We study statistical estimators computed using iterative optimization methods that are not run until completion. Classical results on maximum likelihood estimators (MLEs) assert that a one-step estimator (OSE), in which a single Newton-Raphson iteration is performed from a starting point with certain properties, is asymptotically equivalent to the MLE. We further develop these early-stopping results by deriving properties of one-step estimators defined by a single iteration of scaled proximal methods. Our main results show the asymptotic equivalence of the likelihood-based estimator and various one-step estimators defined by scaled proximal methods. By interpreting OSEs as the last of a sequence of iterates, our results provide insight on scaling numerical tolerance with sample size. Our setting contains scaled proximal gradient descent applied to certain composite models as a special case, making our results applicable to many problems of practical interest. Additionally, our results provide support for the utility of the scaled Moreau envelope as a statistical smoother by interpreting scaled proximal descent as a quasi-Newton method applied to the scaled Moreau envelope.
Erasmus School of Economics, Erasmus University Rotterdam（伊拉斯姆斯大学鹿特丹经济学院）；Tinbergen Institute（丁伯根研究所）；Geneva Finance Research Institute, University of Geneva（日内瓦大学日内瓦金融研究院）；Swiss Finance Institute（瑞士金融研究院）；Collegio Carlo Alberto, University of Turin（都灵大学卡洛阿尔贝托学院）
The question of fast convergence in the classical problem of high dimensional linear regression has been extensively studied. Arguably, one of the fastest procedures in practice is Iterative Hard Thresholding (IHT). Still, IHT relies strongly on the knowledge of the true sparsity parameter $s$. In this paper, we present a novel fast procedure for estimation in the high dimensional linear regression. Taking advantage of the interplay between estimation, support recovery and optimization we achieve both optimal statistical accuracy and fast convergence. The main advantage of our procedure is tha…
This paper presents a detailed theoretical analysis of the three stochastic approximation proximal gradient algorithms proposed in our companion paper [49] to set regularization parameters by marginal maximum likelihood estimation. We prove the convergence of a more general stochastic approximation scheme that includes the three algorithms of [49] as special cases. This includes asymptotic and non-asymptotic convergence results with natural and easily verifiable conditions, as well as explicit bounds on the convergence rates. Importantly, the theory is also general in that it can be applied to…
Nonconvex Penalized LAD Estimation in Partial Linear Models with DNNs: Asymptotic Analysis and Proximal Algorithms
Department of Applied Mathematics, The Hong Kong Polytechnic University（应用数学系，香港理工大学）；College of Computing, Data Science, and Society, University of California, Berkeley（计算、数据科学与社会学院，加州大学伯克利分校）
We explore a method of statistical estimation called Maximum Entropy on the Mean (MEM) which is based on an information-driven criterion that quantifies the compliance of a given point with a reference prior probability measure. At the core of this approach lies the MEM function which is a partial minimization of the Kullback-Leibler divergence over a linear constraint. In many cases, it is known that this function admits a simpler representation (known as the Cramér rate function). Via the connection to exponential families of probability distributions, we study general conditions under which…
Many imaging problems require solving an inverse problem that is ill-conditioned or ill-posed. Imaging methods typically address this difficulty by regularising the estimation problem to make it well-posed. This often requires setting the value of the so-called regularisation parameters that control the amount of regularisation enforced. These parameters are notoriously difficult to set a priori, and can have a dramatic impact on the recovered estimates. In this work, we propose a general empirical Bayesian method for setting regularisation parameters in imaging problems that are convex w.r.t.…
Kakao Entertainment Corp.（Kakao娱乐公司）；Seoul National University（首尔大学）
The implicit stochastic gradient descent (ISGD), a proximal version of SGD, is gaining interest in the literature due to its stability over (explicit) SGD. In this paper, we conduct an in-depth analysis of the two modes of ISGD for smooth convex functions, namely proximal Robbins-Monro (proxRM) and proximal Poylak-Ruppert (proxPR) procedures, for their use in statistical inference on model parameters. Specifically, we derive non-asymptotic point estimation error bounds of both proxRM and proxPR iterates and their limiting distributions, and propose on-line estimators of their asymptotic covari…
Parameter Estimation for Time-Scaled Inhomogeneous Phase-Type Distributions from Discrete Observations
Facultad de Ciencias, Universidad Nacional Autónoma de México（墨西哥国立自治大学理学院）；Department of Statistics, University of Wisconsin, Madison（威斯康星大学麦迪逊分校统计学系）
University of Michigan（密歇根大学）；School of Information, University of Michigan（密歇根大学信息学院）；Harvard University（哈佛大学）
Due to the recent advancements in wearables and sensing technology, health scientists are increasingly developing mobile health (mHealth) interventions. In mHealth interventions, mobile devices are used to deliver treatment to individuals as they go about their daily lives. These treatments are generally designed to impact a near time, proximal outcome such as stress or physical activity. The mHealth intervention policies, often called just-in-time adaptive interventions, are decision rules that map an individual's current state (e.g., individual's past behaviors as well as current observation…
A Decomposed Bilevel Search for Variable-Metric Proximal Gradient Methods
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院、中国科学院大学）；State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）；Air Force Research Laboratory（美国空军研究实验室）
Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is to faithfully recover the tensor from highly incomplete measurements in a statistically and computationally efficient manner. Harnessing the low-rank structure of tensors in the Tucker decomposition, this paper develops a scaled gradient descent (ScaledGD) algorithm to directly recover the tensor factors with tailored spectral initializations, and shows tha…
In sparse estimation, such as fused lasso and convex clustering, we apply either the proximal gradient method or the alternating direction method of multipliers (ADMM) to solve the problem. It takes time to include matrix division in the former case, while an efficient method such as FISTA (fast iterative shrinkage-thresholding algorithm) has been developed in the latter case. This paper proposes a general method for converting the ADMM solution to the proximal gradient method, assuming that assumption that the derivative of the objective function is Lipschitz continuous. Then, we apply it to …
Univ. Lille, CNRS, Inria, Centrale Lille, UMR 9189 CRIStAL（里尔大学、法国国家科学研究中心、法国国家信息与自动化研究所、里尔中央理工学院）；Université de Lyon, (Université Claude Bernard, CNRS UMR 5007)（里昂大学（克莱贝尔·伯纳德大学、法国国家科学研究中心））；Univ. Gustave Eiffel, Inria（格扎维埃·埃菲尔大学、法国国家信息与自动化研究所）
A Semismooth Newton-Based Proximal Augmented Lagrangian Method for Joint Estimation of Multiple Gaussian Graphical Models with Clustered Structure
School of Mathematics and Statistics, Fuzhou University（福州大学数学与统计学院）；Center for Applied Mathematics of Fujian Province, School of Mathematics and Statistics, Fuzhou University（福建省应用数学中心，福州大学数学与统计学院）
Low-rank matrix estimation is a canonical problem that finds numerous applications in signal processing, machine learning and imaging science. A popular approach in practice is to factorize the matrix into two compact low-rank factors, and then optimize these factors directly via simple iterative methods such as gradient descent and alternating minimization. Despite nonconvexity, recent literatures have shown that these simple heuristics in fact achieve linear convergence when initialized properly for a growing number of problems of interest. However, upon closer examination, existing approach…
Coupled Tensor-Matrix Recovery via Proximal Alternating Linearized Minimization, with an Application to Workforce Skill and Small-Business Health Estimation
This paper is concerned with the computation of the high-dimensional zero-norm penalized quantile regression estimator, defined as a global minimizer of the zero-norm penalized check loss function. To seek a desirable approximation to the estimator, we reformulate this NP-hard problem as an equivalent augmented Lipschitz optimization problem, and exploit its coupled structure to propose a multi-stage convex relaxation approach (MSCRA\_PPA), each step of which solves inexactly a weighted $\ell_1$-regularized check loss minimization problem with a proximal dual semismooth Newton method. Under a …
