Online Matrix Completion: A Collaborative Approach with Hott Items
This work addresses the problem of online recommendation systems for users by providing incremental improvements in regret guarantees under structured assumptions.
The paper tackles the online low-rank matrix completion problem by proposing two computationally efficient algorithms, PhasedClusterElim and DeterminantElim, that leverage user collaboration to recommend items under different assumptions, achieving near-optimal per-user regret bounds with specific gap parameters.
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.