Stefania Bellavia

Stefania Bellavia, Natasa Krejic, Natasa Krklec Jerinkic

This paper deals with the minimization of large sum of convex functions by Inexact Newton (IN) methods employing subsampled functions, gradients and Hessian approximations. The Conjugate Gradient method is used to compute the inexact Newton step and global convergence is enforced by a nonmonotone line search procedure. The aim is to obtain methods with affordable costs and fast convergence. Assuming strongly convex functions, R-linear convergence and worst-case iteration complexity of the procedure are investigated when functions and gradients are approximated with increasing accuracy. A set of rules for the forcing parameters and subsample Hessian sizes are derived that ensure local q-linear/superlinear convergence of the proposed method. The random choice of the Hessian subsample is also considered and convergence in the mean square, both for finite and infinite sums of functions, is proved. Finally, global convergence with asymptotic R-linear rate of IN methods is extended to the case of sum of convex function and strongly convex objective function. Numerical results on well known binary classification problems are also given. Adaptive strategies for selecting forcing terms and Hessian subsample size, streaming out of the theoretical analysis, are employed and the numerical results showed that they yield effective IN methods.

Stefania Bellavia, Nicolò Fiorini, Adriano Milazzo et al.

Textile drying is a key operation in the textile production cycle as it represents one of the most energy-intensive stages and plays a critical role in determining both product quality and overall process efficiency. In this work we propose a mathematical model for the drying process of a generic textile material using heated cylinders, operating under low-pressure conditions. The model's parameters are estimated by nonlinear least squares regression. Given a specific fabric, the developed model allows to predict the drying time and the residual moisture content. The model is validated using real world data provided by a major Italian textile company.

Stefania Bellavia, Francesco Della Santa, Alessandra Papini

This paper introduces novel alternate training procedures for hard-parameter sharing Multi-Task Neural Networks (MTNNs). Traditional MTNN training faces challenges in managing conflicting loss gradients, often yielding sub-optimal performance. The proposed alternate training method updates shared and task-specific weights alternately through the epochs, exploiting the multi-head architecture of the model. This approach reduces computational costs per epoch and memory requirements. Convergence properties similar to those of the classical stochastic gradient method are established. Empirical experiments demonstrate enhanced training regularization and reduced computational demands. In summary, our alternate training procedures offer a promising advancement for the training of hard-parameter sharing MTNNs.

S. Bellavia, G. Gurioli, B. Morini et al.

A regularization algorithm using inexact function values and inexact derivatives is proposed and its evaluation complexity analyzed. This algorithm is applicable to unconstrained problems and to problems with inexpensive constraints (that is constraints whose evaluation and enforcement has negligible cost) under the assumption that the derivative of highest degree is $β$-Hölder continuous. It features a very flexible adaptive mechanism for determining the inexactness which is allowed, at each iteration, when computing objective function values and derivatives. The complexity analysis covers arbitrary optimality order and arbitrary degree of available approximate derivatives. It extends results of Cartis, Gould and Toint (2018) on the evaluation complexity to the inexact case: if a $q$th order minimizer is sought using approximations to the first $p$ derivatives, it is proved that a suitable approximate minimizer within $ε$ is computed by the proposed algorithm in at most $O(ε^{-\frac{p+β}{p-q+β}})$ iterations and at most $O(|\log(ε)|ε^{-\frac{p+β}{p-q+β}})$ approximate evaluations. An algorithmic variant, although more rigid in practice, can be proved to find such an approximate minimizer in $O(|\log(ε)|+ε^{-\frac{p+β}{p-q+β}})$ evaluations.While the proposed framework remains so far conceptual for high degrees and orders, it is shown to yield simple and computationally realistic inexact methods when specialized to the unconstrained and bound-constrained first- and second-order cases. The deterministic complexity results are finally extended to the stochastic context, yielding adaptive sample-size rules for subsampling methods typical of machine learning.

S. Bellavia, V. De Simone, D. di Serafino et al.

This work focuses on the iterative solution of sequences of KKT linear systems arising in interior point methods applied to large convex quadratic programming problems. This task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method. Constraint preconditioners are very effective in this context; nevertheless, their computation may be very expensive for large-scale problems, and resorting to approximations of them may be convenient. Here we propose a procedure for building inexact constraint preconditioners by updating a "seed" constraint preconditioner computed for a KKT matrix at a previous interior point iteration. These updates are obtained through low-rank corrections of the Schur complement of the (1,1) block of the seed preconditioner. The updated preconditioners are analyzed both theoretically and computationally. The results obtained show that our updating procedure, coupled with an adaptive strategy for determining whether to reinitialize or update the preconditioner, can enhance the performance of interior point methods on large problems.

Stefania Bellavia, Benedetta Morini

In this paper we address the numerical solution of nonlinear ill-posed systems by iterative regularization methods in the classes of Levenberg-Marquardt, trust-region and adaptive quadratic regularization procedures. Both with exact and noisy data, our focus is on the potential to approach a solution of the unperturbed systems without assumptions on its vicinity to the initial guess. Regularizing properties of the methods proposed are shown theoretically and validated numerically along with enhanced convergence.

Stefania Bellavia, Benedetta Morini, Elisa Riccietti

In this paper we address the stable numerical solution of nonlinear ill-posed systems by a trust-region method. We show that an appropriate choice of the trust-region radius gives rise to a procedure that has the potential to approach a solution of the unperturbed system. This regularizing property is shown theoretically and validated numerically.