 That tireless teacher who gets to class early and stays late and dips into her own pocket to buy supplies because she believes that every child is her.

 That tireless teacher who gets to class early and stays late and dips into her own pocket to buy supplies because she believes that every child is her charge -- she’s marching. (Applause.) That successful businessman who doesn’t have to, but pays his workers a fair wage and then offers a shot to a man, maybe an ex-con, who’s down on his luck -- he’s marching.  (Cheers, applause.) The mother who pours her love into her daughter so that she grows up with the confidence to walk through the same doors as anybody’s son -- she’s marching.  (Cheers, applause.) The father who realizes the most important job he’ll ever have is raising his boy right, even if he didn’t have a father, especially if he didn’t have a father at home -- he’s marching.  The graduate student who not only posts his comment, but comments on other peoples comments on Piazza, he is marching. 1 8/28/2013 50 years since the man’s dream

 That tireless teacher who gets to class early and stays late and dips into his own pocket to buy ice cream sandwich because he believes that every grad is his charge -- he’s marching. (Applause.) That successful businessman who doesn’t have to, but pays his workers a fair wage and then offers a shot to a man, maybe an ex-con, who’s down on his luck -- he’s marching.  (Cheers, applause.) The mother who pours her love into her daughter so that she grows up with the confidence to walk through the same doors as anybody’s son -- she’s marching.  (Cheers, applause.) The father who realizes the most important job he’ll ever have is raising his boy right, even if he didn’t have a father, especially if he didn’t have a father at home -- he’s marching.  The graduate student who not only posts his comment, but comments on other peoples comments on Piazza, he is marching. 4 8/28/2013 50 years since the man’s dream

8/28/63 50 years since the man’s dream

A: A Unified Brand-name-Free Introduction to Planning Subbarao Kambhampati Environment action perception Goals (Static vs. Dynamic) (Observable vs. Partially Observable) (perfect vs. Imperfect) (Deterministic vs. Stochastic) What action next? (Instantaneous vs. Durative) (Full vs. Partial satisfaction) The $$$$$$ Question

Representation Mechanisms: Logic (propositional; first order) Probabilistic logic Learning the models Search Blind, Informed SAT; Planning Inference Logical resolution Bayesian inference How the course topics stack up…

Topics Covered in CSE471 Table of Contents (Full Version)Full Version Preface (html); chapter map Part I Artificial Intelligence 1 Introduction 2 Intelligent Agents Part II Problem Solving 3 Solving Problems by Searching 4 Informed Search and Exploration 5 Constraint Satisfaction Problems 6 Adversarial Search Part III Knowledge and Reasoning 7 Logical Agents 8 First-Order Logic 9 Inference in First-Order Logic 10 Knowledge Representation Part IV Planning 11 Planning (pdf) 12 Planning and Acting in the Real Worldhtmlchapter mappdf Part V Uncertain Knowledge and Reasoning 13 Uncertainty 14 Probabilistic Reasoning 15 Probabilistic Reasoning Over Time 16 Making Simple Decisions 17 Making Complex Decisions Part VI Learning 18 Learning from Observations 19 Knowledge in Learning 20 Statistical Learning Methods 21 Reinforcement Learning Part VII Communicating, Perceiving, and Acting 22 Communication 23 Probabilistic Language Processing 24 Perception 25 Robotics Part VIII Conclusions 26 Philosophical Foundations 27 AI: Present and Future

Agent Classification in Terms of State Representations TypeState representationFocus AtomicStates are indivisible; No internal structure Search on atomic states; Propositional (aka Factored) States are made of state variables that take values (Propositional or Multi- valued or Continuous) Search+inference in logical (prop logic) and probabilistic (bayes nets) representations RelationalStates describe the objects in the world and their inter-relations Search+Inference in predicate logic (or relational prob. Models) First-order+functions over objectsSearch+Inference in first order logic (or first order probabilistic models)

Pendulum Swings in AI Top-down vs. Bottom-up Ground vs. Lifted representation – The longer I live the farther down the Chomsky Hierarchy I seem to fall [Fernando Pereira] Pure Inference and Pure Learning vs. Interleaved inference and learning Knowledge Engineering vs. Model Learning vs. Data-driven Inference Human-aware vs. Stand-Alone vs. Human- driven(!)

Discussion What are the current controversies in AI? What are the hot topics in AI?

CLASS OF 9/4  Class forum

16 Markov Decision Processes Atomic Model for stochastic environments with generalized rewards Based in part on slides by Alan Fern, Craig Boutilier and Daniel Weld Some slides from Mausam/Kolobov Tutorial; and a couple from Terran Lane

Atomic Model for Deterministic Environments and Goals of Attainment Deterministic worlds + goals of attainment  Atomic model: Graph search  Propositional models: The PDDL planning that we discussed..  What is missing?  Rewards are only at the end (and then you die).  What about “the Journey is the reward” philosophy?  Dynamics are assumed to be Deterministic  What about stochastic dynamics? 17

Atomic Model for stochastic environments with generalized rewards Stochastic worlds +generalized rewards  An action can take you to any of a set of states with known probability  You get rewards for visiting each state  Objective is to increase your “cumulative” reward…  What is the solution? 18

A: A Unified Brand-name-Free Introduction to Planning Subbarao Kambhampati Environment action perception Goals (Static vs. Dynamic) (Observable vs. Partially Observable) (perfect vs. Imperfect) (Deterministic vs. Stochastic) What action next? (Instantaneous vs. Durative) (Full vs. Partial satisfaction) The $$$$$$ Question

Optimal Policies depend on horizon, rewards.. --- -

22 Percepts Actions ???? World perfect fully observable instantaneous deterministic Classical Planning Assumptions sole source of change

23 Percepts Actions ???? World perfect fully observable instantaneous stochastic Stochastic/Probabilistic Planning: Markov Decision Process (MDP) Model sole source of change

24 Types of Uncertainty  Disjunctive (used by non-deterministic planning) Next state could be one of a set of states.  Stochastic/Probabilistic Next state is drawn from a probability distribution over the set of states. How are these models related?

25 Markov Decision Processes  An MDP has four components: S, A, R, T:  (finite) state set S (|S| = n)  (finite) action set A (|A| = m)  (Markov) transition function T(s,a,s’) = Pr(s’ | s,a)  Probability of going to state s’ after taking action a in state s  How many parameters does it take to represent?  bounded, real-valued (Markov) reward function R(s)  Immediate reward we get for being in state s  For example in a goal-based domain R(s) may equal 1 for goal states and 0 for all others  Can be generalized to include action costs: R(s,a)  Can be generalized to be a stochastic function  Can easily generalize to countable or continuous state and action spaces (but algorithms will be different)

26 Graphical View of MDP StSt RtRt S t+1 AtAt R t+1 S t+2 A t+1 R t+2

27 Assumptions  First-Order Markovian dynamics (history independence)  Pr(S t+1 |A t,S t,A t-1,S t-1,..., S 0 ) = Pr(S t+1 |A t,S t )  Next state only depends on current state and current action  First-Order Markovian reward process  Pr(R t |A t,S t,A t-1,S t-1,..., S 0 ) = Pr(R t |A t,S t )  Reward only depends on current state and action  As described earlier we will assume reward is specified by a deterministic function R(s)  i.e. Pr(R t =R(S t ) | A t,S t ) = 1  Stationary dynamics and reward  Pr(S t+1 |A t,S t ) = Pr(S k+1 |A k,S k ) for all t, k  The world dynamics do not depend on the absolute time  Full observability  Though we can’t predict exactly which state we will reach when we execute an action, once it is realized, we know what it is

28 Policies (“plans” for MDPs)  Nonstationary policy [Even though we have stationary dynamics and reward??]  π :S x T → A, where T is the non-negative integers  π (s,t) is action to do at state s with t stages-to-go  What if we want to keep acting indefinitely?  Stationary policy  π: S → A  π (s) is action to do at state s (regardless of time)  specifies a continuously reactive controller  These assume or have these properties:  full observability  history-independence  deterministic action choice Why not just consider sequences of actions? Why not just replan? If you are 20 and are not a liberal, you are heartless If you are 40 and not a conservative, you are mindless -Churchill # non-stationary policies: |A| |S|*T # stationary policies: |A| |S|

29 Value of a Policy  How good is a policy π ?  How do we measure “accumulated” reward?  Value function V: S →ℝ associates value with each state (or each state and time for non-stationary π)  V π (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 (method depends on how we define value)

30 Finite-Horizon Value Functions  We first consider maximizing total reward over a finite horizon  Assumes the agent has n time steps to live  To act optimally, should the agent use a stationary or non-stationary policy?  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?

31 Finite Horizon Problems  Value (utility) depends on stage-to-go  hence so should policy: nonstationary π( s,k )  is k-stage-to-go value function for π  expected total reward after executing π for k time steps (for k=0?)  Here R t and s t are random variables denoting the reward received and state at stage t respectively

32 Computing Finite-Horizon Value  Can use dynamic programming to compute  Markov property is critical for this (a) (b) V k-1 VkVk 0.7 0.3 π(s,k) immediate reward expected future payoff with k-1 stages to go

33 Bellman Backup a1a1 a2a2 How can we compute optimal V t+1 (s) given optimal V t ? s4 s1 s3 s2 V t 0.7 0.3 0.4 0.6 0.4 V t (s2) + 0.6 V t (s3) Compute Expectations 0.7 V t (s1) + 0.3 V t (s4) V t+1 (s) s Compute Max V t+1 (s) = R(s)+max { }

34 Value Iteration: Finite Horizon Case  Markov property allows exploitation of DP principle for optimal policy construction  no need to enumerate |A| Tn possible policies  Value Iteration V k is optimal k-stage-to-go value function Π*(s,k) is optimal k-stage-to-go policy Bellman backup

35 Value Iteration 0.3 0.7 0.4 0.6 s4 s1 s3 s2 V0V0 V1V1 0.4 0.3 0.7 0.6 0.3 0.7 0.4 0.6 V2V2 V3V3 0.7 V 0 (s1) + 0.3 V 0 (s4) 0.4 V 0 (s2) + 0.6 V 0 (s3) V 1 (s4) = R(s4)+max { } Optimal value depends on stages-to-go (independent in the infinite horizon case)

36 Value Iteration s4 s1 s3 s2 0.3 0.7 0.4 0.6 0.3 0.7 0.4 0.6 0.3 0.7 0.4 0.6 V0V0 V1V1 V2V2 V3V3  * (s4,t) = max { }

9/10/2012 37

38 Value Iteration  Note how DP is used  optimal soln to k-1 stage problem can be used without modification as part of optimal soln to k-stage problem  Because of finite horizon, policy nonstationary  What is the computational complexity?  T iterations  At each iteration, each of n states, computes expectation for |A| actions  Each expectation takes O(n) time  Total time complexity: O(T|A|n 2 )  Polynomial in number of states. Is this good?

39 Summary: Finite Horizon  Resulting policy is optimal  convince yourself of this  Note: optimal value function is unique, but optimal policy is not  Many policies can have same value

40 Discounted Infinite Horizon MDPs  Defining value as total reward is problematic with infinite horizons  many or all policies have infinite expected reward  some MDPs are ok (e.g., zero-cost absorbing states)  “Trick”: introduce discount factor 0 ≤ β < 1  future rewards discounted by β per time step  Note:  Motivation: economic? failure prob? convenience?

41 Notes: Discounted Infinite Horizon  Optimal policy maximizes value at each state  Optimal policies guaranteed to exist (Howard60)  Can restrict attention to stationary policies  I.e. there is always an optimal stationary policy  Why change action at state s at new time t?  We define for some optimal π

42 Computing an Optimal Value Function  Bellman equation for optimal value function  Bellman proved this is always true  How can we compute the optimal value function?  The MAX operator makes the system non-linear, so the problem is more difficult than policy evaluation  Notice that the optimal value function is a fixed-point of the Bellman Backup operator B  B takes a value function as input and returns a new value function

43 Value Iteration  Can compute optimal policy using value iteration, just like finite-horizon problems (just include discount term)  Will converge to the optimal value function as k gets large. Why?

44 Convergence  B[V] is a contraction operator on value functions  For any V and V’ we have || B[V] – B[V’] || ≤ β || V – V’ ||  Here ||V|| is the max-norm, which returns the maximum element of the vector  So applying a Bellman backup to any two value functions causes them to get closer together in the max-norm sense.  Convergence is assured  any V: || V* - B[V] || = || B[V*] – B[V] || ≤ β|| V* - V ||  so applying Bellman backup to any value function brings us closer to V* by a factor β  thus, Bellman fixed point theorems ensure convergence in the limit  When to stop value iteration? when ||V k - V k-1 ||≤ ε  this ensures ||V k – V*|| ≤ εβ /1-β

Contraction property proof sketch  Note that for any functions f and g  We can use this to show that  |B[V]-B[V’]| <=  |V – V’| 45 f g

46 How to Act  Given a V k from value iteration that closely approximates V*, what should we use as our policy?  Use greedy policy:  Note that the value of greedy policy may not be equal to V k  Let V G be the value of the greedy policy? How close is V G to V*?

47 How to Act  Given a V k from value iteration that closely approximates V*, what should we use as our policy?  Use greedy policy:  We can show that greedy is not too far from optimal if V k is close to V *  In particular, if V k is within ε of V*, then V G within 2εβ /1-β of V* (if ε is 0.001 and β is 0.9, we have 0.018)  Furthermore, there exists a finite ε s.t. greedy policy is optimal  That is, even if value estimate is off, greedy policy is optimal once it is close enough

Improvements to Value Iteration  Initialize with a good approximate value function  Instead of R(s), consider something more like h(s)  Well defined only for SSPs  Asynchronous value iteration  Can use the already updated values of neighors to update the current node  Prioritized sweeping  Can decide the order in which to update states  As long as each state is updated infinitely often, it doesn’t matter if you don’t update them  What are good heuristics for Value iteration? 48

9/14 (make-up for 9/12)  Policy Evaluation for Infinite Horizon MDPS  Policy Iteration  Why it works  How it compares to Value Iteration  Indefinite Horizon MDPs  The Stochastic Shortest Path MDPs  With initial state  Value Iteration works; policy iteration?  Reinforcement Learning start 49

50 Policy Evaluation  Value equation for fixed policy   Notice that this is stage-indepedent  How can we compute the value function for a policy?  we are given R and Pr  simple linear system with n variables (each variables is value of a state) and n constraints (one value equation for each state)  Use linear algebra (e.g. matrix inverse)

51 Policy Iteration  Given fixed policy, can compute its value exactly:  Policy iteration exploits this: iterates steps of policy evaluation and policy improvement 1. Choose a random policy π 2. Loop: (a) Evaluate V π (b) For each s in S, set (c) Replace π with π’ Until no improving action possible at any state Policy improvement

P.I. in action PolicyValue Iteration 0 The PI in action slides from Terran Lane’s Notes

P.I. in action PolicyValue Iteration 1

P.I. in action PolicyValue Iteration 6: done

59 Policy Iteration Notes  Each step of policy iteration is guaranteed to strictly improve the policy at some state when improvement is possible  [Why? The same contraction property of Bellman Update. Note that when you go from value to policy to value to policy, you are effectively, doing a DP update on the value]  Convergence assured (Howard)  intuitively: no local maxima in value space, and each policy must improve value; since finite number of policies, will converge to optimal policy  Gives exact value of optimal policy  Complexity:  There are at most exp(n) policies, so PI is no worse than exponential time in number of states  Empirically O(n) iterations are required  Still no polynomial bound on the number of PI iterations (open problem)!

Improvements to Policy Iteration  Find the value of the policy approximately (by value iteration) instead of exactly solving the linear equations  Can run just a few iterations of the value iteration  This can be asynchronous, prioritized etc. 60

61 Value Iteration vs. Policy Iteration  Which is faster? VI or PI  It depends on the problem  VI takes more iterations than PI, but PI requires more time on each iteration  PI must perform policy evaluation on each step which involves solving a linear system  Can be done approximately  Also, VI can be done with asynchronous and prioritized update fashion..  ***Value Iteration is more robust—it’s convergence is guaranteed for many more types of MDPs..

Need for Indefinite Horizon MDPs  We have see  Finite horizon MDPs  Infinite horizon MDPs  In many cases, we neither have finite nor infinite horizon, but rather some indefinite horizon  Need to model MDPs without discount factor, knowing only that the behavior sequences will be finite 62

Stochastic Shortest-Path MDPs: Definition SSP MDP is a tuple, where: S is a finite state space (D is an infinite sequence (1,2, …)) A is a finite action set T: S x A x S  [0, 1] is a stationary transition function C: S x A x S  R is a stationary cost function (= -R: S x A x S  R ) G is a set of absorbing cost-free goal states Under two conditions: There is a proper policy (reaches a goal with P= 1 from all states) – No sink states allowed.. Every improper policy incurs a cost of ∞ from every state from which it does not reach the goal with P=1 63 Bertsekas, 1995 [SSP slides from Mausam/Kolobov Tutorial]

SSP MDP Details In SSP, maximizing ELAU = minimizing exp. cost Every cost-minimizing policy is proper! Thus, an optimal policy = cheapest way to a goal 64

Not an SSP MDP Example 65 S1S1 S2S2 a1a1 C(s 2, a 1, s 1 ) = -1 C(s 1, a 1, s 2 ) = 1 a2a2 a2a2 C(s 1, a 2, s 1 ) = 7.2 C(s 2, a 2, s G ) = 1 SGSG C(s G, a 2, s G ) = 0 C(s G, a 1, s G ) = 0 C(s 2, a 2, s 2 ) = -3 T(s 2, a 2, s G ) = 0.3 T(s 2, a 2, s G ) = 0.7 S3S3 C(s 3, a 2, s 3 ) = 0.8C(s 3, a 1, s 3 ) = 2.4 a1a1 a2a2 a1a1 No dead ends allowed! a1a1 a2a2 No cost-free “loops” allowed!

SSP MDPs: Optimality Principle For an SSP MDP, let: – V π (s,t) = E s,t [C 1 + C 2 + …] for all s, t Then: – V* exists, π* exists, both are stationary – For all s: V*(s) = min a in A [ ∑ s’ in S T(s, a, s’) [ C(s, a, s’) + V*(s’) ] ] π*(s) = argmin a in A [ ∑ s’ in S T(s, a, s’) [ C(s, a, s’) + V*(s’) ] ] 66 π Exp. Lin. Add. Utility Every policy either takes a finite exp. # of steps to reach a goal, or has an infinite cost. For every s,t, the value of a policy is well-defined!

SSP and Other MDP Classes SSP is an “indefinite-horizon” MDP Can compile FH by considering a new state space that is (state at epoch) Can compile IHDR to SSP by introducing a “goal” node and adding a  probability transition from any state to goal state. 67 SSP IHDRFH

Algorithms for SSP Value Iteration works without change for SSPs – (as long as they *are* SSPs—i.e., have proper policies and infinite costs for all improper ones) – Instead of Max operations, the Bellman update does min operations Policy iteration works *iff* we start with a proper policy (otherwise, it diverges) – It is not often clear how to pick a proper policy though

SSPs and A* Search The SSP model, being based on absorbing goals and action costs, is very close to the A* search model – Identical if you have deterministic actions and start the SSP with a specific initial state – For this case, the optimal value function of SSP is the perfect heuristic for the corresponding A* search – An admissible heuristic for A* will be a “lower bound” on V* for the SSP Start value iteration by initializing with an admissible heuristic!..and since SSP theoretically subsumes Finite Horizon and Infinite Horizon models, you get an effective bridge between MDPs and A* search The bridge also allows us to “solve” MDPs using advances in deterministic planning/A* search..

Summary of MDPs There are many MDPs, we looked at those that – aim to maximize expected cumulative reward (as against, say, average reward) – Finite horizon, Infinite Horizon and SSP MDPs We looked at Atomic MDPs—states are atomic We looked at exact methods for solving MDPs – Value Iteration (and improvements including asynchronous and prioritized sweeping) – Policy Iteration (and improvements including modified PI) We looked at connections to A* search

Other topics in MDPs (that we will get back to) Approximate solutions for MDPs – E.g. Online solutions based on determinizations Factored representations for MDPs – States in terms of state variables – Actions in terms of either Probabilistic STRIPS or Dynamic Bayes Net representations – Value and Reward functions in terms of decision diagrams

Heuristic Search vs. Dynamic Programming (Value/Policy Iteration) VI and PI approaches use Dynamic Programming Update Set the value of a state in terms of the maximum expected value achievable by doing actions from that state. They do the update for every state in the state space –Wasteful if we know the initial state(s) that the agent is starting from Heuristic search (e.g. A*/AO*) explores only the part of the state space that is actually reachable from the initial state Even within the reachable space, heuristic search can avoid visiting many of the states. –Depending on the quality of the heuristic used.. But what is the heuristic? –An admissible heuristic is a lowerbound on the cost to reach goal from any given state –It is a lowerbound on J*!

Modeling Softgoal problems as deterministic MDPs Consider the net-benefit problem, where actions have costs, and goals have utilities, and we want a plan with the highest net benefit How do we model this as MDP? –(wrong idea): Make every state in which any subset of goals hold into a sink state with reward equal to the cumulative sum of utilities of the goals. Problem—what if achieving g1 g2 will necessarily lead you through a state where g1 is already true? –(correct version): Make a new fluent called “done” dummy action called Done-Deal. It is applicable in any state and asserts the fluent “done”. All “done” states are sink states. Their reward is equal to sum of rewards of the individual states.

 That tireless teacher who gets to class early and stays late and dips into her own pocket to buy supplies because she believes that every child is her.

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 That tireless teacher who gets to class early and stays late and dips into her own pocket to buy supplies because she believes that every child is her.

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