Near-optimal discrete optimization for experimental design: a regret minimization approach

The experimental design problem concerns the selection of k points from a potentially large design pool of p -dimensional vectors, so as to maximize the statistical efficiency regressed on the selected k design points. Statistical efficiency is measured by optimality criteria , including A(verage), D(eterminant), T(race), E(igen), V(ariance) and G-optimality. Except for the T-optimality, exact optimization is challenging, and for certain instances of D/E-optimality exact or even approximate optimization is proven to be NP-hard. We propose a polynomial-time regret minimization framework to achieve a ( 1 + ε ) approximation with only O ( p / ε 2 ) design points, for all the optimality criteria above. In contrast, to the best of our knowledge, before our work, no polynomial-time algorithm achieves ( 1 + ε ) approximations for D/E/G-optimality, and the best poly-time algorithm achieving ( 1 + ε ) -approximation for A/V-optimality requires k = Ω ( p 2 / ε ) design points.