FLOPs as a Discriminant for Dense Linear Algebra Algorithms

TL;DR

This paper investigates whether FLOP counts reliably predict the fastest dense linear algebra algorithms, finding that anomalies exist and FLOPs alone are insufficient for optimal algorithm selection.

Contribution

The study demonstrates that FLOP counts are unreliable discriminants for dense linear algebra algorithms and highlights the need for combined performance models.

Findings

Anomalies in FLOP-based algorithm selection exist and cluster in regions.

FLOP count is not a dependable predictor of algorithm performance.

Most anomalies persist even after filtering cache effects.

Abstract

Expressions that involve matrices and vectors, known as linear algebra expressions, are commonly evaluated through a sequence of invocations to highly optimised kernels provided in libraries such as BLAS and LAPACK. A sequence of kernels represents an algorithm, and in general, because of associativity, algebraic identities, and multiple kernels, one expression can be evaluated via many different algorithms. These algorithms are all mathematically equivalent (i.e., in exact arithmetic, they all compute the same result), but often differ noticeably in terms of execution time. When faced with a decision, high-level languages, libraries, and tools such as Julia, Armadillo, and Linnea choose by selecting the algorithm that minimises the FLOP count. In this paper, we test the validity of the FLOP count as a discriminant for dense linear algebra algorithms, analysing "anomalies": problem instances for which the fastest algorithm does not perform the least number of FLOPs. To do so, we focused on relatively simple expressions and analysed when and why anomalies occurred. We found that anomalies exist and tend to cluster into large contiguous regions. For one expression anomalies were rare, whereas for the other they were abundant. We conclude that FLOPs is not a sufficiently dependable discriminant even when building algorithms with highly optimised kernels. Plus, most of the anomalies remained as such even after filtering out the inter-kernel cache effects. We conjecture that combining FLOP counts with kernel performance models will significantly improve our ability to choose optimal algorithms.