Sequential change detection is a classical problem with a variety of applications. However, the majority of prior work has been parametric, for example, focusing on exponential families. We develop a fundamentally new and general framework for sequential change detection when the pre- and post-change distributions are nonparametrically specified (and thus composite). Our procedures come with clean, nonasymptotic bounds on the average run length (frequency of false alarms). In certain nonparametric cases (like sub-Gaussian or sub-exponential), we also provide near-optimal bounds on the detection delay following a changepoint. The primary technical tool that we introduce is called an \emph{e-detector}, which is composed of sums of e-processes -- a fundamental generalization of nonnegative supermartingales -- that are started at consecutive times. We first introduce simple Shiryaev-Roberts and CUSUM-style e-detectors, and then show how to design their mixtures in order to achieve both statistical and computational efficiency. Our e-detector framework can be instantiated to recover classical likelihood-based procedures for parametric problems, as well as yielding the first change detection method for many nonparametric problems. As a running example, we tackle the problem of detecting changes in the mean of a bounded random variable without i.i.d. assumptions, with an application to tracking the performance of a basketball team over multiple seasons.

It is well known that in stochastic multi-armed bandits (MAB), the sample mean of an arm is typically not an unbiased estimator of its true mean. In this paper, we decouple three different sources of this selection bias: adaptive \emph{sampling} of arms, adaptive \emph{stopping} of the experiment and adaptively \emph{choosing} which arm to study. Through a new notion called ``optimism'' that captures certain natural monotonic behaviors of algorithms, we provide a clean and unified analysis of how optimistic rules affect the sign of the bias. The main takeaway message is that optimistic sampling induces a negative bias, but optimistic stopping and optimistic choosing both induce a positive bias. These results are derived in a general stochastic MAB setup that is entirely agnostic to the final aim of the experiment (regret minimization or best-arm identification or anything else). We provide examples of optimistic rules of each type, demonstrate that simulations confirm our theoretical predictions, and pose some natural but hard open problems.

In the classic stochastic multi-armed bandit problem, it is well known that the sample mean for a chosen arm is a biased estimator of its true mean. In this paper, we characterize the effect of four sources of this selection bias: adaptively \emph{sampling} an arm at each step, adaptively \emph{stopping} the data collection, adaptively \emph{choosing} which arm to target for mean estimation, and adaptively \emph{rewinding} the clock to focus on the sample mean of the chosen arm at some past time. We qualitatively characterize data collecting strategies for which the bias induced by adaptive sampling and stopping can be negative or positive. For general parametric and nonparametric classes of distributions with varying tail decays, we provide bounds on the risk (expected Bregman divergence between the sample and true mean) that hold for arbitrary rules for sampling, stopping, choosing and rewinding. These risk bounds are minimax optimal up to log factors, and imply tight bounds on the selection bias and sufficient conditions for their consistency.