Lecture 2: Markov Decision Processes Lecture 2: Markov Decision Processes David Silver Lecture 2: Markov Decision Processes 1 Markov Processes 2 Markov Reward Processes 3 Markov Decision Processes 4 Extensions to MDPs Lecture 2: Markov Decision Processes Markov Processes Introduction Introduction to MDPs Markov decision processes formally describe an environment for reinforcement learning Where the environment is fully observable i.e. The current state completely characterises the process Almost all RL problems can be formalised as MDPs, e.g. Optimal control primarily deals with continuous MDPs Partially observable problems can be converted into MDPs Bandits are MDPs with one state Lecture 2: Markov Decision Processes Markov Processes Markov Property Markov Property \The future is independent of the past given the present" Definition A state St is Markov if and only if P [St+1 St ] = P [St+1 S1; :::; St ] j j The state captures all relevant information from the history Once the state is known, the history may be thrown away i.e. The state is a sufficient statistic of the future Lecture 2: Markov Decision Processes Markov Processes Markov Property State Transition Matrix For a Markov state s and successor state s0, the state transition probability is defined by 0 ss0 = P St+1 = s St = s P j State transition matrix defines transition probabilities from all states s to all successorP states s0, to 2 3 11 ::: 1n P. P = from 6 . 7 P 4 5 n1 ::: nn P P where each row of the matrix sums to 1. Lecture 2: Markov Decision Processes Markov Processes Markov Chains Markov Process A Markov process is a memoryless random process, i.e. a sequence of random states S1; S2; ::: with the Markov property. Definition A Markov Process (or Markov Chain) is a tuple ; hS Pi is a (finite) set of states S is a state transition probability matrix, P 0 ss0 = P [St+1 = s St = s] P j Lecture 2: Markov Decision Processes Markov Processes Markov Chains Example: Student Markov Chain 0.9 Facebook Sleep 0.1 0.5 0.2 1.0 0.8 0.6 Class 1 0.5 Class 2 Class 3 Pass 0.4 0.4 0.2 0.4 Pub Lecture 2: Markov Decision Processes Markov Processes Markov Chains Example: Student Markov Chain Episodes Sample episodes for Student Markov 0.9 Chain starting from S1 = C1 Facebook Sleep 0.1 S1; S2; :::; ST 0.5 0.2 C1 C2 C3 Pass Sleep1.0 0.5 0.8 0.6 Class 1 Class 2 C1 FBClass FB 3 C1 C2 SleepPass 0.4 C1 C2 C3 Pub C2 C3 Pass Sleep C1 FB FB C1 C2 C3 Pub C1 FB FB 0.4 FB C1 C2 C3 Pub C2 Sleep 0.2 0.4 Pub Lecture 2: Markov Decision Processes Markov Processes Markov Chains Example: Student Markov Chain Transition Matrix 0.9 Facebook Sleep 0.1 C1 C2 C3 Pass Pub FB Sleep C1 2 0:5 0:5 3 C2 6 0:8 0:2 7 6 7 C3 6 0:6 0:4 7 6 1.0 7 0.5 0.2 P = Pass 6 1:0 7 Pub 6 0:2 0:4 0:4 7 0.8 6 0.6 7 Class 1 0.5 Class 2 FB 4Class0:1 3 Pass 0:9 5 Sleep 1 0.4 0.4 0.2 0.4 Pub Lecture 2: Markov Decision Processes Markov Reward Processes MRP Markov Reward Process A Markov reward process is a Markov chain with values. Definition A Markov Reward Process is a tuple ; ; ; γ hS P R i is a finite set of states S is a state transition probability matrix, P 0 ss0 = P [St+1 = s St = s] P j is a reward function, s = E [Rt+1 St = s] R R j γ is a discount factor, γ [0; 1] 2 Lecture 2: Markov Decision Processes Markov Reward Processes MRP Example: Student MRP 0.9 Facebook Sleep 0.1 R = -1 R = 0 0.5 0.2 1.0 0.8 0.6 Class 1 0.5 Class 2 Class 3 Pass R = -2 R = -2 R = -2 0.4 R = +10 0.4 0.2 0.4 Pub R = +1 Lecture 2: Markov Decision Processes Markov Reward Processes Return Return Definition The return Gt is the total discounted reward from time-step t. 1 X k Gt = Rt+1 + γRt+2 + ::: = γ Rt+k+1 k=0 The discount γ [0; 1] is the present value of future rewards 2 The value of receiving reward R after k + 1 time-steps is γk R. This values immediate reward above delayed reward. γ close to 0 leads to "myopic" evaluation γ close to 1 leads to "far-sighted" evaluation Lecture 2: Markov Decision Processes Markov Reward Processes Return Why discount? Most Markov reward and decision processes are discounted. Why? Mathematically convenient to discount rewards Avoids infinite returns in cyclic Markov processes Uncertainty about the future may not be fully represented If the reward is financial, immediate rewards may earn more interest than delayed rewards Animal/human behaviour shows preference for immediate reward It is sometimes possible to use undiscounted Markov reward processes (i.e. γ = 1), e.g. if all sequences terminate. Lecture 2: Markov Decision Processes Markov Reward Processes Value Function Value Function The value function v(s) gives the long-term value of state s Definition The state value function v(s) of an MRP is the expected return starting from state s v(s) = E [Gt St = s] j Lecture 2: Markov Decision Processes Markov Reward Processes Value Function Example: Student MRP Returns Sample returns for Student MRP: 1 Starting from S1 = C1 with γ = 2 T −2 G1 = R2 + γR3 + ::: + γ RT 1 1 1 C1 C2 C3 Pass Sleep v1 = −2 − 2 ∗ 2 − 2 ∗ 4 + 10 ∗ 8 = −2:25 1 1 1 1 C1 FB FB C1 C2 Sleep v1 = −2 − 1 ∗ 2 − 1 ∗ 4 − 2 ∗ 8 − 2 ∗ 16 = −3:125 1 1 1 1 C1 C2 C3 Pub C2 C3 Pass Sleep v1 = −2 − 2 ∗ 2 − 2 ∗ 4 + 1 ∗ 8 − 2 ∗ 16 ::: = −3:41 1 1 1 1 C1 FB FB C1 C2 C3 Pub C1 ::: v1 = −2 − 1 ∗ − 1 ∗ − 2 ∗ − 2 ∗ ::: 2 4 8 16 = −3:20 FB FB FB C1 C2 C3 Pub C2 Sleep Lecture 2: Markov Decision Processes Markov Reward Processes Value Function Example: State-Value Function for Student MRP (1) v(s) for γ =0 0.9 -1 0 0.1 R = -1 R = 0 0.5 0.2 1.0 0.8 0.6 -2 0.5 -2 -2 10 R = -2 R = -2 R = -2 0.4 R = +10 0.4 0.2 0.4 +1 R = +1 Lecture 2: Markov Decision Processes Markov Reward Processes Value Function Example: State-Value Function for Student MRP (2) v(s) for γ =0.9 0.9 -7.6 0 0.1 R = -1 R = 0 0.5 0.2 1.0 0.8 0.6 -5.0 0.5 0.9 4.1 10 R = -2 R = -2 R = -2 0.4 R = +10 0.4 0.2 0.4 1.9 R = +1 Lecture 2: Markov Decision Processes Markov Reward Processes Value Function Example: State-Value Function for Student MRP (3) v(s) for γ =1 0.9 -23 0 0.1 R = -1 R = 0 0.5 0.2 1.0 0.8 0.6 -13 0.5 1.5 4.3 10 R = -2 R = -2 R = -2 0.4 R = +10 0.4 0.2 0.4 +0.8 R = +1 Lecture 2: Markov Decision Processes Markov Reward Processes Bellman Equation Bellman Equation for MRPs The value function can be decomposed into two parts: immediate reward Rt+1 discounted value of successor state γv(St+1) v(s) = E [Gt St = s] j 2 = E Rt+1 + γRt+2 + γ Rt+3 + ::: St = s j = E [Rt+1 + γ (Rt+2 + γRt+3 + :::) St = s] j = E [Rt+1 + γGt+1 St = s] j = E [Rt+1 + γv(St+1) St = s] j Lecture 2: Markov Decision Processes Markov Reward Processes Bellman Equation Bellman Equation for MRPs (2) v(s) = E [Rt+1 + γv(St+1) St = s] j v(s) s 7! r v(s0) s0 7! X 0 v(s) = s + γ ss0 v(s ) R P s02S Lecture 2: Markov Decision Processes Markov Reward Processes Bellman Equation Example: Bellman Equation for Student MRP 4.3 = -2 + 0.6*10 + 0.4*0.8 0.9 -23 0 0.1 R = -1 R = 0 0.5 0.2 1.0 0.8 0.6 -13 0.5 1.5 4.3 10 R = -2 R = -2 R = -2 0.4 R = +10 0.4 0.2 0.4 0.8 R = +1 Lecture 2: Markov Decision Processes Markov Reward Processes Bellman Equation Bellman Equation in Matrix Form The Bellman equation can be expressed concisely using matrices, v = + γ v R P where v is a column vector with one entry per state 2 3 2 3 2 3 2 3 v(1) 1 11 ::: 1n v(1) R P P 6 . 7 6 . 7 6 . 7 6 . 7 4 . 5 = 4 . 5 + γ 4 . 5 4 . 5 v(n) n 11 ::: nn v(n) R P P Lecture 2: Markov Decision Processes Markov Reward Processes Bellman Equation Solving the Bellman Equation The Bellman equation is a linear equation It can be solved directly: v = + γ v R P (I γ ) v = − P R v = (I γ )−1 − P R Computational complexity is O(n3) for n states Direct solution only possible for small MRPs There are many iterative methods for large MRPs, e.g.