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

• URL: http://arxiv.org/abs/2102.03183v1
• Date: Fri, 5 Feb 2021 14:02:20 GMT
• Title: Last iterate convergence of SGD for Least-Squares in the Interpolation regime
• Authors: Aditya Varre, Loucas Pillaud-Vivien, Nicolas Flammarion
• Abstract summary: 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.
• Score: 19.05750582096579
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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