Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Super-efficiency of automatic differentiation for functions defined as a
  minimum   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.

13
The categories of this paper are: quant-ph
25
The categories of this paper are: patt-sol
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
2
The categories of this paper are: math
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph