Agnostic federated learning (AFL) seeks a model that performs reliably across $m$ heterogeneous workers, but communication remains a bottleneck. We improve communication efficiency by reducing the number of synchronization rounds via faster convergence and the communication cost per round via compression.

We first propose AFL-BR, which updates the dual weights over workers using online mirror ascent with KL divergence and blockwise restarts. It achieves an $O((\log m)^{1/4}T^{-1/8})$ stationarity rate after $T$ update rounds, reducing the $m$-dependence of the synchronization rounds required for convergence from polynomial to logarithmic order.

Building on AFL-BR, we develop AFL-Com by applying bidirectional compression with error feedback (EF). Instead of compressing local gradients, workers apply EF to their dual-weighted gradients, enabling direct control of the aggregated compression error under time-varying weights.

We then establish an $O((δ^{-1}+(\log m)^{1/4})T^{-1/8})$ stationarity rate for AFL-Com under general $δ$-approximate compressors and improve the $δ$-dependence from $δ^{-1}$ to $δ^{-1/2}$ for additive-and-idempotent compressors with shared randomness (SR). With suitable compression levels, AFL-Com retains the same convergence rate as AFL-BR at a lower per-round communication cost, yielding reductions in total communication complexity by factors of $(\log m)^{1/4}$ with Top-$k$ and $(\log m)^{1/2}$ with Rand-$k$ and SR.

Experiments validate the improved synchronization and communication efficiency of our methods.