Abstract:Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are effective in the small-noise regime are suboptimal when large noise is present. To address this issue, we propose a hybrid approach that combines Christoffel sampling with optimal experimental design. We show that the proposed algorithm enjoys appropriate optimality properties for both sample point generation and noise mollification, leading to improved computational efficiency and sample complexity compared to existing methods. We also extend the algorithm to convexity-constrained settings with similar theoretical guarantees. When the target function is defined as the expectation of a random field, we further extend our approach to leverage adaptive random subspaces and establish results on the approximation capacity of the adaptive procedure. Our theoretical findings are supported by numerical studies on both synthetic data and on a more challenging stochastic simulation problem in computational finance.
| Comments: | 30 pages |
| Subjects: | Machine Learning (stat.ML); Machine Learning (cs.LG); Numerical Analysis (math.NA) |
| Cite as: | arXiv:2507.02215 [stat.ML] |
| (or arXiv:2507.02215v2 [stat.ML] for this version) | |
| https://doi.org/10.48550/arXiv.2507.02215 arXiv-issued DOI via DataCite |
From: Yiming Xu [view email]
[v1]
Thu, 3 Jul 2025 00:31:29 UTC (1,236 KB)
[v2]
Mon, 25 May 2026 01:57:42 UTC (898 KB)
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