Enlarged Krylov Subspace Conjugate Gradient Methods for Reducing Communication

In this paper we introduce a new approach for reducing communication in Krylov subspace methods that consists of enlarging the Krylov subspace by a maximum of t vectors per iteration, based on the domain decomposition of the graph of A. The obtained enlarged Krylov subspace is a superset of the Krylov subspace. Thus it is possible to search for the solution of the system Ax=b in the enlarged Krylov subspace instead of the Krylov subspace. Moreover, we show in this paper that the enlarged Krylov projection subspace methods lead to faster convergence in terms of iterations and parallelizable algorithms with less communication, with respect to Krylov methods. In this paper we focus on Conjugate Gradient (CG), a Krylov projection method for symmetric (Hermitian) positive definite matrices. We discuss two new versions of Conjugate Gradient. The first method, multiple search direction with orthogonalization CG (MSDO-CG), is an adapted version of MSD-CG with the A-orthonormalization of the search directions to obtain a projection method that guarentees convergence at least as fast as CG. The second projection method that we propose here, long recurrence enlarged CG (LRE-CG), is similar to GMRES in that we build an orthonormal basis for the enlarged Krylov subspace rather than finding search directions. Then, we use the whole basis to update the solution and the residual. Both methods converge faster than CG in terms of iterations, but LRE-CG converges faster than MSDO-CG since it uses the whole basis to update the solution rather than only t search directions. And the more subdomains are introduced or the larger t is, the faster is the convergence of both methods with respect to CG in terms of iterations. For example, for t = 64 the MSDO-CG and LRE-CG methods converge in 75% up to 98% less iteration with respect to CG for the different test matrices. But increasing t also means increasing the memory requirements. Thus, in practice, t should be relatively small, depending on the available memory, on the size of the matrix, and on the number of iterations needed for convergence. We also present the parallel algorithms along with their expected performance based on the estimated run times, and the preconditioned versions with their convergence behavior

Enlarged Krylov Subspace Conjugate Gradient Methods for Reducing Communication

International audienceIn this paper we introduce a new approach for reducing communication in Krylov subspace methods that consists of enlarging the Krylov subspace by a maximum of $t$ vectors per iteration, based on a domain decomposition of the graph of $A$. The obtained enlarged Krylov subspace $\mathscr{K}_{k,t}(A,r_0)$ is a superset of the Krylov subspace $\mathcal{K}_k(A,r_0)$, $\mathcal{K}_k(A,r_0) \subset \mathscr{K}_{k,t}(A,r_0)$. Thus, we search for the solution of the system $Ax=b$ in $\mathscr{K}_{k,t}(A,r_0)$ instead of $\mathcal{K}_k(A,r_0)$. Moreover, we show in this paper that the enlarged Krylov projection subspace methods lead to faster convergence in terms of iterations and parallelizable algorithms with less communication, with respect to Krylov methods

Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method

Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient method outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus hiding global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by two-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace methods.Comment: 15 pages, 5 figures, 1 table, 2 algorithm

Réduction des coûts de communication et de calcul du Gradient Conjugué dans les sous-espaces de Krylov Élargi

In this paper we propose an algebraic method in order to reduce dynamically the number of search directions during block Conjugate Gradient iterations. Indeed, by monitoring the rank of the optimal step α k it is possible to detect inexact breakdowns and remove the corresponding search directions. We also propose an algebraic criterion that ensures in theory the equivalence between our method with dynamic reduction of the search directions and the classical block Conjugate Gradient. Numerical experiments show that the method is both stable, the number of iterations with or without reduction is of the same order, and effective, the search space is significantly reduced. We use this approach in the context of enlarged Krylov subspace methods which reduce communication when implemented on large scale machines. The reduction of the number of search directions further reduces the computation cost and the memory usage of those methods.Dans ce papier, nous proposons une méthode algébrique pour réduire dynamiquement le nombre de directions de recherche pendant les itérations du Gradient Conjugué par bloc. En effet, en mesurant la perte de rang numérique du pas optimal α k, il est possible d'enlever les directions de recherche superflues. Nous proposons aussi un critère algébrique qui assure en théorie l'équivalence entre notre méthode avec réduction dynamique des directions de recherche et le Gradient Conjugué par bloc classique. Les résultats numériques obtenus montrent que la méthode est à la fois stable, le nombre d'itérations est du même ordre avec ou sans la réduction, et efficace, l'espace de recherche est significativement réduit. Nous utilisons cette approche dans le contexte des méthodes de Krylov élargis qui réduisent les communications lorsqu'elles sont utilisées sur des machines parallèle à grande échelle. La réduction du nombre de directions de recherche réduit encore plus le coût de calcul et l'occupation mémoire de ces méthodes

Communication reduction techniques in numerical methods and deep neural networks

Inter-node communication has turned out to be one of the determining factors of the performance on modern HPC systems. Furthermore, the situation only gets worse with the ever-incresing size of the cores involved. Hence, this thesis explore the various possible techniques to reduce the communication during the execution of a parallel program. It turned out that there is no one-size-fit-all approach to the challenge. Despite this, the problems in each field, due to their unique characteristics, dispose of distinct opportunities for the communication reduction. The thesis, first devles into numerical linear algebra, develops an evolution of the Pipelined CG called IFCG. It eliminates the synchronizations normally take place towards the end of each iteration to increase the parallelism. Secondly, the thesis draws its attention on reducing the necessity to transfer the parameters between the CPU host and GPUs during a neural network training. It develops two routines: ADT and AWP in order to compress and decompress the weights with a reduced data representation format prior and right after the data transfer takes place. The compress rate is adjusted vis-à-vis the L2-norm of the weights of every layer. In the third contribution, the thesis diminish the communication in model parallelizing a deep neural network. Instead of splitting and distributing the neurons of each layer to the available processes on the system, now it is done every other layers. This results in a 50% percent reduction of the communication whereas it introduces 50% of extra local FP computation.La comunicació entre els nodes de computació multi-core sorgeix com un dels factors principals que impacta el rendiment d’un sistema HPC d’avui en dia. I més, mentre més core es pusa, pitjor la situació. Per tant aquesta tesi explora les possibles tècniques per a reduir la comunicació en l’execució d’un programa paral·lel. Tot i això, resulta que no existeix una sola tècnica que pugui resoldre aquest obstacle. Tot i que els problemes en cada àmbit, com que té els seus propis caracristics, disposa variosos oportunitats per la reducció de comunicació. La tesi, en primer lloc, dins de l’àmbit de l’àlgebra lineal numèriques desenvolupa un algoritme IFCG que és una evolució de Pipelined CG. IFCG elimina les sincronitzacions normalment posa cap al final de cada iteració per augmentar el paral·lelisme. En la segona contribució, la tesi dirigeix l’atenció a reduir la necessitat de transferir els paràmetres entre el CPU i els GPUs durant l’entrenament d’una xarxa neuronal. Desenvolupa rutines ADT i AWP per comprimir i descomprimir els pesos amb una representació de dades reduïda abans i just desprès de la transferència. La representació es decideix dinàmicament segons el L2-norm dels pesos a cada capa. Al final la tesi disminueix la comunicació en paral·lelitzar el model duna xarxa neurona. En lloc de distribuir les neurones de cada capa als processos disponibles en el sistema, es fa cada dues capes. Així que corta com mitja de la comunicació. En canvi, com que distribueix només cada dues capes, les capes restes es repliquen, resulta que incorre en una augmenta de 50% de computació local.Postprint (published version

Communication reduction techniques in numerical methods and deep neural networks

Inter-node communication has turned out to be one of the determining factors of the performance on modern HPC systems. Furthermore, the situation only gets worse with the ever-incresing size of the cores involved. Hence, this thesis explore the various possible techniques to reduce the communication during the execution of a parallel program. It turned out that there is no one-size-fit-all approach to the challenge. Despite this, the problems in each field, due to their unique characteristics, dispose of distinct opportunities for the communication reduction. The thesis, first devles into numerical linear algebra, develops an evolution of the Pipelined CG called IFCG. It eliminates the synchronizations normally take place towards the end of each iteration to increase the parallelism. Secondly, the thesis draws its attention on reducing the necessity to transfer the parameters between the CPU host and GPUs during a neural network training. It develops two routines: ADT and AWP in order to compress and decompress the weights with a reduced data representation format prior and right after the data transfer takes place. The compress rate is adjusted vis-à-vis the L2-norm of the weights of every layer. In the third contribution, the thesis diminish the communication in model parallelizing a deep neural network. Instead of splitting and distributing the neurons of each layer to the available processes on the system, now it is done every other layers. This results in a 50% percent reduction of the communication whereas it introduces 50% of extra local FP computation.La comunicació entre els nodes de computació multi-core sorgeix com un dels factors principals que impacta el rendiment d’un sistema HPC d’avui en dia. I més, mentre més core es pusa, pitjor la situació. Per tant aquesta tesi explora les possibles tècniques per a reduir la comunicació en l’execució d’un programa paral·lel. Tot i això, resulta que no existeix una sola tècnica que pugui resoldre aquest obstacle. Tot i que els problemes en cada àmbit, com que té els seus propis caracristics, disposa variosos oportunitats per la reducció de comunicació. La tesi, en primer lloc, dins de l’àmbit de l’àlgebra lineal numèriques desenvolupa un algoritme IFCG que és una evolució de Pipelined CG. IFCG elimina les sincronitzacions normalment posa cap al final de cada iteració per augmentar el paral·lelisme. En la segona contribució, la tesi dirigeix l’atenció a reduir la necessitat de transferir els paràmetres entre el CPU i els GPUs durant l’entrenament d’una xarxa neuronal. Desenvolupa rutines ADT i AWP per comprimir i descomprimir els pesos amb una representació de dades reduïda abans i just desprès de la transferència. La representació es decideix dinàmicament segons el L2-norm dels pesos a cada capa. Al final la tesi disminueix la comunicació en paral·lelitzar el model duna xarxa neurona. En lloc de distribuir les neurones de cada capa als processos disponibles en el sistema, es fa cada dues capes. Així que corta com mitja de la comunicació. En canvi, com que distribueix només cada dues capes, les capes restes es repliquen, resulta que incorre en una augmenta de 50% de computació local