Recent advancements in machine learning have improved performance while also increasing computational demands. While federated and distributed setups address these issues, their structures remain vulnerable to malicious influences. In this paper, we address a specific threat: Byzantine attacks, wherein compromised clients inject adversarial updates to derail global convergence. We combine the concept of trust scores with trial function methodology to dynamically filter outliers. Our methods address the critical limitations of previous approaches, allowing operation even when Byzantine nodes are in the majority. Moreover, our algorithms adapt to widely used scaled methods such as Adam and RMSProp, as well as practical scenarios, including local training and partial participation. We validate the robustness of our methods by conducting extensive experiments on both public datasets and private ECG data collected from medical institutions. Furthermore, we provide a broad theoretical analysis of our algorithms and their extensions to the aforementioned practical setups. The convergence guaranties of our methods are comparable to those of classical algorithms developed without Byzantine interference.

Variational inequalities (VIs) have emerged as a universal framework for solving a wide range of problems. A broad spectrum of applications include optimization, equilibrium analysis, reinforcement learning, and the rapidly evolving field of generative adversarial networks (GANs). Stochastic methods have proven to be powerful tools for addressing such problems, but they often suffer from irreducible variance, necessitating the development of variance reduction techniques. Among these, SARAH-based algorithms have demonstrated remarkable practical effectiveness. In this work, we propose a new stochastic variance reduced algorithm for solving stochastic variational inequalities. We push the boundaries of existing methodologies by leveraging PAGE method to solve VIs. Unlike prior studies, which lacked theoretical guarantees under general assumptions, we establish rigorous convergence rates, thus closing a crucial gap in the literature. Our contributions extend both theoretical understanding and practical advancements in solving variational inequalities. To substantiate our claims, we conduct extensive experiments across diverse benchmarks including widely studied denoising task. The results consistently showcase the superior efficiency of our approach, underscoring its potential for real-world applications.