General Multi-User Distributed Computing: A Learning-Theoretic RKHS Framework for Generic Nonlinear Target Functions with Topology-Aware Risk Analysis

This paper studies multi-user distributed computation over shared real-valued subfunctions under computation and communication constraints. We consider a \emph{General Multi-User Distributed Computing (GMUDC)} model in which different users request heterogeneous target functions represented in the reproducing-kernel Hilbert space of a shift-invariant kernel, thereby covering generic nonlinear target mappings beyond linearly separable tasks. Unlike tessellated distributed computing frameworks that rely on disjoint-support topologies in their native setting, the GMUDC model allows arbitrary task-assignment and connectivity topologies subject to per-server computation and communication budgets~$\Gamma$ and~$\Delta$. We analyze two complementary regimes. In the \emph{quenched} regime, the assignment and communication topology are fixed, and we derive upper and lower bounds on the resulting reconstruction risk that separate a spectral approximation term from a topology-dependent coverage term. In the \emph{annealed} regime, the assignment and links are drawn uniformly at random from a prescribed ensemble, and we characterize the corresponding average-risk scaling together with a topology-dependent coverage threshold. These results provide a topology-aware characterization of the computation--communication--accuracy trade-off for approximate multi-user distributed computation. They identify fundamental limits, up to constants and logarithmic factors, under the model assumptions adopted in the paper. In the shared linear/isotropic comparison regime, the framework also recovers the relevant tessellated distributed computing benchmark, while the broader GMUDC formulation applies to generic nonlinear target functions in kernel RKHSs and accommodates a wider range of task-assignment and communication topologies than disjoint-support constructions alone.