We study online estimation for high-dimensional generalized linear models with streaming data. First, for the non-distributed setting, we propose a gradient-enhanced surrogate loss that approximates the cumulative loss using only historical summaries, which modifies and improves upon the existing renewable estimation approach for the same model in the high-dimensional setting, and removes the batch-number constraint in previous studies. We then extend the method to distributed streaming data under the master-client architecture, where batches are partitioned across sites and only summaries (gradient vectors) are exchanged. Instead of directing applying the popular method of Jordan et al. (2019) to the surrogate quadratic loss, our adjusted approach does not require the clients to compute the full surrogate loss. We derive non-asymptotic error bounds under the high-dimensional scaling, without the stringent constraint on the number of batches in the previous studies. Simulation results under linear and logistic models, together with a real-data application, show improved accuracy over existing renewable estimators.

翻译：我们研究面向流式数据的高维广义线性模型的在线估计。首先，针对非分布式场景，我们提出一种梯度增强替代损失函数，仅利用历史摘要信息近似累积损失，该方法改进并优化了高维场景下现有可再生估计方法，并消除了先前研究中所需的批次数量约束。随后，我们将该方法扩展至主从架构下的分布式流式数据场景，其中批次被划分至不同站点，仅交换梯度向量摘要。与直接对替代二次损失函数应用Jordan等人(2019)的流行方法不同，我们的修正方法无需客户端计算完整替代损失。我们在高维尺度下推导了非渐近误差界，无需先前研究中对批次数量的严格约束。在线性模型与逻辑模型上的仿真结果以及真实数据应用表明，该方法较现有可再生估计器具有更高的精度。