Factored State Spaces 3/2/178
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1 Factored State Spaces 3/2/178
2 Converting POMDPs to MDPs In a POMDP: Action + observation updates beliefs Value is a function of beliefs. Instead we can view this as an MDP where: There is a state for every possible belief. Beliefs are probabilities, so we have a continuum. There are infinitely many belief-states. Taking an action transitions to another belief-state. Observations are random, so this transition is random.
3 Value Iteration in POMDPs Value iteration in a finite MDP: 1. Initialize each state s value to Compute the greedy policy for each state. 3. Update the value of each state based on this policy. 4. Goto step 2; repeat until converged. In a POMDP, there are infinitely many states. We can t loop through them. Value is a piecewise-linear function of belief. We can do value iteration over a finite set of linear functions. For a description of the algorithms continue reading pomdp.org.
4 POMDP value iteration is impractical In principle, we can iterate over a finite set of linear functions to update values. In practice, this gets out of hand very quickly. This example is from a 2-state, 2-action MDP, computing values at horizon 3.
5 What can we do instead? Approximation Don t find optimal values for every state. Instead solve for pretty-good values for groups of states. Online planning Don t bother devising a complete policy. Instead come up with a pretty-good policy for the short term, and re-plan in the future. Reinforcement learning Combines aspects of approximation and online planning. We ll focus on RL after spring break.
6 What is the running time of Value Iteration on MDPs? repeat until convergence: (update values table)... for each state: (update state s value)... for each action: (find the best action)... for each next state: (update expected value)...
7 <latexit sha1_base64="r0rtd7kbj5xxwjrmrjkcyez+9gi=">aaaccxicbvdnssnagnz4w+tf1kox1sj4kokikigu9ocxgrgfjobnztmu3wzc7kyoiwcvvooxdypefqnvvo2bngdthvgyzubj22+clfgplovbmjtfwfxarq3uv9fwnzbnre07mwqcewcnlbhdaenckceoooqrbioiigngoshwsvq7d0rimvbbnuqjf6m+pxhfsgnjn/fcvcshn9mlu7jphwv0my4reovt6iqs+mbdalpjwfliv6qbkrr988sne5zfhcvmkjq920qvlyohkgakqluzjcncq9qnpu05ion08vepbtzqsgijrojhfryrvydyfes5igodjjeaygmvfp/zepmktr2c8jrthopjoihjucww7awgvbcs2egthaxvf4v4gatcsrdx1yxy0yfpeueoeda0b44brfoqjrrybfvgenjgbltanwgdb2dwcj7bk3gznowx4934metnjgpmb/yb8fkdboiarw==</latexit> <latexit sha1_base64="r0rtd7kbj5xxwjrmrjkcyez+9gi=">aaaccxicbvdnssnagnz4w+tf1kox1sj4kokikigu9ocxgrgfjobnztmu3wzc7kyoiwcvvooxdypefqnvvo2bngdthvgyzubj22+clfgplovbmjtfwfxarq3uv9fwnzbnre07mwqcewcnlbhdaenckceoooqrbioiigngoshwsvq7d0rimvbbnuqjf6m+pxhfsgnjn/fcvcshn9mlu7jphwv0my4reovt6iqs+mbdalpjwfliv6qbkrr988sne5zfhcvmkjq920qvlyohkgakqluzjcncq9qnpu05ion08vepbtzqsgijrojhfryrvydyfes5igodjjeaygmvfp/zepmktr2c8jrthopjoihjucww7awgvbcs2egthaxvf4v4gatcsrdx1yxy0yfpeueoeda0b44brfoqjrrybfvgenjgbltanwgdb2dwcj7bk3gznowx4934metnjgpmb/yb8fkdboiarw==</latexit> <latexit sha1_base64="r0rtd7kbj5xxwjrmrjkcyez+9gi=">aaaccxicbvdnssnagnz4w+tf1kox1sj4kokikigu9ocxgrgfjobnztmu3wzc7kyoiwcvvooxdypefqnvvo2bngdthvgyzubj22+clfgplovbmjtfwfxarq3uv9fwnzbnre07mwqcewcnlbhdaenckceoooqrbioiigngoshwsvq7d0rimvbbnuqjf6m+pxhfsgnjn/fcvcshn9mlu7jphwv0my4reovt6iqs+mbdalpjwfliv6qbkrr988sne5zfhcvmkjq920qvlyohkgakqluzjcncq9qnpu05ion08vepbtzqsgijrojhfryrvydyfes5igodjjeaygmvfp/zepmktr2c8jrthopjoihjucww7awgvbcs2egthaxvf4v4gatcsrdx1yxy0yfpeueoeda0b44brfoqjrrybfvgenjgbltanwgdb2dwcj7bk3gznowx4934metnjgpmb/yb8fkdboiarw==</latexit> <latexit sha1_base64="r0rtd7kbj5xxwjrmrjkcyez+9gi=">aaaccxicbvdnssnagnz4w+tf1kox1sj4kokikigu9ocxgrgfjobnztmu3wzc7kyoiwcvvooxdypefqnvvo2bngdthvgyzubj22+clfgplovbmjtfwfxarq3uv9fwnzbnre07mwqcewcnlbhdaenckceoooqrbioiigngoshwsvq7d0rimvbbnuqjf6m+pxhfsgnjn/fcvcshn9mlu7jphwv0my4reovt6iqs+mbdalpjwfliv6qbkrr988sne5zfhcvmkjq920qvlyohkgakqluzjcncq9qnpu05ion08vepbtzqsgijrojhfryrvydyfes5igodjjeaygmvfp/zepmktr2c8jrthopjoihjucww7awgvbcs2egthaxvf4v4gatcsrdx1yxy0yfpeueoeda0b44brfoqjrrybfvgenjgbltanwgdb2dwcj7bk3gznowx4934metnjgpmb/yb8fkdboiarw==</latexit> Running time of Value Iteration Value iteration isn t fast, but if the state space is small, it s manageable. The problem is state spaces can be exponentially larger than they seem. If the agent s state has k variables with domains ky D 1,, D k to keep track of, there can be D i distinct states. i=1 Even in the best case (binary variables), this is expontntial (2 k states).
8 How big is the state space? In gridworld, we can track the state space with two variables row and column. This is quite manageable. How big is the state space for realistic MDPs? Search and rescue robot Mars rover What variables do these agents need to keep track of?
9 Mars rover state space Some possible state space variables: position: x,y,z (large domains) environmental factors sunlight wind temperature robot controls battery level arm position solar panel angle drill operation mission objectives current experiment samples being analyzed etc. This state space is gigantic!
10 MDPs vs. State Space Search In state space search, we might also have state variables and factored state spaces. Why didn t we worry about the fact that the state space could be exponentially large then?
11 Handling huge state spaces Full value iteration is implausible. We could try to reason about the factored state space. Plan in terms of variables instead of states. One could spend an entire month on this topic (we won t). Approximation Online planning Reinforcement learning
12 Online Planning Key idea: plan for one state at a time. We may be able to come up with a good action for the current state without solving the entire MDP. We can plan ahead some, but we know that we re going to stop and think about plans again later. Where have we seen something like this before?
13 Thinking about online planning. How can we use ideas we ve already seen to help with online planning? Heuristics? Iterative deepening? Monte Carlo simulations? Other ideas?
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