  We consider the differentiation of the value function for parametric
optimization problems. Such problems are ubiquitous in Machine Learning
applications such as structured support vector machines, matrix factorization
and min-min or minimax problems in general. Existing approaches for computing
the derivative rely on strong assumptions of the parametric function.
Therefore, in several scenarios there is no theoretical evidence that a given
algorithmic differentiation strategy computes the true gradient information of
the value function. We leverage a well known result from convex duality theory
to relax the conditions and to derive convergence rates of the derivative
approximation for several classes of parametric optimization problems in
Machine Learning. We demonstrate the versatility of our approach in several
experiments, including non-smooth parametric functions. Even in settings where
other approaches are applicable, our duality based strategy shows a favorable
performance.
