# Entropy-Regularized Federated Optimization for Non-IID Data
## 3 Methodology
### ERFO Variants: Fixed vs. Decayed Regularization  
We consider two variants of our entropy-regularized federated optimization algorithm, differing in how the regularization coefficient \(\lambda_{t}\) is scheduled over the course of training:  
ERFO-Fixed: This variant uses a constant regularization weight throughout training, \(\lambda_{t}=\lambda_{0}\) for all rounds \(t=1,\ldots,T\). In this case, the influence of the entropy term remains the same from start to finish. We apply ERFO-Fixed in our PneumoniaMNIST experiments, where a fixed moderate \(\lambda\) was sufficient to improve generalization without needing to taper it off. The algorithm in this case simplifies to using \(\lambda\) in every round's local updates (as in Algorithm 1 with the first option for \(\lambda_{t}\)). One advantage of a fixed \(\lambda\) is its simplicity and predictability: the regularizer continuously guides local training to distribute updates, which can enhance stability on non-IID data. However, a potential drawback is that as the global model gets closer to the optimum, a nonzero \(\lambda\) might introduce a persistent bias (preventing full convergence to the exact minimizer of the original loss, as analyzed later).  
ERFO-Decayed: In this variant, the regularization weight is time-dependent, starting from an initial value \(\lambda_{0}\) and gradually decaying to \(0\) by the end of training. We specifically use a linear decay schedule,  
\[\lambda_{t}\coloneqq\lambda_{0}\bigg{(}1-\frac{t}{T}\bigg{)},\qquad t=0,1, \ldots,T, \tag{7}\]  
for round \(t=1,2,\ldots,T\) (so \(\lambda_{T}=0\) at the final round). This schedule linearly anneals the entropy regularization, meaning that early in training the updates are strongly regularized (promoting high-entropy, diffuse updates), whereas in later rounds the regularization diminishes, allowing the model to fine-tune more freely on the actual task loss. We employ ERFO-Decayed in the UNSW-NB15 experiments, where we found that using a high \(\lambda\) initially helps navigate the complex, heterogeneous feature space of the intrusion detection data, while reducing \(\lambda\) to \(0\) by the end ensures that the model can converge to a solution optimized purely for classification performance. In conjunction with this, we found it beneficial to use learning rate warmup for the first few local epochs to mitigate any instability from the initially large \(\lambda_{0}\). After warmup, we keep the learning rate constant (or optionally employ a standard decay rate for the learning rate as well, though in our UNSW-NB15 runs the primary schedule of interest was the decay of \(\lambda_{t}\)).  
In summary, ERFO-Fixed and ERFO-Decayed share the same algorithmic core (Algorithm 1) but differ in the choice of \(\lambda_{t}\). ERFO-Fixed is appropriate when constant regularization provides the best bias-variance trade-off for the problem at hand (and when we are not concerned about a small asymptotic bias in convergence). ERFO-Decayed is useful when one wants to reap the benefits of entropy regularization in early training (improved stability and coverage of the solution space) but still ensure that the final model is not biased by the regularizer. By annealing \(\lambda_{t}\) to zero, ERFO-Decayed aims to combine the best of both worlds: strong regularization initially and exact convergence to the federated objective at the end. We formally analyze these convergence behaviors next.