Paper:
Clustering is an important part of many modern data analysis pipelines, including network analysis and data retrieval. There are many different clustering algorithms developed by various communities, and it is often not clear which algorithm will give the best performance on a specific clustering task. Similarly, we often have multiple ways to measure distances between data points, and the best clustering performance might require a non-trivial combination of those metrics. In this work, we study data-driven algorithm selection and metric learning for clustering problems, where the goal is to simultaneously learn the best algorithm and metric for a specific application. The family of clustering algorithms we consider is parameterized linkage based procedures that includes single and complete linkage. The family of distance functions we learn over are convex combinations of base distance functions. We design efficient learning algorithms which receive samples from an application-specific distribution over clustering instances and learn a near-optimal distance and clustering algorithm from these classes. We also carry out a comprehensive empirical evaluation of our techniques showing that they can lead to significantly improved clustering performance on real-world datasets.

Review:
In this paper, the authors propose an approach to learning combinations of (instance-wise) distance metrics and (cluster-wise) merge functions to optimally cluster instances from a  particular data distribution. In particular, given a set of clustering instances (each of which is a set of instances from the domain and their cluster assignment), a set of distance metrics, and a set of merge functions, the proposed approach aims to learn a convex combination of the distance metrics and merge functions to reconstruct the given clusterings.
The paper has two main contributions. First, a PAC learning type of guarantee is given on the quality of the learned clustering approach. Second, an efficient data structure for identifying the convex combinations is given. A small set of experiments suggests that, in practice, the learned combinations can outperform using single distance metrics and merge functions.
Comments
I am not an expert in this area; I had trouble following the details of the theoretical developments. However, I appreciated that intuition was given on both what the theorems and lemmas were showing as well as the main steps of the proofs.
Concerning Theorem 1, it is not exactly clear to me what the contribution is on top of [Balcan et al., 2019]. The text mentions that they already give sample complexity guarantees in what seems like the same setting (piecewise-structured cost function).
The authors point out that depth-first traversal is a good choice here due to its memory efficiency. However, in cases where the search space is a graph rather than a tree (i.e., there are multiple paths to some nodes), then DFS can exponentially increase the work compared to breadth-first or other search strategies (e.g., [Edelkamp and Schroedl, 2012]). While the name suggests that the “execution tree” is, indeed, a tree, is this guaranteed to be the case? or could multiple paths lead to the same partition?
For the experimental evaluation, it seems as though there is no “test” set of clustering instances. It would be helpful to also include performance of the learned combinations on some test clustering instances to give an idea of how generalizable to approach is to other instances within the data distribution. (Of course, the main contributions of this work are the theoretical developments, so just one or two examples would be sufficient.)
For motivation, it would be helpful to give some examples where the prerequisites of this work are actually met; that is, cases where sufficiently large number of labeled cluster instances are available, but the generative mechanism of the clusters is not.
For context, it could be helpful to briefly mention how, if at all, the current results apply to widely-used clustering algorithms such as k-means or Gaussian mixture models.
Typos, etc.
The references are somewhat inconsistently formatted. Also, some proper nouns in titles are not capitalized (e.g., “lloyd’s families”).
“leaves correspond to” -> “leaves corresponding to”
What does the “big-Oh tilde” notation in Theorem 1 mean?