A Typed Tensor Language for Federated Learning
Abstract
Federated learning and analytics are often described as collections of separate protocols, even when they share the same mathematical form: client-local tensor computation, mergeable aggregation into shared state, and shared-only post-processing. We introduce a typed tensor language that formalizes this structure. The language distinguishes federated tensors, whose records are partitioned across clients along a tracked record axis, from shared tensors, which are available globally. Its semantics are defined by comparison with a virtual global tensor, used only as a reference object. The main result is a shared-state factorization theory. We show that typed one-round programs factor through fixed-dimensional shared state whose size is independent of the number of clients and records, computed from client-local tensor expressions and merged across clients. We also prove a converse representability result; factorizations whose encoders and decoders are expressible in the language are realized by typed one-round programs, and the correspondence extends to iterative programs whose cross-round state is shared. This gives a formal account of the computations in the language that can be expressed as encode, merge, and decode procedures. We then develop a differentiable fragment for learning. If a per-record loss and its per-record gradient are represented by client-local tensor expressions, the global gradient is represented by record-axis summation of the federated gradient tensor. This yields typed iterative programs for server-side gradient descent and shared-linear-algebra second-order updates. The framework characterizes a broad class of federated learning computations whose communication passes through fixed-dimensional shared state.
Cite
@article{arxiv.2605.21103,
title = {A Typed Tensor Language for Federated Learning},
author = {Theofilos Mailis and Kalliopi-Christina Despotidou and Konstantinos Filippopolitis and Yannis Foufoulas and Thanasis-Michail Karampatsis and Andreas Ktenidis and Evdokia Mailli and Theodore Papamarkou and Yannis Ioannidis},
journal= {arXiv preprint arXiv:2605.21103},
year = {2026}
}