On the Regularization Effect of Stochastic Gradient Descent applied to Least Squares

Abstract: 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.