Consider the logistic regression model, where $X\sim \mathcal{N}(0,\Sigma)$ and unkown parameter $\theta\in \mathbb{R}^d$. Furthermore for the random variable $Y\in \{-1,1\}$ we have $$P(Y=y|X,\theta)=\frac{1}{1+\exp(-yX^T\theta)}$$ where $y\in \{-1,1\}$.
Let's assume that we have estimated $\theta$ by $\hat{\theta}$ and we are interested in calculating the following error probability: $$P(Y\neq \hat{Y})=P(Y=1,\hat{Y}=-1)+P(Y=-1,\hat{Y}=1)$$
How can we calculate this generalization error?