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

TL;DR

This paper introduces a comprehensive RKHS-based framework for multi-user distributed computation of nonlinear functions, analyzing topology-aware risk bounds under fixed and random task assignments.

Contribution

It extends distributed computing models to nonlinear target functions in RKHS, providing topology-dependent risk analysis and fundamental limits for diverse network structures.

Findings

Derived risk bounds for fixed topologies in the quenched regime.

Characterized average risk scaling and coverage thresholds in the annealed regime.

Unified framework recovering tessellated distributed computing as a special case.

Abstract

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.