Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal

• URL: http://arxiv.org/abs/2411.19653v1
• Date: Fri, 29 Nov 2024 12:18:01 GMT
• Title: Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal
• Authors: Dimitri Meunier, Zhu Li, Tim Christensen, Arthur Gretton,
• Abstract summary: We study the kernel instrumental variable algorithm of citetsingh 2019. We show that the kernel NPIV estimator converges to the IV solution with minimum norm. We also improve the original kernel NPIV algorithm by adopting a general spectral regularization in stage 1 regression.
• Score: 28.361133177290657
• License:
• Abstract: We study the kernel instrumental variable algorithm of \citet{singh2019kernel}, a nonparametric two-stage least squares (2SLS) procedure which has demonstrated strong empirical performance. We provide a convergence analysis that covers both the identified and unidentified settings: when the structural function cannot be identified, we show that the kernel NPIV estimator converges to the IV solution with minimum norm. Crucially, our convergence is with respect to the strong $L_2$-norm, rather than a pseudo-norm. Additionally, we characterize the smoothness of the target function without relying on the instrument, instead leveraging a new description of the projected subspace size (this being closely related to the link condition in inverse learning literature). With the subspace size description and under standard kernel learning assumptions, we derive, for the first time, the minimax optimal learning rate for kernel NPIV in the strong $L_2$-norm. Our result demonstrates that the strength of the instrument is essential to achieve efficient learning. We also improve the original kernel NPIV algorithm by adopting a general spectral regularization in stage 1 regression. The modified regularization can overcome the saturation effect of Tikhonov regularization.

Related papers

• Optimality and Adaptivity of Deep Neural Features for Instrumental Variable Regression [57.40108516085593]
  Deep feature instrumental variable (DFIV) regression is a nonparametric approach to IV regression using data-adaptive features learned by deep neural networks. We prove that the DFIV algorithm achieves the minimax optimal learning rate when the target structural function lies in a Besov space.
  arXiv  Detail & Related papers  (2025-01-09T01:22:22Z)
• Tight and Efficient Upper Bound on Spectral Norm of Convolutional Layers [0.0]
  Controlling the spectral norm of the Jacobian matrix has been shown to improve generalization, training stability and robustness in CNNs. Existing methods for computing the norm either tend to overestimate it or their performance may deteriorate quickly with increasing the input and kernel sizes. In this paper, we demonstrate that the tensor version of the spectral norm of a four-dimensional convolution kernel, up to a constant factor, serves as an upper bound for the spectral norm of the Jacobian matrix associated with the convolution operation.
  arXiv  Detail & Related papers  (2024-09-18T10:28:28Z)
• Learning with Norm Constrained, Over-parameterized, Two-layer Neural Networks [54.177130905659155]
  Recent studies show that a reproducing kernel Hilbert space (RKHS) is not a suitable space to model functions by neural networks. In this paper, we study a suitable function space for over- parameterized two-layer neural networks with bounded norms.
  arXiv  Detail & Related papers  (2024-04-29T15:04:07Z)
• Stable Nonconvex-Nonconcave Training via Linear Interpolation [51.668052890249726]
  This paper presents a theoretical analysis of linearahead as a principled method for stabilizing (large-scale) neural network training. We argue that instabilities in the optimization process are often caused by the nonmonotonicity of the loss landscape and show how linear can help by leveraging the theory of nonexpansive operators.
  arXiv  Detail & Related papers  (2023-10-20T12:45:12Z)
• Benign Overfitting in Deep Neural Networks under Lazy Training [72.28294823115502]
  We show that when the data distribution is well-separated, DNNs can achieve Bayes-optimal test error for classification. Our results indicate that interpolating with smoother functions leads to better generalization.
  arXiv  Detail & Related papers  (2023-05-30T19:37:44Z)
• Asymptotically Unbiased Instance-wise Regularized Partial AUC Optimization: Theory and Algorithm [101.44676036551537]
  One-way Partial AUC (OPAUC) and Two-way Partial AUC (TPAUC) measures the average performance of a binary classifier. Most of the existing methods could only optimize PAUC approximately, leading to inevitable biases that are not controllable. We present a simpler reformulation of the PAUC problem via distributional robust optimization AUC.
  arXiv  Detail & Related papers  (2022-10-08T08:26:22Z)
• Neural Optimization Kernel: Towards Robust Deep Learning [13.147925376013129]
  Recent studies show a connection between neural networks (NN) and kernel methods. This paper proposes a novel kernel family named Kernel (NOK) We show that over parameterized deep NN (NOK) can increase the expressive power to reduce empirical risk and reduce the bound generalization at the same time.
  arXiv  Detail & Related papers  (2021-06-11T00:34:55Z)
• Sparse Representations of Positive Functions via First and Second-Order Pseudo-Mirror Descent [15.340540198612823]
  We consider expected risk problems when the range of the estimator is required to be nonnegative. We develop first and second-order variants of approximation mirror descent employing emphpseudo-gradients. Experiments demonstrate favorable performance on ingeneous Process intensity estimation in practice.
  arXiv  Detail & Related papers  (2020-11-13T21:54:28Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.