$E_\gamma$ Mixing Time

Behnoosh Zamanlooy, Shahab Asoodeh, Mario Diaz, Flavio Calmon

Abstract: We investigate the mixing times of Markov kernels under $E_\gamma$ - divergence. We demonstrate that the zero-error $E_\gamma$ - mixing time, for any $\gamma>1$, of irreducible and aperiodic Markov chains, is bounded, a property that is not shared by the TV-mixing time. We further obtain upper bounds on the $E_\gamma$ - mixing times for a broad family of contractive Markov kernels via a new non-linear strong data processing inequality for the $E_\gamma$ - divergence. We apply our results to derive new bounds for the local differential privacy guarantees offered by the sequential application of a privacy mechanism to data.