Stabilizing Bandits using Regularization: Precise Regret and A Quantitative Central Limit Theorem

Bandit algorithms generate data for downstream inference, but adaptive sampling biases post-bandit sample means. We analyze this bias for stable index algorithms, including UCB1 and its generalizations, and derive sharp leading-order expressions for the sample-mean bias and expected $Z$-statistic, in bandit experiments of fixed horizon $T$. Our characterization reveals the algorithmic origin of bias through a key index-function-dependent quantity, which we term effective exploration rate. For example, under UCB1, the effective exploration rate is of order $\sqrt{\log T}$, and the standardized bias of any arm (that is not uniquely optimal) decays at the extremely slow rate $1/\sqrt{\log T}$. We also show how the choice of the index function affects both regret and bias, which reveals a regret-bias trade-off: more exploratory algorithm reduces bias but increases regret. We further show how bias most severely distorts confidence intervals and hypothesis tests when the tested arm is one of the tied-optimal arms. Our sharp characterization for bias uses a novel empirical fluid approximation of the algorithm's sampling dynamics, which may be of independent interest.

We study inference on scalar-valued pathwise differentiable targets after adaptive data collection, such as a bandit algorithm. We introduce a novel target-specific condition, directional stability, which is strictly weaker than previously imposed target-agnostic stability conditions. Under directional stability, we show that estimators that would have been efficient under i.i.d. data remain asymptotically normal and semiparametrically efficient when computed from adaptively collected trajectories. The canonical gradient has a martingale form, and directional stability guarantees stabilization of its predictable quadratic variation, enabling high-dimensional asymptotic normality. We characterize efficiency using a convolution theorem for the adaptive-data setting, and give a condition under which the one-step estimator attains the efficiency bound. We verify directional stability for LinUCB, yielding the first semiparametric efficiency guarantee for a regular scalar target under LinUCB sampling.

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.