Budget Feasible Mechanisms for Experimental Design

In the classical experimental design setting, an experimenter E has access toa population of n potential experiment subjects i ∈ {1, . . . , n}, each associatedwith a vector of features xi ∈ R. Conducting an experiment with subject ireveals an unknown value yi ∈ R to E. E typically assumes some hypotheti-cal relationship between xi’s and yi’s, e.g., yi ≈ βxi, and estimates β fromexperiments, e.g., through linear regression. As a proxy for various practicalconstraints, E may select only a subset of subjects on which to conduct theexperiment.We initiate the study of budgeted mechanisms for experimental design. Inthis setting, E has a budget B. Each subject i declares an associated costci > 0 to be part of the experiment, and must be paid at least her cost. Inparticular, the Experimental Design Problem (EDP) is to find a set S of subjectsfor the experiment that maximizes V (S) = log det(Id +∑ i∈S xixTi ) under theconstraint∑i∈S ci ≤ B; our objective function corresponds to the informationgain in parameter β that is learned through linear regression methods, andis related to the so-called D-optimality criterion. Further, the subjects arestrategic and may lie about their costs. Thus, we need to design a mechanismfor EDP with suitable properties.We present a deterministic, polynomial time, budget feasible mechanismscheme, that is approximately truthful and yields a constant (≈ 12.98) factorapproximation to EDP. By applying previous work on budget feasible mech-anisms with a submodular objective, one could only have derived either anexponential time deterministic mechanism or a randomized polynomial timemechanism. We also establish that no truthful, budget-feasible mechanism ispossible within a factor 2 approximation, and show how to generalize our ap-proach to a wide class of learning problems, beyond linear regression.