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Online Matrix Completion: A Collaborative Approach with Hott Items

Dheeraj Baby, Soumyabrata Pal

TL;DR

This work addresses online matrix completion for multiple users and items under a low-rank reward structure with hott-item separability. It introduces two scalable algorithms: PhasedClusterElim (S=1) and DeterminantElim (S=r), delivering improved regret bounds that exploit cross-user collaboration and phase-based elimination. The PhasedClusterElim analysis hinges on identifying opinionated users and progressively refining user-group labels to jointly prune suboptimal items, while DeterminantElim leverages determinant-based gaps to prune r-column subsets, achieving good rates and recovering the rank-1 case of prior work. Empirical results on synthetic data corroborate the theoretical gains, and the authors discuss the feasibility and scope of the hott-items assumptions, along with directions to enhance computational efficiency and relax structural requirements.

Abstract

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.

Online Matrix Completion: A Collaborative Approach with Hott Items

TL;DR

This work addresses online matrix completion for multiple users and items under a low-rank reward structure with hott-item separability. It introduces two scalable algorithms: PhasedClusterElim (S=1) and DeterminantElim (S=r), delivering improved regret bounds that exploit cross-user collaboration and phase-based elimination. The PhasedClusterElim analysis hinges on identifying opinionated users and progressively refining user-group labels to jointly prune suboptimal items, while DeterminantElim leverages determinant-based gaps to prune r-column subsets, achieving good rates and recovering the rank-1 case of prior work. Empirical results on synthetic data corroborate the theoretical gains, and the authors discuss the feasibility and scope of the hott-items assumptions, along with directions to enhance computational efficiency and relax structural requirements.

Abstract

We investigate the low rank matrix completion problem in an online setting with users, items, rounds, and an unknown rank- reward matrix . This problem has been well-studied in the literature and has several applications in practice. In each round, we recommend carefully chosen distinct items to every user and observe noisy rewards. In the regime where , 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 , under additional incoherence/smoothness assumptions on , we propose the phased algorithm \textsc{PhasedClusterElim}. Our algorithm obtains a near-optimal per-user regret of where are problem-dependent gap parameters with almost always. 2) Second, we consider a simplified setting with where we make significantly milder assumptions on . Here, we introduce another phased algorithm, \textsc{DeterminantElim}, to derive a regret guarantee of where 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.
Paper Structure (16 sections, 26 theorems, 64 equations, 1 figure, 4 algorithms)

This paper contains 16 sections, 26 theorems, 64 equations, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation and Results
  3. Formulation
  4. Main Results
  5. Algorithm PhasedClusterElim and Analysis of $\mathsf{Reg}$ (Theorem\ref{['thm:first']})
  6. Algorithm DeterminantElim and Guarantees on $\mathsf{Reg}_{\mathsf{Simple}}$ (Theorem\ref{['thm:second']})
  7. Conclusion and Future Works
  8. Other Related Work
  9. Experiments
  10. Simplified Algorithm:
  11. Missing Algorithmic Modules and Proofs
  12. Offline Low Rank Matrix Completion Oracle (MCO)
  13. Proofs for generalized regret setting (Setting 1 from Section \ref{['sec:problem']})
  14. Proof of Lemma \ref{['lem:prelim_1']}
  15. Proof of Theorem \ref{['thm:second']}
  16. ...and 1 more sections

Key Result

Lemma 1

For any $u\in [\mathsf{M}]$, it must happen that $\pi_u(1),\pi_u(\mathsf{N}) \in \mathcal{A}$.

Figures (1)

  • Figure 1: Comparison of regret incurred by Algorithm \ref{['algo:phased_elim_simplified']} (a simplified version of our proposed phased elimination algorithm namely Algorithm PhasedClusterElim) with baselines such as Explore-Then-Commit (Greedy) algorithm and Alternating Minimization (AM) algorithm. Clearly, Algorithm \ref{['algo:phased_elim_simplified']} outperforms the other baselines significantly.

Theorems & Definitions (53)

  • Lemma 1
  • Definition 1: Hott item gaps
  • Definition 2: Minimum Reward gap
  • Theorem 1
  • Remark 1
  • Remark 2
  • Definition 3: Determinant gap
  • Theorem 2
  • Remark 3
  • Remark 4
  • ...and 43 more