In this paper I propose a concept of a correct loss function in a generative model of supervised learning for an input space $\mathcal{X}$ and a label space $\mathcal{Y}$, both of which are measurable spaces. A correct loss function in a generative model of supervised learning must accurately measure the discrepancy between elements of a hypothesis space $\mathcal{H}$ of possible predictors and the supervisor operator, even when the supervisor operator does not belong to $\mathcal{H}$. To define correct loss functions, I propose a characterization of a regular conditional probability measure $\mu_{\mathcal{Y}|\mathcal{X}}$ for a probability measure $\mu$ on $\mathcal{X} \times \mathcal{Y}$ relative to the projection $\Pi_{\mathcal{X}}: \mathcal{X}\times\mathcal{Y}\to \mathcal{X}$ as a solution of a linear operator equation. If $\mathcal{Y}$ is a separable metrizable topological space with the Borel $\sigma$-algebra $ \mathcal{B} (\mathcal{Y})$, I propose an additional characterization of a regular conditional probability measure $\mu_{\mathcal{Y}|\mathcal{X}}$ as a minimizer of mean square error on the space of Markov kernels, referred to as probabilistic morphisms, from $\mathcal{X}$ to $\mathcal{Y}$. This characterization utilizes kernel mean embeddings. Building upon these results and employing inner measure to quantify the generalizability of a learning algorithm, I extend a result due to Cucker-Smale, which addresses the learnability of a regression model, to the setting of a conditional probability estimation problem. Additionally, I present a variant of Vapnik's regularization method for solving stochastic ill-posed problems, incorporating inner measure, and showcase its applications.

翻译：本文提出了一种监督学习生成模型中正确损失函数的概念，其中输入空间$\mathcal{X}$和标签空间$\mathcal{Y}$均为可测空间。在监督学习的生成模型中，正确损失函数需准确度量假设空间$\mathcal{H}$（由潜在预测器构成）与监督算子之间的差异，即使监督算子不属于$\mathcal{H}$。为定义正确损失函数，本文提出了一种关于$\mathcal{X} \times \mathcal{Y}$上概率测度$\mu$相对于投影映射$\Pi_{\mathcal{X}}: \mathcal{X}\times\mathcal{Y}\to \mathcal{X}$的正则条件概率测度$\mu_{\mathcal{Y}|\mathcal{X}}$的表征方法，该表征以线性算子方程的解形式呈现。当$\mathcal{Y}$为具有Borel $\sigma$-代数$\mathcal{B}(\mathcal{Y})$的可分可度量化拓扑空间时，本文进一步提出将正则条件概率测度$\mu_{\mathcal{Y}|\mathcal{X}}$表征为从$\mathcal{X}$到$\mathcal{Y}$的马尔可夫核空间（即概率态射）中的均方误差极小化器。该表征利用了核均值嵌入方法。基于上述结果，通过引入内测度量化学习算法的泛化能力，本文将Cucker-Smale关于回归模型可学习性的结论推广至条件概率估计问题场景。此外，本文提出了一种融合内测度的Vapnik正则化方法变体，用于求解随机不适定问题，并阐述了其应用实例。