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.

研究了神经网络在最小二乘设置中的应用，讨论了随机梯度下降与最终迭代的相关性，并在统计和优化双重视角下给出了多项式瞬时收敛率的解读，建立与再生核希尔伯特空间的联系。

插值方法下最小二乘SGD的最终收敛迭代