  Constructing asymptotically valid confidence intervals through a valid
central limit theorem is crucial for A/B tests, where a classical goal is to
statistically assert whether a treatment plan is significantly better than a
control plan. In some emerging applications for online platforms, the treatment
plan is not a single plan, but instead encompasses an infinite continuum of
plans indexed by a continuous treatment parameter. As such, the experimenter
not only needs to provide valid statistical inference, but also desires to
effectively and adaptively find the optimal choice of value for the treatment
parameter to use for the treatment plan. However, we find that classical
optimization algorithms, despite of their fast convergence rates under
convexity assumptions, do not come with a central limit theorem that can be
used to construct asymptotically valid confidence intervals. We fix this issue
by providing a new optimization algorithm that on one hand maintains the same
fast convergence rate and on the other hand permits the establishment of a
valid central limit theorem. We discuss practical implementations of the
proposed algorithm and conduct numerical experiments to illustrate the
theoretical findings.
