Doubly High-Dimensional Contextual Bandits: An Interpretable Model for
  Joint Assortment-Pricing
Junhui Cai Ran Chen Martin J. Wainwright Linda Zhao
Key challenges in running a retail business include how to select products to
present to consumers (the assortment problem), and how to price products (the
pricing problem) to maximize revenue or profit. Instead of considering these
problems in isolation, we propose a joint approach to assortment-pricing based
on contextual bandits. Our model is doubly high-dimensional, in that both
context vectors and actions are allowed to take values in high-dimensional
spaces. In order to circumvent the curse of dimensionality, we propose a simple
yet flexible model that captures the interactions between covariates and
actions via a (near) low-rank representation matrix. The resulting class of
models is reasonably expressive while remaining interpretable through latent
factors, and includes various structured linear bandit and pricing models as
particular cases. We propose a computationally tractable procedure that
combines an exploration/exploitation protocol with an efficient low-rank matrix
estimator, and we prove bounds on its regret. Simulation results show that this
method has lower regret than state-of-the-art methods applied to various
standard bandit and pricing models. Real-world case studies on the
assortment-pricing problem, from an industry-leading instant noodles company to
an emerging beauty start-up, underscore the gains achievable using our method.
In each case, we show at least three-fold gains in revenue or profit by our
bandit method, as well as the interpretability of the latent factor models that
are learned.