Reweighting for Transfer Classification: Exact Risk Conversion and Minimax Adaptation

Abstract

A source can make a training mixture easier to learn while making its accuracy less valuable for the target. We study how the training problem a source creates and the finite-sample error of learning it jointly determine target-risk guarantees under reweighting. For binary classification, for each pair of laws and each positive target weight, we derive the sharpest target excess hinge-risk bound implied by mixture excess alone, uniformly over clipped predictors. This exact conversion curve is determined by density-weighted local votes. A two-sample oracle inequality for weighted support vector machines supplies the mixture excess, and the composition becomes explicit for Gaussian kernels. On a tilted-boundary class, we establish the all-procedure minimax classification rate $[n+s_+^2/(m^{-1}+τ^2)]^{-1}$ in the transferable strength $s_+$ and the tilt $τ$, separating source sampling uncertainty ($m^{-1}$) from relation uncertainty ($τ^2$). A target-checked Gaussian-SVM selector adapts to unknown $s_+$ and $τ$ up to a logarithmic factor for fixed dimension, and retains the target-only rate up to the same factor against arbitrary independent sources under the stated target model. On a separate smooth opposition class, the minimax risk within the family of exact Gaussian SVMs fitted to the original labels with nonnegative weights is exponentially larger than the unrestricted minimax risk whenever the target sample is at most half the source sample, despite arbitrary data-dependent weighting and tuning. The value of a source thus depends on the training problem it creates, on what accuracy there means for the target, and on whether the permitted use of the source can implement the needed correction.