In this manuscript, we extend our previous work on privacy-preserving regression to address multi-output regression problems using data encrypted under a fully homomorphic encryption scheme. We build upon the simplified fixed Hessian approach for linear and ridge regression and adapt our novel LFFR algorithm, initially designed for single-output logistic regression, to handle multiple outputs. We further refine the constant simplified Hessian method for the multi-output context, ensuring computational efficiency and robustness. Evaluations on multiple real-world datasets demonstrate the effectiveness of our multi-output LFFR algorithm, highlighting its capability to maintain privacy while achieving high predictive accuracy. Normalizing both data and target predictions remains essential for optimizing homomorphic encryption parameters, confirming the practicality of our approach for secure and efficient multi-output regression tasks.

翻译：在本手稿中，我们将先前关于隐私保护回归的研究扩展至解决多输出回归问题，所采用的数据是在全同态加密方案下加密的。我们基于用于线性回归和岭回归的简化固定Hessian方法，并将我们新颖的LFFR算法（最初为单输出逻辑回归设计）进行适配以处理多输出问题。我们进一步为多输出场景改进了常数简化Hessian方法，确保了计算效率与鲁棒性。在多个真实世界数据集上的评估证明了我们多输出LFFR算法的有效性，突显了其在保持隐私的同时实现高预测精度的能力。对数据和目标预测值进行归一化对于优化同态加密参数仍然至关重要，这证实了我们的方法在安全高效的多输出回归任务中的实用性。