# Minimum KL-divergence on complements of L1balls Academic Article

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### abstract

• Pinsker's widely used inequality upper-bounds the total variation distance $\left\Vert P-Q\right\Vert_{1}$ in terms of the Kullback–Leibler divergence $D(P\Vert Q)$ . Although, in general, a bound in the reverse direction is impossible, in many applications the quantity of interest is actually $D^{\ast}({v},Q)$ —defined, for an arbitrary fixed $Q$ , as the infimum of $D(P\Vert Q)$ over all distributions $P$ that are at least ${v}$ -far away from $Q$ in total variation. We show that $D^{\ast}({v},Q)\leq C{v}^{2}+O({v}^{3})$ , 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.

### publication date

• January 1, 2014