$$ \begin{gather} J(\Theta) = - \frac{1}{m} \sum_{i=1}^m \sum_{k=1}^K \left[y^{(i)}_k \log ((h_\Theta (x^{(i)}))_k) + (1 - y^{(i)}_k)\log (1 - (h_\Theta(x^{(i)}))_k)\right] + \frac{\lambda}{2m}\sum_{l=1}^{L-1} \sum_{i=1}^{s_l} \sum_{j=1}^{s_{l+1}} ( \Theta_{j,i}^{(l)})^2 \end{gather} $$

If create a hypothesis with r polynominal terms from $n$ features, then there will be $\frac{(n+r-1)!}{r!(n-1)!}$. For quadratic terms, the time complexity is $O(n^{2}/2)$. Not pratical to compute.

Now we are switching from regression problems to classification problems. Don’t be confused by the name “Logistic Regression”; it is named that way for historical reasons and is actually an approach to classification problems, not regression problems.