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).

Image for the Neural network model

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

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


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

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.)



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