  Adding domain knowledge to a learning system is known to improve results. In
multi-parameter Bayesian frameworks, such knowledge is incorporated as a prior.
On the other hand, various model parameters can have different learning rates
in real-world problems, especially with skewed data. Two often-faced challenges
in Operation Management and Management Science applications are the absence of
informative priors, and the inability to control parameter learning rates. In
this study, we propose a hierarchical Empirical Bayes approach that addresses
both challenges, and that can generalize to any Bayesian framework. Our method
learns empirical meta-priors from the data itself and uses them to decouple the
learning rates of first-order and second-order features (or any other given
feature grouping) in a Generalized Linear Model. As the first-order features
are likely to have a more pronounced effect on the outcome, focusing on
learning first-order weights first is likely to improve performance and
convergence time. Our Empirical Bayes method clamps features in each group
together and uses the deployed model's observed data to empirically compute a
hierarchical prior in hindsight. We report theoretical results for the
unbiasedness, strong consistency, and optimal frequentist cumulative regret
properties of our meta-prior variance estimator. We apply our method to a
standard supervised learning optimization problem, as well as an online
combinatorial optimization problem in a contextual bandit setting implemented
in an Amazon production system. Both during simulations and live experiments,
our method shows marked improvements, especially in cases of small traffic. Our
findings are promising, as optimizing over sparse data is often a challenge.
