We study contextual dynamic pricing with arbitrary covariate sequences and bounded, possibly nonbinary purchase quantities. Demand follows a semiparametric surplus-index model with an unknown linear valuation parameter and an unknown Hölder-smooth response. We impose neither concavity nor strong unimodality on revenue and allow nonunique optimal prices. We develop a pilot-corrected layered decision-partitioning policy that combines directional pilot estimation, local polynomial learning, predictable data assignment, and global action elimination. Pilot correction removes the first-order effect of valuation-parameter error, while permanent labels enable concentration under adaptive sampling. The policy attains the minimax smoothness-dependent horizon rate up to logarithmic factors; a matching lower bound already holds for a constant-context binary-demand subclass.
We study the contextual dynamic pricing problem under non-stationarity, where a firm sells products to $T$ sequentially arriving consumers that behave according to an unknown demand model that can change over time. The demand model is assumed to be a generalized linear model (GLM), allowing for a feature vector in $\mathbb{R}^d$ that encodes products and consumer information. To achieve optimal revenue (i.e., least regret), the firm needs to learn and exploit the unknown GLMs while monitoring for potential changes. We propose a multiscale change-point detection based algorithm that achieves a regret of order $\widetilde{O}(\sqrt{s_TdT}\wedge\{V_T^{1/3}d^{1/3}T^{2/3}+\sqrt{dT}\})$, where $s_T$ is the number of piecewise stationary segments and $V_T$ is a newly defined notion of design-adjusted variation budget of model parameters. Our algorithm is adaptive and does not require knowing $s_T$ or $V_T$. Moreover, to our knowledge, this is the first dynamic pricing algorithm that is adaptive to the nature of changes and achieves the best-of-both-worlds rate, thus closing a long-standing gap in the literature. We remark that, due to the varying contexts, existing works in the adaptive non-stationary bandit literature cannot be applied to achieve optimality for contextual dynamic pricing. The regret is further accompanied with a newly constructed minimax lower bound, confirming the optimality of our algorithm (up to logarithmic factors). Extensive numerical experiments are conducted to illustrate the efficiency and robustness of the proposed algorithm in non-stationary dynamic pricing.
Contextual bandit algorithms have transformed modern experimentation by enabling real-time adaptation for personalized treatment. Yet these advantages create challenges for statistical inference due to adaptivity. We study inference with contextual-bandit data without assuming a well-specified outcome model. In this setting, we show a previously overlooked issue: standard algorithms such as LinUCB may fail to stabilize under misspecified working models, leading to non-Gaussian estimator behavior and invalid inference. This issue is practically important, as misspecified working models -- such as approximations of complex dynamical systems -- are often employed by online agents in real-world adaptive experiments to balance reward, computational tractability, and robustness. We develop an inverse-probability-weighted Z-estimation framework for a broad class of marginal moment targets, including projection parameters, structural parameters with noisy contexts, and off-policy values. We identify a stability condition tailored to this framework, scaled inverse-propensity convergence, under which the IPW-Z estimator is consistent and asymptotically normal with a consistent sandwich variance estimator. We further establish sufficient conditions for scaled inverse-propensity convergence for several policy classes, including multi-armed bandit algorithms and smooth contextual allocation policies. Simulations and a HeartSteps V1 real-data-calibrated application show reliable coverage and competitive performance across multiple targets. Overall, our results highlight the importance of stability-aware adaptive design for valid post-experiment inference.
We develop an anytime-valid framework for optimal policy identification from logged contextual bandit data. In many applied settings, the analyst wants to select the optimal policy from a candidate policy class $Π$, but data are generated by an externally determined logging policy that they do not control. The analyst may also wish to monitor evidence continuously, stopping once the optimal policy is clear rather than committing to a fixed sample size in advance. This paper addresses these challenges by constructing a time-indexed set $S_t$ that retains the true optimal policy set uniformly over time with high probability. The resulting procedure allows the analyst to monitor policy values, eliminate clearly suboptimal policies, and stop at data-dependent times without invalidating inference. When the optimal policy is unique, we define a stopping time for its identification and derive a sample-complexity bound scaling as $O\!\left(\frac{\log |Π|+\log\log(1/Δ_{\min})}{Δ_{\min}^2}\right)$, where $Δ_{\min}$ is the gap between the best and second-best policy values. Simulations demonstrate that the anytime-valid approach can yield substantial sample savings relative to fixed-$N$ designs. An application to a large adaptive experiment on reducing misinformation online illustrates how the method provides a dynamic view as evidence on the optimal policy accumulates.
The challenge with active learning algorithms is the uncertainty of the statistical distribution of unlabeled data, making it difficult to choose the best hand-crafted strategy. To address this, we introduced Contextual Adaptive Active Learning (CAAL). In CAAL, each "arm" represents a hand-crafted strategy. Unlike existing frameworks that select strategies based only on feedback from labeled data, we dynamically choose strategies for labeling batches of data using reward prediction with external context information. This general framework allows for customization with domain knowledge to design more effective rewards and context candidates. In addition, we experimentally show that CAAL outperforms the existing baseline adaptive strategy on public datasets using our reward and context design. Our results are consistent regardless of batch size in each iteration.