Analysing heavy-tail properties of Stochastic Gradient Descent by means of Stochastic Recurrence Equations

In recent works on the theory of machine learning, it has been observed that heavy tail properties of Stochastic Gradient Descent (SGD) can be studied in the probabilistic framework of stochastic recursions. In particular, G\"{u}rb\"{u}zbalaban et al. (arXiv:2006.04740) considered a setup corresponding to linear regression for which iterations of SGD can be modelled by a multivariate affine stochastic recursion $X_k=A_k X_{k-1}+B_k$, for independent and identically distributed pairs $(A_k, B_k)$, where $A_k$ is a random symmetric matrix and $B_k$ is a random vector. In this work, we will answer several open questions of the quoted paper and extend their results by applying the theory of irreducible-proximal (i-p) matrices.