Debarghya Ghoshdastidar

Debarghya Ghoshdastidar, Ambedkar Dukkipati, Shalabh Bhatnagar

The q-Gaussian distribution results from maximizing certain generalizations of Shannon entropy under some constraints. The importance of q-Gaussian distributions stems from the fact that they exhibit power-law behavior, and also generalize Gaussian distributions. In this paper, we propose a Smoothed Functional (SF) scheme for gradient estimation using q-Gaussian distribution, and also propose an algorithm for optimization based on the above scheme. Convergence results of the algorithm are presented. Performance of the proposed algorithm is shown by simulation results on a queuing model.

Luca Rendsburg, Leena Chennuru Vankadara, Debarghya Ghoshdastidar et al.

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 Schölkopf (2018) and show that it is generally not consistent. We then use tools from random matrix theory to derive an adapted, consistent estimator.

Nil Ayday, Mahalakshmi Sabanayagam, Debarghya Ghoshdastidar

Graph Neural Networks (GNN) are currently the most popular approach for learning and prediction on graph-structured data and are deployed in various fields, from social network analysis to drug discovery. However, there is limited mathematical understanding of the performance of GNNs. We discuss the various perspectives used to study statistical generalisation in GNNs. We identify three broad frameworks. The first approach, rooted in learning theory, relies on uniform convergence bounds and the complexity of the hypothesis class of specific GNN architectures. This approach also builds on the expressivity of GNNs, typically studied through the lens of graph isomorphism tests. The second principle is to simplify the neural architecture by analysing GNNs under the asymptotics of infinitely many parameters or infinite graph size. This approach approximates GNNs using Gaussian processes, neural tangent kernels or graphon neural network operators, which allow studying the generalisation or stability of trained GNNs. The third framework studies GNNs under random graph models, often the contextual stochastic block model, and derives non-asymptotic error rates using tools from high-dimensional statistics. We highlight some key theoretical results and discuss a few limitations and open research questions for each perspective.

Anurag Singh, Mahalakshmi Sabanayagam, Krikamol Muandet et al.

Test-time defenses are used to improve the robustness of deep neural networks to adversarial examples during inference. However, existing methods either require an additional trained classifier to detect and correct the adversarial samples, or perform additional complex optimization on the model parameters or the input to adapt to the adversarial samples at test-time, resulting in a significant increase in the inference time compared to the base model. In this work, we propose a novel test-time defense strategy called Robust Feature Inference (RFI) that is easy to integrate with any existing (robust) training procedure without additional test-time computation. Based on the notion of robustness of features that we present, the key idea is to project the trained models to the most robust feature space, thereby reducing the vulnerability to adversarial attacks in non-robust directions. We theoretically characterize the subspace of the eigenspectrum of the feature covariance that is the most robust for a generalized additive model. Our extensive experiments on CIFAR-10, CIFAR-100, tiny ImageNet and ImageNet datasets for several robustness benchmarks, including the state-of-the-art methods in RobustBench show that RFI improves robustness across adaptive and transfer attacks consistently. We also compare RFI with adaptive test-time defenses to demonstrate the effectiveness of our proposed approach.

Ajinkya Mohgaonkar, Lukas Gosch, Mahalakshmi Sabanayagam et al.

Label-flipping attacks, which corrupt training labels to induce misclassifications at inference, remain a major threat to supervised learning models. This drives the need for robustness certificates that provide formal guarantees about a model's robustness under adversarially corrupted labels. Existing certification frameworks rely on ensemble techniques such as smoothing or partition-aggregation, but treat the corresponding base classifiers as black boxes, yielding overly conservative guarantees. We introduce EnsembleCert, the first certification framework for partition-aggregation ensembles that utilizes white-box knowledge of the base classifiers. Concretely, EnsembleCert yields tighter guarantees than black-box approaches by aggregating per-partition white-box certificates to compute ensemble-level guarantees in polynomial time. To extract white-box knowledge from the base classifiers efficiently, we develop ScaLabelCert, a method that leverages the equivalence between sufficiently wide neural networks and kernel methods using the neural tangent kernel. ScaLabelCert yields the first exact, polynomial-time calculable certificate for neural networks against label-flipping attacks. EnsembleCert is either on par, or significantly outperforms the existing partition-based black box certificates. Exemplary, on CIFAR-10, our method can certify upto +26.5% more label flips in median over the test set compared to the existing black-box approach while requiring 100 times fewer partitions, thus, challenging the prevailing notion that heavy partitioning is a necessity for strong certified robustness.

Pascal Esser, Maximilian Fleissner, Debarghya Ghoshdastidar

Unsupervised and self-supervised representation learning has become popular in recent years for learning useful features from unlabelled data. Representation learning has been mostly developed in the neural network literature, and other models for representation learning are surprisingly unexplored. In this work, we introduce and analyze several kernel-based representation learning approaches: Firstly, we define two kernel Self-Supervised Learning (SSL) models using contrastive loss functions and secondly, a Kernel Autoencoder (AE) model based on the idea of embedding and reconstructing data. We argue that the classical representer theorems for supervised kernel machines are not always applicable for (self-supervised) representation learning, and present new representer theorems, which show that the representations learned by our kernel models can be expressed in terms of kernel matrices. We further derive generalisation error bounds for representation learning with kernel SSL and AE, and empirically evaluate the performance of these methods in both small data regimes as well as in comparison with neural network based models.

Lukas Gosch, Mahalakshmi Sabanayagam, Debarghya Ghoshdastidar et al.

Generalization of machine learning models can be severely compromised by data poisoning, where adversarial changes are applied to the training data. This vulnerability has led to interest in certifying (i.e., proving) that such changes up to a certain magnitude do not affect test predictions. We, for the first time, certify Graph Neural Networks (GNNs) against poisoning attacks, including backdoors, targeting the node features of a given graph. Our certificates are white-box and based upon $(i)$ the neural tangent kernel, which characterizes the training dynamics of sufficiently wide networks; and $(ii)$ a novel reformulation of the bilevel optimization problem describing poisoning as a mixed-integer linear program. Consequently, we leverage our framework to provide fundamental insights into the role of graph structure and its connectivity on the worst-case robustness behavior of convolution-based and PageRank-based GNNs. We note that our framework is more general and constitutes the first approach to derive white-box poisoning certificates for NNs, which can be of independent interest beyond graph-related tasks.

Aishik Mandal, Michaël Perrot, Debarghya Ghoshdastidar

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.

Mahalakshmi Sabanayagam, Pascal Esser, Debarghya Ghoshdastidar

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' in conjunction with a suitable choice for the network architecture, such as depth and activation functions. Therefore, understanding the influence of each of the design choice 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 as the most widely adopted one. However, some empirical studies show that row normalization of the adjacency matrix outperforms it in node classification. Despite the widespread use of GNNs, there is no rigorous theoretical study on the representation power of these convolutions, that could explain this behavior. Similarly, the empirical observation of the linear GNNs performance being on par with non-linear ReLU GNNs lacks rigorous theory. In this work, we theoretically analyze the influence of different aspects of the GNN architecture using the Graph Neural Tangent Kernel in a semi-supervised node classification setting. Under the population Degree Corrected Stochastic Block Model, we prove that: (i) linear networks capture the class information as good as ReLU networks; (ii) row normalization preserves the underlying class structure better than other convolutions; (iii) performance degrades with network depth due to over-smoothing, but the loss in class information is the slowest in row normalization; (iv) skip connections retain the class information even at infinite depth, thereby eliminating over-smoothing. We finally validate our theoretical findings numerically and on real datasets such as Cora and Citeseer.

Pascal Esser, Satyaki Mukherjee, Debarghya Ghoshdastidar

Self-Supervised Learning (SSL) is an important paradigm for learning representations from unlabelled data, and SSL with neural networks has been highly successful in practice. However current theoretical analysis of SSL is mostly restricted to generalisation error bounds. In contrast, learning dynamics often provide a precise characterisation of the behaviour of neural networks based models but, so far, are mainly known in supervised settings. In this paper, we study the learning dynamics of SSL models, specifically representations obtained by minimising contrastive and non-contrastive losses. We show that a naive extension of the dymanics of multivariate regression to SSL leads to learning trivial scalar representations that demonstrates dimension collapse in SSL. Consequently, we formulate SSL objectives with orthogonality constraints on the weights, and derive the exact (network width independent) learning dynamics of the SSL models trained using gradient descent on the Grassmannian manifold. We also argue that the infinite width approximation of SSL models significantly deviate from the neural tangent kernel approximations of supervised models. We numerically illustrate the validity of our theoretical findings, and discuss how the presented results provide a framework for further theoretical analysis of contrastive and non-contrastive SSL.

Pascal Mattia Esser, Satyaki Mukherjee, Mahalakshmi Sabanayagam et al.

The central question in representation learning is what constitutes a good or meaningful representation. In this work we argue that if we consider data with inherent cluster structures, where clusters can be characterized through different means and covariances, those data structures should be represented in the embedding as well. While Autoencoders (AE) are widely used in practice for unsupervised representation learning, they do not fulfil the above condition on the embedding as they obtain a single representation of the data. To overcome this we propose a meta-algorithm that can be used to extend an arbitrary AE architecture to a tensorized version (TAE) that allows for learning cluster-specific embeddings while simultaneously learning the cluster assignment. For the linear setting we prove that TAE can recover the principle components of the different clusters in contrast to principle component of the entire data recovered by a standard AE. We validated this on planted models and for general, non-linear and convolutional AEs we empirically illustrate that tensorizing the AE is beneficial in clustering and de-noising tasks.

Satyaki Mukherjee, Soumendu Sundar Mukherjee, Debarghya Ghoshdastidar

We consider the general dimensionality reduction problem of locating in a high-dimensional data cloud, a $k$-dimensional non-Gaussian subspace of interesting features. We use a projection pursuit approach -- we search for mutually orthogonal unit directions which maximise the 2-Wasserstein distance of the empirical distribution of data-projections along these directions from a standard Gaussian. Under a generative model, where there is a underlying (unknown) low-dimensional non-Gaussian subspace, we prove rigorous statistical guarantees on the accuracy of approximating this unknown subspace by the directions found by our projection pursuit approach. Our results operate in the regime where the data dimensionality is comparable to the sample size, and thus supplement the recent literature on the non-feasibility of locating interesting directions via projection pursuit in the complementary regime where the data dimensionality is much larger than the sample size.

Nil Ayday, Lingchu Yang, Debarghya Ghoshdastidar

Graph transformers are the state-of-the-art for learning from graph-structured data and are empirically known to avoid several pitfalls of message-passing architectures. However, there is limited theoretical analysis on why these models perform well in practice. In this work, we prove that attention-based architectures have structural benefits over graph convolutional networks in the context of node-level prediction tasks. Specifically, we study the neural network gaussian process limits of graph transformers (GAT, Graphormer, Specformer) with infinite width and infinite heads, and derive the node-level and edge-level kernels across the layers. Our results characterise how the node features and the graph structure propagate through the graph attention layers. As a specific example, we prove that graph transformers structurally preserve community information and maintain discriminative node representations even in deep layers, thereby preventing oversmoothing. We provide empirical evidence on synthetic and real-world graphs that validate our theoretical insights, such as integrating informative priors and positional encoding can improve performance of deep graph transformers.

Sara Taheri, Mahalakshmi Sabanayagam, Debarghya Ghoshdastidar et al.

The increasing use of machine learning in safety-critical domains amplifies the risk of adversarial threats, especially data poisoning attacks that corrupt training data to degrade performance or induce unsafe behavior. Most existing defenses lack formal guarantees or rely on restrictive assumptions about the model class, attack type, extent of poisoning, or point-wise certification, limiting their practical reliability. This paper introduces a principled formal robustness certification framework that models gradient-based training as a discrete-time dynamical system (dt-DS) and formulates poisoning robustness as a formal safety verification problem. By adapting the concept of barrier certificates (BCs) from control theory, we introduce sufficient conditions to certify a robust radius ensuring that the terminal model remains safe under worst-case ${\ell}_p$-norm based poisoning. To make this practical, we parameterize BCs as neural networks trained on finite sets of poisoned trajectories. We further derive probably approximately correct (PAC) bounds by solving a scenario convex program (SCP), which yields a confidence lower bound on the certified robustness radius generalizing beyond the training set. Importantly, our framework also extends to certification against test-time attacks, making it the first unified framework to provide formal guarantees in both training and test-time attack settings. Experiments on MNIST, SVHN, and CIFAR-10 show that our approach certifies non-trivial perturbation budgets while being model-agnostic and requiring no prior knowledge of the attack or contamination level.

Victor Thuot, Sebastian Vogt, Debarghya Ghoshdastidar et al.

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.

Mahalakshmi Sabanayagam, Lukas Gosch, Stephan Günnemann et al.

Machine learning models are highly vulnerable to label flipping, i.e., the adversarial modification (poisoning) of training labels to compromise performance. Thus, deriving robustness certificates is important to guarantee that test predictions remain unaffected and to understand worst-case robustness behavior. However, for Graph Neural Networks (GNNs), the problem of certifying label flipping has so far been unsolved. We change this by introducing an exact certification method, deriving both sample-wise and collective certificates. Our method leverages the Neural Tangent Kernel (NTK) to capture the training dynamics of wide networks enabling us to reformulate the bilevel optimization problem representing label flipping into a Mixed-Integer Linear Program (MILP). We apply our method to certify a broad range of GNN architectures in node classification tasks. Thereby, concerning the worst-case robustness to label flipping: $(i)$ we establish hierarchies of GNNs on different benchmark graphs; $(ii)$ quantify the effect of architectural choices such as activations, depth and skip-connections; and surprisingly, $(iii)$ uncover a novel phenomenon of the robustness plateauing for intermediate perturbation budgets across all investigated datasets and architectures. While we focus on GNNs, our certificates are applicable to sufficiently wide NNs in general through their NTK. Thus, our work presents the first exact certificate to a poisoning attack ever derived for neural networks, which could be of independent interest. The code is available at https://github.com/saper0/qpcert.

Maximilian Fleissner, Leena Chennuru Vankadara, Debarghya Ghoshdastidar

Despite the growing popularity of explainable and interpretable machine learning, there is still surprisingly limited work on inherently interpretable clustering methods. Recently, there has been a surge of interest in explaining the classic k-means algorithm, leading to efficient algorithms that approximate k-means clusters using axis-aligned decision trees. However, interpretable variants of k-means have limited applicability in practice, where more flexible clustering methods are often needed to obtain useful partitions of the data. In this work, we investigate interpretable kernel clustering, and propose algorithms that construct decision trees to approximate the partitions induced by kernel k-means, a nonlinear extension of k-means. We further build on previous work on explainable k-means and demonstrate how a suitable choice of features allows preserving interpretability without sacrificing approximation guarantees on the interpretable model.

Anna Van Elst, Debarghya Ghoshdastidar

Contrastive representation learning is a modern paradigm for learning representations of unlabeled data via augmentations -- precisely, contrastive models learn to embed semantically similar pairs of samples (positive pairs) closer than independently drawn samples (negative samples). In spite of its empirical success and widespread use in foundation models, statistical theory for contrastive learning remains less explored. Recent works have developed generalization error bounds for contrastive losses, but the resulting risk certificates are either vacuous (certificates based on Rademacher complexity or $f$-divergence) or require strong assumptions about samples that are unreasonable in practice. The present paper develops non-vacuous PAC-Bayesian risk certificates for contrastive representation learning, considering the practical considerations of the popular SimCLR framework. Notably, we take into account that SimCLR reuses positive pairs of augmented data as negative samples for other data, thereby inducing strong dependence and making classical PAC or PAC-Bayesian bounds inapplicable. We further refine existing bounds on the downstream classification loss by incorporating SimCLR-specific factors, including data augmentation and temperature scaling, and derive risk certificates for the contrastive zero-one risk. The resulting bounds for contrastive loss and downstream prediction are much tighter than those of previous risk certificates, as demonstrated by experiments on CIFAR-10.

Shlomo Libo Feigin, Maximilian Fleissner, Debarghya Ghoshdastidar

Data augmentations play an important role in the recent success of self-supervised learning (SSL). While augmentations are commonly understood to encode invariances between different views into the learned representations, this interpretation overlooks the impact of the pretraining architecture and suggests that SSL would require diverse augmentations which resemble the data to work well. However, these assumptions do not align with empirical evidence, encouraging further theoretical understanding to guide the principled design of augmentations in new domains. To this end, we use kernel theory to derive analytical expressions for data augmentations that achieve desired target representations after pretraining. We consider non-contrastive and contrastive losses, namely VICReg, Barlow Twins and the Spectral Contrastive Loss, and provide an algorithm to construct such augmentations. Our analysis shows that augmentations need not be similar to the data to learn useful representations, nor be diverse, and that the architecture has a significant impact on the optimal augmentations.

Maximilian Fleissner, Gautham Govind Anil, Debarghya Ghoshdastidar

The NTK is a widely used tool in the theoretical analysis of deep learning, allowing us to look at supervised deep neural networks through the lenses of kernel regression. Recently, several works have investigated kernel models for self-supervised learning, hypothesizing that these also shed light on the behavior of wide neural networks by virtue of the NTK. However, it remains an open question to what extent this connection is mathematically sound -- it is a commonly encountered misbelief that the kernel behavior of wide neural networks emerges irrespective of the loss function it is trained on. In this paper, we bridge the gap between the NTK and self-supervised learning, focusing on two-layer neural networks trained under the Barlow Twins loss. We prove that the NTK of Barlow Twins indeed becomes constant as the width of the network approaches infinity. Our analysis technique is a bit different from previous works on the NTK and may be of independent interest. Overall, our work provides a first justification for the use of classic kernel theory to understand self-supervised learning of wide neural networks. Building on this result, we derive generalization error bounds for kernelized Barlow Twins and connect them to neural networks of finite width.

Alexandru Crăciun, Debarghya Ghoshdastidar

A vast literature on convergence guarantees for gradient descent and derived methods exists at the moment. However, a simple practical situation remains unexplored: when a fixed step size is used, can we expect gradient descent to converge starting from any initialization? We provide fundamental impossibility results showing that convergence becomes impossible no matter the initialization if the step size gets too big. Looking at the asymptotic value of the gradient norm along the optimization trajectory, we see that there is a sharp transition as the step size crosses a critical value. This has been observed by practitioners, yet the true mechanisms through which this happens remain unclear beyond heuristics. Using results from dynamical systems theory, we provide a proof of this in the case of linear neural networks with a squared loss. We also prove the impossibility of convergence for more general losses without requiring strong assumptions such as Lipschitz continuity for the gradient. We validate our findings through experiments with non-linear networks.

Maximilian Fleissner, Maedeh Zarvandi, Debarghya Ghoshdastidar

We study the explainable clustering problem first posed by Moshkovitz, Dasgupta, Rashtchian, and Frost (ICML 2020). The goal of explainable clustering is to fit an axis-aligned decision tree with $K$ leaves and minimal clustering cost (where every leaf is a cluster). The fundamental theoretical question in this line of work is the \textit{price of explainability}, defined as the ratio between the clustering cost of the tree and the optimal cost. Numerous papers have provided worst-case guarantees on this quantity. For $K$-medians, it has recently been shown that the worst-case price of explainability is $Θ(\log K)$. While this settles the matter from a data-agnostic point of view, two important questions remain unanswered: Are tighter guarantees possible for well-clustered data? And can we trust decision trees to recover underlying cluster structures? In this paper, we place ourselves in a statistical setting of mixture models to answer both questions. We prove that better guarantees are indeed feasible for well-clustered data. Our algorithm takes as input a mixture model and constructs a tree in data-independent time. We then extend our analysis to kernel clustering, deriving new guarantees that significantly improve over existing worst-case bounds.

Gautham Govind Anil, Pascal Esser, Debarghya Ghoshdastidar

Contrastive learning is a paradigm for learning representations from unlabelled data that has been highly successful for image and text data. Several recent works have examined contrastive losses to claim that contrastive models effectively learn spectral embeddings, while few works show relations between (wide) contrastive models and kernel principal component analysis (PCA). However, it is not known if trained contrastive models indeed correspond to kernel methods or PCA. In this work, we analyze the training dynamics of two-layer contrastive models, with non-linear activation, and answer when these models are close to PCA or kernel methods. It is well known in the supervised setting that neural networks are equivalent to neural tangent kernel (NTK) machines, and that the NTK of infinitely wide networks remains constant during training. We provide the first convergence results of NTK for contrastive losses, and present a nuanced picture: NTK of wide networks remains almost constant for cosine similarity based contrastive losses, but not for losses based on dot product similarity. We further study the training dynamics of contrastive models with orthogonality constraints on output layer, which is implicitly assumed in works relating contrastive learning to spectral embedding. Our deviation bounds suggest that representations learned by contrastive models are close to the principal components of a certain matrix computed from random features. We empirically show that our theoretical results possibly hold beyond two-layer networks.

Yaomengxi Han, Debarghya Ghoshdastidar

Transformers, with their attention mechanisms, have emerged as the state-of-the-art architectures of sequential modeling and empirically outperform feed-forward neural networks (FFNNs) across many fields, such as natural language processing and computer vision. However, their generalization ability, particularly for low-sensitivity functions, remains less studied. We bridge this gap by analyzing transformers on the $k$-parity problem. Daniely and Malach (NeurIPS 2020) show that FFNNs with one hidden layer and $O(nk^7 \log k)$ parameters can learn $k$-parity, where the input length $n$ is typically much larger than $k$. In this paper, we prove that FFNNs require at least $Ω(n)$ parameters to learn $k$-parity, while transformers require only $O(k)$ parameters, surpassing the theoretical lower bound needed by FFNNs. We further prove that this parameter efficiency cannot be achieved with fixed attention heads. Our work establishes transformers as theoretically superior to FFNNs in learning parity function, showing how their attention mechanisms enable parameter-efficient generalization in functions with low sensitivity.

Alexandru Crăciun, Debarghya Ghoshdastidar

The theory of training deep networks has become a central question of modern machine learning and has inspired many practical advancements. In particular, the gradient descent (GD) optimization algorithm has been extensively studied in recent years. A key assumption about GD has appeared in several recent works: the \emph{GD map is non-singular} -- it preserves sets of measure zero under preimages. Crucially, this assumption has been used to prove that GD avoids saddle points and maxima, and to establish the existence of a computable quantity that determines the convergence to global minima (both for GD and stochastic GD). However, the current literature either assumes the non-singularity of the GD map or imposes restrictive assumptions, such as Lipschitz smoothness of the loss (for example, Lipschitzness does not hold for deep ReLU networks with the cross-entropy loss) and restricts the analysis to GD with small step-sizes. In this paper, we investigate the neural network map as a function on the space of weights and biases. We also prove, for the first time, the non-singularity of the gradient descent (GD) map on the loss landscape of realistic neural network architectures (with fully connected, convolutional, or softmax attention layers) and piecewise analytic activations (which includes sigmoid, ReLU, leaky ReLU, etc.) for almost all step-sizes. Our work significantly extends the existing results on the convergence of GD and SGD by guaranteeing that they apply to practical neural network settings and has the potential to unlock further exploration of learning dynamics.

Maedeh Zarvandi, Michael Timothy, Theresa Wasserer et al.

Kernel methods provide a theoretically grounded framework for non-linear and non-parametric learning, with strong analytic foundations and statistical guarantees. Yet, their scalability has long been limited by prohibitive time and memory costs. While progress has been made in scaling kernel regression, no framework exists for scalable kernel-based representation learning, restricting their use in the era of foundation models where representations are learned from massive unlabeled data. We introduce KREPES -- a unified, scalable framework for kernel-based representation learning via Nyström approximation. KREPES accommodates a wide range of unsupervised and self-supervised losses, and experiments on large image and tabular datasets demonstrate its efficiency. Crucially, KREPES enables principled interpretability of the learned representations, an immediate benefit over deep models, which we substantiate through dedicated analysis.

Pascal Esser, Maximilian Fleissner, Debarghya Ghoshdastidar

Representation learning from unlabeled data has been extensively studied in statistics, data science and signal processing with a rich literature on techniques for dimension reduction, compression, multi-dimensional scaling among others. However, current deep learning models use new principles for unsupervised representation learning that cannot be easily analyzed using classical theories. For example, visual foundation models have found tremendous success using self-supervision or denoising/masked autoencoders, which effectively learn representations from massive amounts of unlabeled data. However, it remains difficult to characterize the representations learned by these models and to explain why they perform well for diverse prediction tasks or show emergent behavior. To answer these questions, one needs to combine mathematical tools from statistics and optimization. This paper provides an overview of recent theoretical advances in representation learning from unlabeled data and mentions our contributions in this direction.

Nil Ayday, Mahalakshmi Sabanayagam, Debarghya Ghoshdastidar

Graph Neural Networks (GNNs) are widely used in learning on graph-structured data, yet a principled understanding of why they succeed or fail remains elusive. While prior works have examined architectural limitations such as over-smoothing and over-squashing, these do not explain what enables GNNs to extract meaningful representations or why performance varies drastically between similar architectures. These questions are related to the role of generalisation: the ability of a model to make accurate predictions on unlabelled data. Although several works have derived generalisation error bounds for GNNs, these are typically loose, restricted to a single architecture, and offer limited insight into what governs generalisation in practice. In this work, we take a different approach by deriving the exact generalisation error for GNNs in a transductive fixed-design setting through the lens of signal processing. From this viewpoint, GNNs can be interpreted as graph filter operators that act on node features via the graph structure. By focusing on linear GNNs while allowing non-linearity in the graph filters, we derive the first exact generalisation error for a broad range of GNNs, including convolutional, PageRank-based, and attention-based models. The exact characterisation of the generalisation error reveals that only the aligned information between node features and graph structure contributes to generalisation. Furthermore, we quantify the effect of homophily on generalisation. Our work provides a framework that explains when and why GNNs can effectively leverage structural and feature information, offering practical guidance for model selection.

Jonghyun Ham, Maximilian Fleissner, Debarghya Ghoshdastidar

Modern deep neural networks exhibit strong generalization even in highly overparameterized regimes. Significant progress has been made to understand this phenomenon in the context of supervised learning, but for unsupervised tasks such as denoising, several open questions remain. While some recent works have successfully characterized the test error of the linear denoising problem, they are limited to linear models (one-layer network). In this work, we focus on two-layer linear denoising autoencoders trained under gradient flow, incorporating two key ingredients of modern deep learning architectures: A low-dimensional bottleneck layer that effectively enforces a rank constraint on the learned solution, as well as the possibility of a skip connection that bypasses the bottleneck. We derive closed-form expressions for all critical points of this model under product regularization, and in particular describe its global minimizer under the minimum-norm principle. From there, we derive the test risk formula in the overparameterized regime, both for models with and without skip connections. Our analysis reveals two interesting phenomena: Firstly, the bottleneck layer introduces an additional complexity measure akin to the classical bias-variance trade-off -- increasing the bottleneck width reduces bias but introduces variance, and vice versa. Secondly, skip connection can mitigate the variance in denoising autoencoders -- especially when the model is mildly overparameterized. We further analyze the impact of skip connections in denoising autoencoder using random matrix theory and support our claims with numerical evidence.

Mahalakshmi Sabanayagam, Russell Tsuchida, Cheng Soon Ong et al.

Adversarial robustness of machine learning models is critical to ensuring reliable performance under data perturbations. Recent progress has been on point estimators, and this paper considers distributional predictors. First, using the link between exponential families and Bregman divergences, we formulate an adversarial Bregman divergence loss as an adversarial negative log-likelihood. Using the geometric properties of Bregman divergences, we compute the adversarial perturbation for such models in closed-form. Second, under such losses, we introduce \emph{adversarially robust posteriors}, by exploiting the optimization-centric view of generalized Bayesian inference. Third, we derive the \emph{first} rigorous generalization certificates in the context of an adversarial extension of Bayesian linear regression by leveraging the PAC-Bayesian framework. Finally, experiments on real and synthetic datasets demonstrate the superior robustness of the derived adversarially robust posterior over Bayes posterior, and also validate our theoretical guarantees.

Martin Eppert, Satyaki Mukherjee, Debarghya Ghoshdastidar

Projection Pursuit is a classic exploratory technique for finding interesting projections of a dataset. We propose a method for recovering projections containing either Imbalanced Clusters or a Bernoulli-Rademacher distribution using a gradient-based technique to optimize the projection index. As sample complexity is a major limiting factor in Projection Pursuit, we analyze our algorithm's sample complexity within a Planted Vector setting where we can observe that Imbalanced Clusters can be recovered more easily than balanced ones. Additionally, we give a generalized result that works for a variety of data distributions and projection indices. We compare these results to computational lower bounds in the Low-Degree-Polynomial Framework. Finally, we experimentally evaluate our method's applicability to real-world data using FashionMNIST and the Human Activity Recognition Dataset, where our algorithm outperforms others when only a few samples are available.

Maximilian Fleissner, Pascal Esser, Debarghya Ghoshdastidar

Self-supervised learning (SSL) aims to find meaningful representations from unlabeled data by encoding semantic similarities through data augmentations. Despite its current popularity, theoretical insights about SSL are still scarce. For example, it is not yet known whether commonly used SSL loss functions can be related to a statistical model, much in the same as OLS, generalized linear models or PCA naturally emerge as maximum likelihood estimates of an underlying generative process. In this short paper, we consider a latent variable statistical model for SSL that exhibits an interesting property: Depending on the informativeness of the data augmentations, the MLE of the model either reduces to PCA, or approaches a simple non-contrastive loss. We analyze the model and also empirically illustrate our findings.

Leena Chennuru Vankadara, Luca Rendsburg, Ulrike von Luxburg et al.

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.

Pascal Mattia Esser, Leena Chennuru Vankadara, Debarghya Ghoshdastidar

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.

Leena Chennuru Vankadara, Philipp Michael Faller, Michaela Hardt et al.

Despite the increasing relevance of forecasting methods, 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 \textit{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 \textit{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 under the VAR model). 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.

Leena Chennuru Vankadara, Sebastian Bordt, Ulrike von Luxburg et al.

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.

Mahalakshmi Sabanayagam, Pascal Esser, Debarghya Ghoshdastidar

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.

Mahalakshmi Sabanayagam, Leena Chennuru Vankadara, Debarghya Ghoshdastidar

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.

Michaël Perrot, Pascal Mattia Esser, Debarghya Ghoshdastidar

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.

Leena Chennuru Vankadara, Debarghya Ghoshdastidar

This paper studies the optimality of kernel methods in high-dimensional data clustering. Recent works have studied the large sample performance of kernel clustering in the high-dimensional regime, where Euclidean distance becomes less informative. However, it is unknown whether popular methods, such as kernel k-means, are optimal in this regime. We consider the problem of high-dimensional Gaussian clustering and show that, with the exponential kernel function, the sufficient conditions for partial recovery of clusters using the NP-hard kernel k-means objective matches the known information-theoretic limit up to a factor of $\sqrt{2}$ for large $k$. It also exactly matches the known upper bounds for the non-kernel setting. We also show that a semi-definite relaxation of the kernel k-means procedure matches up to constant factors, the spectral threshold, below which no polynomial-time algorithm is known to succeed. This is the first work that provides such optimality guarantees for the kernel k-means as well as its convex relaxation. Our proofs demonstrate the utility of the less known polynomial concentration results for random variables with exponentially decaying tails in a higher-order analysis of kernel methods.

Debarghya Ghoshdastidar, Ulrike von Luxburg

Hypothesis testing for graphs has been an important tool in applied research fields for more than two decades, and still remains a challenging problem as one often needs to draw inference from few replicates of large graphs. Recent studies in statistics and learning theory have provided some theoretical insights about such high-dimensional graph testing problems, but the practicality of the developed theoretical methods remains an open question. In this paper, we consider the problem of two-sample testing of large graphs. We demonstrate the practical merits and limitations of existing theoretical tests and their bootstrapped variants. We also propose two new tests based on asymptotic distributions. We show that these tests are computationally less expensive and, in some cases, more reliable than the existing methods.

Debarghya Ghoshdastidar, Michaël Perrot, Ulrike von Luxburg

We address the classical problem of hierarchical clustering, but in a framework where one does not have access to a representation of the objects or their pairwise similarities. Instead, we assume that only a set of comparisons between objects is available, that is, statements of the form "objects $i$ and $j$ are more similar than objects $k$ and $l$." Such a scenario is commonly encountered in crowdsourcing applications. The focus of this work is to develop comparison-based hierarchical clustering algorithms that do not rely on the principles of ordinal embedding. We show that single and complete linkage are inherently comparison-based and we develop variants of average linkage. We provide statistical guarantees for the different methods under a planted hierarchical partition model. We also empirically demonstrate the performance of the proposed approaches on several datasets.

Debarghya Ghoshdastidar, Maurilio Gutzeit, Alexandra Carpentier et al.

The study of networks leads to a wide range of high dimensional inference problems. In many practical applications, one needs to draw inference from one or few large sparse networks. The present paper studies hypothesis testing of graphs in this high-dimensional regime, where the goal is to test between two populations of inhomogeneous random graphs defined on the same set of $n$ vertices. The size of each population $m$ is much smaller than $n$, and can even be a constant as small as 1. The critical question in this context is whether the problem is solvable for small $m$. We answer this question from a minimax testing perspective. Let $P,Q$ be the population adjacencies of two sparse inhomogeneous random graph models, and $d$ be a suitably defined distance function. Given a population of $m$ graphs from each model, we derive minimax separation rates for the problem of testing $P=Q$ against $d(P,Q)>ρ$. We observe that if $m$ is small, then the minimax separation is too large for some popular choices of $d$, including total variation distance between corresponding distributions. This implies that some models that are widely separated in $d$ cannot be distinguished for small $m$, and hence, the testing problem is generally not solvable in these cases. We also show that if $m>1$, then the minimax separation is relatively small if $d$ is the Frobenius norm or operator norm distance between $P$ and $Q$. For $m=1$, only the latter distance provides small minimax separation. Thus, for these distances, the problem is solvable for small $m$. We also present near-optimal two-sample tests in both cases, where tests are adaptive with respect to sparsity level of the graphs.

Debarghya Ghoshdastidar, Maurilio Gutzeit, Alexandra Carpentier et al.

We consider a two-sample hypothesis testing problem, where the distributions are defined on the space of undirected graphs, and one has access to only one observation from each model. A motivating example for this problem is comparing the friendship networks on Facebook and LinkedIn. The practical approach to such problems is to compare the networks based on certain network statistics. In this paper, we present a general principle for two-sample hypothesis testing in such scenarios without making any assumption about the network generation process. The main contribution of the paper is a general formulation of the problem based on concentration of network statistics, and consequently, a consistent two-sample test that arises as the natural solution for this problem. We also show that the proposed test is minimax optimal for certain network statistics.

Siavash Haghiri, Debarghya Ghoshdastidar, Ulrike von Luxburg

We consider machine learning in a comparison-based setting where we are given a set of points in a metric space, but we have no access to the actual distances between the points. Instead, we can only ask an oracle whether the distance between two points $i$ and $j$ is smaller than the distance between the points $i$ and $k$. We are concerned with data structures and algorithms to find nearest neighbors based on such comparisons. We focus on a simple yet effective algorithm that recursively splits the space by first selecting two random pivot points and then assigning all other points to the closer of the two (comparison tree). We prove that if the metric space satisfies certain expansion conditions, then with high probability the height of the comparison tree is logarithmic in the number of points, leading to efficient search performance. We also provide an upper bound for the failure probability to return the true nearest neighbor. Experiments show that the comparison tree is competitive with algorithms that have access to the actual distance values, and needs less triplet comparisons than other competitors.

Debarghya Ghoshdastidar, Ambedkar Dukkipati

In a series of recent works, we have generalised the consistency results in the stochastic block model literature to the case of uniform and non-uniform hypergraphs. The present paper continues the same line of study, where we focus on partitioning weighted uniform hypergraphs---a problem often encountered in computer vision. This work is motivated by two issues that arise when a hypergraph partitioning approach is used to tackle computer vision problems: (i) The uniform hypergraphs constructed for higher-order learning contain all edges, but most have negligible weights. Thus, the adjacency tensor is nearly sparse, and yet, not binary. (ii) A more serious concern is that standard partitioning algorithms need to compute all edge weights, which is computationally expensive for hypergraphs. This is usually resolved in practice by merging the clustering algorithm with a tensor sampling strategy---an approach that is yet to be analysed rigorously. We build on our earlier work on partitioning dense unweighted uniform hypergraphs (Ghoshdastidar and Dukkipati, ICML, 2015), and address the aforementioned issues by proposing provable and efficient partitioning algorithms. Our analysis justifies the empirical success of practical sampling techniques. We also complement our theoretical findings by elaborate empirical comparison of various hypergraph partitioning schemes.

Debarghya Ghoshdastidar, Ambedkar Dukkipati

Hypergraph partitioning lies at the heart of a number of problems in machine learning and network sciences. Many algorithms for hypergraph partitioning have been proposed that extend standard approaches for graph partitioning to the case of hypergraphs. However, theoretical aspects of such methods have seldom received attention in the literature as compared to the extensive studies on the guarantees of graph partitioning. For instance, consistency results of spectral graph partitioning under the stochastic block model are well known. In this paper, we present a planted partition model for sparse random non-uniform hypergraphs that generalizes the stochastic block model. We derive an error bound for a spectral hypergraph partitioning algorithm under this model using matrix concentration inequalities. To the best of our knowledge, this is the first consistency result related to partitioning non-uniform hypergraphs.

Debarghya Ghoshdastidar, Ambedkar Dukkipati, Ajay P. Adsul et al.

Motivated by multi-distribution divergences, which originate in information theory, we propose a notion of `multi-point' kernels, and study their applications. We study a class of kernels based on Jensen type divergences and show that these can be extended to measure similarity among multiple points. We study tensor flattening methods and develop a multi-point (kernel) spectral clustering (MSC) method. We further emphasize on a special case of the proposed kernels, which is a multi-point extension of the linear (dot-product) kernel and show the existence of cubic time tensor flattening algorithm in this case. Finally, we illustrate the usefulness of our contributions using standard data sets and image segmentation tasks.

Debarghya Ghoshdastidar, Ambedkar Dukkipati, Shalabh Bhatnagar

Smoothed functional (SF) schemes for gradient estimation are known to be efficient in stochastic optimization algorithms, specially when the objective is to improve the performance of a stochastic system. However, the performance of these methods depends on several parameters, such as the choice of a suitable smoothing kernel. Different kernels have been studied in literature, which include Gaussian, Cauchy and uniform distributions among others. This paper studies a new class of kernels based on the q-Gaussian distribution, that has gained popularity in statistical physics over the last decade. Though the importance of this family of distributions is attributed to its ability to generalize the Gaussian distribution, we observe that this class encompasses almost all existing smoothing kernels. This motivates us to study SF schemes for gradient estimation using the q-Gaussian distribution. Using the derived gradient estimates, we propose two-timescale algorithms for optimization of a stochastic objective function in a constrained setting with projected gradient search approach. We prove the convergence of our algorithms to the set of stationary points of an associated ODE. We also demonstrate their performance numerically through simulations on a queuing model.

Ambedkar Dukkipati, Gaurav Pandey, Debarghya Ghoshdastidar et al.

Maximum entropy approach to classification is very well studied in applied statistics and machine learning and almost all the methods that exists in literature are discriminative in nature. In this paper, we introduce a maximum entropy classification method with feature selection for large dimensional data such as text datasets that is generative in nature. To tackle the curse of dimensionality of large data sets, we employ conditional independence assumption (Naive Bayes) and we perform feature selection simultaneously, by enforcing a `maximum discrimination' between estimated class conditional densities. For two class problems, in the proposed method, we use Jeffreys ($J$) divergence to discriminate the class conditional densities. To extend our method to the multi-class case, we propose a completely new approach by considering a multi-distribution divergence: we replace Jeffreys divergence by Jensen-Shannon ($JS$) divergence to discriminate conditional densities of multiple classes. In order to reduce computational complexity, we employ a modified Jensen-Shannon divergence ($JS_{GM}$), based on AM-GM inequality. We show that the resulting divergence is a natural generalization of Jeffreys divergence to a multiple distributions case. As far as the theoretical justifications are concerned we show that when one intends to select the best features in a generative maximum entropy approach, maximum discrimination using $J-$divergence emerges naturally in binary classification. Performance and comparative study of the proposed algorithms have been demonstrated on large dimensional text and gene expression datasets that show our methods scale up very well with large dimensional datasets.