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

• URL: http://arxiv.org/abs/2007.13288v2
• Date: Tue, 1 Sep 2020 20:34:27 GMT
• Title: On the Regularization Effect of Stochastic Gradient Descent applied to Least Squares
• Authors: Stefan Steinerberger
• Abstract summary: We study the behavior of gradient descent applied to $|Ax -b |2 rightarrow min$ for invertible $A in mathbbRn times n$. We show that there is an explicit constant $c_A$ depending (mildly) on $A$ such that $$ mathbbE left| Ax_k+1-bright|2_2 leq.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the behavior of stochastic gradient descent applied to $\|Ax -b \|_2^2 \rightarrow \min$ for invertible $A \in \mathbb{R}^{n \times n}$. We show that there is an explicit constant $c_{A}$ depending (mildly) on $A$ such that $$ \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.

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