1CS 188: Artificial IntelligenceFall 2011Lecture 9: MDPs9/22/2011Dan Klein – UC BerkeleyMany slides over the course adapted from either Stuart Russell or Andrew Moore2Grid 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*2Recap: MDPs Markov decision processes: States S Actions A Transitions P(s’|s,a) (or T(s,a,s’)) Rewards R(s,a,s’) (and discount γ) Start state s0 Quantities: Policy = map of states to actions Episode = one run of an MDP Utility = sum of discounted rewards Values = expected future utility from a state Q-Values = expected future utility from a q-stateass, as,a,s’s’4[DEMO – MDP Quantities]Optimal Utilities The utility of a state s:V*(s) = expected utility starting in s and acting optimally The utility of a q-state (s,a):Q*(s,a) = expected utility starting out having taken action a from state s and (thereafter) acting optimally The optimal policy:π*(s) = optimal action from state s5ass’s, a(s,a,s’) is a transitions,a,s’s is a state(s, a) is a q-state3Bellman Equations Definition of utility leads to a simple one-step lookahead relationship amongst optimal utility values:Total optimal rewards = maximize over choice of (first action plus optimal future) Formally:ass, as,a,s’s’6Value Estimates Calculate estimates Vk*(s) Not the optimal value of s! The optimal value considering only next k time steps (k rewards) What you’d get with depth-k expectimax* As k → ∞, it approaches the optimal value* Almost solution: recursion (i.e. expectimax) Correct solution: dynamic programming7[DEMO -- Vk]4Value Iteration Idea: Start with V0*(s) = 0 for all s, which we know is right (why?) Given Vi*, calculate the values for all states for depth i+1: Throw out old vector Vi* Repeat until convergence This is called a value update or Bellman update Theorem: will converge to unique optimal values Basic idea: approximations get refined towards optimal values Policy may converge long before values do8Example: Bellman Updates9max happens for a=right, other actions not shownExample: γ=0.9, living reward=0, noise=0.25Example: Value Iteration Information propagates outward from terminal states and eventually all states have correct value estimatesV2V310Convergence* Define the max-norm: Theorem: For any two approximations U and V I.e. any distinct approximations must get closer to each other, so, in particular, any approximation must get closer to the true U and value iteration converges to a unique, stable, optimal solution Theorem: I.e. once the change in our approximation is small, it must also be close to correct116Practice: Computing Actions Which action should we chose from state s: Given optimal values V? Given optimal q-values Q? Lesson: actions are easier to select from Q’s!12[DEMO – MDP action selection]Utilities for a Fixed Policy Another basic operation: compute the utility of a state s under a fixed (generally non-optimal) policy Define the utility of a state s, under a fixed policy π:Vπ(s) = expected total discounted rewards (return) starting in s and following π Recursive relation (one-step look-ahead / Bellman equation):π(s)ss, π(s)s, π(s),s’s’13[DEMO – Right-Only Policy]7Policy Evaluation How do we calculate the V’s for a fixed policy? Idea one: turn recursive equations into updates Idea two: it’s just a linear system, solve with Matlab (or whatever)14Policy Iteration Alternative approach for optimal values: Step 1: Policy evaluation: calculate utilities for some fixed policy (not optimal utilities!) until convergence Step 2: Policy improvement: update policy using one-step look-ahead with resulting converged (but not optimal!) utilities as future values Repeat steps until policy converges This is policy iteration It’s still optimal! Can converge faster under some conditions158Policy Iteration Policy evaluation: with fixed current policy π, find values with simplified Bellman updates: Iterate until values converge Policy improvement: with fixed utilities, find the best action according to one-step look-ahead16Comparison Both VI and PI compute the same thing (optimal values for all states) In value iteration: Every pass (or “backup”) updates both utilities (explicitly, based on current utilities) and policy (implicitly, based on current utilities) Tracking the policy isn’t necessary; we take the max In policy iteration: Several passes to update utilities with fixed policy After policy is evaluated, a new policy is chosen Both are dynamic programs for solving MDPs179Asynchronous Value Iteration* In value iteration, we update every state in each iteration Actually, any sequences of Bellman updates will converge if every state is visited infinitely often In fact, we can update the policy as seldom or often as we like, and we will still converge Idea: Update states whose value we expect to change:If is large then update predecessors of
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