I ntroduction to Mobile Robotics Gaussian Processes Wolfram Burgard Cyrill Stachniss Giorgio Grisetti Maren Bennewitz Christian Plagemann SS08, University of Freiburg, Department for Computer Science
Announcem ent • Sommercampus 2008 will feature a hands-on course on “Gaussian processes” • Topics: Understanding and applying GPs • Pre-requisites: Programming, basic maths • Tutor: Sebastian Mischke • Duration: 4 sessions 2 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Overview • The regression problem • Gaussian process models • Learning GPs • Applications • Summary Some figures were taken from Carl E. Rasmussen: NIPS 2006 Tutorial 3 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Given n observed points • how can we recover the dependency • to predict new points ? 4 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Given n observed points 5 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Solution 1: Parametric models • Linear • Quadratic • Higher order polynomials • … • Learning: optimizing the parameters 6 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Solution 1: Parametric models 7 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Solution 1: Parametric models 8 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Solution 2: Non-parametric models • Radial Basis Functions • Neural Networks • Splines, Support Vector Machines • Histograms, … • Learning: finding the model structure and optimize the parameters 9 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Given n observed points 10 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Solution 3: Express the model directly in terms of the data points • Idea: Any finite set of values sampled from • has a joint Gaussian distribution • with a covariance matrix given by 11 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Gaussian process m odels • Then, the n+ 1 dimensional vector which includes the new target that is to be predicted, comes from an n+ 1 dimensional normal distribution. • The predictive distribution for this new target is a 1-dimensional Gaussian. 12 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The regression problem • Given n observed points 13 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Gaussian process m odels • Example • Given the n observed points • and the squared exponential covariance function • with • and a noise level 14 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Gaussian process m odels • Example 15 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • The squared exponential covariance function: index/ input distance amplitude characteristic noise level lengthscale • The parameters are easy to interpret! 16 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Example: low noise 17 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Example: medium noise 18 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Example: high noise 19 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Example: small lengthscale 20 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Example: large lengthscale 21 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Covariance function specifies the prior prior posterior 22 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Gaussian process m odels • Recall, the n+ 1 dimensional vector which includes the new target that is to be predicted, comes from an n+ 1 dimensional normal distribution. • The predictive distribution for this new target is a 1-dimensional Gaussian. • Why? 23 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The Gaussian Distribution • Recall the 2-dimensional joint Gaussian: • The conditionals and the marginals Figure taken from Carl E. Rasmussen: are also Gaussians NIPS 2006 Tutorialc 24 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The Gaussian Distribution • Simple bivariate example: 25 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The Gaussian Distribution • Simple bivariate example: joint marginal conditional 26 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The Gaussian Distribution • Marginalization: 27 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The Gaussian Distribution • The conditional: 28 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The Gaussian Distribution • Slightly more complicated in the general case: • The conditionals and the marginals Figure taken from Carl E. Rasmussen: are also Gaussians NIPS 2006 Tutorialc 29 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The Gaussian Distribution • Conditioning the joint Gaussian in general • With zero mean: 30 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Gaussian process m odels • Recall the GP assumption 31 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Gaussian process m odels • Mean and variance of the predictive distribution have the simple form • with 32 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Learning a Gaussian process means • choosing a covariance function • finding its parameters and the noise level • How / to what objective? 33 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • The hyperparameters can be found by maximizing the likelihood of the training data e.g., using gradient methods 34 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Or, for a fully Bayesian treatment, by integrating over the hyperparameters using their priors • This integral can be approximated numerically using Markov chain sampling. 35 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning GPs • Objective: high data likelihood data fit complexity const penalty • Due to the Gaussian assumption, GPs have Occam’s razor built in. 36 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Occam ‘s razor • “use the simplest explanation for the data” 37 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Too short Lengthscale just right Too long
Understanding Gaussian processes • GP mean prediction can be seen as weighted summation over the data weights • Thus, for every mean prediction, there exist an equivalent kernel the produces the same result • But: hard to compute • Purpose: Visualization / understanding 39 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The GP Equivalent Kernel • For different lengthscales 40 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
The GP Equivalent Kernel • At different locations 41 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Classification • Task: • Predict discrete (e.g. binary) target values • Learn the class probabilies from observed cases • Problem: Noise is not Gaussian • Approach: • Introduce latent real-valued variables for each (discrete) target such that 42 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Classification • Problem: Integration over latent variables intractable • � Approximations necessary 43 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Advanced Topics / Extensions • Classification / non-Gaussian noise • Sparse GPs : Approximations for large data sets • Heteroscedastic GPs : Modeling non- constant noise • Nonstationary GPs : Modeling varying smoothness (lengthscales) • Mixtures of GPs • Uncertain inputs • Kernels for non-vectorial inputs 44 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Further Reading 45 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Applications 1. Learning sampling models for DBNs 2. Body scheme learning for manipulation 3. Learning to control an aerial blimp 4. Monocular range sensing 46 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Applications ( 1 ) Learning sam pling m odels for dynam ic Bayesian netw orks ( DBNs) Joint work with Dieter Fox and Wolfram Burgard 47 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
Learning sam pling m odels • A mobile robot collides with obstacles 48 Mobile Robotics, SS08, Universität Freiburg, Gaussian Processes
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