Hironori Fujisawa

Sparse linear regression methods such as Lasso require a tuning parameter that depends on the noise variance, which is typically unknown and difficult to estimate in practice. In the presence of heavy-tailed noise or adversarial outliers, this problem becomes more challenging. In this paper, we propose an estimator for robust and sparse linear regression that eliminates the need for explicit prior knowledge of the noise scale. Our method builds on the Huber loss and incorporates an iterative scheme that alternates between coefficient estimation and adaptive noise calibration via median-of-means. The approach is theoretically grounded and achieves sharp non-asymptotic error bounds under both sub-Gaussian and heavy-tailed noise assumptions. Moreover, the proposed method accommodates arbitrary outlier contamination in the response without requiring prior knowledge of the number of outliers or the sparsity level. While previous robust estimators avoid tuning parameters related to the noise scale or sparsity, our procedure achieves comparable error bounds when the number of outliers is unknown, and improved bounds when it is known. In particular, the improved bounds match the known minimax lower bounds up to constant factors.

We investigate the problem of estimating sparse linear regression coefficients in high-dimensional settings where both covariates and noise variables may exhibit heavy-tailed behavior and be corrupted by outliers. These conditions commonly arise in modern data analysis but pose challenges to standard methods such as the Lasso, which are typically designed for Gaussian or sub-Gaussian settings. Our theoretical analysis covers two regimes. First, in the absence of outliers, we derive non-asymptotic error bounds under a finite-kurtosis assumption on the covariates. Although these bounds are not as tight as the optimal rates achieved by the Lasso under Gaussian or sub-Gaussian designs, they remain near-optimal, indicating that accurate estimation is still feasible under significantly weakened distributional assumptions. Second, in the presence of adversarial contamination, we show that the error bound includes an additional term proportional to the square root of the outlier fraction. This form of dependence is known to be near-optimal under some conditions. We also validate the effectiveness of the proposed estimators through numerical experiments.

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.

This paper presents a comprehensive exploration of the theoretical properties inherent in the Adaptive Lasso and the Transfer Lasso. The Adaptive Lasso, a well-established method, employs regularization divided by initial estimators and is characterized by asymptotic normality and variable selection consistency. In contrast, the recently proposed Transfer Lasso employs regularization subtracted by initial estimators with the demonstrated capacity to curtail non-asymptotic estimation errors. A pivotal question thus emerges: Given the distinct ways the Adaptive Lasso and the Transfer Lasso employ initial estimators, what benefits or drawbacks does this disparity confer upon each method? This paper conducts a theoretical examination of the asymptotic properties of the Transfer Lasso, thereby elucidating its differentiation from the Adaptive Lasso. Informed by the findings of this analysis, we introduce a novel method, one that amalgamates the strengths and compensates for the weaknesses of both methods. The paper concludes with validations of our theory and comparisons of the methods via simulation experiments.

We consider outlier-robust and sparse estimation of linear regression coefficients, when the covariates and the noises are contaminated by adversarial outliers and noises are sampled from a heavy-tailed distribution. Our results present sharper error bounds under weaker assumptions than prior studies that share similar interests with this study. Our analysis relies on some sharp concentration inequalities resulting from generic chaining.

We consider a robust estimation of linear regression coefficients. In this note, we focus on the case where the covariates are sampled from an $L$-subGaussian distribution with unknown covariance, the noises are sampled from a distribution with a bounded absolute moment and both covariates and noises may be contaminated by an adversary. We derive an estimation error bound, which depends on the stable rank and the condition number of the covariance matrix of covariates with a polynomial computational complexity of estimation.

The inverse probability (IPW) and doubly robust (DR) estimators are often used to estimate the average causal effect (ATE), but are vulnerable to outliers. The IPW/DR median can be used for outlier-resistant estimation of the ATE, but the outlier resistance of the median is limited and it is not resistant enough for heavy contamination. We propose extensions of the IPW/DR estimators with density power weighting, which can eliminate the influence of outliers almost completely. The outlier resistance of the proposed estimators is evaluated through the unbiasedness of the estimating equations. Unlike the median-based methods, our estimators are resistant to outliers even under heavy contamination. Interestingly, the naive extension of the DR estimator requires bias correction to keep the double robustness even under the most tractable form of contamination. In addition, the proposed estimators are found to be highly resistant to outliers in more difficult settings where the contamination ratio depends on the covariates. The outlier resistance of our estimators from the viewpoint of the influence function is also favorable. Our theoretical results are verified via Monte Carlo simulations and real data analysis. The proposed methods were found to have more outlier resistance than the median-based methods and estimated the potential mean with a smaller error than the median-based methods.