Minimum KL-Divergence on Complements of $L_{1}$ Balls

Pinsker's widely used inequality upper-bounds the total variation distance ∥P - Q∥
₁
in terms of the Kullback-Leibler divergence D(P∥Q). Although, in general, a bound in the reverse direction is impossible, in many applications the quantity of interest is actually D*(v, Q)-defined, for an arbitrary fixed Q, as the infimum of D(P∥Q) over all distributions P that are at least v-far away from Q in total variation. We show that D*(v, Q) ≤ Cv
²
+ O(v
³
), where C = C(Q) = 1/2 for balanced distributions, thereby providing a kind of reverse Pinsker inequality. Some of the structural results obtained in the course of the proof may be of independent interest. An application to large deviations is given.