SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Lemma 9. Condition on the events  $\mathcal{E}_2^{(\ell)},\mathcal{E}_3^{(\ell)}$  being true. Consider a nice subset of users  $\mathcal{M}^{(\ell ,i)}\in$ $\mathcal{M}'(\ell)$  and their corresponding set of active items  $\mathcal{N}^{(\ell ,i)}$  for which guarantees in eq. 11 holds. Consider a subset of users  $\mathcal{G}\in \mathcal{M}^{(\ell ,i)}$  forming a connected component. Fix any set  $\mathcal{V} = \bigcup_{u\in \mathcal{G}}\mathcal{T}_g^{(\ell)}\backslash$ $\mathcal{I}$  for some  $\mathcal{J}$  such that for every user  $u\in \mathcal{G}$ , we have  $\widetilde{\pi}_u(s')\mid \mathcal{N}^{(\ell ,i)}\equiv \widetilde{\pi}_u(s)\mid \mathcal{Y}$  for  $s' =$ $\mathrm{TB}^{-1} - \left|\mathcal{O}_{\mathcal{M}^{(\ell ,i)}}^{(\ell)}\right|$  and some common index  $s$ . In that case, we must have
$$
\max  _ {v \in \mathcal {G}} \left(\max  _ {x, y \in \mathcal {Y}} \widetilde {\mathbf {P}} _ {v x} - \widetilde {\mathbf {P}} _ {v y}\right) \leq \max  _ {u \in \mathcal {G}} \left(\widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s ^ {\prime}) | \mathcal {N} (\ell , i)}\right) + 2 4 \mathrm {C} \Delta_ {\ell + 1}
$$
Proof. Let us fix a user  $v \in \mathcal{G}$ . We have  $\max_{x, y \in \mathcal{Y}} \widetilde{\mathbf{P}}_{vx} - \widetilde{\mathbf{P}}_{vy} = \widetilde{\mathbf{P}}_{v\widetilde{\pi}_{v}(1)|\mathcal{Y}} - \min_{y \in \mathcal{Y}} \widetilde{\mathbf{P}}_{vy}$ . Now, there are two possibilities for any  $y \in \mathcal{Y}$ : first, suppose that  $y \in T_v^{(\ell)}$ . In that case, we have
$$
\widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v y} = \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s ^ {\prime}) | \mathcal {T} _ {v} ^ {(\ell)}} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s ^ {\prime}) | \mathcal {T} _ {v} ^ {(\ell)}} - \widetilde {\mathbf {P}} _ {v y}
$$
Recall that the set  $\mathcal{T}_v^{(\ell)}$  was constructed as
$$
\mathcal {T} _ {v} ^ {(\ell)} \equiv \left\{j \in \mathcal {N} ^ {(\ell , i)} \mid \widetilde {\mathbf {P}} _ {v j} \geq \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} \left(s ^ {\prime}\right) \mid \mathcal {N} ^ {(\ell , i)}} - 2 \Delta_ {\ell + 1} \right\} \text {w h e r e} s ^ {\prime} = \mathrm {T B} ^ {- 1} - \left| \mathcal {O} _ {\mathcal {M} ^ {(\ell , i)}} ^ {(\ell)} \right| \tag {14}
$$
Since  $y \in \mathcal{T}_v^{(\ell)}$ , we can bound  $\widetilde{\mathbf{P}}_{v\widetilde{\pi}_v(s')|\mathcal{T}_v^{(\ell)}} - \widetilde{\mathbf{P}}_{vy} \leq 2\Delta_{\ell +1}$ . Also, from the construction of  $\mathcal{T}_v^{(\ell)}$ ,  $\{\widetilde{\pi}_v(r) \mid \mathcal{N}^{(\ell ,i)}\}_{r = 1}^s$  is present in the set  $\mathcal{T}_v^{(\ell)}$ . Also, since  $\mathcal{T}_u^{(\ell)}$  is just a subset of  $\mathcal{N}^{(\ell ,i)}$ , the positions of the items  $\{\widetilde{\pi}_v(r) \mid \mathcal{N}^{(\ell ,i)}\}_{r = 1}^s$  do not change in the permutation corresponding to the items in  $\mathcal{T}_v^{(\ell)}$  sorted by estimated reward for  $v$  in decreasing order i.e.  $\widetilde{\pi}_v(r) \mid \mathcal{N}^{(\ell ,i)} = \widetilde{\pi}_v(r) \mid \mathcal{T}_v^{(\ell)}$  for any  $r \in [s]$ . Hence  $\widetilde{\mathbf{P}}_{v\widetilde{\pi}_v(s)|\mathcal{Y}} - \widetilde{\mathbf{P}}_{v\widetilde{\pi}_v(s')|\mathcal{T}_v^{(\ell)}} = 0$ . Hence, by combining the above, we have
$$
\widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v y} \leq \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} + 2 \Delta_ {\ell + 1}.
$$
Next, consider the case when  $y \notin \mathcal{T}_v^{(\ell)}$ . Hence, there must exist an user  $u$  such that  $y \in \mathcal{T}_u^{(\ell)} \setminus \mathcal{T}_v^{(\ell)}$  and  $u$  is connected to  $v$  via a path of length  $L \leq 2C - 1$  (since there are  $C$  clusters and users in the same cluster form a clique as proved in Lemma 7). In that case, we have the following decomposition:
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v y} = \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} \\ + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s)} | \mathcal {Y} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s ^ {\prime})} | \mathcal {T} _ {u} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s ^ {\prime})} | \mathcal {T} _ {u} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u y} + \widetilde {\mathbf {P}} _ {u y} - \widetilde {\mathbf {P}} _ {v y} \\ \end{array}
$$
Now, let us consider the terms pairwise. Due to our construction of the graph  $\mathcal{G}^{(\ell ,i)}$ , for two users  $u,v$  connected via a path of length L, we must have  $\left|\widetilde{\mathbf{P}}_{ux}^{(\ell)} - \widetilde{\mathbf{P}}_{vx}^{(\ell)}\right|\leq 2\mathsf{L}\Delta_{\ell +1}$ . Note that
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} = \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \mathbf {P} _ {u \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} \\ - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} + \mathbf {P} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} \leq 4 L \Delta_ {\ell + 1} \\ \end{array}
$$
where we used that  $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_v(1)|\mathcal{Y}}^{(\ell)} - \widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(1)|\mathcal{Y}}^{(\ell)}\leq 0$  Next, we have  $\widetilde{\mathbf{P}}_{v\widetilde{\pi}_u(1)|\mathcal{Y}} - \widetilde{\mathbf{P}}_{v\widetilde{\pi}_u(s)|\mathcal{Y}}\leq \widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(1)|\mathcal{Y}}-$ $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(s)|\mathcal{Y}} + 4L\Delta_{\ell +1}$  . Furthermore, again we have  $\widetilde{\pi}_u(r)\mid \mathcal{N}^{(\ell ,i)} = \widetilde{\pi}_u(r)\mid \mathcal{T}_u^{(\ell)}$  for any  $r\in [s]$  and therefore  $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_{u}(s)|\mathcal{Y}} - \widetilde{\mathbf{P}}_{u\widetilde{\pi}_{u}(s^{\prime})|\mathcal{T}_{v}^{(\ell)}} = 0$  Finally,  $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(s')|\mathcal{T}_u^{(\ell)}} - \widetilde{\mathbf{P}}_{uy} + \widetilde{\mathbf{P}}_{uy} - \widetilde{\mathbf{P}}_{vy}\leq 4L\Delta_{\ell +1}$  (since  $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(s')|\mathcal{T}_u^{(\ell)}} - \widetilde{\mathbf{P}}_{uy}\leq 2\Delta_{\ell +1})$  . Hence by combining, we have that
$$
\widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \mathcal {Y}} \leq \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} + 1 2 L \Delta_ {\ell + 1}.
$$
Hence, we complete the proof of the lemma (by substituting  $\mathsf{L} \leq 2\mathsf{C}$ ).
Lemma 10. Condition on the events  $\mathcal{E}_2^{(\ell)},\mathcal{E}_3^{(\ell)}$  being true. Consider a nice subset of users  $\mathcal{M}^{(\ell ,i)}\in$ $\mathcal{M}'^{(\ell)}$  and their corresponding set of active items  $\mathcal{N}^{(\ell ,i)}$  for which guarantees in eq. 11 holds. Consider two users  $u,v\in \mathcal{M}^{(\ell ,i)}$  having an edge i.e.  $\left|\widetilde{\mathbf{P}}_{ux}^{(\ell)} - \widetilde{\mathbf{P}}_{vx}^{(\ell)}\right|\leq 2\Delta_{\ell +1}$  for all  $x\in \mathcal{N}^{(\ell ,i)}$ . In that case, for any subset  $\mathcal{V}\subseteq \mathcal{N}^{(\ell ,i)}$  and any  $s\in [|\mathcal{V}|]$ , for every user  $u\in \mathcal{M}^{(\ell ,i)}$ , we must have
$$
\widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} - 1 2 \Delta_ {\ell + 1} \leq \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} \leq \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} + 1 2 \Delta_ {\ell + 1}
$$
Proof. We can decompose the term being studied in the following manner:
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} \\ = \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)}. \\ \end{array}
$$
Let us bound the quantity  $\widetilde{\mathbf{P}}_{v\widetilde{\pi}_v(s)|\mathcal{Y}}^{(\ell)} - \widetilde{\mathbf{P}}_{v\widetilde{\pi}_u(s)|\mathcal{Y}}^{(\ell)}$ . In order to analyze this quantity, we will consider a few cases. In the first case, suppose that  $\widetilde{\pi}_u(s) \mid \mathcal{Y} = \widetilde{\pi}_v(t_1) \mid \mathcal{Y}$  for  $t_1 \geq s$ . In that case, we will have that  $\widetilde{\mathbf{P}}_{v\widetilde{\pi}_v(s)|\mathcal{Y}}^{(\ell)} - \widetilde{\mathbf{P}}_{v\widetilde{\pi}_u(s)|\mathcal{Y}}^{(\ell)} \geq 0$ . In the second case, suppose  $\widetilde{\pi}_u(s) \mid \mathcal{Y} = \widetilde{\pi}_v(t_2) \mid \mathcal{Y}$  for  $t_2 < s$  and  $\widetilde{\pi}_u(t_3) \mid \mathcal{Y} = \widetilde{\pi}_v(s) \mid \mathcal{Y}$  for  $t_3 < s$ . In that case, we have
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} \\ = \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} \geq - 4 \Delta_ {\ell + 1} \\ \end{array}
$$
where we used the fact  $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_v(s)|\mathcal{Y}}^{(\ell)} - \widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(s)|\mathcal{Y}}^{(\ell)} = \widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(t_3)|\mathcal{Y}}^{(\ell)} - \widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(s)|\mathcal{Y}}^{(\ell)}\geq 0$  In the final case, we assume that  $\widetilde{\pi}_u(s)\mid \mathcal{V} = \widetilde{\pi}_v(t_2)\mid \mathcal{V}$  for  $s > t_2$  and  $\widetilde{\pi}_u(t_3)\mid \mathcal{V} = \widetilde{\pi}_v(s)\mid \mathcal{V}$  for  $t_3 > s$  . This means that both the items  $\widetilde{\pi}_u(s)\mid \mathcal{V},\widetilde{\pi}_u(t_3)\mid \mathcal{V}$  have been shifted to the left in the permutation  $\widetilde{\pi}_v\mid \mathcal{V}$  Hence,
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} \\ = \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {v} (s) | \mathcal {Y}} + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {v} (s) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} \\ + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} \geq - 4 \Delta_ {\ell + 1} + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {v} (s) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} \\ = - 4 \Delta_ {\ell + 1} + \tilde {\mathbf {P}} _ {u \tilde {\pi} _ {u} (t _ {3}) | \mathcal {Y}} - \tilde {\mathbf {P}} _ {u \tilde {\pi} _ {u} (s) | \mathcal {Y}}. \\ \end{array}
$$
Hence there must exist an element  $\widetilde{\pi}_u(t_4) \mid \mathcal{V}$  such that  $t_4 < s$  and  $\widetilde{\pi}_v(t_5) \mid \mathcal{V} = \widetilde{\pi}_u(t_4) \mid \mathcal{V}$  for  $t_5 > s$ . In that case, we must have
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (t _ {3}) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} \geq \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (t _ {3}) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (t _ {4}) | \mathcal {Y}} \\ = \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (t _ {3}) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (t _ {3}) | \mathcal {Y}} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (t _ {3}) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (t _ {4}) | \mathcal {Y}} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (t _ {4}) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (t _ {4}) | \mathcal {Y}} \\ \geq - 4 \Delta_ {\ell + 1} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (t _ {3}) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (t _ {4}) | \mathcal {Y}} ^ {(\ell)} = - 4 \Delta_ {\ell + 1} + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (t _ {5}) | \mathcal {Y}} ^ {(\ell)} \geq - 4 \Delta_ {\ell + 1}. \\ \end{array}
$$
Therefore, in this case, we get that  $\widetilde{\mathbf{P}}_{v\widetilde{\pi}_v(s)}^{(\ell)} - \widetilde{\mathbf{P}}_{v\pi_u(s)}^{(\ell)} \geq -8\Delta_{\ell + 1}$ . Hence we get
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} = \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} \\ + \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} \geq \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - 8 \Delta_ {\ell + 1}. \\ \end{array}
$$
Also, we will have that
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} = \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \mathbf {P} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} \\ + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} + \mathbf {P} _ {u \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} \geq - 4 \Delta_ {\ell + 1}. \\ \end{array}
$$
$$
\widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} \geq \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} - 4 \Delta_ {\ell + 1} \geq \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} - 1 2 \Delta_ {\ell + 1}.
$$
$$
\widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (s) | \mathcal {Y}} ^ {(\ell)} \leq \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (1) | \mathcal {Y}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {v \widetilde {\pi} _ {v} (s) | \mathcal {Y}} ^ {(\ell)} + 1 2 \Delta_ {\ell + 1}.
$$
Now, for any fixed subset  $\mathcal{Y} \subseteq \mathcal{N}^{(\ell, i)}$ , let us define the set  $\mathcal{R}_u^{(\ell)} \mid \mathcal{Y}$  for user  $u$  below:
$$
\mathcal {R} _ {u} ^ {(\ell)} \mid \mathcal {Y} = \{j \in \mathcal {Y} \mid \widetilde {\mathbf {P}} _ {u j} \geq \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} - 2 \Delta_ {\ell + 1} \}
$$
Hence,  $\mathcal{R}_u^{(\ell)}\mid \mathcal{V}$  corresponds to the set of items for user  $u$  that is close to the item with the highest estimated reward for user  $u$  restricted to the set  $\mathcal{V}$  at the end of the explore component of phase  $\ell$ .
Lemma 11. Condition on the events  $\mathcal{E}_2^{(\ell)},\mathcal{E}_3^{(\ell)}$  being true. Consider a nice subset of users  $\mathcal{M}^{(\ell ,i)}\in$ $\mathcal{M}'(\ell)$  and their corresponding set of active items  $\mathcal{N}^{(\ell ,i)}$  for which guarantees in eq. 11 holds. Fix any subset  $\mathcal{V}\subseteq \mathcal{N}^{(\ell ,i)}$ . In that case, for every user  $u\in \mathcal{M}^{(\ell ,i)}$ , the item with the highest reward  $\pi_u(1)\mid \mathcal{V}$  in the set  $\mathcal{V}$  must belong to the set  $\mathcal{R}_u^{(\ell)}\mid \mathcal{V}$ . Moreover,  $\max_{s,s'\in \mathcal{R}_u^{(\ell)}|\mathcal{V}}|\mathbf{P}_{us} - \mathbf{P}_{us'}|\leq 4\Delta_{\ell +1}$ .
Proof. Let us fix a user  $u \in \mathcal{M}^{(\ell, i)}$  with active set of arms  $\mathcal{N}^{(\ell, i)}$ . Now, we will have
$$
\begin{array}{l} \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \pi_ {u} (1) | \mathcal {Y}} ^ {(\ell)} = \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} ^ {(\ell)} \\ - \mathbf {P} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} + \mathbf {P} _ {u \widetilde {\pi} _ {u} (1) | \mathcal {Y}} - \mathbf {P} _ {u \pi_ {u} (1) | \mathcal {Y}} + \mathbf {P} _ {u \pi_ {u} (1) | \mathcal {Y}} - \widetilde {\mathbf {P}} _ {u \pi_ {u} (1) | \mathcal {Y}} ^ {(\ell)} \leq 2 \Delta_ {\ell + 1} \\ \end{array}
$$
which implies that  $\pi_u(1) \mid \mathcal{Y} \in \mathcal{R}_u^{(\ell)} \mid \mathcal{Y}$ . Here we used the fact that  $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(1)|\mathcal{Y}}^{(\ell)} - \mathbf{P}_{u\widetilde{\pi}_u(1)|\mathcal{Y}} \leq \Delta_{\ell + 1}$ ,  $\mathbf{P}_{u|\mathcal{Y}} - \widetilde{\mathbf{P}}_{u\pi_u(1)|\mathcal{Y}}^{(\ell)} \leq \Delta_{\ell + 1}$  and  $\mathbf{P}_{u\widetilde{\pi}_u(1)|\mathcal{Y}} - \mathbf{P}_{u\pi_u(1)|\mathcal{Y}} \leq 0$ . Next, notice that for any  $s, s' \in \mathcal{R}_u^{(\ell)} \mid \mathcal{Y}$
$$
\mathbf {P} _ {u s} - \mathbf {P} _ {u s ^ {\prime}} = \mathbf {P} _ {u s} - \widetilde {\mathbf {P}} _ {u s} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {u s} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u t _ {1}} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {u t _ {1}} ^ {(\ell)} - \widetilde {\mathbf {P}} _ {u s ^ {\prime}} ^ {(\ell)} + \widetilde {\mathbf {P}} _ {u s ^ {\prime}} ^ {(\ell)} - \mathbf {P} _ {u s ^ {\prime}} \leq 4 \Delta_ {\ell + 1}.
$$
Lemma 12. Condition on the events  $\mathcal{E}_2^{(\ell)},\mathcal{E}_3^{(\ell)}$  being true. Consider a nice subset of users  $\mathcal{M}^{(\ell ,i)}\in$ $\mathcal{M}'(\ell)$  and their corresponding set of active items  $\mathcal{N}^{(\ell ,i)}$  for which guarantees in eq. 11 holds. Fix any subset  $\mathcal{V}\subseteq \mathcal{N}^{(\ell ,i)}$  . Consider two users  $u,v\in \mathcal{M}^{(\ell ,i)}$  having an edge in the graph  $\mathcal{G}^{(\ell ,i)}$  Conditioned on the events  $\mathcal{E}_2^{(\ell)},\mathcal{E}_3^{(\ell)}$  , we must have
$$
\max_{x\in \mathcal{R}_{u}^{(\ell)}|\mathcal{Y},y\in \mathcal{R}_{v}^{(\ell)}|\mathcal{Y}}\left|\mathbf{P}_{ux} - \mathbf{P}_{uy}\right|\leq 16\Delta_{\ell +1}  and\max_{x\in \mathcal{R}_{u}^{(\ell)}|\mathcal{Y},y\in \mathcal{R}_{v}^{(\ell)}|\mathcal{Y}}\left|\mathbf{P}_{vx} - \mathbf{P}_{vy}\right|\leq 16\Delta_{\ell +1}
$$
Proof. From the construction of  $\mathcal{G}^{(\ell, i)}$ , we know that users  $u, v \in \mathcal{M}^{(\ell, i)}$  have an edge if  $\left|\widetilde{\mathbf{P}}_{ux}^{(\ell)} - \widetilde{\mathbf{P}}_{vx}^{(\ell)}\right| \leq 2\Delta_{\ell+1}$  (and therefore  $\left|\mathbf{P}_{ux} - \mathbf{P}_{vx}\right| \leq \left|\widetilde{\mathbf{P}}_{ux}^{(\ell)} - \mathbf{P}_{ux}\right| + \left|\mathbf{P}_{vx} - \widetilde{\mathbf{P}}_{vx}^{(\ell)}\right| + \left|\widetilde{\mathbf{P}}_{ux}^{(\ell)} - \widetilde{\mathbf{P}}_{vx}^{(\ell)}\right| \leq 4\Delta_{\ell+1}$ ) for all  $x \in \mathcal{N}^{(\ell, i)}$ . For simplicity of notation, let us denote  $\mathcal{R}_u$  to
be the set  $\mathcal{R}_u^{(\ell)}\mid \mathcal{Y}$ . Suppose  $\mathbf{P}_{u\pi_u(1)|\mathcal{Y}} = a$  and  $\mathbf{P}_{v\pi_v(1)|\mathcal{Y}} = b$ . Consider any pair of items  $x\in \mathcal{R}_u,y\in \mathcal{R}_v$  respectively. Therefore, for  $x\in \mathcal{R}_u$  this must mean that  $\mathbf{P}_{ux}\geq a - 4\Delta_{\ell +1}$  (note that  $\pi_u(1)\mid \mathcal{V}\in \mathcal{R}_u$ ). For  $y\in \mathcal{R}_v$ , we must similarly have  $\mathbf{P}_{vy}\geq b - 4\Delta_{\ell +1}$ . Since  $u,v$  are connected by an edge, we must have  $\mathbf{P}_{vx}\geq a - 8\Delta_{\ell +1}$  and  $\mathbf{P}_{uy}\geq b - 8\Delta_{\ell +1}$ . Now, we have  $\mathbf{P}_{v\pi_v(1)|\mathcal{Y}}\geq \mathbf{P}_{vx}$  implying that  $b\geq a - 8\Delta_{\ell +1}$ ; hence
$$
\mathbf {P} _ {u x} - \mathbf {P} _ {u y} \leq \mathbf {P} _ {u \pi_ {u} (1) | \mathcal {R} _ {u}} - \mathbf {P} _ {u y} \leq a - (b - 8 \Delta_ {\ell + 1}) \leq 1 6 \Delta_ {\ell + 1}.
$$
A similar analysis for  $v$  shows that  $\mathbf{P}_{vy} - \mathbf{P}_{vx} \leq 16\Delta_{\ell + 1}$ . This completes the proof of the lemma.
Lemma 13. Condition on the events  $\mathcal{E}_2^{(\ell)},\mathcal{E}_3^{(\ell)}$  being true. Consider a nice subset of users  $\mathcal{M}^{(\ell ,i)}\in$ $\mathcal{M}'(\ell)$  and their corresponding set of active items  $\mathcal{N}^{(\ell ,i)}$  for which guarantees in eq. 11 holds. Fix any subset  $\mathcal{V}\subseteq \mathcal{N}^{(\ell ,i)}$  . Consider two users  $u,v\in \mathcal{M}^{(\ell ,i)}$  having a path of length L in the graph  $\mathcal{G}^{(\ell ,i)}$  . Conditioned on the events  $\mathcal{E}_2^{(\ell)},\mathcal{E}_3^{(\ell)}$  , we must have
$$
\max _ {x \in \mathcal {R} _ {u} ^ {(\ell)} | \mathcal {Y}, y \in \mathcal {R} _ {v} ^ {(\ell)} | \mathcal {Y}} \left| \mathbf {P} _ {u x} - \mathbf {P} _ {u y} \right| \leq 8 (\mathsf {L} + 1) \Delta_ {\ell + 1}
$$
$$
a n d \max  _ {x \in \mathcal {R} _ {u} ^ {(\ell)} | \mathcal {Y}, y \in \mathcal {R} _ {v} ^ {(\ell)} | \mathcal {Y}} | \mathbf {P} _ {v x} - \mathbf {P} _ {v y} | \leq 8 (\mathrm {L} + 1) \Delta_ {\ell + 1}
$$