Synergistic Learning in Domain Adaptation with a Known Link: A Minimax Understanding

When can combining source and target data yield a much better performance than using either one alone? Statistically, we address this issue from a minimax viewpoint: writing $R(n,n_T)$ for the minimax risk attainable from $n$ source and $n_T$ target observations, we ask when $R(n,n_T)$ decays *strictly faster* than both $R(n,0)$ and $R(0,n_T)$. When this happens, we call it the *Synergistic Learning Phenomenon* (SLP) and quantify it by the *Synergistic Advantage Ratio* $\mathrm{SAR}=\min (R(n,0),R(0,n_T))/R(n,n_T)$. In this work, we focus on domain adaptation with a *known link*: the source regression function is $f$, the target regression function is $g=\mathcal{A}f$ for a known operator $\mathcal{A}$, and the two domains may also differ in their designs. Three links—the identity, integration, and differentiation—together with the behaviour of the density ratio $\rho =h_T/h_S$ 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 a uniform target, SLP is possible only if $a\ge 2+1/(2\beta )$, where $\beta$ is the smoothness of $f$ (Result I; Zhou et al., 2022). An integral link $g(x)={\int }_0^{x}f(t),dt$ instead can yield the joint rate $(n{n}_T{)}^{-1/2}$ even for a bounded ratio (Result II), and under a Beta$(a,1)$ source it 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 directional 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 the design. We characterize the sample-size windows in which combining source and target data improves on both individual minimax rates, and how the SAR varies with the source and target sample sizes. Simulations support the predicted phase structure, and a mirrored integration–differentiation analysis of real vehicle and smartphone signals illustrates the directional asymmetry.