  We study the behavior of stochastic gradient descent applied to $\|Ax -b
\|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$. We
show that there is an explicit constant $c_{A}$ depending (mildly) on $A$ such
that $$ \mathbb{E} ~\left\| Ax_{k+1}-b\right\|^2_{2} \leq \left(1 +
\frac{c_{A}}{\|A\|_F^2}\right) \left\|A x_k -b \right\|^2_{2} -
\frac{2}{\|A\|_F^2} \left\|A^T A (x_k - x)\right\|^2_{2}.$$ This is a curious
inequality: the last term has one more matrix applied to the residual $u_k - u$
than the remaining terms: if $x_k - x$ is mainly comprised of large singular
vectors, stochastic gradient descent leads to a quick regularization. For
symmetric matrices, this inequality has an extension to higher-order Sobolev
spaces. This explains a (known) regularization phenomenon: an energy cascade
from large singular values to small singular values smoothes.

Numerical Analysis;Machine Learning;Numerical Analysis;Optimization and Control;Machine Learning