or of $p$, ignoring poly-logarithmic factors.

Tighter Analysis for the Special Case of Top-$\ell$ Separators Scenario {#sec:topl}
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The main result in Theorem \[thm:main2\] is general in the sense that it applies to any partial ranking data that is represented by positions of the separators. However, the bound can be quite loose, especially when $\gamma$ is small, i.e. $p_{j,\ell_j}$ is close to $\kappa_j$. For some special cases, we can tighten the analysis to get a sharper bound. One caveat is that we use a slightly sub-optimal choice of parameters $\lambda_{j,a} = 1/\kappa_j$ instead of $1/(\kappa_j - a)$, to simplify the analysis and still get the order optimal error bound we want. Concretely, we consider a special case of top-$\ell$ separators scenario, where each agent gives a ranked list of her most preferred $\ell_j$ alternatives among $\kappa_j$ offered set of items. Precisely, the locations of the separators are $(p_{j,1},p_{j,2},\ldots,p_{j,\ell_j})=(1,2,\ldots,\ell_j)$.

\[thm:topl\_upperbound\] Under the PL model, $n$ partial orderings are sampled over $d$ items parametrized by $\theta^* \in \Omega_b$, where the $j$-th sample is a ranked list of the top-$\ell_j$ items among the $\kappa_j$ items offered to the agent. If $$\begin{aligned}
 \label{eq:topl1}
\sum_{j = 1}^n \ell_j \;\; \geq \;\; \frac{2^{12}e^{6b}}{\beta\alpha^2} d\log d\,,\end{aligned}$$ where $b \equiv \max_{i,\i} |\theta^*_i - \theta^*_{\i}|$ and $\alpha,\beta$ are defined in and , then the [*rank-breaking estimator*]{} in with the choice of $\lambda_{j,a} =  1/{\kappa_j}$ for all $a\in[\ell_j]$ and $j\in[n]$ achieves $$\begin{aligned}
    \label{eq:main_topl} 
    \frac{1}{\sqrt{d}}\big\|\widehat{\theta} - \theta^* \big\|_2 \;\; \leq  \;\; \frac{16(1+ e^{2b})^2}{\alpha} \sqrt{\frac{d\, \log d}{\sum_{j=1}^n \ell_j}} \;,
    \end{aligned}$$ with probability at least $ 1- 3e^3 d^{-3}$.

A proof is provided in Section \[sec:proof\_topl\_upperbound\]. In comparison to 