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CS 182/Ling109/CogSci110Spring 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
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 f
1 1/6 1
2 1/6 2
3 1/6 3
4 1/6 4
5 1/6 5
6 1/6 6
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 correlatesSkill learningNatural rewards Reward pathway? Learning?Intracranial self-
stimulation;Drug addiction;
Parkinson’s Disease Motor control + initiation?
Also involved in: Working memory Novel situations ADHD Schizophrenia …
Dorsal Striatum (Caudate, Putamen)
Ventral TegmentalArea
Substantia Nigra
Amygdala
Nucleus Accumbens(Ventral Striatum)
Prefrontal CortexDorsal Striatum (Caudate, Putamen)
Ventral TegmentalArea
Substantia Nigra
Amygdala
Nucleus Accumbens(Ventral Striatum)
Prefrontal CortexDorsal Striatum (Caudate, Putamen)
Ventral TegmentalArea
Substantia Nigra
Amygdala
Nucleus Accumbens(Ventral Striatum)
Prefrontal CortexDorsal Striatum (Caudate, Putamen)
Ventral TegmentalArea
Substantia Nigra
Amygdala
Nucleus Accumbens(Ventral Striatum)
Prefrontal Cortex
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)
(Montague et al. 1996)Wolfram Schultz Lab 1990-1996
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: 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
DEMO
Elements of RL
Transition model, how action influences states Reward R, immediate value of state-action transition Policy , maps states to actions
Agent
Environment
State Reward Action
Policy
sss 221100 r a2
r a1
r a0 :::
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) = -2.0R(s) = -0.4
R(s) = -0.03R(s) = -0.01
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