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This paper investigates the impact of pre-existing offline data on onlinelearning, in the context of dynamic pricing. We study a single-product dynamicpricing problem over a selling horizon of $T$ periods. The demand in eachperiod is determined by the price of the product according to a linear demandmodel with unknown parameters. We assume that before the start of the sellinghorizon, the seller already has some pre-existing offline data. The offlinedata set contains $n$ samples, each of which is an input-output pair consistingof a historical price and an associated demand observation. The seller wants toutilize both the pre-existing offline data and the sequential online data tominimize the regret of the online learning process.  We characterize the joint effect of the size, location and dispersion of theoffline data on the optimal regret of the online learning process.Specifically, the size, location and dispersion of the offline data aremeasured by the number of historical samples $n$, the distance between theaverage historical price and the optimal price $\delta$, and the standarddeviation of the historical prices $\sigma$, respectively. We show that theoptimal regret is $\widetilde \Theta\left(\sqrt{T}\wedge \frac{T}{(n\wedgeT)\delta^2+n\sigma^2}\right)$, and design a learning algorithm based on the"optimism in the face of uncertainty" principle, whose regret is optimal up toa logarithmic factor. Our results reveal surprising transformations of theoptimal regret rate with respect to the size of the offline data, which werefer to as phase transitions. In addition, our results demonstrate that thelocation and dispersion of the offline data also have an intrinsic effect onthe optimal regret, and we quantify this effect via the inverse-square law.

Online Pricing with Offline Data: Phase Transition and Inverse Square Law