Meta Dynamic Pricing: Learning Across Experiments

We study the problem of learning \emph{across} a sequence of price experiments for related products, focusing on implementing the Thompson sampling algorithm for dynamic pricing. We consider a practical formulation of this problem where the unknown parameters of the demand function for each product come from a prior that is shared across products, but is unknown a priori. Our main contribution is a meta dynamic pricing algorithm that learns this prior online while solving a sequence of non-overlapping pricing experiments (each with horizon $T$) for $N$ different products. Our algorithm addresses two challenges: (i) balancing the need to learn the prior (\emph{meta-exploration}) with the need to leverage the current estimate of the prior to achieve good performance (\emph{meta-exploitation}), and (ii) accounting for uncertainty in the estimated prior by appropriately "widening" the prior as a function of its estimation error, thereby ensuring convergence of each price experiment. We prove that the price of an unknown prior for Thompson sampling is negligible in experiment-rich environments (large $N$). In particular, our algorithm's meta regret can be upper bounded by $\widetilde{O}\left(\sqrt{NT}\right)$ when the covariance of the prior is known, and $\widetilde{O}\left(N^{\frac{3}{4}}\sqrt{T}\right)$ otherwise. Numerical experiments on synthetic and real auto loan data demonstrate that our algorithm significantly speeds up learning compared to prior-independent algorithms or a naive approach of greedily using the updated prior across products.