arXiv subjects

Koulik Khamaru

Publications and source records attributed to Koulik Khamaru.

At least 19 recordsLinked to original sources

arXiv2026

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

Statistical inference with bandit data presents fundamental challenges owing to adaptive sampling, which violates the independence assumptions underlying classical asymptotic theory. Recent work has identified stability~\citep{laiwei82} as a sufficient condition for valid inference under adaptivity. This paper first provides a refined stability condition, stated in terms of the iterates of an online algorithm, and shows that a large class of regularized stochastic-mirror-descent-style algorithms satisfy it. This refined condition allows us to strengthen the asymptotic results of~\citet{laiwei82} in several ways. First, we derive a non-asymptotic Berry--Esseen bound for the empirical reward estimates under adaptive sampling. Second, we derive matching non-asymptotic upper and lower bounds on the regret of the proposed algorithm, yielding a precise characterization of its regret. Third, we show that these regularized algorithms preserve asymptotic normality and valid inference under a prescribed level of adversarial corruption. Finally, we show that regularization is necessary rather than incidental: Lai--Wei stability is incompatible with the optimal $O(\sqrt{T})$ regret rate -- the rate attained by unregularized algorithms such as EXP3 -- so that a controlled, polylogarithmic inflation in regret is the price of valid inference.

stat.ML↗

arXiv2026

Bandit Simulation for Average Reward Inference

Multi-arm bandit algorithms are increasingly used in online platforms, clinical trials, and social science experiments, but valid statistical inference on their performance remains an open challenge. After deploying bandits, a natural question is whether one can construct a confidence interval for its mean reward and assess whether it reliably outperforms a baseline policy. The total reward achieved in any single bandit deployment is random, and deploying a bandit twice on the same population typically yields different reward trajectories due to stochastic rewards. Standard statistical inference methods cannot be used because bandit algorithms introduce complex dependencies in the collected data, which violate the i.i.d. assumption underlying many classical approaches. Moreover, existing inference methods for adaptively collected data only apply to estimands that do not depend on the data-collection algorithm (such as the mean reward under a fixed action). We propose Bandit Simulation for Inference (BSI), a framework that fits a simulator of the bandit environment from observed data--either on-policy or off-policy--and uses it to estimate the mean reward under any evaluation policy, including adaptive blackbox algorithms. BSI formally propagates uncertainty in the estimated simulator parameters into the confidence interval construction. Furthermore, for BSI to be valid, it requires only weak exploration assumptions on the behavior policy and avoids importance weighting. We prove that BSI yields asymptotically valid confidence intervals, and demonstrate empirically that it maintains nominal coverage in settings where standard off-policy evaluation methods fail.

stat.ML↗

arXiv2026

Uncertainty Quantification With Multiple Sources

Weighted conformal prediction (WCP) has been commonly used to quantify prediction uncertainty under covariate shift. However, the effectiveness of WCP relies heavily on the degree of overlap between the training and test covariate distributions. This challenge is exacerbated in multi-source settings with varying covariate distributions, where direct application of WCP can be impractical. In this paper, we address the multi-source setup by leveraging WCP under the assumption of a shared conditional distribution. We investigate two extensions of WCP: (i) a merge-based aggregation of source-specific weighted conformal prediction sets, and (ii) a data-pooling strategy that jointly reweights samples across all sources. Theoretical guarantees are provided for the proposed approaches, and experiments are conducted based on a synthetic regression task and a multi-domain image classification benchmark to validate our proposed methods.

stat.ME↗

arXiv2026

Design Stability in Adaptive Experiments: Implications for Treatment Effect Estimation

We study the problem of estimating the average treatment effect (ATE) under sequentially adaptive treatment assignment mechanisms. In contrast to classical completely randomized designs, we consider a setting in which the probability of assigning treatment to each experimental unit may depend on prior assignments and observed outcomes. Within the potential outcomes framework, we propose and analyze two natural estimators for the ATE: the inverse propensity weighted (IPW) estimator and an augmented IPW (AIPW) estimator. The cornerstone of our analysis is the concept of design stability, which requires that as the number of units grows, either the assignment probabilities converge, or sample averages of the inverse propensity scores and of the inverse complement propensity scores converge in probability to fixed, non-random limits. Our main results establish central limit theorems for both the IPW and AIPW estimators under design stability and provide explicit expressions for their asymptotic variances. We further propose estimators for these variances, enabling the construction of asymptotically valid confidence intervals. Finally, we illustrate our theoretical results in the context of Wei's adaptive coin design and Efron's biased coin design, highlighting the applicability of the proposed methods to sequential experimentation with adaptive randomization.

math.ST↗

arXiv2026

Stable Thompson Sampling: Valid Inference via Variance Inflation

We consider the problem of statistical inference when the data is collected via a Thompson Sampling-type algorithm. While Thompson Sampling (TS) is known to be both asymptotically optimal and empirically effective, its adaptive sampling scheme poses challenges for constructing confidence intervals for model parameters. We propose and analyze a variant of TS, called Stable Thompson Sampling, in which the posterior variance is inflated by a logarithmic factor. We show that this modification leads to asymptotically normal estimates of the arm means, despite the non-i.i.d. nature of the data. Importantly, this statistical benefit comes at a modest cost: the variance inflation increases regret by only a logarithmic factor compared to standard TS. Our results reveal a principled trade-off: by paying a small price in regret, one can enable valid statistical inference for adaptive decision-making algorithms.

stat.ML↗

arXiv2026

Efficient Inference after Directionally Stable Adaptive Experiments

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.

stat.ML↗

arXiv2026

Avoiding the Price of Adaptivity: Inference in Linear Contextual Bandits via Stability

Statistical inference in contextual bandits is challenging due to the adaptive, non-i.i.d. nature of the data. A growing body of work shows that classical least-squares inference can fail under adaptive sampling, and that valid confidence intervals for linear functionals typically require an inflation of order $\sqrt{d \log T}$. This phenomenon -- often termed the price of adaptivity -- reflects the intrinsic difficulty of reliable inference under general contextual bandit policies. A key structural condition that overcomes this limitation is the stability condition of Lai and Wei, which requires the empirical feature covariance to converge to a deterministic limit. When stability holds, the ordinary least-squares estimator satisfies a central limit theorem, and classical Wald-type confidence intervals remain asymptotically valid under adaptation, without incurring the $\sqrt{d \log T}$ price of adaptivity. In this paper, we propose and analyze a regularized 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. We additionally provide quantitative rates of convergence in the associated central limit theorem. 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. As an application of our theory, we show how it can be used to construct confidence intervals for the conditional average treatment effect (CATE) under adaptively collected data. Finally, we complement our theory with simulations illustrating the empirical normality of the resulting estimators and the sharpness of the corresponding confidence intervals.

stat.ML↗

arXiv2025

On Instability of Minimax Optimal Optimism-Based Bandit Algorithms

Statistical inference from data generated by multi-armed bandit (MAB) algorithms is challenging due to their adaptive, non-i.i.d. nature. A classical manifestation is that sample averages of arm rewards under bandit sampling may fail to satisfy a central limit theorem. Lai and Wei's stability condition provides a sufficient, and essentially necessary criterion, for asymptotic normality in bandit problems. While the celebrated Upper Confidence Bound (UCB) algorithm satisfies this stability condition, it is not minimax optimal, raising the question of whether minimax optimality and statistical stability can be achieved simultaneously. In this paper, we analyze the stability properties of a broad class of bandit algorithms that are based on the optimism principle. We establish general structural conditions under which such algorithms violate the Lai-Wei stability criterion. As a consequence, we show that widely used minimax-optimal UCB-style algorithms, including MOSS, Anytime-MOSS, Vanilla-MOSS, ADA-UCB, OC-UCB, KL-MOSS, KL-UCB++, KL-UCB-SWITCH, and Anytime KL-UCB-SWITCH, are unstable. We further complement our theoretical results with numerical simulations demonstrating that, in all these cases, the sample means fail to exhibit asymptotic normality. Overall, our findings suggest a fundamental tension between stability and minimax optimal regret, raising the question of whether it is possible to design bandit algorithms that achieve both. Understanding whether such simultaneously stable and minimax optimal strategies exist remains an important open direction.

stat.ML↗

arXiv2025

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.

math.ST↗

arXiv2024

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 $σ\sqrt{K\log T/T}$. 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.

math.ST↗

arXiv2024

PICS: A sequential approach to obtain optimal designs for non-linear models leveraging closed-form solutions for faster convergence

D-Optimal designs for estimating parameters of response models are derived by maximizing the determinant of the Fisher information matrix. For non-linear models, the Fisher information matrix depends on the unknown parameter vector of interest, leading to a weird situation that in order to obtain the D-optimal design, one needs to have knowledge of the parameter to be estimated. One solution to this problem is to choose the design points sequentially, optimizing the D-optimality criterion using parameter estimates based on available data, followed by updating the parameter estimates using maximum likelihood estimation. On the other hand, there are many non-linear models for which closed-form results for D-optimal designs are available, but because such solutions involve the parameters to be estimated, they can only be used by substituting "guestimates" of parameters. In this paper, a hybrid sequential strategy called PICS (Plug into closed-form solution) is proposed that replaces the optimization of the objective function at every single step by a draw from the probability distribution induced by the known optimal design by plugging in the current estimates. Under regularity conditions, asymptotic normality of the sequence of estimators generated by this approach are established. Usefulness of this approach in terms of saving computational time and achieving greater efficiency of estimation compared to the standard sequential approach are demonstrated with simulations conducted from two different sets of models.

stat.ME↗

arXiv2024

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.

stat.ML↗

arXiv2024

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 Hàjek and Le Cam up to universal constants and a logarithmic factor of the sample size.

math.OC↗

arXiv2023

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.

math.ST↗

arXiv2023

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.

math.ST↗

arXiv2023

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 a family of online debiasing estimators to correct these distributional anomalies in least squares estimation. Our proposed methods take advantage of the covariance structure present in the dataset and provide sharper estimates in directions for which more information has accrued. We establish an asymptotic normality property for our proposed online debiasing estimators 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 estimators achieve the minimax lower bound. We demonstrate the usefulness of our theory via applications to multi-armed bandit, autoregressive time series estimation, and active learning with exploration.

math.ST↗

arXiv2022

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.

math.ST↗

arXiv2022

Instability, Computational Efficiency and Statistical Accuracy

Many statistical estimators are defined as the fixed point of a data-dependent operator, with estimators based on minimizing a cost function being an important special case. The limiting performance of such estimators depends on the properties of the population-level operator in the idealized limit of infinitely many samples. We develop a general framework that yields bounds on statistical accuracy based on the interplay between the deterministic convergence rate of the algorithm at the population level, and its degree of (in)stability when applied to an empirical object based on $n$ samples. Using this framework, we analyze both stable forms of gradient descent and some higher-order and unstable algorithms, including Newton's method and its cubic-regularized variant, as well as the EM algorithm. We provide applications of our general results to several concrete classes of models, including Gaussian mixture estimation, non-linear regression models, and informative non-response models. We exhibit cases in which an unstable algorithm can achieve the same statistical accuracy as a stable algorithm in exponentially fewer steps -- namely, with the number of iterations being reduced from polynomial to logarithmic in sample size $n$.

cs.LG↗