CS489/698 Lecture 11: Feb 8, 2017 Gaussian Processes [B] Section - PowerPoint PPT Presentation

CS489/698 Lecture 11: Feb 8, 2017 Gaussian Processes [B] Section 6.4 [M] Chap. 15 [HTF] Sec. 8.3 CS489/698 (c) 2017 P. Poupart 1

Gaussian Process Regression • Idea: distribution over functions CS489/698 (c) 2017 P. Poupart 2

Bayesian Linear Regression • Setting: and unknown • Weight space view: – Prior: – Posterior: Gaussian Gaussian Gaussian CS489/698 (c) 2017 P. Poupart 3

Bayesian Linear Regression • Setting: and unknown • Function space view: – Prior: Gaussian Gaussian Deterministic – Posterior: Deterministic Gaussian Gaussian CS489/698 (c) 2017 P. Poupart 4

Gaussian Process • According to the function view, there is a Gaussian at for every . Those Gaussians are correlated through . • What is the general form of (i.e., distribution over functions)? • Answer: Gaussian Process (infinite dimensional Gaussian distribution) CS489/698 (c) 2017 P. Poupart 5

Gaussian Process • Distribution over functions: • Where is the mean and is the kernel covariance function CS489/698 (c) 2017 P. Poupart 6

Mean function • Compute the mean function as follows: • Let with • Then CS489/698 (c) 2017 P. Poupart 7

Kernel covariance function • Compute kernel covariance as follows: • � � • In some cases we can use domain knowledge to specify directly. CS489/698 (c) 2017 P. Poupart 8

Examples • Sampled functions from a Gaussian Process Gaussian kernel Exponential kernel � (Brownian motion) � � � CS489/698 (c) 2017 P. Poupart 9

Gaussian Process Regression • Gaussian Process Regression corresponds to kernelized Bayesian Linear Regression • Bayesian Linear Regression: – Weight space view – Goal: (posterior over ) – Complexity: cubic in # of basis functions • Gaussian Process Regression: – Function space view – Goal: (posterior over ) – Complexity: cubic in # of training points CS489/698 (c) 2017 P. Poupart 10

Recap: Bayesian Linear Regression • Prior: • Likelihood: • Posterior: where • Prediction: • Complexity: inversion of is cubic in # of basis functions CS489/698 (c) 2017 P. Poupart 11

Gaussian Process Regression • Prior: • Likelihood: • Posterior: where • Prediction: • Complexity: inversion of is cubic in # of training points CS489/698 (c) 2017 P. Poupart 12

Case Study: AIBO Gait Optimization CS489/698 (c) 2017 P. Poupart 13

Gait Optimization • Problem: find best parameter setting of the gait controller to maximize walking speed – Why?: Fast robots have a better chance of winning in robotic soccer • Solutions: – Stochastic hill climbing – Gaussian Processes • Lizotte, Wang, Bowling, Schuurmans (2007) Automatic Gait Optimization with Gaussian Processes, International Joint Conferences on Artificial Intelligence (IJCAI) . CS489/698 (c) 2017 P. Poupart 14

Search Problem • Let , be a vector of 15 parameters that defines a controller for gait • Let be a mapping from controller parameters to gait speed • Problem: find parameters that yield highest speed. But is unknown… CS489/698 (c) 2017 P. Poupart 15

Approach • Picture CS489/698 (c) 2017 P. Poupart 16

Approach • Initialize • Repeat: – Select new � ��∈� – Evaluate by observing speed of robot with parameters set to – Update Gaussian process: • and ��� ��� �� �� • � �� • CS489/698 (c) 2017 P. Poupart 17

Results Gaussian kernel: �� � ��� � � � ��� � � � � CS489/698 (c) 2017 P. Poupart 18