the network, or assume the over-parameterized property of the model and they study convergence, while our paper makes no such assumptions on the model and we study variance with respect to the mini-batch size. The proof also reveals the structure of the coefficients of the polynomial, and thus serving as a tool for future work on proving other properties of the stochastic gradient estimators.

The proofs are involved and require several key ideas. The main one is to show a more general result than it is necessary in order to carry out the induction. The induction is not only on time step $t$ but also on the batch size with the latter one being tricky to handle. New concepts and definitions are introduced in order to handle the more general case. Along the way we show a result of general interest establishing expectation of several rank one matrices sampled from a normal distribution intertwined with constant matrices.

In conclusion, we study the dynamics of SGD under linear regression and a two-layer linear network setting by focusing on the decreasing property of the variance of stochastic gradient estimators with respect to the mini-batch size. The proof techniques can also be used 