作为回归量的条件平均嵌入-补充
Conditional mean embeddings as regressors - supplementary
摘要：
我们证明了再生核希尔伯特空间（RKHS）嵌入的条件分布和向量值回归之间的等价性。这种连接引入了一个自然的正则化损失函数，RKHS嵌入最小化，提供了一个直观的理解嵌入和使用的理由。此外，等价允许应用向量值回归方法和结果的学习条件分布的问题。使用这个链接，我们得到一个稀疏版本的嵌入考虑替代配方。此外，通过将向量值回归的收敛结果应用于嵌入问题，我们得到了O（\log（n）/n）的极大极小收敛速度-与当前最新的O（n^{-1/4}）速度相比-并且在更温和和更直观的假设下是有效的。这些最小最大上限率与较低的利率达到对数因子一致，表明嵌入方法达到了接近最优的利率。我们研究了我们的稀疏嵌入算法在强化学习任务中，该算法显示出显着改善稀疏不完整的Cholesky分解。
英文摘要：
We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vector-valued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings minimise, providing an intuitive understanding of the embeddings and a justification for their use. Furthermore, the equivalence allows the application of vector-valued regression methods and results to the problem of learning conditional distributions. Using this link we derive a sparse version of the embedding by considering alternative formulations. Further, by applying convergence results for vector-valued regression to the embedding problem we derive minimax convergence rates which are O(\log(n)/n) -- compared to current state of the art rates of O(n^{-1/4}) -- and are valid under milder and more intuitive assumptions. These minimax upper rates coincide with lower rates up to a logarithmic factor, showing that the embedding method achieves nearly optimal rates. We study our sparse embedding algorithm in a reinforcement learning task where the algorithm shows significant improvement in sparsity over an incomplete Cholesky decomposition.
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