A Note on the Performance Distribution of Affine Schedules

Iterative optimization has been shown to improve the performance of benchmarks significantly, but its application involves challenges such as the requirement of an expressive search space and the design of efficient search techniques. In this paper, we apply iterative optimization to the problem of optimizing in the polyhedral model, a powerful algebraic representation of any static control program, by using affine multidimensional schedules to represent arbitrarily complex transformation sequences. We propose to study the performance distribution of the search space of affine multidimensional schedules built specifically to guarantee legality and uniqueness of each program version. We extensively study the optimization of 5 representative benchmarks in this representation, and highlight a series of static and dynamic characteristics of the search space. We show how the space can be decoupled into subspaces, which can be statically ordered with respect to their impact on performance. Finally, we present a practical search method leveraging these properties to traverse the search space, yielding a 32.56% speedup on eight representative kernels.