Dr. Itamar Arel College of Engineering Department of Electrical Engineering and Computer Science

1. ECE-517: Reinforcement Learningin 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. 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. T T T T T T T T T T T T T T T T T T T T Simple Monte Carlo

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

6. T T T T T T T T T T Dynamic Programming 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. Returns from each state rewards Example: Driving Home

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 orleft with equal probability • Termination: R = +1, L = 0 • True values:V(C)=1/2, V(A)=1/6, V(B)=2/6V(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. 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?