{
  "abstract": "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 $\\mathrm{SAR}=\\min(R(n,0),R(0,n_T))/R(n,n_T)$. We study domain adaptation with a known link: source regression $f$, target regression $g=\\mathcal{A}f$ for a known operator $\\mathcal{A}$, 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 $a\\geq2+1/(2β)$, where $β$ is the smoothness of $f$ (Result I). An integral link $g(x)=\\int_0^xf(t)\\,dt$ 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 $g=\\dot f$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15576",
  "author_total": 3,
  "authors": [
    {
      "name": "Pengcheng Xiang"
    },
    {
      "name": "Ling Zhou"
    },
    {
      "name": "Yuhong Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2306vpq2f22pg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15576",
  "fulltext_url": null,
  "id": "2306vpq2f22pg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15576",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.ST"
    ],
    [
      "arxiv",
      "stat.TH"
    ],
    [
      "openalex",
      "1702|Domain Adaptation and Few-Shot Learning"
    ]
  ],
  "openalex": "W7221752607",
  "pdf_url": "https://arxiv.org/pdf/2610.15576",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2306vpq2f22pg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15576",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15576"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Synergistic Learning in Domain Adaptation with a Known Link: A Minimax Understanding",
  "title_text": "Synergistic Learning in Domain Adaptation with a Known Link: A Minimax Understanding",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:26Z",
  "url": "https://research.draconitic.com/paper/2306vpq2f22pg",
  "venue": "arXiv",
  "versions": []
}