My partner and I were doing a basic least squares regression line on some dataset. We got an intercept and coefficient that were different, but our MSE on the test dataset was the same. Is this a thing? For some reason I had always thought there was only one regression line that minimized MSE, but I guess I was wrong?
Are you certain you both performed your regression and testing on the same subsets of data?
Linear regression with least squares is a convex optimisation problem, so you should get the same line.
An exception would be performing regression on a single data point (two parameters, slope and intercept, but only one observation). In this case any line that passes through the point would be optimal.