Training deep neural networks on large datasets can often be accelerated by using multiple compute nodes. This approach, known as distributed training, can utilize hundreds of computers via specialized message-passing protocols such as Ring All-Reduce. However, running these protocols at scale requires reliable high-speed networking that is only available in dedicated clusters. In contrast, many real-world applications, such as federated learning and cloud-based distributed training, operate on unreliable devices with unstable network bandwidth. As a result, these applications are restricted to using parameter servers or gossip-based averaging protocols. In this work, we lift that restriction by proposing Moshpit All-Reduce — an iterative averaging protocol that exponentially converges to the global average. We demonstrate the efficiency of our protocol for distributed optimization with strong theoretical guarantees. The experiments show 1.3x speedup for ResNet-50 training on ImageNet compared to competitive gossip-based strategies and 1.5x speedup when training ALBERT-large on preemptible compute nodes.