## Two Abstractions

- “temporal abstractions”: making decisions without consideration / abstracting away time (MDP)
- “state abstractions”: making decisions about groups of states at once

## Graph

MaxQ formulates a policy as a graph, which formulates a set of \(n\) policies

### Max Node

This is a “policy node”, connected to a series of \(Q\) nodes from which it takes the max and propegate down. If we are at a leaf max-node, the actual action is taken and control is passed back t to the top of the graph

### Q Node

each node computes \(Q(S,A)\) for a value at that action

## Hierachical Value Function

\begin{equation} Q(s,a) = V_{a}(s) + C_{i}(s,a) \end{equation}

the value function of the root node is the value obtained over all nodes in the graph

where:

\begin{equation} C_{i}(s,a) = \sum_{s’}^{} P(s’|s,a) V(s’) \end{equation}

## Learning MaxQ

- maintain two tables \(C_{i}\) and \(\tilde{C}_{i}(s,a)\) (which is a special completion function which corresponds to a special reward \(\tilde{R}\) which prevents the model from doing egregious ending actions)
- choose \(a\) according to exploration strategy
- execute \(a\), observe \(s’\), and compute \(R(s’|s,a)\)

Then, update: