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 $\widehat{\boldsymbol{p}}$ and contains the unknown $\boldsymbol{p}$ with a specified confidence. This paper shows how one can construct minimum average volume confidence regions, answering a long standing question. We also show the optimality of the regions directly translates to optimal confidence intervals of linear functionals such as the mean, implying sample complexity and regret improvements for adaptive machine learning algorithms.
Linear combinations of multinomial probabilities, such as those resulting from contingency tables, are of use when evaluating classification system performance. While large sample inference methods for these combinations exist, small sample methods exist only for regions on the multinomial parameter space instead of the linear combinations. However, in medical classification problems it is common to have small samples necessitating a small sample confidence interval on linear combinations of multinomial probabilities. Therefore, in this paper we derive an exact confidence interval, through the…
Non-asymptotic approximations for Pearson's chi-square statistic and its application to confidence intervals for strictly convex functions of the probability weights of discrete distributions
We consider multinomial goodness-of-fit tests in the high-dimensional regime where the number of bins increases with the sample size. In this regime, Pearson's chi-squared test can suffer from low power due to the substantial bias as well as high variance of its statistic. To resolve these issues, we introduce a family of U-statistic for multinomial goodness-of-fit and study their asymptotic behaviors in high-dimensions. Specifically, we establish conditions under which the considered U-statistic is asymptotically Poisson or Gaussian, and investigate its power function under each asymptotic re…
Multinomial processing tree (MPT) models are tools for disentangling the contributions of latent cognitive processes in a given experimental paradigm. The present note analyzes MPT models subject to order constraints on subsets of its parameters. The constraints that we consider frequently arise in cases where the response categories are ordered in some sense such as in confidence-rating data, Likert scale data, where graded guessing tendencies or response biases are created via base-rate or payoff manipulations, in the analysis of contingency tables with order constraints, and in many other c…
We introduce a semiparametric ``tubular neighborhood'' of a parametric model in the multinomial setting. It consists of all multinomial distributions lying in a distance-based neighborhood of the parametric model of interest. Fitting such a tubular model allows one to use a parametric model while treating it as an approximation to the true distribution. In this paper, the Kullback--Leibler distance is used to build the tubular region. Based on this idea one can define the distance between the true multinomial distribution and the parametric model to be the index of fit. The paper develops a li…
Enjoying Non-linearity in Multinomial Logistic Bandits: A Minimax-Optimal Algorithm
INRIA（法国国家信息与自动化研究所）；École Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立高等物理、数学与信息学院）；SDA Bocconi School of Management（博科尼管理学院）
Finding anonymization mechanisms to protect personal data is at the heart of recent machine learning research. Here, we consider the consequences of local differential privacy constraints on goodness-of-fit testing, i.e. the statistical problem assessing whether sample points are generated from a fixed density $f_0$, or not. The observations are kept hidden and replaced by a stochastic transformation satisfying the local differential privacy constraint. In this setting, we propose a testing procedure which is based on an estimation of the quadratic distance between the density $f$ of the unobs…
In Quantum Non Demolition measurements, the sequence of observations is distributed as a mixture of multinomial random variables. Parameters of the dynamics are naturally encoded into this family of distributions. We show the local asymptotic mixed normality of the underlying statistical model and the consistency of the maximum likelihood estimator. Furthermore, we prove the asymptotic optimality of this estimator as it saturates the usual Cramér Rao bound.
Combinatorial Inference on the Optimal Assortment in Multinomial Logit Models
Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）；Leonard N. Stern School of Business, New York University（纽约大学伦纳德·N·斯特恩商学院）；Duke University（杜克大学）
$\ell_1$-penalized Multinomial Regression: Estimation, inference, and prediction, with an application to risk factor identification for different dementia subtypes
Leonard N. Stern School of Business, New York University（纽约大学伦纳德·N·斯特恩商学院）；Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Computer Science Department, Indiana University at Bloomington（印第安纳大学伯明顿分校计算机科学系）
We study the dynamic assortment planning problem, where for each arriving customer, the seller offers an assortment of substitutable products and customer makes the purchase among offered products according to an uncapacitated multinomial logit (MNL) model. Since all the utility parameters of MNL are unknown, the seller needs to simultaneously learn customers' choice behavior and make dynamic decisions on assortments based on the current knowledge. The goal of the seller is to maximize the expected revenue, or equivalently, to minimize the expected regret. Although dynamic assortment planning …
University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；University of British Columbia（不列颠哥伦比亚大学）
Out of the rich family of generalized linear bandits, perhaps the most well studied ones are logisitc bandits that are used in problems with binary rewards: for instance, when the learner/agent tries to maximize the profit over a user that can select one of two possible outcomes (e.g., `click' vs `no-click'). Despite remarkable recent progress and improved algorithms for logistic bandits, existing works do not address practical situations where the number of outcomes that can be selected by the user is larger than two (e.g., `click', `show me later', `never show again', `no click'). In this pa…
We consider the goodness-of-fit testing problem of distinguishing whether the data are drawn from a specified distribution, versus a composite alternative separated from the null in the total variation metric. In the discrete case, we consider goodness-of-fit testing when the null distribution has a possibly growing or unbounded number of categories. In the continuous case, we consider testing a Lipschitz density, with possibly unbounded support, in the low-smoothness regime where the Lipschitz parameter is not assumed to be constant. In contrast to existing results, we show that the minimax r…
We consider the identity testing problem - or goodness-of-fit testing problem - in multivariate binomial families, multivariate Poisson families and multinomial distributions. Given a known distribution $p$ and $n$ iid samples drawn from an unknown distribution $q$, we investigate how large $ρ>0$ should be to distinguish, with high probability, the case $p=q$ from the case $d(p,q) \geq ρ$, where $d$ denotes a specific distance over probability distributions. We answer this question in the case of a family of different distances: $d(p,q) = \|p-q\|_t$ for $t \in [1,2]$ where $\|\cdot\|_t$ is the…
Yale University（耶鲁大学）；University of Michigan（密歇根大学）
Information theoretic measures (e.g. the Kullback Liebler divergence and Shannon mutual information) have been used for exploring possibly nonlinear multivariate dependencies in high dimension. If these dependencies are assumed to follow a Markov factor graph model, this exploration process is called structure discovery. For discrete-valued samples, estimates of the information divergence over the parametric class of multinomial models lead to structure discovery methods whose mean squared error achieves parametric convergence rates as the sample size grows. However, a naive application of thi…
Fast Model Selection and Stable Optimization for Softmax-Gated Multinomial-Logistic Mixture of Experts Models
School of Computing, National University of Singapore, Singapore.（新加坡国立大学计算机学院）；ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems（细胞系统数学分析卓越研究中心）；School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.（昆士兰科技大学数学科学学院）；Department of Statistics and Data Science, University of Texas at Austin, Austin, USA（德克萨斯大学奥斯汀分校统计与数据科学系）
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…
An $(n,k)$-Poisson Multinomial Distribution (PMD) is the distribution of the sum of $n$ independent random vectors supported on the set ${\cal B}_k=\{e_1,\ldots,e_k\}$ of standard basis vectors in $\mathbb{R}^k$. We show that any $(n,k)$-PMD is ${\rm poly}\left({k\over σ}\right)$-close in total variation distance to the (appropriately discretized) multi-dimensional Gaussian with the same first two moments, removing the dependence on $n$ from the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is obtained by bootstrapping the Valiant-Valiant CLT itself through the structura…
Many inference problems, such as sequential decision problems like A/B testing, adaptive sampling schemes like bandit selection, are often online in nature. The fundamental problem for online inference is to provide a sequence of confidence intervals that are valid uniformly over the growing-into-infinity sample sizes. To address this question, we provide a near-optimal confidence sequence for bounded random variables by utilizing Bentkus' concentration results. We show that it improves on the existing approaches that use the Cram{é}r-Chernoff technique such as the Hoeffding, Bernstein, and Be…
In this paper, we develop an approach for optimizing the explicit binomial confidence interval recently derived by Chen et al. The optimization reduces conservativeness while guaranteeing prescribed coverage probability.
One-sided confidence intervals are presented for the average of non-identical Bernoulli parameters. These confidence intervals are expressed as analytical functions of the total number of Bernoulli games won, the number of rounds and the confidence level. Tightness of these bounds in the sense of Buehler, i.e. as the strictest possible monotonic intervals, is demonstrated for all confidence levels. A simple interval valid for all confidence levels is also provided with a tightness guarantee. Finally, an application of the proposed confidence intervals to sequential sampling is discussed.
Learning in Position-Aware Multinomial Logit Bandits: From Multiplicative to General Position Effects
Leonard N. Stern School of Business, New York University（纽约大学勒纳商学院）；Qiuzhen College, Tsinghua University（清华大学齐臻学院）；Department of Management Science, School of Management, Fudan University（复旦大学管理学院管理科学系）；Yau Mathematical Sciences Center & Department of Mathematical Sciences, Tsinghua University（清华大学尤数学科学中心及数学科学系）
University of Edinburgh（爱丁堡大学）；University of Southern California（南加州大学）；University of California, San Diego（加州大学圣地亚哥分校）
An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ In this paper, we obtain a refined structural understanding of PMDs by analyzing their Fourier transform. As our core structural result, we prove that the Fourier transform of PMDs is {\em approximately sparse}, i.e., roughly speaking, its $L_1$-norm is small outside a small set. By building on this result, we obtain the following applications: {\bf Learning Theory.} We desi…
We study an extension of the classic stochastic multi-armed bandit problem which involves multiple plays and Markovian rewards in the rested bandits setting. In order to tackle this problem we consider an adaptive allocation rule which at each stage combines the information from the sample means of all the arms, with the Kullback-Leibler upper confidence bound of a single arm which is selected in round-robin way. For rewards generated from a one-parameter exponential family of Markov chains, we provide a finite-time upper bound for the regret incurred from this adaptive allocation rule, which …
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…
We present an efficient method of calculating exact confidence intervals for the hypergeometric parameter representing the number of "successes," or "special items," in the population. The method inverts minimum-width acceptance intervals after shifting them to make their endpoints nondecreasing while preserving their level. The resulting set of confidence intervals achieves minimum possible average size, and even in comparison with confidence sets not required to be intervals it attains the minimum possible cardinality most of the time, and always within $1$. The method compares favorably wit…
The University of Iowa（爱荷华大学）；University of Central Florida（中佛罗里达大学）
Dropout has been witnessed with great success in training deep neural networks by independently zeroing out the outputs of neurons at random. It has also received a surge of interest for shallow learning, e.g., logistic regression. However, the independent sampling for dropout could be suboptimal for the sake of convergence. In this paper, we propose to use multinomial sampling for dropout, i.e., sampling features or neurons according to a multinomial distribution with different probabilities for different features/neurons. To exhibit the optimal dropout probabilities, we analyze the shallow l…
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 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…
Computation of confidence sets is central to data science and machine learning, serving as the workhorse of A/B testing and underpinning the operation and analysis of reinforcement learning algorithms. This paper studies the geometry of the minimum-volume confidence sets for the multinomial parameter. When used in place of more standard confidence sets and intervals based on bounds and asymptotic approximation, learning algorithms can exhibit improved sample complexity. Prior work showed the minimum-volume confidence sets are the level-sets of a discontinuous function defined by an exact p-val…
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…
This paper studies higher-order inference properties of nonparametric local polynomial regression methods under random sampling. We prove Edgeworth expansions for $t$ statistics and coverage error expansions for interval estimators that (i) hold uniformly in the data generating process, (ii) allow for the uniform kernel, and (iii) cover estimation of derivatives of the regression function. The terms of the higher-order expansions, and their associated rates as a function of the sample size and bandwidth sequence, depend on the smoothness of the population regression function, the smoothness ex…
We construct honest confidence regions for a Hilbert space-valued parameter in various statistical models. The confidence sets can be centered at arbitrary adaptive estimators, and have diameter which adapts optimally to a given selection of models. The latter adaptation is necessarily limited in scope. We review the notion of adaptive confidence regions, and relate the optimal rates of the diameter of adaptive confidence regions to the minimax rates for testing and estimation. Applications include the finite normal mean model, the white noise model, density estimation and regression with rand…
This paper is motivated by the need to quantify human immune responses to environmental challenges. Specifically, the genome of the selected cell population from a blood sample is amplified by the well-known PCR process of successive heating and cooling, producing a large number of reads. They number roughly 30,000 to 300,000. Each read corresponds to a particular rearrangement of so-called V(D)J sequences. In the end, the observation consists of a set of numbers of reads corresponding to different V(D)J sequences. The underlying relative frequencies of distinct V(D)J sequences can be summariz…
We propose a new procedure for inference on optimal treatment regimes in the model-free setting, which does not require to specify an outcome regression model. Existing model-free estimators for optimal treatment regimes are usually not suitable for the purpose of inference, because they either have nonstandard asymptotic distributions or do not necessarily guarantee consistent estimation of the parameter indexing the Bayes rule due to the use of surrogate loss. We first study a smoothed robust estimator that directly targets the parameter corresponding to the Bayes decision rule for optimal t…
