Chapter 6: Temporal Difference Learning
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1 Chapter 6: emporal Difference Learning Objectives of this chapter: Introduce emporal Difference (D) learning Focus first on policy evaluation, or prediction, methods hen extend to control methods R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1
2 D Prediction Policy Evaluation (the prediction problem): for a given policy π, compute the state-value function V π Recall: Simple every - visit Monte Carlo method : V( s ) V( s ) + R V( s ) α [ ] t t t t target: the actual return after time t he simplest D method, D(0) : [ ] V( s ) V( s ) + α r + γ V( s ) V( s ) t t t + 1 t + 1 t target: an estimate of the return R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 2
3 Simple Monte Carlo V( s ) V( s ) + R V( s ) t t t t where R is the actual return following state s. t α [ ] s t t R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 3
4 Simplest D Method [ ] V( s ) V( s ) + α r + γ V( s ) V( s ) t t t + 1 t + 1 t s t s t +1 r t +1 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 4
5 cf. Dynamic Programming { } V( s ) E r + γ V( s ) t π t + 1 t s t r t +1 s t +1 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 5
6 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 6
7 Example: 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 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 7
8 Driving Home Changes recommended by Monte Carlo methods (α=1) Changes recommended by D methods (α=1) Predicted total travel time actual outcome Predicted total travel time actual outcome leaving office reach exiting 2ndary home arrive car highway road street home Situation leaving office reach exiting 2ndary home arrive car highway road street home Situation R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 8
9 Advantages of D Learning 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? R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 9
10 Random Walk Example A B C D E start 1 Estimated value Values learned by D(0) after various numbers of episodes A B C D E State true values R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 10
11 D and MC on the Random Walk α=.01 RMS error, averaged over states α=.15 D α=.02 MC α= Walks / Episodes Data averaged over 100 sequences of episodes α=.1 α=.04 α=.03 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 11
12 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 α. Constant-α MC also converges under these conditions, but to a difference answer! R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 12
13 Random Walk under Batch Updating BACH RAINING RMS error, averaged over states D MC Walks / Episodes 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
14 You are the Predictor Suppose you observe the following 8 episodes: A, 0, B, 0 B, 1 B, 1 B, 1 B, 1 B, 1 B, 1 B, 0 V( A )? V( B)? R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 14
15 You are the Predictor A r = 0 100% B r = 1 75% r = 0 25% V( A )? R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 15
16 You are the Predictor 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 16
17 Learning An Action-Value Function Estimate Q π for the current behavior policy π. r t+1 st s t+1 r t+2 s t+1,a t+1 s t,a t s t+2 s t+2,a t+2 After every transition from a nonterminal state s, do this : Q s, a Q s, a α r γ Q s, a Q s, a If ( ) ( ) + + ( ) ( ) t t t t t + 1 t + 1 t + 1 t t s is terminal, then Q( s, a ) =. t + 1 t + 1 t [ ] t R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 17
18 Sarsa: On-Policy D Control urn this into a control method by always updating the policy to be greedy with respect to the current estimate: R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 18
19 Windy Gridworld S G standard moves king's moves undiscounted, episodic, reward = 1 until goal R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 19
20 Results of Sarsa on the Windy Gridworld S G Episodes ime steps R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 20
21 Q-Learning: Off-Ploicy D Control One - step Q - learning : [ ] ( ) ( ) + + ( ) ( ) Q s, a Q s, a α r γ max Q s, a Q s, a t t t t t + 1 t + 1 t t a R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 21
22 Cliffwalking r = 1 safe path S h e C l i f f G optimal path r = 100 ε greedy, ε = Sarsa Reward per epsiode Q-learning Episodes R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 22
23 Actor-Critic Methods state Critic Actor Value Function Policy D error action 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 reward Appealing as psychological and neural models Environment R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 23
24 Actor-Critic Details D - error is used to evaluate actions : δ = r + γ V( s ) V( s ) t t + 1 t + 1 t If actions are determined by preferences, p( s, a), as follows : { } = π ( s, a) = Pr a = a s = s t t t e e then you can update the preferences like this : p( s, a ) p( s, a ) + β δ t t t t t b p( s, a) p( s, b), R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 24
25 Dopamine Neurons and D Error W. Schultz et al. Universite de Fribourg R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 25
26 Average Reward Per ime Step Average expected reward per time step under policy π : ρ π 1 n = lim E { r } n n t = 1 π t the same for each state if ergodic Value of a state relative to ρ π { t + k t } π π V s E r ρ s s ( ) = = k = 1 π Value of a state - action pair relative to ρ : π { t + k t t } π π Q s, a E r ρ s s, a a ( ) = = = k = 1 π : R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 26
27 R-Learning R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 27
28 Access-Control Queuing ask 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 Value of best action Apply R-learning n=10, h=.5, p= Priority 0 1 REJEC POLICY ACCEP Number of free servers priority 8 priority 4 priority 2 priority Number of free servers VALUE FUNCION R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 28
29 Afterstates 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? X O + X O X + X X O X What is this in general? R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 29
30 Summary 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 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 30
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