Even Sharper Bounds for Transductive Learning and Its Applications

2026-09-23

AI Summary

本文通过引入更尖锐的传递局部复杂性方法，解决了在无放回均匀采样下传递学习的问题，并提供了改进的风险界。

Abstract

We introduce Sharper Transductive Local Complexity (STLC), a localized complexity method for transductive learning under uniform sampling without replacement. The construction starts from a Bernstein-type concentration inequality for the supremum of the test--train empirical process. Its proof uses the modified log-Sobolev inequality for the swap walk and a two-parameter entropy closure. A peeling argument with a surrogate localization functional then gives excess-risk bounds with the same fixed-point and confidence terms as the classical inductive local Rademacher-complexity bounds, without the additional logarithmic confidence factor in earlier transductive results. For realizable learning over a binary class of VC dimension $\dVC$, with training size $m$, test size $u$, and $u\ge m\ge\dVC$, STLC yields $\cO\{\dVC\log(me/\dVC)/m\}$. This matches the standard inductive rate and, when $m\ge9$, is within a logarithmic factor of the transductive minimax lower bound of order $\dVC/m$. For transductive kernel learning, STLC gives a spectrum-adaptive excess-risk bound without the multiplicative imbalance factors appearing in the earlier local-complexity bound.

Problem

• transductive learning
• uniform sampling without replacement
• local complexity
• excess-risk bounds

Innovation

• Sharper Transductive Local Complexity
• transductive learning
• Bernstein-type concentration inequality
• uniform sampling without replacement
• spectrum-adaptive excess-risk bound