On Differentiating Parameterized Argmin and Argmax Problems with   Application to Bi-level Optimization
  Some recent works in machine learning and computer vision involve the solution of a bi-level optimization problem. Here the solution of a parameterized lower-level problem binds variables that appear in the objective of an upper-level problem. The lower-level problem typically appears as an argmin or argmax optimization problem. Many techniques have been proposed to solve bi-level optimization problems, including gradient descent, which is popular with current end-to-end learning approaches. In this technical report we collect some results on differentiating argmin and argmax optimization problems with and without constraints and provide some insightful motivating examples. 
Stephen Gould and Basura Fernando and Anoop Cherian and Peter Anderson   and Rodrigo Santa Cruz and Edison Guo
cs.CV math.OC