首次证明神经网络在非参数工具变量回归中两阶段最小二乘的全局收敛性。
Towards a Unified Analysis of Neural Networks in Nonparametric Instrumental Variable Regression: Optimization and Generalization
- 通过平均场朗之万动力学建立双层优化框架，解决2SLS的复杂结构。
- 提出新型一阶算法F²BMLD，实现优化与泛化性能的理论平衡。
- 适用于离线强化学习等需稳健因果推断的场景。
我们首次建立了神经网络在非参数工具变量回归（NPIV）的两阶段最小二乘（2SLS）方法中的全局收敛性。不同于标准平均场朗之万动力学（MFLD），本工作中的2SLS在概率测度空间中构成一个双层优化问题。为此，我们采用近期提出的惩罚梯度法，将双层优化转化为拉格朗日形式，从而导出一种全新的全一阶算法，称为\texttt{F$^2$BMLD}。除了提供收敛界外，还给出了泛化界，揭示了拉格朗日乘子的选择在优化与统计保证之间存在固有权衡。最后，我们在一个离线强化学习基准上实证验证了所提方法的有效性。
原文摘要 · Abstract (English)
We establish the first global convergence result of neural networks for two stage least squares (2SLS) approach in nonparametric instrumental variable regression (NPIV). This is achieved by adopting a lifted perspective through mean-field Langevin dynamics (MFLD), unlike standard MFLD, however, our setting of 2SLS entails a \emph{bilevel} optimization problem in the space of probability measures. To address this challenge, we leverage the penalty gradient approach recently developed for bilevel optimization which formulates bilevel optimization as a Lagrangian problem. This leads to a novel fully first-order algorithm, termed \texttt{F$^2$BMLD}. Apart from the convergence bound, we further provide a generalization bound, revealing an inherent trade-off in the choice of the Lagrange multiplier between optimization and statistical guarantees. Finally, we empirically validate the effectiveness of the proposed method on an offline reinforcement learning benchmark.
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