Dynamic Pricing in High-dimensions

We study the pricing problem faced by a firm that sells a large number of products, described via a wide range of features, to customers that arrive over time. Customers independently make purchasing decisions according to a general choice model that includes products features and customers' characteristics, encoded as d-dimensional numerical vectors, as well as the price offered. The parameters of the choice model are a priori unknown to the firm, but can be learned as the (binary-valued) sales data accrues over time. The firm's objective is to minimize the regret, i.e., the expected revenue loss against a clairvoyant policy that knows the parameters of the choice model in advance, and always offers the revenue-maximizing price. This setting is motivated in part by the prevalence of online marketplaces that allow for real-time pricing. We assume a structured choice model, parameters of which depend on s0 out of the d product features. We propose a dynamic policy, called Regularized Maximum Likelihood Pricing (RMLP) that leverages the (sparsity) structure of the high-dimensional model and obtains a logarithmic regret in T. More specifically, the regret of our algorithm is of O(s0 log d · log T). Furthermore, we show that no policy can obtain regret better than O(s0 (log d + log T)).