Report

Markov Decision Processes Basics Concepts Alan Fern 1 Some AI Planning Problems Fire & Rescue Response Planning Helicopter Control Solitaire Real-Time Strategy Games Legged Robot Control Network Security/Control Some AI Planning Problems Health Care Personalized treatment planning Hospital Logistics/Scheduling Transportation Autonomous Vehicles Supply Chain Logistics Air traffic control Assistive Technologies Dialog Management Automated assistants for elderly/disabled Household robots Sustainability Smart grid Forest fire management ….. 3 Common Elements We have a systems that changes state over time Can (partially) control the system state transitions by taking actions Problem gives an objective that specifies which states (or state sequences) are more/less preferred Problem: At each moment must select an action to optimize the overall (long-term) objective Produce most preferred state sequences 4 Observe-Act Loop of AI Planning Agent Observations State of world/system world/ system Actions action ???? Goal maximize expected reward over lifetime 5 Stochastic/Probabilistic Planning: Markov Decision Process (MDP) Model Markov Decision Process World State ???? Action from finite set Goal maximize expected reward over lifetime 6 Example MDP State describes all visible info about dice ???? Action are the different choices of dice to roll Goal maximize score or score more than opponent 7 Markov Decision Processes An MDP has four components: S, A, R, T: finite state set S (|S| = n) finite action set A (|A| = m) transition function T(s,a,s’) = Pr(s’ | s,a) Probability of going to state s’ after taking action a in state s bounded, real-valued reward function R(s,a) Immediate reward we get for being in state s and taking action a Roughly speaking the objective is to select actions in order to maximize total reward For example in a goal-based domain R(s,a) may equal 1 for goal states and 0 for all others (or -1 reward for non-goal states) 8 State action = roll the two 5’s … 1,1 1,2 6,6 Probabilistic state transition What is a solution to an MDP? MDP Planning Problem: Input: an MDP (S,A,R,T) Output: ???? Should the solution to an MDP be just a sequence of actions such as (a1,a2,a3, ….) ? Consider a single player card game like Blackjack/Solitaire. No! In general an action sequence is not sufficient Actions have stochastic effects, so the state we end up in is uncertain This means that we might end up in states where the remainder of the action sequence doesn’t apply or is a bad choice A solution should tell us what the best action is for any possible situation/state that might arise 10 Policies (“plans” for MDPs) A solution to an MDP is a policy Two types of policies: nonstationary and stationary Nonstationary policies are used when we are given a finite planning horizon H I.e. we are told how many actions we will be allowed to take Nonstationary policies are functions from states and times to actions π:S x T → A, where T is the non-negative integers π(s,t) tells us what action to take at state s when there are t stages-to-go (note that we are using the convention that t represents stages/decisions to go, rather than the time step) 11 Policies (“plans” for MDPs) What if we want to continue taking actions indefinately? Use stationary policies A Stationary policy is a mapping from states to actions π:S → A π(s) is action to do at state s (regardless of time) specifies a continuously reactive controller Note that both nonstationary and stationary policies assume or have these properties: full observability of the state history-independence deterministic action choice 12 What is a solution to an MDP? MDP Planning Problem: Input: an MDP (S,A,R,T) Output: a policy such that ???? We don’t want to output just any policy We want to output a “good” policy One that accumulates lots of reward 13 Value of a Policy How good is a policy π? How do we measure reward “accumulated” by π? Value function V: S →ℝ associates value with each state (or each state and time for non-stationary π) Vπ(s) denotes value of policy at state s Depends on immediate reward, but also what you achieve subsequently by following π An optimal policy is one that is no worse than any other policy at any state The goal of MDP planning is to compute an optimal policy 14 What is a solution to an MDP? MDP Planning Problem: Input: an MDP (S,A,R,T) Output: a policy that achieves an “optimal value” This depends on how we define the value of a policy There are several choices and the solution algorithms depend on the choice We will consider finite-horizon value 15 Finite-Horizon Value Functions We first consider maximizing expected total reward over a finite horizon Assumes the agent has H time steps to live To act optimally, should the agent use a stationary or non-stationary policy? I.e. Should the action it takes depend on absolute time? Put another way: If you had only one week to live would you act the same way as if you had fifty years to live? 16 Finite Horizon Problems Value (utility) depends on stage-to-go hence use a nonstationary policy k V (s)is k-stage-to-go value function for π expected total reward for executing π starting in s for k time steps k V ( s) E [ R | , s ] k t t 0 k E [ R( s t ) | a t ( s t , k t ), s 0 s ] t 0 Here Rt and st are random variables denoting the reward received and state at time-step t when starting in s These are random variables since the world is stochastic 17 18 Computational Problems There are two problems that are of typical interest Policy evaluation: Given an MDP and a policy π Compute finite-horizon value function V k (s) for any k and s Policy optimization: Given an MDP and a horizon H Compute the optimal finite-horizon policy How many finite horizon policies are there? |A|Hn So can’t just enumerate policies for efficient optimization 19 Computational Problems Dynamic programming techniques can be used for both policy evaluation and optimization Polynomial time in # of states and actions http://web.engr.oregonstate.edu/~afern/classes/cs533/ Is polytime in # of states and actions good? Not when these numbers are enormous! As is the case for most realistic applications Consider Klondike Solitaire, Computer Network Control, etc Enters Monte-Carlo Planning 20 Approaches for Large Worlds: Monte-Carlo Planning Often a simulator of a planning domain is available or can be learned from data Fire & Emergency Response Klondike Solitaire 21 Large Worlds: Monte-Carlo Approach Often a simulator of a planning domain is available or can be learned from data Monte-Carlo Planning: compute a good policy for an MDP by interacting with an MDP simulator World Simulator action Real World State + reward 22 Example Domains with Simulators Traffic simulators Robotics simulators Military campaign simulators Computer network simulators Emergency planning simulators large-scale disaster and municipal Sports domains Board games / Video games Go / RTS In many cases Monte-Carlo techniques yield state-of-the-art performance. Even in domains where model-based planners are applicable. 23 MDP: Simulation-Based Representation A simulation-based representation gives: S, A, R, T, I: finite state set S (|S|=n and is generally very large) finite action set A (|A|=m and will assume is of reasonable size) Stochastic, real-valued, bounded reward function R(s,a) = r Stochastically returns a reward r given input s and a Stochastic transition function T(s,a) = s’ (i.e. a simulator) Stochastically returns a state s’ given input s and a Probability of returning s’ is dictated by Pr(s’ | s,a) of MDP Stochastic initial state function I. Stochastically returns a state according to an initial state distribution These stochastic functions can be implemented in any language! 24 Computational Problems Policy evaluation: Given an MDP simulator, a policy , a state s, and horizon h Compute finite-horizon value function ℎ () Policy optimization: Given an MDP simulator, a horizon H, and a state s Compute the optimal finite-horizon policy at state s 25 Trajectories We can use the simulator to observe trajectories of any policy π from any state s: Let Traj(s, π , h) be a stochastic function that returns a length h trajectory of π starting at s. Traj(s, π , h) s0 = s For i = 1 to h-1 si = T(si-1, π(si-1)) Return s0, s1, …, sh-1 The total reward of a trajectory is given by h 1 R( s0 ,..., sh 1 ) R( si ) i 0 26 Policy Evaluation Given a policy π and a state s, how can we estimate ℎ ()? Simply sample w trajectories starting at s and average the total reward received Select a sampling width w and horizon h. The function EstimateV(s, π , h, w) returns estimate of Vπ(s) EstimateV(s, π , h, w) V=0 For i = 1 to w V = V + R( Traj(s, π , h) ) ; add total reward of a trajectory Return V / w How close to true value ℎ will this estimate be? 27 Sampling-Error Bound h 1 Vh ( s ) E [ t R t | , s ] E[ R(Traj ( s, , h))] t 0 approximation due to sampling w Estim ateV(s, , h, w) w1 ri , ri R(Traj( s, , h)) i 1 • Note that the ri are samples of random variable R(Traj(s, π , h)) • We can apply the additive Chernoff bound which bounds the difference between an expectation and an emprical average 28 Aside: Additive Chernoff Bound • Let X be a random variable with maximum absolute value Z. An let xi i=1,…,w be i.i.d. samples of X • The Chernoff bound gives a bound on the probability that the average of the xi are far from E[X] Let {xi | i=1,…, w} be i.i.d. samples of random variable X, then with probability at least 1 we have that, w E[ X ] w1 xi Z i 1 1 w ln 1 2 1 equivalently Pr E[ X ] w xi exp w Z i 1 w 29 Aside: Coin Flip Example • Suppose we have a coin with probability of heads equal to p. • Let X be a random variable where X=1 if the coin flip gives heads and zero otherwise. (so Z from bound is 1) E[X] = 1*p + 0*(1-p) = p • After flipping a coin w times we can estimate the heads prob. by average of xi. • The Chernoff bound tells us that this estimate converges exponentially fast to the true mean (coin bias) p. w 1 Pr p w xi exp 2 w i 1 30 Sampling Error Bound h 1 V ( s, h) E [ t R t | , s ] E[ R(Traj( s, , h))] t 0 approximation due to sampling w Estim ateV(s, , h, w) w1 ri , ri R(Traj( s, , h)) i 1 We get that, V ( s ) EstimateV ( s, , h, w) Vmax h with probability at least 1 w ln 1 1 31 Two Player MDP (aka Markov Games) • So far we have only discussed single-player MDPs/games • Your labs and competition will be 2-player zero-sum games (zero sum means sum of player rewards is zero) • We assume players take turns (non-simultaneous moves) Player 1 Player 2 Markov Game action action State/ reward State/ reward Simulators for 2-Player Games A simulation-based representation gives: , 1 , 2 , , , : finite state set S (assume state encodes whose turn it is) action sets 1 and 2 for player 1 (called max) and player 2 (called min) Stochastic, real-valued, bounded reward function , Stochastically returns a reward r for max given input s and action a (it is assumed here that reward for min is -r) So min maximizes its reward by minimizing the reward of max Stochastic transition function T(s,a) = s’ (i.e. a simulator) Stochastically returns a state s’ given input s and a Probability of returning s’ is dictated by Pr(s’ | s,a) of game Generally s’ will be a turn for the other player These stochastic functions can be implemented in any language! 33 Finite Horizon Value of Game Given two policies 1 and 2 one for each player we can define the finite horizon value ℎ1 ,2 () is h-horizon value function with respect to player 1 expected total reward for player 1 after executing and starting in s for k steps For zero-sum games the value with respect to player 2 is just − ℎ1 ,2 () Given 1 and 2 we can easily use the simulator to estimate ℎ1 ,2 () Just as we did for the single-player MDP setting 34