The method of stable random projections is a tool for efficiently computing the $l_α$ distances using low memory, where $0<α\leq 2$ is a tuning parameter. The method boils down to a statistical estimation task and various estimators have been proposed, based on the geometric mean, the harmonic mean, and the fractional power etc.
This study proposes the optimal quantile estimator, whose main operation is selecting, which is considerably less expensive than taking fractional power, the main operation in previous estimators. Our experiments report that the optimal quantile estimator is nearly one order of magnitude more computationally efficient than previous estimators. For large-scale learning tasks in which storing and computing pairwise distances is a serious bottleneck, this estimator should be desirable.
In addition to its computational advantages, the optimal quantile estimator exhibits nice theoretical properties. It is more accurate than previous estimators when $α>1$. We derive its theoretical error bounds and establish the explicit (i.e., no hidden constants) sample complexity bound.
We consider the question of learning the natural parameters of a $k$ parameter minimal exponential family from i.i.d. samples in a computationally and statistically efficient manner. We focus on the setting where the support as well as the natural parameters are appropriately bounded. While the traditional maximum likelihood estimator for this class of exponential family is consistent, asymptotically normal, and asymptotically efficient, evaluating it is computationally hard. In this work, we propose a computationally efficient estimator that is consistent as well as asymptotically normal unde…
Massachusetts Institute of Technology（麻省理工学院）；University Toulouse 3（图卢兹第三大学）
In the context of regression, we consider the fundamental question of making an estimator fair while preserving its prediction accuracy as much as possible. To that end, we define its projection to fairness as its closest fair estimator in a sense that reflects prediction accuracy. Our methodology leverages tools from optimal transport to construct efficiently the projection to fairness of any given estimator as a simple post-processing step. Moreover, our approach precisely quantifies the cost of fairness, measured in terms of prediction accuracy.
Georgia Institute of Technology（佐治亚理工学院）；The Ohio State University（俄亥俄州立大学）
We study an estimator with a convex formulation for recovery of low-rank matrices from rank-one projections. Using initial estimates of the factors of the target $d_1\times d_2$ matrix of rank-$r$, the estimator admits a practical subgradient method operating in a space of dimension $r(d_1+d_2)$. This property makes the estimator significantly more scalable than the convex estimators based on lifting and semidefinite programming. Furthermore, we present a streamlined analysis for exact recovery under the real Gaussian measurement model, as well as the partially derandomized measurement model b…
Massachusetts Institute of Technology（麻省理工学院）；Max Plank Institute for Informatics（马克斯·普朗克信息学研究所）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
We provide a computationally and statistically efficient estimator for the classical problem of truncated linear regression, where the dependent variable $y = w^T x + ε$ and its corresponding vector of covariates $x \in R^k$ are only revealed if the dependent variable falls in some subset $S \subseteq R$; otherwise the existence of the pair $(x, y)$ is hidden. This problem has remained a challenge since the early works of [Tobin 1958, Amemiya 1973, Hausman and Wise 1977], its applications are abundant, and its history dates back even further to the work of Galton, Pearson, Lee, and Fisher. Whi…
We provide a new computationally-efficient class of estimators for risk minimization. We show that these estimators are robust for general statistical models: in the classical Huber epsilon-contamination model and in heavy-tailed settings. Our workhorse is a novel robust variant of gradient descent, and we provide conditions under which our gradient descent variant provides accurate estimators in a general convex risk minimization problem. We provide specific consequences of our theory for linear regression, logistic regression and for estimation of the canonical parameters in an exponential f…
University of California, Berkeley（加州大学伯克利分校）；CREST - ENSAE（法国国家统计与经济管理学院 Crest 实验室）；Google Brain（谷歌大脑）
Optimal transport (OT) distances are increasingly used as loss functions for statistical inference, notably in the learning of generative models or supervised learning. Yet, the behavior of minimum Wasserstein estimators is poorly understood, notably in high-dimensional regimes or under model misspecification. In this work we adopt the viewpoint of projection robust (PR) OT, which seeks to maximize the OT cost between two measures by choosing a $k$-dimensional subspace onto which they can be projected. Our first contribution is to establish several fundamental statistical properties of PR Wass…
Yale Institute for Network Science（耶鲁网络科学研究所）；Yale University（耶鲁大学）
In this paper, we propose the first computationally efficient projection-free algorithm for bandit convex optimization (BCO). We show that our algorithm achieves a sublinear regret of $O(nT^{4/5})$ (where $T$ is the horizon and $n$ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against baselines on both synthetic and real data sets for quadratic programming, portfolio selection and matrix completion problems.
Zhejiang University（浙江大学）；University of Pennsylvania（宾夕法尼亚大学）；Tencent AI Lab（腾讯人工智能实验室）；University of Western Ontario（西安大略大学）
This paper focuses on projection-free methods for solving smooth Online Convex Optimization (OCO) problems. Existing projection-free methods either achieve suboptimal regret bounds or have high per-iteration computational costs. To fill this gap, two efficient projection-free online methods called ORGFW and MORGFW are proposed for solving stochastic and adversarial OCO problems, respectively. By employing a recursive gradient estimator, our methods achieve optimal regret bounds (up to a logarithmic factor) while possessing low per-iteration computational costs. Experimental results demonstrate…
We analyze the performance of the Tukey median estimator under total variation (TV) distance corruptions. Previous results show that under Huber's additive corruption model, the breakdown point is 1/3 for high-dimensional halfspace-symmetric distributions. We show that under TV corruptions, the breakdown point reduces to 1/4 for the same set of distributions. We also show that a certain projection algorithm can attain the optimal breakdown point of 1/2. Both the Tukey median estimator and the projection algorithm achieve sample complexity linear in dimension.
In this paper, we propose a random projection approach to estimate variance in kernel ridge regression. Our approach leads to a consistent estimator of the true variance, while being computationally more efficient. Our variance estimator is optimal for a large family of kernels, including cubic splines and Gaussian kernels. Simulation analysis is conducted to support our theory.
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Wisconsin Institute for Discovery（威斯康星发现研究所）
This paper derives a perturbation bound on the optimal subspace estimator obtained from a subset of its canonical projections contaminated by noise. This fundamental result has important implications in matrix completion, subspace clustering, and related problems.
We introduce a new family of estimators for unnormalized statistical models. Our family of estimators is parameterized by two nonlinear functions and uses a single sample from an auxiliary distribution, generalizing Maximum Likelihood Monte Carlo estimation of Geyer and Thompson (1992). The family is such that we can estimate the partition function like any other parameter in the model. The estimation is done by optimizing an algebraically simple, well defined objective function, which allows for the use of dedicated optimization methods. We establish consistency of the estimator family and gi…
University of Waterloo（滑铁卢大学）；Northeastern University（东北大学）；Google Research（谷歌研究院）
We give the first polynomial-time, polynomial-sample, differentially private estimator for the mean and covariance of an arbitrary Gaussian distribution $\mathcal{N}(μ,Σ)$ in $\mathbb{R}^d$. All previous estimators are either nonconstructive, with unbounded running time, or require the user to specify a priori bounds on the parameters $μ$ and $Σ$. The primary new technical tool in our algorithm is a new differentially private preconditioner that takes samples from an arbitrary Gaussian $\mathcal{N}(0,Σ)$ and returns a matrix $A$ such that $A ΣA^T$ has constant condition number.
The method of 1-bit ("sign-sign") random projections has been a popular tool for efficient search and machine learning on large datasets. Given two $D$-dim data vectors $u$, $v\in\mathbb{R}^D$, one can generate $x = \sum_{i=1}^D u_i r_i$, and $y = \sum_{i=1}^D v_i r_i$, where $r_i\sim N(0,1)$ iid. The "collision probability" is ${Pr}\left(sgn(x)=sgn(y)\right) = 1-\frac{\cos^{-1}ρ}π$, where $ρ= ρ(u,v)$ is the cosine similarity. We develop "sign-full" random projections by estimating $ρ$ from (e.g.,) the expectation $E(sgn(x)y)=\sqrt{\frac{2}π} ρ$, which can be further substantially improved by …
We study differentially private mean estimation in a high-dimensional setting. Existing differential privacy techniques applied to large dimensions lead to computationally intractable problems or estimators with excessive privacy loss. Recent work in high-dimensional robust statistics has identified computationally tractable mean estimation algorithms with asymptotic dimension-independent error guarantees. We incorporate these results to develop a strict bound on the global sensitivity of the robust mean estimator. This yields a computationally tractable algorithm for differentially private me…
The method of random projections has become a standard tool for machine learning, data mining, and search with massive data at Web scale. The effective use of random projections requires efficient coding schemes for quantizing (real-valued) projected data into integers. In this paper, we focus on a simple 2-bit coding scheme. In particular, we develop accurate nonlinear estimators of data similarity based on the 2-bit strategy. This work will have important practical applications. For example, in the task of near neighbor search, a crucial step (often called re-ranking) is to compute or estima…
Dimensionality reduction methods, also known as projections, are frequently used for exploring multidimensional data in machine learning, data science, and information visualization. Among these, t-SNE and its variants have become very popular for their ability to visually separate distinct data clusters. However, such methods are computationally expensive for large datasets, suffer from stability problems, and cannot directly handle out-of-sample data. We propose a learning approach to construct such projections. We train a deep neural network based on a collection of samples from a given dat…
Federated Learning on Riemannian Manifolds: A Gradient-Free Projection-Based Approach
School of Information Management and Engineering, Shanghai University of Finance and Economics（上海金融学院信息管理与工程学院）；School of Information Management and Engineering, Shanghai University of Finance and Economics（上海金融学院信息管理与工程学院；明尼苏达大学工业与系统工程系）；Department of Industrial and System Engineering, University of Minnesota（明尼苏达大学电气与计算机工程系）；Department of Electrical and Computer Engineering, University of Minnesota
We study the use of very sparse random projections for compressed sensing (sparse signal recovery) when the signal entries can be either positive or negative. In our setting, the entries of a Gaussian design matrix are randomly sparsified so that only a very small fraction of the entries are nonzero. Our proposed decoding algorithm is simple and efficient in that the major cost is one linear scan of the coordinates. We have developed two estimators: (i) the {\em tie estimator}, and (ii) the {\em absolute minimum estimator}. Using only the tie estimator, we are able to recover a $K$-sparse sign…
In this paper, we develop connections between two seemingly disparate, but central, models in robust statistics: Huber's epsilon-contamination model and the heavy-tailed noise model. We provide conditions under which this connection provides near-statistically-optimal estimators. Building on this connection, we provide a simple variant of recent computationally-efficient algorithms for mean estimation in Huber's model, which given our connection entails that the same efficient sample-pruning based estimators is simultaneously robust to heavy-tailed noise and Huber contamination. Furthermore, w…
GraDE: A Graph Diffusion Estimator for Frequent Subgraph Discovery in Neural Architectures
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；BenchCouncil (International Open Benchmark Council)（国际开放基准委员会）；University of Chinese Academy of Sciences（中国科学院大学）；Northwestern University（西北大学）
Inria（法国国家信息与自动化研究所）；Ecole Normale Supérieure（巴黎高等师范学院）；PSL Research University（巴黎文理研究大学）
We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first consider the ridge regression estimator and review earlier results using classical notions from non-parametric statistics, namely degrees of freedom, also known as effective dimensionality. We then compute asymptotic equivalents of the generalization performance (in terms of squared bias and variance) of the minimum norm least-squares fit with random projectio…
