# Towards Federated Learning with Byzantine-Robust Client Weighting
## 2 Materials and Methods
### Preprocessing Client-Declared Sample Sizes
#### 2.2.4 Convergence Analysis
Additionally, note that this formulation demonstrates how a single Byzantine client can increase this difference arbitrarily by increasing its \(\dot{n}_{i}\). Lastly, observe how both terms vanish as \(U\) increases, which motivates our selection of \(U^{*}\) as the _maximal_ truncation threshold for any given \(\alpha\) and \(\alpha^{*}\). Convergence Guarantees  
Having bounded the difference between the pre-truncation and post-truncation training objectives, we can now utilize known convergence guarantees. Such guarantees and their respective proofs depend on the specific underlying _Aggregate_ procedures and are orthogonal to our work. Nevertheless, we include two examples of how to account for our preprocessing step in the convergence analysis of specific underlying algorithms. In the first example, which we defer to Appendix C, we restate and adapt for truncation the convergence analysis given by [PERSON] et al. [16] for the standard, non-Byzantine-tolerant, \(FedAvg\) algorithm. In the second example, we do the same for the Byzantine-robust trimmed mean aggregation procedure by [PERSON] et al. [12]. This establishes that convergence is still guaranteed, even though the weights of some clients are truncated. The following definitions and assumptions are required for the analysis of [PERSON] et al. [12]:  
**Definition 2** (Coordinate-wise trimmed mean). : _For \(\beta\in[0,\frac{1}{2})\) and vectors \(x^{i}\in\mathbb{R}^{d}\), \(i\in[m]\), the coordinate-wise \(\beta\)-trimmed mean \(\mathbf{g}:=\mathsf{trmean}_{\beta}\{x^{i}:i\in[m]\}\) is a vector with its \(k\)-th coordinate being \(g_{k}=\frac{\sum_{i\in k_{k}}n_{i}x_{k}^{i}}{\sum_{i\in k_{k}}n_{i}}\) for each \(k\in[d]\). Here, \(R_{k}\) is a subset of indices from \(\{x_{k}^{1},\ldots,x_{k}^{m}\}\) obtained by removing the largest and smallest \(\beta\) fraction of its elements._  
**Definition 3** (Sub-exponential random variables). : _A random variable \(X\) with \(\mathbb{E}[X]=\mu\) is called \(v\)-sub-exponential if \(\mathbb{E}\!\left[e^{\lambda(X-\mu)}\right]\leq e^{\frac{1}{2}\
u^{2}\lambda^ {2}},\forall|\lambda|<\frac{1}{9}\)._  
**Definition 4** (Lipschitz). : _h is \(L\)-Lipschitz if \(|h(w)-h(w^{\prime})|\leq L\|w-w^{\prime}\|,\forall w,w^{\prime}\)._  
**Definition 5** (Smoothness). : _h is \(L^{\prime}\)-smooth if \(\|\
abla h(w)-\
abla h(w^{\prime})\|\leq L^{\prime}\|w-w^{\prime}\|,\forall w,w^{\prime}\)._  
**Definition 6** (Strong convexity). : _h is \(\lambda\)-strongly convex if \(h(w^{\prime})\geq h(w)+\langle\
abla h(w),w^{\prime}-w\rangle+\frac{\lambda}{ 2}\|w^{\prime}-w\|^{2},\forall w,w^{\prime}\)._  
**Assumption 3** (Parameter space convexity and compactness). : _We assume that the parameter space \(\mathcal{W}\) is convex and compact with diameter \(D\), i.e., \(\|w-w^{\prime}\|\leq D,\forall w,w^{\prime}\in\mathcal{W}\)._  
**Assumption 4** (Smoothness of \(\ell\) and \(F\)). : _We assume that for any \(z\in\mathbf{Z}\), the partial derivative of \(\ell(\cdot;z)\) with respect to the \(k\)-th coordinate of its first argument, denoted by \(\partial_{k}\ell(\cdot;z)\), is \(L_{k}\)-Lipschitz for each \(k\in[d]\), and the function \(\ell(\cdot;z)\) is \(L\)-smooth. Let \(\widehat{L}:=\sqrt{\sum_{k=1}^{d}L_{k}^{2}}\). We also assume that the population loss function \(F(\cdot)\) is \(L_{F}\)-smooth._  
**Assumption 5** (Sub-exponential gradients). : _We assume that for all \(k\in[d]\) and \(w\in\mathcal{W}\), the partial derivative of \(\ell(w;z)\) with respect to the \(k\)-th coordinate of \(w\), \(\partial_{k}\ell(w;z)\), is \(v\)-sub-exponential._  
Define  
\[m=\min_{i\in[K]}\{n_{i}\},\bar{m}=\min_{i\in[K]}\{\bar{n}_{i}\}.\]  
In order to prove convergence with truncation, we can adapt Theorems 4-6 from [12], which provide convergence guarantees for the trimmed mean aggregation procedure for the strongly convex, non-strongly convex, and non-convex cases, respectively.