I've been using the $K$-fold cross-validation a few times now to evaluate performance of some learning algorithms, but I've always been puzzled as to how I should choose the value of $K$.
I've often seen and used a value of $K = 10$, but this seems totally arbitrary to me, and I now just use $10$ by habit instead of thinking it over. To me it seems that you're getting a better granularity as you improve the value of $K$, so ideally you should make your $K$ very large, but there is also a risk to be biased.
I'd like to know what the value of $K$ should depend on, and how I should be thinking about this when I evaluate my algorithm. Does it change something if I use the stratified version of the cross-validation or not?