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house price prediction is a classical machine learning framing problem.

collect data: use a single property (“feature”)—size of the house, in square feet; to learn is the output—price

Improving PBVI without sacrificing quality.

Initialization

We first initialize HSVI with a set of alpha vectors \(\Gamma\), representing the lower-bound, and a list of tuples of \((b, U(b))\) named \(\Upsilon\), representing the upper-bound. We call the value functions they generate as \(\bar{V}\) and \(\underline V\).

Lower Bound

Set of alpha vectors: best-action worst-state (HSVI1), blind lower bound (HSVI2)

Calculating \(\underline{V}(b)\)

\begin{equation} \underline{V}_{\Gamma} = \max_{\alpha} \alpha^{\top}b \end{equation}

Upper Bound

Fast Informed Bound

• solving fully-observable MDP
• Project \(b\) into the point-set
• Projected the upper bound onto a convex hull (HSVI2: via approximate convex hull projection)

Calculating \(\bar{V}(b)\)

Recall that though the lower-bound is given by alpha vectors, the upper bound is given in terms of a series of tuples \((b, U(b)) \in \Upsilon\).