arXiv subjects

Sadok Jerad

Publications and source records attributed to Sadok Jerad.

9 recordsLinked to original sources

arXiv2026

An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers. The obstacle is increasingly understood to be one of optimisation, owing to the severely ill-conditioned loss landscape. We present $\textbf{DSGNAR}$: Doubly-Sketched Gauss-Newton with Adaptive Ratio, a scalable second-order optimisation framework that confronts this ill-conditioning and, in doing so, obtains unprecedented accuracy and speed. $\textbf{DSGNAR}$ couples a doubly-sketched Gauss-Newton model with a novel strategy that carefully controls both regularisation and step length. Across a suite of problems spanning nonlinear, chaotic, multi-scale, high-dimensional, and Navier-Stokes, the framework greatly improves on the state of the art: able to attain relative $\ell_2$ errors as low as $3\times10^{-16}$ in double precision, improve contemporary results by five orders of magnitude on the canonical Burgers' equation, and as much as eight orders on a high-dimensional Poisson problem, while remaining markedly faster. We further show that, in single precision, solutions at the limit of round-off error can be obtained very quickly: Burgers' equation to $\ell_2^{\text{rel}} = 4.75 \times 10^{-7}$ in under ten seconds. The framework is also robust to the choice of architecture, arithmetic precision, and initial hyperparameters. The code is available at https://www.github.com/wephy/physics-informed-neural-networks

cs.LG↗

arXiv2026

Fast Adaptive Tensor Methods Under Local Smoothness

A new, fast adaptive regularization methods is proposed and analyzed under local Lipschitz smoothness of the $p$-th order tensor. For nonconvex problems, it achieves the optimal $\mathcal{O}\!\left(|\log(\epsilon)|\epsilon^{-(p+1)/p}\right)$ complexity to obtain first-order $\epsilon$-stationary points and in the convex case, it yields $\mathcal{O}\!\left(|\log(\epsilon)|\epsilon^{-1/p}\right)$ iterations to drive the optimality gap below $\epsilon$, thus matching the complexity bounds of standard tensor methods under global Lipschitz smoothness yp to logarithmic terms. The proposed algorithm follows the line of standard tensor methods with an appropriately chosen regularization and suitable modifications. Initial numerical experiments and comparisons for some nonconvex regression problems are made with the standard adaptive cubic regularization where we showcase some potential of the proposed method.

math.OC↗

arXiv2026

A Parameter-Free First-Order Algorithm for Non-Convex Optimization with $\tilde{\mkern1mu O}(\epsilon^{-5/3})$ Global Rate

We introduce PF-AGD, the first parameter-free, deterministic, accelerated first-order method to achieve $O(\epsilon^{-5/3}\log(1/\epsilon))$ oracle complexity bound when minimizing sufficiently smooth, non-convex functions; this is the best-known bound for first-order methods on smooth non-convex objectives. Unlike existing methods possessing this rate that require a priori knowledge of smoothness constants, we use an adaptive backtracking scheme and a gradient-based restart mechanism to estimate local curvature. This yields a practical algorithm that matches best-known theoretical rates. Empirically, PF-AGD outperforms the practical variant of AGD-Until-Guilty (Carmon et al., 2017), as well as other parameter-free variants, and is a viable alternative to nonlinear conjugate gradient methods.

math.OC↗

arXiv2025

On Global Rates for Regularization Methods based on Secant Derivative Approximations

An inexact and globally convergent framework for high-order adaptive regularization methods is presented, in which approximations may be used for the $p$th-order tensor, based on lower-order derivatives. Between each recalculation of the $p$th-order derivative approximation, a high-order secant equation can be used to update the $p$th-order tensor as proposed in (Karl Welzel and Raphael A Hauser, Approximating higher-order derivative tensors using secant updates, SIAM J.Optim, 34(1), 2024) or the approximation can be kept constant in a lazy manner. When refreshing the $p$th-order tensor approximation after $m$ steps, an exact evaluation of the tensor or a finite difference approximation can be used with an explicit discretization stepsize. For all the newly adaptive regularization variants, we prove an $\mathcal{O}\left( \max[ \epsilon_1^{-(p+1)/p}, \, {\epsilon_2^{-(p+1)/(p-1)}} ] \right)$ bound on the number of iterations needed to reach an $(\epsilon_1, \, \epsilon_2)$ second-order stationary points. Discussions on the number of oracle calls for each introduced variant are also provided. When $p=2$, we obtain a second-order method that uses quasi-Newton approximations with an $\mathcal{O}\left(\max[\epsilon_1^{-3/2}, \, \, \epsilon_2^{-3}]\right)$ iteration bound to achieve approximate second-order stationarity. Numerical illustrations for the case $p=3$ are provided in both the deterministic and noisy settings showcasing the merits of secant updates for approximating third-order information, as well as the robustness of our proposed method even in noisy cases.

math.OC↗

arXiv2025

A Fast Newton Method Under Local Lipschitz Smoothness

A new, fast second-order method is proposed that achieves the optimal $\mathcal{O}\left(|\log(\epsilon)|\epsilon^{-3/2}\right)$ complexity to obtain first-order $\epsilon$-stationary points. Crucially, this is deduced without assuming the standard global Lipschitz Hessian continuity condition, but only using an appropriate local smoothness requirement. The algorithm exploits Hessian information to compute a Newton step and a negative curvature step when needed, in an approach similar to that of the AN2C method.Inexact versions of the Newton step and negative curvature are proposed in order to reduce the cost of evaluating second-order information. Details are given of such an iterative implementation using Krylov subspaces. An extended algorithm for finding second-order critical points is also developed and its complexity is again shown to be within a log factor of the optimal one. Initial numerical experiments are discussed for both factorised and Krylov variants, which demonstrate the competitiveness of the proposed algorithm.

math.OC↗

arXiv2024

A Stochastic Objective-Function-Free Adaptive Regularization Method with Optimal Complexity

A fully stochastic second-order adaptive-regularization method for unconstrained nonconvex optimization is presented which never computes the objective-function value, but yet achieves the optimal $\mathcal{O}(\epsilon^{-3/2})$ complexity bound for finding first-order critical points. The method is noise-tolerant and the inexactness conditions required for convergence depend on the history of past steps. Applications to cases where derivative evaluation is inexact and to minimization of finite sums by sampling are discussed. Numerical experiments on large binary classification problems illustrate the potential of the new method.

math.OC↗

arXiv2024

Complexity of Adagrad and other first-order methods for nonconvex optimization problems with bounds constraints

A parametric class of trust-region algorithms for constrained nonconvex optimization is analyzed, where the objective function is never computed. By defining appropriate first-order stationarity criteria, we are able to extend the Adagrad method to the newly considered problem and retrieve the standard complexity rate of the projected gradient method that uses both the gradient and objective function values. Furthermore, we propose an additional iteration-dependent scaling with slightly inferior theoretical guarantees. In both cases, the bounds are essentially sharp, and curvature information can be used to compute the stepsize. Initial experimental results for noisy bound-constrained instances illustrate the benefits of the objective-free approach.

math.OC↗

arXiv2023

Yet another fast variant of Newton's method for nonconvex optimization

A class of second-order algorithms is proposed for minimizing smooth nonconvex functions that alternates between regularized Newton and negative curvature steps in an iteration-dependent subspace. In most cases, the Hessian matrix is regularized with the square root of the current gradient and an additional term taking moderate negative curvature into account, a negative curvature step being taken only exceptionally. Practical variants have been detailed where the subspaces are chosen to be the full space, or Krylov subspaces. In the first case, the proposed method only requires the solution of a single linear system at nearly all iterations. We establish that at most $\mathcal{O}\left(|\logε|\,ε^{-3/2}\right)$ evaluations of the problem's objective function and derivatives are needed for algorithms in the new class to obtain an $ε$-approximate first-order minimizer, and at most $\mathcal{O}\left(|\logε|\,ε^{-3}\right)$ to obtain a second-order one. Encouraging initial numerical experiments with two full-space and two Krylov-subspaces variants are finally presented.

math.OC↗

arXiv2021

H\"older Gradient Descent and Adaptive Regularization Methods in Banach Spaces for First-Order Points

This paper considers optimization of smooth nonconvex functionals in smooth infinite dimensional spaces. A H\"older gradient descent algorithm is first proposed for finding approximate first-order points of regularized polynomial functionals. This method is then applied to analyze the evaluation complexity of an adaptive regularization method which searches for approximate first-order points of functionals with $\beta$-H\"older continuous derivatives. It is shown that finding an $\epsilon$-approximate first-order point requires at most $O(\epsilon^{-\frac{p+\beta}{p+\beta-1}})$ evaluations of the functional and its first $p$ derivatives.

math.OC↗