  A dynamic treatment regimen (DTR) is a set of decision rules to personalize
treatments for an individual using their medical history. The Q-learning based
Q-shared algorithm has been used to develop DTRs that involve decision rules
shared across multiple stages of intervention. We show that the existing
Q-shared algorithm can suffer from non-convergence due to the use of linear
models in the Q-learning setup, and identify the condition in which Q-shared
fails. Leveraging properties from expansion-constrained ordinary least-squares,
we give a penalized Q-shared algorithm that not only converges in settings that
violate the condition, but can outperform the original Q-shared algorithm even
when the condition is satisfied. We give evidence for the proposed method in a
real-world application and several synthetic simulations.
