Most algorithms for solving optimization problems or finding saddle points of
convex-concave functions are fixed-point algorithms. In this work we consider
the generic problem of finding a fixed point of an average of operators, or an
approximation thereof, in a distributed setting. Our work is motivated by the
needs of federated learning. In this context, each local operator models the
computations done locally on a mobile device. We investigate two strategies to
achieve such a consensus: one based on a fixed number of local steps, and the
other based on randomized computations. In both cases, the goal is to limit
communication of the locally-computed variables, which is often the bottleneck
in distributed frameworks. We perform convergence analysis of both methods and
conduct a number of experiments highlighting the benefits of our approach.

本研究探讨了在分布式环境中使用固定步数和随机计算两种策略来实现平均操作符的固定点，以适应联邦学习等需要，经过收敛性分析和实验证明了我们方法的优势。

从本地 SGD 到本地定点方法的联邦学习

From Local SGD to Local Fixed-Point Methods for Federated Learning

It is known that training deep neural networks, in particular, deep
convolutional networks, with aggressively reduced numerical precision is
challenging. The stochastic gradient descent algorithm becomes unstable in the
presence of noisy gradient updates resulting from arithmetic with limited
numeric precision. One of the well-accepted solutions facilitating the training
of low precision fixed point networks is stochastic rounding. However, to the
best of our knowledge, the source of the instability in training neural
networks with noisy gradient updates has not been well investigated. This work
is an attempt to draw a theoretical connection between low numerical precision
and training algorithm stability. In doing so, we will also propose and verify
through experiments methods that are able to improve the training performance
of deep convolutional networks in fixed point.

低数字精度如何影响神经网络的稳定性，以及如何通过随机舍入等方法提高固定点深度卷积网络的训练性能。

深度卷积网络定点训练中的挑战及其解决方法

Overcoming Challenges in Fixed Point Training of Deep Convolutional  Networks

We study trade networks with a tree structure, where a seller with a single
indivisible good is connected to buyers, each with some value for the good, via
a unique path of intermediaries. Agents in the tree make multiplicative revenue
share offers to their parent nodes, who choose the best offer and offer part of
it to their parent, and so on; the winning path is determined by who finally
makes the highest offer to the seller. In this paper, we investigate how these
revenue shares might be set via a natural bargaining process between agents on
the tree, specifically, egalitarian bargaining between endpoints of each edge
in the tree. We investigate the fixed point of this system of bargaining
equations and prove various desirable for this solution concept, including (i)
existence, (ii) uniqueness, (iii) efficiency, (iv) membership in the core, (v)
strict monotonicity, (vi) polynomial-time computability to any given accuracy.
Finally, we present numerical evidence that asynchronous dynamics with randomly
ordered updates always converges to the fixed point, indicating that the fixed
point shares might arise from decentralized bargaining amongst agents on the
trade network.

本文通过自然的交易流程，研究了具有树形结构的贸易网络中收益份额如何被设置，并证明了这种解决方案概念具有存在性、唯一性、效率性、核心成员身份、严格单调性和可任意精度的多项式时间可计算性。最后，本文提供了数值证据，随机更新的异步动态始终收敛于固定点，表明固定点份额可能来自于贸易网络上代理商之间的分散谈判。