R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1 Chapter 4: Dynamic Programming Overview of a collection of classical solution methods for MDPs known as dynamic programming (DP) Show how DP can be used to compute value functions, and hence, optimal policies Discuss efficiency and utility of DP jectives of this chapter:
Chapter 4: Dynamic Programming. Overview of a collection of classical solution methods for MDPs known as dynamic programming (DP) Show how DP can be used to compute value functions , and hence, optimal policies Discuss efficiency and utility of DP. Objectives of this chapter:. - PowerPoint PPT Presentation
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R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 1
Chapter 4: Dynamic Programming
Overview of a collection of classical solution methods for MDPs known as dynamic programming (DP)
Show how DP can be used to compute value functions, and hence, optimal policies
Discuss efficiency and utility of DP
Objectives of this chapter:
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 2
Policy Evaluation
State - value function for policy :
V (s) E Rt st s E krt k 1 st sk 0
Bellman equation for V :
V (s) (s,a) Ps s a Rs s
a V ( s ) s
a
— a system of S simultaneous linear equations
Policy Evaluation: for a given policy , compute the state-value function V
Recall:
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 3
Iterative Methods
V0 V1 Vk Vk1 V
Vk1 (s) (s,a) Ps s a Rs s
a Vk ( s ) s
a
a “sweep”
A sweep consists of applying a backup operation to each state.
A full policy evaluation backup:
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 4
Iterative Policy Evaluation
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 5
A Small Gridworld
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
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 6
Iterative Policy Eval for the Small Gridworld
random (uniform) action choices
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 7
Policy Improvement
Suppose we have computed for a deterministic policy .V
For a given state s, would it be better to do an action ? a (s)
Q (s,a) E rt1 V (st 1 ) st s, at a Ps s
a
s Rs s
a V ( s )
The value of doing a in state s is :
It is better to switch to action a for state s if and only if
Q (s,a) V (s)
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 8
Policy Improvement Cont.
(s) argmaxa
Q (s,a)
argmaxa
Ps s a
s R
s s a V ( s )
Do this for all states to get a new policy that is
greedy with respect to V :
Then V V
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 9
Policy Improvement Cont.
What if V V ?
i.e., for all s S, V (s) maxa
Ps s a
s R
s s a V ( s ) ?
But this is the Bellman Optimality Equation.
So V V and both and are optimal policies.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 10
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 11
Policy Iteration
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 12
Jack owns two locations for car rental.
Customers come to both locations at random with poison distribution
Suppose that =3 and 4 for rent and 3 and 2 for return in two locations
He gets $10 if he rents a car.
Moving the car from one location to the second one costs $2
Maximum 20 cars at each location
Maximum 5 cars can be moved overnight
Use MDP with
States are number of cars
Actions are number of cars moved each night
Starting policy never moves any car
Jack’s Cars Rental
en
Xn
!
9.0
0
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 13
Jack’s Cars Rental
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 14
Value Iteration
Vk1 (s) (s,a) Ps s a Rs s
a Vk ( s ) s
a
Recall the full policy evaluation backup:
Vk1 (s) maxa
Ps s a Rs s
a Vk ( s ) s
Here is the full value iteration backup:
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 15
Value Iteration Cont.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 16
Gambler’s Problem
A gambler plys coins if it is heads he wins
As much as he gambled.
The game ends when he gets $100 or loses
His money
The state is his capital
And actions are his bets
The optimum policy maximizes the
probability of reaching the goal.
If the probability of coins coming
up heads is p=0.4 then the optimum
policy is as shown
}99,...,2,1{s
)}100,min(,...,2,1{ ssa
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 17
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.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 18
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:
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 19
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.
R. S. Sutton and A. G. Barto: Reinforcement Learning: An Introduction 20
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 Full backups (to be contrasted later with sample backups) Generalized Policy Iteration (GPI) Asynchronous DP: a way to avoid exhaustive sweeps Bootstrapping: updating estimates based on other