The Markov Decision Process (MDP) model

Transcription

1 Decision Making in Robots and Autonomous Agents The Markov Decision Process (MDP) model Subramanian Ramamoorthy School of Informatics 25 January, 2013

2 In the MAB Model We were in a single casino and the only decision is to pull from a set of n arms except perhaps in the very last slides, exactly one state! We asked the following, What if there is more than one state? So, in this state space, what is the effect of the distribution of payout changing based on how you pull arms? What happens if you only obtain a net reward corresponding to a long sequence of arm pulls (at the end)? 25/01/2013 2

3 Decision Making Agent-Environment Interface... Agent and environment interact at discrete time steps: t 0,1, 2, Agent observes state at step t : s t S produces action at step t : a t A(s t ) gets resulting reward : r t 1 and resulting next state: s t 1 s t a t r t +1 s a t +1 t +1 r t +2 s a t +2 t +2 r t +3 s... t +3 a t +3 25/01/2013 3

4 Markov Decision Processes A model of the agent-environment system Markov property = history doesn t matter, only current state 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: a P ss Pr s t 1 s s t s,a t a for all s,s S, a A(s). reward probabilities: a R ss E r t 1 s t s,a t a,s t 1 s for all s,s S, a A(s). 25/01/2013 4

5 Recycling Robot An Example Finite MDP 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 25/01/2013 5

6 Recycling Robot MDP S high, low A(high ) search, wait A(low ) search, wait, recharge R search R wait expected no. of cans while searching expected no. of cans while waiting R search R wait 25/01/2013 6

7 Enumerated In Tabular Form If you were given this much, what can you say about the behaviour (over time) of the system? 25/01/2013 7

8 A Very Brief Primer on Markov Chains and Decisions A model, as originally developed in Operations Research/Stochastic Control theory 25/01/2013 8

9 Stochastic Processes A stochastic process is an indexed collection of random variables. e.g., collection of weekly demands for a product One type: At a particular time t, labelled by integers, system is found in exactly one of a finite number of mutually exclusive and exhaustive categories or states, labelled by integers too Process could be imbedded in that time points correspond to occurrence of specific events (or time may be equi-spaced) Random variables may depend on others, e.g., 25/01/2013 9

10 Markov Chains The stochastic process is said to have a Markovian property if Markovian probability means that the conditional probability of a future event given any past events and current state, is independent of past states and depends only on present The conditional probabilities are transition probabilities, These are stationary if time invariant, called p ij, 25/01/

11 Markov Chains Looking forward in time, n-step transition probabilities, p ij (n) One can write a transition matrix, A stochastic process is a finite-state Markov chain if it has, Finite number of states Markovian property Stationary transition probabilities A set of initial probabilities P{X 0 = i} for all i 25/01/

12 Markov Chains n-step transition probabilities can be obtained from 1-step transition probabilities recursively (Chapman-Kolmogorov) We can get this via the matrix too First Passage Time: number of transitions to go from i to j for the first time If i = j, this is the recurrence time In general, this itself is a random variable 25/01/

13 Markov Chains n-step recursive relationship for first passage time For fixed i and j, these f ij (n) are nonnegative numbers so that If,, that state is a recurrent state, absorbing if n=1 25/01/

14 Markov Chains: Long-Run Properties Consider the 8-step transition matrix of the inventory example: Interesting property: probability of being in state j after 8 weeks appears independent of initial level of inventory. For an irreducible ergodic Markov chain, one has limiting probability Reciprocal gives you recurrence time jj 25/01/

15 Markov Decision Model Consider the following application: machine maintenance A factory has a machine that deteriorates rapidly in quality and output and is inspected periodically, e.g., daily Inspection declares the machine to be in four possible states: 0: Good as new 1: Operable, minor deterioration 2: Operable, major deterioration 3: Inoperable Let X t denote this observed state evolves according to some law of motion, so it is a stochastic process Furthermore, assume it is a finite state Markov chain 25/01/

16 Markov Decision Model Transition matrix is based on the following: Once the machine goes inoperable, it stays there until repairs If no repairs, eventually, it reaches this state which is absorbing! Repair is an action a very simple maintenance policy. e.g., machine from from state 3 to state 0 25/01/

17 Markov Decision Model There are costs as system evolves: State 0: cost 0 State 1: cost 1000 State 2: cost 3000 Replacement cost, taking state 3 to 0, is 4000 (and lost production of 2000), so cost = 6000 The modified transition probabilities are: 25/01/

18 Markov Decision Model Simple question: What is the average cost of this maintenance policy? Compute the steady state probabilities: How? (Long run) expected average cost per day, 25/01/

19 Markov Decision Model Consider a slightly more elaborate policy: Repair when inoperable or needing major repairs, replace Transition matrix now changes a little bit Permit one more thing: overhaul Go back to minor repairs state (1) for the next time step Not possible if truly inoperable, but can go from major to minor Key point about the system behaviour. It evolves according to Laws of motion Sequence of decisions made (actions from {1: none,2:overhaul,3: replace}) Stochastic process is now defined in terms of {X t } and { t } Policy, R, is a rule for making decisions Could use all history, although popular choice is (current) state-based 25/01/

20 Markov Decision Model There is a space of potential policies, e.g., Each policy defines a transition matrix, e.g., for R b 0 0 Which policy is best? Need costs. 25/01/

21 Markov Decision Model C ik = expected cost incurred during next transition if system is in state i and decision k is made State Dec The long run average expected cost for each policy may be computed using R b is best 25/01/

22 Markov Decision Processes Solution using Dynamic Programming (*some notation changes upcoming) 25/01/

23 The RL Problem Main Elements: States, s Actions, a State transition dynamics - often, stochastic & unknown Reward (r) process - possibly stochastic Objective: Policy t(s,a) probability distribution over actions given current state Assumption: Environment defines a finite-state MDP 25/01/

24 Back to Our Recycling Robot MDP S high, low A(high ) search, wait A(low ) search, wait, recharge R search R wait expected no. of cans while searching expected no. of cans while waiting R search R wait 25/01/

25 Given an enumeration of transitions and corresponding costs/rewards, what is the best sequence of actions? We want to maximize the criterion: k R t rt k 1 k 0 So, what must one do? 25/01/

26 The Shortest Path Problem 25/01/

27 Finite-State Systems and Shortest Paths state space s k is a finite set for each k a k can get you from s k to f k (s k, a k ) at a cost g k (x k, u k ) Length Cost Sum of length of arcs Solve this first V k (i) = min j [a k ij + V k+1 (j)] 25/01/

28 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 : V (s) E R t s t s E k 0 k r t k 1 s t s 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 : Action - value function for policy : Q (s, a) E R t s t s, a t a E k r t k 1 s t s,a t a k 0 25/01/

29 Recursive Equation for Value The basic idea: R t r t 1 r t 2 2 r t 3 3 r t 4 r t 1 r t 2 r t 3 2 r t 4 r t 1 R t 1 So: V ( s) E R t s t s E r t 1 V s t 1 s t s 25/01/

30 Optimality in MDPs Bellman Equation 25/01/

31 Policy Evaluation How to compute V(s) for an arbitrary policy? (Prediction problem) For a given MDP, this yields a system of simultaneous equations as many unknowns as states (BIG, S linear system!) Solve iteratively, with a sequence of value functions, 3/02/

32 Policy Improvement Does it make sense to deviate from (s) at any state (following the policy everywhere else)? Let us for now assume deterministic (s) - Policy Improvement Theorem [Howard/Blackwell] 3/02/

33 Computing Better Policies Starting with an arbitrary policy, we d like to approach truly optimal policies. So, we compute new policies using the following, Are we restricted to deterministic policies? No. With stochastic policies, 3/02/

34 Grid-World Example 25/01/

35 Iterative Policy Evaluation in Grid World Note: The value function can be searched greedily to find long-term optimal actions 25/01/