UCB algorithms for multi-armed bandits: Precise regret and adaptive inference

Upper Confidence Bound (UCB) algorithms are a widely-used class of sequential algorithms for the $K$-armed bandit problem. Despite extensive research over the past decades aimed at understanding their asymptotic and (near) minimax optimality properties, a precise understanding of their regret behavior remains elusive. This gap has not only hindered the evaluation of their actual algorithmic efficiency, but also limited further developments in statistical inference in sequential data collection. This paper bridges these two fundamental aspects--precise regret analysis and adaptive statistical inference--through a deterministic characterization of the number of arm pulls for an UCB index algorithm [Lai87, Agr95, ACBF02]. Our resulting precise regret formula not only accurately captures the actual behavior of the UCB algorithm for finite time horizons and individual problem instances, but also provides significant new insights into the regimes in which the existing theory remains informative. In particular, we show that the classical Lai-Robbins regret formula is exact if and only if the sub-optimality gaps exceed the order $\sigma\sqrt{K\log T/T}$. We also show that its maximal regret deviates from the minimax regret by a logarithmic factor, and therefore settling its strict minimax optimality in the negative. The deterministic characterization of the number of arm pulls for the UCB algorithm also has major implications in adaptive statistical inference. Building on the seminal work of [Lai82], we show that the UCB algorithm satisfies certain stability properties that lead to quantitative central limit theorems in two settings including the empirical means of unknown rewards in the bandit setting. These results have an important practical implication: conventional confidence sets designed for i.i.d. data remain valid even when data are collected sequentially.

Inference with the Upper Confidence Bound Algorithm

In this paper, we discuss the asymptotic behavior of the Upper Confidence Bound (UCB) algorithm in the context of multiarmed bandit problems and discuss its implication in downstream inferential tasks. While inferential tasks become challenging when data is collected in a sequential manner, we argue that this problem can be alleviated when the sequential algorithm at hand satisfies certain stability property. This notion of stability is motivated from the seminal work of Lai and Wei (1982). Our first main result shows that such a stability property is always satisfied for the UCB algorithm, and as a result the sample means for each arm are asymptotically normal. Next, we examine the stability properties of the UCB algorithm when the number of arms $K$ is allowed to grow with the number of arm pulls $T$. We show that in such a case the arms are stable when $\frac{\log K}{\log T} \rightarrow 0$, and the number of near-optimal arms are large.

Informativeness of Weighted Conformal Prediction

Weighted conformal prediction (WCP), a recently proposed framework, provides uncertainty quantification with the flexibility to accommodate different covariate distributions between training and test data. However, it is pointed out in this paper that the effectiveness of WCP heavily relies on the overlap between covariate distributions; insufficient overlap can lead to uninformative prediction intervals. To enhance the informativeness of WCP, we propose two methods for scenarios involving multiple sources with varied covariate distributions. We establish theoretical guarantees for our proposed methods and demonstrate their efficacy through simulations.

Stochastic Optimization with Constraints: A Non-asymptotic Instance-Dependent Analysis

We consider the problem of stochastic convex optimization under convex constraints. We analyze the behavior of a natural variance reduced proximal gradient (VRPG) algorithm for this problem. Our main result is a non-asymptotic guarantee for VRPG algorithm. Contrary to minimax worst case guarantees, our result is instance-dependent in nature. This means that our guarantee captures the complexity of the loss function, the variability of the noise, and the geometry of the constraint set. We show that the non-asymptotic performance of the VRPG algorithm is governed by the scaled distance (scaled by $\sqrt{N}$) between the solutions of the given problem and that of a certain small perturbation of the given problem -- both solved under the given convex constraints; here, $N$ denotes the number of samples. Leveraging a well-established connection between local minimax lower bounds and solutions to perturbed problems, we show that as $N \rightarrow \infty$, the VRPG algorithm achieves the renowned local minimax lower bound by H\`{a}jek and Le Cam up to universal constants and a logarithmic factor of the sample size.

Statistical Limits of Adaptive Linear Models: Low-Dimensional Estimation and Inference

Estimation and inference in statistics pose significant challenges when data are collected adaptively. Even in linear models, the Ordinary Least Squares (OLS) estimator may fail to exhibit asymptotic normality for single coordinate estimation and have inflated error. This issue is highlighted by a recent minimax lower bound, which shows that the error of estimating a single coordinate can be enlarged by a multiple of $\sqrt{d}$ when data are allowed to be arbitrarily adaptive, compared with the case when they are i.i.d. Our work explores this striking difference in estimation performance between utilizing i.i.d. and adaptive data. We investigate how the degree of adaptivity in data collection impacts the performance of estimating a low-dimensional parameter component in high-dimensional linear models. We identify conditions on the data collection mechanism under which the estimation error for a low-dimensional parameter component matches its counterpart in the i.i.d. setting, up to a factor that depends on the degree of adaptivity. We show that OLS or OLS on centered data can achieve this matching error. In addition, we propose a novel estimator for single coordinate inference via solving a Two-stage Adaptive Linear Estimating equation (TALE). Under a weaker form of adaptivity in data collection, we establish an asymptotic normality property of the proposed estimator.

Adaptive Linear Estimating Equations

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. For instance, the ordinary least squares (OLS) estimator in an adaptive linear regression model can exhibit non-normal asymptotic behavior, posing challenges for accurate inference and interpretation. In this paper, we propose a general method for constructing debiased estimator which remedies this issue. It makes use of the idea of adaptive linear estimating equations, and we establish theoretical guarantees of asymptotic normality, supplemented by discussions on achieving near-optimal asymptotic variance. A salient feature of our estimator is that in the context of multi-armed bandits, our estimator retains the non-asymptotic performance of the least square estimator while obtaining asymptotic normality property. Consequently, this work helps connect two fruitful paradigms of adaptive inference: a) non-asymptotic inference using concentration inequalities and b) asymptotic inference via asymptotic normality.

Semi-parametric inference based on adaptively collected data

Many standard estimators, when applied to adaptively collected data, fail to be asymptotically normal, thereby complicating the construction of confidence intervals. We address this challenge in a semi-parametric context: estimating the parameter vector of a generalized linear regression model contaminated by a non-parametric nuisance component. We construct suitably weighted estimating equations that account for adaptivity in data collection, and provide conditions under which the associated estimates are asymptotically normal. Our results characterize the degree of "explorability" required for asymptotic normality to hold. For the simpler problem of estimating a linear functional, we provide similar guarantees under much weaker assumptions. We illustrate our general theory with concrete consequences for various problems, including standard linear bandits and sparse generalized bandits, and compare with other methods via simulation studies.

Instance-Dependent Confidence and Early Stopping for Reinforcement Learning

Various algorithms for reinforcement learning (RL) exhibit dramatic variation in their convergence rates as a function of problem structure. Such problem-dependent behavior is not captured by worst-case analyses and has accordingly inspired a growing effort in obtaining instance-dependent guarantees and deriving instance-optimal algorithms for RL problems. This research has been carried out, however, primarily within the confines of theory, providing guarantees that explain \textit{ex post} the performance differences observed. A natural next step is to convert these theoretical guarantees into guidelines that are useful in practice. We address the problem of obtaining sharp instance-dependent confidence regions for the policy evaluation problem and the optimal value estimation problem of an MDP, given access to an instance-optimal algorithm. As a consequence, we propose a data-dependent stopping rule for instance-optimal algorithms. The proposed stopping rule adapts to the instance-specific difficulty of the problem and allows for early termination for problems with favorable structure.

Optimal variance-reduced stochastic approximation in Banach spaces

We study the problem of estimating the fixed point of a contractive operator defined on a separable Banach space. Focusing on a stochastic query model that provides noisy evaluations of the operator, we analyze a variance-reduced stochastic approximation scheme, and establish non-asymptotic bounds for both the operator defect and the estimation error, measured in an arbitrary semi-norm. In contrast to worst-case guarantees, our bounds are instance-dependent, and achieve the local asymptotic minimax risk non-asymptotically. For linear operators, contractivity can be relaxed to multi-step contractivity, so that the theory can be applied to problems like average reward policy evaluation problem in reinforcement learning. We illustrate the theory via applications to stochastic shortest path problems, two-player zero-sum Markov games, as well as policy evaluation and $Q$-learning for tabular Markov decision processes.

Near-optimal inference in adaptive linear regression

When data is collected in an adaptive manner, even simple methods like ordinary least squares can exhibit non-normal asymptotic behavior. As an undesirable consequence, hypothesis tests and confidence intervals based on asymptotic normality can lead to erroneous results. We propose an online debiasing estimator to correct these distributional anomalies in least squares estimation. Our proposed method takes advantage of the covariance structure present in the dataset and provides sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimator under mild conditions on the data collection process, and provide asymptotically exact confidence intervals. We additionally prove a minimax lower bound for the adaptive linear regression problem, thereby providing a baseline by which to compare estimators. There are various conditions under which our proposed estimator achieves the minimax lower bound up to logarithmic factors. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.