RKHS回归与序列模型在Lipschitz谱算法下的风险等价性
Risk Equivalence between RKHS Regression and Sequence Models for Lipschitz Spectral Algorithms
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中文总结 AI 辅助
本文证明核谱估计器与高斯序列模型估计器风险渐近等价，覆盖多种谱滤波器，并应用于恢复极小极大速率和确立Pinsker常数。
中文摘要 AI 辅助
核谱算法通常通过收敛速率来概括，这掩盖了其风险如何联合依赖于正则化、噪声、总体谱、目标系数以及所选滤波器。高斯序列模型作为一个简化但具有代表性的设置出现，用于研究这些因素之间的相互作用，其中核谱算法对应于坐标wise收缩估计器。在温和假设下，我们证明核谱估计器的风险与具有相同滤波器的相应高斯序列模型估计器的风险渐近等价。显式的序列模型风险随后提供了核谱算法风险在总体谱、目标系数和滤波器方面的完整刻画。我们为广泛的谱滤波器类别建立了这种等价性，涵盖核岭回归、广义岭回归、迭代核岭回归、梯度流、稳定梯度下降、平滑谱截断、谱裁剪和Pinsker收缩。我们的风险等价性不仅在经典的固定维度机制下成立，而且适用于输入维度随样本量缩放的高维机制。作为应用，我们的风险等价性恢复了极小极大上界速率，并在RKHS回归中确立了精确的Pinsker常数。
英文摘要
Kernel spectral algorithms are often summarized by convergence rates, which hide how their risk depends jointly on regularization, noise, the population spectrum, target coefficients, and the chosen filter. Gaussian sequence models arise as a simplified but characteristic setting for studying the interplay of these factors, where the kernel spectral algorithm corresponds to a coordinatewise shrinkage estimator. Under mild assumptions, we show that the risk of a kernel spectral estimator is asymptotically equivalent to that of the corresponding Gaussian sequence model estimator with the same filter. The explicit sequence model risk then yields a full characterization of the risk of kernel spectral algorithms in terms of the population spectrum, target coefficients, and filter. We establish this equivalence for a broad class of spectral filters, covering kernel ridge regression, generalized ridge regression, iterated kernel ridge regression, gradient flow, stable gradient descent, smoothed spectral cutoff, spectral clipping, and Pinsker shrinkage. Our risk equivalence not only holds in the classical fixed-dimensional regime but also applies to the high dimensional regime where the input dimension scales with the sample size. As applications, our risk equivalence recovers the minimax upper rates, and establishes the exact Pinsker constant in RKHS regression.
