Mattia Silvestri

Mattia Silvestri, Senne Berden, Jayanta Mandi et al.

Many real-world optimization problems contain parameters that are unknown before deployment time, either due to stochasticity or to lack of information (e.g., demand or travel times in delivery problems). A common strategy in such cases is to estimate said parameters via machine learning (ML) models trained to minimize the prediction error, which however is not necessarily aligned with the downstream task-level error. The decision-focused learning (DFL) paradigm overcomes this limitation by training to directly minimize a task loss, e.g. regret. Since the latter has non-informative gradients for combinatorial problems, state-of-the-art DFL methods introduce surrogates and approximations that enable training. But these methods exploit specific assumptions about the problem structures (e.g., convex or linear problems, unknown parameters only in the objective function). We propose an alternative method that makes no such assumptions, it combines stochastic smoothing with score function gradient estimation which works on any task loss. This opens up the use of DFL methods to nonlinear objectives, uncertain parameters in the problem constraints, and even two-stage stochastic optimization. Experiments show that it typically requires more epochs, but that it is on par with specialized methods and performs especially well for the difficult case of problems with uncertainty in the constraints, in terms of solution quality, scalability, or both.

Mattia Silvestri, Allegra De Filippo, Michele Lombardi et al.

The interplay between Machine Learning (ML) and Constrained Optimization (CO) has recently been the subject of increasing interest, leading to a new and prolific research area covering (e.g.) Decision Focused Learning and Constrained Reinforcement Learning. Such approaches strive to tackle complex decision problems under uncertainty over multiple stages, involving both explicit (cost function, constraints) and implicit knowledge (from data), and possibly subject to execution time restrictions. While a good degree of success has been achieved, the existing methods still have limitations in terms of both applicability and effectiveness. For problems in this class, we propose UNIFY, a unified framework to design a solution policy for complex decision-making problems. Our approach relies on a clever decomposition of the policy in two stages, namely an unconstrained ML model and a CO problem, to take advantage of the strength of each approach while compensating for its weaknesses. With a little design effort, UNIFY can generalize several existing approaches, thus extending their applicability. We demonstrate the method effectiveness on two practical problems, namely an Energy Management System and the Set Multi-cover with stochastic coverage requirements. Finally, we highlight some current challenges of our method and future research directions that can benefit from the cross-fertilization of the two fields.

Mattia Silvestri, Federico Baldo, Eleonora Misino et al.

In the last decade, the scientific community has devolved its attention to the deployment of data-driven approaches in scientific research to provide accurate and reliable analysis of a plethora of phenomena. Most notably, Physics-informed Neural Networks and, more recently, Universal Differential Equations (UDEs) proved to be effective both in system integration and identification. However, there is a lack of an in-depth analysis of the proposed techniques. In this work, we make a contribution by testing the UDE framework in the context of Ordinary Differential Equations (ODEs) discovery. In our analysis, performed on two case studies, we highlight some of the issues arising when combining data-driven approaches and numerical solvers, and we investigate the importance of the data collection process. We believe that our analysis represents a significant contribution in investigating the capabilities and limitations of Physics-informed Machine Learning frameworks.

Mattia Silvestri, Michele Lombardi, Michela Milano

Given enough data, Deep Neural Networks (DNNs) are capable of learning complex input-output relations with high accuracy. In several domains, however, data is scarce or expensive to retrieve, while a substantial amount of expert knowledge is available. It seems reasonable that if we can inject this additional information in the DNN, we could ease the learning process. One such case is that of Constraint Problems, for which declarative approaches exists and pure ML solutions have obtained mixed success. Using a classical constrained problem as a case study, we perform controlled experiments to probe the impact of progressively adding domain and empirical knowledge in the DNN. Our results are very encouraging, showing that (at least in our setup) embedding domain knowledge at training time can have a considerable effect and that a small amount of empirical knowledge is sufficient to obtain practically useful results.