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Saddle Avoidance, Asymptotic Normality, and Exponential Acceleration in Nonsmooth Optimization
Saddle Avoidance, Asymptotic Normality, and Exponential Acceleration in Nonsmooth Optimization
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20250211151353
- ISBN
- 9798382842714
- DDC
- 519
- 저자명
- Jiang, Liwei.
- 서명/저자
- Saddle Avoidance, Asymptotic Normality, and Exponential Acceleration in Nonsmooth Optimization
- 발행사항
- [Sl] : Cornell University, 2024
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2024
- 형태사항
- 311 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 85-12, Section: B.
- 주기사항
- Advisor: Davis, Damek.
- 학위논문주기
- Thesis (Ph.D.)--Cornell University, 2024.
- 초록/해제
- 요약Optimization-based algorithms are the foundation for empirically successful methods in modern fields, such as artificial intelligence and data science. Although classical optimization theory provides guarantees for functions with smoothness or convexity, a significant portion of modern problems do not possess any of these. Despite the worst-case examples where efficient algorithms are unavailable, typical nonsmoothness arises with a "partly smooth" structure, meaning that they are well-behaved relative to a smooth "active manifold."This thesis develops and analyzes first-order algorithms based on the aforementioned nonsmooth structure. We first develop two regularity conditions describing how sub-gradients interact with active manifolds and then show that they hold for a broad and generic class of functions. With these cornerstones, we demonstrate that when randomly perturbed or equipped with stochastic noise, subgradient methods only converge to minimizers of generic, Clarke regular semialgebraic problems. When convergence to a certain minimizer is known, we demonstrate that stochastic (projected) subgradient methods have asymptotic normality, making them asymptotically optimal algorithms in the locally minimax sense of Hajek and Le Cam.These findings culminate with a new first-order algorithm-NTDescent-which exhibits local nearly linear convergence on typical nonsmooth functions with quadratic growth. The convergence rate of NTDescent depends only on the function's intrinsic quantities but not the problem's underlying dimension.
- 일반주제명
- Applied mathematics
- 일반주제명
- Statistics
- 키워드
- Parameter-free
- 기타저자
- Cornell University Operations Research and Information Engineering
- 기본자료저록
- Dissertations Abstracts International. 85-12B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■00520250211151353
■006m o d
■007cr#unu||||||||
■020 ▼a9798382842714
■035 ▼a(MiAaPQ)AAI31243432
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a519
■1001 ▼aJiang, Liwei.▼0(orcid)0009-0005-3287-9966
■24510▼aSaddle Avoidance, Asymptotic Normality, and Exponential Acceleration in Nonsmooth Optimization
■260 ▼a[Sl]▼bCornell University▼c2024
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2024
■300 ▼a311 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 85-12, Section: B.
■500 ▼aAdvisor: Davis, Damek.
■5021 ▼aThesis (Ph.D.)--Cornell University, 2024.
■520 ▼aOptimization-based algorithms are the foundation for empirically successful methods in modern fields, such as artificial intelligence and data science. Although classical optimization theory provides guarantees for functions with smoothness or convexity, a significant portion of modern problems do not possess any of these. Despite the worst-case examples where efficient algorithms are unavailable, typical nonsmoothness arises with a "partly smooth" structure, meaning that they are well-behaved relative to a smooth "active manifold."This thesis develops and analyzes first-order algorithms based on the aforementioned nonsmooth structure. We first develop two regularity conditions describing how sub-gradients interact with active manifolds and then show that they hold for a broad and generic class of functions. With these cornerstones, we demonstrate that when randomly perturbed or equipped with stochastic noise, subgradient methods only converge to minimizers of generic, Clarke regular semialgebraic problems. When convergence to a certain minimizer is known, we demonstrate that stochastic (projected) subgradient methods have asymptotic normality, making them asymptotically optimal algorithms in the locally minimax sense of Hajek and Le Cam.These findings culminate with a new first-order algorithm-NTDescent-which exhibits local nearly linear convergence on typical nonsmooth functions with quadratic growth. The convergence rate of NTDescent depends only on the function's intrinsic quantities but not the problem's underlying dimension.
■590 ▼aSchool code: 0058.
■650 4▼aApplied mathematics
■650 4▼aStatistics
■653 ▼aAsymptotic normality
■653 ▼aFirst-order method
■653 ▼aLinear convergence
■653 ▼aNonsmooth optimization
■653 ▼aParameter-free
■653 ▼aSaddle point avoidance
■690 ▼a0796
■690 ▼a0364
■690 ▼a0463
■71020▼aCornell University▼bOperations Research and Information Engineering.
■7730 ▼tDissertations Abstracts International▼g85-12B.
■790 ▼a0058
■791 ▼aPh.D.
■792 ▼a2024
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17161412▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
