Transductive Rademacher Complexity and its Applications

We develop a technique for deriving data-dependent error bounds for transductive learning algorithms based on transductive Rademacher complexity. Our technique is based on a novel general error bound for transduction in terms of transductive Rademacher complexity, together with a novel bounding technique for Rademacher averages for particular algorithms, in terms of their “unlabeled-labeled” representation. This technique is relevant to many advanced graph-based transductive algorithms and we demonstrate its effectiveness by deriving error bounds to three well known algorithms. Finally, we present a new PAC-Bayesian bound for mixtures of transductive algorithms based on our Rademacher bounds.

💡 Research Summary

The paper introduces a novel theoretical tool tailored for transductive learning, called transductive Rademacher complexity, and demonstrates how it can be used to derive data‑dependent generalization bounds for a variety of graph‑based transductive algorithms. In the transductive setting, the entire dataset is fixed and only a subset receives labels; consequently, the classical i.i.d. assumptions underlying standard Rademacher complexity do not hold. To address this, the authors partition the data into a labeled set L and an unlabeled set U and define a “transductive Rademacher process” that assigns independent Rademacher signs to both parts while preserving the dependence structure imposed by the fixed overall sample. This leads to a definition of transductive Rademacher complexity (\mathcal{R}_T(H)) that naturally captures the influence of the unlabeled points.

The first major contribution is a general error bound (Theorem 1) that links the expected risk (R(h)) of any hypothesis (h) to its empirical risk on the labeled portion (\hat R_L(h)), the transductive Rademacher complexity of the hypothesis class, and a standard concentration term. Formally, with probability at least (1-\delta):
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