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. This is motivated in part by the prevalence of online marketplaces that allow for real-time pricing. We propose a dynamic policy, called Regularized Maximum Likelihood Pricing (RMLP), that obtains asymptotically optimal revenue. Our policy leverages the structure (sparsity) of a high-dimensional demand space in order to obtain a logarithmic regret compared to the clairvoyant policy that knows the parameters of the demand in advance. More specifically, the regret of our algorithm is of $O(s_0 \log T (\log d + \log T))$, where $d$ and $s_0$ correspond to the dimension of the demand space and its sparsity. Furthermore, we show that no policy can obtain regret better than $O(s_0 (\log d + \log T))$.