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About this paper
Dimitri Meunier, Zhu Li, Tim Christensen, Arthur Gretton
KIV estimates a nonparametric structural relationship when an endogenous input is confounded, using an instrument to condition kernel features of the input before predicting outcomes. It proves strong L2(X) convergence and minimax rates under polynomial-link assumptions, including nonidentified cases via a minimum-RKHS-norm target. In a Sobolev example, its squared-risk rate is n^{-s/(s+ξ+d/2)}, with an extra ξ penalty.
Key insights
- The KIV estimator converges in the strong L2(X) norm to a unique minimum-norm solution in the RKHS, providing a robust and well-defined target even when the structural function is not uniquely identified in L2(X).
- The introduced polynomial 'link condition' (LINK) provides an interpretable ill-posedness parameter ($\gamma$) that quantifies how instrument strength directly affects the unavoidable slowdown in strong L2(X) learning rates compared to standard regression.
- Utilizing general spectral regularization in Stage 1 successfully avoids the saturation effect of Tikhonov regularization, enabling the KIV estimator to achieve faster convergence rates for smoother first-stage targets.
Summary by alphaXiv.
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