Xiaoye Sherry Li

Hengrui Luo, Younghyun Cho, James W. Demmel et al.

This paper presents a new type of hybrid model for Bayesian optimization (BO) adept at managing mixed variables, encompassing both quantitative (continuous and integer) and qualitative (categorical) types. Our proposed new hybrid models (named hybridM) merge the Monte Carlo Tree Search structure (MCTS) for categorical variables with Gaussian Processes (GP) for continuous ones. hybridM leverages the upper confidence bound tree search (UCTS) for MCTS strategy, showcasing the tree architecture's integration into Bayesian optimization. Our innovations, including dynamic online kernel selection in the surrogate modeling phase and a unique UCTS search strategy, position our hybrid models as an advancement in mixed-variable surrogate models. Numerical experiments underscore the superiority of hybrid models, highlighting their potential in Bayesian optimization.

Xiaoye Sherry Li, Yang Liu

Efficient solutions of large-scale, ill-conditioned and indefinite algebraic equations are ubiquitously needed in numerous computational fields, including multiphysics simulations, machine learning, and data science. Because of their robustness and accuracy, direct solvers are crucial components in building a scalable solver toolchain. In this chapter, we will review recent advances of sparse direct solvers along two axes: 1) reducing communication and latency costs in both task- and data-parallel settings, and 2) reducing computational complexity via low-rank and other compression techniques such as hierarchical matrix algebra. In addition to algorithmic principles, we also illustrate the key parallelization challenges and best practices to deliver high speed and reliability on modern heterogeneous parallel machines.

Navjot Singh, Kipton Barros, Xiaoye Sherry Li

Objectives involving bilinear forms $u^\top f(A(θ))v$ for Hermitian $A$ arise widely in scientific computing and probabilistic machine learning. For large matrices, Lanczos efficiently approximates these quantities, but differentiating them with respect to $θ$ is challenging. Existing approaches either backpropagate through the Lanczos recurrence, requiring reorthogonalization for stability, or apply Arnoldi to an augmented block matrix of twice the original size. Both introduce extra computation and orthogonalization costs that can limit performance on modern hardware. We propose a forward-only gradient approximation that reuses the Lanczos pass and adds very minimal overhead in most cases. We prove that its error is proportional to the Lanczos residual norm, the same quantity controlling the forward approximation. Whereas a traditional adjoint-based calculation would be unstable without reorthogonalization, the new method appears unconditionally stable in our tests. It is also faster than existing state-of-the-art approaches.

Navjot Singh, Edgar Solomonik, Xiaoye Sherry Li et al.

This paper presents low-complexity tensor completion algorithms and their efficient implementation to reconstruct highly oscillatory operators discretized as $n\times n$ matrices. The underlying tensor decomposition is based on the reshaping of the input matrix and its butterfly decomposition into an order $O (\log n)$ tensor. The reshaping of the input matrix into a tensor allows for representation of the butterfly decomposition as a tensor decomposition with dense tensors. This leads to efficient utilization of the existing software infrastructure for dense and sparse tensor computations. We propose two tensor completion algorithms in the butterfly format, using alternating least squares and gradient-based optimization, as well as a novel strategy that uses low-rank matrix completion to efficiently generate an initial guess for the proposed algorithms. To demonstrate the efficiency and applicability of our proposed algorithms, we perform three numerical experiments using simulated oscillatory operators in seismic applications. In these experiments, we use $O (n \log n)$ observed entries in the input matrix and demonstrate an $O(n\log^3 n)$ computational cost of the proposed algorithms, leading to a speedup of orders of magnitudes per iteration for large matrices compared to the low-rank matrix and quantized tensor-train completion. Moreover, the proposed butterfly completion algorithms, equipped with the novel initial guess generation strategy, achieve reconstruction errors that are smaller by an order of magnitude, enabling accurate recovery of the underlying structure compared to the state-of-the-art completion algorithms.

Anil Gaihre, Da Zheng, Scott Weitze et al.

Recent top-$k$ computation efforts explore the possibility of revising various sorting algorithms to answer top-$k$ queries on GPUs. These endeavors, unfortunately, perform significantly more work than needed. This paper introduces Dr. Top-k, a Delegate-centric top-$k$ system on GPUs that can reduce the top-$k$ workloads significantly. Particularly, it contains three major contributions: First, we introduce a comprehensive design of the delegate-centric concept, including maximum delegate, delegate-based filtering, and $β$ delegate mechanisms to help reduce the workload for top-$k$ up to more than 99%. Second, due to the difficulty and importance of deriving a proper subrange size, we perform a rigorous theoretical analysis, coupled with thorough experimental validations to identify the desirable subrange size. Third, we introduce four key system optimizations to enable fast multi-GPU top-$k$ computation. Taken together, this work constantly outperforms the state-of-the-art.

Hengrui Luo, James W. Demmel, Younghyun Cho et al.

Building surrogate models is one common approach when we attempt to learn unknown black-box functions. Bayesian optimization provides a framework which allows us to build surrogate models based on sequential samples drawn from the function and find the optimum. Tuning algorithmic parameters to optimize the performance of large, complicated "black-box" application codes is a specific important application, which aims at finding the optima of black-box functions. Within the Bayesian optimization framework, the Gaussian process model produces smooth or continuous sample paths. However, the black-box function in the tuning problem is often non-smooth. This difficult tuning problem is worsened by the fact that we usually have limited sequential samples from the black-box function. Motivated by these issues encountered in tuning, we propose a novel additive Gaussian process model called clustered Gaussian process (cGP), where the additive components are induced by clustering. In the examples we studied, the performance can be improved by as much as 90% among repetitive experiments. By using this surrogate model, we want to capture the non-smoothness of the black-box function. In addition to an algorithm for constructing this model, we also apply the model to several artificial and real applications to evaluate it.

Wissam M. Sid-Lakhdar, Mohsen Mahmoudi Aznaveh, Xiaoye S. Li et al.

Multitask learning and transfer learning have proven to be useful in the field of machine learning when additional knowledge is available to help a prediction task. We aim at deriving methods following these paradigms for use in autotuning, where the goal is to find the optimal performance parameters of an application treated as a black-box function. We show comparative results with state-of-the-art autotuning techniques. For instance, we observe an average $1.5x$ improvement of the application runtime compared to the OpenTuner and HpBandSter autotuners. We explain how our approaches can be more suitable than some state-of-the-art autotuners for the tuning of any application in general and of expensive exascale applications in particular.

Hongyuan Zhan, Gabriel Gomes, Xiaoye S. Li et al.

Computational efficiency is an important consideration for deploying machine learning models for time series prediction in an online setting. Machine learning algorithms adjust model parameters automatically based on the data, but often require users to set additional parameters, known as hyperparameters. Hyperparameters can significantly impact prediction accuracy. Traffic measurements, typically collected online by sensors, are serially correlated. Moreover, the data distribution may change gradually. A typical adaptation strategy is periodically re-tuning the model hyperparameters, at the cost of computational burden. In this work, we present an efficient and principled online hyperparameter optimization algorithm for Kernel Ridge regression applied to traffic prediction problems. In tests with real traffic measurement data, our approach requires as little as one-seventh of the computation time of other tuning methods, while achieving better or similar prediction accuracy.

Christopher Gorman, Gustavo Chávez, Pieter Ghysels et al.

We present new algorithms for the randomized construction of hierarchically semi-separable matrices, addressing several practical issues. The HSS construction algorithms use a partially matrix-free, adaptive randomized projection scheme to determine the maximum off-diagonal block rank. We develop both relative and absolute stopping criteria to determine the minimum dimension of the random projection matrix that is sufficient for the desired accuracy. Two strategies are discussed to adaptively enlarge the random sample matrix: repeated doubling of the number of random vectors, and iteratively incrementing the number of random vectors by a fixed number. The relative and absolute stopping criteria are based on probabilistic bounds for the Frobenius norm of the random projection of the Hankel blocks of the input matrix. We discuss parallel implementation and computation and communication cost of both variants. Parallel numerical results for a range of applications, including boundary element method matrices and quantum chemistry Toeplitz matrices, show the effectiveness, scalability and numerical robustness of the proposed algorithms.

Elizaveta Rebrova, Gustavo Chavez, Yang Liu et al.

We present memory-efficient and scalable algorithms for kernel methods used in machine learning. Using hierarchical matrix approximations for the kernel matrix the memory requirements, the number of floating point operations, and the execution time are drastically reduced compared to standard dense linear algebra routines. We consider both the general $\mathcal{H}$ matrix hierarchical format as well as Hierarchically Semi-Separable (HSS) matrices. Furthermore, we investigate the impact of several preprocessing and clustering techniques on the hierarchical matrix compression. Effective clustering of the input leads to a ten-fold increase in efficiency of the compression. The algorithms are implemented using the STRUMPACK solver library. These results confirm that --- with correct tuning of the hyperparameters --- classification using kernel ridge regression with the compressed matrix does not lose prediction accuracy compared to the exact --- not compressed --- kernel matrix and that our approach can be extended to $\mathcal{O}(1M)$ datasets, for which computation with the full kernel matrix becomes prohibitively expensive. We present numerical experiments in a distributed memory environment up to 1,024 processors of the NERSC's Cori supercomputer using well-known datasets to the machine learning community that range from dimension 8 up to 784.