LearningTheoretic Foundations of Algorithm Configuration for Combinatorial Partitioning Problems
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Proceedings of the 2017 Conference on Learning Theory, PMLR 65:213274, 2017.
Abstract
Maxcut, clustering, and many other partitioning problems that are of significant importance to machine learning and other scientific fields are NPhard, 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 worstcase analysis is often used to compare algorithms. This may be misleading if worstcase 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. Recently, Gupta and Roughgarden introduced the first learningtheoretic framework to rigorously study this problem, using it to analyze classes of greedy heuristics, parameter tuning in gradient descent, and other problems. We study this algorithm configuration problem for clustering, maxcut, and other partitioning problems, such as integer quadratic programming, by designing computationally efficient and sample efficient learning algorithms which receive samples from an applicationspecific 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 pseudodimension 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 multistage optimization procedures and are significantly more complex than classes typically studied in learning theory.
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