Many convex optimization problems with important applications in machine learning are formulated as empirical risk minimization (ERM). There are several examples: linear and logistic regression, LASSO, kernel regression, quantile regression, $p$-norm regression, support vector machines (SVM), and mean-field variational inference. To improve data privacy, federated learning is proposed in machine learning as a framework for training deep learning models on the network edge without sharing data between participating nodes. In this work, we present an interior point method (IPM) to solve a general ERM problem under the federated learning setting. We show that the communication complexity of each iteration of our IPM is $\tilde{O}(d^{3/2})$, where $d$ is the dimension (i.e., number of features) of the dataset.

翻译：许多在机器学习中具有重要应用的凸优化问题被表述为经验风险最小化（ERM）。例如：线性回归与逻辑回归、LASSO、核回归、分位数回归、$p$范数回归、支持向量机（SVM）以及平均场变分推断。为提升数据隐私性，联邦学习被提出作为一种机器学习框架，用于在网络边缘训练深度学习模型，而无需在参与节点之间共享数据。本文提出一种内点法（IPM），用于求解联邦学习设定下的广义ERM问题。我们证明，该IPM每次迭代的通信复杂度为$\tilde{O}(d^{3/2})$，其中$d$为数据集的维度（即特征数量）。