arXiv subjects

Tommaso Giovannelli

Publications and source records attributed to Tommaso Giovannelli.

11 recordsLinked to original sources

arXiv2026

Non-smooth stochastic gradient descent using smoothing functions

In this paper, we address stochastic optimization problems involving a composition of a non-smooth outer function and a smooth inner function, a formulation frequently encountered in machine learning and operations research. To deal with the non-differentiability of the outer function, we approximate the original non-smooth function using smoothing functions, which are continuously differentiable and approach the original function as a smoothing parameter goes to zero (at the price of increasingly higher Lipschitz constants). The proposed smoothing stochastic gradient method iteratively drives the smoothing parameter to zero at a designated rate. We establish convergence guarantees under strongly convex, convex, and non-convex settings, proving convergence rates that match known results for non-smooth stochastic compositional optimization. In particular, for convex objectives, smoothing stochastic gradient achieves a 1/T^(1/4) rate in terms of the number of stochastic gradient evaluations. We further show how general compositional and finite-sum compositional problems (widely used frameworks in large-scale machine learning and risk-averse optimization) fit the assumptions needed for the rates (unbiased gradient estimates, bounded second moments, and accurate smoothing errors). We present preliminary numerical results indicating that smoothing stochastic gradient descent can be competitive for certain classes of problems.

math.OC↗

arXiv2026

CHAL: Council of Hierarchical Agentic Language

Multi-agent debate has emerged as a promising approach for improving LLM reasoning on ground-truth tasks, yet current methodologies face certain structural limitations: debate tends to induce a martingale over belief trajectories, majority voting accounts for most observed gains, and LLMs exhibit confidence escalation rather than calibration across rounds. We argue that the genuine value of debate, and dialectic systems as a whole, lies not in ground-truth tasks but in defeasible domains, where every position can in principle be defeated by better reasoning. We present the Council of Hierarchical Agentic Language (CHAL), a multi-agent dialectic framework that treats defeasible argumentation as an engine for belief optimization. Each agent maintains a CHAL Belief Schema (CBS), a graph-structured belief representation with a Bayesian-inspired architecture, that facilitates belief revision through a gradient-informed dynamic mechanism by leveraging the strength of the belief's thesis as a differentiable objective. Meta-cognitive value systems spanning epistemology, logic, and ethics are elevated to configurable hyperparameters governing agent reasoning and adjudication outcomes. We provide a series of ablation experiments that demonstrate systematic and interpretable effects: the adjudicator's value system determines the debate's overall trajectories in latent belief space, council diversity refines beliefs for all participants, and the framework generalizes across broad fields. CHAL is, to our knowledge, the first framework to treat multi-agent debate as structured belief optimization over defeasible domains. Further, the auditable belief artifacts it produces establish the foundation for dedicated evaluation suites for defeasible argumentation, with broader implications for building AI systems whose reasoning and value commitments are transparent, aligned, and subject to human oversight.

cs.AI↗

arXiv2026

Pareto sensitivity, most-changing sub-fronts, and knee solutions

When dealing with a multi-objective optimization problem, obtaining a comprehensive representation of the set of Pareto optimal solutions can be computationally expensive. However, identifying the most representative solutions can be difficult and sometimes ambiguous, since what constitutes a representative solution depends on the decision maker's preferences. A popular selection are the so-called Pareto knee solutions, which correspond to nondominated points on the Pareto front where a small improvement in any objective leads to a large deterioration in at least one other objective. In this paper, using Pareto sensitivity, we show how to compute Pareto knee solutions according to their verbal (informal) definition of least maximal change. We refer to the resulting approach as the sensitivity knee (snee) approach, and we apply it to unconstrained and constrained problems. Pareto sensitivity can also be used to compute local most-changing Pareto sub-fronts around a nondominated point, where points on the sub-fronts are distributed along directions of maximum change. Our approach is still restricted to scalarized methods, in particular to the weighted-sum or epsilon-constrained methods, and requires the computation or approximations of first- and second-order derivatives. We include numerical results from synthetic problems that illustrate the benefits of our approach.

math.OC↗

arXiv2026

Stochastic set-valued optimization and its application to robust learning

In this paper, we develop a stochastic set-valued optimization (SVO) framework tailored for robust machine learning. In the SVO setting, each decision variable is mapped to a set of objective values, and optimality is defined via set relations. We focus on SVO problems with hyperbox sets, which can be reformulated as multi-objective optimization (MOO) problems with finitely many objectives and serve as a foundation for representing or approximating more general mapped sets. Two special cases of hyperbox-valued optimization (HVO) are interval-valued (IVO) and rectangle-valued (RVO) optimization. We construct stochastic IVO/RVO formulations that incorporate subquantiles and superquantiles into the objective functions of the MOO reformulations, providing a new characterization for subquantiles. These formulations provide interpretable trade-offs by capturing both lower- and upper-tail behaviors of loss distributions, thereby going beyond standard empirical risk minimization and classical robust models. To solve the resulting multi-objective problems, we adopt stochastic multi-gradient algorithms and select a Pareto knee solution. In numerical experiments, the proposed algorithms with this selection strategy exhibit improved robustness and reduced variability across test replications under distributional shift compared with empirical risk minimization, while maintaining competitive accuracy.

math.OC↗

arXiv2025

A stochastic gradient method for trilevel optimization

With the success that the field of bilevel optimization has seen in recent years, similar methodologies have started being applied to solving more difficult applications that arise in trilevel optimization. At the helm of these applications are new machine learning formulations that have been proposed in the trilevel context and, as a result, efficient and theoretically sound stochastic methods are required. In this work, we propose the first-ever stochastic gradient descent method for solving unconstrained trilevel optimization problems and provide a convergence theory that covers all forms of inexactness of the trilevel adjoint gradient, such as the inexact solutions of the middle-level and lower-level problems, inexact computation of the trilevel adjoint formula, and noisy estimates of the gradients, Hessians, Jacobians, and tensors of third-order derivatives involved. We also demonstrate the promise of our approach by providing numerical results on both synthetic trilevel problems and trilevel formulations for hyperparameter adversarial tuning.

math.OC↗

arXiv2024

Why is soccer so popular: Understanding underdog achievement and randomness in team ball sports

In this paper, we examine team ball sports to investigate how the likelihood of weaker teams winning against stronger ones, referred to as underdog achievement, is influenced by inherent randomness factors that affect match outcomes in such sports. To address our research question, we collected data on match scores and computed corresponding team rankings from major international competitions for 12 popular team ball sports: basketball, cricket, field hockey, futsal, handball, ice hockey, lacrosse, roller hockey, rugby, soccer, volleyball, and water polo. Then, we developed an underdog achievement score to identify the sports with the highest occurrences of weaker teams prevailing over stronger ones, and we designed a randomness model consisting of factors that contribute to unexpected match outcomes within each sport. Our findings indicate that soccer is among the sports in which a weaker team is most likely to win. Through principal component analysis (PCA) and correlation analysis, we demonstrate that our randomness model can explain such a phenomenon, showing that the underdog achievement can be attributed to numerous factors that can randomly influence match outcomes.

stat.AP↗

arXiv2023

The limitation of neural nets for approximation and optimization

We are interested in assessing the use of neural networks as surrogate models to approximate and minimize objective functions in optimization problems. While neural networks are widely used for machine learning tasks such as classification and regression, their application in solving optimization problems has been limited. Our study begins by determining the best activation function for approximating the objective functions of popular nonlinear optimization test problems, and the evidence provided shows that~SiLU has the best performance. We then analyze the accuracy of function value, gradient, and Hessian approximations for such objective functions obtained through interpolation/regression models and neural networks. When compared to interpolation/regression models, neural networks can deliver competitive zero- and first-order approximations (at a high training cost) but underperform on second-order approximation. However, it is shown that combining a neural net activation function with the natural basis for quadratic interpolation/regression can waive the necessity of including cross terms in the natural basis, leading to models with fewer parameters to determine. Lastly, we provide evidence that the performance of a state-of-the-art derivative-free optimization algorithm can hardly be improved when the gradient of an objective function is approximated using any of the surrogate models considered, including neural networks.

cs.LG↗

arXiv2023

Inexact bilevel stochastic gradient methods for constrained and unconstrained lower-level problems

Two-level stochastic optimization formulations have become instrumental in a number of machine learning contexts such as continual learning, neural architecture search, adversarial learning, and hyperparameter tuning. Practical stochastic bilevel optimization problems become challenging in optimization or learning scenarios where the number of variables is high or there are constraints. In this paper, we introduce a bilevel stochastic gradient method for bilevel problems with nonlinear and possibly nonconvex lower-level constraints. We also present a comprehensive convergence theory that addresses both the lower-level unconstrained and constrained cases and covers all inexact calculations of the adjoint gradient (also called hypergradient), such as the inexact solution of the lower-level problem, inexact computation of the adjoint formula (due to the inexact solution of the adjoint equation or use of a truncated Neumann series), and noisy estimates of the gradients, Hessians, and Jacobians involved. To promote the use of bilevel optimization in large-scale learning, we have developed new low-rank practical bilevel stochastic gradient methods (BSG-N-FD and~BSG-1) that do not require second-order derivatives and, in the lower-level unconstrained case, dismiss any matrix-vector products.

math.OC↗

arXiv2023

Bilevel optimization with a multi-objective lower-level problem: Risk-neutral and risk-averse formulations

In this work, we propose different formulations and gradient-based algorithms for deterministic and stochastic bilevel problems with conflicting objectives in the lower level. Such problems have received little attention in the deterministic case and have never been studied from a stochastic approximation viewpoint despite the recent advances in stochastic methods for single-level, bilevel, and multi-objective optimization. To solve bilevel problems with a multi-objective lower level, different approaches can be considered depending on the interpretation of the lower-level optimality. An optimistic formulation that was previously introduced for the deterministic case consists of minimizing the upper-level function over all non-dominated lower-level solutions. In this paper, we develop new risk-neutral and risk-averse formulations, address their main computational challenges, and develop the corresponding deterministic and stochastic gradient-based algorithms.

math.OC↗

arXiv2022

An integrated assignment, routing, and speed model for roadway mobility and transportation with environmental, efficiency, and service goals

Managing all the mobility and transportation services with autonomous vehicles for users of a smart city requires determining the assignment of the vehicles to the users and their routing in conjunction with their speed. Such decisions must ensure low emission, efficiency, and high service quality by also considering the impact on traffic congestion caused by other vehicles in the transportation network. In this paper, we first propose an abstract trilevel multi-objective formulation architecture to model all vehicle routing problems with assignment, routing, and speed decision variables and conflicting objective functions. Such an architecture guides the development of subproblems, relaxations, and solution methods. We also propose a way of integrating the various urban transportation services by introducing a constraint on the speed variables that takes into account the traffic volume caused across the different services. Based on the formulation architecture, we introduce a (bilevel) problem where assignment and routing are at the upper level and speed is at the lower level. To address the challenge of dealing with routing problems on urban road networks, we develop an algorithm that alternates between the assignment-routing problem on an auxiliary complete graph and the speed optimization problem on the original non-complete graph. The computational experiments show the effectiveness of the proposed approach in determining approximate Pareto fronts among the conflicting objectives.

math.OC↗

arXiv2021

Derivative-free methods for mixed-integer nonsmooth constrained optimization

In this paper, we consider mixed-integer nonsmooth constrained optimization problems whose objective/constraint functions are available only as the output of a black-box zeroth-order oracle (i.e., an oracle that does not provide derivative information) and we propose a new derivative-free linesearch-based algorithmic framework to suitably handle those problems. We first describe a scheme for bound constrained problems that combines a dense sequence of directions (to handle the nonsmoothness of the objective function) with primitive directions (to handle discrete variables). Then, we embed an exact penalty approach in the scheme to suitably manage nonlinear (possibly nonsmooth) constraints. We analyze the global convergence properties of the proposed algorithms toward stationary points and we report the results of an extensive numerical experience on a set of mixed-integer test problems.

math.OC↗