Severin Reiz

Severin Reiz, Tobias Neckel, Hans-Joachim Bungartz

Training deep neural networks consumes increasing computational resource shares in many compute centers. Often, a brute force approach to obtain hyperparameter values is employed. Our goal is (1) to enhance this by enabling second-order optimization methods with fewer hyperparameters for large-scale neural networks and (2) to perform a survey of the performance optimizers for specific tasks to suggest users the best one for their problem. We introduce a novel second-order optimization method that requires the effect of the Hessian on a vector only and avoids the huge cost of explicitly setting up the Hessian for large-scale networks. We compare the proposed second-order method with two state-of-the-art optimizers on five representative neural network problems, including regression and very deep networks from computer vision or variational autoencoders. For the largest setup, we efficiently parallelized the optimizers with Horovod and applied it to a 8 GPU NVIDIA P100 (DGX-1) machine.

Chenhan D. Yu, James Levitt, Severin Reiz et al.

We present GOFMM (geometry-oblivious FMM), a novel method that creates a hierarchical low-rank approximation, "compression," of an arbitrary dense symmetric positive definite (SPD) matrix. For many applications, GOFMM enables an approximate matrix-vector multiplication in $N \log N$ or even $N$ time, where $N$ is the matrix size. Compression requires $N \log N$ storage and work. In general, our scheme belongs to the family of hierarchical matrix approximation methods. In particular, it generalizes the fast multipole method (FMM) to a purely algebraic setting by only requiring the ability to sample matrix entries. Neither geometric information (i.e., point coordinates) nor knowledge of how the matrix entries have been generated is required, thus the term "geometry-oblivious." Also, we introduce a shared-memory parallel scheme for hierarchical matrix computations that reduces synchronization barriers. We present results on the Intel Knights Landing and Haswell architectures, and on the NVIDIA Pascal architecture for a variety of matrices.