Robust Mixing
Murali Ganapathy, Google Inc
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 {At}{t>0} satisfying
πAt = π. An AMMC process involves an application of P
followed by At at time t. The robust mixing time of an
ergodic 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 Lt := t mod n with a random card. Mossel, Peres and
Sinclair (2004) showed a lower bound of (0.0345+o(1))n*log(n) for
the mixing time of the random-to-cyclic transposition process. They
also considered a generalization of this process where the choice of
Lt is adversarial, and proved an upper bound of C n*log(n) +
O(n) (with C ≅ 4*105) on the mixing time. We reduce
the constant to 1 by showing that the random-to-top transposition
chain (a Markov Chain) has robust mixing time ≤ n*log(n) +
O(n) when the adversarial strategies are limited to holomorphic
strategies, i.e. those strategies which preserve the symmetry of
the underlying Markov Chain. We also show a O(n*log2(n)) bound on
the robust mixing time of the lazy random-to-top transposition chain
when the adversary is not limited to holomorphic strategies.
Reversible liftings: Chen, Lovász and Pak showed that
for a reversible ergodic Markov Chain P, any reversible lifting
Q of P must satisfy T(P) ≤ T(Q)*log(1/π*) where
π* is the minimum stationary probability. Looking at a specific
adversarial strategy allows us to show that T(Q) ≥ r(P) where
r(P) is the relaxation time of P. This gives an alternate proof of the
reversible lifting result and helps identify cases where reversible liftings
cannot improve the mixing time by more than a constant factor.
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