Communication-Optimal Tilings for Projective Nested Loops with Arbitrary Bounds

Abstract

Reducing communication -either between levels of a memory hierarchy or between processors over a network -is a key component of performance optimization (in both time and energy) for many problems, including dense linear algebra [BCD + 14], particle interactions [DGK + 13], and machine learning [DD18, GAB + 18]. For these problems, which can be represented as nestedloop computations, previous tiling based approaches [CDK + 13, DR16] have been used to find both lower bounds on the communication required to execute them and optimal rearrangements, or blockings, to attain such lower bounds. However, such general approaches have typically assumed the problem sizes are large, an assumption that is often not met in practice. For instance, the classical (# arithmetic operations)/(cache size) 1/2 lower bound for matrix multiplication [HK81, BCD + 14] is not tight for matrix-vector multiplications, which must read in at least O(# arithmetic operations) words of memory; similar issues occur for almost all convolutions in machine learning applications, which use extremely small filter sizes (and therefore, loop bounds).In this paper, we provide an efficient way to both find and obtain, via an appropriate, efficiently constructible blocking, communication lower bounds and matching tilings which attain these lower bounds for nested loop programs with arbitrary loop bounds that operate on multidimensional arrays in the projective case, where the array indices are subsets of the loop indices. Our approach works on all such problems, regardless of dimensionality, size, memory access patterns, or number of arrays, and directly applies to (among other examples) matrix multiplication and similar dense linear algebra operations, tensor contractions, n-body pairwise interactions, pointwise convolutions, and fully connected layers.

On the parallel I/O optimality of linear algebra kernels

“…An input program is a collection of statements S enclosed in loop nests, each of the following form (we use the loop nest notation introduced by Dinh and Demmel [23]):…”

“…Pebbling [13,26,37,45,56] Projection-based [8,15,20,21,23 However, as the local domains become larger and may be more efficiently pipelined and overlapped using asynchronous MPI routines and intra-node OpenMP parallelism, the advantage becomes significant (Figures 8 and 9). COnf LUX outperforms existing libraries up to three times (for P = 4, N = 4096, second-best library is SLATE -Figure 1) and COnf CHOX achieves up to 1.8 times speedup (e.g., P = 4, N = 4,096, second-best is again SLATE).…”

On the parallel I/O optimality of linear algebra kernels

“…We model the program execution as a computational directed acyclic graph (cDAG, details in Section 2. [23]): where (cf. Figure 3 for a summary) for each innermost loop iteration, statement 𝑆 is an evaluation of some function 𝑓 on 𝑚 inputs, where every input is an element of array 𝐴 𝑗 , 𝑗 = 1, .…”

On the Parallel I/O Optimality of Linear Algebra Kernels: Near-Optimal LU Factorization

“…Pebbling [11,25,36,42,52] Projection-based [7,13,18,20,22,49] Problem specific [1,8,15,45 As such, linear solvers are implemented in various libraries for shared-memory environments [3,4,19,30,33,47,59]. For distributed memory, vendor-optimized libraries [14,33] typically implement the ScaLAPACK interface [9], and are based on 2D decomposition, as we empirically verify (Section 8).…”