Information that is stored in an encrypted format is, by definition, usually
not amenable to statistical analysis or machine learning methods. In this paper
we present detailed analysis of coordinate and accelerated gradient descent
algorithms which are capable of fitting least squares and penalised ridge
regression models, using data encrypted under a fully homomorphic encryption
scheme. Gradient descent is shown to dominate in terms of encrypted
computational speed, and theoretical results are proven to give parameter
bounds which ensure correctness of decryption. The characteristics of encrypted
computation are empirically shown to favour a non-standard acceleration
technique. This demonstrates the possibility of approximating conventional
statistical regression methods using encrypted data without compromising
privacy.

本文中，我们提出了详细的分析方法，并证明了使用全同态加密方案下的数据对最小二乘和惩罚岭回归模型进行拟合的可行性，其中梯度下降算法在加密计算速度方面表现优异，并且理论结果也被证明可以确保解密的正确性，从而展示了使用加密数据近似传统统计回归方法的可能性，而不会泄露隐私。