扩展普遍近似保证：现实世界学习任务连续性的理论证明
Extending Universal Approximation Guarantees: A Theoretical Justification for the Continuity of Real-World Learning Tasks
摘要：
通用逼近定理建立了$C(K, \mathbb{R}^m)$中各类神经网络函数逼近器的密度，其中$K \subset \mathbb{R}^n$是紧致的。在本文中，我们的目标是通过建立学习任务的条件，保证其连续性，以延长这些保证。我们考虑由条件期望$x \mapsto \mathrm{E}\left[Y \mid X = x\right]$给出的学习任务，其中学习目标$Y = f \circ L$是一些底层数据生成过程$L$的潜在病理性转变。在一个分解$L = T \circ W$的数据生成过程中，$T$被认为是一个确定性的映射作用于一些随机输入$W$，我们建立的条件（可能很容易验证使用知识的$T$单独），保证连续性的实际\textit{any}派生的学习任务$x \mapsto \mathrm{E}\left[f \circ L \mid X = x\right]$。我们激励我们的条件的现实主义使用随机稳定匹配的例子，从而为现实世界的学习任务的连续性提供了理论依据。
英文摘要：
Universal Approximation Theorems establish the density of various classes of neural network function approximators in $C(K, \mathbb{R}^m)$, where $K \subset \mathbb{R}^n$ is compact. In this paper, we aim to extend these guarantees by establishing conditions on learning tasks that guarantee their continuity. We consider learning tasks given by conditional expectations $x \mapsto \mathrm{E}\left[Y \mid X = x\right]$, where the learning target $Y = f \circ L$ is a potentially pathological transformation of some underlying data-generating process $L$. Under a factorization $L = T \circ W$ for the data-generating process where $T$ is thought of as a deterministic map acting on some random input $W$, we establish conditions (that might be easily verified using knowledge of $T$ alone) that guarantee the continuity of practically \textit{any} derived learning task $x \mapsto \mathrm{E}\left[f \circ L \mid X = x\right]$. We motivate the realism of our conditions using the example of randomized stable matching, thus providing a theoretical justification for the continuity of real-world learning tasks.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
