A New Framework for Convex Clustering in Kernel Spaces: Finite Sample Bounds, Consistency and Performance Insights

Convex clustering is a well-regarded clustering method, resembling the similar centroid-based approach of Lloyd's $k$-means, without requiring a predefined cluster count. It starts with each data point as its centroid and iteratively merges them. Despite its advantages, this method can fail when dealing with data exhibiting linearly non-separable or non-convex structures. To mitigate the limitations, we propose a kernelized extension of the convex clustering method. This approach projects the data points into a Reproducing Kernel Hilbert Space (RKHS) using a feature map, enabling convex clustering in this transformed space. This kernelization not only allows for better handling of complex data distributions but also produces an embedding in a finite-dimensional vector space. We provide a comprehensive theoretical underpinnings for our kernelized approach, proving algorithmic convergence and establishing finite sample bounds for our estimates. The effectiveness of our method is demonstrated through extensive experiments on both synthetic and real-world datasets, showing superior performance compared to state-of-the-art clustering techniques. This work marks a significant advancement in the field, offering an effective solution for clustering in non-linear and non-convex data scenarios.

Stable Thompson Sampling: Valid Inference via Variance Inflation

We consider the problem of statistical inference when the data is collected via a Thompson Sampling-type algorithm. While Thompson Sampling (TS) is known to be both asymptotically optimal and empirically effective, its adaptive sampling scheme poses challenges for constructing confidence intervals for model parameters. We propose and analyze a variant of TS, called Stable Thompson Sampling, in which the posterior variance is inflated by a logarithmic factor. We show that this modification leads to asymptotically normal estimates of the arm means, despite the non-i.i.d. nature of the data. Importantly, this statistical benefit comes at a modest cost: the variance inflation increases regret by only a logarithmic factor compared to standard TS. Our results reveal a principled trade-off: by paying a small price in regret, one can enable valid statistical inference for adaptive decision-making algorithms.