On the uniform continuity of the rate-distortion function

It is well known that the rate-distortion function for a finite alphabet IID source with distribution p, denoted R(p,D), is uniformly continuous in its arguments. We prove an explicit bound on |R(p,D) - R(q,D)| for distributions p, q in terms of the variational distance parp-qpar₁. A simple and elementary proof shows that |R(p,D) - R(q,D)| = O(-parp - qpar₁ log parp - qpar₁), with constants depending on the distortion measure. The uniform continuity of the rate-distortion function has the same behavior as the uniform continuity of entropy in the order sense. The bounds are used for several applications. First, a simple sampling algorithm is presented to compute the rate-distortion function for an arbitrarily varying source to within a given accuracy. The uniform continuity bound is used here to roughly quantify the tradeoff between complexity and accuracy. Second, we comment on the problem of approximating the rate-distortion function for an unknown IID source to within a desired precision.