We consider the basic problem of learning an unknown partition of $n$ elements into at most $k$ sets using simple queries that reveal information about a small subset of elements. Our starting point is the well-studied pairwise same-set queries which ask if a pair of elements belong to the same class. It is known that non-adaptive algorithms require $Θ(n^2)$ queries, while adaptive algorithms require $Θ(nk)$ queries, and the best known algorithm uses $k-1$ rounds. This problem has been studied extensively over the last two decades in multiple communities due to its fundamental nature and relevance to clustering, active learning, and crowd sourcing. In many applications, it is of high interest to reduce adaptivity while minimizing query complexity. We give a complete characterization of the deterministic query complexity of this problem as a function of the number of rounds, $r$, interpolating between the non-adaptive and adaptive settings: for any constant $r$, the query complexity is $Θ(n^{1+\frac{1}{2^r-1}}k^{1-\frac{1}{2^r-1}})$. Our algorithm only needs $O(\log \log n)$ rounds to attain the optimal $O(nk)$ query complexity. Next, we consider two generalizations of pairwise queries to subsets $S$ of size at most $s$: (1) weak subset queries which return the number of classes intersected by $S$, and (2) strong subset queries which return the entire partition restricted on $S$. Once again in crowd sourcing applications, queries on large sets may be prohibitive. For non-adaptive algorithms, we show $Ω(n^2/s^2)$ strong queries are needed. Perhaps surprisingly, we show that there is a non-adaptive algorithm using weak queries that matches this bound up to log-factors for all $s \leq \sqrt{n}$. More generally, we obtain nearly matching upper and lower bounds for algorithms using subset queries in terms of both the number of rounds, $r$, and the query size bound, $s$.

Recent studies show that pretraining a deep neural network with fine-grained labeled data, followed by fine-tuning on coarse-labeled data for downstream tasks, often yields better generalization than pretraining with coarse-labeled data. While there is ample empirical evidence supporting this, the theoretical justification remains an open problem. This paper addresses this gap by introducing a "hierarchical multi-view" structure to confine the input data distribution. Under this framework, we prove that: 1) coarse-grained pretraining only allows a neural network to learn the common features well, while 2) fine-grained pretraining helps the network learn the rare features in addition to the common ones, leading to improved accuracy on hard downstream test samples.

The minimax sample complexity of group distributionally robust optimization (GDRO) has been determined up to a $\log(K)$ factor, for $K$ the number of groups. In this work, we venture beyond the minimax perspective via a novel notion of sparsity that we dub $(\lambda, \beta)$-sparsity. In short, this condition means that at any parameter $\theta$, there is a set of at most $\beta$ groups whose risks at $\theta$ all are at least $\lambda$ larger than the risks of the other groups. To find an $\epsilon$-optimal $\theta$, we show via a novel algorithm and analysis that the $\epsilon$-dependent term in the sample complexity can swap a linear dependence on $K$ for a linear dependence on the potentially much smaller $\beta$. This improvement leverages recent progress in sleeping bandits, showing a fundamental connection between the two-player zero-sum game optimization framework for GDRO and per-action regret bounds in sleeping bandits. The aforementioned result assumes having a particular $\lambda$ as input. Perhaps surprisingly, we next show an adaptive algorithm which, up to log factors, gets sample complexity that adapts to the best $(\lambda, \beta)$-sparsity condition that holds. Finally, for a particular input $\lambda$, we also show how to get a dimension-free sample complexity result.

This paper addresses the item ranking problem with associate covariates, focusing on scenarios where the preference scores can not be fully explained by covariates, and the remaining intrinsic scores, are sparse. Specifically, we extend the pioneering Bradley-Terry-Luce (BTL) model by incorporating covariate information and considering sparse individual intrinsic scores. Our work introduces novel model identification conditions and examines the regularized penalized Maximum Likelihood Estimator (MLE) statistical rates. We then construct a debiased estimator for the penalized MLE and analyze its distributional properties. Additionally, we apply our method to the goodness-of-fit test for models with no latent intrinsic scores, namely, the covariates fully explaining the preference scores of individual items. We also offer confidence intervals for ranks. Our numerical studies lend further support to our theoretical findings, demonstrating validation for our proposed method

Community detection or clustering refers to the task of gathering similar items into a few groups from the data that, most often, correspond to observations of pair-wise interactions between items Newman and Girvan [2004]. A benchmark commonly used to assess the performance of clustering algorithms is the celebrated Stochastic Block Model (SBM) Holland et al. [1983], where pair-wise interactions are represented by a random graph. In this graph, the vertices correspond to items, and the presence of an edge between two items indicates their interaction. The SBM has been extensively studied over the last two decades; for a recent survey, see Abbe [2018]. However, it provides a relatively simplistic view of how items may interact. In real applications, interactions can be of different types (e.g., represented by ratings in recommender systems or a level of proximity between users in a social network). To capture this richer information about item interactions, the Labeled Stochastic Block Model (LSBM), proposed and analyzed in Heimlicher et al. [2012], Lelarge et al. [2013], Yun and Proutiere [2016], describes interactions by labels drawn from an arbitrary collection. The objective of this paper is to devise a clustering algorithm that, based on the observation of these labels, reconstructs the clusters of items while minimizing the expected number of misclassified items. In the following, we formally introduce LSBMs and outline our results.

The problem of causal inference with panel data is a central econometric question. The following is a fundamental version of this problem: Let $M^*$ be a low rank matrix and $E$ be a zero-mean noise matrix. For a `treatment' matrix $Z$ with entries in $\{0,1\}$ we observe the matrix $O$ with entries $O_{ij} := M^*_{ij} + E_{ij} + \mathcal{T}_{ij} Z_{ij}$ where $\mathcal{T}_{ij} $ are unknown, heterogenous treatment effects. The problem requires we estimate the average treatment effect $\tau^* := \sum_{ij} \mathcal{T}_{ij} Z_{ij} / \sum_{ij} Z_{ij}$. The synthetic control paradigm provides an approach to estimating $\tau^*$ when $Z$ places support on a single row. This paper extends that framework to allow rate-optimal recovery of $\tau^*$ for general $Z$, thus broadly expanding its applicability. Our guarantees are the first of their type in this general setting. Computational experiments on synthetic and real-world data show a substantial advantage over competing estimators.

We consider the multivariate max-linear regression problem where the model parameters $\boldsymbol{\beta}_{1},\dotsc,\boldsymbol{\beta}_{k}\in\mathbb{R}^{p}$ need to be estimated from $n$ independent samples of the (noisy) observations $y = \max_{1\leq j \leq k} \boldsymbol{\beta}_{j}^{\mathsf{T}} \boldsymbol{x} + \mathrm{noise}$. The max-linear model vastly generalizes the conventional linear model, and it can approximate any convex function to an arbitrary accuracy when the number of linear models $k$ is large enough. However, the inherent nonlinearity of the max-linear model renders the estimation of the regression parameters computationally challenging. Particularly, no estimator based on convex programming is known in the literature. We formulate and analyze a scalable convex program as the estimator for the max-linear regression problem. Under the standard Gaussian observation setting, we present a non-asymptotic performance guarantee showing that the convex program recovers the parameters with high probability. When the $k$ linear components are equally likely to achieve the maximum, our result shows that a sufficient number of observations scales as $k^{2}p$ up to a logarithmic factor. This significantly improves on the analogous prior result based on alternating minimization (Ghosh et al., 2019). Finally, through a set of Monte Carlo simulations, we illustrate that our theoretical result is consistent with empirical behavior, and the convex estimator for max-linear regression is as competitive as the alternating minimization algorithm in practice.

We propose a distributed bootstrap method for simultaneous inference on high-dimensional massive data that are stored and processed with many machines. The method produces a $\ell_\infty$-norm confidence region based on a communication-efficient de-biased lasso, and we propose an efficient cross-validation approach to tune the method at every iteration. We theoretically prove a lower bound on the number of communication rounds $\tau_{\min}$ that warrants the statistical accuracy and efficiency. Furthermore, $\tau_{\min}$ only increases logarithmically with the number of workers and intrinsic dimensionality, while nearly invariant to the nominal dimensionality. We test our theory by extensive simulation studies, and a variable screening task on a semi-synthetic dataset based on the US Airline On-time Performance dataset. The code to reproduce the numerical results is available at GitHub: https://github.com/skchao74/Distributed-bootstrap.

In this paper, we propose a bootstrap method applied to massive data processed distributedly in a large number of machines. This new method is computationally efficient in that we bootstrap on the master machine without over-resampling, typically required by existing methods \cite{kleiner2014scalable,sengupta2016subsampled}, while provably achieving optimal statistical efficiency with minimal communication. Our method does not require repeatedly re-fitting the model but only applies multiplier bootstrap in the master machine on the gradients received from the worker machines. Simulations validate our theory.

In this paper we show that the computational complexity of the Iterative Thresholding and K-Residual-Means (ITKrM) algorithm for dictionary learning can be significantly reduced by using dimensionality reduction techniques based on the Johnson-Lindenstrauss Lemma. We introduce the Iterative Compressed-Thresholding and K-Means (IcTKM) algorithm for fast dictionary learning and study its convergence properties. We show that IcTKM can locally recover a generating dictionary with low computational complexity up to a target error $\tilde{\varepsilon}$ by compressing $d$-dimensional training data into $m < d$ dimensions, where $m$ is proportional to $\log d$ and inversely proportional to the distortion level $\delta$ incurred by compressing the data. Increasing the distortion level $\delta$ reduces the computational complexity of IcTKM at the cost of an increased recovery error and reduced admissible sparsity level for the training data. For generating dictionaries comprised of $K$ atoms, we show that IcTKM can stably recover the dictionary with distortion levels up to the order $\delta \leq O(1/\sqrt{\log K})$. The compression effectively shatters the data dimension bottleneck in the computational cost of the ITKrM algorithm. For training data with sparsity levels $S \leq O(K^{2/3})$, ITKrM can locally recover the dictionary with a computational cost that scales as $O(d K \log(\tilde{\varepsilon}^{-1}))$ per training signal. We show that for these same sparsity levels the computational cost can be brought down to $O(\log^5 (d) K \log(\tilde{\varepsilon}^{-1}))$ with IcTKM, a significant reduction when high-dimensional data is considered. Our theoretical results are complemented with numerical simulations which demonstrate that IcTKM is a powerful, low-cost algorithm for learning dictionaries from high-dimensional data sets.