  Privacy-preserving machine learning is one class of cryptographic methods
that aim to analyze private and sensitive data while keeping privacy, such as
homomorphic logistic regression training over large encrypted data. In this
paper, we propose an efficient algorithm for logistic regression training on
large encrypted data using Homomorphic Encryption (HE), which is the mini-batch
version of recent methods using a faster gradient variant called
$\texttt{quadratic gradient}$. It is claimed that $\texttt{quadratic gradient}$
can integrate curve information (Hessian matrix) into the gradient and
therefore can effectively accelerate the first-order gradient (descent)
algorithms. We also implement the full-batch version of their method when the
encrypted dataset is so large that it has to be encrypted in the mini-batch
manner. We compare our mini-batch algorithm with our full-batch implementation
method on real financial data consisting of 422,108 samples with 200 freatures.
%Our experiments show that Nesterov's accelerated gradient (NAG) Given the
inefficiency of HEs, our results are inspiring and demonstrate that the
logistic regression training on large encrypted dataset is of practical
feasibility, marking a significant milestone in our understanding.
