  We study the problem of learning sparse structure changes between two Markov
networks $P$ and $Q$. Rather than fitting two Markov networks separately to two
sets of data and figuring out their differences, a recent work proposed to
learn changes \emph{directly} via estimating the ratio between two Markov
network models. In this paper, we give sufficient conditions for
\emph{successful change detection} with respect to the sample size $n_p, n_q$,
the dimension of data $m$, and the number of changed edges $d$. When using an
unbounded density ratio model we prove that the true sparse changes can be
consistently identified for $n_p = \Omega(d^2 \log \frac{m^2+m}{2})$ and $n_q =
\Omega({n_p^2})$, with an exponentially decaying upper-bound on learning error.
Such sample complexity can be improved to $\min(n_p, n_q) = \Omega(d^2 \log
\frac{m^2+m}{2})$ when the boundedness of the density ratio model is assumed.
Our theoretical guarantee can be applied to a wide range of discrete/continuous
Markov networks.
