On Using the Roofline Model with Lower Bounds on Data Movement

Abstract : The roofline model is a popular approach to ``bounds and bottleneck'' performance analysis. It focuses on the limits to performance of processors because of limited bandwidth to off-chip memory. It models upper bounds on performance as a function of operational intensity, the ratio of computational operations per byte of data moved from/to memory. While operational intensity can be directly measured for a specific implementation of an algorithm on a particular target platform, it is of interest to obtain broader insights on bottlenecks, where various semantically equivalent implementations of an algorithm are considered, along with analysis for variations in architectural parameters. This is currently very cumbersome and requires performance modeling and analysis of many variants. In this paper, we alleviate this problem by using the roofline model in conjunction with upper bounds on the operational intensity of computations as a function of cache capacity, derived using lower bounds on data movement. This enables bottleneck analysis that holds across all dependence-preserving semantically equivalent implementations of an algorithm. We demonstrate the utility of the approach in in assessing fundamental limits to performance and energy efficiency for several benchmark algorithms across a design space of architectural variations.