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