MDPs (cont) & Reinforcement Learning

Tamara Berg

CS 560 Artificial Intelligence

Many slides throughout the course adapted from Svetlana Lazebnik, Dan Klein, Stuart Russell, Andrew Moore, Percy Liang, Luke Zettlemoyer

Announcements

• HW2 online: CSPs and Games– Due Oct 8, 11:59pm (start now!)

• Mid-term exam next Wed, Sept 30– Held during regular class time.– Closed book. You may bring a calculator.– Written questions (no coding).– Ric will lead an in class mid-term review/exercises

session in class on Sept 28.

Exam topics1) Intro to AI, agents and environments Turing test

Rationality

Expected utility maximizationPEASEnvironment characteristics: fully vs. partially observable, deterministic vs. stochastic, episodic vs. sequential, static vs. dynamic, discrete vs. continuous, single-agent vs. multi-agent, known vs. unknown

2) SearchSearch problem formulation: initial state, actions, transition model, goal state, path cost

State space graph

Search tree

Frontier, Explored set

Evaluation of search strategies: completeness, optimality, time complexity, space complexity

Uninformed search strategies: breadth-first search, uniform cost search, depth-first search, iterative deepening search

Informed search strategies: greedy best-first, A*, weighted A*

Heuristics: admissibility, dominance

Exam topics

3) Constraint satisfaction problems Backtracking search

Heuristics: most constrained/most constraining variable, least constraining value Forward checking, constraint propagation, arc consistencyTaking advantage of structure – connected components, Tree-structured CSPsLocal search

Formulating photo ordering as a CSP

4) GamesZero-sum games

Game treeMinimax/Expectimax/Expectiminimax searchAlpha-beta pruningEvaluation functionQuiescence searchHorizon effectStochastic elements in games

Exam topics

5) Markov decision processesMarkov assumption, transition model, policy

Bellman equationValue iterationPolicy iteration

6) Reinforcement learningModel-based vs. model-free approaches

Passive vs Active

Exploration vs. exploitation

Direct Estimation

TD Learning

TD Q-learning

Applications to backgammon, quadruped locomotion, helicopter flying

Stochastic, sequential environments

Image credit: P. Abbeel and D. Klein

Markov Decision Processes

Markov Decision Processes• Components:

– States s, beginning with initial state s0

– Actions a• Each state s has actions A(s) available from it

– Transition model P(s’ | s, a)• Markov assumption: the probability of going to s’ from

s depends only on s and a and not on any other past actions or states

– Reward function R(s)• Policy (s): the action that an agent takes in any given state

– The “solution” to an MDP

Overview

• First, we will look at how to “solve” MDPs, ie find the optimal policy when the transition model and the reward function are known

• Second, we will consider reinforcement learning, where we don’t know the rules of the environment or the consequences of our actions

Grid world

R(s) = -0.04 for every non-terminal state

Transition model:

0.8 0.10.1

Source: P. Abbeel and D. Klein

Goal: Policy

Source: P. Abbeel and D. Klein

Grid world

Optimal policy when R(s) = -0.04 for every non-terminal state

Grid world• Optimal policies for other values of R(s):

Solving MDPs• MDP components:

– States s– Actions a– Transition model P(s’ | s, a)– Reward function R(s)

• The solution:– Policy (s): mapping from states to actions– How to find the optimal policy?

Maximizing expected utility• The optimal policy should maximize the expected

utility over all possible state sequences produced by following that policy:

• How to define the utility of a state sequence?– Sum of rewards of individual states– Problem: infinite state sequences

0sfromstarting

sequencesstate

)sequence()sequence( UP

Utilities of state sequences• Normally, we would define the utility of a state sequence

as the sum of the rewards of the individual states• Problem: infinite state sequences• Solution: discount the individual state rewards by a factor

between 0 and 1:

– Sooner rewards count more than later rewards– Makes sure the total utility stays bounded– Helps algorithms converge

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max

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Utilities of states

• Expected utility obtained by policy starting in state s:

• The “true” utility (value) of a state, is the expected sum of discounted rewards if the agent executes an optimal policy starting in state s

sfromstarting

sequencesstate

)sequence()sequence()( UPsU

Finding the utilities of states

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U(s’)

Max node

Chance node

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P(s’ | s, a)

• What is the expected utility of taking action a in state s?

• How do we choose the optimal action?

• What is the recursive expression for U(s) in terms of the utilities of its successor states?

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a sUassPsRsU

The Bellman equation• Recursive relationship between the utilities of

successive states:

End up here with P(s’ | s, a)Get utility U(s’)

(discounted by )

Receive reward R(s)

Choose optimal action a

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)()'(),|'(max)()(

ssAa

sUassPsRsU

The Bellman equation• Recursive relationship between the utilities of

successive states:

• For N states, we get N equations in N unknowns– Solving them solves the MDP– Two methods: value iteration and policy iteration

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Method 1: Value iteration

• Start out with every U(s) = 0• Iterate until convergence

– During the ith iteration, update the utility of all states (simultaneously) according to this rule:

• In the limit of infinitely many iterations, guaranteed to find the correct utility values– In practice, don’t need an infinite number of iterations…

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1 )'(),|'(max)()(s

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Value iteration

• Run value iteration on the following grid-world:

R(s)= -0.25 for all non-terminal states

Transition model: 0.7 chance of going in desired direction, 0.1 chance of going in any of the other 3 directions. If agent moves into a wall, it stays put.

Value iteration

• Run value iteration on the following grid-world:

R(s)= -0.25 for all non-terminal states

Transition model: 0.7 chance of going in desired direction, 0.1 chance of going in any of the other 3 directions. If agent moves into a wall, it stays put.

0 0 0

0 0

0 0 00

Value iteration

Value iteration demo