Why do round objects with specular shader give white, pixelated edges when viewed from a far distance?

I have this problem where cylinders and rods with specular shaders glistens and shimmers when viewed from a far distance. I have anti-aliasing on, but it feels like the specular shader ignores the anti-aliasing, making it appear like this. I have taken screenshots of this:

And I am tired of pretending that it isn’t there and that it isn’t a problem, it clearly is a problem, so how can fix it?

This is a quite common problem and is usually caused by mipmapping when your texture doesn’t have enough padding around the actual mapped areas.