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ECE-517: Reinforcement LearningECE-517: Reinforcement Learningin Artificial Intelligence in Artificial Intelligence

Lecture 10: Temporal-Difference LearningLecture 10: Temporal-Difference Learning

Dr. Itamar ArelDr. Itamar Arel

College of EngineeringCollege of EngineeringDepartment of Electrical Engineering and Computer ScienceDepartment of Electrical Engineering and Computer Science

The University of TennesseeThe University of TennesseeFall 2011Fall 2011

October 6, 2011October 6, 2011

ECE 517 - Reinforcement Learning in AI 22

Introduction to Temporal Learning (TD) & TD PredictionIntroduction to Temporal Learning (TD) & TD Prediction

If one had to identify one idea as central and novel to If one had to identify one idea as central and novel to RL, it would undoubtedly be RL, it would undoubtedly be temporal-differencetemporal-difference (TD) (TD) learninglearning

Combination of ideas from DP and Monte CarloCombination of ideas from DP and Monte Carlo Learns without a model (like MC), bootstraps (like DP)Learns without a model (like MC), bootstraps (like DP)

Both TD and Monte Carlo methods use experience to Both TD and Monte Carlo methods use experience to solve the solve the prediction problemprediction problem

A simple every-visit MC method may be expressed asA simple every-visit MC method may be expressed as

Let’s call this Let’s call this constant-constant- MC MC

We will focus on the We will focus on the prediction problemprediction problem (a.k.a. policy (a.k.a. policy evaluation) evaluation) evaluating V(s) for a given policy evaluating V(s) for a given policy

)()()( tttt sVRsVsV

target: the actual return after time t

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TD Prediction (cont.)TD Prediction (cont.)

Recall that in MC we need to wait until the end of the Recall that in MC we need to wait until the end of the episode to update the value estimatesepisode to update the value estimates

The idea of TD is to do so every time stepThe idea of TD is to do so every time step

Simplest TD method, TD(0):Simplest TD method, TD(0):

Essentially, we are updating one guess based on Essentially, we are updating one guess based on anotheranother

The idea is that we have a “moving target”The idea is that we have a “moving target”

ttttt sVsVrsVsV 11)()(

target: an estimate of the return

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Simple Monte CarloSimple Monte Carlo

T T T TT

T T T T T

. state following return actual theis where

)()()(

tt

tttt

sR

sVRsVsV

st

T T

T T

TT T

T TT

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Simplest TD MethodSimplest TD Method

T T T TT

T T T T T

st1

rt1

st

)()()()( 11 ttttt sVsVrsVsV

TTTTT

T T T T T

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Dynamic ProgrammingDynamic Programming

)()( 1 ttt sVrEsV

T

T T T

st

rt1

st1

T

TT

T

TT

T

T

T

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Tabular TD(0) for estimating Tabular TD(0) for estimating VV

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TD methods Bootstrap and SampleTD methods Bootstrap and Sample

BootstrappingBootstrapping: update involves an : update involves an estimate estimate (i.e. guess from a guess)(i.e. guess from a guess)

Monte Carlo does not bootstrapMonte Carlo does not bootstrap Dynamic Programming bootstrapsDynamic Programming bootstraps Temporal Different bootstrapsTemporal Different bootstraps

SamplingSampling: update does not involve an : update does not involve an expected valueexpected value

Monte Carlo samplesMonte Carlo samples Dynamic Programming does not sampleDynamic Programming does not sample Temporal Difference samplesTemporal Difference samples

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Example: Driving HomeExample: Driving Home

State Elapsed Time(minutes)

PredictedTime to Go

PredictedTotal Time

leaving office 0 30 30

reach car, raining 5 35 40

exit highway 20 15 35

behind truck 30 10 40

home street 40 3 43

arrive home 43 0 43

rewardsrewardsReturns fromReturns from

each stateeach state

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Example: Driving Home (cont.)Example: Driving Home (cont.)

Changes recommended by Monte Carlo methods =1)

Changes recommendedby TD methods (=1)

Value of each state is its expected time-to-goValue of each state is its expected time-to-go

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Is it really necessary to wait until the end of the Is it really necessary to wait until the end of the episode to start learning?episode to start learning?

Monte Carlo says it isMonte Carlo says it is TD learning argues that learning can occur on-lineTD learning argues that learning can occur on-line

Suppose, on another day, you again estimate when Suppose, on another day, you again estimate when leaving your office that it will take 30 minutes to drive leaving your office that it will take 30 minutes to drive home, but then you get stuck in a massive traffic jamhome, but then you get stuck in a massive traffic jam

Twenty-five minutes after leaving the office you are still Twenty-five minutes after leaving the office you are still bumper-to-bumper on the highwaybumper-to-bumper on the highway

You now estimate that it will take another 25 minutes to You now estimate that it will take another 25 minutes to get home, for a total of 50 minutes get home, for a total of 50 minutes

Must you wait until you get home before increasing your Must you wait until you get home before increasing your estimate for the initial state? estimate for the initial state?

In TD you would be shifting your initial estimate from In TD you would be shifting your initial estimate from 30 minutes toward 50 30 minutes toward 50

Example: Driving Home (cont.)Example: Driving Home (cont.)

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Advantages of TD LearningAdvantages of TD Learning

TD methods TD methods do not requiredo not require a model of the a model of the environment, only experienceenvironment, only experience

TD, but not MC, methods can be TD, but not MC, methods can be fully incrementalfully incremental Agent learns a “guess from a guess”Agent learns a “guess from a guess” Agent can learn Agent can learn beforebefore knowing the final outcome knowing the final outcome

Less memoryLess memory

Reduced peak computationReduced peak computation Agent can learn Agent can learn withoutwithout the final outcome the final outcome

From incomplete sequencesFrom incomplete sequences

Helps with applications that have very long episodesHelps with applications that have very long episodes

Both MC and TD converge (under certain assumptions Both MC and TD converge (under certain assumptions to be detailed later), but which is faster?to be detailed later), but which is faster?

Currently unknown – generally TD does better on Currently unknown – generally TD does better on stochastic tasksstochastic tasks

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Random Walk ExampleRandom Walk Example

In this example we empirically compare the prediction In this example we empirically compare the prediction abilities of abilities of TD(0)TD(0) and and constant-constant- MC MC applied to the applied to the small Markov process:small Markov process:

All episodes starts in state CAll episodes starts in state C

Proceed one state, right orProceed one state, right orleft with equal probabilityleft with equal probability

Termination: R = +1, L = 0Termination: R = +1, L = 0

True values:True values:V(C)=1/2, V(A)=1/6, V(B)=2/6V(C)=1/2, V(A)=1/6, V(B)=2/6V(D)=4/6, V(E)=5/6V(D)=4/6, V(E)=5/6

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Random Walk Example (cont.)Random Walk Example (cont.)

Data averaged over100 sequences of episodes

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Optimality of TD(0)Optimality of TD(0)

Suppose only a finite amount of experience is Suppose only a finite amount of experience is available, say 10 episodes or 100 time stepsavailable, say 10 episodes or 100 time steps

Intuitively, we repeatedly present the experience until Intuitively, we repeatedly present the experience until convergence is achievedconvergence is achieved

Updates are made after a Updates are made after a batchbatch of training data of training data Also called batch updatingAlso called batch updating

For any finite Markov prediction task, under batch For any finite Markov prediction task, under batch updating, TD(0) converges for sufficiently small updating, TD(0) converges for sufficiently small MC method also converges deterministically but to a MC method also converges deterministically but to a different answerdifferent answer

To better understand the different between MC and To better understand the different between MC and TD(0), we’ll consider TD(0), we’ll consider the batch random walkthe batch random walk

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Optimality of TD(0) (cont.)Optimality of TD(0) (cont.)

After each new episode, all previous episodes were treated as a batch, and the algorithm was trained until convergence. All repeated 100 times.A key question is what would explain these two curves?

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You are the PredictorYou are the Predictor

Suppose you observe the following 8 episodesSuppose you observe the following 8 episodes

QQ: : What would you guess V(A) and V(B) to be?What would you guess V(A) and V(B) to be?

1) A, 0, B, 02) B, 13) B, 14) B, 15) B, 16) B, 17) B, 18) B, 0

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You are the Predictor (cont.)You are the Predictor (cont.)

V(A) = ¾ is the answer that batch TD(0) givesV(A) = ¾ is the answer that batch TD(0) givesThe other reasonable answer is simply to say that A(0)=0 The other reasonable answer is simply to say that A(0)=0 ((Why?Why?))

This is the answer that MC givesThis is the answer that MC gives

If the process is Markovian, we expect that the TD(0) If the process is Markovian, we expect that the TD(0) answer will produce lower error on answer will produce lower error on futurefuture data, even data, even though the Monte Carlo answer is better on the existing though the Monte Carlo answer is better on the existing datadata

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TD(0) vs. MCTD(0) vs. MC

For MC, the prediction that best matches the For MC, the prediction that best matches the training data is V(A)=0training data is V(A)=0

This This minimizes the mean-square-errorminimizes the mean-square-error on the training set on the training set This is what a batch Monte Carlo method getsThis is what a batch Monte Carlo method gets

If we consider the sequentiality of the problem, If we consider the sequentiality of the problem, then we would set V(A)=.75then we would set V(A)=.75 This is correct for the This is correct for the maximum likelihoodmaximum likelihood

estimate of a Markov model generating the data estimate of a Markov model generating the data i.e, if we do a best fit Markov model, and assume it is i.e, if we do a best fit Markov model, and assume it is exactly correct, and then compute what it predicts exactly correct, and then compute what it predicts

This is called the This is called the certainty-equivalence estimatecertainty-equivalence estimate It is what TD(0) yieldsIt is what TD(0) yields

ECE 517 - Reinforcement Learning in AI 2020

Learning An Action-Value FunctionLearning An Action-Value Function

We now consider the use of TD methods for the We now consider the use of TD methods for the control problemcontrol problem

As with MC, we need to balance exploration and As with MC, we need to balance exploration and exploitationexploitation

Again, two schemes: Again, two schemes: on-policyon-policy and and off-policyoff-policy

We’ll start with on-policy, and learn action-value We’ll start with on-policy, and learn action-value functionfunction

.0),( then terminal,is If

,,,,

: thisdo ,state terminal-nona from sitionevery tranAfter

111

111

ttt

ttttttttt

t

asQs

asQasQrasQasQ

s

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SARSA: On-Policy TD(0) LearningSARSA: On-Policy TD(0) Learning

One can easily turn this into a control method by One can easily turn this into a control method by always updating the policy to be greedy with respect always updating the policy to be greedy with respect to the current estimate of Q(s,a)to the current estimate of Q(s,a)

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Q-Learning: Off-Policy TD ControlQ-Learning: Off-Policy TD Control

One of the most important breakthroughs in RL was One of the most important breakthroughs in RL was the development of Q-Learning - an off-policy TD the development of Q-Learning - an off-policy TD control algorithm (1989)control algorithm (1989)

ttta

ttttt asQasQrasQasQ ,,max,, 11

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Q-Learning: Off-Policy TD Control (cont.)Q-Learning: Off-Policy TD Control (cont.)

The learned action-value function, Q, directly The learned action-value function, Q, directly approximates the optimal action-value function, Q*approximates the optimal action-value function, Q*

Converges as long as all states are visited and state-Converges as long as all states are visited and state-action values updatedaction values updated

Why is it considered an Why is it considered an off-policyoff-policy control method? control method?

How expensive is it to implement?How expensive is it to implement?