This tutorial aims to make GP-MPC accessible to
researchers and practitioners, enriching the learning-based control field with
in-depth theoretical and practical insights and fostering further innovations
in complex system control.

高斯过程学习模型预测控制（GP-MPC）系统地介绍了一种先进的方法，将高斯过程（GP）与模型预测控制（MPC）相结合，以提高复杂系统中的控制效果。它从 GP 回归基础知识开始，说明了它如何提高 MPC 的预测准确性和鲁棒性处理。本教程的一个重点贡献是对 GP-MPC 进行了首次详细、系统的数学形式化，重点关注了推导用于 GP 多步预测的均值和方差传播近似方法。通过讨论在机器人控制中的实际应用，如移动机器人在具有挑战性的地形中的路径跟随和混合车辆编队等，展示了 GP-MPC 的实际有效性和适应性。本教程旨在使 GP-MPC 对研究人员和实践者更加易于理解，为学习控制领域提供深入的理论和实践洞察，并促进复杂系统控制领域的进一步创新。

基于高斯过程学习的模型预测控制教程

A Tutorial on Gaussian Process Learning-based Model Predictive Control

In some settings neural networks exhibit a phenomenon known as grokking,
where they achieve perfect or near-perfect accuracy on the validation set long
after the same performance has been achieved on the training set. In this
paper, we discover that grokking is not limited to neural networks but occurs
in other settings such as Gaussian process (GP) classification, GP regression
and linear regression. We also uncover a mechanism by which to induce grokking
on algorithmic datasets via the addition of dimensions containing spurious
information. The presence of the phenomenon in non-neural architectures
provides evidence that grokking is not specific to SGD or weight norm
regularisation. Instead, grokking may be possible in any setting where solution
search is guided by complexity and error. Based on this insight and further
trends we see in the training trajectories of a Bayesian neural network (BNN)
and GP regression model, we make progress towards a more general theory of
grokking. Specifically, we hypothesise that the phenomenon is governed by the
accessibility of certain regions in the error and complexity landscapes.

网络神经元及其他建模方法中存在着 “不可解释” 的现象，其精确度远超过训练集的表现，本文试图寻找并研究这种神奇现象背后的机制。

超越神经网络的理解：模型复杂性的实证探索

Grokking Beyond Neural Networks: An Empirical Exploration with Model  Complexity

The accurate predictions and principled uncertainty measures provided by GP
regression incur O(n^3) cost which is prohibitive for modern-day large-scale
applications. This has motivated extensive work on computationally efficient
approximations. We introduce a new perspective by exploring robustness
properties and limiting behaviour of GP nearest-neighbour (GPnn) prediction. We
demonstrate through theory and simulation that as the data-size n increases,
accuracy of estimated parameters and GP model assumptions become increasingly
irrelevant to GPnn predictive accuracy. Consequently, it is sufficient to spend
small amounts of work on parameter estimation in order to achieve high MSE
accuracy, even in the presence of gross misspecification. In contrast, as n
tends to infinity, uncertainty calibration and NLL are shown to remain
sensitive to just one parameter, the additive noise-variance; but we show that
this source of inaccuracy can be corrected for, thereby achieving both
well-calibrated uncertainty measures and accurate predictions at remarkably low
computational cost. We exhibit a very simple GPnn regression algorithm with
stand-out performance compared to other state-of-the-art GP approximations as
measured on large UCI datasets. It operates at a small fraction of those other
methods' training costs, for example on a basic laptop taking about 30 seconds
to train on a dataset of size n = 1.6 x 10^6.

本研究提出了一种基于最近邻预测的高斯过程回归算法，相比于现有算法，该算法具有更快的计算速度和更准确的预测结果，并且其对数据集大小和样本误差的鲁棒性更强。

利用局部性和稳健性实现大规模高斯过程回归

Leveraging Locality and Robustness to Achieve Massively Scalable  Gaussian Process Regression

Off-the-shelf Gaussian Process (GP) covariance functions encode smoothness
assumptions on the structure of the function to be modeled. To model complex
and non-differentiable functions, these smoothness assumptions are often too
restrictive. One way to alleviate this limitation is to find a different
representation of the data by introducing a feature space. This feature space
is often learned in an unsupervised way, which might lead to data
representations that are not useful for the overall regression task. In this
paper, we propose Manifold Gaussian Processes, a novel supervised method that
jointly learns a transformation of the data into a feature space and a GP
regression from the feature space to observed space. The Manifold GP is a full
GP and allows to learn data representations, which are useful for the overall
regression task. As a proof-of-concept, we evaluate our approach on complex
non-smooth functions where standard GPs perform poorly, such as step functions
and robotics tasks with contacts.

本研究提出了一种新的监督学习方法，Manifold Gaussian Processes，该方法通过联合学习将数据转换为特征空间，并从特征空间到观测空间建立了 GP 回归模型，实验结果表明该方法适用于包括非光滑函数和机器人任务等不规则函数。