Constructing nonasymptotic confidence intervals (CIs) for the mean of a univariate distribution from independent and identically distributed (i.i.d.) observations is a fundamental task in statistics. For bounded observations, a classical nonparametric approach proceeds by inverting standard concentration bounds, such as Hoeffding's or Bernstein's inequalities. Recently, an alternative betting-based approach for defining CIs and their time-uniform variants called confidence sequences (CSs), has been shown to be empirically superior to the classical methods. In this paper, we provide theoretical justification for this improved empirical performance of betting CIs and CSs.
Our main contributions are as follows: (i) We first compare CIs using the values of their first-order asymptotic widths (scaled by $\sqrt{n}$), and show that the betting CI of Waudby-Smith and Ramdas (2023) has a smaller limiting width than existing empirical Bernstein (EB)-CIs. (ii) Next, we establish two lower bounds that characterize the minimum width achievable by any method for constructing CIs/CSs in terms of certain inverse information projections. (iii) Finally, we show that the betting CI and CS match the fundamental limits, modulo an additive logarithmic term and a multiplicative constant. Overall these results imply that the betting CI~(and CS) admit stronger theoretical guarantees than the existing state-of-the-art EB-CI~(and CS); both in the asymptotic and finite-sample regimes.
This paper derives confidence intervals (CI) and time-uniform confidence sequences (CS) for the classical problem of estimating an unknown mean from bounded observations. We present a general approach for deriving concentration bounds, that can be seen as a generalization and improvement of the celebrated Chernoff method. At its heart, it is based on a class of composite nonnegative martingales, with strong connections to testing by betting and the method of mixtures. We show how to extend these ideas to sampling without replacement, another heavily studied problem. In all cases, our bounds ar…
We use a logical device called the Dutch Book to establish epistemic confidence, defined as the sense of confidence \emph{in an observed} confidence interval. This epistemic property is unavailable -- or even denied -- in orthodox frequentist inference. In financial markets, including the betting market, the Dutch Book is also known as arbitrage or risk-free profitable transaction. A numerical confidence is deemed epistemic if its use as a betting price is protected from the Dutch Book by an external agent. Theoretically, to construct the Dutch Book, the agent must exploit unused information a…
Boston University（波士顿大学）；University of Arizona（亚利桑那大学）
A classic problem in statistics is the estimation of the expectation of random variables from samples. This gives rise to the tightly connected problems of deriving concentration inequalities and confidence sequences, that is confidence intervals that hold uniformly over time. Previous work has shown how to easily convert the regret guarantee of an online betting algorithm into a time-uniform concentration inequality. In this paper, we show that we can go even further: We show that the regret of universal portfolio algorithms give rise to new implicit time-uniform concentrations and state-of-t…
We study the distribution and uncertainty of nonconvex optimization for noisy tensor completion -- the problem of estimating a low-rank tensor given incomplete and corrupted observations of its entries. Focusing on a two-stage estimation algorithm proposed by Cai et al. (2019), we characterize the distribution of this nonconvex estimator down to fine scales. This distributional theory in turn allows one to construct valid and short confidence intervals for both the unseen tensor entries and the unknown tensor factors. The proposed inferential procedure enjoys several important features: (1) it…
We consider the problem of finding a proper confidence interval for the mean based on a single observation from a normal distribution with both mean and variance unknown. Portnoy (2017) characterizes the scale-sign invariant rules and shows that the Hunt-Stein construction provides a randomized invariant rule that improves on any given randomized rule in the sense that it has greater minimal coverage among all procedures with a fixed expected length. Mathematical results here provide a specific mixture of two non-randomized invariant rules that achieve the minimax optimality. A multivariate co…
Berry et al. (1997) initiated the development of the infinite arms bandit problem. They derived a regret lower bound of all allocation strategies for Bernoulli rewards with uniform priors, and proposed strategies based on success runs. Bonald and Proutière (2013) proposed a two-target algorithm that achieves the regret lower bound, and extended optimality to Bernoulli rewards with general priors. We present here a confidence bound target (CBT) algorithm that achieves optimality for rewards that are bounded above. For each arm we construct a confidence bound and compare it against each other an…
We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs characterized by simple linear constraints. Employing this characterization, we obtain a class of E-optimal exact designs for treatment-control comparisons for unequal block sizes. In the studied model, we justify the use of E-optimality by providing a statistical interpretation for all E-optimal approximate designs and for the known classes of E-optimal exact designs. Moreover, we consider the R-optimality criterion, which minimizes th…
A standard approach to confidence intervals for quantiles requires good estimates of the quantile density. The optimal bandwidth for kernel estimation of the quantile density depends on an underlying location-scale family only through the quantile optimality ratio (QOR), which is the starting point for our results. While the QOR is not distribution-free, it turns out that what is optimal for one family often works quite well for families having similar shape. This allows one to rely on a single representative QOR if one has a rough idea of the distributional shape. Another option that we explo…
Tightening optimality gap with confidence through conformal prediction
H. Milton Stewart School of Industrial and Systems Engineering（H. Milton Stewart工业与系统工程学院）；Georgia Institute of Technology（佐治亚理工学院）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Missouri University of Science & Technology（密苏里科技大学）
Construction of tight confidence regions and intervals is central to statistical inference and decision making. This paper develops new theory showing minimum average volume confidence regions for categorical data. More precisely, consider an empirical distribution $\widehat{\boldsymbol{p}}$ generated from $n$ iid realizations of a random variable that takes one of $k$ possible values according to an unknown distribution $\boldsymbol{p}$. This is analogous to a single draw from a multinomial distribution. A confidence region is a subset of the probability simplex that depends on $\widehat{\bol…
This paper proposes a decorrelation-based approach to test hypotheses and construct confidence intervals for the low dimensional component of high dimensional proportional hazards models. Motivated by the geometric projection principle, we propose new decorrelated score, Wald and partial likelihood ratio statistics. Without assuming model selection consistency, we prove the asymptotic normality of these test statistics, establish their semiparametric optimality. We also develop new procedures for constructing pointwise confidence intervals for the baseline hazard function and baseline survival…
We demonstrate and discuss nonasymptotic bounds in probability for the cost of a regression scheme with a general loss function from the perspective of the Rademacher theory, and for the optimality with respect to the average $L^{2}$-distance to the underlying conditional expectations of least squares regression outcomes from the perspective of the Vapnik-Chervonenkis theory. The results follow from an analysis involving independent but possibly nonstationary training samples and can be extended, in a manner that we explain and illustrate, to relevant cases in which the training sample exhibit…
We study a statistical method to estimate the optimal value, and the optimality gap of a given solution for stochastic optimization as an assessment of the solution quality. Our approach is based on bootstrap aggregating, or bagging, resampled sample average approximation (SAA). We show how this approach leads to valid statistical confidence bounds for non-smooth optimization. We also demonstrate its statistical efficiency and stability that are especially desirable in limited-data situations, and compare these properties with some existing methods. We present our theory that views SAA as a ke…
This paper develops a new mathematical-statistical approach to analyze a class of Flajolet-Martin algorithms (FMa), and provides analytical confidence intervals for the number F0 of distinct elements in a stream, based on Chernoff bounds. The class of FMa has reached a significant popularity in bigdata stream learning, and the attention of the literature has mainly been based on algorithmic aspects, basically complexity optimality, while the statistical analysis of these class of algorithms has been often faced heuristically. The analysis provided here shows deep connections with mathematical …
Department of Mathematics, University of Maryland, College Park（马里兰大学帕克分校数学系）；DEVCOM Army Research Laboratory（DEVCOM陆军研究实验室）
Contextual multi-armed bandits (CMAB) have been widely used for learning to filter and prioritize information according to a user's interest. In this work, we analyze top-K ranking under the CMAB framework where the top-K arms are chosen iteratively to maximize a reward. The context, which represents a set of observable factors related to the user, is used to increase prediction accuracy compared to a standard multi-armed bandit. Contextual bandit methods have mostly been studied under strict linearity assumptions, but we drop that assumption and learn non-linear stochastic reward functions wi…
The paper considers the problem of global optimization in the setup of stochastic process bandits. We introduce an UCB algorithm which builds a cascade of discretization trees based on generic chaining in order to render possible his operability over a continuous domain. The theoretical framework applies to functions under weak probabilistic smoothness assumptions and also extends significantly the spectrum of application of UCB strategies. Moreover generic regret bounds are derived which are then specialized to Gaussian processes indexed on infinite-dimensional spaces as well as to quadratic …
Well-recommended methods of forming `confidence intervals' for a binomial proportion give interval estimates that do not actually meet the definition of a confidence interval, in that their coverages are sometimes lower than the nominal confidence level. The methods are favoured because their intervals have a shorter average length than the Clopper-Pearson (gold-standard) method, whose intervals really are confidence intervals. Comparison of such methods is tricky -- the best method should perhaps be the one that gives the shortest intervals (on average), but when is the coverage of a method s…
We propose a general method for constructing confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in a high-dimensional model. It can be easily adjusted for multiplicity taking dependence among tests into account. For linear models, our method is essentially the same as in Zhang and Zhang [J. R. Stat. Soc. Ser. B Stat. Methodol. 76 (2014) 217-242]: we analyze its asymptotic properties and establish its asymptotic optimality in terms of semiparametric efficiency. Our method naturally extends to generalized linear models with convex loss…
This paper develops new methodology, together with related theories, for combining information from independent studies through confidence distributions. A formal definition of a confidence distribution and its asymptotic counterpart (i.e., asymptotic confidence distribution) are given and illustrated in the context of combining information. Two general combination methods are developed: the first along the lines of combining p-values, with some notable differences in regard to optimality of Bahadur type efficiency; the second by multiplying and normalizing confidence densities. The latter app…
Minimax Optimality in Contextual Dynamic Pricing with General Valuation Models
School of Management, Xiamen University（厦门大学管理学院）；Department of Industrial Engineering and Decision Analytics, The Hong Kong University of Science and Technology（香港科技大学工业工程与决策分析系）
We study three fundamental statistical-learning problems: distribution estimation, property estimation, and property testing. We establish the profile maximum likelihood (PML) estimator as the first unified sample-optimal approach to a wide range of learning tasks. In particular, for every alphabet size $k$ and desired accuracy $\varepsilon$: $\textbf{Distribution estimation}$ Under $\ell_1$ distance, PML yields optimal $Θ(k/(\varepsilon^2\log k))$ sample complexity for sorted-distribution estimation, and a PML-based estimator empirically outperforms the Good-Turing estimator on the actual dis…
Gini index is a widely used measure of economic inequality. This article develops a general theory for constructing a confidence interval for Gini index with a specified confidence coefficient and a specified width. Fixed sample size methods cannot simultaneously achieve both the specified confidence coefficient and specified width. We develop a purely sequential procedure for interval estimation of Gini index with a specified confidence coefficient and a fixed margin of error. Optimality properties of the proposed method, namely first order asymptotic efficiency and asymptotic consistency are…
This article studies local and global inference for smoothing spline estimation in a unified asymptotic framework. We first introduce a new technical tool called functional Bahadur representation, which significantly generalizes the traditional Bahadur representation in parametric models, that is, Bahadur [Ann. Inst. Statist. Math. 37 (1966) 577-580]. Equipped with this tool, we develop four interconnected procedures for inference: (i) pointwise confidence interval; (ii) local likelihood ratio testing; (iii) simultaneous confidence band; (iv) global likelihood ratio testing. In particular, our…
In this paper, we propose a new framework to construct confidence sets for a $d$-dimensional unknown sparse parameter $θ$ under the normal mean model $X\sim N(θ,σ^2I)$. A key feature of the proposed confidence set is its capability to account for the sparsity of $θ$, thus named as {\em sparse} confidence set. This is in sharp contrast with the classical methods, such as Bonferroni confidence intervals and other resampling based procedures, where the sparsity of $θ$ is often ignored. Specifically, we require the desired sparse confidence set to satisfy the following two conditions: (i) uniforml…
The theory of statistical inference along with the strategy of divide-and-conquer for large- scale data analysis has recently attracted considerable interest due to great popularity of the MapReduce programming paradigm in the Apache Hadoop software framework. The central analytic task in the development of statistical inference in the MapReduce paradigm pertains to the method of combining results yielded from separately mapped data batches. One seminal solution based on the confidence distribution has recently been established in the setting of maximum likelihood estimation in the literature.…
Confidence intervals for maximum unseen probabilities, with application to sequential sampling design
Bocconi Institute for Data Science and Analytics（博科尼数据科学与分析研究所）；Bocconi University（博科尼大学）；Department of Economics and Statistics, University of Torino（都灵大学经济与统计系）；Collegio Carlo Alberto（卡罗·阿尔贝托学院）
On Pareto Optimality for Parametric Choice Bandits
Department of Management Science and Engineering, Tsinghua University, Beijing, China（清华大学管理科学与工程系，北京，中国）；Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore（新加坡国立大学工业系统工程与管理系，新加坡）
On the Suboptimality of GP-UCB under Polynomial Effective Optimism
Department of Industrial Systems Engineering and Management, National University of Singapore（新加坡国立大学工业系统工程与管理系）；Department of Industrial Engineering and Decision Analytics, The Hong Kong University of Science and Technology（香港科学与技术大学工业工程与决策分析系）
CNRS（法国国家科学研究中心）；Univ. Lille（里尔大学）；Centrum Wiskunde & Informatica（荷兰数学与计算机科学研究院）；Institut de Mathématiques de Toulouse（图卢兹数学研究所）；Université de Toulouse（图卢兹大学）
Learning the minimum/maximum mean among a finite set of distributions is a fundamental sub-task in planning, game tree search and reinforcement learning. We formalize this learning task as the problem of sequentially testing how the minimum mean among a finite set of distributions compares to a given threshold. We develop refined non-asymptotic lower bounds, which show that optimality mandates very different sampling behavior for a low vs high true minimum. We show that Thompson Sampling and the intuitive Lower Confidence Bounds policy each nail only one of these cases. We develop a novel appr…
We propose methodology for estimation of sparse precision matrices and statistical inference for their low-dimensional parameters in a high-dimensional setting where the number of parameters $p$ can be much larger than the sample size. We show that the novel estimator achieves minimax rates in supremum norm and the low-dimensional components of the estimator have a Gaussian limiting distribution. These results hold uniformly over the class of precision matrices with row sparsity of small order $\sqrt{n}/\log p$ and spectrum uniformly bounded, under a sub-Gaussian tail assumption on the margins…
Starting from the observation of an R^n-Gaussian vector of mean f and covariance matrix σ^2 I_n (I_n is the identity matrix), we propose a method for building a Euclidean confidence ball around f, with prescribed probability of coverage. For each n, we describe its nonasymptotic property and show its optimality with respect to some criteria.
Multiclass classification problems such as image annotation can involve a large number of classes. In this context, confusion between classes can occur, and single label classification may be misleading. We provide in the present paper a general device that, given an unlabeled dataset and a score function defined as the minimizer of some empirical and convex risk, outputs a set of class labels, instead of a single one. Interestingly, this procedure does not require that the unlabeled dataset explores the whole classes. Even more, the method is calibrated to control the expected size of the out…
