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AbstractThis paper studies noisy low-rank matrix completion in the presence of heavy-tailed and possibly asymmetric noise, where we aim to estimate an underlying low-rank matrix given a set of highly incomplete noisy entries. Though the matrix completion problem has attracted much attention in the past decade, there is still lack of theoretical understanding when the observations are contaminated by heavy-tailed noises. Prior theory falls short of explaining the empirical results and is unable to capture the optimal dependence of the estimation error on the noise level. In this paper, we adopt an adaptive Huber loss to accommodate heavy-tailed noise, which is robust against large and possibly asymmetric errors when the parameter in the Huber loss function is carefully designed to balance the Huberization biases and robustness to outliers. Then, we propose an efficient nonconvex algorithm via a balanced low-rank Burer-Monteiro matrix factorization and gradient descent with robust spectral initialization. We prove that under merely a bounded second-moment condition on the error distributions, rather than the sub-Gaussian assumption, the Euclidean errors of the iterates generated by the proposed algorithm decrease geometrically fast until achieving a minimax-optimal statistical estimation error, which has the same order as that in the sub-Gaussian case. The key technique behind this significant advancement is a powerful leave-one-out analysis framework. The theoretical results are corroborated by our numerical studies. [ABSTRACT FROM AUTHOR]

AbstractThe response envelope model provides substantial efficiency gains over the standard multivariate linear regression by identifying the material part of the response to the model and by excluding the immaterial part. In this paper, we propose the enhanced response envelope by incorporating a novel envelope regularization term based on a nonconvex manifold formulation. It is shown that the enhanced response envelope can yield better prediction risk than the original envelope estimator. The enhanced response envelope naturally handles high-dimensional data for which the original response envelope is not serviceable without necessary remedies. In an asymptotic high-dimensional regime where the ratio of the number of predictors over the number of samples converges to a non-zero constant, we characterize the risk function and reveal an interesting double descent phenomenon for the envelope model. A simulation study confirms our main theoretical findings. Simulations and real data applications demonstrate that the enhanced response envelope does have significantly improved prediction performance over the original envelope method, especially when the number of predictors is close to or moderately larger than the number of samples. Proofs and additional simulation results are shown in the supplementary file to this paper. [ABSTRACT FROM AUTHOR]

AbstractIn many domains, data measurements can naturally be associated with the leaves of a tree, expressing the relationships among these measurements. For example, companies belong to industries, which in turn belong to ever coarser divisions such as sectors; microbes are commonly arranged in a taxonomic hierarchy from species to kingdoms; street blocks belong to neighborhoods, which in turn belong to larger-scale regions. The problem of tree-based aggregation that we consider in this paper asks which of these tree-defined subgroups of leaves should really be treated as a single entity and which of these entities should be distinguished from each other.We introduce the false split rate, an error measure that describes the degree to which subgroups have been split when they should not have been. While expressible as the false discovery rate in a special case, we show that these measures can be quite different for the general tree structures common in our setting. We then propose a multiple hypothesis testing algorithm for tree-based aggregation, which we prove controls this error measure. We focus on two main examples of tree-based aggregation, one which involves aggregating means and the other which involves aggregating regression coefficients. [ABSTRACT FROM AUTHOR]

AbstractPersonalized pricing, which involves tailoring prices based on individual characteristics, is commonly used by firms to implement a consumer-specific pricing policy. In this process, buyers can also strategically manipulate their feature data to obtain a lower price, incurring certain manipulation costs. Such strategic behavior can hinder firms from maximizing their profits. In this paper, we study the contextual dynamic pricing problem with strategic buyers. The seller does not observe the buyer’s true feature, but a manipulated feature according to buyers’ strategic behavior. In addition, the seller does not observe the buyers’ valuation of the product, but only a binary response indicating whether a sale happens or not. Recognizing these challenges, we propose a strategic dynamic pricing policy that incorporates the buyers’ strategic behavior into the online learning to maximize the seller’s cumulative revenue. We first prove that existing non-strategic pricing policies that neglect the buyers’ strategic behavior result in a linear Ω(T) regret with T the total time horizon, indicating that these policies are not better than a random pricing policy. We then establish an O(T) regret upper bound of our proposed policy and an Ω(T) regret lower bound for any pricing policy within our problem setting. This underscores the rate optimality of our policy. Importantly, our policy is not a mere amalgamation of existing dynamic pricing policies and strategic behavior handling algorithms. Our policy can also accommodate the scenario when the marginal cost of manipulation is unknown in advance. To account for it, we simultaneously estimate the valuation parameter and the cost parameter in the online pricing policy, which is shown to also achieve an O(T) regret bound. Extensive experiments support our theoretical developments and demonstrate the superior performance of our policy compared to other pricing policies that are unaware of the strategic behaviors. [ABSTRACT FROM AUTHOR]

AbstractThis paper proposes a novel dynamic forecasting method using a new supervised Principal Component Analysis (PCA) when a large number of predictors are available. The new supervised PCA provides an effective way to bridge the gap between predictors and the target variable of interest by scaling and combining the predictors and their lagged values, resulting in an effective dynamic forecasting. Unlike the traditional diffusion-index approach, which does not learn the relationships between the predictors and the target variable before conducting PCA, we first re-scale each predictor according to their significance in forecasting the targeted variable in a dynamic fashion, and a PCA is then applied to a re-scaled and additive panel, which establishes a connection between the predictability of the PCA factors and the target variable. We also propose to use penalized methods such as the LASSO to select the significant factors that have superior predictive power over the others. Theoretically, we show that our estimators are consistent and outperform the traditional methods in prediction under some mild conditions. We conduct extensive simulations to verify that the proposed method produces satisfactory forecasting results and outperforms most of the existing methods using the traditional PCA. An example of predicting U.S. macroeconomic variables using a large number of predictors showcases that our method fares better than most of the existing ones in applications. [ABSTRACT FROM AUTHOR]

Abstract–Regression adjustment is widely used in the analysis of randomized experiments to improve the estimation efficiency of the treatment effect. This paper reexamines a weighted regression adjustment method termed tyranny-of-the-minority (ToM), wherein units in the minority group are given greater weights. We demonstrate that ToM regression adjustment is more robust than Lin (2013)’s regression adjustment with treatment-covariate interactions, even though these two regression adjustment methods are asymptotically equivalent in completely randomized experiments. Moreover, ToM regression adjustment can be easily extended to stratified randomized experiments and completely randomized survey experiments. We obtain the design-based properties of the ToM regression-adjusted average treatment effect estimator under such designs. In particular, we show that the ToM regression-adjusted estimator improves the asymptotic estimation efficiency compared to the unadjusted estimator, even when the regression model is misspecified, and is optimal in the class of linearly adjusted estimators. We also study the asymptotic properties of various heteroscedasticity-robust standard errors and provide recommendations for practitioners. Simulation studies and real data analysis demonstrate ToM regression adjustment’s superiority over existing methods. [ABSTRACT FROM AUTHOR]

AbstractOur goal is to develop a general strategy to decompose a random variable X into multiple independent random variables, without sacrificing any information about unknown parameters. A recent paper showed that for some well-known natural exponential families, X can be thinned into independent random variables X(1),…,X(K) , such that X=∑k=1KX(k) . These independent random variables can then be used for various model validation and inference tasks, including in contexts where traditional sample splitting fails. In this article, we generalize their procedure by relaxing this summation requirement and simply asking that some known function of the independent random variables exactly reconstruct X. This generalization of the procedure serves two purposes. First, it greatly expands the families of distributions for which thinning can be performed. Second, it unifies sample splitting and data thinning, which on the surface seem to be very different, as applications of the same principle. This shared principle is sufficiency. We use this insight to perform generalized thinning operations for a diverse set of families. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

Recommender systems are the engine of online advertising. Not only do they suggest movies, music, or romantic partners, but they also are used to select which advertisements to show to users. This paper reviews the basics of recommender system methodology and then looks at the emerging arena of active recommender systems. [ABSTRACT FROM AUTHOR]

The probabilistic topic model imposes a low-rank structure on the expectation of the corpus matrix. Therefore, singular value decomposition (SVD) is a natural tool of dimension reduction. We propose an SVD-based method for estimating a topic model. Our method constructs an estimate of the topic matrix from only a few leading singular vectors of the data matrix, and has a great advantage in memory use and computational cost for large-scale corpora. The core ideas behind our method include a pre-SVD normalization to tackle severe word frequency heterogeneity, a post-SVD normalization to create a low-dimensional word embedding that manifests a simplex geometry, and a post-SVD procedure to construct an estimate of the topic matrix directly from the embedded word cloud. We provide the explicit rate of convergence of our method. We show that our method attains the optimal rate in the case of long and moderately long documents, and it improves the rates of existing methods in the case of short documents. The key of our analysis is a sharp row-wise large-deviation bound for empirical singular vectors, which is technically demanding to derive and potentially useful for other problems. We apply our method to a corpus of Associated Press news articles and a corpus of abstracts of statistical papers. for this article are available online. [ABSTRACT FROM AUTHOR]