1CS 188: Artificial IntelligenceFall 2009Lecture 9: MDPs9/24/2009Dan Klein – UC BerkeleyMany slides over the course adapted from either Stuart Russell or Andrew Moore1Announcements Assignments W1 due today (drop box in 283 Soda or after lecture) P2 due on 9/30 (Wednesday) P3 out now, due 10/12 Readings: For MDPs / reinforcement learning, we’re using an online reading Different treatment and notation than the R&N book, beware! Lecture version is the standard for this class Contest is live!2Reinforcement Learning Basic idea: Receive feedback in the form of rewards Agent’s utility is defined by the reward function Must learn to act so as to maximize expected rewards[DEMOS]Grid World The agent lives in a grid Walls block the agent’s path The agent’s actions do not always go as planned: 80% of the time, the action North takes the agent North (if there is no wall there) 10% of the time, North takes the agent West; 10% East If there is a wall in the direction the agent would have been taken, the agent stays put Small “living” reward each step Big rewards come at the end Goal: maximize sum of rewards*[DEMO – Gridworld Intro]Markov Decision Processes An MDP is defined by: A set of states s ∈ S A set of actions a ∈ A A transition function T(s,a,s’) Prob that a from s leads to s’ i.e., P(s’ | s,a) Also called the model A reward function R(s, a, s’) Sometimes just R(s) or R(s’) A start state (or distribution) Maybe a terminal state MDPs are a family of non-deterministic search problems Reinforcement learning: MDPs where we don’t know the transition or reward functions5What is Markov about MDPs? Andrey Markov (1856-1922) “Markov” generally means that given the present state, the future and the past are independent For Markov decision processes, “Markov” means:2Solving MDPs In deterministic single-agent search problems, want an optimal plan, or sequence of actions, from start to a goal In an MDP, we want an optimal policy π*: S → A A policy π gives an action for each state An optimal policy maximizes expected utility if followed Defines a reflex agentOptimal policy when R(s, a, s’) = -0.03 for all non-terminals s[Demo]Example Optimal PoliciesR(s) = -2.0R(s) = -0.4R(s) = -0.03R(s) = -0.018Example: High-Low Three card types: 2, 3, 4 Infinite deck, twice as many 2’s Start with 3 showing After each card, you say “high” or “low” New card is flipped If you’re right, you win the points shown on the new card Ties are no-ops If you’re wrong, game ends Differences from expectimax: #1: get rewards as you go #2: you might play forever!39High-Low as an MDP States: 2, 3, 4, done Actions: High, Low Model: T(s, a, s’): P(s’=4 | 4, Low) = 1/4 P(s’=3 | 4, Low) = 1/4 P(s’=2 | 4, Low) = 1/2 P(s’=done | 4, Low) = 0 P(s’=4 | 4, High) = 1/4 P(s’=3 | 4, High) = 0 P(s’=2 | 4, High) = 0 P(s’=done | 4, High) = 3/4 … Rewards: R(s, a, s’): Number shown on s’ if s ≠ s’ 0 otherwise Start: 33Example: High-LowLowHighHigh LowHigh LowHighLow, Low, HighT = 0.5, R = 2T = 0.25, R = 3T = 0, R = 4T = 0.25, R = 011MDP Search Trees Each MDP state gives an expectimax-like search treeass’s, a(s,a,s’) called a transitionT(s,a,s’) = P(s’|s,a)R(s,a,s’)s,a,s’s is a state(s, a) is a q-state123Utilities of Sequences In order to formalize optimality of a policy, need to understand utilities of sequences of rewards Typically consider stationary preferences: Theorem: only two ways to define stationary utilities Additive utility: Discounted utility:13Infinite Utilities?! Problem: infinite state sequences have infinite rewards Solutions: Finite horizon: Terminate episodes after a fixed T steps (e.g. life) Gives nonstationary policies (π depends on time left) Absorbing state: guarantee that for every policy, a terminal state will eventually be reached (like “done” for High-Low) Discounting: for 0 < γ < 1 Smaller γ means smaller “horizon” – shorter term focus14Discounting Typically discount rewards by γ < 1 each time step Sooner rewards have higher utility than later rewards Also helps the algorithms converge15Recap: Defining MDPs Markov decision processes: States S Start state s0 Actions A Transitions P(s’|s,a) (or T(s,a,s’)) Rewards R(s,a,s’) (and discount γ) MDP quantities so far: Policy = Choice of action for each state Utility (or return) = sum of discounted rewardsass, as,a,s’s’16Optimal Utilities Fundamental operation: compute the values (optimal expectimaxutilities) of states s Why? Optimal values define optimal policies! Define the value of a state s:V*(s) = expected utility starting in s and acting optimally Define the value of a q-state (s,a):Q*(s,a) = expected utility starting in s, taking action a and thereafter acting optimally Define the optimal policy:π*(s) = optimal action from state sass, as,a,s’s’17[DEMO – Grid Values]The Bellman Equations Definition of “optimal utility” leads to a simple one-step lookahead relationship amongst optimal utility values:Optimal rewards = maximize over first action and then follow optimal policy Formally:ass, as,a,s’s’184Solving MDPs We want to find the optimal policy π* Proposal 1: modified expectimax search, starting from each state s:ass, as,a,s’s’19Why Not Search Trees? Why not solve with expectimax? Problems: This tree is usually infinite (why?) Same states appear over and over (why?) We would search once per state (why?) Idea: Value iteration Compute optimal values for all states all at once using successive approximations Will be a bottom-up dynamic program similar in cost to memoization Do all planning offline, no replanning needed!20Value Estimates Calculate estimates Vk*(s) Not the optimal value of s! The optimal value considering only next k time steps (k rewards) As k → ∞, it approaches the optimal value Why: If discounting, distant rewards become negligible If terminal states reachable from everywhere, fraction of episodes not ending becomes negligible Otherwise, can get infinite expected utility and then this approach actually won’t work21Value Iteration Idea: Start
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