  Many modern data analyses benefit from explicitly modeling dependence
structure in data -- such as measurements across time or space, ordered words
in a sentence, or genes in a genome. A gold standard evaluation technique is
structured cross-validation (CV), which leaves out some data subset (such as
data within a time interval or data in a geographic region) in each fold. But
CV here can be prohibitively slow due to the need to re-run already-expensive
learning algorithms many times. Previous work has shown approximate
cross-validation (ACV) methods provide a fast and provably accurate alternative
in the setting of empirical risk minimization. But this existing ACV work is
restricted to simpler models by the assumptions that (i) data across CV folds
are independent and (ii) an exact initial model fit is available. In structured
data analyses, both these assumptions are often untrue. In the present work, we
address (i) by extending ACV to CV schemes with dependence structure between
the folds. To address (ii), we verify -- both theoretically and empirically --
that ACV quality deteriorates smoothly with noise in the initial fit. We
demonstrate the accuracy and computational benefits of our proposed methods on
a diverse set of real-world applications.
