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

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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

SLIDE 1

CS489/698 Lecture 11: Feb 8, 2017

Gaussian Processes [B] Section 6.4 [M] Chap. 15 [HTF]

Sec. 8.3

SLIDE 2

Gaussian Process Regression

Idea: distribution over functions

SLIDE 3

Bayesian Linear Regression

Setting:

and

Weight space view:

– Prior: – Posterior:

unknown Gaussian Gaussian Gaussian

SLIDE 4

Bayesian Linear Regression

Setting:

and

Function space view:

– Prior: – Posterior:

unknown Gaussian Gaussian Deterministic Gaussian Gaussian Deterministic

SLIDE 5

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

ver functions)?
Answer: Gaussian Process (infinite dimensional

Gaussian distribution)

SLIDE 6

Gaussian Process

Distribution over functions:
Where

is the mean and is the kernel covariance function

SLIDE 7

Mean function

Compute the mean function

as follows:

with

Then

SLIDE 8

Kernel covariance function

Compute kernel covariance

as follows:

In some cases we can use domain knowledge to

specify directly.

SLIDE 9

Examples

Sampled functions from a Gaussian Process

Gaussian kernel

Exponential kernel

(Brownian motion)

SLIDE 10

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

SLIDE 11

Recap: Bayesian Linear Regression

Prior:
Likelihood:
Posterior:

where

Prediction:
Complexity: inversion of

is cubic in # of basis functions

SLIDE 12

Gaussian Process Regression

Prior:
Likelihood:
Posterior:

where

Prediction:
Complexity: inversion of

is cubic in # of training points

SLIDE 13

Case Study: AIBO Gait Optimization

SLIDE 14

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).

SLIDE 15

Search Problem

, be a vector of 15 parameters that defines a controller for gait

be a mapping from controller parameters to gait speed

Problem: find parameters

that yield highest speed.

But is unknown…

SLIDE 16

Approach

Picture

SLIDE 17

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

SLIDE 18

Results

Gaussian kernel: