1. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1 Chapter 4: Dynamic Programming pOverview of a collection of classical solution methods for MDPs known as dynamic programming (DP) pShow how DP can be used to compute value functions, and hence, optimal policies pDiscuss efficiency and utility of DP Objectives of this chapter:

2. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 2 Policy Evaluation Policy Evaluation: for a given policy , compute the state-value function Recall:

3. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 3 Iterative Methods a “sweep” A sweep consists of applying a backup operation to each state. A full policy evaluation backup:

4. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 4 Iterative Policy Evaluation

5. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 5 A Small Gridworld pAn undiscounted episodic task pNonterminal states: 1, 2,..., 14; pOne terminal state (shown twice as shaded squares) pActions that would take agent off the grid leave state unchanged pReward is –1 until the terminal state is reached

6. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 6 Iterative Policy Eval for the Small Gridworld

7. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 7 Policy Improvement Suppose we have computed for a deterministic policy . For a given state s, would it be better to do an action ?

8. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 8 Policy Improvement Cont.

9. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 9 Policy Improvement Cont.

10. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 10 Policy Iteration policy evaluationpolicy improvement “greedification”

11. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 11 Policy Iteration

12. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 12 Jack owns two locations for car rental. Customers come to both locations at random with poison distribution Suppose that =3 and 4 for rent and 3 and 2 for return in two locations He gets $10 if he rents a car. Moving the car from one location to the second one costs $2 Maximum 20 cars at each location Maximum 5 cars can be moved overnight Use MDP with States are number of cars Actions are number of cars moved each night Starting policy never moves any car Jack’s Cars Rental

13. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 13 Jack’s Cars Rental

14. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 14 Value Iteration Recall the full policy evaluation backup: Here is the full value iteration backup:

15. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 15 Value Iteration Cont.

16. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 16 Gambler’s Problem A gambler plys coins if it is heads he wins As much as he gambled. The game ends when he gets $100 or loses His money The state is his capital And actions are his bets The optimum policy maximizes the probability of reaching the goal. If the probability of coins coming up heads is p=0.4 then the optimum policy is as shown

17. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 17 Asynchronous DP pAll the DP methods described so far require exhaustive sweeps of the entire state set. pAsynchronous DP does not use sweeps. Instead it works like this: Repeat until convergence criterion is met: –Pick a state at random and apply the appropriate backup pStill need lots of computation, but does not get locked into hopelessly long sweeps pCan you select states to backup intelligently? YES: an agent’s experience can act as a guide.

18. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 18 Generalized Policy Iteration Generalized Policy Iteration (GPI): any interaction of policy evaluation and policy improvement, independent of their granularity. A geometric metaphor for convergence of GPI:

19. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 19 Efficiency of DP pTo find an optimal policy is polynomial in the number of states… pBUT, the number of states is often astronomical, e.g., often growing exponentially with the number of state variables (what Bellman called “the curse of dimensionality”). pIn practice, classical DP can be applied to problems with a few millions of states. pAsynchronous DP can be applied to larger problems, and appropriate for parallel computation. pIt is surprisingly easy to come up with MDPs for which DP methods are not practical.

20. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 20 Summary pPolicy evaluation: backups without a max pPolicy improvement: form a greedy policy, if only locally pPolicy iteration: alternate the above two processes pValue iteration: backups with a max pFull backups (to be contrasted later with sample backups) pGeneralized Policy Iteration (GPI) pAsynchronous DP: a way to avoid exhaustive sweeps pBootstrapping: updating estimates based on other estimates