We propose Federated Preconditioned Mixing (FedPM), a novel Federated Learning (FL) method that leverages second-order optimization. Prior methods - such as LocalNewton, LTDA, and FedSophia - have incorporated second-order optimization in FL by performing iterative local updates on clients and applying simple mixing of local parameters on the server. However, these methods often suffer from drift in local preconditioners, which significantly disrupts the convergence of parameter training, particularly in heterogeneous data settings. To overcome this issue, we refine the update rules by decomposing the ideal second-order update - computed using globally preconditioned global gradients - into parameter mixing on the server and local parameter updates on clients. As a result, our FedPM introduces preconditioned mixing of local parameters on the server, effectively mitigating drift in local preconditioners. We provide a theoretical convergence analysis demonstrating a superlinear rate for strongly convex objectives in scenarios involving a single local update. To demonstrate the practical benefits of FedPM, we conducted extensive experiments. The results showed significant improvements with FedPM in the test accuracy compared to conventional methods incorporating simple mixing, fully leveraging the potential of second-order optimization.

Local learning, which trains a network through layer-wise local targets and losses, has been studied as an alternative to backpropagation (BP) in neural computation. However, its algorithms often become more complex or require additional hyperparameters due to the locality, making it challenging to identify desirable settings where the algorithm progresses in a stable manner. To provide theoretical and quantitative insights, we introduce maximal update parameterization ($\mu$P) in the infinite-width limit for two representative designs of local targets: predictive coding (PC) and target propagation (TP). We verify that $\mu$P enables hyperparameter transfer across models of different widths. Furthermore, our analysis reveals unique and intriguing properties of $\mu$P that are not present in conventional BP. By analyzing deep linear networks, we find that PC's gradients interpolate between first-order and Gauss-Newton-like gradients, depending on the parameterization. We demonstrate that, in specific standard settings, PC in the infinite-width limit behaves more similarly to the first-order gradient. For TP, even with the standard scaling of the last layer differing from classical $\mu$P, its local loss optimization favors the feature learning regime over the kernel regime.

The Mixture of Experts (MoE) architecture reduces the training and inference cost significantly compared to a dense model of equivalent capacity. Upcycling is an approach that initializes and trains an MoE model using a pre-trained dense model. While upcycling leads to initial performance gains, the training progresses slower than when trained from scratch, leading to suboptimal performance in the long term. We propose Drop-Upcycling - a method that effectively addresses this problem. Drop-Upcycling combines two seemingly contradictory approaches: utilizing the knowledge of pre-trained dense models while statistically re-initializing some parts of the weights. This approach strategically promotes expert specialization, significantly enhancing the MoE model's efficiency in knowledge acquisition. Extensive large-scale experiments demonstrate that Drop-Upcycling significantly outperforms previous MoE construction methods in the long term, specifically when training on hundreds of billions of tokens or more. As a result, our MoE model with 5.9B active parameters achieves comparable performance to a 13B dense model in the same model family, while requiring approximately 1/4 of the training FLOPs. All experimental resources, including source code, training data, model checkpoints and logs, are publicly available to promote reproducibility and future research on MoE.

Graph Convolutional Networks (GCNs), particularly for largescale graphs, are crucial across numerous domains.However, training distributed full-batch GCNs on large-scale graphs suffers from inefficient memory access patterns and high communication overhead.To address these challenges, we introduce SuperGCN, an efficient and scalable distributed GCN

Variational Learning (VL) has recently gained popularity for training deep neural networks. Part of its empirical success can be explained by theories such as PAC-Bayes bounds, minimum description length and marginal likelihood, but little has been done to unravel the implicit regularization in play. Here, we analyze the implicit regularization of VL through the Edge of Stability (EoS) framework. EoS has previously been used to show that gradient descent can find flat solutions and we extend this result to show that VL can find even flatter solutions. This result is obtained by controlling the shape of the variational posterior as well as the number of posterior samples used during training. The derivation follows in a similar fashion as in the standard EoS literature for deep learning, by first deriving a result for a quadratic problem and then extending it to deep neural networks. We empirically validate these findings on a wide variety of large networks, such as ResNet and ViT, to find that the theoretical results closely match the empirical ones. Ours is the first work to analyze the EoS dynamics of~VL.

Gated Linear Units (GLUs) have become essential components in the feed-forward networks of state-of-the-art Large Language Models (LLMs). However, they require twice as many memory reads compared to feed-forward layers without gating, due to the use of separate weight matrices for the gate and value streams. To address this bottleneck, we introduce Masked Gated Linear Units (MGLUs), a novel family of GLUs with an efficient kernel implementation. The core contribution of MGLUs include: (1) the Mixture of Element-wise Gating (MoEG) architecture that learns multiple binary masks, each determining gate or value assignments at the element level on a single shared weight matrix resulting in reduced memory transfer, and (2) FlashMGLU, a hardware-friendly kernel that yields up to a 19.7$\times$ inference-time speed-up over a na\"ive PyTorch MGLU and is 47\% more memory-efficient and 34\% faster than standard GLUs despite added architectural complexity on an RTX5090 GPU. In LLM experiments, the Swish-activated variant SwiMGLU preserves its memory advantages while matching—or even surpassing—the downstream accuracy of the SwiGLU baseline.

Graph Convolutional Networks (GCNs) are widely used in various domains. However, training distributed full-batch GCNs on large-scale graphs poses challenges due to inefficient memory access patterns and high communication overhead. This paper presents a general and efficient GCN training framework on CPU supercomputers. It comprises a general aggregation kernel designed to optimize irregular memory access and a quantization method with label propagation to reduce communication overhead. Experimental results show that our method achieves a speedup of up to 4.1× compared with the SoTA implementations.

We give extensive empirical evidence against the common belief that variational learning is ineffective for large neural networks. We show that an optimizer called Improved Variational Online Newton (IVON) consistently matches or outperforms Adam for training large networks such as GPT-2 and ResNets from scratch. IVON's computational costs are nearly identical to Adam but its predictive uncertainty is better. We show several new use cases of IVON where we improve finetuning and model merging in Large Language Models, accurately predict generalization error, and faithfully estimate sensitivity to data. We find overwhelming evidence that variational learning is effective. Code is available at https://github.com/team-approx-bayes/ivon.

Formula-driven supervised learning (FDSL) has been shown to be an effective method for pre-training vision transformers, where ExFractalDB-21k was shown to exceed the pre-training effect of ImageNet-21k. These studies also indicate that contours mattered more than textures when pre-training vision transformers. However, the lack of a systematic investigation as to why these contour-oriented synthetic datasets can achieve the same accuracy as real datasets leaves much room for skepticism. In the present work, we develop a novel methodology based on circular harmonics for systematically investigating the design space of contour-oriented synthetic datasets. This allows us to efficiently search the optimal range of FDSL parameters and maximize the variety of synthetic images in the dataset, which we found to be a critical factor. When the resulting new dataset VisualAtom-21k is used for pre-training ViT-Base, the top-1 accuracy reached 83.7% when fine-tuning on ImageNet-1k. This is only 0.5% difference from the top-1 accuracy (84.2%) achieved by the JFT-300M pre-training, even though the scale of images is 1/14. Unlike JFT-300M which is a static dataset, the quality of synthetic datasets will continue to improve, and the current work is a testament to this possibility. FDSL is also free of the common issues associated with real images, e.g. privacy/copyright issues, labeling costs/errors, and ethical biases.

Matrix-matrix multiplication is used for various linear algebra algorithms such as matrix decomposition and tensor contraction. NVIDIA Tensor Core is a mixed-precision matrix-matrix multiplication and addition computing unit, where the theoretical peak performance is more than 300 TFlop/s on NVIDIA A100 GPU. NVIDIA provides WMMA API for using Tensor Cores in custom kernel functions. The most common way to use Tensor Core is to supply the input matrices from shared memory, which has higher bandwidth than global memory. However, the Bytes-per-Flops (B/F) ratio of the shared memory and Tensor Cores is small since the performance of Tensor Cores is high. Thus, it is important to reduce the shared memory footprint for efficient Tensor Cores usage. In this paper, we analyze the simple matrix-matrix multiplication on Tensor Cores by the roofline model and figure out that the bandwidth of shared memory might be a limitation of the performance when using WMMA API. To alleviate this issue, we provide a WMMA API extension library to boost the throughput of the computation, which has two components. The first one allows for manipulating the array of registers input to Tensor Cores flexibly. We evaluate the performance improvement of this library. The outcome of our evaluation shows that our library reduces the shared memory footprint and speeds up the computation using Tensor Cores. The second one is an API for the SGEMM emulation on Tensor Cores without additional shared memory usage. We have demonstrated that the single-precision emulating batch SGEMM implementation on Tensor Cores using this library achieves 54.2 TFlop/s on A100 GPU, which outperforms the theoretical peak performance of FP32 SIMT Cores while achieving the same level of accuracy as cuBLAS. The achieved throughput can not be achieved without reducing the shared memory footprint done by our library with the same amount of register usage.

Formula-driven supervised learning (FDSL) is a pre-training method that relies on synthetic images generated from mathematical formulae such as fractals. Prior work on FDSL has shown that pre-training vision transformers on such synthetic datasets can yield competitive accuracy on a wide range of downstream tasks. These synthetic images are categorized according to the parameters in the mathematical formula that generate them. In the present work, we hypothesize that the process for generating different instances for the same category in FDSL, can be viewed as a form of data augmentation. We validate this hypothesis by replacing the instances with data augmentation, which means we only need a single image per category. Our experiments shows that this one-instance fractal database (OFDB) performs better than the original dataset where instances were explicitly generated. We further scale up OFDB to 21,000 categories and show that it matches, or even surpasses, the model pre-trained on ImageNet-21k in ImageNet-1k fine-tuning. The number of images in OFDB is 21k, whereas ImageNet-21k has 14M. This opens new possibilities for pre-training vision transformers with much smaller datasets.

Pre-training is a strong strategy for enhancing visual models to efficiently train them with a limited number of labeled images. In semantic segmentation, creating annotation masks requires an intensive amount of labor and time, and therefore, a large-scale pre-training dataset with semantic labels is quite difficult to construct. Moreover, what matters in semantic segmentation pre-training has not been fully investigated. In this paper, we propose the Segmentation Radial Contour DataBase (SegRCDB), which for the first time applies formula-driven supervised learning for semantic segmentation. SegRCDB enables pre-training for semantic segmentation without real images or any manual semantic labels. SegRCDB is based on insights about what is important in pre-training for semantic segmentation and allows efficient pre-training. Pre-training with SegRCDB achieved higher mIoU than the pre-training with COCO-Stuff for fine-tuning on ADE-20k and Cityscapes with the same number of training images. SegRCDB has a high potential to contribute to semantic segmentation pre-training and investigation by enabling the creation of large datasets without manual annotation. The SegRCDB dataset will be released under a license that allows research and commercial use. Code is available at: https://github.com/dahlian00/SegRCDB

The numerical solution of eigenvalue problems is essential in various application areas of scientific and engineering domains. In many problem classes, the practical interest is only a small subset of eigenvalues so it is unnecessary to compute all of the eigenvalues. Notable examples are the electronic structure problems where the k-th smallest eigenvalue is closely related to the electronic properties of materials. In this paper, we consider the k-th eigenvalue problems of symmetric dense matrices with low-rank off-diagonal blocks. We present a linear time generalized LDL decomposition of matrices and combine it with the bisection eigenvalue algorithm to compute the k-th eigenvalue with controllable accuracy. In addition, if more than one eigenvalue is required, some of the previous computations can be reused to compute the other eigenvalues in parallel. Numerical experiments show that our method is more efficient than the state-of-the-art dense eigenvalue solver in LAPACK/ScaLAPACK and ELPA. Furthermore, tests on electronic state calculations of carbon nanomaterials demonstrate that our method outperforms the existing HSS-based bisection eigenvalue algorithm on 3D problems.

Structured dense matrices result from boundary integral problems in electrostatics and geostatistics, and also Schur complements in sparse preconditioners such as multi-frontal methods. Exploiting the structure of such matrices can reduce the time for dense direct factorization from O(N3) to O(N). The Hierarchically Semi-Separable (HSS) matrix is one such low rank matrix format that can be factorized using a Cholesky-like algorithm called ULV factorization. The HSS-ULV algorithm is highly parallel because it removes the dependency on trailing sub-matrices at each HSS level. However, a key merge step that links two successive HSS levels remains a challenge for efficient parallelization. In this paper, we use an asynchronous runtime system PaRSEC with the HSS-ULV algorithm. We compare our work with STRUMPACK and LORAPO, both state-of-the-art implementations of dense direct low rank factorization, and achieve up to 2x better factorization time for matrices arising from a diverse set of applications on up to 128 nodes of Fugaku for similar or better accuracy for all the problems that we survey.

Symmetric eigenvalue decomposition (EVD) is a fundamental analytic and numerical tool used in many scientific areas. The state-of-the-art algorithm in terms of performance is typically the two-stage tridiagonalization method. The first stage in the two-stage tridiagonalization is called successive band reduction (SBR), which reduces a symmetric matrix to a band form, and its computational cost usually dominates. When Tensor Core (specialized matrix computational accelerator) is used to accelerate the expensive EVD, the conventional ZY-representation-based method results in suboptimal performance due to unfavorable shapes of the matrix computations. In this paper, we propose a new method that uses WY representation instead of ZY representation (see Section 3.2 for details), which can provide a better combination of locality and parallelism so as to perform better on Tensor Cores. Experimentally, the proposed method can bring up to 3.7x speedup in SBR and 2.3x in the entire EVD compared to state-of-the-art implementations.

Factorization and multiplication of dense matrices and tensors are critical, yet extremely expensive pieces of the scientific toolbox. Careful use of low rank approximation can drastically reduce the computation and memory requirements of these operations. In addition to a lower arithmetic complexity, such methods can, by their structure, be designed to efficiently exploit modern hardware architectures. The majority of existing work relies on batched BLAS libraries to handle the computation of many small dense matrices. We show that through careful analysis of the cache utilization, register accumulation using SIMD registers and a redesign of the implementation, one can achieve significantly higher throughput for these types of batched low-rank matrices across a large range of block and batch sizes. We test our algorithm on three CPUs using diverse ISAs – the Fujitsu A64FX using ARM SVE, the Intel Xeon 6148 using AVX-512, and AMD EPYC 7502 using AVX-2, and show that our new batching methodology is able to obtain more than twice the throughput of vendor optimized libraries for all CPU architectures and problem sizes.

In the present work, we show that the performance of formula-driven supervised learning (FDSL) can match or even exceed that of ImageNet-21k without the use of real images, human-, and self-supervision during the pre-training of Vision Transformers (ViTs). For example, ViT-Base pre-trained on ImageNet-21k shows 81.8% top-1 accuracy when fine-tuned on ImageNet-1k and FDSL shows 82.7% top-1 accuracy when pre-trained under the same conditions (number of images, hyperparameters, and number of epochs). Images generated by formulas avoid the privacy/copyright issues, labeling cost and errors, and biases that real images suffer from, and thus have tremendous potential for pre-training general models. To understand the performance of the synthetic images, we tested two hypotheses, namely (i) object contours are what matter in FDSL datasets and (ii) increased number of parameters to create labels affects performance improvement in FDSL pre-training. To test the former hypothesis, we constructed a dataset that consisted of simple object contour combinations. We found that this dataset can match the performance of fractals. For the latter hypothesis, we found that increasing the difficulty of the pre-training task generally leads to better fine-tuning accuracy.

Inverse Reinforcement Learning (IRL) is attractive in scenarios where reward engineering can be tedious. However, prior IRL algorithms use on-policy transitions, which require intensive sampling from the current policy for stable and optimal performance. This limits IRL applications in the real world, where environment interactions can become highly expensive. To tackle this problem, we present Off-Policy Inverse Reinforcement Learning (OPIRL), which (1) adopts off-policy data distribution instead of on-policy and enables significant reduction of the number of interactions with the environment, (2) learns a reward function that is transferable with high generalization capabilities on changing dynamics, and (3) leverages mode-covering behavior for faster convergence. We demonstrate that our method is considerably more sample efficient and generalizes to novel environments through the experiments. Our method achieves better or comparable results on policy performance baselines with significantly fewer interactions. Furthermore, we empirically show that the recovered reward function generalizes to different tasks where prior arts are prone to fail.

Among various supervised deep metric learning methods proxy-based approaches have achieved high retrieval accuracies. Proxies, which are class-representative points in an embedding space, receive updates based on proxy-sample similarities in a similar manner to sample representations. In existing methods, a relatively small number of samples can produce large gradient magnitudes (i.e., hard samples), and a relatively large number of samples can produce small gradient magnitudes (i.e., easy samples); these can play a major part in updates. Assuming that acquiring too much sensitivity to such extreme sets of samples would deteriorate the generalizability of a method, we propose a novel proxy-based method called Informative Sample-Aware Proxy (Proxy-ISA), which directly modifies a gradient weighting factor for each sample using a scheduled threshold function, so that the model is more sensitive to the informative samples. Extensive experiments on the CUB-200-2011, Cars-196, Stanford Online Products and In-shop Clothes Retrieval datasets demonstrate the superiority of Proxy-ISA compared with the state-of-the-art methods.

Factorization of large dense matrices are ubiquitous in engineering and data science applications, e.g. preconditioners for iterative boundary integral solvers, frontal matrices in sparse multifrontal solvers, and computing the determinant of covariance matrices. HSS and$\mathcal{H}^{2}$-matrices are hierarchical low-rank matrix formats that can reduce the complexity of factorizing such dense matrices from$\mathrm{O}(N^{3})$to$\mathrm{O}(N)$. For HSS matrices, it is possible to remove the dependency on the trailing matrices during Cholesky/LU factorization, which results in a highly parallel algorithm. However, the weak admissibility of HSS causes the rank of off-diagonal blocks to grow for 3-D problems, and the method is no longer$\mathrm{O}(N)$. On the other hand, the strong admissibility of$\mathcal{H}^{2}$-matrices allows it to handle 3-D problems in$\mathcal{O}(N)$, but introduces a dependency on the trailing matrices. In the present work, we pre-compute the fill-ins and integrate them into the shared basis, which allows us to remove the dependency on trailing-matrices even for$\mathcal{H}^{2}$-matrices. Comparisons with a block low-rank factorization code LORAPO showed a maximum speed up of 4,700x for a 3-D problem with complex geometry.

Large-scale distributed training of deep neural networks results in models with worse generalization performance as a result of the increase in the effective mini-batch size. Previous approaches attempt to address this problem by varying the learning rate and batch size over epochs and layers, or ad hoc modifications of batch normalization. We propose scalable and practical natural gradient descent (SP-NGD), a principled approach for training models that allows them to attain similar generalization performance to models trained with first-order optimization methods, but with accelerated convergence. Furthermore, SP-NGD scales to large mini-batch sizes with a negligible computational overhead as compared to first-order methods. We evaluated SP-NGD on a benchmark task where highly optimized first-order methods are available as references: training a ResNet-50 model for image classification on ImageNet. We demonstrate convergence to a top-1 validation accuracy of 75.4 percent in 5.5 minutes using a mini-batch size of 32,768 with 1,024 GPUs, as well as an accuracy of 74.9 percent with an extremely large mini-batch size of 131,072 in 873 steps of SP-NGD.

We present two new algorithms for Householder QR factorization of Block Low-Rank (BLR) matrices: one that performs block-column-wise QR and another that is based on tiled QR. We show how the block-column-wise algorithm exploits BLR structure to achieve arithmetic complexity of 𝒪( mn ), while the tiled BLR-QR exhibits 𝒪( mn 1.5 complexity. However, the tiled BLR-QR has finer task granularity that allows parallel task-based execution on shared memory systems. We compare the block-column-wise BLR-QR using fork-join parallelism with tiled BLR-QR using task-based parallelism. We also compare these two implementations of Householder BLR-QR with a block-column-wise Modified Gram–Schmidt (MGS) BLR-QR using fork-join parallelism and a state-of-the-art vendor-optimized dense Householder QR in Intel MKL. For a matrix of size 131k × 65k, all BLR methods are more than an order of magnitude faster than the dense QR in MKL. Our methods are also robust to ill conditioning and produce better orthogonal factors than the existing MGS-based method. On a CPU with 64 cores, our parallel tiled Householder and block-column-wise Householder algorithms show a speedup of 50 and 37 times, respectively.

We present RePOSE, a fast iterative refinement method for 6D object pose estimation. Prior methods perform refinement by feeding zoomed-in input and rendered RGB images into a CNN and directly regressing an update of a refined pose. Their runtime is slow due to the computational cost of CNN, which is especially prominent in multiple-object pose refinement. To overcome this problem, RePOSE leverages image rendering for fast feature extraction using a 3D model with a learnable texture. We call this deep texture rendering, which uses a shallow multilayer perceptron to directly regress a view-invariant image representation of an object. Furthermore, we utilize differentiable Levenberg-Marquardt (LM) optimization to refine a pose fast and accurately by minimizing the distance between the input and rendered image representations without the need of zooming in. These image representations are trained such that differentiable LM optimization converges within few iterations. Consequently, RePOSE runs at 92 FPS and achieves state-of-the-art accuracy of 51.6% on the Occlusion LineMOD dataset - a 4.1% absolute improvement over the prior art, and comparable result on the YCB-Video dataset with a much faster runtime. The code is available at https://github.com/sh8/repose.

Hierarchical Matrix (H-matrix) is an approximation technique which splits a target dense matrix into multiple submatrices, and where a selected portion of submatrices are low-rank approximated. The technique substantially reduces both time and space complexity of dense matrix vector multiplication, and hence has been applied to numerous practical problems.

Rich information matrices from first and second-order derivatives have many potential applications in both theoretical and practical problems in deep learning. However, computing these information matrices is extremely expensive and this enormous cost is currently limiting its application to important problems regarding generalization, hyperparameter tuning, and optimization of deep neural networks. One of the most challenging use cases of information matrices is their use as a preconditioner for the optimizers, since the information matrices need to be updated every step. In this work, we conduct a step-by-step performance analysis when computing the Fisher information matrix during training of ResNet-50 on ImageNet, and show that the overhead can be reduced to the same amount as the cost of performing a single SGD step. We also show that the resulting Fisher preconditioned optimizer can converge in 1/3 the number of epochs compared to SGD, while achieving the same Top-1 validation accuracy. This is the first work to achieve such accuracy with K-FAC while reducing the training time to match that of SGD.

Classical learning theory states that when the number of parameters of the model is too large compared to the data, the model will overfit and the generalization performance deteriorates. However, it has been empirically shown that deep neural networks (DNN) can achieve high generalization capability by training with extremely large amount of data and model parameters, which exceeds the predictions of classical learning theory. One drawback of this is that training of DNN requires enormous calculation time. Therefore, it is necessary to reduce the training time through large scale parallelization. Straightforward data-parallelization of DNN degrades convergence and generalization. In the present work, we investigate the possibility of using second order methods to solve this generalization gap in large-batch training. This is motivated by our observation that each mini-batch becomes more statistically stable, and thus the effect of considering the curvature plays a more important role in large-batch training. We have also found that naively adapting the natural gradient method causes the generalization performance to deteriorate further due to the lack of regularization capability. We propose an improved second order method by smoothing the loss function, which allows second-order methods to generalize as well as mini-batch SGD.

Data-parallel distributed deep learning requires an AllReduce operation between all GPUs with message sizes in the order of hundreds of megabytes. The popular implementation of AllReduce for deep learning is the Ring-AllReduce, but this method suffers from latency when using thousands of GPUs. There have been efforts to reduce this latency by combining the ring with more latency-optimal hierarchical methods. In the present work, we consider these hierarchical communication methods as a general hierarchical Ring-AllReduce with a pure Ring-AllReduce on one end and Rabenseifner's algorithm on the other end of the spectrum. We exhaustively test the various combinations of hierarchical partitioning of processes on the ABCI system in Japan on up to 2048 GPUs. We develop a performance model for this generalized hierarchical Ring-AllReduce and show the lower-bound of the effective bandwidth achievable for the hierarchical NCCL communication on thousands of GPUs. Our measurements agree well with our performance model. We also find that the optimal large-scale process hierarchy contains the optimal small-scale process hierarchy so the search space for the optimal communication will be reduced.

Large-scale distributed training of deep neural networks suffers from the generalization gap caused by the increase in the effective mini-batch size. Previous approaches try to solve this problem by varying the learning rate and batch size over epochs and layers, or some ad hoc modification of the batch normalization. We propose an alternative approach using a second order optimization method that shows similar generalization capability to first order methods, but converges faster and can handle larger mini-batches. To test our method on a benchmark where highly optimized first order methods are available as references, we train ResNet-50 on ImageNet-1K. We converged to 75% Top-1 validation accuracy in 35 epochs for mini-batch sizes under 16,384, and achieved 75% even with a mini-batch size of 131,072, which took only 978 iterations.

Bayesian methods promise to fix many shortcomings of deep learning, but they are impractical and rarely match the performance of standard methods, let alone improve them. In this paper, we demonstrate practical training of deep networks with natural-gradient variational inference. By applying techniques such as batch normalisation, data augmentation, and distributed training, we achieve similar performance in about the same number of epochs as the Adam optimiser, even on large datasets such as ImageNet. Importantly, the benefits of Bayesian principles are preserved: predictive probabilities are well-calibrated, uncertainties on out-of-distribution data are improved, and continual-learning performance is boosted. This work enables practical deep learning while preserving benefits of Bayesian principles. A PyTorch implementation is available as a plug-and-play optimiser.

HACApK is a software package for solving dense linear systems of equations and is used in other software packages, like ppohBEM for solving boundary integral equations. To enable the solution of large-scale boundary value problems, HACApK hierarchically compresses the coefficient matrix and uses the BiConjugate Gradient Stabilized (BiCGStab) method for solving the linear system. To extend HACApK's capability, this paper outlines how we ported the HACApK linear solver onto GPU clusters. Though the potential of GPUs has been widely accepted in high-performance computing, it is still a challenge to utilize the GPUs for a solver, like HACApK, that requires fine-grained irregular computation and global communication. To utilize the GPUs, we integrated the variable-size batched GPU kernel that was recently released in the MAGMA software package. This is the first time the variable-size batched kernels were used in a solver or application code. We discuss several techniques to improve the performance of the batched kernel and demonstrate the effects of these techniques on two state-of-the-art GPU clusters. For instance, with two 14-core Intel Xeon CPUs and four NVIDIA P100 GPUs per node, the GPU kernel obtained a solver speedup of 8× on one node and 4× on eight nodes. We also show that when the inter-GPU communication becomes significant, the solution time can be further reduced by a factor of 2× by carefully designing the communication layer with the underlying node architecture in mind.

Tapas is our new C++ programming framework for hierarchical algorithms such as N-body, on large scale heterogeneous supercomputers. Although N-body and their variants are widely used in scientific applications, their correct implementations are often difficult on such modern machines, as the algorithms are irregular, complex, and involve explicit task parallel programming over distributed nodes. Encapsulating the complexities in a library or a framework has been challenging due to irregular data access over massively distributed memory. Tapas solves this by converting the users clean implicit-style parallel program into an inspector-executor style code on heterogeneous multi-core, multi-node environment solely by the use of C++ template metaprogramming. A prototype implementation of the Fast Multipole Method on Tapas demonstrates a comparable performance and scaling as ExaFMM, the fastest hand-tuned implementation of FMM, as well as efficient usage of hundreds of GPUs. Specifically, the serial performance is 95% of ExaFMM, whereas the distributed-memory strong-scaling evaluation using up to 1500 CPU cores demonstrates 64% to 81% of the ExaFMM performance. The multi-GPU version of the Tapas-based FMM achieves a 5.15x speedup when executed on 100 nodes of TSUBAME2.5 with 300 GPUs.

SUMMARY Fast multipole methods (FMMs) have complexity, are compute bound, and require very little synchronization, which makes them a favorable algorithm on next‐generation supercomputers. Their most common application is to accelerateN‐body problems, but they can also be used to solve boundary integral equations. When the particle distribution is irregular and the tree structure is adaptive, load balancing becomes a non‐trivial question. A common strategy for load balancing FMMs is to use the work load from the previous step as weights to statically repartition the next step. The authors discuss in the paper another approach based on data‐driven execution to efficiently tackle this challenging load balancing problem. The core idea consists of breaking the most time‐consuming stages of the FMMs into smaller tasks. The algorithm can then be represented as a directed acyclic graph where nodes represent tasks and edges represent dependencies among them. The execution of the algorithm is performed by asynchronously scheduling the tasks using the queueing and runtime for kernels runtime environment, in a way such that data dependencies are not violated for numerical correctness purposes. This asynchronous scheduling results in an out‐of‐order execution. The performance results of the data‐driven FMM execution outperform the previous strategy and show linear speedup on a quad‐socket quad‐core Intel Xeon system.Copyright © 2013 John Wiley & Sons, Ltd.

We present an extension to ExaFMM, a Fast Multipole Method library, as a generalized approach for fast and scalable execution of the Force-Directed Graph Layout algorithm. The Force-Directed Graph Layout algorithm is a physics-based approach to graph layout that treats the vertices V as repelling charged particles with the edges E connecting them acting as springs. Traditionally, the amount of work required in applying the Force-Directed Graph Layout algorithm is O(|V|2+ |E|) using direct calculations and O(|V| log |V| + |E|) using truncation, filtering, and/or multi-level techniques. Correct application of the Fast Multipole Method allows us to maintain a lower complexity of O(|V| + |E|) while regaining most of the precision lost in other techniques. Solving layout problems for truly large graphs with millions of vertices still requires a scalable algorithm and implementation. We have been able to leverage the scalability and architectural adaptability of the ExaFMM library to create a Force-Directed Graph Layout implementation that runs efficiently on distributed multicore and multi-GPU architectures.

As an entry for the 2009 Gordon Bell price/performance prize, we present the results of two different hierarchical N-body simulations on a cluster of 256 graphics processing units (GPUs). Unlike many previous N-body simulations on GPUs that scale as O(N2), the present method calculates the O(N log N) treecode and O(N) fast multipole method (FMM) on the GPUs with unprecedented efficiency. We demonstrate the performance of our method by choosing one standard application --a gravitational N-body simulation-- and one non-standard application --simulation of turbulence using vortex particles. The gravitational simulation using the treecode with 1,608,044,129 particles showed a sustained performance of 42.15 TFlops. The vortex particle simulation of homogeneous isotropic turbulence using the periodic FMM with 16,777,216 particles showed a sustained performance of 20.2 TFlops. The overall cost of the hardware was 228,912 dollars. The maximum corrected performance is 28.1TFlops for the gravitational simulation, which results in a cost performance of 124 MFlops/$. This correction is performed by counting the Flops based on the most efficient CPU algorithm. Any extra Flops that arise from the GPU implementation and parameter differences are not included in the 124 MFlops/$.