Last iterate convergence of SGD for Least-Squares in the Interpolation regime

TL;DR

This paper demonstrates that the last iterate of SGD converges in a noiseless least-squares setting with over-parameterization, providing explicit non-asymptotic rates and a novel analysis for non-strongly convex problems.

Contribution

It shows the convergence of SGD's last iterate in a non-strongly convex setting and provides explicit polynomial convergence rates in over-parameterized models.

Findings

SGD last iterate converges in a noiseless least-squares model.

Explicit non-asymptotic convergence rates are derived.

Polynomial rates can outperform standard $O(1/T)$ bounds.

Abstract

Motivated by the recent successes of neural networks that have the ability to fit the data perfectly and generalize well, we study the noiseless model in the fundamental least-squares setup. We assume that an optimum predictor fits perfectly inputs and outputs $\langle \theta_* , \phi(X) \rangle = Y$, where $\phi(X)$ stands for a possibly infinite dimensional non-linear feature map. To solve this problem, we consider the estimator given by the last iterate of stochastic gradient descent (SGD) with constant step-size. In this context, our contribution is two fold: (i) from a (stochastic) optimization perspective, we exhibit an archetypal problem where we can show explicitly the convergence of SGD final iterate for a non-strongly convex problem with constant step-size whereas usual results use some form of average and (ii) from a statistical perspective, we give explicit non-asymptotic convergence rates in the over-parameterized setting and leverage a fine-grained parameterization of the problem to exhibit polynomial rates that can be faster than $O(1/T)$. The link with reproducing kernel Hilbert spaces is established.