Parameter Tuning

If observations from the joint distribution of (A,Y,Z,W) are available in both stages, we can tune the regularization parameters λ1,λ2 using the approach proposed in Singh et al. [30], Xu et al. [35]. Let the complete data of stage 1 and stage 2 be denoted as (ai,yi,zi,wi) and ( ai, yi, zi, wi). Then, we can use the data not used in each stage to evaluate the out-of-sample performance of the other stage. A(2), ˆV(T),u(T) are the learned parameters by Algorithm 1. In this appendix, we prove propositions given in the main text. In the following, we assume that the spaces U, A, Z,W are separable and completely metrizable topological spaces and equipped with Borel σ-algebras. In this section, we use the notation PA|Z=z to express the distribution of a random variable Agiven another variable Z = z.