Graph Expansion and Communication Costs of Fast Matrix Multiplication

The communication cost of algorithms (also known as I/O-complexity) is shown to be closely related to the expansion properties of the corresponding computation graphs. We demonstrate this on Strassen's and other fast matrix multiplication algorithms, and obtain first lower bounds on their communication costs. In the sequential case, where the processor has a fast memory of size $M$, too small to store three $n$-by-$n$ matrices, the lower bound on the number of words moved between fast and slow memory is, for many of the matrix multiplication algorithms, $\Omega((\frac{n}{\sqrt M})^{\omega_0}\cdot M)$, where $\omega_0$ is the exponent in the arithmetic count (e.g., $\omega_0 = \lg 7$ for Strassen, and $\omega_0 = 3$ for conventional matrix multiplication). With $p$ parallel processors, each with fast memory of size $M$, the lower bound is $p$ times smaller. These bounds are attainable both for sequential and for parallel algorithms and hence optimal. These bounds can also be attained by many fast algorithms in linear algebra (e.g., algorithms for LU, QR, and solving the Sylvester equation)

Exploiting Multiple Levels of Parallelism in Sparse Matrix-Matrix Multiplication

Sparse matrix-matrix multiplication (or SpGEMM) is a key primitive for many high-performance graph algorithms as well as for some linear solvers, such as algebraic multigrid. The scaling of existing parallel implementations of SpGEMM is heavily bound by communication. Even though 3D (or 2.5D) algorithms have been proposed and theoretically analyzed in the flat MPI model on Erdos-Renyi matrices, those algorithms had not been implemented in practice and their complexities had not been analyzed for the general case. In this work, we present the first ever implementation of the 3D SpGEMM formulation that also exploits multiple (intra-node and inter-node) levels of parallelism, achieving significant speedups over the state-of-the-art publicly available codes at all levels of concurrencies. We extensively evaluate our implementation and identify bottlenecks that should be subject to further research

Strong Scaling of Matrix Multiplication Algorithms and Memory-Independent Communication Lower Bounds

A parallel algorithm has perfect strong scaling if its running time on P processors is linear in 1/P, including all communication costs. Distributed-memory parallel algorithms for matrix multiplication with perfect strong scaling have only recently been found. One is based on classical matrix multiplication (Solomonik and Demmel, 2011), and one is based on Strassen's fast matrix multiplication (Ballard, Demmel, Holtz, Lipshitz, and Schwartz, 2012). Both algorithms scale perfectly, but only up to some number of processors where the inter-processor communication no longer scales. We obtain a memory-independent communication cost lower bound on classical and Strassen-based distributed-memory matrix multiplication algorithms. These bounds imply that no classical or Strassen-based parallel matrix multiplication algorithm can strongly scale perfectly beyond the ranges already attained by the two parallel algorithms mentioned above. The memory-independent bounds and the strong scaling bounds generalize to other algorithms.Comment: 4 pages, 1 figur

Algebraic Methods in the Congested Clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an $O(n^{1-2/\omega})$ round matrix multiplication algorithm, where $\omega < 2.3728639$ is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection algorithm of Dolev et al. [DISC 2012], -- a $(1 + o(1))$-approximation of all-pairs shortest paths in $O(n^{0.158})$ rounds, improving upon the $\tilde{O} (n^{1/2})$-round $(2 + o(1))$-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in $O(n^{0.158})$ rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266

Towards an Adaptive Skeleton Framework for Performance Portability

The proliferation of widely available, but very different, parallel architectures makes the ability to deliver good parallel performance on a range of architectures, or performance portability, highly desirable. Irregularly-parallel problems, where the number and size of tasks is unpredictable, are particularly challenging and require dynamic coordination. The paper outlines a novel approach to delivering portable parallel performance for irregularly parallel programs. The approach combines declarative parallelism with JIT technology, dynamic scheduling, and dynamic transformation. We present the design of an adaptive skeleton library, with a task graph implementation, JIT trace costing, and adaptive transformations. We outline the architecture of the protoype adaptive skeleton execution framework in Pycket, describing tasks, serialisation, and the current scheduler.We report a preliminary evaluation of the prototype framework using 4 micro-benchmarks and a small case study on two NUMA servers (24 and 96 cores) and a small cluster (17 hosts, 272 cores). Key results include Pycket delivering good sequential performance e.g. almost as fast as C for some benchmarks; good absolute speedups on all architectures (up to 120 on 128 cores for sumEuler); and that the adaptive transformations do improve performance

Chameleon: A Hybrid Secure Computation Framework for Machine Learning Applications

We present Chameleon, a novel hybrid (mixed-protocol) framework for secure function evaluation (SFE) which enables two parties to jointly compute a function without disclosing their private inputs. Chameleon combines the best aspects of generic SFE protocols with the ones that are based upon additive secret sharing. In particular, the framework performs linear operations in the ring $\mathbb{Z}_{2^l}$ using additively secret shared values and nonlinear operations using Yao's Garbled Circuits or the Goldreich-Micali-Wigderson protocol. Chameleon departs from the common assumption of additive or linear secret sharing models where three or more parties need to communicate in the online phase: the framework allows two parties with private inputs to communicate in the online phase under the assumption of a third node generating correlated randomness in an offline phase. Almost all of the heavy cryptographic operations are precomputed in an offline phase which substantially reduces the communication overhead. Chameleon is both scalable and significantly more efficient than the ABY framework (NDSS'15) it is based on. Our framework supports signed fixed-point numbers. In particular, Chameleon's vector dot product of signed fixed-point numbers improves the efficiency of mining and classification of encrypted data for algorithms based upon heavy matrix multiplications. Our evaluation of Chameleon on a 5 layer convolutional deep neural network shows 133x and 4.2x faster executions than Microsoft CryptoNets (ICML'16) and MiniONN (CCS'17), respectively

Distributed-Memory Breadth-First Search on Massive Graphs

This chapter studies the problem of traversing large graphs using the breadth-first search order on distributed-memory supercomputers. We consider both the traditional level-synchronous top-down algorithm as well as the recently discovered direction optimizing algorithm. We analyze the performance and scalability trade-offs in using different local data structures such as CSR and DCSC, enabling in-node multithreading, and graph decompositions such as 1D and 2D decomposition.Comment: arXiv admin note: text overlap with arXiv:1104.451