Successor Representation

Dayan (1992) proposed an alternative formulation of the state value function \(V^{\pi}(s)\) that enables rapidly adapting if reward probabilities or transition probabilities change. The idea is to separate the state value function into a product of rewards and future discounted state occupancies, which Dayan termed the successor representation (SR) and which I denote \(M\). Then, if either the rewards or the transitions change, recomputing \(V^{\pi}(s)\) is simple.

The successor representation \(M\) can also be learnt using temporal difference-like algorithms.

Definition

Let \(S\) denote the state space, let \(R \in \mathbb{R}^{|S|}\) be a vector denoting the reward in each state i.e. \(R_s = R(s)\) and let \(T^{\pi} \in \mathbb{R}^{|S| \times |S|}\) be a matrix denoting the one-step transition matrix under the policy \(\pi\).

Recall that the state-value function is defined as the expected sum of future rewards:

\[V^{\pi}(s_t) := \mathbb{E}_{\pi}[\sum_{k=0}^{k=\infty} \gamma^{k-1} r(s_{t + k}) \lvert S_t = s_t]\]

If the state space \(\mathcal{X}\) is discrete and the rewards are deterministic, then the state value function can be written as:

\[V^{\pi}(s_t) = r^T \mathbb{E}_{\pi}[\sum_{k=0} (\gamma T^{\pi})^{k}] s\]

where \(r \in \mathbb{R}^{\lvert S \lvert}\) is a vector of the reward in each state. That matrix, the future sum of discounted state occupancies, is called the success representation. It is a Neumann series and can therefore be expressed as:

\[S^{\pi} := \sum_{k=0}^{\infty} (\gamma T^{\pi})^k = (I - \gamma T^{\pi})^{-1}\]

Learning the Successor Representation

One can learn the successor representation from samples with TD-learning. Let \(M^{\pi}_{(n)}\) denote the SR after the \(n\)th update and let \(\eta\) denote the learning rate. The Bellman update is:

\[M_{(n+1)}^{\pi} \leftarrow M_{(n)}^{\pi} + \eta (\mathbb{I}() + \gamma M_{(n)}^{\pi} - M_{(n)}^{\pi})\]