# Is there a “flaw” in the backpropagation algorithm?

While trying to find a better backpropagation algorithm, I came across a paradox in my algorithm and then I found out this also happens in the usual backpropagation algorithm.

Our neural network looks like this:

In this simple example, we have no biases. The activiation function is the usual $f(x)=\frac{1}{1+e^{-x}}$

Our training set is just one datapoint: input = 0.9 and target_output = 0.9. The learning factor is set to 0.1.

The paradox happens when w2 is randomly initialized to a negative value. First, let's take a look at the "normal" situation (where the paradox does not happen). Let's say w1 and w2 are both positive, but the output is less than 0.9. w1 and w2 are both initialized at 0.1. Then to reduce the error, both w1 and w2 are increased, so the the output will get closer to 0.9. This is visible in the next graph (x is the training iteration, each training iteration contains one backpropagation).

Now let's take a look at the situation that w1 is intialized at 0.1 and w2 is initialized with a negative value (-0.5). Initially, an increase in w1 would cause a more negative value for the output layer, which increases the error. However, that's only the case because w2 is negative. Although w1 will decrease in the beginning, it will start increasing as soon as w2 becomes positive. The blue line (w1) is clearly decreasing in the beginning in the below graph from my experiment:

This "wrong" learning in the beginning of the blue line w1, is a consequence of using the derivative of the error function to induce small learning steps. Would it be better to somehow anticipate a positive w2 that will be needed anyway and so increase w1 from the beginning instead? Can we make a better backpropagation algorithm?