Total variation distance

The total variation (TV) distance is half the L1 distance between the two distributions:

The second form gives its meaning: it is the largest possible difference in probability that P and Q can assign to the same event. It is a true metric — symmetric, satisfies the triangle inequality — and bounded in $[0,1]$.

TV is both an f-divergence (with $f(t)=\frac{1}{2}|t-1|$) and an IPM (with test functions bounded by 1). Like JS, it saturates at 1 for non-overlapping distributions, so it gives weak gradients when the model is far from the data.