Logistic regression training over encrypted data has been an attractive idea to security concerns for years. In this paper, we propose a faster gradient variant called $\texttt{quadratic gradient}$ to implement logistic regression training in a homomorphic encryption domain, the core of which can be seen as an extension of the simplified fixed Hessian. We enhance Nesterov's accelerated gradient (NAG) and Adaptive Gradient Algorithm (Adagrad) respectively with this gradient variant and evaluate the enhanced algorithms on several datasets. Experimental results show that the enhanced methods have a state-of-the-art performance in convergence speed compared to the naive first-order gradient methods. We then adopt the enhanced NAG method to implement homomorphic logistic regression training and obtain a comparable result by only $3$ iterations.

翻译：多年来，对加密数据进行逻辑回归训练一直是解决安全问题的有吸引力的思路。本文提出了一种名为$\texttt{quadratic gradient}$的更快梯度变体，用于在同态加密域中实现逻辑回归训练，其核心可视为简化固定黑塞矩阵的扩展。我们分别利用该梯度变体增强了Nesterov加速梯度（NAG）和自适应梯度算法（Adagrad），并在多个数据集上评估了这些增强算法。实验结果表明，与朴素的一阶梯度方法相比，增强方法在收敛速度上达到了当前最优性能。随后，我们采用增强的NAG方法实现同态逻辑回归训练，仅需$3$次迭代即可获得可比的结果。