| Copyright | Patrick Steele 2015 |
|---|---|
| License | MIT (see the LICENSE file) |
| Maintainer | prs233@cornell.edu |
| Safe Haskell | None |
| Language | Haskell98 |
Algorithms.MDP
Description
Algorithms and data structures for expressing and solving Markov decision processes (MDPs).
See the following for references on the algorithms implemented, along with general terminology.
- "Dynamic Programmand and Optimal Control, Vol. II", by Dimitri P. Bertsekas, Athena Scientific, Belmont, Massachusetts.
- "Stochastic Dynamic Programming and the Control of Queueing Systems", by Linn I. Sennott, A Wiley- Interscience Publication, New York.
The module Algorithms.MDP.Examples contains implementations of several example problems from these texts.
To actually solve an MDP, use (for example) the
valueIteration function from the
Algorithms.MDP.ValueIteration module.
- data MDP a b t = MDP {}
- mkDiscountedMDP :: Eq b => [a] -> [b] -> Transitions a b t -> Costs a b t -> ActionSet a b -> t -> MDP a b t
- mkUndiscountedMDP :: (Eq b, Num t) => [a] -> [b] -> Transitions a b t -> Costs a b t -> ActionSet a b -> MDP a b t
- type Transitions a b t = b -> a -> a -> t
- type Costs a b t = b -> a -> t
- type ActionSet a b = a -> [b]
- type CF a b t = Vector (a, b, t)
- data CFBounds a b t = CFBounds {}
- cost :: Eq a => a -> CF a b t -> t
- action :: Eq a => a -> CF a b t -> b
- optimalityGap :: Num t => CFBounds a b t -> t
- verifyStochastic :: (Ord t, Num t) => MDP a b t -> t -> Either (MDPError a b t) ()
- data MDPError a b t = MDPError {
- _negativeProbability :: [(b, a, a, t)]
- _notOneProbability :: [(b, a, t)]
Markov decision processes
A Markov decision process.
An MDP consists of a state space, an action space, state- and action-dependent costs, and state- and action-dependent transition probabilities. The goal is to compute a policy -- a mapping from states to actions -- which minimizes the total discounted cost of the problem, assuming a given discount factor in the range (0, 1].
Here the type variable a represents the type of the states, b
represents the type of the actions, and t represents the numeric
type used in computations. Generally choosing t to be a Double is
fine, although there is no reason a higher-precision type cannot be
used.
This type should not be constructed directly; use the
mkDiscountedMDP or mkUndiscountedMDP constructors instead.
Arguments
| :: Eq b | |
| => [a] | The state space |
| -> [b] | The action space |
| -> Transitions a b t | The transition probabilities |
| -> Costs a b t | The action-dependent costs |
| -> ActionSet a b | The state-dependent actions |
| -> t | The discount factor |
| -> MDP a b t | The resulting DiscountedMDP |
Creates a discounted MDP.
Arguments
| :: (Eq b, Num t) | |
| => [a] | The state space |
| -> [b] | The action space |
| -> Transitions a b t | The transition probabilities |
| -> Costs a b t | The action-dependent costs |
| -> ActionSet a b | The state-dependent actions |
| -> MDP a b t | The resulting DiscountedMDP |
Creates an undiscounted MDP.
Types
type Transitions a b t = b -> a -> a -> t Source
A type representing an action- and state-dependent probablity vector.
type CF a b t = Vector (a, b, t) Source
A cost function is a vector containing (state, action, cost) triples. Each triple describes the cost of taking the action in that state.
A cost function with error bounds. The cost in a (state, action, cost) triple is guaranteed to be in the range [cost + lb, cost + ub]
Utility functions
cost :: Eq a => a -> CF a b t -> t Source
Get the cost associated with a state.
This function is only defined over the state values passed in to the original MDP.
action :: Eq a => a -> CF a b t -> b Source
Get the action associated with a state.
This function is only defined over the state values passed in to the original MDP.
optimalityGap :: Num t => CFBounds a b t -> t Source
Compute the optimality gap associated with a CFBounds.
This error is absolute, not relative.
Validation
verifyStochastic :: (Ord t, Num t) => MDP a b t -> t -> Either (MDPError a b t) () Source
Returns the non-stochastic (action, state) pairs in an MDP.
An (action, state) pair is not stochastic if any transitions out of the state occur with negative probability, or if the total probability all possible transitions is not 1 (within the given tolerance).
Verifies that the MDP is stochastic.
An MDP is stochastic if all transition probabilities are non-negative, and the total sum of transitions out of a state under a legal action sum to one.
We verify sums to within the given tolerance.
An error describing the ways an MDP can be poorly-defined.
An MDP can be poorly defined by having negative transition probabilities, or having the total probability associated with a state and action exceeding one.
Constructors
| MDPError | |
Fields
| |