arXiv subjects

Frederik Mallmann-Trenn

Publications and source records attributed to Frederik Mallmann-Trenn.

At least 19 recordsLinked to original sources

arXiv2026

Degree-Mass Message Passing for Betweenness Ranking in Directed and Undirected Networks

Computing the importance of nodes in networks is a long-standing fundamental problem that has driven extensive study of various centrality measures. A particularly well-known centrality measure is betweenness centrality, whose exact computation becomes prohibitive on large-scale networks. Graph Neural Network (GNN) models have thus been proposed to predict the ranking of nodes by betweenness centrality. However, existing GNN-based methods either have graph-size-dependent parameter counts or are limited to undirected graphs. We propose a lightweight GNN architecture that exploits the empirically observed relationship between betweenness centrality and multi-hop degree mass. This motivates the use of degree masses as size-invariant node features. To improve generalization, we train on synthetic graphs whose degree distributions more closely match those of real-world networks, including directed and undirected scale-free graphs and uniformly directed hyperbolic random graphs. We evaluate our model on 14 real-world networks spanning eight domains, including social, email, and citation networks, across both directed and undirected regimes. The experiments show that our model improves the Kendall $τ_b$ correlation by up to 24.6\% on undirected and 10.9\% on directed graphs, while using 56$\times$ fewer parameters than the lightest competing GNN baseline and achieving competitive inference time, with up to a 24.5$\times$ speedup on selected directed graphs.

cs.LG↗

arXiv2026

Probably Approximately Correct Maximum A Posteriori Inference

Computing the conditional mode of a distribution, better known as the maximum a posteriori (MAP) assignment, is a fundamental task in probabilistic inference. However, MAP is generally intractable, and remains hard even under many common structural constraints and approximation schemes. We take a novel approach inspired by multi-armed bandits, recasting MAP as a best arm identification task. We introduce probably approximately correct (PAC) algorithms for MAP that provide provably optimal solutions in both the fixed-confidence and fixed-budget regimes. We characterize tractability conditions using information theoretic measures that can be estimated from finite samples. Our PAC-MAP solvers are efficiently implemented using probabilistic circuits and graphical models with appropriate architectures. The algorithms we develop can be used either as standalone MAP estimators or to improve on standard heuristics, fortifying their solutions with rigorous guarantees. Experiments confirm the benefits of our method in a range of benchmarks.

cs.LG↗

arXiv2026

Decision-Making Under Complete Uncertainty: You Will Not Regret Being Greedy

In this paper, we propose a game-theoretic model to study the properties of the worst-case regret of the greedy strategy under complete (Knightian) uncertainty. In a game between a decision-maker (DM) and an adversarial agent (Nature), Nature chooses an unknown state determining the distribution of ratings for each product. The DM observes a realization of product ratings and then chooses a product according to a strategy. For arbitrary numbers of products and ratings, we first study the equal-observations case in which every product has the same number of observations. In this benchmark, we establish matching upper and lower bounds on the worst-case regret, showing that the regret vanishes as the number of observations increases and that the greedy strategy is rate-optimal up to universal constants. In the special case with two products and two ratings, we show that with one observation per product the greedy strategy is minimax-optimal with respect to worst-case regret. We then allow products to have different numbers of observations. Greedy remains robust in a conservative sense: its worst-case regret is controlled by the least-reviewed product. However, unequal numbers of observations can also change greedy's exact worst-case behavior. In particular, adding observations for only one product can increase greedy's worst-case regret. Finally, we test the model on data collected from Google reviews for restaurants, showing that the greedy strategy's empirical performance closely aligns with the theoretical findings.

cs.GT↗

arXiv2026

Structured vs. Unstructured Pruning: An Exponential Gap

The Strong Lottery Ticket Hypothesis (SLTH) states that large, randomly initialized neural networks contain sparse subnetworks capable of approximating a target function at initialization without training, suggesting that pruning alone is sufficient. Pruning methods are typically classified as unstructured, where individual weights can be removed from the network, and structured, where parameters are removed according to specific patterns, as in neuron pruning. Existing theoretical results supporting the SLTH rely almost exclusively on unstructured pruning, showing that logarithmic overparameterization suffices to approximate simple target networks. In contrast, neuron pruning has received limited theoretical attention, despite its practical appeal for direct hardware speedups. In this work, we consider the problem of approximating a single bias-free ReLU neuron by pruning hidden units of a randomly initialized two-layer ReLU network, effectively isolating the intrinsic limitations of neuron pruning. We show that achieving an $\varepsilon$-approximation requires a starting network size of $Ω(1/\varepsilon)$ for neuron pruning, whereas weight pruning succeeds with only $O(\log(1/\varepsilon))$ hidden units, revealing an exponential separation between the two approaches.

cs.LG↗

arXiv2026

Beyond Fixed Points: Superpolynomial Capacity of Asymmetric Hopfield Networks

Classical Hopfield networks are limited to static patterns due to symmetric weights, whereas asymmetric networks can encode temporal sequences via limit-cycle attractors. Achieving high-capacity storage of long sequences in classical synchronous asymmetric networks, however, has remained a challenge. We present a simple and robust construction within the classical asymmetric Hopfield model with binary neurons and synchronous updates, that allows $n$ neurons to support $\exp\!\big(Ω(n/(\log n)^2)\big)$ distinct limit-cycle attractors, each with period $\exp\!\big(Ω(\sqrt n/\log n)\big)$ and robust to random noise with flip probability up to $\frac12-o(1)$, yielding superpolynomial capacity in both the number and length of stored sequences. This is the first demonstration of such capacity for asymmetric Hopfield networks, which we obtain by combining results from combinatorics, number theory and the analysis of opinion dynamics. Our findings show that synchronous asymmetric Hopfield networks possess a sequence-memory capacity which is larger and more robust than previously recognized, demonstrating that, in both biological and artificial neural systems, robust sequence representation can be achieved through coarse architectural motifs rather than complex nonlinearities.

cs.LG↗

arXiv2026

Voter Model Meets Rumour Spreading: an FPRAS for Consensus Probabilities on Voter Models with Agnostic Nodes

Problems of consensus in multi-agent systems are often viewed as a series of independent, simultaneous local decisions made between a limited set of options, all aimed at reaching a global agreement. Key challenges in these protocols include estimating the likelihood of various outcomes and finding bounds for how long it may take to achieve consensus, if it occurs at all. To date, little attention has been given to the case where some agents have no initial opinion. In this paper, we introduce a variant of the consensus problem which includes what we call `agnostic' nodes and frame it as a combination of two known and well-studied processes: voter model and rumour spreading. We show (1) a martingale that describes the probability of consensus for a given colour, (2) bounds on the number of steps for the process to end using results from rumour spreading and voter models, (3) closed formulas for the probability of consensus in a few special cases, along with a polynomial-time algorithm for the case where the number of agnostic vertices is at most logarithmic and (4) that the computational complexity of estimating the probability with a Markov chain Monte Carlo process is $O(n^2 \log n)$ for general graphs and $O(n\log n)$ for Erdős-Rényi graphs, resulting in a fully polynomial-time randomized approximation scheme (FPRAS) for estimating the probabilities of consensus. Furthermore, we present experimental results suggesting that the number of runs needed for a given standard error decreases when the number of nodes increases.

cs.MA↗

arXiv2026

Undecided State Dynamics with Many Opinions

We study the Undecided-State Dynamics (USD), a fundamental consensus process in which each vertex holds one of $k$ decided opinions or the undecided state. We consider both the gossip model and the population protocol model. Prior work established tight bounds on the consensus time of this process only for the regime $k = O(\sqrt{n}/(\log n)^2)$ (for the population protocol model) and $k = O((n/\log n)^{1/3})$ (for the gossip model), often under restrictive assumptions on the initial configuration. In this paper, we obtain the first consensus-time guarantees for USD that hold for \emph{arbitrary} $2\le k\le n$ and for \emph{arbitrary} initial configurations in both the gossip model and the population protocol model. In the gossip model, USD reaches consensus within $\widetilde O(\min\{k,\sqrt n\})$ synchronous rounds with probability $1-p_{\bot}-n^{-c}$, where $p_{\bot}$ is the gossip-specific probability of collapsing to the all-undecided state in the first round. In the population protocol model, USD reaches consensus within $\widetilde O(\min\{kn,n^{3/2}\})$ asynchronous interactions with high probability. We also present lower bounds that match the upper bounds up to polylogarithmic factors for a specific initial configuration and show that our upper bounds are essentially optimal.

cs.DC↗

arXiv2025

Modeling Feasible Locomotion of Nanobots for Cancer Detection and Treatment

Deploying motile nanosized particles, also known as ``nanobots'', in the human body promises to improve selectivity in drug delivery and reduce side effects. We consider a swarm of nanobots locating a single cancerous region and treating it by releasing an onboard payload of drugs at the site. At nanoscale, the computation, communication, sensing, and locomotion capabilities of individual agents are extremely limited, noisy, and/or nonexistent. We present a general model to formally describe the individual and collective behavior of agents in a colloidal environment, such as the bloodstream, for cancer detection and treatment by nanobots. This includes a feasible and precise model of agent locomotion, inspired by actual nanoparticles that, in the presence of an external chemical gradient, move towards areas of higher concentration by means of self-propulsion. We present two variants of our general model: The first assumes an endogenous chemical gradient that is fixed over time and centered at the targeted cancer site; the second is a more speculative and dynamic variant in which agents themselves create and amplify a chemical gradient centered at the cancer site. In both settings, agents can sense the gradient and ascend it noisily, locating the cancer site more quickly than via simple Brownian motion. For the first variant of the model, we present simulation results to show the behavior of agents under our locomotion model, as well as {analytical results} to bound the time it takes for the agents to reach the cancer site. For the second variant, simulation results highlight the collective benefit in having agents issue their own chemical signal. While arguably more speculative in its agent capability assumptions, this variant shows a significant improvement in runtime performance over the first variant, resulting from its chemical signal amplification mechanism.

cs.MA↗

arXiv2025

IID Prophet Inequality with Random Horizon: Going Beyond Increasing Hazard Rates

Prophet inequalities are a central object of study in optimal stopping theory. In the iid model, a gambler sees values in an online fashion, sampled independently from a given distribution. Upon observing each value, the gambler either accepts it as a reward or irrevocably rejects it and proceeds to observe the next value. The goal of the gambler, who cannot see the future, is maximising the expected value of the reward while competing against the expectation of a prophet (the offline maximum). In other words, one seeks to maximise the gambler-to-prophet ratio of the expectations. This model has been studied with infinite, finite and unknown number of values. When the gambler faces a random number of values, the model is said to have random horizon. We consider the model in which the gambler is given a priori knowledge of the horizon's distribution. Alijani et al. (2020) designed a single-threshold algorithm achieving a ratio of $1/2$ when the random horizon has an increasing hazard rate and is independent of the values. We prove that with a single threshold, a ratio of $1/2$ is actually achievable for several larger classes of horizon distributions, with the largest being known as the $\mathcal{G}$ class in reliability theory. Moreover, we show that this does not extend to its dual, the $\overline{\mathcal{G}}$ class (which includes the decreasing hazard rate class), while it can be extended to low-variance horizons. Finally, we construct the first example of a family of horizons, for which multiple thresholds are necessary to achieve a nonzero ratio. We establish that the Secretary Problem optimal stopping rule provides one such algorithm, paving the way towards the study of the model beyond single-threshold algorithms.

cs.DS↗

arXiv2025

Trading-off Accuracy and Communication Cost in Federated Learning

Leveraging the training-by-pruning paradigm introduced by Zhou et al. and Isik et al. introduced a federated learning protocol that achieves a 34-fold reduction in communication cost. We achieve a compression improvements of orders of orders of magnitude over the state-of-the-art. The central idea of our framework is to encode the network weights $\vec w$ by a the vector of trainable parameters $\vec p$, such that $\vec w = Q\cdot \vec p$ where $Q$ is a carefully-generate sparse random matrix (that remains fixed throughout training). In such framework, the previous work of Zhou et al. [NeurIPS'19] is retrieved when $Q$ is diagonal and $\vec p$ has the same dimension of $\vec w$. We instead show that $\vec p$ can effectively be chosen much smaller than $\vec w$, while retaining the same accuracy at the price of a decrease of the sparsity of $Q$. Since server and clients only need to share $\vec p$, such a trade-off leads to a substantial improvement in communication cost. Moreover, we provide theoretical insight into our framework and establish a novel link between training-by-sampling and random convex geometry.

cs.LG↗

arXiv2024

Prophet Inequalities: Separating Random Order from Order Selection

Prophet inequalities are a central object of study in optimal stopping theory. A gambler is sent values in an online fashion, sampled from an instance of independent distributions, in an adversarial, random or selected order, depending on the model. When observing each value, the gambler either accepts it as a reward or irrevocably rejects it and proceeds to observe the next value. The goal of the gambler, who cannot see the future, is maximising the expected value of the reward while competing against the expectation of a prophet (the offline maximum). In other words, one seeks to maximise the gambler-to-prophet ratio of the expectations. The model, in which the gambler selects the arrival order first, and then observes the values, is known as Order Selection. In this model a ratio of $0.7251$ is attainable for any instance. Recently, this has been improved up to $0.7258$ by Bubna and Chiplunkar (2023). If the gambler chooses the arrival order (uniformly) at random, we obtain the Random Order model. The worst case ratio over all possible instances has been extensively studied for at least $40$ years. In a computer-assisted proof, Bubna and Chiplunkar (2023) also showed that this ratio is at most $0.7254$ for the Random Order model, thus establishing for the first time that carefully choosing the order, instead of simply taking it at random, benefits the gambler. We give an alternative, non-simulation-assisted proof of this fact, by showing mathematically that in the Random Order model, no algorithm can achieve a ratio larger than $0.7235$. This sets a new state-of-the-art hardness for this model, and establishes more formally that there is a real benefit in choosing the order.

cs.DS↗

arXiv2024

On the Sparsity of the Strong Lottery Ticket Hypothesis

Considerable research efforts have recently been made to show that a random neural network $N$ contains subnetworks capable of accurately approximating any given neural network that is sufficiently smaller than $N$, without any training. This line of research, known as the Strong Lottery Ticket Hypothesis (SLTH), was originally motivated by the weaker Lottery Ticket Hypothesis, which states that a sufficiently large random neural network $N$ contains \emph{sparse} subnetworks that can be trained efficiently to achieve performance comparable to that of training the entire network $N$. Despite its original motivation, results on the SLTH have so far not provided any guarantee on the size of subnetworks. Such limitation is due to the nature of the main technical tool leveraged by these results, the Random Subset Sum (RSS) Problem. Informally, the RSS Problem asks how large a random i.i.d. sample $Ω$ should be so that we are able to approximate any number in $[-1,1]$, up to an error of $ ε$, as the sum of a suitable subset of $Ω$. We provide the first proof of the SLTH in classical settings, such as dense and equivariant networks, with guarantees on the sparsity of the subnetworks. Central to our results, is the proof of an essentially tight bound on the Random Fixed-Size Subset Sum Problem (RFSS), a variant of the RSS Problem in which we only ask for subsets of a given size, which is of independent interest.

stat.ML↗

arXiv2024

Asynchronous 3-Majority Dynamics with Many Opinions

We consider 3-Majority, a probabilistic consensus dynamics on a complete graph with $n$ vertices, each vertex starting with one of $k$ initial opinions. At each discrete time step, a vertex $u$ is chosen uniformly at random. The selected vertex $u$ chooses three neighbors $v_1,v_2,v_3$ uniformly at random with replacement and takes the majority opinion held by the three, where ties are broken in favor of the opinion of $v_3$. The main quantity of interest is the consensus time, the number of steps required for all vertices to hold the same opinion. This asynchronous version turns out to be considerably harder to analyze than the synchronous version and so far results have only been obtained for $k=2$. Even in the synchronous version the results for large $k$ are far from tight. In this paper we prove that the consensus time is $\tildeΘ( \min(nk,n^{1.5}) )$ for all $k$. These are the first bounds for all $k$ that are tight up to a polylogarithmic factor.

cs.DC↗

arXiv2024

Community Consensus: Converging Locally despite Adversaries and Heterogeneous Connectivity

We introduce the concept of community consensus in the presence of malicious agents using a well-known median-based consensus algorithm. We consider networks that have multiple well-connected regions that we term communities, characterized by specific robustness and minimum degree properties. Prior work derives conditions on properties that are necessary and sufficient for achieving global consensus in a network. This, however, requires the minimum degree of the network graph to be proportional to the number of malicious agents in the network, which is not very practical in large networks. In this work, we present a natural generalization of this previous result. We characterize cases where, although global consensus is not reached, some subsets of agents $V_i$ will still converge to the same values $M_i$ among themselves. To reach this new type of consensus, we define more relaxed requirements in terms of the number of malicious agents in each community, and the number $k$ of edges connecting an agent in a community to agents external to the community.

cs.MA↗

arXiv2024

Sorting in One and Two Rounds using $t$-Comparators

We examine sorting algorithms for $n$ elements whose basic operation is comparing $t$ elements simultaneously (a $t$-comparator). We focus on algorithms that use only a single round or two rounds -- comparisons performed in the second round depend on the outcomes of the first round comparators. We design deterministic and randomized algorithms. In the deterministic case, we show an interesting relation to design theory (namely, to 2-Steiner systems), which yields a single-round optimal algorithm for $n=t^{2^k}$ with any $k\ge 1$ and a variety of possible values of $t$. For some values of $t$, however, no algorithm can reach the optimal (information-theoretic) bound on the number of comparators. For this case (and any other $n$ and $t$), we show an algorithm that uses at most three times as many comparators as the theoretical bound. We also design a randomized Las-Vegas two-rounds sorting algorithm for any $n$ and $t$. Our algorithm uses an asymptotically optimal number of $O(\max(\frac{n^{3/2}}{t^2},\frac{n}{t}))$ comparators, with high probability, i.e., with probability at least $1-1/n$. The analysis of this algorithm involves the gradual unveiling of randomness, using a novel technique which we coin the binary tree of deferred randomness.

cs.DS↗

arXiv2024

Learning Hierarchically Structured Concepts

We study the question of how concepts that have structure get represented in the brain. Specifically, we introduce a model for hierarchically structured concepts and we show how a biologically plausible neural network can recognize these concepts, and how it can learn them in the first place. Our main goal is to introduce a general framework for these tasks and prove formally how both (recognition and learning) can be achieved. We show that both tasks can be accomplished even in presence of noise. For learning, we analyze Oja's rule formally, a well-known biologically-plausible rule for adjusting the weights of synapses. We complement the learning results with lower bounds asserting that, in order to recognize concepts of a certain hierarchical depth, neural networks must have a corresponding number of layers.

cs.AI↗

arXiv2023

Learning Hierarchically-Structured Concepts II: Overlapping Concepts, and Networks With Feedback

We continue our study from Lynch and Mallmann-Trenn (Neural Networks, 2021), of how concepts that have hierarchical structure might be represented in brain-like neural networks, how these representations might be used to recognize the concepts, and how these representations might be learned. In Lynch and Mallmann-Trenn (Neural Networks, 2021), we considered simple tree-structured concepts and feed-forward layered networks. Here we extend the model in two ways: we allow limited overlap between children of different concepts, and we allow networks to include feedback edges. For these more general cases, we describe and analyze algorithms for recognition and algorithms for learning.

cs.NE↗

arXiv2023

Distributed Averaging in Opinion Dynamics

We consider two simple asynchronous opinion dynamics on arbitrary graphs where every node $u$ has an initial value $ξ_u(0)$. In the first process, the NodeModel, at each time step $t\ge 0$, a random node $u$ and a random sample of $k$ of its neighbours $v_1,v_2,\cdots,v_k$ are selected. Then, $u$ updates its current value $ξ_u(t)$ to $ξ_u(t+1) = αξ_u(t) + \frac{(1-α)}{k} \sum_{i=1}^k ξ_{v_i}(t)$, where $α\in (0,1)$ and $k\ge 1$ are parameters of the process. In the second process, the EdgeModel, at each step a random pair of adjacent nodes $(u,v)$ is selected, and then node $u$ updates its value equivalently to the NodeModel with $k=1$ and $v$ as the selected neighbour. For both processes, the values of all nodes converge to $F$, a random variable depending on the random choices made in each step. For the NodeModel and regular graphs, and for the EdgeModel and arbitrary graphs, the expectation of $F$ is the average of the initial values $\frac{1}{n}\sum_{u\in V} ξ_u(0)$. For the NodeModel and non-regular graphs, the expectation of $F$ is the degree-weighted average of the initial values. Our results are two-fold. We consider the concentration of $F$ and show tight bounds on the variance of $F$ for regular graphs. We show that, when the initial values do not depend on the number of nodes, then the variance is negligible, hence the nodes are able to estimate the initial average of the node values. Interestingly, this variance does not depend on the graph structure. For the proof we introduce a duality between our processes and a process of two correlated random walks. We also analyse the convergence time for both models and for arbitrary graphs, showing bounds on the time $T_\varepsilon$ required to make all node values `$\varepsilon$-close' to each other. Our bounds are asymptotically tight under assumptions on the distribution of the initial values.

math.PR↗