Reading Response: Due Wednesday. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1

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1 Reading Response: Due Wednesday R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1

2 Another Example Get to the top of the hill as quickly as possible. reward = 1 for each step where not at top of hill return = number of steps before reaching top of hill Return is maximized by minimizing number of steps to reach the top of the hill. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 2

3 The Markov Property By the state at step t, the book means whatever information is available to the agent at step t about its environment. The state can include immediate sensations, highly processed sensations, and structures built up over time from sequences of sensations. Ideally, a state should summarize past sensations so as to retain all essential information, i.e., it should have the Markov Property: Pr{ s t +1 = s,r t +1 = r s t,a t,r t, s t 1,a t 1,,r 1,s 0,a 0 }= Pr{ s t +1 = s,r t +1 = r s t,a } t for all s, r, and histories s t,a t,r t, s t 1,a t 1,,r 1, s 0,a 0. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 3

4 Markov Decision Processes If a reinforcement learning task has the Markov Property, it is basically a Markov Decision Process (MDP). If state and action sets are finite, it is a finite MDP. To define a finite MDP, you need to give: state and action sets one-step dynamics defined by transition probabilities: P s s { } for all s, a = Pr s t +1 = s s t = s,a t = a s S, a A(s). reward probabilities: a R s s = E{ r t +1 s t = s,a t = a,s t +1 = s } for all s, s S, a A(s). R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 4

5 An Example Finite MDP Recycling Robot At 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. Searching 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. Reward = number of cans collected R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 5

6 Recycling Robot MDP S = { high, low} A(high) = { search, wait} A(low) = { search, wait, recharge } R search = expected no. of cans while searching R wait = expected no. of cans while waiting R search > R wait R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 6

7 Value Functions The value of a state is the expected return starting from that state; depends on the agent s policy: State- value function for policy π : V π (s) = E { π R t s t = s}= E π γ k r t +k +1 s t = s The value of taking an action in a state under policy π is the expected return starting from that state, taking that action, and thereafter following π : R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 7 k =0 Action - value function for policy π : Q π (s, a) = E { π R t s t = s, a t = a}= E π γ k r t + k +1 s t = s,a t = a k = 0

8 Bellman Equation for a Policy π The basic idea: R t = r t +1 + γ r t +2 +γ 2 r t + 3 +γ 3 r t + 4 = r t +1 + γ( r t +2 + γ r t +3 + γ 2 r t + 4 ) = r t +1 + γ R t +1 So: V π (s) = E π R t s t = s { } { } = E π r t +1 + γ V( s t +1 ) s t = s Or, without the expectation operator: V π a (s) = π(s,a) P s a s s [ R a + γv π ( s s s )] R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 8

9 More on the Bellman Equation V π a (s) = π(s,a) P s a s s [ R a + γv π ( s s s )] This is a set of equations (in fact, linear), one for each state. The value function for π is its unique solution. Backup diagrams: for V π for Q π R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 9

10 Gridworld Actions: north, south, east, west; deterministic. If would take agent off the grid: no move but reward = 1 Other actions produce reward = 0, except actions that move agent out of special states A and B as shown. State-value function for equiprobable random policy; γ = 0.9 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 10

11 Golf State is ball location Reward of 1 for each stroke until the ball is in the hole Value of a state? Actions: putt (use putter) driver (use driver) putt succeeds anywhere on the green R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 11

12 Optimal Value Functions For finite MDPs, policies can be partially ordered: π π if and only if V π (s) V π (s) for all s S There are always one or more policies that are better than or equal to all the others. These are the optimal policies. We denote them all π *. Optimal policies share the same optimal state-value function: V (s) = max V π (s) for all s S π Optimal policies also share the same optimal action-value function: Q (s,a) = max Q π (s, a) for all s S and a A(s) π This is the expected return for taking action a in state s and thereafter following an optimal policy. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 12

13 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 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 13

14 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: V (s) = max Q π (s,a) a A(s) { } = max E r + γv t +1 (s t +1 ) s t = s,a t = a a A(s) = max P a [ s s R a + γv ( s s s )] a A(s) The relevant backup diagram: s V is the unique solution of this system of nonlinear equations. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 14

15 Bellman Optimality Equation for V* V (s) = max a A(s) Qπ (s,a) What is V* for the recycling robot? { } = max E r + γv t +1 (s t +1 ) s t = s,a t = a a A(s) = max P a [ s s R a + γv ( s s s )] a A(s) s R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 15

16 R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 16

17 Bellman Optimality Equation for Q* { } Q (s,a) = E r t +1 + γ maxq (s t +1, a ) s t = s,a t = a a s a = P s s [ R a s s + γ max s, a )] a Q ( The relevant backup diagram: Q * is the unique solution of this system of nonlinear equations. R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 17

Lecture 3: The Reinforcement Learning Problem

Lecture 3: The Reinforcement Learning Problem Objectives of this lecture: describe the RL problem we will be studying for the remainder of the course present idealized form of the RL problem for which