How sensitive are the parameters obtained from OLS, logistic or other regression methods to noise ?
By noise, I mean minor changes. For e.g. adding a small noise $-1<\Delta<1$ to $\beta_1$ in $y= \beta_0 + \beta_1 x_1 + \beta_2 x_2$ to get $y= \beta_0 + (\beta_1 +\Delta) x_1 + \beta_2 x_2$
How drastically would the conclusions drawn initially w.r.t the model metrics (AIC, BIC, $R^2$ hypothesis tests associated with the $x_i$ )without the noise added change ?
I understand this may change from case to case depending on the model at hand. I want to know if there is any existing theory about sensitivity.
This is in the context of security where we need to protect our models from being maliciously attacked.
Edit: By noise, I meant a situation where an attacker adds a delta($\Delta$) to the found beta $\beta$.