1. ECE-517: Reinforcement Learning in Artificial Intelligence Lecture 10: Temporal-Difference Learning
October 6, 2011
Dr. Itamar Arel
College of Engineering
Department of Electrical Engineering and Computer Science
The University of Tennessee
Fall 2011

2. Introduction to Temporal Learning (TD) & TD Prediction
If one had to identify one idea as central and novel to RL, it would undoubtedly be temporal-difference (TD) learning
Combination of ideas from DP and Monte Carlo
Learns without a model (like MC), bootstraps (like DP)
Both TD and Monte Carlo methods use experience to solve the prediction problem
A simple every-visit MC method may be expressed as
Let’s call this constant-a MC
We will focus on the prediction problem (a.k.a. policy evaluation)  evaluating V(s) for a given policy
target: the actual return after time t

3. target: an estimate of the return
TD Prediction (cont.)
Recall that in MC we need to wait until the end of the episode to update the value estimates
The idea of TD is to do so every time step
Simplest TD method, TD(0):
Essentially, we are updating one guess based on another
The idea is that we have a “moving target”
target: an estimate of the return

4. Simple Monte Carlo T T T T T T T T T T T

5. Simplest TD Method T T T T T T T T T T T

6. Dynamic Programming T T T T T T T T T T T T T

7. Tabular TD(0) for estimating Vp

8. TD methods Bootstrap and Sample
Bootstrapping: update involves an estimate (i.e. guess from a guess)
Monte Carlo does not bootstrap
Dynamic Programming bootstraps
Temporal Different bootstraps
Sampling: update does not involve an expected value
Monte Carlo samples
Dynamic Programming does not sample
Temporal Difference samples

9. Example: Driving Home
rewards
Returns from
each state

10. Example: Driving Home (cont.)
Value of each state is its expected time-to-go
Changes recommended by Monte Carlo methods (a=1)
Changes recommended
by TD methods (a=1)

11. Example: Driving Home (cont.)
Is it really necessary to wait until the end of the episode to start learning?
Monte Carlo says it is
TD learning argues that learning can occur on-line
Suppose, on another day, you again estimate when leaving your office that it will take 30 minutes to drive home, but then you get stuck in a massive traffic jam
Twenty-five minutes after leaving the office you are still bumper-to-bumper on the highway
You now estimate that it will take another 25 minutes to get home, for a total of 50 minutes
Must you wait until you get home before increasing your estimate for the initial state?
In TD you would be shifting your initial estimate from 30 minutes toward 50

12. Advantages of TD Learning
TD methods do not require a model of the environment, only experience
TD, but not MC, methods can be fully incremental
Agent learns a “guess from a guess”
Agent can learn before knowing the final outcome
Less memory
Reduced peak computation
Agent can learn without the final outcome
From incomplete sequences
Helps with applications that have very long episodes
Both MC and TD converge (under certain assumptions to be detailed later), but which is faster?
Currently unknown – generally TD does better on stochastic tasks

13. Random Walk Example
In this example we empirically compare the prediction abilities of TD(0) and constant-a MC applied to the small Markov process:
All episodes starts in state C
Proceed one state, right or left with equal probability
Termination: R = +1, L = 0
True values: V(C)=1/2, V(A)=1/6, V(B)=2/6 V(D)=4/6, V(E)=5/6

14. Random Walk Example (cont.)
Data averaged over
100 sequences of episodes

15. Optimality of TD(0)
Suppose only a finite amount of experience is available, say 10 episodes or 100 time steps
Intuitively, we repeatedly present the experience until convergence is achieved
Updates are made after a batch of training data
Also called batch updating
For any finite Markov prediction task, under batch updating, TD(0) converges for sufficiently small a
MC method also converges deterministically but to a different answer
To better understand the different between MC and TD(0), we’ll consider the batch random walk

16. 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?

17. You are the Predictor
Suppose you observe the following 8 episodes
Q: What would you guess V(A) and V(B) to be?
1) A, 0, B, 0
2) B, 1
3) B, 1
4) B, 1
5) B, 1
6) B, 1
7) B, 1
8) B, 0

18. You are the Predictor (cont.)
V(A) = ¾ is the answer that batch TD(0) gives
The other reasonable answer is simply to say that A(0)=0 (Why?)
This is the answer that MC gives
If the process is Markovian, we expect that the TD(0) answer will produce lower error on future data, even though the Monte Carlo answer is better on the existing data

19. For MC, the prediction that best matches the training data is V(A)=0
TD(0) vs. MC
For MC, the prediction that best matches the training data is V(A)=0
This minimizes the mean-square-error on the training set
This is what a batch Monte Carlo method gets
If we consider the sequentiality of the problem, then we would set V(A)=.75
This is correct for the maximum likelihood estimate of a Markov model generating the data
i.e, if we do a best fit Markov model, and assume it is exactly correct, and then compute what it predicts
This is called the certainty-equivalence estimate
It is what TD(0) yields

20. Learning An Action-Value Function
We now consider the use of TD methods for the control problem
As with MC, we need to balance exploration and exploitation
Again, two schemes: on-policy and off-policy
We’ll start with on-policy, and learn action-value function

21. SARSA: On-Policy TD(0) Learning
One can easily turn this into a control method by always updating the policy to be greedy with respect to the current estimate of Q(s,a)

22. Q-Learning: Off-Policy TD Control
One of the most important breakthroughs in RL was the development of Q-Learning - an off-policy TD control algorithm (1989)

23. Q-Learning: Off-Policy TD Control (cont.)
The learned action-value function, Q, directly approximates the optimal action-value function, Q*
Converges as long as all states are visited and state-action values updated
Why is it considered an off-policy control method?
How expensive is it to implement?