In min-min optimization or max-min optimization, one has to compute the gradient of a function defined as a minimum. In most cases, the minimum has no closed-form, and an approximation is obtained via an iterative algorithm. There are two usual ways of estimating the gradient of the function: using either an analytic formula obtained by assuming exactness of the approximation, or automatic differentiation through the algorithm. In this paper, we study the asymptotic error made by these estimators as a function of the optimization error. We find that the error of the automatic estimator is close to the square of the error of the analytic estimator, reflecting a super-efficiency phenomenon. The convergence of the automatic estimator greatly depends on the convergence of the Jacobian of the algorithm. We analyze it for gradient descent and stochastic gradient descent and derive convergence rates for the estimators in these cases. Our analysis is backed by numerical experiments on toy problems and on Wasserstein barycenter computation. Finally, we discuss the computational complexity of these estimators and give practical guidelines to chose between them.

本文研究了用于最小化函数的梯度估计方法的渐近误差，并找到了自动估计器误差接近于解析估计器误差平方的超高效现象，分析了这些估计器的收敛率和计算复杂度，并给出了实际指南以在它们之间进行选择。

自动微分在最小值函数中的超高效性