Towards Efficient Stochastic Optimization of Functions of Convex Sets by Christian J. Convey a Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Computer Science University of Rhode Island
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چکیده
Let D be a DAG and let X be any non-empty subset of D’s vertices. X is a convex set of D if D contains no path that originates in X , then visits one or more vertices not in X , and then re-enters X . This work presents basic convexity algorithms for creating, growing, and shrinking convex sets using two different approaches: predecessor and successor sets, and topological sorts. It shows that the algorithms based on predecessor and successor sets typically have higher asymptotic running times than those based on topological sorts. However, when creating a convex set based upon a potentially non-convex set of “seed” vertices, the use of predecessor and successor sets permits the creation of a convex set which is the uniquely smallest superset of the seeds. This work also considers the problem of stochastically searching for a global minimum over all convex sets of a given DAG, using the basic convexity algorithms described above. This work demonstrates the existence of such an algorithm that, when run for an unbounded finite number of iterations, the probability of it covering the entire search space approaches one. This work presents one possible mapping which extends this work’s optimization of a single parallel task to the optimization of task-parallel programs with multiple tasks. This mapping is studied experimentally. The results demonstrate a significant (32%) speed improvement over a human-crafted parallelization of the same program, suggesting the possible merit in this work’s approach to automated parallel-program optimization.
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