arXiv subjects

Titus Pinta

Publications and source records attributed to Titus Pinta.

5 recordsLinked to original sources

arXiv2026

First-Order Analysis of Optimization in Uniformly Convex Metric Spaces: Directional Subderivatives and Basic Descent

We develop tools for the analysis and implementation of explicit first-order methods for minimizing functions in uniformly convex metric spaces. We formulate sufficient conditions for convergence of descent sequences in terms of directional subderivatives, function values and iterates under regularity assumptions including boundedness, geodesic smoothness and a metric Polyak-{\L}ojasiewicz property. We show that there exists a steepest descent direction in which the assumptions for convergence are satisfied. This work provides a foundation for first-order explicit algorithms for locally smooth, nonconvex optimization.

math.OC↗

arXiv2025

A Newton-Kantorovich Inverse Function Theorem in Quasi-Metric Spaces

The purpose of this work is to investigate root finding problems defined on (quasi-)metric spaces, and ranging in Euclidean spaces. The motivation for this line of inquiry stems from recent models in biology and phylogenetics, where problems of great practical significance are cast as optimization problems on (quasi-)metric spaces. We investigate a minimal algebraic setup that allows us to study a notion of differentiability suitable for Newton-type methods, called Newton differentiability. This notion of differentiability benefits from calculus rules and is sufficient to prove superlinear convergence of a Newton-type method. Finally, a Newton-Kantorovich-type theorem provides an inverse function result, applicable on (quasi-)metric spaces.

math.OC↗

arXiv2025

A Newton-type Method for Non-smooth Under-determined Systems of Equations

We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion of Newton differentiability does not suffice for our purpose, due to the existence of multiple zeros and as such we extend it to uniform Newton differentiability. In this context, we can show that the distance between the iterates and the set of zeros of the system decreases super-linearly. For the special case of smooth equations, the assumptions of our algorithm are simplified. Finally, we provide some numerical examples to showcase the behavior of our proposed method. The key example is a toy model of complementarity constraint problems, showing that our method has great application potential across engineering fields.

math.OC↗

arXiv2025

A Stochastic Newton-type Method for Non-smooth Optimization

We introduce a new framework for analyzing (Quasi-}Newton type methods applied to non-smooth optimization problems. The source of randomness comes from the evaluation of the (approximation) of the Hessian. We derive, using a variant of Chernoff bounds for stopping times, expectation and probability bounds for the random variable representing the number of iterations of the algorithm until approximate first order optimality conditions are validated. As an important distinction to previous results in the literature, we do not require that the estimator is unbiased or that it has finite variance. We then showcase our theoretical results in a stochastic Quasi-Newton method for X-ray free electron laser orbital tomography and in a sketched Newton method for image denoising.

math.OC↗

arXiv2020

New optimization algorithms for neural network training using operator splitting techniques

In the following paper we present a new type of optimization algorithms adapted for neural network training. These algorithms are based upon sequential operator splitting technique for some associated dynamical systems. Furthermore, we investigate through numerical simulations the empirical rate of convergence of these iterative schemes toward a local minimum of the loss function, with some suitable choices of the underlying hyper-parameters. We validate the convergence of these optimizers using the results of the accuracy and of the loss function on the MNIST, MNIST-Fashion and CIFAR 10 classification datasets.

cs.LG↗