Markov Decision Process (MDP)

S: A set of states A: A set of actions Pr(s’|s,a): transition model

• (aka Mas,s’)

C(s,a,s’): cost model G: set of goals s0: start state : discount factor R(s,a,s’): reward model

Value function: expected long term reward from the state

Q values: Expected long term reward of doing a in s V(s) = max Q(s,a)

Greedy Policy w.r.t. a value function

Value of a policy

Optimal value function

Examples of MDPs

Goal-directed, Indefinite Horizon, Cost Minimization MDP• <S, A, Pr, C, G, s0>• Most often studied in planning community

Infinite Horizon, Discounted Reward Maximization MDP• <S, A, Pr, R, >• Most often studied in reinforcement learning

Goal-directed, Finite Horizon, Prob. Maximization MDP• <S, A, Pr, G, s0, T>• Also studied in planning community

Oversubscription Planning: Non absorbing goals, Reward Max. MDP• <S, A, Pr, G, R, s0>• Relatively recent model

SSPP—Stochastic Shortest Path Problem An MDP with Init and Goal states

• MDPs don’t have a notion of an “initial” and “goal” state. (Process orientation instead of “task” orientation)

– Goals are sort of modeled by reward functions

• Allows pretty expressive goals (in theory)

– Normal MDP algorithms don’t use initial state information (since policy is supposed to cover the entire search space anyway).

• Could consider “envelope extension” methods

– Compute a “deterministic” plan (which gives the policy for some of the states; Extend the policy to other states that are likely to happen during execution

– RTDP methods

• SSSP are a special case of MDPs where

– (a) initial state is given– (b) there are absorbing goal states– (c) Actions have costs. All states

have zero rewards• A proper policy for SSSP is a policy

which is guaranteed to ultimately put the agent in one of the absorbing states

• For SSSP, it would be worth finding a partial policy that only covers the “relevant” states (states that are reachable from init and goal states on any optimal policy)

– Value/Policy Iteration don’t consider the notion of relevance

– Consider “heuristic state search” algorithms

• Heuristic can be seen as the “estimate” of the value of a state.

<S, A, Pr, C, G, s0> Define J*(s) {optimal cost} as the

minimum expected cost to reach a goal from this state.

J* should satisfy the following equation:

Bellman Equations for Cost Minimization MDP(absorbing goals)[also called Stochastic Shortest Path]

Q*(s,a)

<S, A, Pr, R, s0, > Define V*(s) {optimal value} as the

maximum expected discounted reward from this state.

V* should satisfy the following equation:

Bellman Equations for infinite horizon discounted reward maximization MDP

<S, A, Pr, G, s0, T> Define P*(s,t) {optimal prob.} as the

maximum probability of reaching a goal from this state at tth timestep.

P* should satisfy the following equation:

Bellman Equations for probability maximization MDP

Modeling Softgoal problems as deterministic MDPs

• Consider the net-benefit problem, where actions have costs, and goals have utilities, and we want a plan with the highest net benefit

• How do we model this as MDP?– (wrong idea): Make every state in which any subset of goals

hold into a sink state with reward equal to the cumulative sum of utilities of the goals.

• Problem—what if achieving g1 g2 will necessarily lead you through a state where g1 is already true?

– (correct version): Make a new fluent called “done” dummy action called Done-Deal. It is applicable in any state and asserts the fluent “done”. All “done” states are sink states. Their reward is equal to sum of rewards of the individual states.

An eye for an eye only ends up making the whole world blind. -Mohandas Karamchand Gandhi, born October 2nd, 1869.

Lecture of October 2nd, 2009

Ideas for Efficient Algorithms..• Use heuristic search

(and reachability information)– LAO*, RTDP

• Use execution and/or Simulation– “Actual Execution”

Reinforcement learning (Main motivation for RL is to

“learn” the model)– “Simulation” –simulate the

given model to sample possible futures

• Policy rollout, hindsight optimization etc.

• Use “factored” representations– Factored representations

for Actions, Reward Functions, Values and Policies

– Directly manipulating factored representations during the Bellman update

Heuristic Search vs. Dynamic Programming (Value/Policy Iteration)• VI and PI approaches use

Dynamic Programming Update• Set the value of a state in

terms of the maximum expected value achievable by doing actions from that state.

• They do the update for every state in the state space– Wasteful if we know the initial

state(s) that the agent is starting from

• Heuristic search (e.g. A*/AO*) explores only the part of the state space that is actually reachable from the initial state

• Even within the reachable space, heuristic search can avoid visiting many of the states. – Depending on the quality of

the heuristic used.. • But what is the heuristic?

– An admissible heuristic is a lowerbound on the cost to reach goal from any given state

– It is a lowerbound on V*!

Real Time Dynamic Programming [Barto, Bradtke, Singh’95]

Trial: simulate greedy policy starting from start state;

perform Bellman backup on visited states

RTDP: repeat Trials until cost function converges

RTDP was originally introduced for Reinforcement Learning For RL, instead of “simulate” you “execute” You also have to do “exploration” in addition to “exploitation”

with probability p, follow the greedy policy with 1-p pick a random action

What if we simulate the action’s effect with noise (rather than exactly wrt its transition probabilities)

Min

?

?s0

Jn

Jn

Jn

Jn

Jn

Jn

Jn

Qn+1(s0,a)

Jn+1(s0)

agreedy = a2

Goala1

a2

a3

RTDP Trial

?

Note that the value function is being updated per each level. How about waiting until you hit goal and then update everyone?

Greedy “On-Policy” RTDP without execution

Using the current utility values, select the action with the highest expected utility (greedy action) at each state, until you reach a terminating state. Update the values along this path. Loop back—until the values stabilize

Labeled RTDP [Bonet&Geffner’03]

Initialise J0 with an admissible heuristic

• ⇒ Jn monotonically increases

Label a state as solved • if the Jn for that state has converged

Backpropagate ‘solved’ labeling Stop trials when they reach any solved state Terminate when s0 is solved

s G

high Q

costs

best action

) J(s) won’t change!

s G?

t

both s and tget solved together

high Q

costs

Properties

admissible J0 ⇒ optimal J*

heuristic-guided• explores a subset of reachable state space

anytime • focusses attention on more probable states

fast convergence • focusses attention on unconverged states

terminates in finite time

Other Advances

Ordering the Bellman backups to maximise information flow.• [Wingate & Seppi’05]• [Dai & Hansen’07]

Partition the state space and combine value iterations from different partitions.• [Wingate & Seppi’05]• [Dai & Goldsmith’07]

External memory version of value iteration• [Edelkamp, Jabbar & Bonet’07]

…