# Why isn’t this formula used at all?

$$w_{n+1} = w_n - \alpha \frac{f \nabla f}{||\nabla f||^2}$$ It calculates where the line that’s in the same direction as gradient and has $$f$$’s slope in that direction becomes zero. In one dimensional case it becomes: $$[x_{n+1}] = [x_n] - \alpha f(x_n) \frac{[f’(x_n)]}{||[f’(x_n)]||^2}$$

$$x_{n+1} = x_n - \alpha \frac{f(x_n)}{f’(x_n)}$$

Which is just Newton’s root finding method if $$\alpha = 1$$.

• Used where? Can you be more specific? Right now, this could not even be related to AI.
– nbro
Commented Aug 1 at 14:07

Your formula appears designed to find a point where a scalar function $$f$$ of several variables is equal to zero. However, this goal is not particularly useful. Generically, a function of $$n$$ variables equals zero on some $$(n - 1)$$-dimensional space. Your formula may converge to some point in this space. But why does one want to find such a point?
If the goal is fitting a model, and $$f$$ is a loss function, one wants to find a point where $$f$$ is minimized, not where $$f = 0$$. (Note, even if somehow one knew a priori that $$f$$'s minimum value is zero, Newton's method would be an inefficient way to find it.)
One can also frame minimization as finding a point where $$\nabla f = 0$$, but this is finding a root of a vector function (i.e., simultaneous root of $$n$$ scalar functions), not of a single scalar function.