Sharper Sub-Weibull Concentrations

• Published in: Mathematics (Basel) 2022-07, Vol.10 (13), p.2252
• Main Authors: Zhang, Huiming, Wei, Haoyu
• Format: Article
• Language: English
• Subjects: Constants; constants-specified concentration inequalities; Deviation; Eigenvalues; exponential tail bounds; Food science; heavy-tailed random variables; Independent variables; Inequalities; Inequality; lower bounds on the least singular value; Machine learning; Norms; Random variables; sub-Weibull parameter
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math10132252

Constant-specified and exponential concentration inequalities play an essential role in the finite-sample theory of machine learning and high-dimensional statistics area. We obtain sharper and constants-specified concentration inequalities for the sum of independent sub-Weibull random variables, which leads to a mixture of two tails: sub-Gaussian for small deviations and sub-Weibull for large deviations from the mean. These bounds are new and improve existing bounds with sharper constants. In addition, a new sub-Weibull parameter is also proposed, which enables recovering the tight concentration inequality for a random variable (vector). For statistical applications, we give an ℓ2-error of estimated coefficients in negative binomial regressions when the heavy-tailed covariates are sub-Weibull distributed with sparse structures, which is a new result for negative binomial regressions. In applying random matrices, we derive non-asymptotic versions of Bai-Yin’s theorem for sub-Weibull entries with exponential tail bounds. Finally, by demonstrating a sub-Weibull confidence region for a log-truncated Z-estimator without the second-moment condition, we discuss and define the sub-Weibull type robust estimator for independent observations {Xi}i=1n without exponential-moment conditions.