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Gennaro Auricchio

Publications and source records attributed to Gennaro Auricchio.

At least 19 recordsLinked to original sources

I-SAFE: Wasserstein Coherence Metrics for Structural Auditing of Scientific AI Models

Deep learning models are increasingly used in scientific prediction tasks where strong benchmark performance is often interpreted as evidence of scientifically meaningful behavior. This interpretation is fragile, as models may exploit shortcut features, dataset-specific regularities, or distributional biases that are predictive on held-out data but not aligned with domain-relevant structure. To address this limitation, we introduce the \textsc{I-SAFE} (Interventional Secure, Accurate, Fair and Explainable) framework, a post-hoc distributional auditing framework for scientific AI models centered on the Wasserstein Coherence Metric (WCM). Given a trained black-box predictor and an external structural prior encoding domain knowledge about task-relevant input structure, \textsc{I-SAFE} evaluates raw model outputs under structurally guided perturbations of the input. The proposed audit measures output-distribution coherence through three complementary metrics: a Quantile-Based Metric (QBM) for location-level coherence, the WCM for ordinal coherence, and a translation-invariant WCM variant for shape coherence. We instantiate \textsc{I-SAFE} on drug--target interaction (DTI) prediction using the Davis kinase benchmark, KLIFS (Kinase--Ligand Interaction Fingerprints and Structures) binding-pocket annotations, and three sequence-based DTI models: DeepConvDTI, DeepDTA, and TAPB. Although the models operate in a comparable predictive regime, \textsc{I-SAFE} reveals substantially different distributional response profiles, a distinction invisible to accuracy-based evaluation. The framework is model-agnostic and applicable to any domain where inputs admit a structured decomposition and an external prior is available.

cs.LG↗

On Rank Graduation Metrics for High Dimensional Ordinal Data

Evaluating the reliability of machine learning classifications remains a fundamental challenge in Artificial Intelligence (AI), particularly when the target variable is multidimensional. Classification variables can be expressed by means of a categorical scale which, at best, is ordinal. Because ordinal data lack a natural metric structure in their underlying space, most conventional distance measures aimed at assessing the accuracy of machine learning classifications cannot be directly or meaningfully applied. In this paper, we develop a mathematical framework for comparing ordinal data based on a family of Rank Graduation $(\mathrm{RGX}_p)$ \emph{metrics}. We demonstrate that these metrics can quantify the proportion of variability of the response explained by the predictions, in a similar manner as the predictive $R^2$ for continuous response variables. After establishing theoretical connections between the $\mathrm{RGX}_p$ family and other prominent metrics in AI, we conduct extensive experiments across diverse datasets and learning tasks to evaluate their empirical performance. The results underscore the versatility, interpretability, and robustness of the $\mathrm{RGX}_p$ metrics as a principled foundation for developing trustworthy and SAFE AI systems.

math.OC↗

Energy distance and evolution problems: a promising tool for kinetic equations

We study the rate of convergence to equilibrium of the solutions to Fokker-Planck type equations with linear drift by means of Cramér and Energy distances, which have been recently widely used in problems related to AI, in particular for tasks related to machine learning. In all cases in which the Fokker-Planck type equations can be treated through these distances, it is shown that the rate of decay is improved with respect to known results which are based on the decay of relative entropy.

math.AP↗

On the computation of the infinity Wasserstein distance and the Wasserstein Projection Problem

Computing the infinity Wasserstein distance and retrieving projections of a probability measure onto a closed subset of probability measures are critical sub-problems in various applied fields. However, the practical applicability of these objects is limited by two factors: either the associated quantities are computationally prohibitive or there is a lack of available algorithms capable of calculating them. In this paper, we propose a novel class of Linear Programming problems and a routine that allows us to compute the infinity Wasserstein distance and to compute a projection of a probability measure over a generic subset of probability measures with respect to any $p$-Wasserstein distance with $p\in[1,\infty]$.

math.OC↗

From Kinetic Theory to AI: a Rediscovery of High-Dimensional Divergences and Their Properties

Selecting an appropriate divergence measure is a critical aspect of machine learning, as it directly impacts model performance. Among the most widely used, we find the Kullback-Leibler (KL) divergence, originally introduced in kinetic theory as a measure of relative entropy between probability distributions. Just as in machine learning, the ability to quantify the proximity of probability distributions plays a central role in kinetic theory. In this paper, we present a comparative review of divergence measures rooted in kinetic theory, highlighting their theoretical foundations and exploring their potential applications in machine learning and artificial intelligence.

math-ph↗

On the Distortion of Multi-winner Election Using Single-Candidate Ballots

In this paper, we study the distortion bounds for voting mechanisms in multi-winner elections in general metric spaces. Our study pertains to the case in which each voter only reports her favorite candidate amongst $m$ possible choices. Given that candidates' locations are undisclosed to the mechanism, the mechanism has to form a $w-$winner committee based solely on the number of votes received by candidates. We establish distortion bounds for both truthful and non-truthful mechanisms. Our research highlights the significance of the $σ$ parameter, which represents the ratio between maximum and minimum distances among all candidate pairs. We show that the distortion is linear in $σ$. First, we demonstrate that all mechanisms possess a distortion greater than $1+\frac{w-1}{w+1}(σ-1)$. To give an upper bound, we study the Single Non-Transferable Vote (SNTV) mechanism, whose distortion is at most $1+2σ$. Second, we retrieve the upper bounds for strategyproof mechanisms. In particular, we infer an upper bound by examining the Random Sequential Dictator mechanism that achieves a distortion less than $1+4σ$ when $w=2$.

cs.GT↗

Designing Optimal Mechanisms to Locate Facilities with Insufficient Capacity for Bayesian Agents

In this paper, we study the Facility Location Problem with Scarce Resources (FLPSR) under the assumption that agents' type follow a probability distribution. In the FLPSR, the objective is to identify the optimal locations for one or more capacitated facilities to maximize Social Welfare (SW), defined as the sum of the utilities of all agents. The total capacity of the facilities, however, is not enough to accommodate all the agents, who thus compete in a First-Come-First-Served game to determine whether they get accommodated and what their utility is. The main contribution of this paper ties Optimal Transport theory to the problem of determining the best truthful mechanism for the FLPSR tailored to the agents' type distributions. Owing to this connection, we identify the mechanism that maximizes the expected SW as the number of agents goes to infinity. For the case of a single facility, we show that an optimal mechanism always exists. We examine three classes of probability distributions and characterize the optimal mechanism either analytically represent the optimal mechanism or provide a routine to numerically compute it. We then extend our results to the case in which we have two capacitated facilities to place. While we initially assume that agents are independent and identically distributed, we show that our techniques are applicable to scenarios where agents are not identically distributed. Finally, we validate our findings through several numerical experiments, including: (i) deriving optimal mechanisms for the class of beta distributions, (ii) assessing the Bayesian approximation ratio of these mechanisms for small numbers of agents, and (iii) assessing how quickly the expected SW attained by the mechanism converges to its limit.

cs.GT↗

How to measure multidimensional variation?

The coefficient of variation, which measures the variability of a distribution from its mean, is not uniquely defined in the multidimensional case, and so is the multidimensional Gini index, which measures the inequality of a distribution in terms of the mean differences among its observations. In this paper, we connect these two notions of sparsity, and propose a multidimensional coefficient of variation based on a multidimensional Gini index. We demonstrate that the proposed coefficient possesses the properties of the univariate coefficient of variation. We also show its connection with the Voinov-Nikulin coefficient of variation, and compare it with the other multivariate coefficients available in the literature.

math.ST↗

Multivariate Gini-type discrepancies

Measuring distances in a multidimensional setting is a challenging problem, which appears in many fields of science and engineering. In this paper, to measure the distance between two multivariate distributions, we introduce a new measure of discrepancy which is scale invariant and which, in the case of two independent copies of the same distribution, and after normalization, coincides with the scaling invariant multidimensional version of the Gini index recently proposed in [34]. A byproduct of the analysis is an easy-to-handle discrepancy metric, obtained by application of the theory to a pair of Gaussian multidimensional densities. The obtained metric does improve the standard metrics, based on the mean squared error, as it is scale invariant. The importance of this theoretical finding is illustrated by means of a real problem that concerns measuring the importance of Environmental, Social and Governance factors for the growth of small and medium enterprises.

stat.ME↗

Facility Location Problem with Aleatory Agents

In this paper, we introduce and study the Facility Location Problem with Aleatory Agents (FLPAA), where the facility accommodates n agents larger than the number of agents reporting their preferences, namely n_r. The spare capacity is used by n_u=n-n_r aleatory agents sampled from a probability distribution μ. The goal of FLPAA is to find a location that minimizes the ex-ante social cost, which is the expected cost of the n_u agents sampled from μplus the cost incurred by the agents reporting their position. We investigate the mechanism design aspects of the FLPAA under the assumption that the Mechanism Designer (MD) lacks knowledge of the distribution $μ$ but can query k quantiles of μ. We explore the trade-off between acquiring more insights into the probability distribution and designing a better-performing mechanism, which we describe through the strong approximation ratio (SAR). The SAR of a mechanism measures the highest ratio between the cost of the mechanisms and the cost of the optimal solution on the worst-case input x and worst-case distribution μ, offering a metric for efficiency that does not depend on μ. We divide our study into four different information settings: the zero information case, in which the MD has access to no quantiles; the median information case, in which the MD has access to the median of μ; the n_u-quantile information case, in which the MD has access to n_u quantiles of its choice, and the k-quantile information case, in which the MD has access to k<n_u quantiles of its choice. For all frameworks, we propose a mechanism that is optimal or achieves a small constant SAR and pairs it with a lower bound on the SAR. In most cases, the lower bound matches the upper bound, thus no truthful mechanism can achieve a lower SAR. Lastly, we extend the FLPAA to include instances in which we must locate two facilities.

cs.GT↗

Extending the Gini Index to Higher Dimensions via Whitening Processes

Measuring the degree of inequality expressed by a multivariate statistical distribution is a challenging problem, which appears in many fields of science and engineering. In this paper, we propose to extend the well known univariate Gini coefficient to multivariate distributions, by maintaining most of its properties. Our extension is based on the application of whitening processes that possess the property of scale stability.

math.ST↗

Mechanism Design for Locating Facilities with Capacities with Insufficient Resources

This paper explores the Mechanism Design aspects of the $m$-Capacitated Facility Location Problem where the total facility capacity is less than the number of agents. Following the framework outlined by Aziz et al., the Social Welfare of the facility location is determined through a First-Come-First-Served (FCFS) game, in which agents compete once the facility positions are established. When the number of facilities is $m > 1$, the Nash Equilibrium (NE) of the FCFS game is not unique, making the utility of the agents and the concept of truthfulness unclear. To tackle these issues, we consider absolutely truthful mechanisms, i.e. mechanisms that prevent agents from misreporting regardless of the strategies used during the FCFS game. We combine this stricter truthfulness requirement with the notion of Equilibrium Stable (ES) mechanisms, which are mechanisms whose Social Welfare does not depend on the NE of the FCFS game. We demonstrate that the class of percentile mechanisms is absolutely truthful and identify the conditions under which they are ES. We also show that the approximation ratio of each ES percentile mechanism is bounded and determine its value. Notably, when all the facilities have the same capacity and the number of agents is sufficiently large, it is possible to achieve an approximation ratio smaller than $1+\frac{1}{2m-1}$. Finally, we extend our study to encompass higher-dimensional problems. Within this framework, we demonstrate that the class of ES percentile mechanisms is even more restricted and characterize the mechanisms that are both ES and absolutely truthful. We further support our findings by empirically evaluating the performance of the mechanisms when the agents are the samples of a distribution.

cs.GT↗

The k-Facility Location Problem Via Optimal Transport: A Bayesian Study of the Percentile Mechanisms

In this paper, we investigate the $k$-Facility Location Problem ($k$-FLP) within the Bayesian Mechanism Design framework, in which agents' preferences are samples of a probability distributed on a line. Our primary contribution is characterising the asymptotic behavior of percentile mechanisms, which varies according to the distribution governing the agents' types. To achieve this, we connect the $k$-FLP and projection problems in the Wasserstein space. Owing to this relation, we show that the ratio between the expected cost of a percentile mechanism and the expected optimal cost is asymptotically bounded. Furthermore, we characterize the limit of this ratio and analyze its convergence speed. Our asymptotic study is complemented by deriving an upper bound on the Bayesian approximation ratio, applicable when the number of agents $n$ exceeds the number of facilities $k$. We also characterize the optimal percentile mechanism for a given agent's distribution through a system of $k$ equations. Finally, we estimate the optimality loss incurred when the optimal percentile mechanism is derived using an approximation of the agents' distribution rather than the actual distribution.

cs.GT↗

From Optimal Transport to Efficient Mechanisms for the $m$-Capacitated Facilities Location Problem with Bayesian Agent

In this paper, we study of the $m$-Capacitated Facility Location Problem ($m$-CFLP) on the line from a Bayesian Mechanism Design perspective and propose a novel class of mechanisms: the \textit{Extended Ranking Mechanisms} (ERMs). We first show that an ERM is truthful if and only if it satisfies a system of inequalities that depends on the capacities of the facilities we need to place. We then establish a connection between the $m$-CFLP and a norm minimization problem in the Wasserstein space, which enables us to show that if the number of agent goes to infinity the limit of the ratio between the expected Social Cost of an ERM and the expected optimal Social Cost is finite and characterize its value. Noticeably, our method generalizes to encompass other truthful mechanisms and other metrics, such as the $l_p$ and Maximum costs. We conclude our theoretical analysis by characterizing the optimal ERM tailored to a $m$-CFLP and a distribution $μ$, that is the ERM whose limit Bayesian approximation ratio is the lowest compared to all other feasible ERMs. We consider mainly two frameworks: (i) in the first framework, the total facility capacity matches the number of agents, (ii) in the second framework, $m=2$. When we consider the Maximum Cost, we retrieve the optimal ERM for every $μ$, while for the Social Cost, we characterize the solution when the measure $μ$ is symmetric. Lastly, we numerically compare the performance of the ERMs against other truthful mechanisms and evaluate how quickly the Bayesian approximation ratio converges to its limit.

cs.GT↗

Facility Location Problems with Capacity Constraints: Two Facilities and Beyond

In this paper, we investigate the Mechanism Design aspects of the $m$-Capacitated Facility Location Problem ($m$-CFLP) on a line. We focus on two frameworks. In the first framework, the number of facilities is arbitrary, all facilities have the same capacity, and the number of agents is equal to the total capacity of all facilities. In the second framework, we aim to place two facilities, each with a capacity of at least half of the total agents. For both of these frameworks, we propose truthful mechanisms with bounded approximation ratios with respect to the Social Cost (SC) and the Maximum Cost (MC). When $m>2$, the result sharply contrasts with the impossibility results known for the classic $m$-Facility Location Problem \cite{fotakis2014power}, where capacity constraints are not considered. Furthermore, all our mechanisms are (i) optimal with respect to the MC (ii) optimal or nearly optimal with respect to the SC among anonymous mechanisms. For both frameworks, we provide a lower bound on the approximation ratio that any truthful and deterministic mechanism can achieve with respect to the SC and MC.

cs.GT↗

Edge Manipulations for the Maximum Vertex-Weighted Bipartite b-matching

In this paper, we explore the Mechanism Design aspects of the Maximum Vertex-weighted $b$-Matching (MVbM) problem on bipartite graphs $(A\cup T, E)$. The set $A$ comprises agents, while $T$ represents tasks. The set $E$ is the private information of either agents or tasks. In this framework, we investigate three mechanisms - $\MB$, $\MD$, and $\MG$ - that, given an MVbM problem as input, return a $b$-matching. We examine scenarios in which either agents or tasks are strategic and report their adjacent edges to one of the three mechanisms. In both cases, we assume that the strategic entities are bounded by their statements: they can hide edges, but they cannot report edges that do not exist. First, we consider the case in which agents can manipulate. In this framework, $\MB$ and $\MD$ are optimal but not truthful. By characterizing the Nash Equilibria induced by $\MB$ and $\MD$, we reveal that both mechanisms have a Price of Anarchy ($PoA$) and Price of Stability ($PoS$) of $2$. These efficiency guarantees are tight; no deterministic mechanism can achieve a lower $PoA$ or $PoS$. In contrast, the third mechanism, $\MG$, is not optimal, but truthful and its approximation ratio is $2$. We demonstrate that this ratio is optimal; no deterministic and truthful mechanism can outperform it. We then shift our focus to scenarios where tasks can exhibit strategic behaviour. In this case, $\MB$, $\MD$, and $\MG$ all maintain truthfulness, making $\MB$ and $\MD$ truthful and optimal mechanisms. In conclusion, we investigate the manipulability of $\MB$ and $\MD$ through experiments on randomly generated graphs. We observe that (1) $\MB$ is less prone to be manipulated by the first agent than $\MD$ (2) $\MB$ is more manipulable on instances in which the total capacity of the agents is equal to the number of tasks (3) randomizing the agents' order reduces the agents' ability to manipulate $\MB$.

cs.GT↗

A Bilevel Formalism for the Peer-Reviewing Problem

Due to the large number of submissions that more and more conferences experience, finding an automatized way to well distribute the submitted papers among reviewers has become necessary. We model the peer-reviewing matching problem as a {\it bilevel programming (BP)} formulation. Our model consists of a lower-level problem describing the reviewers' perspective and an upper-level problem describing the editors'. Every reviewer is interested in minimizing their overall effort, while the editors are interested in finding an allocation that maximizes the quality of the reviews and follows the reviewers' preferences the most. To the best of our knowledge, the proposed model is the first one that formulates the peer-reviewing matching problem by considering two objective functions, one to describe the reviewers' viewpoint and the other to describe the editors' viewpoint. We demonstrate that both the upper-level and lower-level problems are feasible and that our BP model admits a solution under mild assumptions. After studying the properties of the solutions, we propose a heuristic to solve our model and compare its performance with the relevant state-of-the-art methods. Extensive numerical results show that our approach can find fairer solutions with competitive quality and less effort from the reviewers.

cs.MA↗

On the Manipulability of Maximum Vertex-Weighted Bipartite $b$-matching Mechanisms

In this paper, we study the Maximum Vertex-weighted $b$-Matching (MVbM) problem on bipartite graphs in a new game-theoretical environment. In contrast to other game-theoretical settings, we consider the case in which the value of the tasks is public and common to every agent so that the private information of every agent consists of edges connecting them to the set of tasks. In this framework, we study three mechanisms. Two of these mechanisms, namely $\MB$ and $\MD$, are optimal but not truthful, while the third one, $\MG$, is truthful but sub-optimal. Albeit these mechanisms are induced by known algorithms, we show $\MB$ and $\MD$ are the best possible mechanisms in terms of Price of Anarchy and Price of Stability, while $\MG$ is the best truthful mechanism in terms of approximated ratio. Furthermore, we characterize the Nash Equilibria of $\MB$ and $\MD$ and retrieve sets of conditions under which $\MB$ acts as a truthful mechanism, which highlights the differences between $\MB$ and $\MD$. Finally, we extend our results to the case in which agents' capacity is part of their private information.

cs.GT↗