Split federated learning (SFL) is a recent distributed approach for
collaborative model training among multiple clients. In SFL, a global model is
typically split into two parts, where clients train one part in a parallel
federated manner, and a main server trains the other. Despite the recent
research on SFL algorithm development, the convergence analysis of SFL is
missing in the literature, and this paper aims to fill this gap. The analysis
of SFL can be more challenging than that of federated learning (FL), due to the
potential dual-paced updates at the clients and the main server. We provide
convergence analysis of SFL for strongly convex and general convex objectives
on heterogeneous data. The convergence rates are $O(1/T)$ and
$O(1/\sqrt[3]{T})$, respectively, where $T$ denotes the total number of rounds
for SFL training. We further extend the analysis to non-convex objectives and
where some clients may be unavailable during training. Numerical experiments
validate our theoretical results and show that SFL outperforms FL and split
learning (SL) when data is highly heterogeneous across a large number of
clients.

对于高度异构的数据，在并行联邦式 DE in the literature, and this paper aims to fill this gap. The analysis of SFL can be more challenging than that of federated learning (FL), due to the potential dual-paced updates at the clients and the main server. We provide convergence analysis of SFL for strongly convex and general convex objectives on heterogeneous data.

异构数据下分裂联邦学习的收敛分析

Convergence Analysis of Split Federated Learning on Heterogeneous Data

We propose the algorithms for online convex optimization which lead to
cumulative squared constraint violations of the form
$\sum\limits_{t=1}^T\big([g(x_t)]_+\big)^2=O(T^{1-\beta})$, where
$\beta\in(0,1)$. Previous literature has focused on long-term constraints of
the form $\sum\limits_{t=1}^Tg(x_t)$. There, strictly feasible solutions can
cancel out the effects of violated constraints. In contrast, the new form
heavily penalizes large constraint violations and cancellation effects cannot
occur.
Furthermore, useful bounds on the single step constraint violation
$[g(x_t)]_+$ are derived.
For convex objectives, our regret bounds generalize existing bounds, and for
strongly convex objectives we give improved regret bounds.
In numerical experiments, we show that our algorithm closely follows the
constraint boundary leading to low cumulative violation.

该研究提出了一种在线凸优化算法，其可以处理特定类型的累计平方约束违规问题，以及为凸目标导出了另类的后悔边界，并针对强凸目标提出了改进的后悔边界，并在数值实验中说明了该算法的效果。

累积约束的在线凸优化

Online convex optimization for cumulative constraints

Many classical algorithms are found until several years later to outlive the
confines in which they were conceived, and continue to be relevant in
unforeseen settings. In this paper, we show that SVRG is one such method: being
originally designed for strongly convex objectives, it is also very robust in
non-strongly convex or sum-of-non-convex settings.
More precisely, we provide new analysis to improve the state-of-the-art
running times in both settings by either applying SVRG or its novel variant.
Since non-strongly convex objectives include important examples such as Lasso
or logistic regression, and sum-of-non-convex objectives include famous
examples such as stochastic PCA and is even believed to be related to training
deep neural nets, our results also imply better performances in these
applications.

该研究提出了一种名为 SVRG 的算法，可用于处理强凸目标以外的任务并提供更高效的解决方案，并且该算法在许多著名的实际应用中都具有较好的表现。