`

Timezone: »

 
Poster
Last iterate convergence of SGD for Least-Squares in the Interpolation regime.
Aditya Vardhan Varre · Loucas Pillaud-Vivien · Nicolas Flammarion

Fri Dec 10 08:30 AM -- 10:00 AM (PST) @ Virtual #None
Motivated by the recent successes of neural networks that have the ability to fit the data perfectly \emph{and} generalize well, we study the noiseless model in the fundamental least-squares setup. We assume that an optimum predictor perfectly fits the 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) \emph{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) \emph{from a statistical perspective}, we give explicit non-asymptotic convergence rates in the over-parameterized setting and leverage a \emph{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.

Author Information

Aditya Vardhan Varre (EPFL)
Loucas Pillaud-Vivien (EPFL)
Nicolas Flammarion (EPFL)

More from the Same Authors