We introduce a new tool, Transductive Local Complexity (TLC), designed to analyze the generalization performance of transductive learning methods and inspire the development of new algorithms in this domain. Our work extends the concept of the popular Local Rademacher Complexity (LRC) to the transductive setting, incorporating significant and novel modifications compared to the typical analysis of LRC methods in the inductive setting. While LRC has been widely used as a powerful tool for analyzing inductive models, providing sharp generalization bounds for classification and minimax rates for nonparametric regression, it remains an open question whether a localized Rademacher complexity-based tool can be developed for transductive learning. Our goal is to achieve sharp bounds for transductive learning that align with the inductive excess risk bounds established by LRC. We provide a definitive answer to this open problem with the introduction of TLC. We construct TLC by first establishing a novel and sharp concentration inequality for the supremum of a test-train empirical processes. Using a peeling strategy and a new surrogate variance operator, we derive the a novel excess risk bound in the transductive setting which is consistent with the classical LRC-based excess risk bound in the inductive setting. As an application of TLC, we employ this new tool to analyze the Transductive Kernel Learning (TKL) model, deriving sharper excess risk bounds than those provided by the current state-of-the-art under the same assumptions. Additionally, the concentration inequality for the test-train process is employed to derive a sharp concentration inequality for the general supremum of empirical processes involving random variables in the setting of uniform sampling without replacement. The sharpness of our derived bound is compared to existing concentration inequalities under the same conditions.

翻译：本文提出了一种新工具——直推局部复杂度（TLC），旨在分析直推学习方法的泛化性能，并推动该领域新算法的开发。我们的工作将流行的局部Rademacher复杂度（LRC）概念扩展至直推学习场景，相较于归纳场景中LRC方法的典型分析，引入了重要且新颖的改进。尽管LRC已被广泛用作分析归纳模型的强大工具，为分类问题提供锐利的泛化界，并为非参数回归提供极小极大速率，但能否为直推学习开发基于局部Rademacher复杂度的工具仍是一个开放性问题。我们的目标是获得与LRC所建立的归纳超额风险界相一致的直推学习锐利界。通过引入TLC，我们为这一开放问题给出了明确解答。我们首先为测试-训练经验过程的上确界建立了一个新颖且锐利的集中不等式，进而构建TLC。通过使用分层策略和新的代理方差算子，我们推导出直推场景下与经典LRC归纳超额风险界一致的新超额风险界。作为TLC的应用，我们运用这一新工具分析直推核学习（TKL）模型，在相同假设下获得了比当前最优方法更锐利的超额风险界。此外，测试-训练过程的集中不等式被用于推导无放回均匀抽样场景下涉及随机变量的一般经验过程上确界的锐利集中不等式。在相同条件下，我们将所得界的锐利性与现有集中不等式进行了比较。