# Should the minimum value of a cost (loss) function be equal to zero?

We know optimization techniques search in the space of all the possible parameters for a parameter set that minimizes the cost function of the model. The most well-known loss functions, like MSE or Categorical Cross Entropy, has a global minimum value equal to zero, in the ideal case.

For example, the Gradient Descent, $$\theta_j \leftarrow \theta_j - \alpha \frac{\partial}{\partial \theta_j}J(\theta)$$, updates parameters based on the derivation of the calculated cost function value, $$J(\theta)$$.

I was wondering what will happen if we design a cost function that has a non-zero global minimum in its ideal case. Does it make a difference, e.g. in the convergence rate or other aspects of the optimization process, or not?

We need to minimise $J(\theta)$ so that the predictions can be as close to the actuals as possible . For that the derivative of $J(\theta)$ should be zero . It doesn't matter if $J(\theta)$ is zero or non zero . Graphically , for a cost function like this , you want to reach the point $J_{min}(w)$ where the derivative is zero.