  Optimal design for linear regression is a fundamental task in statistics. For
finite design spaces, recent progress has shown that random designs drawn using
proportional volume sampling (PVS) lead to approximation guarantees for
A-optimal design. PVS strikes the balance between design nodes that jointly
fill the design space, while marginally staying in regions of high mass under
the solution of a relaxed convex version of the original problem. In this
paper, we examine some of the statistical implications of a new variant of PVS
for (possibly Bayesian) optimal design. Using point process machinery, we treat
the case of a generic Polish design space. We show that not only are the
A-optimality approximation guarantees preserved, but we obtain similar
guarantees for D-optimal design that tighten recent results. Moreover, we show
that PVS can be sampled in polynomial time. Unfortunately, in spite of its
elegance and tractability, we demonstrate on a simple example that the
practical implications of general PVS are likely limited. In the second part of
the paper, we focus on applications and investigate the use of PVS as a
subroutine for stochastic search heuristics. We demonstrate that PVS is a
robust addition to the practitioner's toolbox, especially when the regression
functions are nonstandard and the design space, while low-dimensional, has a
complicated shape (e.g., nonlinear boundaries, several connected components).
