1. Partially Observable Markov Decision Process (POMDP)
by Ye Fang
Department of Computer Science
Rice University
Douglas Aberdeen, National ICT Australia 2003

2. Overview Recap POMDP and the exact solution Heuristic methods Solution
Heuristics for Exact methods
Grid methods
Factored belief states
Simulation
Methods for continuous state and action space
Solution

3. Learning with a Model The agent knows the model , ,
Observation/action history:
Belief state
1/3
1/3
1/3
Goal
1/2
1/2 1

4. Learning with a Model Update beliefs:
Long-term value of a belief state
Define:

5. Complexity of Exact Methods
Exponential number of state variables:
Updating believe state is expensive.
Believe-state monitoring is hard.
Exponential number of belief states:
PSPACE-Hard for simplified finite-horizon POMDP.
NP-Hard to find a policy.

6. How to make POMDP feasible?
Almost impossible to find a exact solution for POMDP model
Where does the complexity of exact solution come from?
Infinite believe states
Updating believe states and their value functions
Introduce heuristic methods for exact methods

7. How to make POMDP feasible?
Why can Heuristics work?
Simplify the representation of value function by assuming the system is an MDP.
Replace the believe state b with real world state
Then, we have finite many states

8. Heuristic for Exact Methods
The intuition behind these heuristics is to assume the system as an MDP by finding an approximate projection from belief state to world state.
In exact solution,
(Instead of using all possible world state in a belief state b, and their corresponding possible transitions, heuristics are used for exact solution.)
Different approximation method needs 1) belief state  state in MDP 2) update scheme 3) appropriate value function

9. Heuristic for Exact Methods
Goal:
Find an good approximation of projection from belief state to world state.
Find a good policy for each believe state.

10. Heuristic for Exact Methods
MSL(most likely state)
Voting Heuristics
QMDP Heuristic
Heuristic using the uncertainty of belief state

11. MLS Heuristic
We can assume the system is in the most likely world state(MLS) i at time t. The policy executed at that state is the transition with largest Q-value at state i.
Therefore, the state in the model is world state which is the same as MDP model. It ignores the agents confusion about the current world state it in. Then we can use the MDP method learned before.

12. MLS Heuristic
This method neglects all possible world states but the MSL state at belief state b.
EX: Given optimal action in a world with three states and two actions,
u(s0) = a0, u(s1) = a0, u(s2) = a1
b = [0.3, 0.3, 0.4]
The probability of not being at state s2 is 0.4 and the best action a0

13. Voting Heuristic
The voting heuristic assigns a probability distribution over the actions instead of over the states.
Given:
The action for each belief state:
(Probability at state j) * (best action at that state)

14. Voting Heuristic
EX: Given optimal action in a world with three states and two actions,
u(s0) = a0, u(s1) = a0, u(s2) = a1
b = [0.3, 0.3, 0.4],
V(s0, a0)=5, V(s0, a1)=4,
V(s1,a0)=5, V(s1,a1)=4,
V(s2,a0)=0, V(s2,a1)= 10.
At state s2,
expectedR(a0) =3, expectedR(a1) =6.4
The most likely best action is a0 (0.6). A1 maybe better given the reward

15. Voting Heuristic
This method does not take the reward of an action into account.
Introduce QMDP, which emphasize the Q- function of the optimal policy rather than the policy itself.

16. QMDP Heuristic
QMDP only takes into account the belief state at first step.
What if this action does not do much to disambiguate the state, this method cannot improve the action over time.
Find the action with largest Q-value.
This method takes account the belief state for one step and assume the state is entirely known.

17. Shortcomings of the Heuristics
What if the belief states is close to uniform?
Ex: a robot trying to reach the other end of a futureless desert. By observation, it has almost same belief of it is at everywhere.
What if there is a lot of uncertainty in the information state?
Consider the uncertainty when taking action

18. Formal measurement of Uncertainty
Entropy is the measure of a probability distribution that reflects how spiked or spread out the probability mass is, essentially capturing the amount of uncertainty with a single number.
f(.) is a discrete probability mass function.

19. Two Objectives When choosing actions, we want to:
To take actions that will yield the highest rewards.
To reduce the entropy of information state.

20. Weighted Entropy Control
Intuition: relate the entropy to the rewards to give some rough measure of the value of information.

21. Weighted Entropy Control
When the entropy is near 1, it means the environment is totally unobservable.
When the entropy near 0, it means the model is almost a MDP.

22. Weighted Entropy Control
Define VL to be the lower bound for POMDP value function.
The value at each belief state is:
The control strategy will be:
Vco means completely observable. VL is the lower bound for any state in POMDP.

23. Other Heuristics for POMDP
Grid method
Factored belief method
Simulation

24. Grid method
Instead of compose the world state from belief state, it picks the real world states.
How to choose the set of real world states (a interesting region of each belief state)?
How to interpolate?

25. How to choose grid points?
Simulation to find useful points
Adding points where the value differed a lot though with similar observation.

26. How to interpolate? Maintain the convex nature of the value function:
f(g,u) is the value grid point g under action u.
Example: nearest neighbors, linear interpolations, etc.
MSL heuristic can be thought of as a grid method with points at the belief-simplex corners and a simple 1 NN interpolation
A value function over a continuous belief space can be approximated by a finite set of grid points G and an interpolation-extrapolation rule that estimates the value of an arbiturary point of the belief space by relying on the points of the grid and their associate values.

27. Factored Belief State
Intuition: learn the dependency of state variables
Ex: at time t: the state of raining is true
at time t+1: the state of “ground is dry” is not very likely to be true.
Two-slice temporal Bayes network showing dependencies between state variables over successive time steps.

28. Factored Belief State
We can use a subset of state variables to construct a Bayes network(BN).
Belief-state projection can be searched to find a suitable BN for a specific problem（belief monitoring）. It is a learning of adjusting the belief network parametrized by ϕ.
Factored linear value function: weighted linear combinations of polynomial basis functions.
Search procedure is provided to determine belief-state projection with bounded errors.

29. Simulation and Belief State
Concentrate learning effort on the states that are most likely to be encountered.
In terms of Q-learning, we can simulate a path in POMDP and perform iteration of the value function on the monitored current belief states.
Not good for POMDP with more than hundreds of states.
The full DP update is not efficient.

30. Simulation and Belief State
Learn Q-function that generalize to all belief states
Artificial neural network can also be used to approximate the value function of the full belief states.

31. Continuous State and Action Spaces
Sampled belief states.
Use particle filters to update the belief state.
The value function is approximated using the average of k nearest neighbors.
Filtering refers to determine the distribution of some variable at a certain time given the observation up to that time.
Approximate continuous belief states are formed from the sampling using Gaussian Kernels.
1.Infinite memory is needed to represent continuous state space
2. Balance running time and accuracy with the number of samples n.
3. Hyperline methods cannot be used because the state space is continuous.

32. Policy search Vs Value search
It is simpler to just determine how to act instead of the value of acting.
Approximate value function method usually produce deterministic policies.
The heuristic methods are approximated projections from belief states to world states.
It is better to introduce randomness in policy.

33. Policy search Vs Value search
Policy search can be very difficult.
Value search can be better for small POMDP .
Value search imposes Bellman equations as constrains.
Hybrid

34. Policy search Policy search can be implemented by policy iteration.
Step1: Evaluate current policy
Step2: Improve policy

35. Recap Different Heuristics Projecting a belief state to a world state
Evaluating the values for belief states
Finding good policy