Sharper Sub-Weibull Concentrations
This work addresses theoretical challenges in machine learning and high-dimensional statistics by offering improved concentration bounds, but it is incremental as it builds on existing sub-Weibull frameworks.
The authors tackled the problem of deriving sharper concentration inequalities for sums of independent sub-Weibull random variables, resulting in new bounds with improved constants and a mixture of sub-Gaussian and sub-Weibull tails. They applied these results to statistical problems, such as providing new error bounds for negative binomial regressions and non-asymptotic versions of Bai-Yin's theorem for random matrices.
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 if the italic should be retained. Please check the whole text. is also proposed, which enables recovering the tight concentration inequality for a random variable (vector). For statistical applications, we give an $\ell_2$-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 $\{X_i\}_{i=1}^{n}$ without exponential-moment conditions.