# How do we minimize loss for a single neuron with a feedback?

Suppose we had a series of single-dimensional data points $$X = \{x_1, x_2, \dots, x_n \}$$, where $$n$$ is the number of data points and there corresponding output values $$T = \{t_1, t_2, \dots, t_n \}$$.

Now, I want to train a single neuron network given below to learn from the data (the model is bad, but I just wanted to try it out as an exercise).

The output function of this neuron would be a recursive function as:

$$y = f(a_0 + a_1x + a_2 y)$$

where

$$f(x) = \frac{1}{1 + e^{-x}}$$

for a given $$x$$.

The error function for such a model would be:

$$e = \sum_{i=1}^N (y_i - t_i)^2$$

How should I minimise this loss function? What are the derivatives that I need to use to update the parameters?

(Also, I am new to this problem, therefore it would be really helpful if you tell me sources/books to read about such problems.)