2023 Fiscal Year Final Research Report
Riemannian Fixed Point Optimization Algorithm and Its Application to Machine Learning
| Project/Area Number |
21K11773
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 60020:Mathematical informatics-related
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| Research Institution | Meiji University |
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Principal Investigator |
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| Project Period (FY) |
2021-04-01 – 2024-03-31
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| Keywords | リーマン多様体 / 不動点最適化 / 機械学習 / 適応手法 / リーマン不動点最適化アルゴリズム |
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| Outline of Final Research Achievements |
We consider a stochastic optimization problem over the fixed point sets of nonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian optimization problems over complicated sets, such as the intersection of closed convex sets. For such a problem, we propose a Riemannian fixed point optimization algorithm, which combines fixed point approximation methods on Riemannian manifolds with adaptive learning rate optimization algorithms. We also give convergence analyses of the proposed algorithm. The analysis results indicate that, with small constant step-sizes, the proposed algorithm approximates a solution to the problem. Consideration of the case in which step-size sequences are diminishing demonstrates that the proposed algorithm solves the problem. We also provide numerical comparisons that demonstrate the effectiveness of the proposed algorithms.
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| Free Research Field |
最適化
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| Academic Significance and Societal Importance of the Research Achievements |
複雑な機械学習においては、リーマン多様体上の大規模かつ複雑な確率的最適化問題を解く必要がある。しかしながら、従来の機械学習法は、その問題の緩和やその問題の解へ収束する保証がないアルゴリズムに基づいており、本来達成すべき機械学習法の性能を満たしていない。本研究での提案手法は、その問題に直接適用できる不動点最適化アルゴリズムに基づく機械学習法であり、世界的に例のない新解法である。本研究の成果は、従来機械学習法の適用範囲に関する改善に多大な貢献ができることから応用数学的観点のみならず、工学的観点から見ても意義があるといえる。
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