# XOR Neural Network gets stuck in training

I'm trying to create a neural network to simulate a XOR gate.

Here's my dataset:

╔════════╦════════╗
║ x1, x2 ║ y1, y2 ║
╠════════╬════════╣
║  0, 0  ║  0, 1  ║
║  0, 1  ║  1, 0  ║
║  1, 0  ║  1, 0  ║
║  1, 1  ║  0, 1  ║
╚════════╩════════╝


And my neural network: I use logistic loss to get the error between target $$y_{k}$$ and output $$\hat{y}_{k}$$:

$$E(y_{k}, \hat{y}_{k}) = - y_{k} \cdot log(\hat{y}_{k}) + (1 - y_{k}) \cdot log(1 - \hat{y}_{k})$$

And then use the chain rule to update the weights. For example weight $$w_{3}$$'s contribution to the error is:

$$\sum_{k=1}^{2} \frac{\partial E(y_{k}, \hat{y}_{k})}{\partial w_{3}} = \sum_{k=1}^{2} \left(\frac{\partial E(y_{k}, \hat{y}_{k})}{\partial \hat{y}_{k}} \cdot \frac{\partial s_{k}}{\partial c_{1}}\right) \cdot \frac{\partial c_{1}}{\partial w_{3}}$$

Which in developed form is:

$$\sum_{k=1}^{2} \frac{\partial E(y_{k}, \hat{y}_{k})}{\partial w_{3}} = \left( \left( - \frac{y_{1}}{\hat{y}_{1}} + \frac{1 - y_{1}}{1 - \hat{y}_{1}} \right) \cdot \hat{y}_{1} \cdot (1 - \hat{y}_{1}) + \left( - \frac{y_{2}}{\hat{y}_{2}} + \frac{1 - y_{2}}{1 - \hat{y}_{2}} \right) \cdot (- \hat{y}_{2}) \cdot\frac{c_{1}}{c_{1} + c_{2}} \right) \cdot s_{0}$$

My issue is that after a couple epochs of training on the entire dataset, the network always outputs: $$\hat{y}_{1} = \hat{y}_{2} = 0.5$$

What am I doing wrong?