arXiv subjects

Alessandro Scagliotti

Publications and source records attributed to Alessandro Scagliotti.

18 recordsLinked to original sources

arXiv2026

Trade-off invariance for weighted scalarizations in multi-objective optimization

We consider weighted-sum scalarizations for an abstract multi-objective minimization problem defined by the vector-valued map $U\ni u\mapsto ( f_1(u),\ldots, f_N(u))$, where $U$ is an arbitrary nonempty set and no topology, convexity, compactness, or lower semicontinuity assumption is imposed. Using the open simplex as parameter space for positive weights, we show that the Trade-off Invariance Principle for scalarizations yields a generic uniqueness property in the objective space. Namely, for almost every weight vector, all minimizers of the corresponding weighted-sum scalarization have the same objective vector. Moreover, excluding again a null-measure subset, all minimizing sequences determine the same limiting objective vector, independently of the chosen sequence. We also give a geometric interpretation of these results in the attainable objective set: for almost every positive weight vector, the scalarization exposes at most one nondominated point. Moreover, minimizing sequences determine at most one asymptotically exposed objective vector in the closure of the attainable set.

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arXiv2026

Risk-Averse Ensemble Control for Control-Affine Systems

A number of important modern applications in optimal control can be formulated as open loop control problems in which the underlying dynamical systems are subject to random inputs. These so-called ensemble control problems require the corresponding optimal control to be deterministic, as it must be computed before the realization of uncertainty and the passage of time. Practical applications of ensemble control include quantum control and the training of Neural ODEs. However, the standard approach to ensemble control treats the uncertainty in the objective function via the expectation, which provides optimal controls that only work well on average while ignoring critical outlier phenomena. This study provides a comprehensive mathematical treatment of risk-averse ensemble control. Within this setting, we adopt a control-affine structure that ensures the lower semi-continuity needed for proving the existence of optimal solutions. The central analytical contribution of this paper is a rigorous characterization of the control-to-state mapping in which we establish weak-to-strong continuity, continuous Fr\'echet differentiability, and weak-to-strong continuity of the derivative operator. Furthermore, this regularity yields primal and dual first-order optimality conditions characterized by an adjoint state of bounded variation, and it fulfills the functional prerequisites required for the convergence of infinite dimensional optimization algorithms. We conclude by validating these theoretical developments through a numerical experiment in quantum control.

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arXiv2025

A case study in ensemble optimal control for Bayesian input design

We discuss the problem of input design for uncertainty reduction in a parameter estimation procedure. Assuming a linear continuous-time control system with noisy measurements, we formulate an objective of variance reduction in a Bayesian Gaussian setting as an optimal control problem and analyze it from a geometric control perspective. The resulting cost functional depends on the unknown parameter, we compare the optimal control approach with a non-standard alternative inspired by ensemble control, where the cost is averaged over the prior distribution after computation, rather than before. This requires the statement of a generalized Pontryagin's maximum principle adapted to Gaussian distributions.

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arXiv2025

Towards optimal control of ensembles of discrete-time systems

The control of ensembles of dynamical systems is an intriguing and challenging problem, arising for example in quantum control. We initiate the investigation of optimal control of ensembles of discrete-time systems, focusing on minimising the average finite horizon cost over the ensemble. For very general nonlinear control systems and stage and terminal costs, we establish existence of minimisers under mild assumptions. Furthermore, we provide a $\Gamma$-convergence result which enables consistent approximation of the challenging ensemble optimal control problem, for example, by using empirical probability measures over the ensemble. Our results form a solid foundation for discrete-time optimal control of ensembles, with many interesting avenues for future research.

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arXiv2025

Balanced quasistatic evolutions of critical points in metric spaces

Quasistatic evolutions of critical points of time-dependent energies exhibit piecewise smooth behavior, making them useful for modeling continuum mechanics phenomena like elastic-plasticity and fracture. Traditionally, such evolutions have been derived as vanishing viscosity and inertia limits, leading to balanced viscosity solutions. However, for nonconvex energies, these constructions have been realized in Euclidean spaces and assume non-degenerate critical points. In this paper, we take a different approach by decoupling the time scales of the energy evolution and of the transition to equilibria. Namely, starting from an equilibrium configuration, we let the energy evolve, while keeping frozen the system state; then, we update the state by freezing the energy, while letting the system transit via gradient flow or an approximation of it (e.g., minimizing movement or backward differentiation schemes). This approach has several advantages. It aligns with the physical principle that systems transit through energy-minimizing steady states. It is also fully constructive and computationally implementable, with physical and computational costs governed by appropriate action functionals. Additionally, our analysis is simpler and more general than previous formulations in the literature, as it does not require non-degenerate critical points. Finally, this approach extends to evolutions in locally compact metric path spaces, and our axiomatic presentation allows for various realizations.

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arXiv2025

Approximation of diffeomorphisms for quantum state transfers

In this paper, we seek to combine two emerging standpoints in control theory. On the one hand, recent advances in infinite-dimensional geometric control have unlocked a method for controlling (with arbitrary precision and in arbitrarily small times) state transfers for bilinear Schrödinger PDEs posed on a Riemannian manifold $M$. In particular, these arguments rely on controllability results in the group of the diffeomorphisms of $M$. On the other hand, using tools of $Γ$-convergence, it has been proved that we can phrase the retrieval of a diffeomorphism of $M$ as an ensemble optimal control problem. More precisely, this is done by employing a control-affine system for \emph{simultaneously} steering a finite swarm of points towards the respective targets. Here we blend these two theoretical approaches and numerically find control laws driving state transitions (such as eigenstate transfers) in small time in a bilinear Schrödinger PDE posed on the torus. Such systems have experimental relevance and are currently used to model rotational dynamics of molecules, and cold atoms trapped in periodic optical lattices.

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arXiv2025

Ensemble optimal control for managing drug resistance in cancer therapies

In this paper, we explore the application of ensemble optimal control to derive enhanced strategies for pharmacological cancer treatment, and we tackle the problem of the long-term management of the disease, i.e., when the complete eradication of the tumor is not achievable. In particular, we focus on moving beyond the classical clinical approach of giving the patient the maximal tolerated drug dose (MTD), which does not properly exploit the fight among sensitive and resistant cells for the available resources. Here, we employ a Lotka-Volterra model to describe the competing subpopulations, and we enclose this system within the ensemble control framework. In the first part, we establish general results suitable for application to various cancers. Then, we carry out numerical simulations in the setting of prostate cancer treated with androgen deprivation therapy, yielding a computed policy that is reminiscent of the medical `active surveillance' paradigm. Finally, inspired by the numerical evidence, we propose a variant of the celebrated adaptive therapy (AT), which we call `Off-On' AT.

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arXiv2025

Trade-off Invariance Principle for minimizers of regularized functionals

In this paper, we consider functionals of the form $H_α(u)=F(u)+αG(u)$ with $α\in[0,+\infty)$, where $u$ varies in a set $U\neq\emptyset$ (without further structure). We first revisit a result stating that, excluding at most countably many values of $α$, we have $\inf_{H_α^\star}G= \sup_{H_α^\star}G$, where $H_α^\star := \arg\min_UH_α$, which is assumed to be non-empty. Then, we prove a stronger result that concerns the invariance of the limiting value of the functional $G$ along minimizing sequences for $H_α$, which extends the above Principle to the case $H_α^\star= \emptyset$. Moreover, we show to what extent these findings generalize to multi-regularized functionals and -- in the presence of an underlying differentiable structure -- to critical points. Finally, the main result implies an unexpected consequence for functionals regularized with uniformly convex norms: excluding again at most countably many values of $α$, it turns out that for a minimizing sequence, convergence to a minimizer in the weak or strong sense is equivalent.

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arXiv2024

Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs

In this paper, we consider the problem of recovering the $W_2$-optimal transport map T between absolutely continuous measures $μ,ν\in\mathcal{P}(\mathbb{R}^n)$ as the flow of a linear-control neural ODE, where the control depends only on the time variable and takes values in a finite-dimensional space. We first show that, under suitable assumptions on $μ,ν$ and on the controlled vector fields governing the neural ODE, the optimal transport map is contained in the $C^0_c$-closure of the flows generated by the system. Then, we tackle the problem under the assumption that only discrete approximations of $μ_N,ν_N$ of the original measures $μ,ν$ are available: we formulate approximated optimal control problems, and we show that their solutions give flows that approximate the original optimal transport map $T$. In the framework of generative models, the approximating flow constructed here can be seen as a `Normalizing Flow', which usually refers to the task of providing invertible transport maps between probability measures by means of deep neural networks. We propose an iterative numerical scheme based on the Pontryagin Maximum Principle for the resolution of the optimal control problem, resulting in a method for the practical computation of the approximated optimal transport map, and we test it on a two-dimensional example.

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arXiv2024

Losing momentum in continuous-time stochastic optimisation

The training of modern machine learning models often consists in solving high-dimensional non-convex optimisation problems that are subject to large-scale data. In this context, momentum-based stochastic optimisation algorithms have become particularly widespread. The stochasticity arises from data subsampling which reduces computational cost. Both, momentum and stochasticity help the algorithm to converge globally. In this work, we propose and analyse a continuous-time model for stochastic gradient descent with momentum. This model is a piecewise-deterministic Markov process that represents the optimiser by an underdamped dynamical system and the data subsampling through a stochastic switching. We investigate longtime limits, the subsampling-to-no-subsampling limit, and the momentum-to-no-momentum limit. We are particularly interested in the case of reducing the momentum over time. Under convexity assumptions, we show convergence of our dynamical system to the global minimiser when reducing momentum over time and letting the subsampling rate go to infinity. We then propose a stable, symplectic discretisation scheme to construct an algorithm from our continuous-time dynamical system. In experiments, we study our scheme in convex and non-convex test problems. Additionally, we train a convolutional neural network in an image classification problem. Our algorithm {attains} competitive results compared to stochastic gradient descent with momentum.

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arXiv2024

Minimax problems for ensembles of control-affine systems

In this paper, we consider ensembles of control-affine systems in $\mathbb{R}^d$, and we study simultaneous optimal control problems related to the worst-case minimization. After proving that such problems admit solutions, denoting with $(Θ^N)_N$ a sequence of compact sets that parametrize the ensembles of systems, we first show that the corresponding minimax optimal control problems are $Γ$-convergent whenever $(Θ^N)_N$ has a limit with respect to the Hausdorff distance. Besides its independent interest, the previous result plays a crucial role for establishing the Pontryagin Maximum Principle (PMP) when the ensemble is parametrized by a set $Θ$ consisting of infinitely many points. Namely, we first approximate $Θ$ by finite and increasing-in-size sets $(Θ^N)_N$ for which the PMP is known, and then we derive the PMP for the $Γ$-limiting problem. The same strategy can be pursued in applications, where we can reduce infinite ensembles to finite ones to compute the minimizers numerically. We bring as a numerical example the Schrödinger equation for a qubit with uncertain resonance frequency.

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arXiv2024

Shortest-path recovery from signature with an optimal control approach

In this paper, we consider the signature-to-path reconstruction problem from the control theoretic perspective. Namely, we design an optimal control problem whose solution leads to the minimal-length path that generates a given signature. In order to do that, we minimize a cost functional consisting of two competing terms, i.e., a weighted final-time cost combined with the $L^2$-norm squared of the controls. Moreover, we can show that, by taking the limit to infinity of the parameter that tunes the final-time cost, the problem $Γ$-converges to the problem of finding a sub-Riemannian geodesic connecting two signatures. Finally, we provide an alternative reformulation of the latter problem, which is particularly suitable for the numerical implementation.

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arXiv2024

A minimax optimal control approach for robust neural ODEs

In this paper, we address the adversarial training of neural ODEs from a robust control perspective. This is an alternative to the classical training via empirical risk minimization, and it is widely used to enforce reliable outcomes for input perturbations. Neural ODEs allow the interpretation of deep neural networks as discretizations of control systems, unlocking powerful tools from control theory for the development and the understanding of machine learning. In this specific case, we formulate the adversarial training with perturbed data as a minimax optimal control problem, for which we derive first order optimality conditions in the form of Pontryagin's Maximum Principle. We provide a novel interpretation of robust training leading to an alternative weighted technique, which we test on a low-dimensional classification task.

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arXiv2023

From NeurODEs to AutoencODEs: a mean-field control framework for width-varying Neural Networks

The connection between Residual Neural Networks (ResNets) and continuous-time control systems (known as NeurODEs) has led to a mathematical analysis of neural networks which has provided interesting results of both theoretical and practical significance. However, by construction, NeurODEs have been limited to describing constant-width layers, making them unsuitable for modeling deep learning architectures with layers of variable width. In this paper, we propose a continuous-time Autoencoder, which we call AutoencODE, based on a modification of the controlled field that drives the dynamics. This adaptation enables the extension of the mean-field control framework originally devised for conventional NeurODEs. In this setting, we tackle the case of low Tikhonov regularization, resulting in potentially non-convex cost landscapes. While the global results obtained for high Tikhonov regularization may not hold globally, we show that many of them can be recovered in regions where the loss function is locally convex. Inspired by our theoretical findings, we develop a training method tailored to this specific type of Autoencoders with residual connections, and we validate our approach through numerical experiments conducted on various examples.

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arXiv2023

A subgradient method with constant step-size for $\ell_1$-composite optimization

Subgradient methods are the natural extension to the non-smooth case of the classical gradient descent for regular convex optimization problems. However, in general, they are characterized by slow convergence rates, and they require decreasing step-sizes to converge. In this paper we propose a subgradient method with constant step-size for composite convex objectives with $\ell_1$-regularization. If the smooth term is strongly convex, we can establish a linear convergence result for the function values. This fact relies on an accurate choice of the element of the subdifferential used for the update, and on proper actions adopted when non-differentiability regions are crossed. Then, we propose an accelerated version of the algorithm, based on conservative inertial dynamics and on an adaptive restart strategy, that is guaranteed to achieve a linear convergence rate in the strongly convex case. Finally, we test the performances of our algorithms on some strongly and non-strongly convex examples.

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arXiv2023

Optimal Control of ensembles of dynamical systems

In this paper we consider the problem of the optimal control of an ensemble of affine-control systems. After proving the well-posedness of the minimization problem under examination, we establish a $Γ$-convergence result that allows us to substitute the original (and usually infinite) ensemble with a sequence of finite increasing-in-size sub-ensembles. The solutions of the optimal control problems involving these sub-ensembles provide approximations in the $L^2$-strong topology of the minimizers of the original problem. Using again a $Γ$-convergence argument, we manage to derive a Maximum Principle for ensemble optimal control problems with end-point cost. Moreover, in the case of finite sub-ensembles, we can address the minimization of the related cost through numerical schemes. In particular, we propose an algorithm that consists of a subspace projection of the gradient field induced on the space of admissible controls by the approximating cost functional. In addition, we consider an iterative method based on the Pontryagin Maximum Principle. Finally, we test the algorithms on an ensemble of linear systems in $\mathbb{R}^2$.

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arXiv2022

Deep Learning Approximation of Diffeomorphisms via Linear-Control Systems

In this paper we propose a Deep Learning architecture to approximate diffeomorphisms diffeotopic to the identity. We consider a control system of the form $\dot x = \sum_{i=1}^lF_i(x)u_i$, with linear dependence in the controls, and we use the corresponding flow to approximate the action of a diffeomorphism on a compact ensemble of points. Despite the simplicity of the control system, it has been recently shown that a Universal Approximation Property holds. The problem of minimizing the sum of the training error and of a regularizing term induces a gradient flow in the space of admissible controls. A possible training procedure for the discrete-time neural network consists in projecting the gradient flow onto a finite-dimensional subspace of the admissible controls. An alternative approach relies on an iterative method based on Pontryagin Maximum Principle for the numerical resolution of Optimal Control problems. Here the maximization of the Hamiltonian can be carried out with an extremely low computational effort, owing to the linear dependence of the system in the control variables.

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arXiv2022

A gradient flow equation for optimal control problems with end-point cost

In this paper we consider a control system of the form $\dot x = F(x)u$, linear in the control variable $u$. Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an end-point cost and the squared $2$-norm of the control. This functional induces a gradient flow on the Hilbert space of admissible controls, and we prove a convergence result by means of the Lojasiewicz-Simon inequality. Finally, we show that, if we let the weight of the end-point cost tend to infinity, the resulting family of functionals is $Γ$-convergent, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.

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