Many data analysis problems are NP-hard, a reality that has motivated researchers to develop a wealth of approximation algorithms and heuristics over the past few decades. Max-cut, clustering, and many other widely applicable partitioning problems fall into this camp. We focus on two common classes of algorithms that are used for clustering and partitioning problems, including SDP rounding algorithms and agglomerative clustering algorithms with dynamic programming. The best algorithm to use typically depends on the specific application or distribution over instances. A worst-case analysis is often used to compare algorithms, but worst-case instances may not be present in the particular problem domain, and thus may be misleading when determining which algorithm to apply. Therefore, it is necessary to develop optimization methods which return the algorithms and parameters best suited for typical inputs from the application at hand. We address this problem for integer quadratic programming and clustering within a learning-theoretic framework where the learning algorithm is given samples from an application-specific distribution over problem instances and learns an algorithm which performs well over the distribution. We provide sample complexity guarantees and computationally efficient algorithms which optimize an algorithm family's parameters to best suit typical inputs from a particular application. We analyze these algorithms using the learning-theoretic notion of pseudo-dimension. Along with upper bounds, we provide the first pseudo-dimension lower bounds in this line of work, which require an involved analysis of each algorithm family's performance on carefully constructed problem instances. Our bounds are tight and therefore nail down the intrinsic complexity of the algorithm classes we analyze, which are significantly more complex than classes commonly used in learning theory.
['Maria-Florina Balcan', 'Vaishnavh Nagarajan', 'Ellen Vitercik', 'Colin White']
Learning the best algorithm for max-cut, clustering, and other partitioning problems