  In this paper, we introduce analytic federated learning (AFL), a new training
paradigm that brings analytical (i.e., closed-form) solutions to the federated
learning (FL) community. Our AFL draws inspiration from analytic learning -- a
gradient-free technique that trains neural networks with analytical solutions
in one epoch. In the local client training stage, the AFL facilitates a
one-epoch training, eliminating the necessity for multi-epoch updates. In the
aggregation stage, we derive an absolute aggregation (AA) law. This AA law
allows a single-round aggregation, removing the need for multiple aggregation
rounds. More importantly, the AFL exhibits a \textit{weight-invariant}
property, meaning that regardless of how the full dataset is distributed among
clients, the aggregated result remains identical. This could spawn various
potentials, such as data heterogeneity invariance, client-number invariance,
absolute convergence, and being hyperparameter-free (our AFL is the first
hyperparameter-free method in FL history). We conduct experiments across
various FL settings including extremely non-IID ones, and scenarios with a
large number of clients (e.g., $\ge 1000$). In all these settings, our AFL
constantly performs competitively while existing FL techniques encounter
various obstacles. Code is available at
\url{https://github.com/ZHUANGHP/Analytic-federated-learning}
