Artificial Intelligence

1. Artificial Intelligence CH 17 Making complex decisions

2. Group (9) • Team Members : • Ahmed HelalEid • Mina Victor William • Supervised by : • Dr. Nevin M. Darwish

3. Agenda • Introduction • Sequential Decision Problems • Optimality in sequential decision problems • Value Iteration • The value iteration algorithm • Policy Iteration

4. Introduction • Previously in ch16 • MAKING SIMPLE DECISION. • Concerned with episodic decision problems, in which the utility of each action's outcome was well known. • Episodic environment: the agent experience is divided into atomic episodes each one consists of the agent perceiving and then performing a single action.

5. Introduction • In This Chapter • The computational issues involved in making decisions in stochastic environment. • Sequential decision problems, • in which the agent's utility depends on a sequence of decisions. • Sequential decision problems, which include utilities, uncertainty, and sensing, generalize the search and planning problems as special cases.

6. Sequential Decision Problems

7. Sequential Decision Problems Unfortunately the environment not go along with this situation 3 +1 -1 2 What if the environment was deterministic ? start 1 1 2 3 4 Actions A(s) in every state are (Up , Down , Left , Right)

8. Sequential Decision Problems 0.8 3 +1 0.8 -1 2 start 0.1 0.1 1 Model for stochastic motion 1 2 3 4 0.1 0.1 • [ Up, Up, Right, Right, Right ]0.8^5 =0.32768 • [ Right, Right, Up, Up, Right ]0.1^4× 0.8 =0.00008

9. Sequential Decision Problems Probability of reaching state S` if action a is done at state S • Transition model • T( S, a, S`) • Markovian transition • Utility function • Reward R(s) Probability of reaching state S` from S depend only on S Depend on a sequence of state environment history 3 -0.04 -0.04 -0.04 +1 Agent receives reward in each state (+ve Or –ve) 2 -0.04 -1 -0.04 1 -0.04 -0.04 -0.04 Utility = ( - 0.04 × 10 )+1=0.6 For 10 steps to the goal 1 2 3 4

10. Markov Decision Process (MDP) • We use MDPs to solve sequential decision problems. • We eventually want to find the best choice of action for each state. • Consists of: • a set of actions A(s) • for actions in each state in state s • transition model P(s' | s, a) • describing the probability of reaching s' using action a in s • transitions are Markovian - only depends on s not previous states • reward function R(s) • the reward an agent receives for arriving in state s

11. Sequential Decision Problems What is the solution to a problem look like ? • Policy (π) • A solution must specify what the agent should do for any state that the agent might reach. • (π(s)) • The action recommended by the policy π for state S • Optimal policy (π*) • Yield the highest expected utility

12. Continue…… 3 +1 -1 2 R (s) < -1.6284 1 3 +1 -1 2 -0.4278 < R (s) < -0.0850 1

13. Continue…… 3 +1 -1 2 -0.0221 < R (s) < 0 1 3 +1 R (s) > 0 -1 2 1

14. The Horizon • Finite horizon: • Fixed time N after which nothing matter (the game is over) • Optimal policy is Non-stationary Is there a finite Or infinite horizon for decision making ?

15. Example of Finite horizon N= 3 3 +1 -1 2 start 1 1 2 3 4 • Optimal action in a given state could change over time

16. Optimality in sequential decision problems • Infinite horizon: • No fixed deadline (time at state doesn’t matter) • Optimal policy is stationary Is there a finite Or infinite horizon for decision making ?

17. Example of Infinite horizon N= 100 3 +1 -1 2 start 1 1 2 3 4 • Optimal action in a given state could not change over time

18. Optimality in sequential decision problems We are mainly going to use infinite horizon utility functions because • there is no reason to behave differently in the same state. • Hence, the optimal action depends only on the current state, and the optimal policy is stationary. Is there a finite Or infinite horizon for decision making ?

19. Optimality in sequential decision problems

20. Optimality in sequential decision problems How to calculate utility of a state Sequence ? • Additive reward: • Discount reward: Discount factor is between 0 & 1

21. Optimality in sequential decision problems What if there isn't terminal State Or agent never reach one? • If the environment doesn’t contain a terminal state, Or if the agent never reach one, then • all environment Histories will be infinitely long, and utilities with Additive rewards will generally be infinite.

22. Optimality in sequential decision problems What if there isn't terminal State Or agent never reach one? Solution • With Discount rewards : the utility of an infinite sequence is finite, if rewards are bounded by Rmaxand γ<1 Uh([S0,S1,…..])= <= =

23. Optimal Policies for utilities of states • Expected utility for some policy π starting in state s • The optimal policy π* has the highest expected utility and will be given by • This sets π*(s) to the argument a of A(s) which gives the highest utility

24. Optimal Policies for utilities of states • Policy is actually independent of start state: • actions will differ but policy will never change • this comes from the nature of a Markovian decision problem with discounted utilities over infinite horizons • U(s) is also independent of start state and current state

25. Optimal Policies for utilities of states The utilities are higher for states closer to the +1 exit. Because fewer steps are required to reach the exit

26. Algorithms for Calculating the Optimal Policy

27. Value Iteration Algorithm • Hard to calculate • because it's non-linear so use an iterating algorithm. • Basic idea • Start at an initial value for all states then • update each state using their neighbours until they hit equilibrium.

28. Value Iteration Algorithm

29. Value Iteration Algorithm

30. Using Value Iteration on the Example

31. Value Iteration Algorithm • When to terminate??! • Bellman update is small. So the error compared with the true utility function is small. • Why use cRmax(1- γ) / γ • Recall: if γ < 1 and infinite-horizon then Uh converges to Rmax / (1 – γ) when summed over infinity If ||Ui+1-Ui|| < ε(1- γ)/ γ then ||Ui+1-U|| < ε

32. Policy iteration • Policy iteration algorithm alternates two steps: • policy evaluation :given policy πi calculate Ui=U πi, the utility of each state if were to be executed. • policy improvement: calculate a new policyΠi+1

33. Policy Iteration • Algorithm start with policy π0 repeat Policy evaluation: for each state calculate Ui given by policy πi • simplified version of Bellman Update eqn – no need for max • check if unchanged • Policy improvement: for each state • if the max utility over each action gives a better result than π(s) • set π(s) to the new policy • until unchanged

34. Policy Iteration Algorithm

35. Questions ???

36. Thanks