  In this paper, we investigate how to manipulate the coefficients obtained via
linear regression by adding carefully designed poisoning data points to the
dataset or modify the original data points. Given the energy budget, we first
provide the closed-form solution of the optimal poisoning data point when our
target is modifying one designated regression coefficient. We then extend the
analysis to the more challenging scenario where the attacker aims to change one
particular regression coefficient while making others to be changed as small as
possible. For this scenario, we introduce a semidefinite relaxation method to
design the best attack scheme. Finally, we study a more powerful adversary who
can perform a rank-one modification on the feature matrix. We propose an
alternating optimization method to find the optimal rank-one modification
matrix. Numerical examples are provided to illustrate the analytical results
obtained in this paper.
