We propose Federated Preconditioned Mixing (FedPM), a novel Federated Learning (FL) method that leverages second-order optimization. Prior methods--such as LocalNewton, LTDA, and FedSophia--have incorporated second-order optimization in FL by performing iterative local updates on clients and applying simple mixing of local parameters on the server. However, these methods often suffer from drift in local preconditioners, which significantly disrupts the convergence of parameter training, particularly in heterogeneous data settings. To overcome this issue, we refine the update rules by decomposing the ideal second-order update--computed using globally preconditioned global gradients--into parameter mixing on the server and local parameter updates on clients. As a result, our FedPM introduces preconditioned mixing of local parameters on the server, effectively mitigating drift in local preconditioners. We provide a theoretical convergence analysis demonstrating a superlinear rate for strongly convex objectives in scenarios involving a single local update. To demonstrate the practical benefits of FedPM, we conducted extensive experiments. The results showed significant improvements with FedPM in the test accuracy compared to conventional methods incorporating simple mixing, fully leveraging the potential of second-order optimization.

One main challenge in federated learning is the large communication cost of exchanging weight updates from clients to the server at each round. While prior work has made great progress in compressing the weight updates through gradient compression methods, we propose a radically different approach that does not update the weights at all. Instead, our method freezes the weights at their initial random values and learns how to sparsify the random network for the best performance. To this end, the clients collaborate in training a stochastic binary mask to find the optimal sparse random network within the original one. At the end of the training, the final model is a sparse network with random weights - or a subnetwork inside the dense random network. We show improvements in accuracy, communication (less than 1 bit per parameter (bpp)), convergence speed, and final model size (less than 1 bpp) over relevant baselines on MNIST, EMNIST, CIFAR-10, and CIFAR-100 datasets, in the low bitrate regime. Federated learning (FL) is a distributed learning framework where clients collaboratively train a model by performing local training on their data and by sharing their local updates with a server every few iterations, which in turn aggregates the local updates to create a global model, that is then transmitted to the clients for the next round of training. While being an appealing approach for enabling model training without the need to collect client data at the server, uplink communication of local updates is a significant bottleneck in FL (Kairouz et al., 2021). In this work, while aiming for communication efficiency in FL, we take a radically different approach from prior work, and propose a strategy that does not require communication of weight updates.

Federated learning refers to a distributed learning scenario in which users/nodes keep their data private but only share intermediate locally computed iterates with the master node. The master, in turn, shares a global aggregate of these iterates with all the nodes at each iteration. In this work, we consider a wireless federated learning scenario where the nodes communicate to and from the master node via a wireless channel. Current and upcoming technologies such as 5G (and beyond) will operate mostly in a non-orthogonal multiple access (NOMA) mode where transmissions from the users occupy the same bandwidth and interfere at the access point. These technologies naturally lend themselves to an "over-the-air" superposition whereby information received from the user nodes can be directly summed at the master node. However, over-the-air aggregation also means that the channel noise can corrupt the algorithm iterates at the time of aggregation at the master. This iteration noise introduces a novel set of challenges that have not been previously studied in the literature. It needs to be treated differently from the well-studied setting of noise or corruption in the dataset itself. In this work, we first study the subspace learning problem in a federated over-the-air setting. Subspace learning involves computing the subspace spanned by the top $r$ singular vectors of a given matrix. We develop a federated over-the-air version of the power method (FedPM) and show that its iterates converge as long as (i) the channel noise is very small compared to the $r$-th singular value of the matrix; and (ii) the ratio between its $(r+1)$-th and $r$-th singular value is smaller than a constant less than one. The second important contribution of this work is to show how over-the-air FedPM can be used to obtain a provably accurate federated solution for subspace tracking in the presence of missing data.