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.
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