We identify fundamental tradeoffs between statistical utility and privacy under local models of privacy in which data is kept private even from the statistician, providing instance-specific bounds for private estimation and learning problems by developing the local minimax risk. In contrast to approaches based on worst-case (minimax) error, which are conservative, this allows us to evaluate the difficulty of individual problem instances and delineate the possibilities for adaptation in private estimation and inference. Our main results show that the local modulus of continuity of the estimand with respect to the variation distance—as opposed to the Hellinger distance central to classical statistics—characterizes rates of convergence under locally private estimation for many notions of privacy, including differential privacy and its relaxations. As consequences of these results, we identify an alternative to the Fisher information for private estimation, giving a more nuanced understanding of the challenges of adaptivity and optimality.

Motivated by applications to single-particle cryo-electron microscopy (cryo-EM), we study several problems of function estimation in a high noise regime, where samples are observed after random rotation and possible linear projection of the function domain. We describe a stratification of the Fisher information eigenvalues according to transcendence degrees of graded pieces of the algebra of group invariants, and we relate critical points of the log-likelihood landscape to a sequence of moment optimization problems, extending previous results for a discrete rotation group without projections.

We then compute the transcendence degrees and forms of these optimization problems for several examples of function estimation under $\mathsf{SO}(2)$ and $\mathsf{SO}(3)$ rotations, including a simplified model of cryo-EM as introduced by Bandeira, Blum-Smith, Kileel, Niles-Weed, Perry and Wein. We affirmatively resolve conjectures that third-order moments are sufficient to locally identify a generic signal up to its rotational orbit in these examples.

For low-dimensional approximations of the electric potential maps of two small protein molecules, we empirically verify that the noise scalings of the Fisher information eigenvalues conform with our theoretical predictions over a range of SNR, in a model of $\mathsf{SO}(3)$ rotations without projections.

Variable selection on the large-scale networks has been extensively studied in the literature. While most of the existing methods are limited to the local functionals especially the graph edges, this paper focuses on selecting the discrete hub structures of the networks. Specifically, we propose an inferential method, called StarTrek filter, to select the hub nodes with degrees larger than a certain thresholding level in the high-dimensional graphical models and control the false discovery rate (FDR). Discovering hub nodes in the networks is challenging: there is no straightforward statistic for testing the degree of a node due to the combinatorial structures; complicated dependence in the multiple testing problem is hard to characterize and control. In methodology, the StarTrek filter overcomes this by constructing p-values based on the maximum test statistics via the Gaussian multiplier bootstrap. In theory, we show that the StarTrek filter can control the FDR by providing accurate bounds on the approximation errors of the quantile estimation and addressing the dependence structures among the maximal statistics.

To this end, we establish novel Cramér-type comparison bounds for the high-dimensional Gaussian random vectors. Compared to the Gaussian comparison bound via the Kolmogorov distance established by Chernozhukov, Chetverikov and Kato (Ann. Statist. 42 (2014) 1787–1818), our Cramér-type comparison bounds establish the relative difference between the distribution functions of two high-dimensional Gaussian random vectors, which is essential in the theoretical analysis of FDR control. Moreover, the StarTrek filter can be applied to general statistical models for FDR control of discovering discrete structures such as simultaneously testing the sparsity levels of multiple high-dimensional linear models. We illustrate the validity of the StarTrek filter in a series of numerical experiments and apply it to the genotype-tissue expression dataset to discover central regulator genes.

In this paper, we introduce a novel class of graphical models for representing time-lag specific causal relationships and independencies of multivariate time series with unobserved confounders. We completely characterize these graphs and show that they constitute proper subsets of the currently employed model classes. As we show, from the novel graphs one can thus draw stronger causal inferences—without additional assumptions. We further introduce a graphical representation of Markov equivalence classes of the novel graphs. This graphical representation contains more causal knowledge than what current state-of-the-art causal discovery algorithms learn.

Tensor PCA is a stylized statistical inference problem introduced by Montanari and Richard to study the computational difficulty of estimating an unknown parameter from higher-order moment tensors. Unlike its matrix counterpart, Tensor PCA exhibits a statistical-computational gap, that is, a sample size regime where the problem is information-theoretically solvable but conjectured to be computationally hard. This paper derives computational lower bounds on the run-time of memory bounded algorithms for Tensor PCA using communication complexity. These lower bounds specify a trade-off among the number of passes through the data sample, the sample size and the memory required by any algorithm that successfully solves Tensor PCA. While the lower bounds do not rule out polynomial-time algorithms, they do imply that many commonly-used algorithms, such as gradient descent and power method, must have a higher iteration count when the sample size is not large enough. Similar lower bounds are obtained for non-Gaussian component analysis, a family of statistical estimation problems in which low-order moment tensors carry no information about the unknown parameter. Finally, stronger lower bounds are obtained for an asymmetric variant of Tensor PCA and related statistical estimation problems. These results explain why many estimators for these problems use a memory state that is significantly larger than the effective dimensionality of the parameter of interest.

This paper studies the semisupervised novelty detection problem where a set of “typical” measurements is available to the researcher. Motivated by recent advances in multiple testing and conformal inference, we propose AdaDetect, a flexible method that is able to wrap around any probabilistic classification algorithm and control the false discovery rate (FDR) on detected novelties in finite samples without any distributional assumption other than exchangeability. In contrast to classical FDR-controlling procedures that are often committed to a pre-specified p-value function, AdaDetect learns the transformation in a data-adaptive manner to focus the power on the directions that distinguish between inliers and outliers. Inspired by the multiple testing literature, we further propose variants of AdaDetect that are adaptive to the proportion of nulls while maintaining the finite-sample FDR control. The methods are illustrated on synthetic datasets and real-world datasets, including an application in astrophysics.

To test independence between two high-dimensional random vectors, we propose three tests based on the rank-based indices derived from Hoeffding’s D, Blum–Kiefer–Rosenblatt’s R and Bergsma–Dassios–Yanagimoto’s ${\mathit{\tau }}^{\ast }$. Under the null hypothesis of independence, we show that the distributions of the proposed test statistics converge to normal ones if the dimensions diverge arbitrarily with the sample size. We further derive an explicit rate of convergence. Thanks to the monotone transformation-invariant property, these distribution-free tests can be readily used to generally distributed random vectors including heavily-tailed ones. We further study the local power of the proposed tests and compare their relative efficiencies with two classic distance covariance/correlation based tests in high-dimensional settings. We establish explicit relationships between D, R, ${\mathit{\tau }}^{\ast }$ and Pearson’s correlation for bivariate normal random variables. The relationships serve as a basis for power comparison. Our theoretical results show that under a Gaussian equicorrelation alternative: (i) the proposed tests are superior to the two classic distance covariance/correlation based tests if the components of random vectors have very different scales; (ii) the asymptotic efficiency of the proposed tests based on D, ${\mathit{\tau }}^{\ast }$ and R are sorted in a descending order.

Motivated by a range of applications, we study in this paper the problem of transfer learning for nonparametric contextual multi-armed bandits under the covariate shift model, where we have data collected from source bandits before the start of the target bandit learning. The minimax rate of convergence for the cumulative regret is established and a novel transfer learning algorithm that attains the minimax regret is proposed. The results quantify the contribution of the data from the source domains for learning in the target domain in the context of nonparametric contextual multi-armed bandits.

In view of the general impossibility of adaptation to unknown smoothness, we develop a data-driven algorithm that achieves near-optimal statistical guarantees (up to a logarithmic factor) while automatically adapting to the unknown parameters over a large collection of parameter spaces under an additional self-similarity assumption. A simulation study is carried out to illustrate the benefits of utilizing the data from the source domains for learning in the target domain.

This paper is concerned with offline reinforcement learning (RL), which learns using precollected data without further exploration. Effective offline RL would be able to accommodate distribution shift and limited data coverage. However, prior results either suffer from suboptimal sample complexities or incur high burn-in cost to reach sample optimality, thus posing an impediment to efficient offline RL in sample-starved applications.

We demonstrate that the model-based (or “plug-in”) approach achieves minimax-optimal sample complexity without any burn-in cost for tabular Markov decision processes (MDPs). Concretely, consider a γ-discounted infinite-horizon (resp., finite-horizon) MDP with S states and effective horizon $\frac{1}{1-\mathit{\gamma }}$ (resp., horizon H), and suppose the distribution shift of data is reflected by some single-policy clipped concentrability coefficient ${\mathit{C}}_{\mathsf{clipped}}^{\star }$. We prove that model-based offline RL yields ε-accuracy with a sample complexity of

$$\left\{\begin{array}{ll}\frac{\mathit{S}{\mathit{C}}_{\mathsf{clipped}}^{\star }}{{(1-\mathit{\gamma })}^3{\mathit{\epsilon }}^2}& (\text{infinite-horizon MDPs}),\\ \frac{{\mathit{H}}^4\mathit{S}{\mathit{C}}_{\mathsf{clipped}}^{\star }}{{\mathit{\epsilon }}^2}& (\text{finite-horizon MDPs}),\end{array}\right.$$

up to log factor, which is minimax optimal for the entire ε-range. The proposed algorithms are “pessimistic” variants of value iteration with Bernstein-style penalties, and do not require sophisticated variance reduction. Our analysis framework is established upon delicate leave-one-out decoupling arguments in conjunction with careful self-bounding techniques tailored to MDPs.

We study the continuous multireference alignment model of estimating a periodic function on the circle from noisy and circularly-rotated observations. Motivated by analogous high-dimensional problems that arise in cryo-electron microscopy, we establish minimax rates for estimating generic signals that are explicit in the dimension K. In a high-noise regime with noise variance ${\mathit{\sigma }}^2\gtrsim \mathit{K}$, for signals with Fourier coefficients of roughly uniform magnitude, the rate scales as ${\mathit{\sigma }}^6$ and has no further dependence on the dimension. This rate is achieved by a bispectrum inversion procedure, and our analyses provide new stability bounds for bispectrum inversion that may be of independent interest. In a low-noise regime where ${\mathit{\sigma }}^2\lesssim \mathit{K}/log\mathit{K}$, the rate scales instead as $\mathit{K}{\mathit{\sigma }}^2$, and we establish this rate by a sharp analysis of the maximum likelihood estimator that marginalizes over latent rotations. A complementary lower bound that interpolates between these two regimes is obtained using Assouad’s hypercube lemma. We extend these analyses also to signals whose Fourier coefficients have a slow power law decay.

Fusing regression coefficients into homogeneous groups can unveil those coefficients that share a common value within each group. Such groupwise homogeneity reduces the intrinsic dimension of the parameter space and unleashes sharper statistical accuracy. We propose and investigate a new combinatorial grouping approach called ${\mathit{L}}_0$-Fusion that is amenable to mixed integer optimization (MIO). On the statistical aspect, we identify a fundamental quantity called MSE grouping sensitivity that underpins the difficulty of recovering the true groups. We show that ${\mathit{L}}_0$-Fusion achieves grouping consistency under the weakest possible requirement of the grouping sensitivity: if this requirement is violated, then the minimax risk of group misspecification will fail to converge to zero. Moreover, we show that in the high-dimensional regime, one can apply ${\mathit{L}}_0$-Fusion with a sure screening set of features without any essential loss of statistical efficiency, while reducing the computational cost substantially. On the algorithmic aspect, we provide an MIO formulation for ${\mathit{L}}_0$-Fusion along with a warm start strategy. Simulation and real data analysis demonstrate that ${\mathit{L}}_0$-Fusion exhibits superiority over its competitors in terms of grouping accuracy.

Testing for pairwise independence for the case where the number of variables may be of the same size or even larger than the sample size has received increasing attention in the recent years. We contribute to this branch of the literature by considering tests that allow to detect higher-order dependencies. The proposed methods are based on connecting the problem to copulas and making use of the Moebius transformation of the empirical copula process; an approach that is related to Lancaster interactions and that has already been used successfully for the case where the number of variables is fixed. Based on a martingale central limit theorem, it is shown that respective test statistics converge to the standard normal distribution, allowing for straightforward definition of critical values. The results are illustrated by a Monte Carlo simulation study.

In this paper, we explore adaptive inference based on variational Bayes. Although several studies have been conducted to analyze the contraction properties of variational posteriors, there is still a lack of a general and computationally tractable variational Bayes method that performs adaptive inference. To fill this gap, we propose a novel adaptive variational Bayes framework, which can operate on a collection of models. The proposed framework first computes a variational posterior over each individual model separately and then combines them with certain weights to produce a variational posterior over the entire model. It turns out that this combined variational posterior is the closest member to the posterior over the entire model in a predefined family of approximating distributions. We show that the adaptive variational Bayes attains optimal contraction rates adaptively under very general conditions. We also provide a methodology to maintain the tractability and adaptive optimality of the adaptive variational Bayes even in the presence of an enormous number of individual models, such as sparse models. We apply the general results to several examples, including deep learning and sparse factor models, and derive new and adaptive inference results. In addition, we characterize an implicit regularization effect of variational Bayes and show that the adaptive variational posterior can utilize this.

The idiosyncratic components of a tensor time series factor model can exhibit serial correlations, (e.g., finance or economic data), ruling out many state-of-the-art methods that assume white/independent idiosyncratic components. While the traditional higher order orthogonal iteration (HOOI) is proved to be convergent to a set of factor loading matrices, the closeness of them to the true underlying factor loading matrices are in general not established, or only under i.i.d. Gaussian noises. Under the presence of serial and cross-correlations in the idiosyncratic components and time series variables with only bounded fourth-order moments, for tensor time series data with tensor order two or above, we propose a pre-averaging procedure that can be considered a random projection method. The estimated directions corresponding to the strongest factors are then used for projecting the data for a potentially improved re-estimation of the factor loading spaces themselves, with theoretical guarantees and rate of convergence spelt out when not all factors are pervasive. We also propose a new rank estimation method, which utilizes correlation information from the projected data. Extensive simulations are performed and compared to other state-of-the-art or traditional alternatives. A set of tensor-valued NYC taxi data is also analyzed.

Optimal estimation and inference for both the minimizer and minimum of a convex regression function under the white noise and nonparametric regression models are studied in a nonasymptotic local minimax framework, where the performance of a procedure is evaluated at individual functions. Fully adaptive and computationally efficient algorithms are proposed and sharp minimax lower bounds are given for both the estimation accuracy and expected length of confidence intervals for the minimizer and minimum.

The nonasymptotic local minimax framework brings out new phenomena in simultaneous estimation and inference for the minimizer and minimum. We establish a novel uncertainty principle that provides a fundamental limit on how well the minimizer and minimum can be estimated simultaneously for any convex regression function. A similar result holds for the expected length of the confidence intervals for the minimizer and minimum.

This erratum offers a correction to Chételat and Wells ((2012) Ann. Statist. 40 3137–3160), following the note of Foroushani and Nkurunziza ((2023) arXiv: 2311.13140).