Vector-valued self-normalized concentration inequalities beyond sub-Gaussianity
This work addresses a gap in concentration theory for vector-valued processes, which is important for applications like sequential decision-making and econometrics, though it appears incremental as it extends existing scalar results to vectors.
The paper tackles the problem of deriving concentration bounds for vector-valued self-normalized processes with light tails beyond sub-Gaussianity, such as Bennett or Bernstein bounds, and demonstrates their application in online linear regression and kernelized linear bandits.
The study of self-normalized processes plays a crucial role in a wide range of applications, from sequential decision-making to econometrics. While the behavior of self-normalized concentration has been widely investigated for scalar-valued processes, vector-valued processes remain comparatively underexplored, especially outside of the sub-Gaussian framework. In this contribution, we provide concentration bounds for self-normalized processes with light tails beyond sub-Gaussianity (such as Bennett or Bernstein bounds). We illustrate the relevance of our results in the context of online linear regression, with applications in (kernelized) linear bandits.