CS 188: Artificial Intelligence
Spring 2007
Lecture 21:Reinforcement Learning: II
MDP
4/12/2007
Srini Narayanan – ICSI and UC Berkeley
Announcements
 Othello tournament signup
 Please send email to
cs188@imail.berkeley.edu
 HW on classification out
 Due 4/23
 Can work in pairs
Reinforcement Learning
 Basic idea:
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Receive feedback in the form of rewards
Agent’s utility is defined by the reward function
Must learn to act so as to maximize expected utility
Change the rewards, change the behavior
 Examples:
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Learning optimal paths
Playing a game, reward at the end for winning / losing
Vacuuming a house, reward for each piece of dirt picked up
Automated taxi, reward for each passenger delivered
Recap: MDPs
 Markov decision processes:
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States S
Actions A
Transitions P(s’|s,a) (or T(s,a,s’))
Rewards R(s,a,s’)
Start state s0
 Examples:
 Gridworld, High-Low, N-Armed Bandit
 Any process where the result of your action is stochastic
 Goal: find the “best” policy 
 Policies are maps from states to actions
 What do we mean by “best”?
 This is like search – it’s planning using a model, not actually
interacting with the environment
MDP Solutions
 In deterministic single-agent search, want an optimal
sequence of actions from start to a goal
 In an MDP, like expectimax, want an optimal policy (s)
 A policy gives an action for each state
 Optimal policy maximizes expected utility (i.e. expected rewards)
if followed
 Defines a reflex agent
Optimal policy when
R(s, a, s’) = -0.04 for all
non-terminals s
Example Optimal Policies
R(s) = -0.01
R(s) = -0.03
R(s) = -0.4
R(s) = -2.0
Stationarity
 In order to formalize optimality of a policy, need to
understand utilities of reward sequences
 Typically consider stationary preferences:
 Theorem: only two ways to define stationary utilities
 Additive utility:
 Discounted utility:
Infinite Utilities?!
 Problem: infinite state sequences with infinite rewards
 Solutions:
 Finite horizon:
 Terminate after a fixed T steps
 Gives nonstationary policy ( depends on time left)
 Absorbing state(s): guarantee that for every policy, agent will
eventually “die” (like “done” for High-Low)
 Discounting: for 0 <  < 1
 Smaller  means smaller horizon
How (Not) to Solve an MDP
 The inefficient way:
 Enumerate policies
 For each one, calculate the expected utility
(discounted rewards) from the start state
 E.g. by simulating a bunch of runs
 Choose the best policy
 We’ll return to a (better) idea like this later
Utility of a State
 Define the utility of a state under a policy:
V(s) = expected total (discounted) rewards starting in s
and following 
 Recursive definition (one-step look-ahead):
Policy Evaluation
 Idea one: turn recursive equations into updates
 Idea two: it’s just a linear system, solve with
Matlab (or Mosek, or Cplex)
Example: High-Low
 Policy: always say “high”
 Iterative updates:
Optimal Utilities
 Goal: calculate the optimal
utility of each state
V*(s) = expected (discounted)
rewards with optimal actions
 Why: Given optimal utilities,
MEU tells us the optimal policy
Bellman’s Equation for Selecting
actions
 Definition of utility leads to a simple relationship
amongst optimal utility values:
Optimal rewards = maximize over first action and then
follow optimal policy
Formally: Bellman’s Equation
That’s my
equation!
Example: GridWorld
Value Iteration
 Idea:
 Start with bad guesses at all utility values (e.g. V0(s) = 0)
 Update all values simultaneously using the Bellman equation
(called a value update or Bellman update):
 Repeat until convergence
 Theorem: will converge to unique optimal values
 Basic idea: bad guesses get refined towards optimal values
 Policy may converge long before values do
Example: Bellman Updates
Example: Value Iteration
 Information propagates outward from terminal
states and eventually all states have correct
value estimates
[DEMO]
Convergence*
 Define the max-norm:
 Theorem: For any two approximations U and V (any two
utility vectors)
 I.e. any distinct approximations must get closer to each other
(after the Bellman update), so, in particular, any approximation
must get closer to the true U (Bellman update is U) and value
iteration converges to a unique, stable, optimal solution
 Theorem:
 I.e. once the change in our approximation is small, it must also
be close to correct
Policy Iteration
 Alternate approach:
 Policy evaluation: calculate utilities for a fixed policy
until convergence (remember the beginning of
lecture)
 Policy improvement: update policy based on resulting
converged utilities
 Repeat until policy converges
 This is policy iteration
 Can converge faster under some conditions
Policy Iteration
 If we have a fixed policy , use simplified
Bellman equation to calculate utilities:
 For fixed utilities, easy to find the best action
according to one-step look-ahead
Comparison
 In value iteration:
 Every pass (or “backup”) updates both utilities (explicitly, based
on current utilities) and policy (possibly implicitly, based on
current policy)
 In policy iteration:
 Several passes to update utilities with frozen policy
 Occasional passes to update policies
 Hybrid approaches (asynchronous policy iteration):
 Any sequences of partial updates to either policy entries or
utilities will converge if every state is visited infinitely often
Reinforcement Learning
 Reinforcement learning:
 Still have an MDP:
 A set of states s  S
 A model T(s,a,s’)
 A reward function R(s)
 Still looking for a policy (s)
 New twist: don’t know T or R
 I.e. don’t know which states are good or what the actions do
 Must actually try actions and states out to learn
Example: Animal Learning
 RL studied experimentally for more than 60
years in psychology
 Rewards: food, pain, hunger, drugs, etc.
 Mechanisms and sophistication debated
 Example: foraging
 Bees learn near-optimal foraging plan in field of
artificial flowers with controlled nectar supplies
 Bees have a direct neural connection from nectar
intake measurement to motor planning area
Example: Backgammon
 Reward only for win / loss in
terminal states, zero
otherwise
 TD-Gammon learns a
function approximation to
U(s) using a neural network
 Combined with depth 3
search, one of the top 3
players in the world
Passive Learning
 Simplified task
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You don’t know the transitions T(s,a,s’)
You don’t know the rewards R(s,a,s’)
You are given a policy (s)
Goal: learn the state values (and maybe the model)
 In this case:
 No choice about what actions to take
 Just execute the policy and learn from experience
 We’ll get to the general case soon
Example: Direct Estimation
y
 Episodes:
+100
(1,1) up -1
(1,1) up -1
(1,2) up -1
(1,2) up -1
(1,2) up -1
(1,3) right -1
(1,3) right -1
(2,3) right -1
(2,3) right -1
(3,3) right -1
(3,3) right -1
(3,2) right -1
(3,2) up -1
(4,2) right -100
(3,3) right +100
(done)
(done)
-100
x
 = 1, R = -1
U(1,1) ~ (93 + -105) / 2 = -6
U(3,3) ~ (100 + 98 + -101) / 3 = 32.3
Model-Based Learning
 Idea:
 Learn the model empirically (rather than values)
 Solve the MDP as if the learned model were correct
 Empirical model learning
 Simplest case:
 Count outcomes for each s,a
 Normalize to give estimate of T(s,a,s’)
 Discover R(s,a,s’) the first time we experience (s,a,s’)
 More complex learners are possible (e.g. if we know
that all squares have related action outcomes, e.g.
“stationary noise”)
Example: Model-Based Learning
y
 Episodes:
+100
(1,1) up -1
(1,1) up -1
(1,2) up -1
(1,2) up -1
(1,2) up -1
(1,3) right -1
(1,3) right -1
(2,3) right -1
(2,3) right -1
(3,3) right -1
(3,3) right -1
(3,2) right -1
(3,2) up -1
(4,2) right -100
(3,3) right +100
(done)
(done)
-100
x
=1
T(<3,3>, right, <4,3>) = 1 / 3
T(<2,3>, right, <3,3>) = 2 / 2
Model-Based Learning
 In general, want to learn the optimal policy, not
evaluate a fixed policy
 Idea: adaptive dynamic programming
 Learn an initial model of the environment:
 Solve for the optimal policy for this model (value or
policy iteration)
 Refine model through experience and repeat
 Crucial: we have to make sure we actually learn
about all of the model