  Max-cut, clustering, and many other partitioning problems that are of
significant importance to machine learning and other scientific fields are
NP-hard, a reality that has motivated researchers to develop a wealth of
approximation algorithms and heuristics. Although the best algorithm to use
typically depends on the specific application domain, a worst-case analysis is
often used to compare algorithms. This may be misleading if worst-case
instances occur infrequently, and thus there is a demand for optimization
methods which return the algorithm configuration best suited for the given
application's typical inputs. We address this problem for clustering, max-cut,
and other partitioning problems, such as integer quadratic programming, by
designing computationally efficient and sample efficient learning algorithms
which receive samples from an application-specific distribution over problem
instances and learn a partitioning algorithm with high expected performance.
Our algorithms learn over common integer quadratic programming and clustering
algorithm families: SDP rounding algorithms and agglomerative clustering
algorithms with dynamic programming. For our sample complexity analysis, we
provide tight bounds on the pseudodimension of these algorithm classes, and
show that surprisingly, even for classes of algorithms parameterized by a
single parameter, the pseudo-dimension is superconstant. In this way, our work
both contributes to the foundations of algorithm configuration and pushes the
boundaries of learning theory, since the algorithm classes we analyze consist
of multi-stage optimization procedures and are significantly more complex than
classes typically studied in learning theory.
