  We study the problem of maximizing a monotone set function subject to a
cardinality constraint $k$ in the setting where some number of elements $\tau$
is deleted from the returned set. The focus of this work is on the worst-case
adversarial setting. While there exist constant-factor guarantees when the
function is submodular, there are no guarantees for non-submodular objectives.
In this work, we present a new algorithm Oblivious-Greedy and prove the first
constant-factor approximation guarantees for a wider class of non-submodular
objectives. The obtained theoretical bounds are the first constant-factor
bounds that also hold in the linear regime, i.e. when the number of deletions
$\tau$ is linear in $k$. Our bounds depend on established parameters such as
the submodularity ratio and some novel ones such as the inverse curvature. We
bound these parameters for two important objectives including support selection
and variance reduction. Finally, we numerically demonstrate the robust
performance of Oblivious-Greedy for these two objectives on various datasets.
