CS 541: Artificial Intelligence
Lecture X: Markov Decision Process
Slides Credit: Peter Norvig and Sebastian Thrun
Schedule
TA: Vishakha Sharma
Email Address: vsharma1@stevens.edu
Week
Date
Topic
Reading
Homework
1
08/30/2011
Introduction & Intelligent Agent
Ch 1 & 2
N/A
2
09/06/2011
Search: search strategy and heuristic search
Ch 3 & 4s
HW1 (Search)
3
09/13/2011
Search: Constraint Satisfaction & Adversarial Search
Ch 4s & 5 & 6
Teaming Due
4
09/20/2011
Logic: Logic Agent & First Order Logic
Ch 7 & 8s
HW1 due, Midterm Project (Game)
5
09/27/2011
Logic: Inference on First Order Logic
Ch 8s & 9
6
10/04/2011
Uncertainty and Bayesian Network
Ch 13 &14s
HW2 (Logic)
10/11/2011
No class (function as Monday)
7
10/18/2011
Midterm Presentation
Midterm Project Due
8
10/25/2011
Inference in Baysian Network
Ch 14s
HW2 Due, HW3 (Probabilistic Reasoning)
9
11/01/2011
Probabilistic Reasoning over Time
Ch 15
10
11/08/2011
TA Session: Review of HW1 & 2
HW3 due, HW4 (MDP)
11
11/15/2011
Supervised Machine Learning
Ch 18
12
11/22/2011
Markov Decision Process
Ch 16
HW4 due
13
11/29/2011
Reinforcement learning
Ch 21
14
12/06/2011
Final Project Presentation
Final Project Due
Re-cap of Lecture IX
Machine Learning
Classification (Naïve Bayes)
Regression (Linear, Smoothing)
Linear Separation (Perceptron, SVMs)
Non-parametric classification (KNN)
Outline
Planning under uncertainty
Planning with Markov Decision Processes
Utilities and Value Iteration
Partially Observable Markov Decision Processes
Planning under uncertainty
Mine Mapping Robot Groundhog
9
10
Planning
Uncertainty
11
Planning: Classical Situation
hell
heaven
• World deterministic
• State observable
MDP-Style Planning
hell
heaven
• World stochastic
• State observable
[Koditschek 87, Barto et al. 89]
• Policy
• Universal Plan
• Navigation function
Stochastic, Partially Observable
sign
hell?
heaven?
[Sondik 72] [Littman/Cassandra/Kaelbling 97]
Stochastic, Partially Observable
sign
hell
heaven
sign
heaven
hell
Stochastic, Partially Observable
sign
heaven
hell
sign
?
?
sign
hell
heaven
start
50%
50%
Stochastic, Partially Observable
sign
?
?
start
sign
heaven
hell
sign
hell
heaven
50%
50%
sign
?
?
start
A Quiz
-dim continuous
stochastic
1-dim
continuous
stochastic
actions
# states
size belief space?
sensors
3: s1, s2, s3
deterministic
3
perfect
3: s1, s2, s3
stochastic
3
perfect
23-1: s1, s2, s3, s12, s13, s23, s123
deterministic
3
abstract states
deterministic
3
stochastic
2-dim continuous: p(S=s1), p(S=s2)
stochastic
3
none
2-dim continuous: p(S=s1), p(S=s2)
-dim continuous
deterministic
1-dim
continuous
stochastic
aargh!
stochastic
-dim
continuous
stochastic
Planning with Markov DecisionProcesses
MPD Planning
Solution for Planning problem
Noisy controls
Perfect perception
Generates “universal plan” (=policy)
Grid World
The agent lives in a grid
Walls block the agent’s path
The agent’s actions do not always go asplanned:
80% of the time, the action Northtakes the agent North(if there is no wall there)
10% of the time, North takes the agentWest; 10% East
If there is a wall in the direction theagent would have been taken, the agentstays put
Small “living” reward each step
Big rewards come at the end
Goal: maximize sum of rewards*
Grid Futures
22
Deterministic Grid World
Stochastic Grid World
X
X
E N S W
X
E N S W
?
X
X
X
Markov Decision Processes
An MDP is defined by:
A set of states s S
A set of actions a A
A transition function T(s,a,s’)
Prob that a from s leads to s’
i.e., P(s’ | s,a)
Also called the model
A reward function R(s, a, s’)
Sometimes just R(s) or R(s’)
A start state (or distribution)
Maybe a terminal state
MDPs are a family of non-deterministicsearch problems
Reinforcement learning: MDPs where we don’tknow the transition or reward functions
23
Solving MDPs
In deterministic single-agent search problems, want an optimal plan, orsequence of actions, from start to a goal
In an MDP, we want an optimal policy *: S → A
A policy gives an action for each state
An optimal policy maximizes expected utility if followed
Defines a reflex agent
Optimal policy when R(s, a,s’) = -0.03 for all non-terminals s
Example Optimal Policies
R(s) = -2.0
R(s) = -0.4
R(s) = -0.03
R(s) = -0.01
MDP Search Trees
Each MDP state gives a search tree
a
s
s’
s, a
(s,a,s’) called a transition
T(s,a,s’) = P(s’|s,a)
R(s,a,s’)
s,a,s’
s is a state
(s, a) is a q-state
26
Why Not Search Trees?
Why not solve with conventional planning?
Problems:
This tree is usually infinite (why?)
Same states appear over and over (why?)
We would search once per state (why?)
27
Utilities
Utilities
Utility = sum of future reward
Problem: infinite state sequences have infinite rewards
Solutions:
Finite horizon:
Terminate episodes after a fixed T steps (e.g. life)
Gives nonstationary policies ( depends on time left)
Absorbing state: guarantee that for every policy, a terminal state willeventually be reached (like “done” for High-Low)
Discounting: for 0 < < 1
Smaller means smaller “horizon” – shorter term focus
29
Discounting
Typically discountrewards by < 1 eachtime step
Sooner rewards havehigher utility than laterrewards
Also helps the algorithmsconverge
30
Recap: Defining MDPs
Markov decision processes:
States S
Start state s0
Actions A
Transitions P(s’|s,a) (or T(s,a,s’))
Rewards R(s,a,s’) (and discount )
MDP quantities so far:
Policy = Choice of action for each state
Utility (or return) = sum of discounted rewards
a
s
s, a
s,a,s’
s’
31
Optimal Utilities
Fundamental operation: compute thevalues (optimal expect-max utilities)of states s
Why? Optimal values define optimalpolicies!
Define the value of a state s:
V*(s) = expected utility starting in s andacting optimally
Define the value of a q-state (s,a):
Q*(s,a) = expected utility starting in s,taking action a and thereafter actingoptimally
Define the optimal policy:
*(s) = optimal action from state s
a
s
s, a
s,a,s’
s’
32
The Bellman Equations
Definition of “optimal utility” leads to asimple one-step lookahead relationshipamongst optimal utility values:
Optimal rewards = maximize over first action andthen follow optimal policy
Formally:
a
s
s, a
s,a,s’
s’
33
Solving MDPs
We want to find the optimal policy *
Proposal 1: modified expect-max search, starting from eachstate s:
a
s
s, a
s,a,s’
s’
34
Value Iteration
Idea:
Start with V0*(s) = 0
Given Vi*, calculate the values for all states for depth i+1:
This is called a value update or Bellman update
Repeat until convergence
Theorem: will converge to unique optimal values
Basic idea: approximations get refined towards optimal values
Policy may converge long before values do
35
Example: Bellman Updates
36
max happens fora=right, other actionsnot shown
Example: =0.9, livingreward=0, noise=0.2
Example: Value Iteration
Information propagates outward from terminal states andeventually all states have correct value estimates
V2
V3
37
Computing Actions
Which action should we chose from state s:
Given optimal values V?
Given optimal q-values Q?
Lesson: actions are easier to select from Q’s!
Asynchronous Value Iteration*
In value iteration, we update every state in each iteration
Actually, any sequences of Bellman updates will converge if everystate is visited infinitely often
In fact, we can update the policy as seldom or often as we like,and we will still converge
Idea: Update states whose value we expect to change:
If is large then update predecessors of s
Value Iteration: Example
Another Example
Value Function and Plan
Map
Another Example
Value Function and Plan
Map
Partially Observable MarkovDecision Processes
Stochastic, Partially Observable
sign
?
?
start
sign
heaven
hell
sign
hell
heaven
50%
50%
sign
?
?
start
Value Iteration in Belief space: POMDPs
Partially Observable Markov Decision Process
Known model (learning even harder!)
Observation uncertainty
Usually also: transition uncertainty
Planning in belief space = space of all probability distributions
Value function: Piecewise linear, convex function over the beliefspace
45
Introduction to POMDPs (1 of 3)
80
100
b
a
0
b
a
40
s2
s1
action a
action b
p(s1)
[Sondik 72, Littman, Kaelbling, Cassandra ‘97]
s2
s1
100
0
100
action a
action b
Introduction to POMDPs (2 of 3)
80
100
b
a
0
b
a
40
s2
s1
80%
c
20%
p(s1)
s2
s1’
s1
s2’
p(s1’)
p(s1)
s2
s1
100
0
100
[Sondik 72, Littman, Kaelbling, Cassandra ‘97]
Introduction to POMDPs (3 of 3)
80
100
b
a
0
b
a
40
s2
s1
80%
c
20%
p(s1)
s2
s1
100
0
100
p(s1)
s2
s1
s1
s2
p(s1’|A)
B
A
50%
50%
30%
70%
B
A
p(s1’|B)
[Sondik 72, Littman, Kaelbling, Cassandra ‘97]
POMDP Algorithm
Belief space = Space of all probability distribution(continuous)
Value function: Max of set of linear functions in beliefspace
Backup: Create new linear functions
Number of linear functions can grow fast!
49
Why is This So Complex?
State Space Planning
(no state uncertainty)
Belief Space Planning
(full state uncertainties)
?
but not usually.
The controller may beglobally uncertain...
Belief Space Structure
Augmented MDPs:
[Roy et al, 98/99]
conventional state space
uncertainty (entropy)
Path Planning with Augmented MDPs
information gain
Conventional planner
Probabilistic Planner
[Roy et al, 98/99]