Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure.

Each block represents atime-invariant iterativeprocess as the first layer in thei-th block,xi(1), is unrolled into a pattern-dependent number,Ki, of processing stages, using weight matricesAi andBi. The skip connections from the input,ui, to all layers in blockimake the process nonautonomous. Blocks can be chained together (each block modeling adifferent latent space) by passing final latentrepresentation,xi(Ki),ofblockiastheinputtoblocki+1.

This progress has led to a revival of robust statistics from an algorithmic perspective (see, e.g., [DK19, DKK+21] for surveys on the topic).

Statistical inference in contextual bandits is complicated by the adaptive, non-i.i.d. nature of the data. A growing body of work has shown that classical least-squares inference may fail under adaptive sampling, and that constructing valid confidence intervals for linear functionals of the model parameter typically requires paying an unavoidable inflation of order $\sqrt{d \log T}$. This phenomenon -- often referred to as the price of adaptivity -- highlights the inherent difficulty of reliable inference under general contextual bandit policies. A key structural property that circumvents this limitation is the \emph{stability} condition of Lai and Wei, which requires the empirical feature covariance to concentrate around a deterministic limit. When stability holds, the ordinary least-squares estimator satisfies a central limit theorem, and classical Wald-type confidence intervals -- designed for i.i.d. data -- become asymptotically valid even under adaptation, \emph{without} incurring the $\sqrt{d \log T}$ price of adaptivity. In this paper, we propose and analyze a penalized EXP4 algorithm for linear contextual bandits. Our first main result shows that this procedure satisfies the Lai--Wei stability condition and therefore admits valid Wald-type confidence intervals for linear functionals. Our second result establishes that the same algorithm achieves regret guarantees that are minimax optimal up to logarithmic factors, demonstrating that stability and statistical efficiency can coexist within a single contextual bandit method. Finally, we complement our theory with simulations illustrating the empirical normality of the resulting estimators and the sharpness of the corresponding confidence intervals.

Recently, several researchers have started to view VDNNs from a dynamical systems perspective.

Uncertainty quantification (UQ) for adaptively collected data, such as that coming from adaptive experiments, bandits, or reinforcement learning, is necessary for critical elements of data collection such as ensuring safety and conducting after-study inference. The data's adaptivity creates significant challenges for frequentist UQ, yet Bayesian UQ remains the same as if the data were independent and identically distributed (i.i.d.), making it an appealing and commonly used approach. Bayesian UQ requires the (correct) specification of a prior distribution while frequentist UQ does not, but for i.i.d. data the celebrated Bernstein-von Mises theorem shows that as the sample size grows, the prior 'washes out' and Bayesian UQ becomes frequentist-valid, implying that the choice of prior need not be a major impediment to Bayesian UQ as it makes no difference asymptotically. This paper for the first time extends the Bernstein-von Mises theorem to adaptively collected data, proving asymptotic equivalence between Bayesian UQ and Wald-type frequentist UQ in this challenging setting. Our result showing this asymptotic agreement does not require the standard stability condition required by works studying validity of Wald-type frequentist UQ; in cases where stability is satisfied, our results combined with these prior studies of frequentist UQ imply frequentist validity of Bayesian UQ. Counterintuitively however, they also provide a negative result that Bayesian UQ is not asymptotically frequentist valid when stability fails, despite the fact that the prior washes out and Bayesian UQ asymptotically matches standard Wald-type frequentist UQ. We empirically validate our theory (positive and negative) via a range of simulations.

Sequential data collection has emerged as a widely adopted technique for enhancing the efficiency of data gathering processes. Despite its advantages, such data collection mechanism often introduces complexities to the statistical inference procedure.