Value and Planning in MDPs

Administrivia•Reading 3 assigned today

•Mahdevan, S., “Representation Policy Iteration”. In Proc. of 21st Conference on Uncertainty in Artificial Intelligence (UAI-2005).

•http://www.cs.umass.edu/~mahadeva/papers/uai-final-paper.pdf

•Due: Apr 20

•Groups assigned this time

Where we are•Last time:

•Expected value of policies

•Principle of maximum expected utility

•The Bellman equation

•Today:

•A little intuition (pictures)

•Finding π*: the policy iteration algorithm

•The Q function

•On to actual learning (maybe?)

The Bellman equation•The final recursive equation is known as the Bellman equation:

•Unique soln to this eqn gives value of a fixed policy π when operating in a known MDP M= 〈 S,A,T,R 〈

•When state/action spaces are discrete, can think of V and R as vectors and Tπ as matrix, and get matrix eqn:

Exercise•Solve the matrix Bellman equation (i.e., find V):

•I formulated the Bellman equations for “state-based” rewards: R(s)

•Formulate & solve the B.E. for:

•“state-action” rewards (R(s,a))

•“state-action-state” rewards (R(s,a,s’))

Exercise•Solve the matrix Bellman equation (i.e., find V):

•Formulate & solve the B.E. for:

•“state-action” rewards (R(s,a))

•“state-action-state” rewards (R(s,a,s’))

Policy values in practice“Robot” navigation in a grid maze

Goalstate

The MDP formulation•State space:

•Action space:

•Reward function:

•Transition function: ...

The MDP formulation•Transition function:

•If desired direction is unblocked

•Move in desired direction with probability 0.7

•Stay in same place w/ prob 0.1

•Move “forward right” w/ prob 0.1

•Move “forward left” w/ prob 0.1

•If desired direction is blocked (wall)

•Stay in same place w/ prob 1.0

Policy values in practice

Optimal policy, π*

EAST

SOUTH

WEST

NORTH

Policy values in practice

Value function for optimal policy, V*

Why does it look like this?

A harder “maze”...

Walls

Doors

A harder “maze”...Optimal policy, π*

A harder “maze”...Value function for optimal policy, V*

A harder “maze”...Value function for optimal policy, V*

Still more complex...

Still more complex...Optimal policy, π*

Still more complex...Value function for optimal policy, V*

Still more complex...Value function for optimal policy, V*

Planning: finding π*•So we know how to evaluate a single policy, π

•How do you find the best policy?

•Remember: still assuming that we know M= 〈 S,A,T,R 〈

Planning: finding π*•So we know how to evaluate a single policy, π

•How do you find the best policy?

•Remember: still assuming that we know M= 〈 S,A,T,R 〈

•Non-solution: iterate through all possible π, evaluating each one; keep best

Policy iteration & friends•Many different solutions available.

•All exploit some characteristics of MDPs:

•For infinite-horizon discounted reward in a discrete, finite MDP, there exists at least one optimal, stationary policy (may exist more than one equivalent policy)

•The Bellman equation expresses recursive structure of an optimal policy

•Leads to a series of closely related policy solutions: policy iteration, value iteration, generalized policy iteration, etc.

The policy iteration alg.Function: policy_iteration

Input: MDP M= 〈 S,A,T,R 〈 discount γ

Output: optimal policy π*; opt. value func. V*Initialization: choose π

0 arbitrarily

Repeat {Vi=eval_policy(M,π

i,γ) // from Bellman eqn

πi+1=local_update_policy(π

i,V

i)

} Until (πi+1==π

i)

Function: π’=local_update_policy(π,V)for i=1..|S| {π’(s

i)=argmax

a∈A( sum

j(T(s

i,a,s

j)*V(s

j)) )

}

Why does this work?•2 explanations:

•Theoretical:

•The local update w.r.t. the policy value is a contractive mapping, ergo a fixed point exists and will be reached

•See, “contraction mapping”, “Banach fixed-point theorem”, etc.•http://math.arizona.edu/~restrepo/475A/Notes/sourcea/node22.html

•http://planetmath.org/encyclopedia/BanachFixedPointTheorem.html

•Contracts w.r.t. the Bellman Error:

Why does this work?•The intuitive explanation

•It’s doing a dynamic-programming “backup” of reward from reward “sources”

•At every step, the policy is locally updated to take advantage of new information about reward that is propagated back by the evaluation step

•Value “propagates away” from sources and the policy is able to say “hey! there’s reward over there! I can get some of that action if I change a bit!”

P.I. in actionPolicy Value

Iteration 0

P.I. in actionPolicy Value

Iteration 1

P.I. in actionPolicy Value

Iteration 2

P.I. in actionPolicy Value

Iteration 3

P.I. in actionPolicy Value

Iteration 4

P.I. in actionPolicy Value

Iteration 5

P.I. in actionPolicy Value

Iteration 6: done

Properties•Policy iteration

•Known to converge (provable)

•Observed to converge exponentially quickly

•# iterations is O(ln(|S|))

•Empirical observation; strongly believed but no proof (yet)

•O(|S|3) time per iteration (policy

evaluation)

Variants

•Other methods possible

•Linear program (poly time soln exists)

•Value iteration

•Generalized policy iter. (often best in practice)