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Clustering Items through Bandit Feedback: Finding the Right Feature out of Many

Maximilian Graf, Victor Thuot, Nicolas Verzelen

TL;DR

The paper tackles clustering of $n$ items described by $d$-dimensional features under sequential bandit feedback, aiming to recover a binary partition with high confidence while minimizing observations. It introduces BanditClustering, a three-part approach that first identifies a discriminative feature via Sequential Halving-based subroutines, then pins down a representative from the other group, and finally clusters all items using the most informative feature. The authors provide tight non-asymptotic upper bounds on the budget, an instance-dependent lower bound, and experimental evidence showing substantial budget savings in sparse regimes compared to uniform sampling and batch methods. The work advances adaptive sensing for clustering, with implications for crowd-sourcing and other settings where costly feature queries must be allocated judiciously.

Abstract

We study the problem of clustering a set of items based on bandit feedback. Each of the $n$ items is characterized by a feature vector, with a possibly large dimension $d$. The items are partitioned into two unknown groups such that items within the same group share the same feature vector. We consider a sequential and adaptive setting in which, at each round, the learner selects one item and one feature, then observes a noisy evaluation of the item's feature. The learner's objective is to recover the correct partition of the items, while keeping the number of observations as small as possible. We provide an algorithm which relies on finding a relevant feature for the clustering task, leveraging the Sequential Halving algorithm. With probability at least $1-δ$, we obtain an accurate recovery of the partition and derive an upper bound on the budget required. Furthermore, we derive an instance-dependent lower bound, which is tight in some relevant cases.

Clustering Items through Bandit Feedback: Finding the Right Feature out of Many

TL;DR

The paper tackles clustering of items described by -dimensional features under sequential bandit feedback, aiming to recover a binary partition with high confidence while minimizing observations. It introduces BanditClustering, a three-part approach that first identifies a discriminative feature via Sequential Halving-based subroutines, then pins down a representative from the other group, and finally clusters all items using the most informative feature. The authors provide tight non-asymptotic upper bounds on the budget, an instance-dependent lower bound, and experimental evidence showing substantial budget savings in sparse regimes compared to uniform sampling and batch methods. The work advances adaptive sensing for clustering, with implications for crowd-sourcing and other settings where costly feature queries must be allocated judiciously.

Abstract

We study the problem of clustering a set of items based on bandit feedback. Each of the items is characterized by a feature vector, with a possibly large dimension . The items are partitioned into two unknown groups such that items within the same group share the same feature vector. We consider a sequential and adaptive setting in which, at each round, the learner selects one item and one feature, then observes a noisy evaluation of the item's feature. The learner's objective is to recover the correct partition of the items, while keeping the number of observations as small as possible. We provide an algorithm which relies on finding a relevant feature for the clustering task, leveraging the Sequential Halving algorithm. With probability at least , we obtain an accurate recovery of the partition and derive an upper bound on the budget required. Furthermore, we derive an instance-dependent lower bound, which is tight in some relevant cases.

Paper Structure

This paper contains 31 sections, 10 theorems, 138 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Problem formulation and notation
  3. Algorithms
  4. Introduction to our method
  5. Warm-up: adaptation of Sequential Halving
  6. First step: CandidateRow
  7. Second step: ClusterByCandidates
  8. Main Algorithm
  9. Lower bounds
  10. Experiments
  11. Experiment 1
  12. Experiment 2
  13. Discussion
  14. Comparison to other active clustering settings and batch clustering.
  15. Extension to a larger number of groups.
  16. ...and 16 more sections

Key Result

Theorem 3.1

For $\delta\in (0,1/e)$, consider Algorithm alg:cluster with entry $\delta$. Define where $s^*$ is the effective sparsity defined in eq:effectivelysparse. With a probability of at least $1-\delta$, Algorithm alg:cluster returns $\hat{g}=g$ with a budget of at most where there exists a numerical constant $C$, and an index $\tilde{s}=s^* \vee (\lceil d/n\rceil \wedge |\{j\in [d] \; , \Delta_j\ne 0

Figures (2)

  • Figure 1: Different budgets for Experiment 1, depending on the sparsity of $\Delta^s$.
  • Figure 2: Different budgets for Experiment 2, depending on the dimensionality of the problem $n$ and $d=10\cdot n$.

Theorems & Definitions (17)

  • Theorem 3.1
  • Corollary 3.2
  • Theorem 4.1
  • Lemma B.1
  • Lemma B.2
  • proof : Proof of Lemma \ref{['lem:upperboundCSH']}
  • proof : Proof of Lemma \ref{['lem:proportionlowerbounded']}
  • Proposition C.1
  • proof : Proof of Proposition \ref{['thm:cr']}
  • Proposition D.1
  • ...and 7 more