On Dynamic Pricing with Covariates
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We consider the dynamic pricing problem with covariates under a generalized linear demand model: a seller can dynamically adjust the price of a product over a horizon of T time periods, and at each time period t, the demand of the product is jointly determined by the price and an observable covariate vector x_t∈ℝ^d through an unknown generalized linear model. Most of the existing literature assumes the covariate vectors x_t's are independently and identically distributed (i.i.d.); the few papers that relax this assumption either sacrifice model generality or yield sub-optimal regret bounds. In this paper we show that a simple pricing algorithm has an O(d√(T)log T) regret upper bound without assuming any statistical structure on the covariates x_t (which can even be arbitrarily chosen). The upper bound on the regret matches the lower bound (even under the i.i.d. assumption) up to logarithmic factors. Our paper thus shows that (i) the i.i.d. assumption is not necessary for obtaining low regret, and (ii) the regret bound can be independent of the (inverse) minimum eigenvalue of the covariance matrix of the x_t's, a quantity present in previous bounds. Furthermore, we discuss a condition under which a better regret is achievable and how a Thompson sampling algorithm can be applied to give an efficient computation of the prices.
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