  Instrumental variables (IVs) are a popular and powerful tool for estimating
causal effects in the presence of unobserved confounding. However, classical
approaches rely on strong assumptions such as the $\textit{exclusion
criterion}$, which states that instrumental effects must be entirely mediated
by treatments. This assumption often fails in practice. When IV methods are
improperly applied to data that do not meet the exclusion criterion, estimated
causal effects may be badly biased. In this work, we propose a novel solution
that provides $\textit{partial}$ identification in linear systems given a set
of $\textit{leaky instruments}$, which are allowed to violate the exclusion
criterion to some limited degree. We derive a convex optimization objective
that provides provably sharp bounds on the average treatment effect under some
common forms of information leakage, and implement inference procedures to
quantify the uncertainty of resulting estimates. We demonstrate our method in a
set of experiments with simulated data, where it performs favorably against the
state of the art. An accompanying $\texttt{R}$ package, $\texttt{leakyIV}$, is
available from $\texttt{CRAN}$.
