Nonparametric Kernel Clustering with Bandit Feedback

TL;DR

This work addresses nonparametric clustering with bandit feedback, where arms correspond to unknown distributions and clusters are determined by equality of their kernel mean embeddings in an RKHS. The Kernel Active Bandit Clustering (KABC) algorithm uses variance-aware kernel two-sample tests based on the maximum mean discrepancy (MMD) and a graph-based clustering step, coupled with a doubling-budget, adaptive strategy to achieve a $\delta$-PAC partition without parametric assumptions. A key contribution is an instance-dependent, non-asymptotic budget bound governed by a variance-aware signal-to-noise ratio $s_*^2(\nu)$, enabling learning that adapts to both the separations and variances of arm distributions. The framework leverages recent variance-aware concentration results for KMEs to provide strong guarantees and practical guidance for active, kernel-based clustering in complex, high-dimensional data contexts. This approach has potential impact for applications in recommendation systems, crowdsourcing, and biomedical settings where parametric assumptions are untenable.

Abstract

Clustering with bandit feedback refers to the problem of partitioning a set of items, where the clustering algorithm can sequentially query the items to receive noisy observations. The problem is formally posed as the task of partitioning the arms of an N-armed stochastic bandit according to their underlying distributions, grouping two arms together if and only if they share the same distribution, using samples collected sequentially and adaptively. This setting has gained attention in recent years due to its applicability in recommendation systems and crowdsourcing. Existing works on clustering with bandit feedback rely on a strong assumption that the underlying distributions are sub-Gaussian. As a consequence, the existing methods mainly cover settings with linearly-separable clusters, which has little practical relevance. We introduce a framework of ``nonparametric clustering with bandit feedback'', where the underlying arm distributions are not constrained to any parametric, and hence, it is applicable for active clustering of real-world datasets. We adopt a kernel-based approach, which allows us to reformulate the nonparametric problem as the task of clustering the arms according to their kernel mean embeddings in a reproducing kernel Hilbert space (RKHS). Building on this formulation, we introduce the KABC algorithm with theoretical correctness guarantees and analyze its sampling budget. We introduce a notion of signal-to-noise ratio for this problem that depends on the maximum mean discrepancy (MMD) between the arm distributions and on their variance in the RKHS. Our algorithm is adaptive to this unknown quantity: it does not require it as an input yet achieves instance-dependent guarantees.

Theorems & Definitions (8)

• Theorem 3.1: KABC$(\delta, K)$ is $\delta$-PAC
• Lemma 1.1
• proof : Proof of Theorem \ref{['theorem:firstAdaptive']}
• proof
• Lemma 1.2: Variance-aware empirical bound wolfer2025variance
• Lemma 1.3: Variance-aware empirical bound for the distance of two arms
• Lemma 1.4
• Lemma 1.5: Bound for empirical Variance wolfer2025variance