Computational Modeling Lab

Wednesday 18 June 2003

Reinforcement Learning an introduction

part 3

Ann NowéAnn.nowe@vub.ac.behttp://como.vub.ac.be

By Sutton and Barto

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Policy and Value Iteration

Evaluate the policy, and locally improve it

V(s)V(s2

’)V(s2)

1π2π3π

1a

2a

3a

Q(s,a)

Q(s2,a2)

Q(s2,a1)

1r

2r

s1

s2

Estimate quality of an action for a state

If model is known use PI of DPIf model is not known use PI of RL

If model is known use VI of DPIf model is not known use VI of RL

),(max)( asQsVa

=

Q(s1,a2)

Q(s1,a1)

V(s3’)

V(s3)

V(s1’)

V(s1)

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Value Functions

The value of a state is the expected return starting from that state; it depends on the agent’s policy:

The value of taking an action in a state under policy π is the expected return starting from that state, taking that action, and thereafter following π :Action-value function for policy π :

Qπ (s,a) =Eπ Rt st =s,at =a{ }=Eπ γkrt+k+1 st =s,at =ak=0

∞

∑⎧ ⎨ ⎩

⎫ ⎬ ⎭

{ }⎭⎬⎫

⎩⎨⎧

==== ∑∞

=++

01)(

ktkt

ktt ssrEssREsV γ

π

πππ

: policyfor function value-State

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Bellman Equation for a Policy π

Rt =rt+1 +γrt+2 +γ2rt+3 +γ3rt+4L

=rt+1 +γ rt+2 +γrt+3 +γ2rt+4L( )

=rt+1 +γRt+1

The basic idea:

So: Vπ (s)=Eπ Rt st =s{ }

=Eπ rt+1 +γV st+1( ) st =s{ }

Or, without the expectation operator:

€

V π (s) = π (s,a) Ps ′ s a Rs ′ s

a + γV π ( ′ s )[ ]′ s

∑a

∑

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Gridworld, example

Actions: north, south, east, west; deterministic.If would take agent off the grid: no move but reward

= –1Other actions produce reward = 0, except actions

that move agent out of special states A and B as shown.

State-value function for equiprobable random policy; = 0.9

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Optimal Value Functions

π ≥ ′ π if and only if Vπ (s) ≥V ′ π (s) for all s∈SFor finite MDPs, policies can be partially ordered:

There are always one or more policies that are better than or equal to all the others. These are the optimal policies. We denote them all π *.

Optimal policies share the same optimal state-value function:

Optimal policies also share the same optimal action-value function:

V∗(s) =maxπ

Vπ (s) for all s∈S

Q∗(s,a)=maxπ

Qπ (s,a) for all s∈S and a∈A(s)

This is the expected return for taking action a in state s and thereafter following an optimal policy.

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Bellman Optimality Equation for V*

€

V ∗(s) = maxa∈A (s)

Qπ ∗

(s,a)

= maxa∈A (s)

E rt +1 + γV ∗(st +1) st = s,at = a{ }

= maxa∈A (s)

Ps ′ s a

′ s

∑ Rs ′ s a + γV ∗( ′ s )[ ]

The value of a state under an optimal policy must equalthe expected return for the best action from that state:

The relevant backup diagram:

is the unique solution of this system of nonlinear equations.V∗

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Bellman Optimality Equation for Q*

€

Q∗(s,a) = E rt +1 + γ max′ a

Q∗(st +1, ′ a ) st = s,at = a{ }

= Ps ′ s a Rs ′ s

a + γ max′ a

Q∗( ′ s , ′ a )[ ]′ s

∑

The relevant backup diagram:

is the unique solution of this system of nonlinear equations.Q*

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Why Optimal State-Value Functions are Useful

Any policy that is greedy with respect to is an optimal policy.

Therefore, given , one-step-ahead search produces the long-term optimal actions.

€

V π (s) = π (s,a) Ps ′ s a Rs ′ s

a + γV π ( ′ s )[ ]′ s

∑a

∑

V∗

V∗

Given , the agent does not even have to do a one-step-ahead search:

π∗(s)=argmaxa∈A(s)

Q∗(s,a)

Q*

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Gridworld example

π*

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What About Optimal Action-Value Functions?

Given , the agent does not evenhave to do a one-step-ahead search:

Q*

π∗(s)=argmaxa∈A(s)

Q∗(s,a)

Computational Modeling LabPolicy Iteration in Dynamic Programming

Policy Evaluation: for a given policy π, compute the state-value function

{ }⎭⎬⎫

⎩⎨⎧

==== ∑∞

=++

01)(

ktkt

ktt ssrEssREsV γ

π

πππ

: policyfor function value-State

[ ]equations linear ussimultaneo of system a S

sVRPassV

V

a s

ass

ass

—

)(),()(

:

∑ ∑′

′′ ′+= ππ

π

γπ

for equation Bellman

Recall:

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Compute iteratively:

Policy Iteration in Dynamic Programming

Policy Evaluation: for a given policy π, compute the state-value function

[ ]∑ ∑′

′′ ′+=a s

ass

ass sVRPassV

V

)(),()(

:ππ

π

γπ

for equation Bellman

Vk+1(s)← π(s,a) Ps ′ s a Rs ′ s

a +γVk( ′ s )[ ]′ s

∑a∑

πV

How to solve?

Dynamic programming operator

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Iterative Policy Evaluation

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Policy Improvement

Suppose we have computed for a deterministic policy π.Vπ

For a given state s, would it be better to do an action ? )(sa π≠

Qπ (s,a) =Eπ rt+1 +γVπ(st+1) st =s,at =a{ }

= Ps ′ s a

′ s ∑ Rs ′ s

a +γVπ ( ′ s )[ ]

:is state in doing of value The sa

)(),( sVasQ

saππ >

ifonly and if state for action to switch to better is It

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Policy Improvement Cont.

′ π (s) =argmaxa

Qπ (s,a)

=argmaxa

Ps ′ s a

′ s ∑ Rs ′ s

a +γVπ ( ′ s )[ ]

:π

πV to respect with

is that policy new a get to states all for this Do

greedy

′

ππ VV ≥′Then

[ ] , all for i.e.,

! get Finally we

)(max)( sVRPsVSs

VVass

s

ass

a′+=∈

=

′′

′′

′

∑ ππ

ππ

γ

policies. optimal are and both and So

Equation. Optimality Bellman the is this But

πππ ′= ∗′ VV

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Policy Iteration in DP

π0 → Vπ0 → π1 → Vπ1 → L π* → V* → π*

policy improvement“greedification”

policy evaluation

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A Small Gridworld example

• An undiscounted episodic task• Nonterminal states: 1, 2, . . ., 14; • One terminal state (shown twice as shaded squares)• Actions that would take agent off the grid leave state unchanged• Reward is –1 until the terminal state is reached• is set to 1

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Iterative Policy Evaluation for the Small Gridworld

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Iterative Policy Evaluation for the Small Gridworld, cont.

Improved policy∞

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Jack’s Car Rental, an example

$10 for each car rented (must be available when request recorded)

Two locations, maximum of 20 cars at eachCars returned and requested randomly

Poisson distribution, n returns/requests with prob 1st location: average requests = 3, average returns

= 3 2nd location: average requests = 4, average returns

= 2Can move up to 5 cars between locations overnight

States, Actions, Rewards?Transition probabilities?

€

λn

n!e−λ

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Jack’s Car Rental, an example (cont.)

€

λn

n!e−λ

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Value Iteration in DP

Vk+1(s)← π(s,a) Ps ′ s a Rs ′ s

a +γVk( ′ s )[ ]′ s

∑a∑

Recall the full policy-evaluation backup:

Vk+1(s)← maxa

Ps ′ s a Rs ′ s

a +γVk( ′ s )[ ]′ s

∑

Here is the full value-iteration backup:

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Value Iteration Cont.

Q(s,a)

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Asynchronous DP

All the DP methods described so far require exhaustive sweeps of the entire state set.

Asynchronous DP does not use sweeps. Instead it works like this:

Repeat until convergence criterion is met:

Pick a state at random and apply the appropriate backup

Still need lots of computation, but does not get locked into hopelessly long sweeps

Can you select states to backup intelligently? YES: an agent’s experience can act as a guide.

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Generalized Policy Iteration

Generalized Policy Iteration (GPI): any interaction of policy evaluation and policy improvement, independent of their granularity.

A geometric metaphor forconvergence of GPI:

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Efficiency of DP

To find an optimal policy is polynomial in the number of states…

BUT, the number of states is often astronomical, e.g., often growing exponentially with the number of state variables (what Bellman called “the curse of dimensionality”).

In practice, classical DP can be applied to problems with a few millions of states.

Asynchronous DP can be applied to larger problems, and appropriate for parallel computation.

It is surprisingly easy to come up with MDPs for which DP methods are not practical.

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Summary

• Policy evaluation: backups without a max

• Policy improvement: form a greedy policy, if only locally

• Policy iteration: alternate the above two processes

• Value iteration: backups with a max• Generalized Policy Iteration (GPI)• Asynchronous DP: a way to avoid

exhaustive sweeps