  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.
