arXiv subjects

Dario Pighin

Publications and source records attributed to Dario Pighin.

12 recordsLinked to original sources

arXiv2026

Towards a Control interpretation of Quantum Advantage

We develop a control-theoretic framework for understanding Quantum Advantage (QA), providing a systematic route to characterize when and how QA can arise. The bilinear controlled Schrödinger equation is the common thread: the target quantum computation is recast as an operator controllability problem on the special unitary group $SU(N)$, and QA is identified with a polynomial-in-$n$ upper bound on the associated minimal-time function. We illustrate the framework on two paradigmatic problems: a) the Quantum Fourier Transform (QFT) on superconducting digital quantum processors (such as IBM's ibm_brisbane), for which we prove operator controllability by a Lie-algebraic argument and derive an $O(n^2)$ upper bound on the minimal time via a gate-concatenation lemma combined with the standard QFT circuit decomposition; b) the Maximum Independent Set (MIS) problem on neutral-atom analog quantum processors (such as Pasqal's hardware), for which we analyze the Rydberg-blockade Hamiltonian as a bilinear control system and reformulate the Quantum Approximate Optimization Algorithm (QAOA) as a continuous-time optimal control problem. By a controllability result, we show how the problem can be solved on Pasqal Quantum Computers and we introduce a control-based definition of Quantum Advantage for MIS. We conclude by outlining several open problems that chart directions for future research at the intersection of Control Theory and Quantum Computing.

math.OC↗

arXiv2026

Sherpa.ai Privacy-Preserving Multi-Party Entity Alignment without Intersection Disclosure for Noisy Identifiers

Federated Learning (FL) enables collaborative model training among multiple parties without centralizing raw data. There are two main paradigms in FL: Horizontal FL (HFL), where all participants share the same feature space but hold different samples, and Vertical FL (VFL), where parties possess complementary features for the same set of samples. A prerequisite for VFL training is privacy-preserving entity alignment (PPEA), which establishes a common index of samples across parties (alignment) without revealing which samples are shared between them. Conventional private set intersection (PSI) achieves alignment but leaks intersection membership, exposing sensitive relationships between datasets. The standard private set union (PSU) mitigates this risk by aligning on the union of identifiers rather than the intersection. However, existing approaches are often limited to two parties or lack support for typo-tolerant matching. In this paper, we introduce the Sherpa.ai multi-party PSU protocol for VFL, a PPEA method that hides intersection membership and enables both exact and noisy matching. The protocol generalizes two-party approaches to multiple parties with low communication overhead and offers two variants: an order-preserving version for exact alignment and an unordered version tolerant to typographical and formatting discrepancies. We prove correctness and privacy, analyze communication and computational (exponentiation) complexity, and formalize a universal index mapping from local records to a shared index space. This multi-party PSU offers a scalable, mathematically grounded protocol for PPEA in real-world VFL deployments, such as multi-institutional healthcare disease detection, collaborative risk modeling between banks and insurers, and cross-domain fraud detection between telecommunications and financial institutions, while preserving intersection privacy.

cs.CR↗

arXiv2025

The Sherpa.ai Blind Vertical Federated Learning Paradigm to Minimize the Number of Communications

Federated Learning (FL) enables collaborative decentralized training across multiple parties (nodes) while keeping raw data private. There are two main paradigms in FL: Horizontal FL (HFL), where all participant nodes share the same feature space but hold different samples, and Vertical FL (VFL), where participants hold complementary features for the same samples. While HFL is widely adopted, VFL is employed in domains where nodes hold complementary features about the same samples. Still, VFL presents a significant limitation: the vast number of communications required during training. This compromises privacy and security, and can lead to high energy consumption, and in some cases, make model training unfeasible due to the high number of communications. In this paper, we introduce Sherpa.ai Blind Vertical Federated Learning (SBVFL), a novel paradigm that leverages a distributed training mechanism enhanced for privacy and security. Decoupling the vast majority of node updates from the server dramatically reduces node-server communication. Experiments show that SBVFL reduces communication by ~99% compared to standard VFL while maintaining accuracy and robustness. Therefore, SBVFL enables practical, privacy-preserving VFL across sensitive domains, including healthcare, finance, manufacturing, aerospace, cybersecurity, and the defense industry.

cs.LG↗

arXiv2022

Turnpike in Lipschitz-nonlinear optimal control

We present a new proof of the turnpike property for nonlinear optimal control problems, when the running target is a steady control-state pair of the underlying system. Our strategy combines the construction of quasi-turnpike controls via controllability, and a bootstrap argument, and does not rely on analyzing the optimality system or linearization techniques. This in turn allows us to address several optimal control problems for finite-dimensional, control-affine systems with globally Lipschitz (possibly nonsmooth) nonlinearities, without any smallness conditions on the initial data or the running target. These results are motivated by applications in machine learning through deep residual neural networks, which may be fit within our setting. We show that our methodology is applicable to controlled PDEs as well, such as the semilinear wave and heat equation with a globally Lipschitz nonlinearity, once again without any smallness assumptions.

math.OC↗

arXiv2021

The turnpike property and the long-time behavior of the Hamilton-Jacobi-Bellman equation for finite-dimensional LQ control problems

We analyze the consequences that the so-called turnpike property has on the long-time behavior of the value function corresponding to a finite-dimensional linear-quadratic optimal control problem with general terminal cost and constrained controls. We prove that, when the time horizon $T$ tends to infinity, the value function asymptotically behaves as $W(x) + c\, T + λ$, and we provide a control interpretation of each of these three terms, making clear the link with the turnpike property. As a by-product, we obtain the long-time behavior of the solution to the associated Hamilton-Jacobi-Bellman equation in a case where the Hamiltonian is not coercive in the momentum variable. As a result of independent interest, we showed that linear-quadratic optimal control problems with constrained control enjoy a turnpike property, also particularly when the steady optimum may saturate the control constraints.

math.AP↗

arXiv2021

Large-time asymptotics in deep learning

We consider the neural ODE perspective of supervised learning and study the impact of the final time $T$ (which may indicate the depth of a corresponding ResNet) in training. For the classical $L^2$--regularized empirical risk minimization problem, whenever the neural ODE dynamics are homogeneous with respect to the parameters, we show that the training error is at most of the order $\mathcal{O}\left(\frac{1}{T}\right)$. Furthermore, if the loss inducing the empirical risk attains its minimum, the optimal parameters converge to minimal $L^2$--norm parameters which interpolate the dataset. By a natural scaling between $T$ and the regularization hyperparameter $λ$ we obtain the same results when $λ\searrow0$ and $T$ is fixed. This allows us to stipulate generalization properties in the overparametrized regime, now seen from the large depth, neural ODE perspective. To enhance the polynomial decay, inspired by turnpike theory in optimal control, we propose a learning problem with an additional integral regularization term of the neural ODE trajectory over $[0,T]$. In the setting of $\ell^p$--distance losses, we prove that both the training error and the optimal parameters are at most of the order $\mathcal{O}\left(e^{-μt}\right)$ in any $t\in[0,T]$. The aforementioned stability estimates are also shown for continuous space-time neural networks, taking the form of nonlinear integro-differential equations. By using a time-dependent moving grid for discretizing the spatial variable, we demonstrate that these equations provide a framework for addressing ResNets with variable widths.

math.OC↗

arXiv2021

The turnpike property in semilinear control

An exponential turnpike property for a semilinear control problem is proved. The state-target is assumed to be small, whereas the initial datum can be arbitrary. Turnpike results are also obtained for large targets, requiring that the control acts everywhere. In this case, we prove the convergence of the infimum of the averaged time-evolution functional towards the steady one. Numerical simulations are performed.

math.OC↗

arXiv2020

Rotor imbalance suppression by optimal control

An imbalanced rotor is considered. A system of moving balancing masses is given. We determine the optimal movement of the balancing masses to minimize the imbalance on the rotor. The optimal movement is given by an open-loop control solving an optimal control problem posed in infinite time. By methods of the Calculus of Variations, the existence of the optimum is proved and the corresponding optimality conditions have been derived. Asymptotic behavior of the control system is studied rigorously. By Łojasiewicz inequality, convergence of the optima as time $t\to +\infty$ towards a steady configuration is ensured. An explicit estimate of the convergence rate is given. This guarantees that the optimal control stabilizes the system. In case the imbalance is below a computed threshold, the convergence occurs exponentially fast. This is proved by the Stable Manifold Theorem applied to the Pontryagin optimality system. Moreover, a closed-loop control strategy based on Reinforcement Learning is proposed. Numerical simulations have been performed, validating the theoretical results.

math.OC↗

arXiv2020

Nonuniqueness of minimizers for semilinear optimal control problems

A counterexample to uniqueness of global minimizers of semilinear optimal control problems is given. The lack of uniqueness occurs for a special choice of the state-target in the cost functional. Our arguments show also that, for some state-targets, there exist local minimizers, which are not global. When this occurs, gradient-type algorithms may be trapped by the local minimizers, thus missing the global ones. Furthermore, the issue of convexity of quadratic functional in optimal control is analyzed in an abstract setting. As a Corollary of the nonuniqueness of the minimizers, a nonuniqueness result for a coupled elliptic system is deduced. Numerical simulations have been performed illustrating the theoretical results. We also discuss the possible impact of the multiplicity of minimizers on the turnpike property in long time horizons.

math.OC↗

arXiv2020

The turnpike with lack of observability

We obtain turnpike results for optimal control problems with lack of stabilizability in the state equation and/or detectability in the state term in the cost functional. We show how, under weakened stabilizability/detectability conditions, terminal conditions may affect turnpike phenomena. Numerical simulations have been performed to illustrate the theoretical results.

math.OC↗

arXiv2019

Controllability under positivity constraints of multi-d wave equations

We consider both the internal and boundary controllability problems for wave equations under non-negativity constraints on the controls. First, we prove the steady state controllability property with nonnegative controls for a general class of wave equations with time-independent coefficients. According to it, the system can be driven from a steady state generated by a strictly positive control to another, by means of nonnegative controls, when the time of control is long enough. Secondly, under the added assumption of conservation and coercivity of the energy, controllability is proved between states lying on two distinct trajectories. Our methods are described and developed in an abstract setting, to be applicable to a wide variety of control systems.

math.OC↗

arXiv2018

Controllability under positivity constraints of semilinear heat equations

In many practical applications of control theory some constraints on the state and/or on the control need to be imposed. In this paper, we prove controllability results for semilinear parabolic equations under positivity constraints on the control, when the time horizon is long enough. As we shall see, in fact, the minimal controllability time turns out to be strictly positive. More precisely, we prove a global steady state constrained controllability result for a semilinear parabolic equation with $C^1$ nonlinearity, without sign or globally Lipschitz assumptions on the nonlinear term. Then, under suitable dissipativity assumptions on the system, we extend the result to any initial datum and any target trajectory. We conclude with some numerical simulations that confirm the theoretical results that provide further information of the sparse structure of constrained controls in minimal time.

math.OC↗