Sharp Concentration Inequalities: Phase Transition and Mixing of Orlicz Tails with Variance

In this work, we investigate how to develop sharp concentration inequalities for sub-Weibull random variables, including sub-Gaussian and sub-exponential distributions. Although the random variables may not be sub-Guassian, the tail probability around the origin behaves as if they were sub-Gaussian, and the tail probability decays align with the Orlicz $\Psi_\alpha$-tail elsewhere. Specifically, for independent and identically distributed (i.i.d.) $\{X_i\}_{i=1}^n$ with finite Orlicz norm $\|X\|_{\Psi_\alpha}$, our theory unveils that there is an interesting phase transition at $\alpha = 2$ in that $\PP\l(\l|\sum_{i=1}^n X_i \r| \geq t\r)$ with $t > 0$ is upper bounded by $2\exp\l(-C\max\l\{\frac{t^2}{n\|X\|_{\Psi_{\alpha}}^2},\frac{t^{\alpha}}{ n^{\alpha-1} \|X\|_{\Psi_{\alpha}}^{\alpha}}\r\}\r)$ for $\alpha\geq 2$, and by $2\exp\l(-C\min\l\{\frac{t^2}{n\|X\|_{\Psi_{\alpha}}^2},\frac{t^{\alpha}}{ n^{\alpha-1} \|X\|_{\Psi_{\alpha}}^{\alpha}}\r\}\r)$ for $1\leq \alpha\leq 2$ with some positive constant $C$. In many scenarios, it is often necessary to distinguish the standard deviation from the Orlicz norm when the latter can exceed the former greatly. To accommodate this, we build a new theoretical analysis framework, and our sharp, flexible concentration inequalities involve the variance and a mixing of Orlicz $\Psi_\alpha$-tails through the min and max functions. Our theory yields new, improved concentration inequalities even for the cases of sub-Gaussian and sub-exponential distributions with $\alpha = 2$ and $1$, respectively. We further demonstrate our theory on martingales, random vectors, random matrices, and covariance matrix estimation. These sharp concentration inequalities can empower more precise non-asymptotic analyses across different statistical and machine learning applications.