arXiv subjects

Gábor Lugosi

Publications and source records attributed to Gábor Lugosi.

At least 19 recordsLinked to original sources

arXiv2026

Finding Adam in noisy trees

We consider the problem of finding the root vertex of a random uniform attachment tree, when the union of the unlabeled tree and an Erdős-Rényi random graph $\mathbb{G}(n,p)$ is observed. We prove that, as long as $p=o(\log n /n)$, for any $\varepsilon>0$, one can construct a confidence set of vertices of size $K(\varepsilon)$ that depends only on $\varepsilon$ and not on $n$, such that it contains the root with probability at least $1-\varepsilon$. This affirms a conjecture of Crane and Xu (2021). Our approach ranks vertices by their Jordan centrality in the largest component of the subgraph spanned by high-degree vertices. We show that the same approach works in other noise models as well.

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arXiv2026

Conformal and kNN Predictive Uncertainty Quantification Algorithms in Metric Spaces

This paper introduces a framework for uncertainty quantification in regression models defined on metric spaces. Using a proposed notion of homoscedasticity, we define a conformal prediction algorithm that provides finite-sample marginal coverage guarantees and fast convergence rates to the oracle prediction region. For heteroscedastic settings, we introduce a kNN procedure that yields locally adaptive prediction radii in general metric spaces. Although this procedure does not provide the same finite-sample guarantees as the conformal algorithm, it is designed to improve local coverage calibration without imposing smoothing assumptions. Both procedures are compatible with a broad range of regression algorithms and scale to large datasets, allowing practitioners to use their preferred models and incorporate domain-specific knowledge. Building on the heteroscedastic $k$NN approach, we also develop a flexible sequential extension for metric-space-valued time series based on nearest-neighbor expert aggregation. We establish the consistency of the proposed estimators under minimal conditions. Finally, we illustrate the practical utility of our framework in personalized medicine applications involving random objects such as probability distributions and graph Laplacians.

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arXiv2026

The temporal stochastic block model

Motivated by the need to understand infection spreading in inhomogeneous populations, we consider a \emph{temporal} version of the stochastic block model, where each edge is equipped with a random label, interpreted as a timestamp. We study the size and structure of reachable sets via increasing paths (that is, paths whose edge timestamps are strictly increasing) where the connections per node are of the order of $\log n$. We prove tight results for the first order asymptotics of the size of the reachable set from a typical vertex, and the proportions of reached vertices in each community. We also determine the typical length of the shortest and longest increasing paths with one or two fixed endpoints. In the regime where the reachable set $S$ of a typical vertex is of sublinear size, we describe the structure of the subgraph induced by $S$ by identifying an almost spanning tree (i.e., spanning all but $o(|S|)$ vertices), that is distributed as a weighted random recursive tree, whose height, profile and degree sequence are known. In particular, when $|S|\ll \sqrt{n}$, the subgraph induced by $S$ is itself distributed as this well-understood weighted random recursive tree.

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arXiv2026

Detecting weighted hidden cliques

We study a generalization of the classical hidden clique problem to graphs with real-valued edge weights. Formally, we define a hypothesis testing problem. Under the null hypothesis, edges of a complete graph on $n$ vertices are associated with independent and identically distributed edge weights from a distribution $P$. Under the alternate hypothesis, $k$ vertices are chosen at random and the edge weights between them are drawn from a distribution $Q$, while the remaining are sampled from $P$. The goal is to decide, upon observing the edge weights, which of the two hypotheses they were generated from. We investigate the problem under two different scenarios: (1) when $P$ and $Q$ are completely known, and (2) when there is only partial information of $P$ and $Q$. In the first scenario, we obtain statistical limits on $k$ when the two hypotheses are distinguishable, and when they are not. Additionally, in each of the scenarios, we provide bounds on the minimal risk of the hypothesis testing problem when $Q$ is not absolutely continuous with respect to $P$. We also provide computationally efficient spectral tests that can distinguish the two hypotheses as long as $k=Ω(\sqrt{n})$ in both the scenarios.

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arXiv2026

Testing properties of trees in graphical models with covariance queries

We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the underlying graph is a tree. The main results of the paper show that, while reconstructing the entire tree may be costly, certain global structural properties can be tested efficiently. In particular, we design randomized tests for global structural properties that use a sub-quadratic number of queries. We develop testing procedures for several fundamental properties, including the number of leaves, the maximum degree, the typical distance, and the diameter of the tree. For each property, we obtain explicit query complexity bounds that depend on the target threshold and tolerance parameters.

stat.ML↗

arXiv2026

Ratio Covers of Convex Sets and Optimal Mixture Density Estimation

We study density estimation in Kullback-Leibler divergence: given an i.i.d. sample from an unknown density $p^\star$, the goal is to construct an estimator $\widehat{p}$ such that $\mathrm{KL}(p^\star,\widehat{p})$ is small with high probability. We consider two fundamental settings involving a finite dictionary of densities: (i) model aggregation, where $p^\star$ belongs to the dictionary, and (ii) convex aggregation (mixture density estimation), where $p^\star$ is a mixture of densities from the dictionary. Crucially, we make no assumption on the base densities: their ratios may be unbounded and their supports may differ. For both problems, we identify the best possible high-probability guarantees in terms of the dictionary size, sample size, and confidence level. These optimal rates are higher than those achievable when density ratios are bounded by absolute constants; for mixture density estimation, they match existing lower bounds in the special case of discrete distributions. Our analysis of the mixture case hinges on two new covering results. First, we provide a sharp, distribution-free upper bound on the local Hellinger entropy of the class of mixtures of $M$ distributions. Second, we prove an optimal ratio covering theorem for convex sets: for every convex compact set $K \subset \mathbb{R}_+^d$, there exists a subset $A \subset K$ with at most $2^{O(d)}$ elements such that each element of $K$ is coordinate-wise dominated by an element of $A$ up to a universal constant factor. This geometric result is of independent interest; notably, it yields new cardinality estimates for $\varepsilon$-approximate Pareto sets in multi-objective optimization with convex feasible set.

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arXiv2026

The statistical threshold for planted matchings and spanning trees

In this paper, we study the problem of detecting the presence of a planted perfect matching or spanning tree in an Erdős--Rényi random graph. More precisely, we study the hypothesis testing problem where the statistician observes a graph on $n$ vertices. Under the null hypothesis, the graph is a realization of an Erdős--Rényi random graph $G(n,q)$, while under the alternative hypothesis, the graph is the union of an Erdős--Rényi random graph and a random perfect matching (or random spanning tree). In order to avoid trivial detection by counting edges, we adjust the alternative hypothesis so that the expected number of edges under both distributions coincides. We prove that in both problems, when $q\gg n^{-1/2}$, no test can perform better than random guessing, while for $q\ll n^{-1/2}$, there exist computationally efficient tests that guess correctly with high probability.

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arXiv2025

Subtractive random forests with two choices

Recommendation systems are pivotal in aiding users amid vast online content. Broutin, Devroye, Lugosi, and Oliveira proposed Subtractive Random Forests (\textsc{surf}), a model that emphasizes temporal user preferences. Expanding on \textsc{surf}, we introduce a model for a multi-choice recommendation system, enabling users to select from two independent suggestions based on past interactions. We evaluate its effectiveness and robustness across diverse scenarios, incorporating heavy-tailed distributions for time delays. By analyzing user topic evolution, we assess the system's consistency. Our study offers insights into the performance and potential enhancements of multi-choice recommendation systems in practical settings.

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arXiv2025

On the quality of randomized approximations of Tukey's depth

Tukey's depth (or halfspace depth) is a widely used measure of centrality for multivariate data. However, exact computation of Tukey's depth is known to be a hard problem in high dimensions. As a remedy, randomized approximations of Tukey's depth have been proposed. In this paper we explore when such randomized algorithms return a good approximation of Tukey's depth. We study the case when the data are sampled from a log-concave isotropic distribution. We prove that, if one requires that the algorithm runs in polynomial time in the dimension, the randomized algorithm correctly approximates the maximal depth $1/2$ and depths close to zero. On the other hand, for any point of intermediate depth, any good approximation requires exponential complexity.

stat.ML↗

arXiv2025

Temporal connectivity of Random Geometric Graphs

A temporal random geometric graph is a random geometric graph in which all edges are endowed with a uniformly random time-stamp, representing the time of interaction between vertices. In such graphs, paths with increasing time stamps indicate the propagation of information. We determine a threshold for the existence of monotone increasing paths between all pairs of vertices in temporal random geometric graphs. The results reveal that temporal connectivity appears at a significantly larger edge density than simple connectivity of the underlying random geometric graph. This is in contrast with Erdős-Rényi random graphs in which the thresholds for temporal connectivity and simple connectivity are of the same order of magnitude. Our results hold for a family of "soft" random geometric graphs as well as the standard random geometric graph.

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arXiv2024

Estimating the history of a random recursive tree

This paper studies the problem of estimating the order of arrival of the vertices in a random recursive tree. Specifically, we study two fundamental models: the uniform attachment model and the linear preferential attachment model. We propose an order estimator based on the Jordan centrality measure and define a family of risk measures to quantify the quality of the ordering procedure. Moreover, we establish a minimax lower bound for this problem, and prove that the proposed estimator is nearly optimal. Finally, we numerically demonstrate that the proposed estimator outperforms degree-based and spectral ordering procedures.

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arXiv2024

Online-to-PAC Conversions: Generalization Bounds via Regret Analysis

We present a new framework for deriving bounds on the generalization bound of statistical learning algorithms from the perspective of online learning. Specifically, we construct an online learning game called the "generalization game", where an online learner is trying to compete with a fixed statistical learning algorithm in predicting the sequence of generalization gaps on a training set of i.i.d. data points. We establish a connection between the online and statistical learning setting by showing that the existence of an online learning algorithm with bounded regret in this game implies a bound on the generalization error of the statistical learning algorithm, up to a martingale concentration term that is independent of the complexity of the statistical learning method. This technique allows us to recover several standard generalization bounds including a range of PAC-Bayesian and information-theoretic guarantees, as well as generalizations thereof.

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arXiv2024

Leaf Stripping on Uniform Attachment Trees

In this note we analyze the performance of a simple root-finding algorithm in uniform attachment trees. The leaf-stripping algorithm recursively removes all leaves of the tree for a carefully chosen number of rounds. We show that, with probability $1 - ε$, the set of remaining vertices contains the root and has a size only depending on $ε$ but not on the size of the tree.

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arXiv2024

Property testing in graphical models: testing small separation numbers

In many statistical applications, the dimension is too large to handle for standard high-dimensional machine learning procedures. This is particularly true for graphical models, where the interpretation of a large graph is difficult and learning its structure is often computationally impossible either because the underlying graph is not sufficiently sparse or the number of vertices is too large. To address this issue, we develop a procedure to test a property of a graph underlying a graphical model that requires only a subquadratic number of correlation queries (i.e., we require that the algorithm only can access a tiny fraction of the covariance matrix). This provides a conceptually simple test to determine whether the underlying graph is a tree or, more generally, if it has a small separation number, a quantity closely related to the treewidth of the graph. The proposed method is a divide-and-conquer algorithm that can be applied to quite general graphical models.

math.ST↗

arXiv2024

Uncertainty quantification in metric spaces

This paper introduces a novel uncertainty quantification framework for regression models where the response takes values in a separable metric space, and the predictors are in a Euclidean space. The proposed algorithms can efficiently handle large datasets and are agnostic to the predictive base model used. Furthermore, the algorithms possess asymptotic consistency guarantees and, in some special homoscedastic cases, we provide non-asymptotic guarantees. To illustrate the effectiveness of the proposed uncertainty quantification framework, we use a linear regression model for metric responses (known as the global Fréchet model) in various clinical applications related to precision and digital medicine. The different clinical outcomes analyzed are represented as complex statistical objects, including multivariate Euclidean data, Laplacian graphs, and probability distributions.

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arXiv2024

The top eigenvalue of uniformly random trees

Let ${\mathbf T}_n$ be a uniformly random tree with vertex set $[n]=\{1,\ldots,n\}$, let $Δ_{{\mathbf T}_n}$ be the largest vertex degree in ${\mathbf T}_n$, and let $λ_1({\mathbf T}_n),\ldots,λ_n({\mathbf T}_n)$ be the eigenvalues of its adjacency matrix, arranged in decreasing order. We prove that $|λ_1({\mathbf T}_n)-\sqrt{Δ_{{\mathbf T}_n}}| \to 0$ in expectation as $n \to \infty$, and additionally prove probability tail bounds for $|λ_1({\mathbf T}_n)-\sqrt{Δ_{{\mathbf T}_n}}|$. Writing $a_n$ for any median of $Δ_{{\mathbf T}_n}$, we also prove that $|λ_k({\mathbf T}_n)-\sqrt{a_n}| \to 0$ in expectation, uniformly over $1 \le k \le e^{\log^β(n)}$, for any fixed $β\in (0,1/2)$. The proof is based on the trace method and thus on counting closed walks in a random tree. To this end, we develop novel combinatorial tools for encoding walks in trees that we expect will find other applications. In order to apply these tools, we show that uniformly random trees -- after appropriate "surgery" -- satisfy, with high probability, the properties required for the combinatorial bounds to be effective.

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arXiv2024

Random friend trees

We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saramäki and Kaski. Vertices are attached in a sequential manner one by one by selecting an existing target vertex and connecting to one of its neighbours (or friends), chosen uniformly at random. This model has interesting emergent properties, such as a highly skewed degree sequence. In contrast to the preferential attachment model, these emergent phenomena stem from a local rather than a global attachment mechanism. The structure of the resulting tree is also strikingly different from both the preferential attachment tree and the uniform random recursive tree: every edge is incident to a macro-hub of asymptotically linear degree, and with high probability all but at most $n^{9/10}$ vertices in a tree of size $n$ are leaves. We prove various results on the neighbourhood of fixed vertices and edges, and we study macroscopic properties such as the diameter and the degree distribution, providing insights into the overall structure of the tree. We also present a number of open questions on this model and related models.

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arXiv2022

Archaeology of random recursive dags and Cooper-Frieze random networks

We study the problem of finding the root vertex in large growing networks. We prove that it is possible to construct confidence sets of size independent of the number of vertices in the network that contain the root vertex with high probability in various models of random networks. The models include uniform random recursive dags and uniform Cooper-Frieze random graphs.

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