110 research outputs found
An Expanded Theoretical Treatment of Iteration-Dependent Majorize-Minimize Algorithms
The majorize-minimize (MM) optimization technique has received considerable attention in signal and image processing applications, as well as in statistics literature. At each iteration of an MM algorithm, one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate. The next iterate is obtained by minimizing this tangent majorant function, resulting in a sequence of iterates that reduces the cost function monotonically. A well-known special case of MM methods are expectation-maximization algorithms. In this paper, we expand on previous analyses of MM, due to Fessler and Hero, that allowed the tangent majorants to be constructed in iteration-dependent ways. Also, this paper overcomes an error in one of those earlier analyses. There are three main aspects in which our analysis builds upon previous work. First, our treatment relaxes many assumptions related to the structure of the cost function, feasible set, and tangent majorants. For example, the cost function can be nonconvex and the feasible set for the problem can be any convex set. Second, we propose convergence conditions, based on upper curvature bounds, that can be easier to verify than more standard continuity conditions. Furthermore, these conditions allow for considerable design freedom in the iteration-dependent behavior of the algorithm. Finally, we give an original characterization of the local region of convergence of MM algorithms based on connected (e.g., convex) tangent majorants. For such algorithms, cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods. This expanded treatment widens the scope of the MM algorithm designs that can be considered for signal and image processing applications, allows us to verify the convergent behavior of previously published algorithms, and gives a fuller understanding overall of how these algorithms behave.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85958/1/Fessler34.pd
๋ณ๋ ฌํ ์ฉ์ดํ ํต๊ณ๊ณ์ฐ ๋ฐฉ๋ฒ๋ก ๊ณผ ํ๋ ๊ณ ์ฑ๋ฅ ์ปดํจํ ํ๊ฒฝ์์ ์ ์ฉ
ํ์๋
ผ๋ฌธ (๋ฐ์ฌ) -- ์์ธ๋ํ๊ต ๋ํ์ : ์์ฐ๊ณผํ๋ํ ํต๊ณํ๊ณผ, 2020. 8. ์์คํธ.Technological advances in the past decade, hardware and software alike, have made access to high-performance computing (HPC) easier than ever. In this dissertation, easily-parallelizable, inversion-free, and variable-separated algorithms and their implementation in statistical computing are discussed. The first part considers statistical estimation problems under structured sparsity posed as minimization of a sum of two or three convex functions, one of which is a composition of non-smooth and linear functions. Examples include graph-guided sparse fused lasso and overlapping group lasso. Two classes of inversion-free primal-dual algorithms are considered and unified from a perspective of monotone operator theory. From this unification, a continuum of preconditioned forward-backward operator splitting algorithms amenable to parallel and distributed computing is proposed. The unification is further exploited to introduce a continuum of accelerated algorithms on which the theoretically optimal asymptotic rate of convergence is obtained. For the second part, easy-to-use distributed matrix data structures in PyTorch and Julia are presented. They enable users to write code once and run it anywhere from a laptop to a workstation with multiple graphics processing units (GPUs) or a supercomputer in a cloud. With these data structures, various parallelizable statistical applications, including nonnegative matrix factorization, positron emission tomography, multidimensional scaling, and โ1-regularized Cox regression, are demonstrated. The examples scale up to an 8-GPU workstation and a 720-CPU-core cluster in a cloud. As a case in point, the onset of type-2 diabetes from the UK Biobank with 400,000 subjects and about 500,000 single nucleotide polymorphisms is analyzed using the HPC โ1-regularized Cox regression. Fitting a half-million variate model took about 50 minutes, reconfirming known associations. To my knowledge, the feasibility of a joint genome-wide association analysis of survival
outcomes at this scale is first demonstrated.์ง๋ 10๋
๊ฐ์ ํ๋์จ์ด์ ์ํํธ์จ์ด์ ๊ธฐ์ ์ ์ธ ๋ฐ์ ์ ๊ณ ์ฑ๋ฅ ์ปดํจํ
์ ์ ๊ทผ์ฅ๋ฒฝ์ ๊ทธ ์ด๋ ๋๋ณด๋ค ๋ฎ์ถ์๋ค. ์ด ํ์๋
ผ๋ฌธ์์๋ ๋ณ๋ ฌํ ์ฉ์ดํ๊ณ ์ญํ๋ ฌ ์ฐ์ฐ์ด ์๋ ๋ณ์ ๋ถ๋ฆฌ ์๊ณ ๋ฆฌ์ฆ๊ณผ ๊ทธ ํต๊ณ๊ณ์ฐ์์์ ๊ตฌํ์ ๋
ผ์ํ๋ค. ์ฒซ ๋ถ๋ถ์ ๋ณผ๋ก ํจ์ ๋ ๊ฐ ๋๋ ์ธ ๊ฐ์ ํฉ์ผ๋ก ๋ํ๋๋ ๊ตฌ์กฐํ๋ ํฌ์ ํต๊ณ ์ถ์ ๋ฌธ์ ์ ๋ํด ๋ค๋ฃฌ๋ค. ์ด ๋ ํจ์๋ค ์ค ํ๋๋ ๋นํํ ํจ์์ ์ ํ ํจ์์ ํฉ์ฑ์ผ๋ก ๋ํ๋๋ค. ๊ทธ ์์๋ก๋ ๊ทธ๋ํ ๊ตฌ์กฐ๋ฅผ ํตํด ์ ๋๋๋ ํฌ์ ์ตํฉ Lasso ๋ฌธ์ ์ ํ ๋ณ์๊ฐ ์ฌ๋ฌ ๊ทธ๋ฃน์ ์ํ ์ ์๋ ๊ทธ๋ฃน Lasso ๋ฌธ์ ๊ฐ ์๋ค. ์ด๋ฅผ ํ๊ธฐ ์ํด ์ญํ๋ ฌ ์ฐ์ฐ์ด ์๋ ๋ ์ข
๋ฅ์ ์์-์๋ (primal-dual) ์๊ณ ๋ฆฌ์ฆ์ ๋จ์กฐ ์ฐ์ฐ์ ์ด๋ก ๊ด์ ์์ ํตํฉํ๋ฉฐ ์ด๋ฅผ ํตํด ๋ณ๋ ฌํ ์ฉ์ดํ precondition๋ ์ ๋ฐฉ-ํ๋ฐฉ ์ฐ์ฐ์ ๋ถํ ์๊ณ ๋ฆฌ์ฆ์ ์งํฉ์ ์ ์ํ๋ค. ์ด ํตํฉ์ ์ ๊ทผ์ ์ผ๋ก ์ต์ ์๋ ด๋ฅ ์ ๊ฐ๋ ๊ฐ์ ์๊ณ ๋ฆฌ์ฆ์ ์งํฉ์ ๊ตฌ์ฑํ๋ ๋ฐ ํ์ฉ๋๋ค. ๋ ๋ฒ์งธ ๋ถ๋ถ์์๋ PyTorch์ Julia๋ฅผ ํตํด ์ฌ์ฉํ๊ธฐ ์ฌ์ด ๋ถ์ฐ ํ๋ ฌ ์๋ฃ ๊ตฌ์กฐ๋ฅผ ์ ์ํ๋ค. ์ด ๊ตฌ์กฐ๋ ์ฌ์ฉ์๋ค์ด ์ฝ๋๋ฅผ ํ ๋ฒ ์์ฑํ๋ฉด
์ด๊ฒ์ ๋
ธํธ๋ถ ํ ๋์์๋ถํฐ ์ฌ๋ฌ ๋์ ๊ทธ๋ํฝ ์ฒ๋ฆฌ ์ฅ์น (GPU)๋ฅผ ๊ฐ์ง ์ํฌ์คํ
์ด์
, ๋๋ ํด๋ผ์ฐ๋ ์์ ์๋ ์ํผ์ปดํจํฐ๊น์ง ๋ค์ํ ์ค์ผ์ผ์์ ์คํํ ์ ์๊ฒ ํด ์ค๋ค. ์์ธ๋ฌ, ์ด ์๋ฃ ๊ตฌ์กฐ๋ฅผ ๋น์ ํ๋ ฌ ๋ถํด, ์์ ์ ๋จ์ธต ์ดฌ์, ๋ค์ฐจ์ ์ฒ
๋๋ฒ, โ1-๋ฒ์ ํ Cox ํ๊ท ๋ถ์ ๋ฑ ๋ค์ํ ๋ณ๋ ฌํ ๊ฐ๋ฅํ ํต๊ณ์ ๋ฌธ์ ์ ์ ์ฉํ๋ค. ์ด ์์๋ค์ 8๋์ GPU๊ฐ ์๋ ์ํฌ์คํ
์ด์
๊ณผ 720๊ฐ์ ์ฝ์ด๊ฐ ์๋ ํด๋ผ์ฐ๋ ์์ ๊ฐ์ ํด๋ฌ์คํฐ์์ ํ์ฅ ๊ฐ๋ฅํ๋ค. ํ ์ฌ๋ก๋ก 400,000๋ช
์ ๋์๊ณผ 500,000๊ฐ์ ๋จ์ผ ์ผ๊ธฐ ๋คํ์ฑ ์ ๋ณด๊ฐ ์๋ UK Biobank ์๋ฃ์์์ ์ 2ํ ๋น๋จ๋ณ (T2D) ๋ฐ๋ณ ๋์ด๋ฅผ โ1-๋ฒ์ ํ Cox ํ๊ท ๋ชจํ์ ํตํด ๋ถ์ํ๋ค. 500,000๊ฐ์ ๋ณ์๊ฐ ์๋ ๋ชจํ์ ์ ํฉ์ํค๋ ๋ฐ 50๋ถ ๊ฐ๋์ ์๊ฐ์ด ๊ฑธ๋ ธ์ผ๋ฉฐ ์๋ ค์ง T2D ๊ด๋ จ ๋คํ์ฑ๋ค์ ์ฌํ์ธํ ์ ์์๋ค. ์ด๋ฌํ ๊ท๋ชจ์ ์ ์ ์ ์ฒด ๊ฒฐํฉ ์์กด ๋ถ์์ ์ต์ด๋ก ์๋๋ ๊ฒ์ด๋ค.Chapter1Prologue 1
1.1 Introduction 1
1.2 Accessible High-Performance Computing Systems 4
1.2.1 Preliminaries 4
1.2.2 Multiple CPU nodes: clusters, supercomputers, and clouds 7
1.2.3 Multi-GPU node 9
1.3 Highly Parallelizable Algorithms 12
1.3.1 MM algorithms 12
1.3.2 Proximal gradient descent 14
1.3.3 Proximal distance algorithm 16
1.3.4 Primal-dual methods 17
Chapter 2 Easily Parallelizable and Distributable Class of Algorithms for Structured Sparsity, with Optimal Acceleration 20
2.1 Introduction 20
2.2 Unification of Algorithms LV and CV (g โก 0) 30
2.2.1 Relation between Algorithms LV and CV 30
2.2.2 Unified algorithm class 34
2.2.3 Convergence analysis 35
2.3 Optimal acceleration 39
2.3.1 Algorithms 40
2.3.2 Convergence analysis 41
2.4 Stochastic optimal acceleration 45
2.4.1 Algorithm 45
2.4.2 Convergence analysis 47
2.5 Numerical experiments 50
2.5.1 Model problems 50
2.5.2 Convergence behavior 52
2.5.3 Scalability 62
2.6 Discussion 63
Chapter 3 Towards Unified Programming for High-Performance Statistical Computing Environments 66
3.1 Introduction 66
3.2 Related Software 69
3.2.1 Message-passing interface and distributed array interfaces 69
3.2.2 Unified array interfaces for CPU and GPU 69
3.3 Easy-to-use Software Libraries for HPC 70
3.3.1 Deep learning libraries and HPC 70
3.3.2 Case study: PyTorch versus TensorFlow 73
3.3.3 A brief introduction to PyTorch 76
3.3.4 A brief introduction to Julia 80
3.3.5 Methods and multiple dispatch 80
3.3.6 Multidimensional arrays 82
3.3.7 Matrix multiplication 83
3.3.8 Dot syntax for vectorization 86
3.4 Distributed matrix data structure 87
3.4.1 Distributed matrices in PyTorch: distmat 87
3.4.2 Distributed arrays in Julia: MPIArray 90
3.5 Examples 98
3.5.1 Nonnegative matrix factorization 100
3.5.2 Positron emission tomography 109
3.5.3 Multidimensional scaling 113
3.5.4 L1-regularized Cox regression 117
3.5.5 Genome-wide survival analysis of the UK Biobank dataset 121
3.6 Discussion 126
Chapter 4 Conclusion 131
Appendix A Monotone Operator Theory 134
Appendix B Proofs for Chapter II 139
B.1 Preconditioned forward-backward splitting 139
B.2 Optimal acceleration 147
B.3 Optimal stochastic acceleration 158
Appendix C AWS EC2 and ParallelCluster 168
C.1 Overview 168
C.2 Glossary 169
C.3 Prerequisites 172
C.4 Installation 173
C.5 Configuration 173
C.6 Creating, accessing, and destroying the cluster 178
C.7 Installation of libraries 178
C.8 Running a job 179
C.9 Miscellaneous 180
Appendix D Code for memory-efficient L1-regularized Cox proportional hazards model 182
Appendix E Details of SNPs selected in L1-regularized Cox regression 184
Bibliography 188
๊ตญ๋ฌธ์ด๋ก 212Docto
Doctor of Philosophy
dissertationX-ray computed tomography (CT) is a widely popular medical imaging technique that allows for viewing of in vivo anatomy and physiology. In order to produce high-quality images and provide reliable treatment, CT imaging requires the precise knowledge of t
Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations
In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations
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