arXiv subjects

Jianqing Fan

Publications and source records attributed to Jianqing Fan.

At least 91 records · Page 5Linked to original sources

arXiv2022

Strategic Decision-Making in the Presence of Information Asymmetry: Provably Efficient RL with Algorithmic Instruments

We study offline reinforcement learning under a novel model called strategic MDP, which characterizes the strategic interactions between a principal and a sequence of myopic agents with private types. Due to the bilevel structure and private types, strategic MDP involves information asymmetry between the principal and the agents. We focus on the offline RL problem, where the goal is to learn the optimal policy of the principal concerning a target population of agents based on a pre-collected dataset that consists of historical interactions. The unobserved private types confound such a dataset as they affect both the rewards and observations received by the principal. We propose a novel algorithm, Pessimistic policy Learning with Algorithmic iNstruments (PLAN), which leverages the ideas of instrumental variable regression and the pessimism principle to learn a near-optimal principal's policy in the context of general function approximation. Our algorithm is based on the critical observation that the principal's actions serve as valid instrumental variables. In particular, under a partial coverage assumption on the offline dataset, we prove that PLAN outputs a $1 / \sqrt{K}$-optimal policy with $K$ being the number of collected trajectories. We further apply our framework to some special cases of strategic MDP, including strategic regression, strategic bandit, and noncompliance in recommendation systems.

stat.ML↗

arXiv2022

Robust Matrix Completion with Heavy-tailed Noise

This paper studies 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 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 decent with robust spectral initialization. We prove that under merely bounded second moment condition on the error distributions, rather than the sub-Gaussian assumption, the Euclidean error 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 simulation studies.

math.ST↗

arXiv2022

Policy Optimization Using Semi-parametric Models for Dynamic Pricing

In this paper, we study the contextual dynamic pricing problem where the market value of a product is linear in its observed features plus some market noise. Products are sold one at a time, and only a binary response indicating success or failure of a sale is observed. Our model setting is similar to Javanmard and Nazerzadeh [2019] except that we expand the demand curve to a semiparametric model and need to learn dynamically both parametric and nonparametric components. We propose a dynamic statistical learning and decision-making policy that combines semiparametric estimation from a generalized linear model with an unknown link and online decision-making to minimize regret (maximize revenue). Under mild conditions, we show that for a market noise c.d.f. $F(\cdot)$ with $m$-th order derivative ($m\geq 2$), our policy achieves a regret upper bound of $\tilde{O}_{d}(T^{\frac{2m+1}{4m-1}})$, where $T$ is time horizon and $\tilde{O}_{d}$ is the order that hides logarithmic terms and the dimensionality of feature $d$. The upper bound is further reduced to $\tilde{O}_{d}(\sqrt{T})$ if $F$ is super smooth whose Fourier transform decays exponentially. In terms of dependence on the horizon $T$, these upper bounds are close to $Ω(\sqrt{T})$, the lower bound where $F$ belongs to a parametric class. We further generalize these results to the case with dynamically dependent product features under the strong mixing condition.

cs.LG↗

arXiv2022

An $\ell_p$ theory of PCA and spectral clustering

Principal Component Analysis (PCA) is a powerful tool in statistics and machine learning. While existing study of PCA focuses on the recovery of principal components and their associated eigenvalues, there are few precise characterizations of individual principal component scores that yield low-dimensional embedding of samples. That hinders the analysis of various spectral methods. In this paper, we first develop an $\ell_p$ perturbation theory for a hollowed version of PCA in Hilbert spaces which provably improves upon the vanilla PCA in the presence of heteroscedastic noises. Through a novel $\ell_p$ analysis of eigenvectors, we investigate entrywise behaviors of principal component score vectors and show that they can be approximated by linear functionals of the Gram matrix in $\ell_p$ norm, which includes $\ell_2$ and $\ell_\infty$ as special examples. For sub-Gaussian mixture models, the choice of $p$ giving optimal bounds depends on the signal-to-noise ratio, which further yields optimality guarantees for spectral clustering. For contextual community detection, the $\ell_p$ theory leads to a simple spectral algorithm that achieves the information threshold for exact recovery. These also provide optimal recovery results for Gaussian mixture and stochastic block models as special cases.

math.ST↗

arXiv2022

The Efficacy of Pessimism in Asynchronous Q-Learning

This paper is concerned with the asynchronous form of Q-learning, which applies a stochastic approximation scheme to Markovian data samples. Motivated by the recent advances in offline reinforcement learning, we develop an algorithmic framework that incorporates the principle of pessimism into asynchronous Q-learning, which penalizes infrequently-visited state-action pairs based on suitable lower confidence bounds (LCBs). This framework leads to, among other things, improved sample efficiency and enhanced adaptivity in the presence of near-expert data. Our approach permits the observed data in some important scenarios to cover only partial state-action space, which is in stark contrast to prior theory that requires uniform coverage of all state-action pairs. When coupled with the idea of variance reduction, asynchronous Q-learning with LCB penalization achieves near-optimal sample complexity, provided that the target accuracy level is small enough. In comparison, prior works were suboptimal in terms of the dependency on the effective horizon even when i.i.d. sampling is permitted. Our results deliver the first theoretical support for the use of pessimism principle in the presence of Markovian non-i.i.d. data.

cs.LG↗

arXiv2022

Are Latent Factor Regression and Sparse Regression Adequate?

We propose the Factor Augmented sparse linear Regression Model (FARM) that not only encompasses both the latent factor regression and sparse linear regression as special cases but also bridges dimension reduction and sparse regression together. We provide theoretical guarantees for the estimation of our model under the existence of sub-Gaussian and heavy-tailed noises (with bounded (1+x)-th moment, for all x>0), respectively. In addition, the existing works on supervised learning often assume the latent factor regression or the sparse linear regression is the true underlying model without justifying its adequacy. To fill in such an important gap, we also leverage our model as the alternative model to test the sufficiency of the latent factor regression and the sparse linear regression models. To accomplish these goals, we propose the Factor-Adjusted de-Biased Test (FabTest) and a two-stage ANOVA type test respectively. We also conduct large-scale numerical experiments including both synthetic and FRED macroeconomics data to corroborate the theoretical properties of our methods. Numerical results illustrate the robustness and effectiveness of our model against latent factor regression and sparse linear regression models.

stat.ME↗

arXiv2022

Do We Exploit all Information for Counterfactual Analysis? Benefits of Factor Models and Idiosyncratic Correction

Optimal pricing, i.e., determining the price level that maximizes profit or revenue of a given product, is a vital task for the retail industry. To select such a quantity, one needs first to estimate the price elasticity from the product demand. Regression methods usually fail to recover such elasticities due to confounding effects and price endogeneity. Therefore, randomized experiments are typically required. However, elasticities can be highly heterogeneous depending on the location of stores, for example. As the randomization frequently occurs at the municipal level, standard difference-in-differences methods may also fail. Possible solutions are based on methodologies to measure the effects of treatments on a single (or just a few) treated unit(s) based on counterfactuals constructed from artificial controls. For example, for each city in the treatment group, a counterfactual may be constructed from the untreated locations. In this paper, we apply a novel high-dimensional statistical method to measure the effects of price changes on daily sales from a major retailer in Brazil. The proposed methodology combines principal components (factors) and sparse regressions, resulting in a method called Factor-Adjusted Regularized Method for Treatment evaluation (\texttt{FarmTreat}). The data consist of daily sales and prices of five different products over more than 400 municipalities. The products considered belong to the \emph{sweet and candies} category and experiments have been conducted over the years of 2016 and 2017. Our results confirm the hypothesis of a high degree of heterogeneity yielding very different pricing strategies over distinct municipalities.

econ.EM↗

arXiv2021

Understanding Implicit Regularization in Over-Parameterized Single Index Model

In this paper, we leverage over-parameterization to design regularization-free algorithms for the high-dimensional single index model and provide theoretical guarantees for the induced implicit regularization phenomenon. Specifically, we study both vector and matrix single index models where the link function is nonlinear and unknown, the signal parameter is either a sparse vector or a low-rank symmetric matrix, and the response variable can be heavy-tailed. To gain a better understanding of the role played by implicit regularization without excess technicality, we assume that the distribution of the covariates is known a priori. For both the vector and matrix settings, we construct an over-parameterized least-squares loss function by employing the score function transform and a robust truncation step designed specifically for heavy-tailed data. We propose to estimate the true parameter by applying regularization-free gradient descent to the loss function. When the initialization is close to the origin and the stepsize is sufficiently small, we prove that the obtained solution achieves minimax optimal statistical rates of convergence in both the vector and matrix cases. In addition, our experimental results support our theoretical findings and also demonstrate that our methods empirically outperform classical methods with explicit regularization in terms of both $\ell_2$-statistical rate and variable selection consistency.

stat.ML↗

arXiv2021

Curriculum Learning for Vision-and-Language Navigation

Vision-and-Language Navigation (VLN) is a task where an agent navigates in an embodied indoor environment under human instructions. Previous works ignore the distribution of sample difficulty and we argue that this potentially degrade their agent performance. To tackle this issue, we propose a novel curriculum-based training paradigm for VLN tasks that can balance human prior knowledge and agent learning progress about training samples. We develop the principle of curriculum design and re-arrange the benchmark Room-to-Room (R2R) dataset to make it suitable for curriculum training. Experiments show that our method is model-agnostic and can significantly improve the performance, the generalizability, and the training efficiency of current state-of-the-art navigation agents without increasing model complexity.

cs.LG↗

arXiv2021

Negative Sample is Negative in Its Own Way: Tailoring Negative Sentences for Image-Text Retrieval

Matching model is essential for Image-Text Retrieval framework. Existing research usually train the model with a triplet loss and explore various strategy to retrieve hard negative sentences in the dataset. We argue that current retrieval-based negative sample construction approach is limited in the scale of the dataset thus fail to identify negative sample of high difficulty for every image. We propose our TAiloring neGative Sentences with Discrimination and Correction (TAGS-DC) to generate synthetic sentences automatically as negative samples. TAGS-DC is composed of masking and refilling to generate synthetic negative sentences with higher difficulty. To keep the difficulty during training, we mutually improve the retrieval and generation through parameter sharing. To further utilize fine-grained semantic of mismatch in the negative sentence, we propose two auxiliary tasks, namely word discrimination and word correction to improve the training. In experiments, we verify the effectiveness of our model on MS-COCO and Flickr30K compared with current state-of-the-art models and demonstrates its robustness and faithfulness in the further analysis. Our code is available in https://github.com/LibertFan/TAGS.

cs.CV↗

arXiv2021

Canonical thresholding for non-sparse high-dimensional linear regression

We consider a high-dimensional linear regression problem. Unlike many papers on the topic, we do not require sparsity of the regression coefficients; instead, our main structural assumption is a decay of eigenvalues of the covariance matrix of the data. We propose a new family of estimators, called the canonical thresholding estimators, which pick largest regression coefficients in the canonical form. The estimators admit an explicit form and can be linked to LASSO and Principal Component Regression (PCR). A theoretical analysis for both fixed design and random design settings is provided. Obtained bounds on the mean squared error and the prediction error of a specific estimator from the family allow to clearly state sufficient conditions on the decay of eigenvalues to ensure convergence. In addition, we promote the use of the relative errors, strongly linked with the out-of-sample $R^2$. The study of these relative errors leads to a new concept of joint effective dimension, which incorporates the covariance of the data and the regression coefficients simultaneously, and describes the complexity of a linear regression problem. Some minimax lower bounds are established to showcase the optimality of our procedure. Numerical simulations confirm good performance of the proposed estimators compared to the previously developed methods.

math.ST↗

arXiv2021

Spectral Methods for Data Science: A Statistical Perspective

Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues (resp. singular values) and eigenvectors (resp. singular vectors) of some properly designed matrices constructed from data. A diverse array of applications have been found in machine learning, data science, and signal processing. Due to their simplicity and effectiveness, spectral methods are not only used as a stand-alone estimator, but also frequently employed to initialize other more sophisticated algorithms to improve performance. While the studies of spectral methods can be traced back to classical matrix perturbation theory and methods of moments, the past decade has witnessed tremendous theoretical advances in demystifying their efficacy through the lens of statistical modeling, with the aid of non-asymptotic random matrix theory. This monograph aims to present a systematic, comprehensive, yet accessible introduction to spectral methods from a modern statistical perspective, highlighting their algorithmic implications in diverse large-scale applications. In particular, our exposition gravitates around several central questions that span various applications: how to characterize the sample efficiency of spectral methods in reaching a target level of statistical accuracy, and how to assess their stability in the face of random noise, missing data, and adversarial corruptions? In addition to conventional $\ell_2$ perturbation analysis, we present a systematic $\ell_{\infty}$ and $\ell_{2,\infty}$ perturbation theory for eigenspace and singular subspaces, which has only recently become available owing to a powerful "leave-one-out" analysis framework.

stat.ML↗

arXiv2021

Constructing Phrase-level Semantic Labels to Form Multi-Grained Supervision for Image-Text Retrieval

Existing research for image text retrieval mainly relies on sentence-level supervision to distinguish matched and mismatched sentences for a query image. However, semantic mismatch between an image and sentences usually happens in finer grain, i.e., phrase level. In this paper, we explore to introduce additional phrase-level supervision for the better identification of mismatched units in the text. In practice, multi-grained semantic labels are automatically constructed for a query image in both sentence-level and phrase-level. We construct text scene graphs for the matched sentences and extract entities and triples as the phrase-level labels. In order to integrate both supervision of sentence-level and phrase-level, we propose Semantic Structure Aware Multimodal Transformer (SSAMT) for multi-modal representation learning. Inside the SSAMT, we utilize different kinds of attention mechanisms to enforce interactions of multi-grain semantic units in both sides of vision and language. For the training, we propose multi-scale matching losses from both global and local perspectives, and penalize mismatched phrases. Experimental results on MS-COCO and Flickr30K show the effectiveness of our approach compared to some state-of-the-art models.

cs.CV↗

arXiv2021

SIMPLE: Statistical Inference on Membership Profiles in Large Networks

Network data is prevalent in many contemporary big data applications in which a common interest is to unveil important latent links between different pairs of nodes. Yet a simple fundamental question of how to precisely quantify the statistical uncertainty associated with the identification of latent links still remains largely unexplored. In this paper, we propose the method of statistical inference on membership profiles in large networks (SIMPLE) in the setting of degree-corrected mixed membership model, where the null hypothesis assumes that the pair of nodes share the same profile of community memberships. In the simpler case of no degree heterogeneity, the model reduces to the mixed membership model for which an alternative more robust test is also proposed. Both tests are of the Hotelling-type statistics based on the rows of empirical eigenvectors or their ratios, whose asymptotic covariance matrices are very challenging to derive and estimate. Nevertheless, their analytical expressions are unveiled and the unknown covariance matrices are consistently estimated. Under some mild regularity conditions, we establish the exact limiting distributions of the two forms of SIMPLE test statistics under the null hypothesis and contiguous alternative hypothesis. They are the chi-square distributions and the noncentral chi-square distributions, respectively, with degrees of freedom depending on whether the degrees are corrected or not. We also address the important issue of estimating the unknown number of communities and establish the asymptotic properties of the associated test statistics. The advantages and practical utility of our new procedures in terms of both size and power are demonstrated through several simulation examples and real network applications.

stat.ME↗

arXiv2021

Best subset selection is robust against design dependence

Best subset selection (BSS) is widely known as the holy grail for high-dimensional variable selection. Nevertheless, the notorious NP-hardness of BSS substantially restricts its practical application and also discourages its theoretical development to some extent, particularly in the current era of big data. In this paper, we investigate the variable selection properties of BSS when its target sparsity is greater than or equal to the true sparsity. Our main message is that BSS is robust against design dependence in terms of achieving model consistency and sure screening, and more importantly, that such robustness can be propagated to the near best subsets that are computationally tangible. Specifically, we introduce an identifiability margin condition that is free of restricted eigenvalues and show that it is sufficient and nearly necessary for BSS to exactly recover the true model. A relaxed version of this condition is also sufficient for BSS to achieve the sure screening property. Moreover, taking optimization error into account, we find that all the established statistical properties for the exact best subset carry over to any near best subset whose residual sum of squares is close enough to that of the best one. In particular, a two-stage fully corrective iterative hard thresholding (IHT) algorithm can provably find a sparse sure screening subset within logarithmic steps; another round of exact BSS within this set can recover the true model. The simulation studies and real data examples show that IHT yields lower false discovery rates and higher true positive rates than the competing approaches including LASSO, SCAD and Sure Independence Screening (SIS), especially under highly correlated design.

stat.ME↗

arXiv2021

Communication-Efficient Accurate Statistical Estimation

When the data are stored in a distributed manner, direct application of traditional statistical inference procedures is often prohibitive due to communication cost and privacy concerns. This paper develops and investigates two Communication-Efficient Accurate Statistical Estimators (CEASE), implemented through iterative algorithms for distributed optimization. In each iteration, node machines carry out computation in parallel and communicate with the central processor, which then broadcasts aggregated information to node machines for new updates. The algorithms adapt to the similarity among loss functions on node machines, and converge rapidly when each node machine has large enough sample size. Moreover, they do not require good initialization and enjoy linear converge guarantees under general conditions. The contraction rate of optimization errors is presented explicitly, with dependence on the local sample size unveiled. In addition, the improved statistical accuracy per iteration is derived. By regarding the proposed method as a multi-step statistical estimator, we show that statistical efficiency can be achieved in finite steps in typical statistical applications. In addition, we give the conditions under which the one-step CEASE estimator is statistically efficient. Extensive numerical experiments on both synthetic and real data validate the theoretical results and demonstrate the superior performance of our algorithms.

stat.ML↗

arXiv2021

Optimal Covariate Balancing Conditions in Propensity Score Estimation

Inverse probability of treatment weighting (IPTW) is a popular method for estimating the average treatment effect (ATE). However, empirical studies show that the IPTW estimators can be sensitive to the misspecification of the propensity score model. To address this problem, researchers have proposed to estimate propensity score by directly optimizing the balance of pre-treatment covariates. While these methods appear to empirically perform well, little is known about how the choice of balancing conditions affects their theoretical properties. To fill this gap, we first characterize the asymptotic bias and efficiency of the IPTW estimator based on the Covariate Balancing Propensity Score (CBPS) methodology under local model misspecification. Based on this analysis, we show how to optimally choose the covariate balancing functions and propose an optimal CBPS-based IPTW estimator. This estimator is doubly robust; it is consistent for the ATE if either the propensity score model or the outcome model is correct. In addition, the proposed estimator is locally semiparametric efficient when both models are correctly specified. To further relax the parametric assumptions, we extend our method by using a sieve estimation approach. We show that the resulting estimator is globally efficient under a set of much weaker assumptions and has a smaller asymptotic bias than the existing estimators. Finally, we evaluate the finite sample performance of the proposed estimators via simulation and empirical studies. An open-source software package is available for implementing the proposed methods.

stat.ME↗

arXiv2021

Convex and Nonconvex Optimization Are Both Minimax-Optimal for Noisy Blind Deconvolution under Random Designs

We investigate the effectiveness of convex relaxation and nonconvex optimization in solving bilinear systems of equations under two different designs (i.e.$~$a sort of random Fourier design and Gaussian design). Despite the wide applicability, the theoretical understanding about these two paradigms remains largely inadequate in the presence of random noise. The current paper makes two contributions by demonstrating that: (1) a two-stage nonconvex algorithm attains minimax-optimal accuracy within a logarithmic number of iterations. (2) convex relaxation also achieves minimax-optimal statistical accuracy vis-à-vis random noise. Both results significantly improve upon the state-of-the-art theoretical guarantees.

stat.ML↗