# Entropy-Regularized Federated Optimization for Non-IID Data
## 3 Methodology
### Convergence Analysis
Although \(\Phi(w)\) is not explicitly given in closed form (due to the interdependence of \(w^{t}\) and local updates, \(\Phi\) would effectively be a function measuring the concentration of \(w-w\)), one can bound the difference between the gradient of \(F\) and the gradient of \(\widehat{F}\). In particular, \(\
abla\widehat{F}(w)=\
abla F(w)+\lambda\
abla\Phi(w)\), and we can show \(|\
abla\Phi(w)|\leq G_{H}\) under our bounded-entropy-gradient assumption. Thus, the update in ERFO-Fixed is an inexact gradient step for minimizing \(F(w)\), with an error proportional to \(\lambda\). Using techniques from inexact or perturbed gradient descent analysis, we obtain that \(F(w^{t})\) converges to within \(O(\lambda)\) of \(F(w)\). When \(\lambda\) is small, this bias is negligible; however, a nonzero \(\lambda\) prevents us from reaching the exact optimum \(w\) in general (unless \(w\) itself happens to have a maximally spread-out update distribution so that \(\
abla\Phi(w^{*})=0\)). In summary, constant entropy regularization yields stable convergence but to a slightly biased solution. Theorem 1 shows that ERFO with a fixed regularisation coefficient attains essentially the same convergence rate as standard FedAvg in convex problems, but converges to a biased limit point (see Appendix A for the full proof). The bias term \(O(\lambda)\) is intuitively expected: the algorithm is effectively optimizing a proxy objective \(F(w)+\lambda\Phi(w)\), so it converges to the minimizer of that proxy, which can differ from the true minimizer \(w^{*}\) by an amount that vanishes as \(\lambda\to 0\). In practice, we choose a \(\lambda\) value that is small enough so that this bias is within acceptable tolerance for the problem, trading off some optimality for improved training stability. We now turn to the decaying regularization scenario. When \(\lambda_{t}\) is gradually reduced to 0, we eliminate the bias in the limit, but at the cost of a more complex time-varying analysis. Fortunately, since \(\lambda_{t}\) shrinks over rounds, the bias effect becomes smaller as training progresses, and in the final rounds the algorithm behaves nearly like FedAvg on the original objective. **Theorem 2** (Convergence with Decaying \(\lambda_{t}\)). : _Assume the same conditions as Theorem 1, but now let \(\lambda_{t}\) be a nonincreasing sequence that tends to 0 as \(t\to T\) (specifically, the linear schedule \(\lambda_{t}=\lambda_{0}(1-t/T)\)). Then Algorithm 1 (ERFO-Decayed) converges to the true optimum \(w^{*}\) of \(F(w)\). Moreover, for sufficiently large \(T\), the convergence rate in the convex setting approaches that of FedAvg without regularization. In particular, for convex \(F_{i}\), one can ensure_  
\[F\big{(}w_{T}\big{)}\ -\ F\big{(}w^{*}\big{)}\ =\ \mathcal{O}\big{(}T^{-1}\big{)}. \tag{9}\]  
_And under strong convexity (and appropriate diminishing step sizes) \(w^{t}\) converges linearly to \(w^{*}\) as \(t\to T\)._  
Proof Sketch. : The proof builds upon Theorem 1. Since \(\lambda_{t}\) decays, for any \(\varepsilon>0\) one can choose a round \(t_{0}\) for all \(t\geq t_{0}\), \(\lambda_{t}<\varepsilon\). Beyond this point, the regularization is very weak, and the algorithm behaves almost like unregularized FedAvg, which we know will converge to \(w\) given convexity. The earlier rounds (\(t<t_{0}\)) with larger \(\lambda_{t}\) do not harm convergence; in fact, they help drive the model quickly into a good region, after which the diminishing \(\lambda_{t}\) ensures that any remaining bias is flushed out. Formally, one can derive error bounds similar to Theorem 1 for each round and sum them telescopically, taking into account the decreasing weight on the perturbation. The linear decay \(\lambda_{t}=\lambda_{0}(1-t/T)\) is gentle enough that the error does not accumulate significantly. In the limit \(T\to\infty\) (infinitely many rounds with infinitesimal decay per round), the method essentially becomes FedAvgin the long run, guaranteeing \(w\).