  In many clinical practices, the goal of medical interventions is often to
maintain clinical measures within a desirable range, rather than maximize or
minimize their values. To achieve this, it may be more practical to recommend a
therapeutic dose interval rather than a single dose for a given patient. Since
each patient may respond differently to the same dose of medication, the
therapeutic dose interval should be personalized based on the unique
characteristics of each patient. This problem is challenging, as it requires
jointly learning the lower and upper bound functions for personalized dose
intervals. Currently, no methods are available to address this challenge
without resorting to iterative procedures to constrain the monotonicity between
lower and upper bound functions. However, these iterative procedures do not
have a theoretical justification. To fill this gap, we propose a method to
learn personalized two-sided dose intervals based on empirical risk
minimization associated with a novel loss function. The proposed loss function
is designed to be well-defined over a tensor product of function spaces, which
eliminates the need for iterative procedures. In addition, our loss function is
doubly-robust to the misspecification of nuisance functions. We establish
statistical properties of the estimated dose interval functions in terms of
excess risk by leveraging approximation theory and reproducing kernel Hilbert
space theory. Our simulation study and a real-world application of personalized
warfarin dose intervals show that our proposed direct estimation method
outperforms competing methods, including indirect regression-based methods.
