I want to train a neural network with pictures of public figures (politicians, singers, etc), but I do not know if it's legal, I do not plan to show them in my project I only want to use them to train the neural network, can this cause legal problems?
If you use any pictures you find online, you can use them as you wish: as long as you don't (re)publish them under your name. Also, if you really want to play safe, never upload them at all. Download them, use them, disregard them.
People can't proof you used certain pictures to train your network by looking at your network data.
And the 'big' companies out there, like Google, train their networks by using public photos.
As you are using it for a personal/educational project, it shouldn't be a problem.
It depends on the country. In France, for example, you've got to have the agreement of the person. Doesn't matter whether he is the president, a singer or... me :)
Doesn't matter what for. Your image belongs to you. Otherwise he (me) can sue you for that and seek redress. Faces of persons are blurred in Google Map Street View. Here is a link, in french.
About re-publication, you'll also have to have the agreement of the copyright holder.