← 研踪 PaperTrailarXiv:2605.21341 · v2
Parametric Rates for Nonparametric Conditional-Moment Bilevel Optimization
首次提交 2026-05-21 00:07
版本更新 2026-10-27 03:29
雷达观测 2026-10-29 00:35
摘要
We study functional bilevel optimization with conditional moments, where plug-in hypergradients can inherit first-order errors from nonparametrically learned conditional expectations. We introduce OBiGrad, an orthogonalized hypergradient estimator whose bias is a product of nuisance errors. Under our regression and complexity conditions, projected gradient descent with the direct plug-in hypergradient yields expected stationarity error $O(T^{-1/2}+n^{-1/2}+n^{-ν/b})$ for the best iterate, whereas OBiGrad achieves $O(T^{-1/2}+n^{-1/2}+n^{-2ν/b})$, where $b=ν(2+p)+γ$ combines the approximation and complexity exponents. Consequently, when $ν(2-p)\geγ$, OBiGrad attains the parametric statistical rate $O(T^{-1/2}+n^{-1/2})$, improving the sufficient sample-size bound relative to the direct plug-in bound for a fixed iteration budget. We illustrate these gains on instrumental-variable and fitted-Q-regression problems, examining stationary-point estimation, returned population stationarity, and regularization sensitivity.
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v2 · 2026-10-27 03:29 · 最新观测版本
Parametric Rates for Nonparametric Conditional-Moment Bilevel Optimization
We study functional bilevel optimization with conditional moments, where plug-in hypergradients can inherit first-order errors from nonparametrically learned conditional expectations. We introduce OBiGrad, an orthogonalized hypergradient estimator whose bias is a product of nuisance errors. Under our regression and complexity conditions, projected gradient descent with the direct plug-in hypergradient yields expected stationarity error $O(T^{-1/2}+n^{-1/2}+n^{-ν/b})$ for the best iterate, whereas OBiGrad achieves $O(T^{-1/2}+n^{-1/2}+n^{-2ν/b})$, where $b=ν(2+p)+γ$ combines the approximation and complexity exponents. Consequently, when $ν(2-p)\geγ$, OBiGrad attains the parametric statistical rate $O(T^{-1/2}+n^{-1/2})$, improving the sufficient sample-size bound relative to the direct plug-in bound for a fixed iteration budget. We illustrate these gains on instrumental-variable and fitted-Q-regression problems, examining stationary-point estimation, returned population stationarity, and regularization sensitivity.
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