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Efficient Low-Rank Matrix Estimation, Experimental Design, and Arm-Set-Dependent Low-Rank Bandits

Kyoungseok Jang, Chicheng Zhang, Kwang-Sung Jun

TL;DR

The paper advances low-rank matrix estimation and bandit optimization by introducing LowPopArt, an operator-norm–oriented estimator whose accuracy depends on a geometry-aware hardness term $B(Q)$. It couples estimation with an experimental design to minimize $B(Q)$ across the measurement set and leverages this to derive two arm-set adaptive bandit algorithms with improved regret bounds in general arm-set scenarios. Theoretical results establish operator-norm recovery guarantees, compare favorably to nuclear-norm approaches, and provide a convex design framework with concrete scaling for common arm-set geometries. Empirical results on synthetic and real data (e.g., Movielens) demonstrate improved nuclear-norm recovery and lower regret relative to baselines. The lower-bound analysis further delineates the fundamental limits of low-rank bandits and highlights the vital role of arm-set geometry in regret behavior.

Abstract

We study low-rank matrix trace regression and the related problem of low-rank matrix bandits. Assuming access to the distribution of the covariates, we propose a novel low-rank matrix estimation method called LowPopArt and provide its recovery guarantee that depends on a novel quantity denoted by B(Q) that characterizes the hardness of the problem, where Q is the covariance matrix of the measurement distribution. We show that our method can provide tighter recovery guarantees than classical nuclear norm penalized least squares (Koltchinskii et al., 2011) in several problems. To perform efficient estimation with a limited number of measurements from an arbitrarily given measurement set A, we also propose a novel experimental design criterion that minimizes B(Q) with computational efficiency. We leverage our novel estimator and design of experiments to derive two low-rank linear bandit algorithms for general arm sets that enjoy improved regret upper bounds. This improves over previous works on low-rank bandits, which make somewhat restrictive assumptions that the arm set is the unit ball or that an efficient exploration distribution is given. To our knowledge, our experimental design criterion is the first one tailored to low-rank matrix estimation beyond the naive reduction to linear regression, which can be of independent interest.

Efficient Low-Rank Matrix Estimation, Experimental Design, and Arm-Set-Dependent Low-Rank Bandits

TL;DR

The paper advances low-rank matrix estimation and bandit optimization by introducing LowPopArt, an operator-norm–oriented estimator whose accuracy depends on a geometry-aware hardness term . It couples estimation with an experimental design to minimize across the measurement set and leverages this to derive two arm-set adaptive bandit algorithms with improved regret bounds in general arm-set scenarios. Theoretical results establish operator-norm recovery guarantees, compare favorably to nuclear-norm approaches, and provide a convex design framework with concrete scaling for common arm-set geometries. Empirical results on synthetic and real data (e.g., Movielens) demonstrate improved nuclear-norm recovery and lower regret relative to baselines. The lower-bound analysis further delineates the fundamental limits of low-rank bandits and highlights the vital role of arm-set geometry in regret behavior.

Abstract

We study low-rank matrix trace regression and the related problem of low-rank matrix bandits. Assuming access to the distribution of the covariates, we propose a novel low-rank matrix estimation method called LowPopArt and provide its recovery guarantee that depends on a novel quantity denoted by B(Q) that characterizes the hardness of the problem, where Q is the covariance matrix of the measurement distribution. We show that our method can provide tighter recovery guarantees than classical nuclear norm penalized least squares (Koltchinskii et al., 2011) in several problems. To perform efficient estimation with a limited number of measurements from an arbitrarily given measurement set A, we also propose a novel experimental design criterion that minimizes B(Q) with computational efficiency. We leverage our novel estimator and design of experiments to derive two low-rank linear bandit algorithms for general arm sets that enjoy improved regret upper bounds. This improves over previous works on low-rank bandits, which make somewhat restrictive assumptions that the arm set is the unit ball or that an efficient exploration distribution is given. To our knowledge, our experimental design criterion is the first one tailored to low-rank matrix estimation beyond the naive reduction to linear regression, which can be of independent interest.
Paper Structure (95 sections, 26 theorems, 149 equations, 5 figures, 1 table, 6 algorithms)

This paper contains 95 sections, 26 theorems, 149 equations, 5 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction and related work
  2. Preliminaries
  3. Basic Notations.
  4. Low-rank bandits.
  5. LowPopArt: A novel low-rank matrix estimator
  6. Analysis of Algorithm \ref{['alg:lowPopart']}
  7. Comparison with nuclear norm penalty methods
  8. Experimental design
  9. Main novelty of LowPopArt compared to PopArt jang22popart.
  10. Low rank bandit algorithms
  11. Explore-then-commit based algorithm.
  12. Explore-Subspace-Then-Refine (ESTR) based algorithm.
  13. Experiments
  14. Low-rank matrix recovery.
  15. Low-rank matrix bandits.
  16. ...and 80 more sections

Key Result

Theorem 3.1

Suppose we run Algorithm alg:lowPopart with the arm set $\mathcal{A}$ which satisfies Assumption assumption:op-bound, sample size $n_0$, population covariance matrix of vectorized matrices $Q$, pilot estimator $\Theta_0$ and pilot estimation error bound $R_0$, such that $\max_{A \in \mathcal{A}} \ab where where $D_i^{(\mathrm{col})}=(Q^{-1})_{[i\cdot d_s+1: (i+1)\cdot d_s],[i\cdot d_s+1: (i+1)\cd

Figures (5)

  • Figure 1: Illustration of $D_i^{(\mathrm{col})}$ and $D_i^{(\mathrm{row})}$
  • Figure 2: Experiment results on nuclear norm error
  • Figure 3: Experiment results on bandits with ETC-based (left) and ESTR-based algorithms (right)
  • Figure 4: Experiment results on nuclear norm error
  • Figure 5: Experiment results on bandit using a real-world dataset

Theorems & Definitions (64)

  • Definition 2.1
  • Remark 1
  • Theorem 3.1
  • Remark 2
  • Theorem 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 4.1: Regret upper bound
  • ...and 54 more