Synergistic Learning in Domain Adaptation with a Known Link: A Minimax Understanding
Abstract
When can combining source and target data yield much better performance than using either alone? From a minimax viewpoint, writing R(n,nT) for the minimax risk from n source and nT target observations, we ask when R(n,nT) decays strictly faster than both R(n,0) and R(0,nT). We call this the Synergistic Learning Phenomenon (SLP) and quantify it by the Synergistic Advantage Ratio SAR=min(R(n,0),R(0,nT))/R(n,nT). We study domain adaptation with a known link: source regression f, target regression g=𝒜f for a known operator 𝒜, and possibly different designs. Three links---identity, integration, and differentiation---and the density ratio ρ=hT/hS determine whether synergy is possible, at what rate, and in which sample-size window. For the identity link (covariate shift) with a Beta(a,1) source and uniform target, a known necessary condition for SLP is a≥2+1/(2β), where β is the smoothness of f (Result I). An integral link g(x)=∫0xf(t) dt can yield the joint rate (nnT)−1/2 even for a bounded ratio (Result II), and under a Beta(a,1) source has a phase diagram with cuts at a=2 and a=3: integral SLP for 1<a<2, an extra level-split rate nT−1 for 2<a<3, and covariate-shift-type spatial SLP for a>3 (Result III). A derivative link g=ḟ synergizes only under a stronger singularity (Result IV), giving a sharp integration--differentiation asymmetry under a bounded ratio. Across all three links, synergy arises from role separation: each dataset resolves a distinct bottleneck associated with the link and designs. We characterize the sample-size windows for synergy and how SAR varies with n and nT. Simulations support the predicted phase structure, and a mirrored integration--differentiation analysis of real vehicle and smartphone signals illustrates the asymmetry.
Related
- Retrieval-Based In-Context Learning: A Domain Adaptation Framework
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- Minimax-Optimal Robust Identification of Continuous-Time Systems: Handling Narrow-Band Disturbances in the Frequency Domain
- A Domain-specific adaptation network for elliptic interface problems
- Language-Induced Priors for Domain Adaptation
BibTeX
@misc{xiang2026synergistic,
title = {{Synergistic Learning in Domain Adaptation with a Known Link: A Minimax Understanding}},
author = {Pengcheng Xiang and Ling Zhou and Yuhong Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15576},
eprint = {2610.15576},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15576},
}
RIS
TY - UNPB
TI - Synergistic Learning in Domain Adaptation with a Known Link: A Minimax Understanding
AU - Pengcheng Xiang
AU - Ling Zhou
AU - Yuhong Yang
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15576
UR - https://arxiv.org/abs/2610.15576
AB - When can combining source and target data yield much better performance than using either alone? From a minimax viewpoint, writing R(n,n_T) for the minimax risk from n source and n_T target observations, we ask when R(n,n_T) decays strictly faster than both R(n,0) and R(0,n_T). We call this the Synergistic Learning Phenomenon (SLP) and quantify it by the Synergistic Advantage Ratio SAR=min(R(n,0),R(0,n_T))/R(n,n_T). We study domain adaptation with a known link: source regression f, target regression g=𝒜f for a known operator 𝒜, and possibly different designs. Three links---identity, integration, and differentiation---and the density ratio ρ=h_T/h_S determine whether synergy is possible, at what rate, and in which sample-size window. For the identity link (covariate shift) with a Beta(a,1) source and uniform target, a known necessary condition for SLP is a≥2+1/(2β), where β is the smoothness of f (Result I). An integral link g(x)=∫_0^xf(t) dt can yield the joint rate (nn_T)^−1/2 even for a bounded ratio (Result II), and under a Beta(a,1) source has a phase diagram with cuts at a=2 and a=3: integral SLP for 1<a<2, an extra level-split rate n_T^−1 for 2<a<3, and covariate-shift-type spatial SLP for a>3 (Result III). A derivative link g=ḟ synergizes only under a stronger singularity (Result IV), giving a sharp integration--differentiation asymmetry under a bounded ratio. Across all three links, synergy arises from role separation: each dataset resolves a distinct bottleneck associated with the link and designs. We characterize the sample-size windows for synergy and how SAR varies with n and n_T. Simulations support the predicted phase structure, and a mirrored integration--differentiation analysis of real vehicle and smartphone signals illustrates the asymmetry.
ER -