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Miscellaneous Aspects of Machine Learning/Everything Else

Kai Klede, Thomas Altstidl, Dario Zanca, Bjoern Eskofier

Popular metrics for clustering comparison, like the Adjusted Rand Index and the Adjusted Mutual Information, are type II biased. The Standardized Mutual Information removes this bias but suffers from counterintuitive non-monotonicity and poor computational efficiency. We introduce the $p$-value adjusted Rand Index ($\operatorname{PMI}_2$), the first cluster comparison method that is type II unbiased and provably monotonous. The $\operatorname{PMI}_2$ has fast approximations that outperform the Standardized Mutual information. We demonstrate its unbiased clustering selection, approximation quality, and runtime efficiency on synthetic benchmarks. In experiments on image and social network datasets, we show how the $\operatorname{PMI}_2$ can help practitioners choose better clustering and community detection algorithms.

Miscellaneous Aspects of Machine Learning/Causality

Panagiotis Misiakos, Chris Wendler, Markus Püschel

We present a novel perspective and algorithm for learning directed acyclic graphs (DAGs) from data generated by a linear structural equation model (SEM). First, we show that a linear SEM can be viewed as a linear transform that, in prior work, computes the data from a dense input vector of random valued root causes (as we will call them) associated with the nodes. Instead, we consider the case of (approximately) few root causes and also introduce noise in the measurement of the data. Intuitively, this means that the DAG data is produced by few data generating events whose effect percolates through the DAG. We prove identifiability in this new setting and show that the true DAG is the global minimizer of the $L^0$-norm of the vector of root causes. For data satisfying the few root causes assumption, we show superior performance compared to prior DAG learning methods.

Miscellaneous Aspects of Machine Learning/Kernel methods

Patrik Robert Gerber, Tianze Jiang, Yury Polyanskiy, Rui Sun

Given $n$ observations from two balanced classes, consider the task of labeling an additional $m$ inputs that are known to all belong to \emph{one} of the two classes. Special cases of this problem are well-known: with completeknowledge of class distributions ($n=\infty$) theproblem is solved optimally by the likelihood-ratio test; when$m=1$ it corresponds to binary classification; and when $m\approx n$ it is equivalent to two-sample testing. The intermediate settings occur in the field of likelihood-free inference, where labeled samples are obtained by running forward simulations and the unlabeled sample is collected experimentally. In recent work it was discovered that there is a fundamental trade-offbetween $m$ and $n$: increasing the data sample $m$ reduces the amount $n$ of training/simulationdata needed. In this work we (a) introduce a generalization where unlabeled samples come from a mixture of the two classes -- a case often encountered in practice; (b) study the minimax sample complexity for non-parametric classes of densities under \textit{maximum meandiscrepancy} (MMD) separation; and (c) investigate the empirical performance of kernels parameterized by neural networks on two tasks: detectionof the Higgs boson and detection of planted DDPM generated images amidstCIFAR-10 images. For both problems we confirm the existence of the theoretically predicted asymmetric $m$ vs $n$ trade-off.

Miscellaneous Aspects of Machine Learning/Unsupervised and Semi-supervised Learning

Annie Gray, Alexander Modell, Patrick Rubin-Delanchy, Nick Whiteley

In this paper we offer a new perspective on the well established agglomerative clustering algorithm, focusing on recovery of hierarchical structure. We recommend a simple variant of the standard algorithm, in which clusters are merged by maximum average dot product and not, for example, by minimum distance or within-cluster variance. We demonstrate that the tree output by this algorithm provides a bona fide estimate of generative hierarchical structure in data, under a generic probabilistic graphical model. The key technical innovations are to understand how hierarchical information in this model translates into tree geometry which can be recovered from data, and to characterise the benefits of simultaneously growing sample size and data dimension. We demonstrate superior tree recovery performance with real data over existing approaches such as UPGMA, Ward's method, and HDBSCAN.

Miscellaneous Aspects of Machine Learning/Representation Learning

David Loiseaux, Mathieu Carrière, Andrew Blumberg

Topological data analysis (TDA) is an area of data science that focuses on using invariants from algebraic topology to provide multiscale shape descriptors for geometric data sets such as point clouds. One of the most important such descriptors is persistent homology, which encodes the change in shape as a filtration parameter changes; a typical parameter is the feature scale. For many data sets, it is useful to simultaneously vary multiple filtration parameters, for example feature scale and density. While the theoretical properties of single parameter persistent homology are well understood, less is known about the multiparameter case. A central question is the problem of representing multiparameter persistent homology by elements of a vector space for integration with standard machine learning algorithms. Existing approaches to this problem either ignore most of the multiparameter information to reduce to the one-parameter case or are heuristic and potentially unstable in the face of noise. In this article, we introduce a new general representation framework that leverages recent results on decompositions of multiparameter persistent homology. This framework is rich in information, fast to compute, and encompasses previous approaches. Moreover, we establish theoretical stability guarantees under this framework as well as efficient algorithms for practical computation, making this framework an applicable and versatile tool for analyzing geometric and point cloud data. We validate our stability results and algorithms with numerical experiments that demonstrate statistical convergence, prediction accuracy, and fast running times on several real data sets.

Miscellaneous Aspects of Machine Learning/Scalable Algorithms

Anders Aamand, Justin Chen, Huy Nguyen, Sandeep Silwal, Ali Vakilian

Estimating frequencies of elements appearing in a data stream is a key task in large-scale data analysis. Popular sketching approaches to this problem (e.g., CountMin and CountSketch) come with worst-case guarantees that probabilistically bound the error of the estimated frequencies for any possible input. The work of Hsu et al.~(2019) introduced the idea of using machine learning to tailor sketching algorithms to the specific data distribution they are being run on. In particular, their learning-augmented frequency estimation algorithm uses a learned heavy-hitter oracle which predicts which elements will appear many times in the stream. We give a novel algorithm, which in some parameter regimes, already theoretically outperforms the learning based algorithm of Hsu et al. *without* the use of any predictions. Augmenting our algorithm with heavy-hitter predictions further reduces the error and improves upon the state of the art. Empirically, our algorithms achieve superior performance in all experiments compared to prior approaches.

Miscellaneous Aspects of Machine Learning

Indradyumna Roy, Rishi Agarwal, Soumen Chakrabarti, Anirban Dasgupta, Abir De

In many search applications related to passage retrieval, text entailment, and subgraph search, the query and each 'document' is a set of elements, with a document being relevant if it contains the query. These elements are not represented by atomic IDs, but by embedded representations, thereby extending set containment to *soft* set containment. Recent applications address soft set containment by encoding sets into fixed-size vectors and checking for elementwise *vector* *dominance*. This 0/1 property can be relaxed to an asymmetric *hinge* *distance* for scoring and ranking candidate documents. Here we focus on data-sensitive, trainable indices for fast retrieval of relevant documents. Existing LSH methods are designed for mostly symmetric or few simple asymmetric distance functions, which are not suitable for hinge distance. Instead, we transform hinge distance into a proposed *dominance* *similarity* measure, to which we then apply a Fourier transform, thereby expressing dominance similarity as an expectation of inner products of functions in the frequency domain. Next, we approximate the expectation with an importance-sampled estimate. The overall consequence is that now we can use a traditional LSH, but in the frequency domain. To ensure that the LSH uses hash bits efficiently, we learn hash functions that are sensitive to both corpus and query distributions, mapped to the frequency domain. Our experiments show that the proposed asymmetric dominance similarity is critical to the targeted applications, and that our LSH, which we call FourierHashNet, provides a better query time vs. retrieval quality trade-off, compared to several baselines. Both the Fourier transform and the trainable hash codes contribute to performance gains.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Muhammad Faaiz Taufiq, Arnaud Doucet, Rob Cornish, Jean-Francois Ton

Off-Policy Evaluation (OPE) in contextual bandits is crucial for assessing new policies using existing data without costly experimentation. However, current OPE methods, such as Inverse Probability Weighting (IPW) and Doubly Robust (DR) estimators, suffer from high variance, particularly in cases of low overlap between target and behaviour policies or large action and context spaces. In this paper, we introduce a new OPE estimator for contextual bandits, the Marginal Ratio (MR) estimator, which focuses on the shift in the marginal distribution of outcomes $Y$ instead of the policies themselves. Through rigorous theoretical analysis, we demonstrate the benefits of the MR estimator compared to conventional methods like IPW and DR in terms of variance reduction. Additionally, we establish a connection between the MR estimator and the state-of-the-art Marginalized Inverse Propensity Score (MIPS) estimator, proving that MR achieves lower variance among a generalized family of MIPS estimators. We further illustrate the utility of the MR estimator in causal inference settings, where it exhibits enhanced performance in estimating Average Treatment Effects (ATE). Our experiments on synthetic and real-world datasets corroborate our theoretical findings and highlight the practical advantages of the MR estimator in OPE for contextual bandits.

Miscellaneous Aspects of Machine Learning/Causality

Trang Nguyen, Amin Mansouri, Kanika Madan, Khuong Duy Nguyen, Kartik Ahuja, Dianbo Liu, Yoshua Bengio

Agents with the ability to comprehend and reason about the dynamics of objects would be expected to exhibit improved robustness and generalization in novel scenarios. However, achieving this capability necessitates not only an effective scene representation but also an understanding of the mechanisms governing interactions among object subsets. Recent studies have made significant progress in representing scenes using object slots. In this work, we introduce Reusable Slotwise Mechanisms, or RSM, a framework that models object dynamics by leveraging communication among slots along with a modular architecture capable of dynamically selecting reusable mechanisms for predicting the future states of each object slot. Crucially, RSM leverages the Central Contextual Information (CCI), enabling selected mechanisms to access the remaining slots through a bottleneck, effectively allowing for modeling of higher order and complex interactions that might require a sparse subset of objects. Experimental results demonstrate the superior performance of RSM compared to state-of-the-art methods across various future prediction and related downstream tasks, including Visual Question Answering and action planning. Furthermore, we showcase RSM’s Out-of-Distribution generalization ability to handle scenes in intricate scenarios.

Miscellaneous Aspects of Machine Learning/Scalable Algorithms

Alaa Maalouf, Murad Tukan, Noel Loo, Ramin Hasani, Mathias Lechner, Daniela Rus

Dataset Distillation is the task of synthesizing small datasets from large ones while still retaining comparable predictive accuracy to the original uncompressed dataset. Despite significant empirical progress in recent years, there is little understanding of the theoretical limitations/guarantees of dataset distillation, specifically, what excess risk is achieved by distillation compared to the original dataset, and how large are distilled datasets? In this work, we take a theoretical view on kernel ridge regression (KRR) based methods of dataset distillation such as Kernel Inducing Points. By transforming ridge regression in random Fourier features (RFF) space, we provide the first proof of the existence of small (size) distilled datasets and their corresponding excess risk for shift-invariant kernels. We prove that a small set of instances exists in the original input space such that its solution in the RFF space coincides with the solution of the original data. We further show that a KRR solution can be generated using this distilled set of instances which gives an approximation towards the KRR solution optimized on the full input data. The size of this set is linear in the dimension of the RFF space of the input set or alternatively near linear in the number of effective degrees of freedom, which is a function of the kernel, number of data points, and the regularization parameter $\lambda$. The error bound of this distilled set is also a function of $\lambda$. We verify our bounds analytically and empirically.

Miscellaneous Aspects of Machine Learning/Kernel methods

Zhaolu Liu, Robert Peach, Pedro A.M Mediano, Mauricio Barahona

Models that rely solely on pairwise relationships often fail to capture the complete statistical structure of the complex multivariate data found in diverse domains, such as socio-economic, ecological, or biomedical systems. Non-trivial dependencies between groups of more than two variables can play a significant role in the analysis and modelling of such systems, yet extracting such high-order interactions from data remains challenging. Here, we introduce a hierarchy of $d$-order ($d \geq 2$) interaction measures, increasingly inclusive of possible factorisations of the joint probability distribution, and define non-parametric, kernel-based tests to establish systematically the statistical significance of $d$-order interactions. We also establish mathematical links with lattice theory, which elucidate the derivation of the interaction measures and their composite permutation tests; clarify the connection of simplicial complexes with kernel matrix centring; and provide a means to enhance computational efficiency. We illustrate our results numerically with validations on synthetic data, and through an application to neuroimaging data.

Miscellaneous Aspects of Machine Learning/Causality

Sina Akbari, Fateme Jamshidi, Ehsan Mokhtarian, Matthew Vowels, Jalal Etesami, Negar Kiyavash

Causal identification is at the core of the causal inference literature, where complete algorithms have been proposed to identify causal queries of interest. The validity of these algorithms hinges on the restrictive assumption of having access to a correctly specified causal structure. In this work, we study the setting where a probabilistic model of the causal structure is available. Specifically, the edges in a causal graph exist with uncertainties which may, for example, represent degree of belief from domain experts. Alternatively, the uncertainty about an edge may reflect the confidence of a particular statistical test. The question that naturally arises in this setting is: Given such a probabilistic graph and a specific causal effect of interest, what is the subgraph which has the highest plausibility and for which the causal effect is identifiable? We show that answering this question reduces to solving an NP-hard combinatorial optimization problem which we call the edge ID problem. We propose efficient algorithms to approximate this problem and evaluate them against both real-world networks and randomly generated graphs.

Miscellaneous Aspects of Machine Learning/Causality

Ahmed Alaa, Zaid Ahmad, Mark van der Laan

We investigate the problem of machine learning-based (ML) predictive inference on individual treatment effects (ITEs). Previous work has focused primarily on developing ML-based “meta-learners” that can provide point estimates of the conditional average treatment effect (CATE)—these are model-agnostic approaches for combining intermediate nuisance estimates to produce estimates of CATE. In this paper, we develop conformal meta-learners, a general framework for issuing predictive intervals for ITEs by applying the standard conformal prediction (CP) procedure on top of CATE meta-learners. We focus on a broad class of meta-learners based on two-stage pseudo-outcome regression and develop a stochastic ordering framework to study their validity. We show that inference with conformal meta-learners is marginally valid if their (pseudo-outcome) conformity scores stochastically dominate “oracle” conformity scores evaluated on the unobserved ITEs. Additionally, we prove that commonly used CATE meta-learners, such as the doubly-robust learner, satisfy a model- and distribution-free stochastic (or convex) dominance condition, making their conformal inferences valid for practically-relevant levels of target coverage. Whereas existing procedures conduct inference on nuisance parameters (i.e., potential outcomes) via weighted CP, conformal meta-learners enable direct inference on the target parameter (ITE). Numerical experiments show that conformal meta-learners provide valid intervals with competitive efficiency while retaining the favorable point estimation properties of CATE meta-learners.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Alexis Bellot, Alan Malek, Silvia Chiappa

A unifying theme in the design of intelligent agents is to efficiently optimize a policy based on what prior knowledge of the problem is available and what actions can be taken to learn more about it. Bandits are a canonical instance of this task that has been intensely studied in the literature. Most methods, however, typically rely solely on an agent's experimentation in a single environment (or multiple closely related environments). In this paper, we relax this assumption and consider the design of bandit algorithms from a combination of batch data and qualitative assumptions about the relatedness across different environments, represented in the form of causal models. In particular, we show that it is possible to exploit invariances across environments, wherever they may occur in the underlying causal model, to consistently improve learning. The resulting bandit algorithm has a sub-linear regret bound with an explicit dependency on a term that captures how informative related environments are for the task at hand; and may have substantially lower regret than experimentation-only bandit instances.

Miscellaneous Aspects of Machine Learning/Representation Learning

David Loiseaux, Luis Scoccola, Mathieu Carrière, Magnus Bakke Botnan, Steve OUDOT

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case---where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest---and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes---a recent family of MPH descriptors---as signed Radon measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

Miscellaneous Aspects of Machine Learning/Supervised Learning

Erik Schultheis, Marek Wydmuch, Wojciech Kotlowski, Rohit Babbar, Krzysztof Dembczynski

Extreme multi-label classification (XMLC) is the task of selecting a small subset of relevant labels from a very large set of possible labels. As such, it is characterized by long-tail labels, i.e., most labels have very few positive instances. With standard performance measures such as precision@k, a classifier can ignore tail labels and still report good performance. However, it is often argued that correct predictions in the tail are more "interesting" or "rewarding," but the community has not yet settled on a metric capturing this intuitive concept. The existing propensity-scored metrics fall short on this goal by confounding the problems of long-tail and missing labels. In this paper, we analyze generalized metrics budgeted "at k" as an alternative solution. To tackle the challenging problem of optimizing these metrics, we formulate it in the expected test utility (ETU) framework, which aims to optimize the expected performance on a given test set. We derive optimal prediction rules and construct their computationally efficient approximations with provable regret guarantees and being robust against model misspecification. Our algorithm, based on block coordinate descent, scales effortlessly to XMLC problems and obtains promising results in terms of long-tail performance.

Miscellaneous Aspects of Machine Learning/Sequential, Network, and Time Series Modeling

Giovanni De Felice, John Goulermas, Vladimir Gusev

Kernel design is a pivotal but challenging aspect of time series analysis, especially in the context of small datasets. In recent years, Reservoir Computing (RC) has emerged as a powerful tool to compare time series based on the underlying dynamics of the generating process rather than the observed data. However, the performance of RC highly depends on the hyperparameter setting, which is hard to interpret and costly to optimize because of the recurrent nature of RC. Here, we present a new kernel for time series based on the recently established equivalence between reservoir dynamics and Nonlinear Vector AutoRegressive (NVAR) processes. The kernel is non-recurrent and depends on a small set of meaningful hyperparameters, for which we suggest an effective heuristic. We demonstrate excellent performance on a wide range of real-world classification tasks, both in terms of accuracy and speed. This further advances the understanding of RC representation learning models and extends the typical use of the NVAR framework to kernel design and representation of real-world time series data.

Miscellaneous Aspects of Machine Learning/Transfer, Multitask and Meta-learning

Tung Nguyen, Sudhanshu Agrawal, Aditya Grover

Experimental design is a fundamental problem in many science and engineering fields. In this problem, sample efficiency is crucial due to the time, money, and safety costs of real-world design evaluations. Existing approaches either rely on active data collection or access to large, labeled datasets of past experiments, making them impractical in many real-world scenarios. In this work, we address the more challenging yet realistic setting of few-shot experimental design, where only a few labeled data points of input designs and their corresponding values are available. We approach this problem as a conditional generation task, where a model conditions on a few labeled examples and the desired output to generate an optimal input design. To this end, we introduce Experiment Pretrained Transformers (ExPT), a foundation model for few-shot experimental design that employs a novel combination of synthetic pretraining with in-context learning. In ExPT, we only assume knowledge of a finite collection of unlabelled data points from the input domain and pretrain a transformer neural network to optimize diverse synthetic functions defined over this domain. Unsupervised pretraining allows ExPT to adapt to any design task at test time in an in-context fashion by conditioning on a few labeled data points from the target task and generating the candidate optima. We evaluate ExPT on few-shot experimental design in challenging domains and demonstrate its superior generality and performance compared to existing methods. The source code is available at https://github.com/tung-nd/ExPT.git.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Yu-Hu Yan, Peng Zhao, Zhi-Hua Zhou

In this paper, we propose an online convex optimization approach with two different levels of adaptivity. On a higher level, our approach is agnostic to the unknown types and curvatures of the online functions, while at a lower level, it can exploit the unknown niceness of the environments and attain problem-dependent guarantees. Specifically, we obtain $\mathcal{O}(\log V_T)$, $\mathcal{O}(d \log V_T)$ and $\hat{\mathcal{O}}(\sqrt{V_T})$ regret bounds for strongly convex, exp-concave and convex loss functions, respectively, where $d$ is the dimension, $V_T$ denotes problem-dependent gradient variations and the $\hat{\mathcal{O}}(\cdot)$-notation omits $\log V_T$ factors. Our result not only safeguards the worst-case guarantees but also directly implies the small-loss bounds in analysis. Moreover, when applied to adversarial/stochastic convex optimization and game theory problems, our result enhances the existing universal guarantees. Our approach is based on a multi-layer online ensemble framework incorporating novel ingredients, including a carefully designed optimism for unifying diverse function types and cascaded corrections for algorithmic stability. Notably, despite its multi-layer structure, our algorithm necessitates only one gradient query per round, making it favorable when the gradient evaluation is time-consuming. This is facilitated by a novel regret decomposition equipped with carefully designed surrogate losses.

Miscellaneous Aspects of Machine Learning/Unsupervised and Semi-supervised Learning

Peter Macgregor

Spectral clustering is a popular and effective algorithm designed to find $k$ clusters in a graph $G$.In the classical spectral clustering algorithm, the vertices of $G$ are embedded into $\mathbb{R}^k$ using $k$ eigenvectors of the graph Laplacian matrix.However, computing this embedding is computationally expensive and dominates the running time of the algorithm.In this paper, we present a simple spectral clustering algorithm based on a vertex embedding with $O(\log(k))$ vectors computed by the power method.The vertex embedding is computed in nearly-linear time with respect to the size of the graph, andthe algorithm provably recovers the ground truth clusters under natural assumptions on the input graph.We evaluate the new algorithm on several synthetic and real-world datasets, finding that it is significantly faster than alternative clustering algorithms,while producing results with approximately the same clustering accuracy.