deletions | additions       diff --git a/paragraph_RTM_runs_in_polynomial__.tex b/paragraph_RTM_runs_in_polynomial__.tex  index eec32e0..6f4d6ba 100644  --- a/paragraph_RTM_runs_in_polynomial__.tex  +++ b/paragraph_RTM_runs_in_polynomial__.tex 
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\paragraph{RTM runs in polynomial time}  The algorithm has nice convergence property: property (Theorem 3.1 and Theoreom 3.2):  for a given accuracy $\| \hat{\mathbf{q}} - \mathbf{q}_{\ast} \|_2 \leq \epsilon$, it takes RTM a polynomial number (in $p$ and $n$) of iterations to converge to $\hat{\mathbf{q}}$. This is proved by showing   \begin{enumerate}  \item in $R_{II}$ and $R_{III}$, each step reduces the objective values by at least a constant (Proposition 3.7), and  \item in $R_{I}$, the algorithm first takes \textit{constrained steps} and followed by \textit{all unconstrained steps}  \item objective is reduced by at least a fixed amount for each constrained step in $R_{I}$, and   \item the distance to the local minimum drops quadratically for unconstrained steps in $R_I$.   \end{enumerate}