r t +1 s t +1 TD Prediction Chapter 6: Temporal Difference Learning [ ] [ ] Simplest TD Method Simple Monte Carlo

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1 Chapter 6: emporal Difference Learning D Prediction Objectives of this chapter: Policy Evaluation (the prediction problem: for a given policy!, compute the state-value function V!! Introduce emporal Difference (D learning! Focus first on policy evaluation, or prediction, methods! hen extend to control methods Recall: Simple every - visit Monte Carlo method : [ ]! +" R t # V (s t target: the actual return after time t he simplest D method, D(0 : [ ]! +" r t +1 + # V (s t+1 target: an estimate of the return R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 2 Simple Monte Carlo! +" [ R t # V (s t ] where R t is the actual return following state s t. s t Simplest D Method! +" [ r t +1 + # V (s t+1 ] s t s t +1 r t +1 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 3 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 4

2 cf. Dynamic Programming { }! E " r t +1 +# s t r t +1 s t +1 D Bootstraps and Samples!Bootstrapping: update involves an estimate! MC does not bootstrap! DP bootstraps! D bootstraps!sampling: update does not involve an expected value! MC samples! DP does not sample! D samples R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 5 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 6 Example: Driving Home Driving Home State Elapsed ime Predicted Predicted (minutes ime to Go otal ime leaving office reach car, raining exit highway behind truck home street arrive home Changes recommended by Monte Carlo methods (!=1 Changes recommended by D methods (!=1 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 7 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 8

3 Advantages of D Learning Random Walk Example! D methods do not require a model of the environment, only experience! D, but not MC, methods can be fully incremental! You can learn before knowing the final outcome Less memory Less peak computation! You can learn without the final outcome From incomplete sequences! Both MC and D converge (under certain assumptions to be detailed later, but which is faster? equiprobable transitions " = 0.1 Values learned by D(0 after various numbers of episodes R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 9 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 10 D and MC on the Random Walk Optimality of D(0 Batch Updating: train completely on a finite amount of data, e.g., train repeatedly on 10 episodes until convergence. Compute updates according to D(0, but only update estimates after each complete pass through the data. For any finite Markov prediction task, under batch updating, D(0 converges for sufficiently small!. Data averaged over 100 sequences of episodes Constant-! MC also converges under these conditions, but to a difference answer! R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 11 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 12

4 Random Walk under Batch Updating You are the Predictor Suppose you observe the following 8 episodes: A, 0, B, 0 B, 0 V(A? V(B? After each new episode, all previous episodes were treated as a batch, and algorithm was trained until convergence. All repeated 100 times. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 13 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 14 You are the Predictor You are the Predictor V(A?! he prediction that best matches the training data is V(A=0! his minimizes the mean-square-error on the training set! his is what a batch Monte Carlo method gets! If we consider the sequentiality of the problem, then we would set V(A=.75! his 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 (how?! his is called the certainty-equivalence estimate! his is what D(0 gets R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 15 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 16

5 Learning An Action-Value Function Sarsa: On-Policy D Control Estimate Q! for the current behavior policy!. urn this into a control method by always updating the policy to be greedy with respect to the current estimate: After every transition from a nonterminal state s t, do this : [ ] Q( s t! Q( s t + " r t +1 +# Q( s t +1,a t +1 Q( s t,a t If s t +1 is terminal, then Q(s t = 0. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 17 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 18 Windy Gridworld Results of Sarsa on the Windy Gridworld undiscounted, episodic, reward = 1 until goal R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 19 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 20

6 Q-Learning: Off-Policy D Control Cliffwalking One - step Q - learning : Q( s t! Q( s t + " r t +1 +# max [ Q ( s t+1, a Q s t a ( ] "#greedy, " = 0.1 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 21 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 22 Actor-Critic Architecture Actor-Critic Methods Situations or States Environment Primary Critic Primary Rewards Adaptive Critic Effective Rewards: (involves values Actions! Explicit representation of policy as well as value function! Minimal computation to select actions! Can learn an explicit stochastic policy! Can put constraints on policies! Appealing as psychological and neural models Actor R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 23 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 24

7 Actor-Critic Details D - error is used to evaluate actions :! t = r t +1 + " V (s t +1 # r D-error " t = r t + V t # V t #1 regular predictors of z over this interval If actions are determined by preferences, p(s, a, as follows : { } = ep( s, a "! t (s, a = Pr a t = a s t = s b p(s,b e then you can update the preferences like this : p(s t # p(s t,a t + % t, early in learning learning complete V V r omitted R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 25 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 26 Dopamine Neurons and D Error W. Schultz et al. Universite de Fribourg Average Reward Per ime Step Average expected reward per time step under policy! : "! n 1 = lim E n# n %! { r t } the same for each state if ergodic t =1 Value of a state relative to! " : V " % { } ( s = E " r t +k #! " s t = s k =1 Value of a state - action pair relative to! " : Q " % { } ( s, a = E " r t + k #! " s t = s,a t = a k =1 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 27 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 28

8 R-Learning R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 29 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 30 Access-Control Queuing ask Afterstates! n servers! Customers have four different priorities, which pay reward of 1, 2, 3, or 4, if served! At each time step, customer at head of queue is accepted (assigned to a server or removed from the queue! Proportion of randomly distributed high priority customers in queue is h! Busy server becomes free with probability p on each time step! Statistics of arrivals and departures are unknown Apply R-learning n=10, h=.5, p=.06! Usually, a state-value function evaluates states in which the agent can take an action.! But sometimes it is useful to evaluate states after agent has acted, as in tic-tac-toe.! Why is this useful?! What is this in general? R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 31 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 32

9 Summary Some Questions! D prediction! Introduced one-step tabular model-free D methods! Extend prediction to control by employing some form of GPI! On-policy control: Sarsa! Off-policy control: Q-learning and R-learning! hese methods bootstrap and sample, combining aspects of DP and MC methods What can I tell you about RL? What is common to all three classes of methods? DP, MC, D What are the principle strengths and weaknesses of each? In what sense is our RL view complete? In what senses is it incomplete? What are the principal things missing? he broad applicability of these ideas What does the term bootstrapping refer to? What is the relationship between DP and learning? R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 33 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 34

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