Robust Mixing

Abstract

In this paper, we develop a new “robust mixing” framework for reasoning about adversarially modified Markov Chains (AMMC). Let ℙ be the transition matrix of an irreducible Markov Chain with stationary distribution π. An adversary announces a sequence of stochastic matrices $\{{\mathbb{A}}_t\}_{t > 0}$ satisfying $\pi{\mathbb{A}}_t = \pi$. An AMMC process involves an application of ℙ followed by ${\mathbb{A}}_t$ at time t. The robust mixing time of an irreducible Markov Chain ℙ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains.

Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time t, exchanges the card at position $t {\pmod n}$ with a random card. Mossel, Peres and Sinclair (2004) showed that the mixing time of this process lies between (0.0345+o(1))nlogn and Cnlogn + O(n) (with C ≈4 ×10⁵). We reduce the constant C to 1 by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time ≤nlogn + O(n) when the adversarial strategies are limited to those which preserve the symmetry of the underlying Markov Chain.

Reversible liftings: Chen, Lovász and Pak showed that for a reversible ergodic Markov Chain ℙ, any reversible lifting ℚ of ℙ must satisfy ${\mathcal{T}}({\mathbb{P}}) \leq {\mathcal{T}}(\mathbb{Q})\log (1/\pi_*)$ where π _* is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that ${\mathcal{T}}(\mathbb{Q}) \geq r({\mathbb{P}})$ where r(ℙ) is the relaxation time of ℙ. This helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.