Rejoinder: New Objectives for Policy Learning

Click to increase image sizeClick to decrease image size

This article is related to:

Introduction to the Theory and Methods Special Issue on Precision Medicine and Individualized Policy Discovery

This article responds to:

Discussion of Kallus and Mo, Qi, and Liu: New Objectives for Policy Learning

Discussion of Kallus (2020) and Mo et al. (2020)

Discussion of Kallus (2020) and Mo, Qi, and Liu (2020): New Objectives for Policy Learning

Notes

1 The derivation is also unclear. The conditional restriction considered in the appendix is equivalent to $\mathbb{E}[Y(+)|X]=g_1(Z)$, that is, the conditional mean of treated outcome given X is a function of only Z. The final loss function, appearing in their last display equation, also appears to me to be in error.

2 Unfortunately, a typo in the Journal version replaced the two $\max$’s in its definition with two $\min$’s. In private correspondence, LLL, who pointed out the typo, explain this typo is the cause for their mistaking this quantity for the value diameter of the policy space (display equation above their Equation (5)) rather than the optimality gap.

3 Both statements depend on the relevant smoothness, of course. More generally and in multivariate settings, this can be phrased in terms of Lipschitz gradient (first statement) and strong convexity (second statement) of the value function. Alternatively, in the finite-policy-space case, we have the argument above using the probability of optimal choice.

Additional information

Funding

This material is based upon work supported by the National Science Foundation under grant no. 1846210.