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  1. Decision Making in Intelligent SystemsLecture 4 BSc course Kunstmatige Intelligentie 2008 Bram Bakker Intelligent Systems Lab Amsterdam Informatics Institute Universiteit van Amsterdam

  2. Overview of this lecture • Solving the full RL problem • Given that we have the MDP model • Dynamic Programming (DP) • Given that we do not have the MDP model • Monte Carlo (MC) methods • Temporal Difference (TD) methods

  3. Summary of last lecture: dynamic programming • Policy evaluation: estimate value of current policy • Policy improvement: determine new current policy by being greedy w.r.t. current value function • Policy iteration: alternate the above two processes • Value iteration: combine the above two processes in 1 step • Bootstrapping: updating estimates based on your own estimates • Full backups (to be contrasted with sample backups) • Generalized Policy Iteration (GPI)

  4. Monte Carlo Methods • Monte Carlo methods learn • Learn from complete sample returns • Only defined for episodic tasks • Monte Carlo methods learn directly from experience • On-line: No model necessary and still attains optimality • Simulated: No need for a full model

  5. 1 2 3 4 5 Monte Carlo Policy Evaluation • Goal: learn Vp(s) • Given: some number of episodes under p which contain s • Idea: Average returns observed after visits to s • Every-Visit MC: average returns for every time s is visited in an episode • First-visit MC: average returns only for first time s is visited in an episode • Both converge asymptotically

  6. T T T T T T T T T T T T T T T T T T T T Monte Carlo

  7. T T T T T T T T T T cf. Dynamic Programming T T T

  8. First-visit Monte Carlo policy evaluation

  9. Blackjack example • Object: Have your card sum be greater than the dealers without exceeding 21. • States (200 of them): • current sum (12-21) • dealer’s showing card (ace-10) • do I have a useable ace? • Reward: +1 for winning, 0 for a draw, -1 for losing • Actions: stick (stop receiving cards), hit (receive another card) • Initial Policy: Stick if my sum is 20 or 21, else hit

  10. Blackjack value functions

  11. Monte Carlo Estimation of Action Values (Q) • Monte Carlo is most useful when a model is not available • We want to values learn Q* : state-action values • Qp(s,a) - average return starting from state s and action a following p • Also converges asymptotically if every state-action pair is visited • Exploring starts: Every state-action pair has a non-zero probability of being the starting pair

  12. Monte Carlo Control • MC policy iteration: Policy evaluation using MC methods followed by policy improvement • Policy improvement step: greedify with respect to value (or action-value) function

  13. Monte Carlo Exploring Starts

  14. Blackjack example continued • Exploring starts • Initial policy as described before

  15. On-policy Monte Carlo Control • On-policy: learn about policy currently executing • So, learn about policy which explores • Simplest: policy with exploring starts • More sophisticated: soft policies: p(s,a) > 0 for all s and a

  16. Off-policy Monte Carlo control • Behavior policy generates behavior in environment • Estimation policy is policy being learned about • Usually: try to estimate best deterministic policy, based on policy which keeps exploring • Weigh returns from behavior policy to do value estimation of states of the estimation policy • Weigh by estimating relative probabilities of this state-action pair occurring

  17. Learning about p while following

  18. Off-policy MC control

  19. Incremental Implementation • MC can be implemented incrementally • saves memory • Compute the weighted average of each return non-incremental incremental equivalent

  20. Simple incremental implementation target: the actual return after time t

  21. Summary of MC • MC has several advantages over DP: • Can learn directly from interaction with environment • No need for full models • No need to learn about ALL states • Less harm by Markovian violations (later in book) • MC methods provide an alternate policy evaluation process • One issue to watch for: maintaining sufficient exploration • exploring starts, soft policies • Introduced distinction between on-policy and off-policy methods • No bootstrapping (as opposed to DP)

  22. Overview of this lecture • Solving the full RL problem • Given that we have the MDP model • Dynamic Programming (DP) • Given that we do not have the MDP model • Monte Carlo (MC) methods • Temporal Difference (TD) methods

  23. Temporal Difference (TD) methods • Learn from experience, like MC • Can learn directly from interaction with environment • No need for full models • Estimate values based on estimated values of next states, like DP • Bootstrapping (like DP) • Issue to watch for: maintaining sufficient exploration

  24. Bootstrapping A full policy-evaluation backup (DP): The new estimated value for each state s is based on the estimated old value of all possible successor states s’ Bootstrapping: estimating values based on your own estimates of values

  25. TD Prediction Policy Evaluation (the prediction problem): for a given policy p, compute the state-value function Recall: target: the actual return after time t target: an estimate of the return

  26. T T T T T T T T T T T T T T T T T T T T Simple Monte Carlo

  27. T T T T T T T T T T Dynamic Programming T T T

  28. T T T T T T T T T T T T T T T T T T T T Simplest TD Method

  29. TD methods bootstrap and sample • Bootstrapping: update involves an estimate • MC does not bootstrap • DP bootstraps • TD bootstraps • Sampling: update does not involve an expected value • MC samples • DP does not sample • TD samples

  30. Driving Home • Changes recommended by Monte Carlo methods (a=1) • Changes recommended • by TD methods (a=1)

  31. 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 • 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 TD converge (under certain assumptions to be detailed later), but which is faster?

  32. Random Walk Example • Values learned by TD(0) after • various numbers of episodes

  33. TD and MC on the Random Walk • Data averaged over • 100 sequences of episodes

  34. Optimality of TD(0) Batch Updating: Compute updates according to TD(0), but only update estimates after each complete pass through the data (end of the episode). Let’s assume we train completely on a finite amount of data, e.g., train repeatedly on 10 episodes until convergence. For any finite Markov prediction task, under batch updating, TD(0) converges for sufficiently small a. Constant-a MC also converges under these conditions, but to a difference answer!

  35. 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

  36. You are the Predictor

  37. You are the Predictor • 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 (how?) • This is called the certainty-equivalence estimate • This is what TD(0) gets

  38. Learning An Action-Value Function

  39. Sarsa: On-Policy TD Control Turn TD into a control method by always updating the policy to be (epsilon-)greedy with respect to the current estimate. S A R S A: State Action Reward State Action

  40. Windy Gridworld undiscounted, episodic, reward = –1 until goal

  41. Results of Sarsa on the Windy Gridworld

  42. Q-Learning: Off-Policy TD Control

  43. Cliffwalking • e-greedy, e = 0.1

  44. The Book • Part I: The Problem • Introduction • Evaluative Feedback • The Reinforcement Learning Problem • Part II: Elementary Solution Methods • Dynamic Programming • Monte Carlo Methods • Temporal Difference Learning • Part III: A Unified View • Eligibility Traces • Generalization and Function Approximation • Planning and Learning • Dimensions of Reinforcement Learning • Case Studies

  45. Unified View

  46. Summary • TD prediction • Introduced one-step tabular model-free TD methods • Extend prediction to control by employing some form of GPI • On-policy control: Sarsa • Off-policy control: Q-learning • These methods bootstrap and sample, combining aspects of DP and MC methods

  47. Questions • What is common to all three classes of methods? – DP, MC, TD • 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? • The broad applicability of these ideas… • What does the term bootstrapping refer to? • What is the relationship between DP and learning?

  48. Next Class • Lecture next Monday, 5 March : • Chapters 6 & 7 of Sutton & Barto