  Present-day federated learning (FL) systems deployed over edge networks
consists of a large number of workers with high degrees of heterogeneity in
data and/or computing capabilities, which call for flexible worker
participation in terms of timing, effort, data heterogeneity, etc. To satisfy
the need for flexible worker participation, we consider a new FL paradigm
called "Anarchic Federated Learning" (AFL) in this paper. In stark contrast to
conventional FL models, each worker in AFL has the freedom to choose i) when to
participate in FL, and ii) the number of local steps to perform in each round
based on its current situation (e.g., battery level, communication channels,
privacy concerns). However, such chaotic worker behaviors in AFL impose many
new open questions in algorithm design. In particular, it remains unclear
whether one could develop convergent AFL training algorithms, and if yes, under
what conditions and how fast the achievable convergence speed is. Toward this
end, we propose two Anarchic Federated Averaging (AFA) algorithms with
two-sided learning rates for both cross-device and cross-silo settings, which
are named AFA-CD and AFA-CS, respectively. Somewhat surprisingly, we show that,
under mild anarchic assumptions, both AFL algorithms achieve the best known
convergence rate as the state-of-the-art algorithms for conventional FL.
Moreover, they retain the highly desirable {\em linear speedup effect} with
respect of both the number of workers and local steps in the new AFL paradigm.
We validate the proposed algorithms with extensive experiments on real-world
datasets.
