DOCSLIB.ORG

Active Regression with Adaptive Huber Loss Jacopo Cavazza and Vittorio Murino, Senior Member, IEEE

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Active Regression with Adaptive Huber Loss Jacopo Cavazza and Vittorio Murino, Senior Member, IEEE 1 Active Regression with Adaptive Huber Loss Jacopo Cavazza and Vittorio Murino, Senior Member, IEEE Abstract—This paper addresses the scalar regression problem construct as many decision functions, finally fused together. through a novel solution to exactly optimize the Huber loss in Also, manifold regularization is another state-of-the-art semi- a general semi-supervised setting, which combines multi-view supervised method where the geometry of the data space learning and manifold regularization. We propose a principled algorithm to 1) avoid computationally expensive iterative schemes shapes the solution, typically by means of the graph Laplacian while 2) adapting the Huber loss threshold in a data-driven operator [3], [4]. fashion and 3) actively balancing the use of labelled data to However, even in the small labelling regime, since anno- remove noisy or inconsistent annotations at the training stage. In tations are usually provided by human operators, they are a wide experimental evaluation, dealing with diverse applications, frequently prone to errors and noisy in general, making them we assess the superiority of our paradigm which is able to combine robustness towards noise with both strong performance rather misleading. Hence, for both theoretical and practical and low computational cost. aspects, it is of utmost importance to devise algorithms which are able to automatically analyse the data as to guarantee Index Terms—Robust Scalar Regression, Learning with Noisy Labels, Huber Loss with Adaptive Threshold, Convex Optimiza- robustness towards outliers. In the literature, several works tion, Crowd Counting. have tackled such a problem [5]: since the archetypal work [6], many robust regression and classification frameworks [7]– [10] successfully leveraged on the (convex and differentiable) I. INTRODUCTION Huber loss function Regression is one of the most widely studied problems in different research disciplines such as statistics, econometrics, ( 2 y if jyj ≤ ξ computational biology and physics to name a few, and is a 2 Hξ : R ! [0; +1);Hξ(y) = ξ2 (1) pillar topic in machine learning. Formally, it addresses the ξjyj − 2 otherwise, problem of inferring a functional input-output relationship. where ξ > 0. However, as the major drawback of H , there Under a classical machine learning perspective, the latter is ξ is no closed-form solution to optimize it and, as a consequence, learnt by means of (training) examples and, to this aim, iterative schemes (such as quadratic programming [8] or self- two mainstream approaches pop up: optimization-based and dual minimization [9]) were previously exploited for either the Bayesian frameworks. In the former, once a suitable hypothesis original Huber loss [7], [9] or its spurious versions (hinge- space is fixed, the goal is minimizing an objective functional, Huber [8] or the huberized Laplacian [10]). Moreover, in where a loss measures how good the learned regressor re- all cases, additional computational efforts have to be spent produces the relationship between input and output variables in order to fix the threshold ξ, such as statistical efficiency inside the training set. Instead, in the Bayesian formalism, a analysis [7]. prior distribution constrains the solution upon some a priori In this work we face all the aforementioned issues trough knowledge, while the final regression map maximizes the the following main contributions. posterior/likelihood probability distribution. Actually, a problem affecting both paradigms lies in the I. We derive a novel theoretical solution to exactly op- scarcity of annotations in the training data, which can seri- timize the Huber loss in a general multi-view, semi- ously impact on the generalizability of the regression method. supervised and manifold regularized setting [11], in arXiv:1606.01568v2 [cs.LG] 26 Jun 2016 Nevertheless, labelled data are always time-consuming, fre- order to guarantee a broad applicability of the developed quently onerous and sometimes difficult to obtain. Thus, in formalism. this respect, semi-supervised approaches play a substantial II. We devise the novel Huber Loss Regression (HLR) role in exploiting unlabelled samples to support the search algorithm to efficiently implement the proposed solution for the solution. Among the most effective semi-supervised and avoid classical iterative schemes [8], [9]. Moreover, algorithms, multi-view learning [1], [2] considers the structure two additional aspects are notable. of the input data as composed by several “views” which Active-learning. While taking advantage of the both la- are associated to different hypothesis spaces employed to belled and unlabelled training samples, the former ones are inspected so that HLR automatically removes those Jacopo Cavazza and Vittorio Murino are with Pattern Analysis & Computer annotations which violate a specific numerical check, Vision, Istituto Italiano di Tecnologia (IIT), Via Morego, 30, 16163, Genova, whenever recognized as either noisy or inconsistent for Italy. Jacopo Cavazza is also with Dipartimento di Ingegneria Navale, Elettrica, learning the regressor. Elettronica e delle Telecomunicazioni (DITEN), University of Genova, Via Adaptive threshold. Differently from all [7]–[10], HLR Opera Pia 11, 16145, Genova, Italy. automatically learns ξ in a data-driven fashion without Vittorio Murino is also with Computer Science Department, University of Verona, Strada Le Grazie 15, 37134, Verona, Italy. increasing the computational complexity of the whole Primary email contact: [email protected]. pipeline. 2 P α α α α α III. Throughout an extensive empirical evaluation, we val- i c κ (z ; xi )ui , for α = 1; : : : ; m. Differently, in [1], an idate the proposed technique, which allows to score analogous result holds for m = 2 only. competitive results in curve fitting, learning with noisy labels, classical regression problems and crowd counting III. HUBER MULTI-VIEW REGRESSION WITH MANIFOLD application. While using variegate types of data and REGULARIZATION addressing diverse problems, HLR is able to outperform Once the hypothesis space H is defined according to Section state-of-the-art regression algorithms. II, we can perform optimization to learn for the regression The paper is outlined as follows. In Sections II and III, map. To this aim, we consider a training set D made of we present our exact optimization for the Huber loss after ` labelled instances (x1; y1);:::; (x`; y`) 2 X × R and u defining the general semi-supervised setting we consider. additional unlabelled inputs x`+1;:::; x`+u 2 X : Among the Section IV broadly discusses the HLR algorithm. As to prove class of functions (2), similarly to [3], [11], the regression map its versatility, once benchmarked with a state-of-the-art convex is found by minimizing the following objective solver in Section V-A, we registered a strong performance ` 1 X when applying HLR to curve fitting and learning with noisy J (f) = H (y − c>f(x )) + λkfk2 + γkfk2 : (3) λ,γ ` ξ i i K M labels (Section V-B), classical regression problems (Section i=1 V-C) and crowd counting application (Section V-D). Finally, 1 P` > Section VI draws the conclusions. In equation (3), ` i=1 Hξ(yi − c f(xi)) represents the empirical risk and is defined by means of the Huber loss (1): it c>f(x ) y : ULTI VIEW SCALAR REGRESSION measures how well i predicts the ground truth output i II. M - 2 For λ > 0, to avoid either under-fitting or over-fitting, kfkK In this Section, we introduce the formalism to model data is a Tichonov regularizer which controls the complexity of the points sampled from a composite input space X which is solution. Instead, γ > 0 weighs divided into multiple substructures. Precisely, we assume that 1 m α m u+` for any x 2 X ; we have x = [x ; : : : ; x ] and x belongs to 2 X X α α α α α the subspace X α; for any α = 1; : : : ; m. This is a very natural kfkM = f (xi )Mijf (xj ); (4) way to model high dimensional data: x is the concatenation α=1 i;j=1 of x1; : : : ; xm; each one representing a particular class of which infers the geometrical information of the feature space. features, that is one out of multiple views [1], [2], [11] in The family fM 1;:::;M mg of symmetric and positive semi- α (u+`)×(u+`) 2 which data may be structured. definite matrices M 2 R specifies kfkM and In order to find the regression map, we assume that it be- ensures its non-negativeness. This setting generalizes [1]–[3] 1 m longs to an hypothesis space H of functions h: X! R whose where M = ··· = M = L; being L is the graph Laplacian construction is investigated below. For any α = 1; : : : ; m; related to the (u + `) × (u + `) adjacency matrix W: The latter α α α let κ : X × X ! R a Mercer kernel [12], that is a captures the interdependencies between f(x1); : : : ; f(xu+`) symmetric and positive semi-definite function. Let us define by measuring their mutual similarity. Then, in this simplified 1 1 1 m m m m×m Pu+` 2 K(x; z) = diag(κ (x ; z ); : : : ; κ (x ; z )) 2 R ; setting, the regularizer rewrites i;j=1 wijkf(xi) − f(xj)k2 1 m 1 m where x = [x ; : : : ; x ]; z = [z ; : : : ; z ] 2 X : Consider if wij denotes the (i; j)-th entry of W: In this particular Pn case, we retrieve the idea of manifold regularization [3] to S0 the space of functions z 7! f(z) = i=1 K(z; xi)ui with x1;:::; xn 2 X and u1; : : : ; un column vectors in enforce nearby patterns in the high-dimensional RKHS to m 2 Pn > share similar labels, so imposing further regularity on the so- R : Define the norm kfkK = i;j=1 ui K(xi; xj)uj: The reproducing kernel Hilbert space (RKHS) SK related to K lution. Conversely, with our generalization, we can consider m 1 m is the completion of S0 with the limits of Cauchy sequences different graph Laplacians L ;:::;L and adjacency matrices m 1 m converging with respect to k·kK [13].
Load more

Recommended publications
  • A Robust Hybrid of Lasso and Ridge Regression
    A robust hybrid of lasso and ridge regression Art B. Owen Stanford University October 2006 Abstract Ridge regression and the lasso are regularized versions of least squares regression using L2 and L1 penalties respectively, on the coefficient vector. To make these regressions more robust we may replace least squares with Huber’s criterion which is a hybrid of squared error (for relatively small errors) and absolute error (for relatively large ones). A reversed version of Huber’s criterion can be used as a hybrid penalty function. Relatively small coefficients contribute their L1 norm to this penalty while larger ones cause it to grow quadratically. This hybrid sets some coefficients to 0 (as lasso does) while shrinking the larger coefficients the way ridge regression does. Both the Huber and reversed Huber penalty functions employ a scale parameter. We provide an objective function that is jointly convex in the regression coefficient vector and these two scale parameters. 1 Introduction We consider here the regression problem of predicting y ∈ R based on z ∈ Rd. The training data are pairs (zi, yi) for i = 1, . , n. We suppose that each vector p of predictor vectors zi gets turned into a feature vector xi ∈ R via zi = φ(xi) for some fixed function φ. The predictor for y is linear in the features, taking the form µ + x0β where β ∈ Rp. In ridge regression (Hoerl and Kennard, 1970) we minimize over β, a criterion of the form n p X 0 2 X 2 (yi − µ − xiβ) + λ βj , (1) i=1 j=1 for a ridge parameter λ ∈ [0, ∞].
    [Show full text]
  • Lecture 22 - 11/19/2019 Lecture 22: Robust Location Estimation Lecturer: Jiantao Jiao Scribe: Vignesh Subramanian
    EE290 Mathematics of Data Science Lecture 22 - 11/19/2019 Lecture 22: Robust Location Estimation Lecturer: Jiantao Jiao Scribe: Vignesh Subramanian In this lecture, we get a historical perspective into the robust estimation problem and discuss Huber's work [1] for robust estimation of a location parameter. The Huber loss function is given by, ( 1 2 2 t ; jtj ≤ k ρHuber(t) = 1 2 : (1) k jtj − 2 k ; jtj > k Here k is a parameter and the idea behind the loss function is to penalize outliers (beyond k) linearly instead of quadratically. Figure 1 shows the Huber loss function for k = 1. In this lecture we will get an intuitive 1 2 Figure 1: The green line plots the Huber-loss function for k = 1, and the blue line plots the quadratic function 2 t . understanding for the reasons behind the particular form of this function, quadratic in interior, linear in exterior and convex and will see that this loss function is optimal for one dimensional robust mean estimation for Gaussian location model. First we describe the problem setting. 1 Problem Setting Suppose we observe X1;X2;:::;Xn where Xi − µ ∼ F 2 F are i.i.d. Here, F = fF j F = (1 − )G + H; H 2 Mg; (2) where G 2 M is some fixed distribution function which is usually assumed to have zero mean, and M denotes the space of all probability measures. This describes the corruption model where the observed distribution is a convex combination of the true distribution G(x) and an arbitrary corruption distribution H.
    [Show full text]
  • Robust Regression Through the Huber's Criterion and Adaptive Lasso
    Robust Regression through the Huber’s criterion and adaptive lasso penalty Sophie Lambert-Lacroix, Laurent Zwald To cite this version: Sophie Lambert-Lacroix, Laurent Zwald. Robust Regression through the Huber’s criterion and adap- tive lasso penalty. Electronic Journal of Statistics , Shaker Heights, OH : Institute of Mathematical Statistics, 2011, 5, pp.1015-1053. 10.1214/11-EJS635. hal-00661864 HAL Id: hal-00661864 https://hal.archives-ouvertes.fr/hal-00661864 Submitted on 20 Jan 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Electronic Journal of Statistics Vol. 5 (2011) 1015–1053 ISSN: 1935-7524 DOI: 10.1214/11-EJS635 Robust regression through the Huber’s criterion and adaptive lasso penalty Sophie Lambert-Lacroix UJF-Grenoble 1 / CNRS / UPMF / TIMC-IMAG UMR 5525, Grenoble, F-38041, France e-mail: [email protected] and Laurent Zwald LJK - Universit´eJoseph Fourier BP 53, Universit´eJoseph Fourier 38041 Grenoble cedex 9, France e-mail: [email protected] Abstract: The Huber’s Criterion is a useful method for robust regression. The adaptive least absolute shrinkage and selection operator (lasso) is a popular technique for simultaneous estimation and variable selection.
    [Show full text]
  • Sketching for M-Estimators and Robust Numerical Linear Algebra
    Sketching for M-Estimators and Robust Numerical Linear Algebra David Woodruff CMU Talk Outline • Regression – Sketching for least squares regression – Sketching for fast robust regression • Low Rank Approximation – Sketching for fast SVD – Sketching for fast robust low rank approximation • Recent sketching/sampling work for robust problems Linear Regression Matrix form Input: nd-matrix A and a vector b=(b1,…, bn) n is the number of examples; d is the number of unknowns Output: x* so that Ax* and b are close • Consider the over-constrained case, when n d Least Squares Regression 2 • Find x* that minimizes |Ax-b|2 •Ax* is the projection of b onto the column span of A • Desirable statistical properties • Closed form solution: x* = (ATA)-1 AT b Sketching to Solve Least Squares Regression . How to find an approximate solution x to minx |Ax-b|2 ? . Goal: output x‘ for which |Ax‘-b|2 ≤ (1+ε) minx |Ax-b|2 with high probability . Draw S from a k x n random family of matrices, for a value k << n . Compute S*A and S*b . Output the solution x‘ to minx‘ |(SA)x-(Sb)|2 How to Choose the Right Sketching Matrix ? . Recall: output the solution x‘ to minx‘ |(SA)x-(Sb)|2 . Lots of matrices work . S is d/ε2 x n matrix of i.i.d. Normal random variables . Computing S*A may be slow… . Can speed up to O(nd log n) time using Fast Johnson Lindenstrauss transforms [Sarlos] . Not sensitive to input sparsity Faster Sketching Matrices [CW] Think of rows . CountSketch matrix as hash buckets .
    [Show full text]
  • A General and Adaptive Robust Loss Function
    A General and Adaptive Robust Loss Function Jonathan T. Barron Google Research Abstract and can be adjusted to model a wider family of functions. This allows us to generalize algorithms built around a fixed We present a generalization of the Cauchy/Lorentzian, robust loss with a new “robustness” hyperparameter that can Geman-McClure, Welsch/Leclerc, generalized Charbon- be tuned or annealed to improve performance. nier, Charbonnier/pseudo-Huber/L1-L2, and L2 loss func- Though new hyperparameters may be valuable to a prac- tions. By introducing robustness as a continuous param- titioner, they complicate experimentation by requiring man- eter, our loss function allows algorithms built around ro- ual tuning or time-consuming cross-validation. However, bust loss minimization to be generalized, which improves by viewing our general loss function as the negative log- performance on basic vision tasks such as registration and likelihood of a probability distribution, and by treating the clustering. Interpreting our loss as the negative log of a robustness of that distribution as a latent variable, we show univariate density yields a general probability distribution that maximizing the likelihood of that distribution allows that includes normal and Cauchy distributions as special gradient-based optimization frameworks to automatically cases. This probabilistic interpretation enables the training determine how robust the loss should be without any manual of neural networks in which the robustness of the loss auto- parameter tuning. This “adaptive” form of our loss is par- matically adapts itself during training, which improves per- ticularly effective in models with multivariate output spaces formance on learning-based tasks such as generative im- (say, image generation or depth estimation) as we can intro- age synthesis and unsupervised monocular depth estima- duce independent robustness variables for each dimension tion, without requiring any manual parameter tuning.
    [Show full text]
  • ERM and RERM Are Optimal Estimators for Regression Problems When Malicious Outliers Corrupt the Labels
    ERM and RERM are optimal estimators for regression problems when malicious outliers corrupt the labels CHINOT Geoffrey ENSAE, 5 avenue Henri Chatelier, 91120, Palaiseau, France e-mail: [email protected] Abstract: We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk Minimizers (RERM) for regression problems with convex and L-Lipschitz loss functions. We consider a setting where jOj malicious outliers contaminate the labels. In that case, under a local Bernstein condi- tion, we show that the L2-error rate is bounded by rN + ALjOj=N, where N is the total number of observations, rN is the L2-error rate in the non- contaminated setting and A is a parameter coming from the local Bernstein condition. When rN is minimax-rate-optimal in a non-contaminated set- ting, the rate rN +ALjOj=N is also minimax-rate-optimal when jOj outliers contaminate the label. The main results of the paper can be used for many non-regularized and regularized procedures under weak assumptions on the noise. We present results for Huber's M-estimators (without penalization or regularized by the `1-norm) and for general regularized learning problems in reproducible kernel Hilbert spaces when the noise can be heavy-tailed. MSC 2010 subject classifications: Primary 62G35 ; secondary 62G08 . Keywords and phrases: Regularized empirical risk minimizers, outliers, robustness, minimax-rate-optimality. 1. Introduction Let (Ω; A;P ) be a probability space where Ω = X ×Y. X denotes the measurable space of the inputs and Y ⊂ R the measurable space of the outputs. Let (X; Y ) be a random variable taking values in Ω with joint distribution P and let µ be the marginal distribution of X.
    [Show full text]
  • Ensemble Methods - Boosting
    Ensemble Methods - Boosting Bin Li IIT Lecture Series 1 / 48 Boosting a weak learner 1 1 I Weak learner L produces an h with error rate β = 2 < 2 , with Pr 1 δ for any . L has access to continuous− stream of training≥ data− and a classD oracle. 1. L learns h1 on first N training points. 2. L randomly filters the next batch of training points, extracting N=2 points correctly classified by h1, N=2 incorrectly classified, and produces h2. 3. L builds a third training set of N points for which h1 and h2 disagree, and produces h3. 4. L ouptuts h = Majority Vote(h1; h2; h3) I Theorem (Schapire, 1990): \The Strength of Weak Learnability" 2 3 errorD(h) 3β 2β < β ≤ − 2 / 48 Plot of error rate 3β2 2β3 − 0.5 0.4 0.3 0.2 0.1 0.0 0.0 0.1 0.2 0.3 0.4 0.5 3 / 48 What is AdaBoost? Figure from HTF 2009. ) x ( m G m α =1 M m ) ) ) ) x ( x x x ( ( ( 3 2 M 1 G G G G )=sign x Final Classifier ( Boosting G Training SampleTraining Sample Training Sample Weighted SampleWeighted Sample Weighted Sample Weighted SampleWeighted Sample Weighted SampleWeighted Sample Weighted Sample Weighted Sample January 2003 Trevor Hastie, Stanford University 11 4 / 48 AdaBoost (Freund & Schapire, 1996) 1. Initialize the observation weigths: wi = 1=N, i = 1; 2;:::; N. 2. For m = 1 to M repeat steps (a)-(d): (a) Fit a classifier Gm(x) to the training data using weigths wi (b) Compute N i=1 wi I (yi = Gm(xi )) errm = N 6 P i=1 wi (c) Compute αm = log((1 errm)=errPm).
    [Show full text]
  • Benchmarking Daily Line Loss Rates of Low Voltage Transformer Regions in Power Grid Based on Robust Neural Network
    applied sciences Article Benchmarking Daily Line Loss Rates of Low Voltage Transformer Regions in Power Grid Based on Robust Neural Network Weijiang Wu 1,2 , Lilin Cheng 3,*, Yu Zhou 1,2, Bo Xu 4, Haixiang Zang 3 , Gaojun Xu 1,2 and Xiaoquan Lu 1,2 1 State Grid Jiangsu Electric Power Company Limited Marketing Service Center, Nanjing 210019, China; [email protected] (W.W.); [email protected] (Y.Z.); [email protected] (G.X.); [email protected] (X.L.) 2 State Grid Key Laboratory of Electric Power Metering, Nanjing 210019, China 3 College of Energy and Electrical Engineering, Hohai University, Nanjing 211100, China; [email protected] 4 Jiangsu Frontier Electric Technology Co., Ltd., Nanjing 211100, China; [email protected] * Correspondence: [email protected]; Tel.: +86-173-7211-8166 Received: 28 September 2019; Accepted: 3 December 2019; Published: 17 December 2019 Abstract: Line loss is inherent in transmission and distribution stages, which can cause certain impacts on the profits of power-supply corporations. Thus, it is an important indicator and a benchmark value of which is needed to evaluate daily line loss rates in low voltage transformer regions. However, the number of regions is usually very large, and the dataset of line loss rates contains massive outliers. It is critical to develop a regression model with both great robustness and efficiency when trained on big data samples. In this case, a novel method based on robust neural network (RNN) is proposed. It is a multi-path network model with denoising auto-encoder (DAE), which takes the advantages of dropout, L2 regularization and Huber loss function.
    [Show full text]
  • Consistent Regression When Oblivious Outliers Overwhelm
    Consistent regression when oblivious outliers overwhelm∗ Tommaso d’Orsi† Gleb Novikov‡ David Steurer§ Abstract We consider a robust linear regression model H = - , where an adversary ∗ + oblivious to the design - ℝ= 3 may choose to corrupt all but an fraction of ∈ × the observations H in an arbitrary way. Prior to our work, even for Gaussian -, no estimator for ∗ was known to be consistent in this model except for quadratic sample size = & 3 2 or for logarithmic inlier fraction > 1 log =. We show that consistent ( / ) / estimation is possible with nearly linear sample size and inverse-polynomial inlier frac- tion. Concretely, we show that the Huber loss estimator is consistent for every sample size = = $ 3 2 and achieves an error rate of $ 3 2= 1 2. Both bounds are optimal ( / ) ( / ) / (up to constantfactors). Our results extendto designsfar beyond the Gaussian case and only require the column span of - to not contain approximately sparse vectors (simi- lar to the kind of assumption commonly made about the kernel space for compressed sensing). We provide two technically similar proofs. One proof is phrased in terms of strong convexity, extending work of [TJSO14], and particularly short. The other proof highlights a connection between the Huber loss estimator and high-dimensional me- dian computations. In the special case of Gaussian designs, this connection leads us to a strikingly simple algorithm based on computing coordinate-wise medians that achieves optimal guarantees in nearly-linear time, and that can exploit sparsity of ∗. The model studied here also captures heavy-tailed noise distributions that may not arXiv:2009.14774v2 [cs.LG] 25 May 2021 even have a first moment.
    [Show full text]
  • Statistica Sinica Preprint No: SS-2019-0324
    Statistica Sinica Preprint No: SS-2019-0324 Title Robustness and Tractability for Non-convex M-estimators Manuscript ID SS-2019-0324 URL http://www.stat.sinica.edu.tw/statistica/ DOI 10.5705/ss.202019.0324 Complete List of Authors Ruizhi Zhang, Yajun Mei, Jianjun Shi and Huan Xu Corresponding Author Ruizhi Zhang E-mail [email protected] Notice: Accepted version subject to English editing. Statistica Sinica: Newly accepted Paper (accepted author-version subject to English editing) Statistica Sinica Robustness and Tractability for Non-convex M-estimators Ruizhi Zhang*, Yajun Mei**, Jianjun Shi**, Huan Xu*** *University of Nebraska-Lincoln, **Georgia Institute of Technology, ***Alibaba Inc. Abstract: We investigate two important properties of M-estimators, namely, ro- bustness and tractability, in the linear regression setting, when the observations are contaminated by some arbitrary outliers. Specifically, robustness means the statistical property that the estimator should always be close to the true under- lying parameters regardless of the distribution of the outliers, and tractability indicates the computational property that the estimator can be computed effi- ciently, even if the objective function of the M-estimator is non-convex. In this article, by learning the landscape of the empirical risk, we show that under some sufficient conditions, many M-estimators enjoy nice robustness and tractability properties simultaneously when the percentage of outliers is small. We further extend our analysis to the high-dimensional setting, where the number of pa- rameters is greater than the number of samples, p n, and prove that when the proportion of outliers is small, the penalized M-estimators with L1 penalty will enjoy robustness and tractability simultaneously.
    [Show full text]
  • Robust Regression Implementation Scottish Hill Races
    Introduction Robust regression Implementation Scottish hill races Robust regression Patrick Breheny December 1 Patrick Breheny BST 764: Applied Statistical Modeling 1/17 Introduction Robust regression Implementation Scottish hill races Belgian phone calls We begin our discussion of robust regression with a simple motivating example dealing with the number of phone calls made per year in Belgium The data set phones.txt contains two columns: year calls: Number of calls made (in millions) As it turns out, there is a flaw in the data { for a period of time from 1964-1969, the total length of calls was recorded instead of the number Patrick Breheny BST 764: Applied Statistical Modeling 2/17 Introduction Robust regression Implementation Scottish hill races Belgian phone calls: Linear vs. robust regression Least squares ● Huber 200 Tukey ● ● 150 ● ● ● 100 Millions of calls 50 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 1950 1955 1960 1965 1970 Year Patrick Breheny BST 764: Applied Statistical Modeling 3/17 Introduction Robust regression Implementation Scottish hill races Robust loss Robust regression methods achieve their robustness by modifying the loss function P 2 The linear regression loss function, l(r) = i ri , increases sharply with the size of the residual One alternative is to use the absolute value as a loss function P instead of squaring the residual: l(r) = i jrij This achieves robustness, but is hard to work with in practice because the absolute value function is not differentiable Patrick Breheny BST 764: Applied Statistical Modeling 4/17
    [Show full text]
  • Generalized Huber Loss for Robust Learning and Its Efficient
    1 Generalized Huber Loss for Robust Learning and its Efficient Minimization for a Robust Statistics Kaan Gokcesu, Hakan Gokcesu Abstract—We propose a generalized formulation of the Huber To create a robust loss with fast convergence, we need loss. We show that with a suitable function of choice, specifically to combine the properties of the absolute and the quadratic the log-exp transform; we can achieve a loss function which loss. The most straightforward approach is to use a piecewise combines the desirable properties of both the absolute and the quadratic loss. We provide an algorithm to find the minimizer of function to combine the quadratic and absolute losses where such loss functions and show that finding a centralizing metric they work the best. As an example, we can straightforwardly is not that much harder than the traditional mean and median. use the following function x2 , x 1 I. INTRODUCTION LP (x)= | |≤ . (1) ( x , x > 1 Many problems in learning, optimization and statistics liter- | | | | ature [1]–[4] require robustness, i.e., that a model trained (or While this function is continuous, the strict cutoff at 1 may optimized) be less influenced by some outliers than by inliers prove to be arbitrary or even useless for certain datasets. We (i.e., the nominal data) [5], [6]. This approach is extremely can bypass this problem by using a free variable as common in parameter estimation and learning tasks, where 1 2 a robust loss (e.g., the absolute error) may be more desirable δ x , x δ LP,δ(x)= | |≤ . (2) over a nonrobust loss (e.g., the quadratic error) due to its insen- ( x , x >δ sitivity to the large errors.
    [Show full text]