CS 182/Ling109/CogSci110
Spring 2008
Reinforcement Learning: Basics
3/20/2008
Srini Narayanan – ICSI and UC Berkeley
Lecture Outline
 Introduction
 Basic Concepts
 Expectation, Utility, MEU
 Neural correlates of reward based learning
 Utility theory from economics
 Preferences, Utilities.
 Reinforcement Learning: AI approach
Models of Learning

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Hebbian ~ coincidence
Recruitment ~ one trial
Supervised ~ correction (backprop)
Reinforcement ~ Reward based
 delayed reward
 Unsupervised ~ similarity
Reinforcement Learning
 Basic idea:
 Receive feedback in the form of rewards
 also called reward based learning in psychology
 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:




Learning coordinated behavior/skills (x-schemas)
Playing a game, reward at the end for winning / losing
Vacuuming robot, reward for each piece of dirt picked up
Automated taxi, reward for each passenger delivered
Coordination: Making Breakfast
 Phil prepares his breakfast. Closely examined, even this apparently
mundane activity reveals a complex web of conditional behavior and
interlocking goal-subgoal relationships: walking to the cupboard,
opening it, selecting a cereal box, then reaching for, grasping, and
retrieving the box. Other complex, tuned, interactive sequences of
behavior are required to obtain a bowl, spoon, and milk jug. Each
step involves a series of eye movements to obtain information and
to guide reaching and locomotion. Rapid judgments are continually
made about how to carry the objects or whether it is better to ferry
some of them to the dining table before obtaining others. Each step
is guided by goals, such as grasping a spoon or getting to the
refrigerator, and is in service of other goals, such as having the
spoon to eat with once the cereal is prepared and ultimately
obtaining nourishment (Sutton and Barto,Section 1.1)
Basic Features
 Interaction between agent and environment.
 Agent seeks to achieve a goal despite uncertainty in
the environment.
 Effects of actions cannot be completely predicted
 Requires monitoring the environment frequently.
 Agent’s actions change the future state of the
environment (opportunities and future options are
impacted)
 Correct choice requires taking into account indirect,
delayed consequences of actions, thus may require
foresight or planning.
Reinforcement Learning
 Multiple fields contribute to the study of
reinforcement learning
 Economics
 Utility theory and preferences, game theory
 Artificial Intelligence
 Machine learning, action and state representation, inference
 Psychology
 Reward based prediction and control, conditioning
 Neuroscience
 Reward related circuits, timing of rewards, neuroeconomics
Basic Ideas
 Utility
 Preferences
 Maximum Expected Utility (MEU)
 Reward
 Immediate and Delayed rewards
 Average Reward
 Discounting
 Learning and Acting
 Prediction error
 Optimal Policy
Basic Idea: Maximum Expected
Utility (MEU)
 MEU: An agent should chose the action
which maximizes its expected utility, given
its knowledge
 General principle for decision making
 Often taken as the definition of rationality
 Let’s decompress this definition…
Reminder: Expectations
 Often a quantity of interest depends on a random
variable
 The expected value of a function is the average
output, weighted by some distribution over inputs
 Example: How late will I be?
 Lateness is a function of traffic:
L(T=none) = -10, L(T=light) = -5, L(T=heavy) = 15
 What is my expected lateness?
 Need to specify some belief over T to weight the outcomes
 Say P(T) = {none: 2/5, light: 2/5, heavy: 1/5}
 The expected lateness:
Expectations
 Real valued functions of random variables:
 Expectation of a function of a random variable
 Example: Expected value of a fair die roll
X
P
1
1/6
1
2
1/6
2
3
1/6
3
4
1/6
4
5
1/6
5
6
1/6
6
f
Utilities
 Utilities are functions from outcomes (states of the world)
to real numbers that describe an agent’s preferences
 Where do utilities come from?
 In a game, may be simple (+1/-1)
 Utilities summarize the agent’s goals
 Theorem: any set of preferences between outcomes can be
summarized as a utility function (provided the preferences meet
certain conditions)
 In general, utilities are determined from rewards and
actions emerge to maximize expected utility.
Lecture Outline
 Introduction
 Basic Concepts
 Expectation, Utility, MEU
 Neural correlates of reward based learning
 Utility theory from economics
 Preferences, Utilities.
 Reinforcement Learning: AI approach
Multiple neurotransmitters are
involved in reinforcement learning
Dopamine based neural correlates
Prefrontal Cortex
Dorsal Striatum (Caudate, Putamen)
Nucleus Accumbens
(Ventral Striatum)
Amygdala
Ventral Tegmental
Area
Skill learning
Natural rewards
 Reward pathway?
 Learning?
Intracranial self-stimulation;
Drug addiction;
Parkinson’s Disease
 Motor control +
initiation?
Also involved in:
 Working memory
 Novel situations
Substantia Nigra  ADHD
 Schizophrenia
 …
Conditioning
Ivan Pavlov
CS
UCS
I rang the
bell!
Dopamine levels track prediction error
Unpredicted reward
(unlearned/no stimulus)
Predicted reward
(learned task)
Omitted reward
(probe trial)
Wolfram Schultz Lab 1990-1996
(Montague et al. 1996)
Basic concept: Prediction Error
 Learning theory suggests that learning
occurs when
 a reward value fails to meet the value
predicted by conditioned stimuli
 The difference between expected and actual
reward is the prediction error.
Ventral Striatum and amount of
reward
Areas that are probably directly
involved in RL
 Basal Ganglia
 Striatum (Ventral/Dorsal), Putamen, Substantia Nigra






Midbrain (VT)
Amygdala
Orbito-Frontal Cortex
Cingulate Circuit (ACC)
Cerebellum
PFC
Current Hypothesis
 Ventral Striatum (Nucleus Accumbens) encodes anticipation of reward.
 Different (overlapping) circuits for reward and punishment (OFC
involvement in punishment).
 Phasic dopamine encodes a reward prediction error
 Evidence
 Monkey single cell recordings
 Human fMRI studies
 Current Research
 Better information processing model
 Other reward/punishment circuits including Amygdala (for visual perception)
 Overall circuit (PFC-Basal Ganglia interaction)
 More in future lectures! Preview Wolfram Schultz’s article at
 http://www.scholarpedia.org/article/Reward_signals
Lecture Outline
 Introduction
 Basic Concepts
 Expectation, Utility, MEU
 Neural correlates of reward based learning
 Utility theory from economics
 Preferences, Utilities.
 Reinforcement Learning: AI approach
Economic Models of Utility
 Preferences
 Rational Preferences
 Axioms for preferences
 Human Rationality?
Preferences
 An agent chooses among:
 Prizes: A, B, etc.
 Lotteries: situations with
uncertain prizes
 Notation:
Rational Preferences
 We want some constraints on
preferences before we call
them rational
 For example: an agent with
intransitive preferences can
be induced to give away all its
money
 If B > C, then an agent with C
would pay (say) 1 cent to get B
 If A > B, then an agent with B
would pay (say) 1 cent to get A
 If C > A, then an agent with A
would pay (say) 1 cent to get C
Rational Preferences
 Preferences of a rational agent must obey constraints.
 These constraints (plus one more) are the axioms of rationality
 Theorem: Rational preferences imply behavior
describable as maximization of expected utility
MEU Principle
 Theorem:
 [Ramsey, 1931; von Neumann & Morgenstern, 1944]
 Given any preferences satisfying these constraints, there exists
a real-valued function U such that:
 Maximum expected likelihood (MEU) principle:
 Choose the action that maximizes expected utility
 Note: an agent can be entirely rational (consistent with MEU)
without ever representing or manipulating utilities and
probabilities
 E.g., a lookup table for perfect tictactoe, reflex vacuum cleaner
Human Utilities
 Utilities map states to real numbers. Which numbers?
 Standard approach to assessment of human utilities:
 Compare a state A to a standard lottery Lp between
 ``best possible prize'' u+ with probability p
 ``worst possible catastrophe'' u- with probability 1-p
 Adjust lottery probability p until A ~ Lp
 Resulting p is a utility in [0,1]
Utility Scales

Normalized utilities: u+ = 1.0, u- = 0.0

Micromorts: one-millionth chance of death, useful for paying to reduce
product risks, etc.

QALYs: quality-adjusted life years, useful for medical decisions involving
substantial risk
 One year with good health = 1 QALY

Note: behavior is invariant under positive linear transformation

With deterministic prizes only (no lottery choices), only ordinal utility can be
determined, i.e., total order on prizes
Example: Insurance
 Consider the lottery [0.5,$1000; 0.5,$0]
 What is its expected monetary value? ($500)
 What is its certainty equivalent?
 Monetary value acceptable in lieu of lottery
 $400 for most people
 Difference of $100 is the insurance premium
 There’s an insurance industry because people will pay to
reduce their risk
 If everyone were risk-prone, no insurance needed!
Example: Human Rationality?
 Famous example of Allais (1953)
 A: [0.8,$4k; 0.2,$0]
 B: [1.0,$3k; 0.0,$0]
 C: [0.2,$4k; 0.8,$0]
 D: [0.25,$3k; 0.75,$0]
 Most people prefer B > A, C > D
 But if U($0) = 0, then
 B > A  U($3k) > 0.8 U($4k)
 C > D  0.8 U($4k) > U($3k)
The Ultimatum Game
 Proposer: receives $x, offers split $k / $(x-k)
 Accepter: either
 Accepts: gets $k, proposer gets $(x-k)
 Rejects: neither gets anything
 Nash equilibrium (MEU play)?
 Any strategy profile where proposer offers $k and accepter will accept
$k or greater
 Issues:
 Why do people tend to reject offers which are very unfair (e.g. $20 from
$100)?
 Irrationality?
 Utility of $20 exceeded by utility of punishing the unfair proposer?
 What about if x is very very large?
 fMRI experiments:
 Dopamine pathways implicated.
 Pleasure from punishment of others or injustice?
 More in coming lectures!
Lecture Outline
 Introduction
 Basic Concepts
 Expectation, Utility, MEU
 Neural correlates of reward based learning
 Utility theory from economics
 Preferences, Utilities.
 Reinforcement Learning: AI approach
 The problem
 Computing total expected value with discounting
 Bellman’s equation
Reinforcement Learning
 Basic idea:




DEMO
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:




Learning your way around, reward for reaching the destination.
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
Elements of RL
Agent
State
Policy
Reward
Action
Environment
0 : r0
1 : r1
2 : r2
s0 a
 s1 a
 s2 a
 
 Transition model, how action influences states
 Reward R, immediate value of state-action transition
 Policy , maps states to actions
Markov Decision Processes
 Markov decision processes (MDPs)
 A set of states s  S
 A model T(s,a,s’) = P(s’ | s,a)
 Probability that action a in state s
leads to s’
 A reward function R(s, a, s’)
(sometimes just R(s) for leaving a
state or R(s’) for entering one)
 A start state (or distribution)
 Maybe a terminal state
 MDPs are the simplest case of
reinforcement learning
 In general reinforcement learning, we
don’t know the model or the reward
function
MDP Solutions
 In deterministic single-agent search, want an optimal
sequence of actions from start to a goal
 In an MDP we 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 a “done” state)
 Discounting: for 0 <  < 1
 Smaller  means smaller horizon
Finding Optimal Policies Demo
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
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]
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