White bumps under my eyes

I think that you mean millia. It's common on upper arms and under the eyes because this skin is thinner than other areas on your face and body. It's essentially a whitehead that has created a thicker set of cells above it.

To prevent and reduce the milia you have now - don't use products that are too rich for your skin. Most of the time oil from products like oily eye makeup remover, creamy eyeshadow and foundation can cause these especially if you are prone to them. To prevent milia and get yours to drastically reduce in number and size, use powder eye shadows (lol I am assuming you are female), oil-free foundation and a non pore clogging eye makeup remover - I have used Neutragena's Oil-free eye makeup remover forever! It's only about $5 for a bottle that will last you a year and it works better than most oily products.

For the ones that are left - don't every pick at them. See a dermatologist because you will no doubt cause scaring on this delicate tissue if you try to get rid of these little cysts yourself.

Good luck and I hope that helps!