1. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1 Chapter 3: The Reinforcement Learning Problem pdescribe the RL problem we will be studying for the remainder of the course ppresent idealized form of the RL problem for which we have precise theoretical results; pintroduce key components of the mathematics: value functions and Bellman equations; pdescribe trade-offs between applicability and mathematical tractability. Objectives of this chapter:

2. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 2 The Agent-Environment Interface t... s t a r t +1 s a r t +2 s a r t +3 s... t +3 a

3. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 3 The Agent Learns a Policy pReinforcement learning methods specify how the agent changes its policy as a result of experience. pRoughly, the agent’s goal is to get as much reward as it can over the long run.

4. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 4 Getting the Degree of Abstraction Right pTime steps need not refer to fixed intervals of real time. pActions can be low level (e.g., voltages to motors), or high level (e.g., accept a job offer), “mental” (e.g., shift in focus of attention), etc. pStates can low-level “sensations”, or they can be abstract, symbolic, based on memory, or subjective (e.g., the state of being “surprised” or “lost”). pAn RL agent is not like a whole animal or robot, which consist of many RL agents as well as other components. pThe environment is not necessarily unknown to the agent, only incompletely controllable. pReward computation is in the agent’s environment because the agent cannot change it arbitrarily.

5. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 5 Goals and Rewards pIs a scalar reward signal an adequate notion of a goal?— maybe not, but it is surprisingly flexible. pA goal should specify what we want to achieve, not how we want to achieve it. pA goal must be outside the agent’s direct control—thus outside the agent. pThe agent must be able to measure success: explicitly; frequently during its lifespan.

6. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 6 Returns Episodic tasks: interaction breaks naturally into episodes, e.g., plays of a game, trips through a maze. where T is a final time step at which a terminal state is reached, ending an episode.

7. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 7 Returns for Continuing Tasks Continuing tasks: interaction does not have natural episodes. Discounted return:

8. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 8 An Example Avoid failure: the pole falling beyond a critical angle or the cart hitting end of track. As an episodic task where episode ends upon failure: As a continuing task with discounted return: In either case, return is maximized by avoiding failure for as long as possible.

9. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 9 Another Example Get to the top of the hill as quickly as possible. Return is maximized by minimizing number of steps reach the top of the hill.

10. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 10 A Unified Notation pIn episodic tasks, we number the time steps of each episode starting from zero. pWe usually do not have to distinguish between episodes, so we write instead of for the state at step t of episode j. pThink of each episode as ending in an absorbing state that always produces reward of zero: pWe can cover all cases by writing

11. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 11 The Markov Property pBy “the state” at step t, the book means whatever information is available to the agent at step t about its environment. pThe state can include immediate “sensations,” highly processed sensations, and structures built up over time from sequences of sensations. pIdeally, a state should summarize past sensations so as to retain all “essential” information, i.e., it should have the Markov Property:

12. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 12 Markov Decision Processes pIf a reinforcement learning task has the Markov Property, it is basically a Markov Decision Process (MDP). pIf state and action sets are finite, it is a finite MDP. pTo define a finite MDP, you need to give: state and action sets one-step “dynamics” defined by transition probabilities: reward probabilities:

13. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 13 Recycling Robot An Example Finite MDP pAt each step, robot has to decide whether it should (1) actively search for a can, (2) wait for someone to bring it a can, or (3) go to home base and recharge. pSearching is better but runs down the battery; if runs out of power while searching, has to be rescued (which is bad).  Decisions made on basis of current energy level: high, low. pReward = number of cans collected

14. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 14 Recycling Robot MDP

15. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 15 Value Functions pThe value of a state is the expected return starting from that state; depends on the agent’s policy:  The value of taking an action in a state under policy  is the expected return starting from that state, taking that action, and thereafter following  :

16. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 16 Bellman Equation for a Policy  The basic idea: So: Or, without the expectation operator (Bellman equation for V  :

17. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 17 More on the Bellman Equation This is a set of equations (in fact, linear), one for each state. The value function for  is its unique solution. Backup diagrams:

18. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 18 Grid world  Actions: north, south, east, west ; deterministic. pIf would take agent off the grid: no move but reward = –1 pOther actions produce reward = 0, except actions that move agent out of special states A (+10) and B (+5) to their goals as shown. State-value function for equiprobable random policy;  = 0.9 Note that learning the policy that maximizes the award is an objective of RL

19. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 19 Grid world % Example 3.8 % matrix is scanned column wise % action - going north clear all; R=zeros(4,25); % next states S_prime(1,1)=1; S_prime(1,2:25)=(1:24); S_prime(1,11)=11; S_prime(1,21)=21; % reward in north direction for i=0:4 R(1,i*5+1)=-1; end; % action - going east S_prime(2,1:20)=(6:25); S_prime(2,21:25)=(21:25); % reward in east direction R(2,21:25)=-1; % action - going south S_prime(3,1:24)=(2:25); S_prime(3,5:5:25)=(5:5:25); % reward in south direction R(3,5:5:25)=-1; % action - going west S_prime(4,6:25)=(1:20); S_prime(4,1:5)=(1:5); % reward in west direction R(4,1:5)=-1; % special states for i=1:4 S_prime(i,6)=10; S_prime(i,16)=18; R(i,6)=10; R(i,16)=5; end; 16111621 27121722 38121823 49141924 510152025

20. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 20 Grid world % initialize variables Veps=0.01; delV=1; gamma=0.9; Vpi=zeros(1,25); % iterate until convergence while delV>Veps % Bellman equation Vpi_new=0.25*sum(R+gamma*Vpi(S_prime)); delV=norm(Vpi_new-Vpi); Vpi=Vpi_new; end; Vpi=reshape(Vpi,5,5) % Vpi = % 3.3253 8.8065 4.4398 5.3320 1.4968 % 1.5362 3.0064 2.2612 1.9157 0.5529 % 0.0652 0.7517 0.6852 0.3681 -0.3946 % -0.9582 -0.4202 -0.3407 -0.5720 -1.1703 % -1.8410 -1.3287 -1.2127 -1.4068 -1.9593 0.25R combines this product

21. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 21 More on the Bellman Equation Since this is a set of linear equations, they can be solved without iterations (and exactly) by direct inverse. Vpi=reshape(Vpi,5,5) % Vpi = % 3.3090 8.7893 4.4276 5.3224 1.4922 % 1.5216 2.9923 2.2501 1.9076 0.5474 % 0.0508 0.7382 0.6731 0.3582 -0.4031 % -0.9736 -0.4355 -0.3549 -0.5856 -1.1831 % -1.8577 -1.3452 -1.2293 -1.4229 -1.9752 % direct solution can be obtained in a linear system T=eye(25); for i=1:25 for j=1:4 T(i,S_prime(j,i)')=T(i,S_prime(j,i)')-0.25*gamma; end; Vpi=inv(T)*0.25*sum(R)'; Diagonal matrix for V  (s), and effect of V  (s’) Rewards on the rhs

22. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 22 Golf pState is ball location pReward of –1 for each stroke until the ball is in the hole pValue of a state? pActions: putt (use putter) driver (use driver)  putt succeeds anywhere on the green

23. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 23 Optimal Value Functions pFor finite MDPs, policies can be partially ordered:  There is always at least one (and possibly many) policies that is better than or equal to all the others. This is an optimal policy. We denote them all  *. pOptimal policies share the same optimal state-value function: pOptimal policies also share the same optimal action-value function: This is the expected return for taking action a in state s and thereafter following an optimal policy.

24. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 24 Optimal Value Function for Golf  We can hit the ball farther with driver than with putter, but with less accuracy  Q*(s, driver ) gives the value or using driver first, then using whichever actions are best

25. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 25 Bellman Optimality Equation for V* The value of a state under an optimal policy must equal the expected return for the best action from that state: The relevant backup diagram: is the unique solution of this system of nonlinear equations.

26. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 26 Bellman Optimality Equation for Q* The relevant backup diagram: is the unique solution of this system of nonlinear equations.

27. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 27 Why Optimal State-Value Functions are Useful Any policy that is greedy with respect to is an optimal policy. Therefore, given, one-step-ahead search produces the long-term optimal actions. E.g., back to the grid world:

28. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 28 What About Optimal Action-Value Functions? Given, the agent does not even have to do a one-step-ahead search: % find the optimum state value while delV>Veps [Vpi_new,I]=max(R+gamma*Vpi(S_prime)); delV=norm(Vpi_new-Vpi); Vpi=Vpi_new; end; % determine the optimum policy Reward=R+gamma*Vpi(S_prime); for i=1:25 Optimum_policy(:,i)=(Reward(:,i)==Reward(I(i),i)); end; norm=sum(Optimum_policy); for i=1:25 Optimum_policy(:,i)=Optimum_policy(:,i)./norm(i); end

29. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 29 What About Optimal Action-Value Functions? Vpi=reshape(Vpi,5,5) % Vpi = % 21.9694 24.4104 21.9694 19.4104 17.4694 % 19.7724 21.9694 19.7724 17.7952 16.0115 % 17.7952 19.7724 17.7952 16.0115 14.4104 % 16.0115 17.7952 16.0115 14.4104 12.9694 % 14.4104 16.0115 14.4104 12.9694 11.6724 Optimum_policy % First 14 columns of Optimum_policy = % Columns 1 through 7 % 0 0.5000 0.5000 0.5000 0.5000 0.2500 1.0000 % 1.0000 0.5000 0.5000 0.5000 0.5000 0.2500 0 % 0 0 0 0 0 0.2500 0 % Columns 8 through 14 % 1.0000 1.0000 1.0000 0 0.5000 0.5000 0.5000 % 0 0 0 0 0 0 0 % 0 0 0 1.0000 0.5000 0.5000 0.5000 16111621 27121722 38121823 49141924 510152025

30. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 30 Solving the Bellman Optimality Equation pFinding an optimal policy by solving the Bellman Optimality Equation requires the following: accurate knowledge of environment dynamics; we have enough space an time to do the computation; the Markov Property. pHow much space and time do we need? polynomial in number of states (via dynamic programming methods; Chapter 4), BUT, number of states is often huge (e.g., backgammon has about 10**20 states). pWe usually have to settle for approximations. pMany RL methods can be understood as approximately solving the Bellman Optimality Equation.

31. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 31 Summary pAgent-environment interaction States Actions Rewards pPolicy: stochastic rule for selecting actions pReturn: the function of future rewards agent tries to maximize pEpisodic and continuing tasks pMarkov Property pMarkov Decision Process Transition probabilities Expected rewards pValue functions State-value function for a policy Action-value function for a policy Optimal state-value function Optimal action-value function pOptimal value functions pOptimal policies pBellman Equations pThe need for approximation