用算子理论分析多任务深度学习泛化能力，提升预测可靠性。
Operator-Based Generalization Bound for Deep Learning: Insights on Multi-Task Learning
- 基于算子理论构建多任务神经网络泛化界，融合Koopman方法。
- 提出压缩技术降低计算成本，实现鲁棒与分位数回归的性能保证。
- 新框架通过核函数优化缓解过拟合与欠拟合，适合多任务建模研究者。
本文针对向量值神经网络与深度核方法，基于算子理论框架为多任务学习提出新型泛化边界。核心创新在于将Koopman方法与现有技术结合，相比传统范数型边界获得更紧的泛化保证。为缓解Koopman方法的计算负担，引入适用于向量值神经网络的压缩技术，在通用Lipschitz损失下获得超额风险界，为鲁棒回归与多分位数回归提供性能保障。进一步提出一种新深度学习框架——深度向量值再生核希尔伯特空间（vvRKHS），利用Perron-Frobenius算子增强深度核方法，并推导出新的Rademacher泛化界，通过核函数精炼策略显式控制过拟合与欠拟合。该工作为深度学习架构下的多任务学习泛化性质提供了新视角，此前该领域研究相对匮乏。
原文摘要 · Abstract (English)
This paper presents novel generalization bounds for vector-valued neural networks and deep kernel methods, focusing on multi-task learning through an operator-theoretic framework. Our key development lies in strategically combining a Koopman based approach with existing techniques, achieving tighter generalization guarantees compared to traditional norm-based bounds. To mitigate computational challenges associated with Koopman-based methods, we introduce sketching techniques applicable to vector valued neural networks. These techniques yield excess risk bounds under generic Lipschitz losses, providing performance guarantees for applications including robust and multiple quantile regression. Furthermore, we propose a novel deep learning framework, deep vector-valued reproducing kernel Hilbert spaces (vvRKHS), leveraging Perron Frobenius (PF) operators to enhance deep kernel methods. We derive a new Rademacher generalization bound for this framework, explicitly addressing underfitting and overfitting through kernel refinement strategies. This work offers novel insights into the generalization properties of multitask learning with deep learning architectures, an area that has been relatively unexplored until recent developments.
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