论文标题
正则条件平均嵌入学习的最佳率
Optimal Rates for Regularized Conditional Mean Embedding Learning
论文作者
论文摘要
我们解决了条件平均嵌入（CME）的内核脊回归估算的一致性，这是给定$ y $ x $的条件分布的嵌入到目标重现内核Hilbert Space $ Mathcal $ \ MATHCAL {H H} _y $的目标。 CME允许我们对目标RKHS功能的有条件期望，并已在非参数因果和贝叶斯推论中使用。我们解决了错误指定的设置，其中目标CME位于Hilbert-Schmidt操作员的空间中，该设置是从$ \ Mathcal {H} _X _x $和$ L_2 $和$ \ MATHCAL {h MATHCAL {H} _y $之间的输入插值空间作用的。该操作员的空间被证明是新定义的矢量值插值空间的同构。使用这种同构，我们得出了在未指定的设置下的经验CME估计器的新颖和适应性统计学习率。我们的分析表明，我们的费率与最佳$ o（\ log n / n）$速率匹配，而无需假设$ \ mathcal {h} _y $是有限维度的。我们进一步建立了学习率的下限，这表明所获得的上限是最佳的。
We address the consistency of a kernel ridge regression estimate of the conditional mean embedding (CME), which is an embedding of the conditional distribution of $Y$ given $X$ into a target reproducing kernel Hilbert space $\mathcal{H}_Y$. The CME allows us to take conditional expectations of target RKHS functions, and has been employed in nonparametric causal and Bayesian inference. We address the misspecified setting, where the target CME is in the space of Hilbert-Schmidt operators acting from an input interpolation space between $\mathcal{H}_X$ and $L_2$, to $\mathcal{H}_Y$. This space of operators is shown to be isomorphic to a newly defined vector-valued interpolation space. Using this isomorphism, we derive a novel and adaptive statistical learning rate for the empirical CME estimator under the misspecified setting. Our analysis reveals that our rates match the optimal $O(\log n / n)$ rates without assuming $\mathcal{H}_Y$ to be finite dimensional. We further establish a lower bound on the learning rate, which shows that the obtained upper bound is optimal.
