Sparse Linear Regression when Noises and Covariates are Heavy-Tailed and Contaminated by Outliers

We investigate a problem estimating coefficients of linear regression under sparsity assumption when covariates and noises are sampled from heavy tailed distributions. Additionally, we consider the situation where not only covariates and noises are sampled from heavy tailed distributions but also contaminated by outliers. Our estimators can be computed efficiently, and exhibit sharp error bounds.

Estimation of sparse linear regression coefficients under $L$-subexponential covariates

We address a task of estimating sparse coefficients in linear regression when the covariates are drawn from an $L$-subexponential random vector, which belongs to a class of distributions having heavier tails than a Gaussian random vector. Prior works have tackled this issue by assuming that the covariates are drawn from an $L$-subexponential random vector and have established error bounds that resemble those derived for Gaussian random vectors. However, these previous methods require stronger conditions to derive error bounds than those employed for Gaussian random vectors. In the present paper, we present an error bound identical to that obtained for Gaussian random vectors, up to constant factors, without requiring stronger conditions, even when the covariates are drawn from an $L$-subexponential random vector. Somewhat interestingly, we utilize an $\ell_1$-penalized Huber regression, that is recognized for its robustness to heavy-tailed random noises, not covariates. We believe that the present paper reveals a new aspect of the $\ell_1$-penalized Huber regression.

Outlier Robust and Sparse Estimation of Linear Regression Coefficients

We consider outlier robust and sparse estimation of linear regression coefficients when covariates and noise are sampled, respectively, from an $\mathfrak{L}$-subGaussian distribution and a heavy-tailed distribution, and additionally, the covariates and noise are contaminated by adversarial outliers. We deal with two cases: known or unknown covariance of the covariates. Particularly, in the former case, our estimator attains nearly information theoretical optimal error bound, and our error bound is sharper than that of earlier studies dealing with similar situations. Our estimator analysis relies heavily on Generic Chaining to derive sharp error bounds.

Robust and Sparse Estimation of Linear Regression Coefficients with Heavy-tailed Noises and Covariates

Robust and sparse estimation of linear regression coefficients is investigated. The situation addressed by the present paper is that covariates and noises are sampled from heavy-tailed distributions, and the covariates and noises are contaminated by malicious outliers. Our estimator can be computed efficiently. Further, our estimation error bound is sharp.

Adversarial robust weighted Huber regression

We propose a novel method to estimate the coefficients of linear regression when outputs and inputs are contaminated by malicious outliers. Our method consists of two-step: (i) Make appropriate weights $\left\{\hat{w}_i\right\}_{i=1}^n$ such that the weighted sample mean of regression covariates robustly estimates the population mean of the regression covariate, (ii) Process Huber regression using $\left\{\hat{w}_i\right\}_{i=1}^n$. When (a) the regression covariate is a sequence with i.i.d. random vectors drawn from sub-Gaussian distribution with unknown mean and known identity covariance and (b) the absolute moment of the random noise is finite, our method attains a faster convergence rate than Diakonikolas, Kong and Stewart (2019) and Cherapanamjeri et al. (2020). Furthermore, our result is minimax optimal up to constant factor. When (a) the regression covariate is a sequence with i.i.d. random vectors drawn from heavy tailed distribution with unknown mean and bounded kurtosis and (b) the absolute moment of the random noise is finite, our method attains a convergence rate, which is minimax optimal up to constant factor.

Adversarial Robust Low Rank Matrix Estimation: Compressed Sensing and Matrix Completion

We consider robust low rank matrix estimation when random noise is heavy-tailed and output is contaminated by adversarial noise. Under the clear conditions, we firstly attain a fast convergence rate for low rank matrix estimation including compressed sensing and matrix completion with convex estimators.

Robust estimation with Lasso when outputs are adversarially contaminated

We consider robust estimation when outputs are adversarially contaminated. Nguyen and Tran (2012) proposed an extended Lasso for robust parameter estimation and then they showed the convergence rate of the estimation error. Recently, Dalalyan and Thompson (2019) gave some useful inequalities and then they showed a sharper convergence rate than Nguyen and Tran (2012) . They focused on the fact that the minimization problem of the extended Lasso can become that of the penalized Huber loss function with $L_1$ penalty. The distinguishing point is that the Huber loss function includes an extra tuning parameter, which is different from the conventional method. However, there is a critical mistake in the proof of Dalalyan and Thompson (2019). We improve the proof and then we give a sharper convergence rate than Nguyen and Tran (2012) , when the number of outliers is larger. The significance of our proof is to use some specific properties of the Huber function. Such techniques have not been used in the past proofs.