  In federated learning, data heterogeneity significantly impacts performance.
A typical solution involves segregating these parameters into shared and
personalized components, a concept also relevant in multi-task learning.
Addressing this, we propose "Loop Improvement" (LI), a novel method enhancing
this separation and feature extraction without necessitating a central server
or data interchange among participants. Our experiments reveal LI's superiority
in several aspects: In personalized federated learning environments, LI
consistently outperforms the advanced FedALA algorithm in accuracy across
diverse scenarios. Additionally, LI's feature extractor closely matches the
performance achieved when aggregating data from all clients. In global model
contexts, employing LI with stacked personalized layers and an additional
network also yields comparable results to combined client data scenarios.
Furthermore, LI's adaptability extends to multi-task learning, streamlining the
extraction of common features across tasks and obviating the need for
simultaneous training. This approach not only enhances individual task
performance but also achieves accuracy levels on par with classic multi-task
learning methods where all tasks are trained simultaneously. LI integrates a
loop topology with layer-wise and end-to-end training, compatible with various
neural network models. This paper also delves into the theoretical
underpinnings of LI's effectiveness, offering insights into its potential
applications. The code is on https://github.com/axedge1983/LI
