A Revenue Function for Comparison-Based Hierarchical Clustering

Comparison-based learning addresses the problem of learning when, instead of explicit features or pairwise similarities, one only has access to comparisons of the form: \emph{Object $A$ is more similar to $B$ than to $C$.} Recently, it has been shown that, in Hierarchical Clustering, single and complete linkage can be directly implemented using only such comparisons while several algorithms have been proposed to emulate the behaviour of average linkage. Hence, finding hierarchies (or dendrograms) using only comparisons is a well understood problem. However, evaluating their meaningfulness when no ground-truth nor explicit similarities are available remains an open question. In this paper, we bridge this gap by proposing a new revenue function that allows one to measure the goodness of dendrograms using only comparisons. We show that this function is closely related to Dasgupta's cost for hierarchical clustering that uses pairwise similarities. On the theoretical side, we use the proposed revenue function to resolve the open problem of whether one can approximately recover a latent hierarchy using few triplet comparisons. On the practical side, we present principled algorithms for comparison-based hierarchical clustering based on the maximisation of the revenue and we empirically compare them with existing methods.

A Consistent Estimator for Confounding Strength

Regression on observational data can fail to capture a causal relationship in the presence of unobserved confounding. Confounding strength measures this mismatch, but estimating it requires itself additional assumptions. A common assumption is the independence of causal mechanisms, which relies on concentration phenomena in high dimensions. While high dimensions enable the estimation of confounding strength, they also necessitate adapted estimators. In this paper, we derive the asymptotic behavior of the confounding strength estimator by Janzing and Sch\"olkopf (2018) and show that it is generally not consistent. We then use tools from random matrix theory to derive an adapted, consistent estimator.

Representation Power of Graph Convolutions : Neural Tangent Kernel Analysis

The fundamental principle of Graph Neural Networks (GNNs) is to exploit the structural information of the data by aggregating the neighboring nodes using a graph convolution. Therefore, understanding its influence on the network performance is crucial. Convolutions based on graph Laplacian have emerged as the dominant choice with the symmetric normalization of the adjacency matrix $A$, defined as $D^{-1/2}AD^{-1/2}$, being the most widely adopted one, where $D$ is the degree matrix. However, some empirical studies show that row normalization $D^{-1}A$ outperforms it in node classification. Despite the widespread use of GNNs, there is no rigorous theoretical study on the representation power of these convolution operators, that could explain this behavior. In this work, we analyze the influence of the graph convolutions theoretically using Graph Neural Tangent Kernel in a semi-supervised node classification setting. Under a Degree Corrected Stochastic Block Model, we prove that: (i) row normalization preserves the underlying class structure better than other convolutions; (ii) performance degrades with network depth due to over-smoothing, but the loss in class information is the slowest in row normalization; (iii) skip connections retain the class information even at infinite depth, thereby eliminating over-smoothing. We finally validate our theoretical findings on real datasets.

Interpolation and Regularization for Causal Learning

We study the problem of learning causal models from observational data through the lens of interpolation and its counterpart -- regularization. A large volume of recent theoretical, as well as empirical work, suggests that, in highly complex model classes, interpolating estimators can have good statistical generalization properties and can even be optimal for statistical learning. Motivated by an analogy between statistical and causal learning recently highlighted by Janzing (2019), we investigate whether interpolating estimators can also learn good causal models. To this end, we consider a simple linearly confounded model and derive precise asymptotics for the *causal risk* of the min-norm interpolator and ridge-regularized regressors in the high-dimensional regime. Under the principle of independent causal mechanisms, a standard assumption in causal learning, we find that interpolators cannot be optimal and causal learning requires stronger regularization than statistical learning. This resolves a recent conjecture in Janzing (2019). Beyond this assumption, we find a larger range of behavior that can be precisely characterized with a new measure of *confounding strength*. If the confounding strength is negative, causal learning requires weaker regularization than statistical learning, interpolators can be optimal, and the optimal regularization can even be negative. If the confounding strength is large, the optimal regularization is infinite, and learning from observational data is actively harmful.

Learning Theory Can (Sometimes) Explain Generalisation in Graph Neural Networks

In recent years, several results in the supervised learning setting suggested that classical statistical learning-theoretic measures, such as VC dimension, do not adequately explain the performance of deep learning models which prompted a slew of work in the infinite-width and iteration regimes. However, there is little theoretical explanation for the success of neural networks beyond the supervised setting. In this paper we argue that, under some distributional assumptions, classical learning-theoretic measures can sufficiently explain generalization for graph neural networks in the transductive setting. In particular, we provide a rigorous analysis of the performance of neural networks in the context of transductive inference, specifically by analysing the generalisation properties of graph convolutional networks for the problem of node classification. While VC Dimension does result in trivial generalisation error bounds in this setting as well, we show that transductive Rademacher complexity can explain the generalisation properties of graph convolutional networks for stochastic block models. We further use the generalisation error bounds based on transductive Rademacher complexity to demonstrate the role of graph convolutions and network architectures in achieving smaller generalisation error and provide insights into when the graph structure can help in learning. The findings of this paper could re-new the interest in studying generalisation in neural networks in terms of learning-theoretic measures, albeit in specific problems.

Causal Forecasting:Generalization Bounds for Autoregressive Models

Despite the increasing relevance of forecasting methods, the causal implications of these algorithms remain largely unexplored. This is concerning considering that, even under simplifying assumptions such as causal sufficiency, the statistical risk of a model can differ significantly from its \textit{causal risk}. Here, we study the problem of *causal generalization* -- generalizing from the observational to interventional distributions -- in forecasting. Our goal is to find answers to the question: How does the efficacy of an autoregressive (VAR) model in predicting statistical associations compare with its ability to predict under interventions? To this end, we introduce the framework of *causal learning theory* for forecasting. Using this framework, we obtain a characterization of the difference between statistical and causal risks, which helps identify sources of divergence between them. Under causal sufficiency, the problem of causal generalization amounts to learning under covariate shifts albeit with additional structure (restriction to interventional distributions). This structure allows us to obtain uniform convergence bounds on causal generalizability for the class of VAR models. To the best of our knowledge, this is the first work that provides theoretical guarantees for causal generalization in the time-series setting.

Recovery Guarantees for Kernel-based Clustering under Non-parametric Mixture Models

Despite the ubiquity of kernel-based clustering, surprisingly few statistical guarantees exist beyond settings that consider strong structural assumptions on the data generation process. In this work, we take a step towards bridging this gap by studying the statistical performance of kernel-based clustering algorithms under non-parametric mixture models. We provide necessary and sufficient separability conditions under which these algorithms can consistently recover the underlying true clustering. Our analysis provides guarantees for kernel clustering approaches without structural assumptions on the form of the component distributions. Additionally, we establish a key equivalence between kernel-based data-clustering and kernel density-based clustering. This enables us to provide consistency guarantees for kernel-based estimators of non-parametric mixture models. Along with theoretical implications, this connection could have practical implications, including in the systematic choice of the bandwidth of the Gaussian kernel in the context of clustering.

New Insights into Graph Convolutional Networks using Neural Tangent Kernels

Graph Convolutional Networks (GCNs) have emerged as powerful tools for learning on network structured data. Although empirically successful, GCNs exhibit certain behaviour that has no rigorous explanation -- for instance, the performance of GCNs significantly degrades with increasing network depth, whereas it improves marginally with depth using skip connections. This paper focuses on semi-supervised learning on graphs, and explains the above observations through the lens of Neural Tangent Kernels (NTKs). We derive NTKs corresponding to infinitely wide GCNs (with and without skip connections). Subsequently, we use the derived NTKs to identify that, with suitable normalisation, network depth does not always drastically reduce the performance of GCNs -- a fact that we also validate through extensive simulation. Furthermore, we propose NTK as an efficient `surrogate model' for GCNs that does not suffer from performance fluctuations due to hyper-parameter tuning since it is a hyper-parameter free deterministic kernel. The efficacy of this idea is demonstrated through a comparison of different skip connections for GCNs using the surrogate NTKs.

Graphon based Clustering and Testing of Networks: Algorithms and Theory

Network-valued data are encountered in a wide range of applications and pose challenges in learning due to their complex structure and absence of vertex correspondence. Typical examples of such problems include classification or grouping of protein structures and social networks. Various methods, ranging from graph kernels to graph neural networks, have been proposed that achieve some success in graph classification problems. However, most methods have limited theoretical justification, and their applicability beyond classification remains unexplored. In this work, we propose methods for clustering multiple graphs, without vertex correspondence, that are inspired by the recent literature on estimating graphons -- symmetric functions corresponding to infinite vertex limit of graphs. We propose a novel graph distance based on sorting-and-smoothing graphon estimators. Using the proposed graph distance, we present two clustering algorithms and show that they achieve state-of-the-art results. We prove the statistical consistency of both algorithms under Lipschitz assumptions on the graph degrees. We further study the applicability of the proposed distance for graph two-sample testing problems.

Near-Optimal Comparison Based Clustering

The goal of clustering is to group similar objects into meaningful partitions. This process is well understood when an explicit similarity measure between the objects is given. However, far less is known when this information is not readily available and, instead, one only observes ordinal comparisons such as "object i is more similar to j than to k." In this paper, we tackle this problem using a two-step procedure: we estimate a pairwise similarity matrix from the comparisons before using a clustering method based on semi-definite programming (SDP). We theoretically show that our approach can exactly recover a planted clustering using a near-optimal number of passive comparisons. We empirically validate our theoretical findings and demonstrate the good behaviour of our method on real data.