dayan1993improving: Improving Generalization for Temporal Difference Learning: The Successor Representation
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source :
- authors
- Dayan, P.
- year
- 1993
This paper discusses temporal-dependent representations for Reinforcement Learning. This is for the discrete state and action case.
The idea is to construct a matrix \(\Xmat\) where each row corresponds to a vector of the effective likelihoods of transitioning to a state (column) starting in a state (row). With \(\eye\) as the identity and \(\Qmat\) as the markov transition matrix
\[ \Xmat_{ij} = [\eye]_{ij} + [\Qmat]_{ij} + [\Qmat^2]_{ij} + \ldots = [(\eye - \Qmat)^\inv]_{ij} \]
This representation can be learned online through a TD prediction algorithm. Finally, when learning the vector of value functions we can easily see the optimal weights for the true value functions
\[\vvec = \Xmat \wvec^*\]
where
\[\wvec^{*}_i = \sum_{k \in T} s_{ik} \bar{r}_k\]
where \(T\) is the set of absorbing states, \(s_{ik}\) is the probability of transitioning from state \(i\) to state \(k\), and \(\bar{r}_k\) is the expected reward at the absorbing state \(k\).
They then go through and show this working well on a navigation tasks as compared to older representational schemes.
Quotes
As briefly reviewed in the next section, TD methods apply to learning framework, which specifies the goal for learning and precisely how the system fails to attain this goal in particular circumstances.
… neighborliness is defined in terms of temporal succession. If the transition matrix of the chain is initially unknown, this representation will have to be learned directly through experience.