SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Proof Sketch of Theorem 1 We condition on the high probability event that the low rank matrix completion estimation step is always successful (Lemma 17). Note that for a fixed user, the items chosen for recommendation in the exploit component of any phase are golden items and costs zero regret if they are unblocked and recommended. Even if it is blocked, we show a swapping argument to a similar effect. That is, with an appropriate permutation of the recommended items, we can ignore the regret incurred from golden-items altogether. Moreover, in the explore component of the  $\ell^{\mathrm{th}}$  phase, we can bound the sub-optimality gap of the surviving active items by some pre-determined  $\epsilon_{\ell}$ . We prove that this holds even with the (chosen) permutation of the recommended items (Lemma 18). We choose  $\epsilon_{\ell}$  to be exponentially decreasing in  $\ell$  and the number of rounds in the explore component of phase  $\ell$  is roughly  $1 / \epsilon_{\ell}^{2}$ . Putting these together, we obtain the regret guarantee in Theorem 1.
We study the problem of Collaborative Bandits in the setting where each item can be recommended to a user a small number of times. Under some standard assumptions and a blocking constraint of  $\Theta (\log T)$ , we show a phased algorithm B-LATTICE with regret guarantees that match the tight results with no blocking constraint [25]. To the best of our knowledge, this is the first regret guarantee for such a general problem with no assumption of item clusters. We also provide novel regret lower bounds that match the upper bound in several regimes. Relaxing the assumptions, extending guarantees to a blocking constraint of  $B = 1$  and tightening the gap between the regret upper and lower bounds are very interesting directions for future work.
We would like to thank Sandeep Juneja for helpful discussions on understanding the information theoretic limits of the problem.
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Limitations: The main contributions of our works are theoretical. From a theoretical point of view, the limitations of our paper are discussed in Sections 3 and 5. In particular, we believe that tightening the gap between the upper and lower bounds in regret will require novel and non-trivial algorithmic ideas - we leave this as an important direction of future work.
Broader Impact: Due to the theoretical nature of the work, we do not foresee any adverse societal impact.
A Low rank matrix completion
Below, we describe the Estimate sub-routine to estimate a  $\mathsf{M} \times \mathsf{N}$  low rank matrix  $\mathbf{Q}$  of rank  $r$  given a partially observed set of noisy entries  $\{\mathbf{Z}_{ij}\}_{(i,j) \in \Omega}$  corresponding to a subset  $\Omega \subseteq [\mathsf{M}] \times [\mathsf{N}]$ . Here  $\mathbb{E}\mathbf{Z}_{ij} = \mathbf{Q}_{ij}$  for all  $(i,j) \in \Omega$  and furthermore,  $\{\mathbf{Z}_{ij}\}_{(i,j) \in \Omega}$  are independent sub-gaussian random variables with variance proxy at most  $\sigma^2$ .
Algorithm 4 Estimate (Low-rank matrix completion) [17]
Require: Matrix dimensions  $(\mathsf{M},\mathsf{N})$ , noise variance  $\sigma^2$ , rank  $r$ , subset of indices that are observed  $\Omega \subseteq [\mathsf{M}] \times [\mathsf{N}]$  and noisy observations  $\{\mathbf{Z}_{ij}\}_{(i,j) \in \Omega}$ .
1: Partition the rectangular matrix into square matrices. Without loss of generality, assume  $\mathsf{M} \leq \mathsf{N}$ . For each  $i \in [\mathsf{N}]$ , randomly set  $\zeta_i$  to be a value in the set  $[\lceil \mathsf{N} / \mathsf{M} \rceil]$  uniformly at random. Partition indices in  $[\mathsf{N}]$  into  $[\mathsf{N}]^{(1)}, [\mathsf{N}]^{(2)}, \ldots, [\mathsf{N}]^{(k)}$  where  $k = \lceil \mathsf{N} / \mathsf{M} \rceil$  and  $[\mathsf{N}]^{(q)} = \{i \in [\mathsf{N}] \mid \zeta_i = q\}$  for each  $q \in [k]$ . Set  $\Omega^{(q)} \gets \Omega \cap ([\mathsf{M} \times [\mathsf{N}]^{(q)})$  for all  $q \in [k]$ .
2: for  $q\in [k]$  do
3: Solve the following convex program with  $\lambda = C_{\lambda}\sigma \sqrt{|\Omega| / \max(M,N)}$ , for some constant  $C_{\lambda} > 0$
$$
\min  _ {\widetilde {\mathbf {Q}} ^ {(q)} \in \mathbb {R} ^ {M \times | [ N ] ^ {(q)} |}} \sum_ {(i, j) \in \Omega^ {(q)}} \frac {\left(\widetilde {\mathbf {Q}} _ {i \gamma_ {u} (j)} ^ {(q)} - \mathbf {Z} _ {i j}\right) ^ {2}}{2} + \lambda \| \widetilde {\mathbf {Q}} ^ {(q)} \| _ {\star}
$$
where  $\| \widetilde{\mathbf{Q}}^{(q)}\|_{\star}$  denotes nuclear norm of matrix  $\widetilde{\mathbf{Q}}^{(q)}$  and  $\gamma_u(j)$  is index of  $j$  in set  $[\mathsf{N}]^{(q)}$ .
4: end for
5: Return  $\widetilde{\mathbf{Q}}\in \mathbb{R}^{M\times N}$  such that  $\widetilde{\mathbf{Q}}_{[M],[N](q)} = \widetilde{\mathbf{Q}}^{(q)}$  for all  $q\in [k]$ .
B Explore-Then-Commit (ETC)
We first present a greedy algorithm in the blocked setting with  $\mathsf{B} = 1$  (no repetition) that uses the Explore-Then-Commit (ETC) framework. Such an algorithm has two disjoint phases - exploration and exploitation. We will first jointly explore the set of items (without repeating same item for any user) for all users for a certain number of rounds and compute an estimate  $\widetilde{\mathbf{P}}$  of the reward matrix  $\mathbf{P}$ . Subsequently, in the exploitation phase, for each user, we recommend the best estimated distinct items (that have not been recommended in the exploration phase to that user) inferred from the estimated reward matrix  $\widetilde{\mathbf{P}}$ . Note that if we explore too less, then our estimate will be poor and hence we will suffer large regret once we commit in the exploitation phase. On the other hand, if we explore too much, then the exploration cost will be high. Our goal is to balance both the exploration length and the exploitation cost under the blocked setting. Thus, we obtain the following result:
Theorem 3. Consider the GBB setting with  $\mathsf{M}$  users,  $\mathsf{C} = O(1)$  clusters,  $\mathsf{N}$  items,  $\mathsf{T}$  recommendation rounds and blocking constraint  $\mathsf{B} = 1$ . Set  $d_{2} = \min (\mathsf{M},\mathsf{N})$ . Let  $\mathbf{R}_{u\rho_u(t)}^{(t)}$  be the reward in each round, defined as in (1). Suppose  $d_{2} = \Omega (\mu r\log (rd_{2}))$ . Let  $\mathbf{P}\in \mathbb{R}^{\mathsf{M}\times \mathsf{N}}$  be the expected reward matrix that satisfies the conditions stated in Lemma 1, and let  $\sigma^2$  be the noise variance in rewards. Then, Algorithm 5, applied to the online rank- $r$  matrix completion problem under the blocked setting guarantees the regret defined as in eq. 2 to be:
$$
\operatorname {R e g} (T) = \widetilde {O} \left(\mu T ^ {2 / 3} \| \mathbf {P} \| _ {\infty} ^ {1 / 3} \max  \left(1, \frac {N}{M}\right) ^ {1 / 3} + \mu^ {2} \| \mathbf {P} \| _ {\infty} \max  \left(1, \frac {N}{M}\right) + \| \mathbf {P} \| _ {\infty} T ^ {- 2}\right). \tag {6}
$$
In order to understand the result, note that the second term in the regret bound stems from the fact that in our algorithmic framework, the low rank matrix completion module needs a certain number of
Algorithm 5 ETC (Explore-Then-Commit Algorithm with Blocking constraint  $\mathsf{B} = 1$ )
Require: users M, items N, rounds T, noise  $\sigma^2$ , rank  $r$  of P.
1: Set  $p = \widetilde{O}\Big((\mathsf{N}\| \mathbf{P}\|_{\infty})^{-2 / 3}\Big(\frac{\mathsf{T}\sigma r}{\sqrt{d_2}}\sqrt{\mu^3}\Big)^{2 / 3}\bigvee \frac{\mu^2}{d_2}\Big)$ . Set  $d_{2} = \min (\mathsf{M},\mathsf{N})$  and  $\lambda = C\sigma \sqrt{d_2p}$  for constant  $C$ .
2: For each tuple of indices  $(i,j) \in [\mathsf{M}] \times [\mathsf{N}]$ , independently set  $\delta_{ij} = 1$  with probability  $p$  and  $\delta_{ij} = 0$  with probability  $1 - p$ .
3: Denote  $\Omega = \{(i,j)\in [\mathsf{M}]\times [\mathsf{N}]\mid \delta_{ij} = 1\}$  and  $m = \max_{i\in [\mathsf{M}]}\mid |j\in [\mathsf{N}]\mid (i,j)\in \Omega |$  to be the maximum number of index tuples in a particular row. For all  $(i,j)\in \Omega$  , set  $\mathsf{Mask}_{ij} = 0$
4: for rounds  $t = 1,2,\ldots ,m$  do
5: For each user  $u \in \mathcal{U}$ , recommend an item  $\rho_u(t)$  in  $\{j \in [\mathbb{N}] \mid (u, j) \in \Omega, \mathsf{Mask}_{uj} = 0\}$  and set  $\mathsf{Mask}_{u\rho_u(t)} = 1$ . If not possible then recommend any item  $\rho_u(t)$  in  $[\mathbb{N}]$  s.t.  $(u, \rho_u(t)) \notin \Omega$  and has not been recommended yet to user  $u$ . Observe  $\mathbf{R}_{u\rho_u(t)}^{(t)}$ .
6: end for
7: Compute the estimate  $\widetilde{\mathbf{P}}\in \mathbb{R}^{M\times N}$  as output of  $\widetilde{\mathbf{P}}$  ← Estimate([M],[N],  $\sigma^2,r,\Omega ,\{\mathbf{R}_{u\rho_u(t)}^{(t)}\}_{t\in [m]}$
8: for each of remaining rounds do
9: Set  $j_u' = \arg \max_{j \in [\mathbb{N}]} \widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(j)}$  for each user  $u$ , s.t.  $\widetilde{\pi}_u(j_u')$  has not been recommended before to  $u$ .
10: For each user  $u \in [\mathsf{M}]$ , recommend the item  $\widetilde{\pi}_u(j_u')$  to user  $u$ .
11: end for
observed indices and therefore a certain number of exploration rounds (note that  $p \geq C\mu^2 d_2^{-1}\log^3 d_2$  for some constant  $C > 0$  in Lemma 1). Similarly, the third term stems from the failure of the estimation module; again, the term  $T^{-2}$  can be replaced by  $T^{-c}$  for any constant  $c > 0$ . The first term in the regret bound captures the dependence on the number of rounds  $T$  - the scaling of  $T^{2/3}$  is sub-optimal and our subsequent goal is to improve this dependence to the rate of  $\sqrt{T}$ .
Proof of Theorem 3. Suppose we explore for a period of  $S$  rounds such that the exploration period succeeds with a probability of  $1 - \nu$ . Conditioned on the event that the exploration period succeeds, we obtain an estimate  $\widetilde{\mathbf{P}}$  of the reward matrix  $\mathbf{P}$  satisfying  $\| \mathbf{P} - \widetilde{\mathbf{P}} \|_{\infty} \leq \rho$ . Recall  $\pi_u : [\mathbb{N}] \to [\mathbb{N}]$  to be the permutation on  $[\mathbb{N}]$  such that for any  $i, j \in [\mathbb{N}]$ ;  $i < j$ , we have  $\mathbf{P}_{u\pi_u(i)} \geq \mathbf{P}_{u\pi_u(j)}$ . Similarly, denote  $\widetilde{\pi}_u : [\mathbb{N}] \to [\mathbb{N}]$  such that for any  $i, j \in [\mathbb{N}]$ ;  $i < j$ , we have  $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(i)} \geq \widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(j)}$ . Now consider any index  $i \in [\mathsf{T}]$  for which we will analyze  $\mathbf{P}_{u\widetilde{\pi}_u(i)} - \mathbf{P}_{u\pi_u(i)}$  which is the error if we choose the  $i^{\text{th}}$  item according to the estimated matrix  $\widetilde{\mathbf{P}}$  (instead of  $\mathbf{P}$ ). There are several cases that we need to consider. First, suppose  $\widetilde{\pi}_u(i) = \pi_u(j)$  where  $j \leq i$ . In that case, we have  $\mathbf{P}_{u\widetilde{\pi}_u(i)} - \mathbf{P}_{u\pi_u(i)} \geq 0$ . Now, consider the other case where  $j > i$  implying that the element in the  $j^{\text{th}}$  position in the permutation  $\pi_u$  has shifted to the left in  $\widetilde{\pi}_u$ . In order for this to happen, there must exist an element  $i_1 \leq i \leq i_2$  for which  $\widetilde{\pi}_u(i_2) = \pi_u(i_1)$  implying that an element  $i_1$  in the permutation  $\pi_u$  has shifted to the right in  $\widetilde{\pi}_u$ . Therefore,
$$
\begin{array}{l} \mathbf {P} _ {u \widetilde {\pi} _ {u} (i)} - \mathbf {P} _ {u \pi_ {u} (i)} = \mathbf {P} _ {u \widetilde {\pi} _ {u} (i)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (i)} + \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (i)} - \widetilde {\mathbf {P}} _ {u \widetilde {\pi} _ {u} (i _ {2})} \\ + \mathbf {P} _ {u \tilde {\pi} _ {u} (i _ {2})} - \mathbf {P} _ {u \tilde {\pi} _ {u} (i _ {2})} + \mathbf {P} _ {u \tilde {\pi} _ {u} (i _ {2})} - \mathbf {P} _ {u \pi_ {u} (i)} \geq - 2 \rho \\ \end{array}
$$
where we used the fact that  $\| \mathbf{P} - \widetilde{\mathbf{P}} \|_{\infty} \leq \rho$ ,  $\widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(i)} - \widetilde{\mathbf{P}}_{u\widetilde{\pi}_u(i_2)} \geq 0$  (since  $i \leq i_2$ ),  $\mathbf{P}_{u\widetilde{\pi}_u(i_2)} - \mathbf{P}_{u\pi_u(i)} = \mathbf{P}_{u\pi_u(i_1)} - \mathbf{P}_{u\pi_u(i)} \geq 0$  (since  $i_1 \leq i$ ). Therefore, at each step of the exploitation stage, for each user  $u$ , we recommend one of the top  $\mathsf{T} - \mathsf{S}$  items (as inferred from  $\widetilde{\mathbf{P}}$ ) with the highest reward that have not been recommended until that round to the user  $u$ ; in each such step, we will suffer a regret of at most  $2\rho$  if we compare with the item at the same index in  $\pi_u$ . As before, conditioned on the event that the exploration fails (and in the exploration stage as well), the regret at each step can be
bounded from above by  $2\| \mathbf{P}\|_{\infty}$ . In that case, we have
$$
\begin{array}{l} \operatorname {R e g} _ {\Pi} (T) = \frac {1}{M} \sum_ {u \in [ M ]} \sum_ {t \in [ T ]} \mathbf {P} _ {u \pi_ {u} (t)} - \sum_ {t \in [ T ]} \mathbf {P} _ {u \rho_ {u} (t)} \leq \max  _ {u \in [ M ]} \left(\sum_ {t \in [ T ]} \mathbf {P} _ {u \pi_ {u} (t)} - \sum_ {t \in [ T ]} \mathbf {P} _ {u \widetilde {\pi} _ {u} (t)}\right) \\ \leq 2 \mathbf {S} \| \mathbf {P} \| _ {\infty} + 2 (\mathsf {T} - \mathsf {S}) \rho \Pr (\text {E x p l o r a t i o n s u c e e d s}) + 2 (\mathsf {T} - \mathsf {S}) \| \mathbf {P} \| _ {\infty} \Pr (\text {E x p l o r a t i o n f a i l s}) \\ \leq 2 \mathsf {S} \| \mathbf {P} \| _ {\infty} + 2 \mathsf {T} \rho + 2 \mathsf {T} \delta \| \mathbf {P} \| _ {\infty}. \\ \end{array}
$$
We use Lemma 1 to set  $S = O\left(\mathsf{N}p + \sqrt{\mathsf{N}p\log\mathsf{M}\delta^{-1}}\right)$  such that  $\rho = O\left(\frac{\sigma r}{\sqrt{d_2}}\sqrt{\frac{\mu^3\log d_2}{p}}\right)$  and  $\nu = 1 - \delta - O(d_2^{-3})$ . We have
$$
\operatorname {R e g} (T) \leq O \left(\mathsf {N} p + \sqrt {\mathsf {N} p \log \mathsf {M} \delta^ {- 1}}\right) \| \mathbf {P} \| _ {\infty} + T O \left(\frac {\sigma r}{\sqrt {d _ {2}}} \sqrt {\frac {\mu^ {3} \log d _ {2}}{p}}\right) + T (\delta + d _ {2} ^ {- 3}) \| \mathbf {P} \| _ {\infty}.
$$
For simplicity, we ignore the logarithmic and lower order terms and attempt to minimize  $\mathsf{N}p\| \mathbf{P}\|_{\infty} + \mathsf{T}\left(\frac{\sigma r}{\sqrt{d_2}}\sqrt{\frac{\mu^3\log d_2}{p}}\right)$  (Simplified Expression) by choosing  $p = (\mathsf{N}\| \mathbf{P}\|_{\infty})^{-2 / 3}\left(\frac{\mathsf{T}\sigma r}{\sqrt{d_2}}\sqrt{\mu^3\log d_1}\right)^{2 / 3},$ $\delta = \mathsf{T}^{-4}$ . If  $p\geq C\mu^{2}d_{2}^{-1}\log^{3}d_{2}$ , then notice that  $\mathsf{N}p\geq 1$  and therefore  $\mathsf{N}p + \sqrt{\mathsf{N}p\log\mathsf{M}\delta^{-1}} = O(\mathsf{N}p\sqrt{\log\mathsf{M}\delta^{-1}})$ . Subsequently, we have
$$
\operatorname {R e g} (T) = O \left(T ^ {2 / 3} \left(\sigma^ {2} r ^ {2} \| \mathbf {P} \| _ {\infty}\right) ^ {1 / 3} \left(\frac {\mu^ {3} N \log d _ {2}}{d _ {2}}\right) ^ {1 / 3} \log \sqrt {M T} + \| \mathbf {P} \| _ {\infty} T ^ {- 2}\right).
$$
There exists an edge case when the value of  $p$  that minimizes the simplified expression satisfies  $p \leq C\mu^2 d_2^{-1}\log^3 d_2$ . Then we can substitute  $p = C\mu^2 d_2^{-1}\log^3 d_2$ . In that case, the second term in the simplified expression will still be bounded as before. On the other hand the first term in the simplified expression will now be bounded by  $O\left(\frac{\mathsf{N}\mu^2}{d_2}\log^3 d_2\log^2 (\mathsf{MNT})||\mathbf{P}||_{\infty}\right)$ . Hence, our regret will be bounded by
$$
\begin{array}{l} \operatorname {R e g} (T) = O \left(T ^ {2 / 3} \left(\sigma^ {2} r ^ {2} \| \mathbf {P} \| _ {\infty}\right) ^ {1 / 3} \left(\frac {\mu^ {3} N \log d _ {2}}{d _ {2}}\right) ^ {1 / 3} \log \sqrt {M T} + \frac {N \mu^ {2}}{d _ {2}} \log^ {5} (M N T) \| \mathbf {P} \| _ {\infty} + \| \mathbf {P} \| _ {\infty} T ^ {- 2}\right) \\ = \widetilde {O} \left(\mu \mathsf {T} ^ {2 / 3} | | \mathbf {P} | | _ {\infty} ^ {1 / 3} \max \left(1, \frac {\mathsf {N}}{\mathsf {M}}\right) ^ {1 / 3} + \mu^ {2} | | \mathbf {P} | | _ {\infty} \max \left(1, \frac {\mathsf {N}}{\mathsf {M}}\right) + | | \mathbf {P} | | _ {\infty} \mathsf {T} ^ {- 2}\right). \\ \end{array}
$$
We conduct detailed synthetic experiments in order to validate the theoretical guarantees/properties of our algorithm. For this purpose we use a simplified version of B-LATTICE described in Alg. 6 (all experiments have been performed on a Google Colab instance with 12GB RAM):
In PB-LATTICE(Alg. 6), we do not use the exploit component for simplicity. Intuitively, if the user is recommended a golden item during the explore component, it is a good event in any case. Furthermore, in Step 14, we use  $k$ -means to cluster the users. Since we obtain a significantly large embedding vector for each user in Step 13,  $k$ -means is quite practical.
We run PB-LATTICEon several synthetic datasets. There are two main baselines for us to consider 1) Greedy Algorithm namely Alg. 5 2) In the setting where the user  $u$ , on being recommended item  $j$  provides a like (+1) with probability  $\mathbf{P}_{uj}$  and a dislike (-1) with probability  $1 - \mathbf{P}_{uj}$ , we also compare with Collaborative-Greedy in [4]. However, recall that [4], even in the restricted setting, only provides theoretical guarantees on the number of likeable items (items with probability of liking > 0.5) recommended during the course of T rounds.
We generate three synthetic datasets to validate our algorithms. For each of them, we take  $M = 150$  users,  $N = 150$  items and  $T = 60$  rounds (hence the total items recommended will be 9000). The reward matrix  $\mathbf{P} = \mathbf{U}\mathbf{V}^{\mathrm{T}}$  ( $\mathbf{U} \in \mathbb{R}^{M \times C}$  and  $\mathbf{V} \in \mathbb{R}^{N \times C}$ ) is generated as in [25] - in the  $i^{\text{th}}$  row of  $\mathbf{U}$ , the  $(i\% C)^{\text{th}}$  entry is set to be 1 while the other entries are 0; the entries of  $\mathbf{V}$  are sampled in the following way for the three datasets.
Algorithm 6 PB-LATTICE (Practical blocked Latent bAndiTs via maTrIx ComplEtion with blocking constraint  $\mathsf{B} = 1$ )
Require: Phase index  $\ell$ , phase length  $m_{\ell}$ , List of disjoint nice subsets of users  $\mathcal{M}^{(\ell)}$ , list of corresponding subsets of active items  $\mathcal{N}^{(\ell)}$ , clusters  $\mathsf{C}$ , rounds  $\mathsf{T}$ , noise  $\sigma^2 > 0$ , round index  $t_0$ , gap factor  $\nu_{\ell}$ .
1: Set  $\lambda = 10\sigma \sqrt{(m_{\ell} / \mathsf{M})}$
2: for rounds  $t_0 + 1, t_0 + 2, \ldots, \min(t_0 + m, \mathsf{T})$  do
3: for  $i^{\mathrm{th}}$  nice subset of users  $\mathcal{M}^{(\ell ,i)}\in \mathcal{M}^{(\ell)}$  with active items  $\mathcal{N}^{(\ell ,i)}$  ( $i^{\mathrm{th}}$  set in list  $\mathcal{N}^{(\ell)}$ ) do
4: Initialize  $\Omega^{(i)} = \Phi$
5: for user  $u$  in  $\mathcal{M}^{(\ell ,i)}$  do