Recall, from conditional plan evaluation, we had that:

\begin{equation} U^{\pi}(b) = \sum_{s}^{} b(s) U^{\pi}(s) \end{equation}

let’s write it as:

\begin{equation} U^{\pi}(b) = \sum_{s}^{} b(s) U^{\pi}(s) = {\alpha_{\pi}}^{\top} b \end{equation}

where \(\U_{\pi}(s)\) is the conditional plan evaluation starting at each of the initial states.

\begin{equation} \alpha_{\pi} = \qty[ U^{\pi}(s_1), U^{\pi}(s_2) ] \end{equation}

You will notice, then the utility of \(b\) is linear on \(b\) for different policies \(\alpha_{\pi}\):

At every belief \(b\), there is a policy which has the highest \(U(b)\) at that \(b\) given be the alpha vector formulation.

## Additional Information

### top action

you can just represent a policy out of alpha vectors by taking the top (root) action of the conditional plan with the alpha vector on top.

### optimal value function for POMDP with alpha vector

Recall:

\begin{equation} U^{*}(b) = \max_{\pi} U^{\pi}(b) = \max_{\pi} \alpha_{\pi}^{\top}b \end{equation}

NOTE! This function (look at the chart above from \(b\) to \(u\)) is:

- piecewise linear
- convex (because the “best” (highest) line) is always curving up

and so, for a policy instantiated by a bunch of alpha vectors \(\Gamma\), we have:

\begin{equation} U^{\Gamma}(b) = \max_{\alpha \in \Gamma} \alpha^{\top} b \end{equation}

To actually extract a policy out of this set of vectors \(\Gamma\), we turn to one-step lookahead in POMDP

### one-step lookahead in POMDP

Say you want to extract a policy out of a bunch of alpha vectors.

Let \(\alpha \in \Gamma\), a set of alpha vectors.

\begin{equation} \pi^{\Gamma}(b) = \arg\max_{a}\qty[R(b,a)+\gamma \qty(\sum_{o}^{}P(o|b,a) U^{\Gamma}(update(b,a,o)))] \end{equation}

where:

\begin{equation} R(b,a) = \sum_{s}^{} R(s,a)b(s) \end{equation}

\begin{align} P(o|b,a) &= \sum_{s}^{} p(o|s,a) b(s) \\ &= \sum_{s}^{} \sum_{s’}^{} T(s’|s,a) O(o|s’,a) b(s) \end{align}

and

\begin{equation} U^{\Gamma}(b) = \max_{\alpha \in \Gamma} \alpha^{\top} b \end{equation}

### alpha vector pruning

Say we had as set of alpha vectors \(\Gamma\):

\(\alpha_{3}\) isn’t all that useful here. So we ask:

“Is alpha dominated by some \(\alpha_{i}\) everywhere?”

We formulate this question in terms of a linear program:

\begin{equation} \max_{\delta, b} \delta \end{equation}

where \(\delta\) is the gap between \(\alpha\) and the utility o

subject to:

\begin{align} &1^{\top} b = 1\ \text{(b adds up to 1)} \\ & b\geq 0 \\ & \alpha^{\top} b \geq \alpha’^{\top} b + \delta, \forall \alpha’ \in \Gamma \end{align}

if \(\delta < 0\), then we can prune \(\alpha\) because it had been dominated.

if each value on the top of the set