Contextual Online Pricing with (Biased) Offline Data

Abstract: We study contextual online pricing with biased offline data. For the scalar price elasticity case, we identify the instance-dependent quantity $\delta^2$ that measures how far the offline data lies from the (unknown) online optimum. We show that the time length $T$, bias bound $V$, size $N$ and dispersion $\lambda_{\min}(\hat{\Sigma})$ of the offline data, and $\delta^2$ jointly determine the statistical complexity. An Optimism-in-the-Face-of-Uncertainty (OFU) policy achieves a minimax-optimal, instance-dependent regret bound $\tilde{\mathcal{O}}\big(d\sqrt{T} \wedge (V^2T + \frac{dT}{\lambda_{\min}(\hat{\Sigma}) + (N \wedge T) \delta^2})\big)$. For general price elasticity, we establish a worst-case, minimax-optimal rate $\tilde{\mathcal{O}}\big(d\sqrt{T} \wedge (V^2T + \frac{dT }{\lambda_{\min}(\hat{\Sigma})})\big)$ and provide a generalized OFU algorithm that attains it. When the bias bound $V$ is unknown, we design a robust variant that always guarantees sub-linear regret and strictly improves on purely online methods whenever the exact bias is small. These results deliver the first tight regret guarantees for contextual pricing in the presence of biased offline data. Our techniques also transfer verbatim to stochastic linear bandits with biased offline data, yielding analogous bounds.