Online Matrix Completion: A Collaborative Approach with Hott Items

We investigate the low rank matrix completion problem in an online setting with ${M}$ users, ${N}$ items, ${T}$ rounds, and an unknown rank-$r$ reward matrix ${R}\in \mathbb{R}^{{M}\times {N}}$. This problem has been well-studied in the literature and has several applications in practice. In each round, we recommend ${S}$ carefully chosen distinct items to every user and observe noisy rewards. In the regime where ${M},{N} >> {T}$, we propose two distinct computationally efficient algorithms for recommending items to users and analyze them under the benign \emph{hott items} assumption.1) First, for ${S}=1$, under additional incoherence/smoothness assumptions on ${R}$, we propose the phased algorithm \textsc{PhasedClusterElim}. Our algorithm obtains a near-optimal per-user regret of $\tilde{O}({N}{M}^{-1}(Δ^{-1}+Δ_{hott}^{-2}))$ where $Δ_{hott},Δ$ are problem-dependent gap parameters with $Δ_{hott} >> Δ$ almost always. 2) Second, we consider a simplified setting with ${S}=r$ where we make significantly milder assumptions on ${R}$. Here, we introduce another phased algorithm, \textsc{DeterminantElim}, to derive a regret guarantee of $\widetilde{O}({N}{M}^{-1/r}Δ_{det}^{-1}))$ where $Δ_{det}$ is another problem-dependent gap. Both algorithms crucially use collaboration among users to jointly eliminate sub-optimal items for groups of users successively in phases, but with distinctive and novel approaches.