arXiv subjects

Nizar Bousselmi

Publications and source records attributed to Nizar Bousselmi.

5 recordsLinked to original sources

arXiv2025

On the Convex Interpolation for Linear Operators

The worst-case performance of an optimization method on a problem class can be analyzed using a finite description of the problem class, known as interpolation conditions. In this work, we study interpolation conditions for linear operators given scalar products between discrete inputs and outputs. First, we show that if only convex constraints on the scalar products of inputs and outputs are allowed,it is only possible to characterize classes of linear operators or symmetric linear operator whose all singular values or eigenvalues belong to some subset of R. Then, we propose new interpolation conditions for linear operators with minimal and maximal singular values and linear operators whose eigenvalues or singular values belong to unions of subsets. Finally, we illustrate the new interpolation conditions through the analysis of the Gradient and Chambolle-Pock methods. It allows to obtain new numerical worst-case guarantees on these methods.

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arXiv2025

Performance Estimation of second-order optimization methods on classes of univariate functions

We develop a principled approach to obtain exact computer-aided worst-case guarantees on the performance of second-order optimization methods on classes of univariate functions. We first present a generic technique to derive interpolation conditions for a wide range of univariate functions, and use it to obtain such conditions for generalized self-concordant functions (including self-concordant and quasi-self-concordant functions) and functions with Lipschitz Hessian (both convex and non-convex). We then exploit these conditions within the Performance Estimation framework to tightly analyze the convergence of second-order methods on univariate functions, including (Cubic Regularized) Newton's method and several of its variants. Thereby, we improve on existing convergence rates, exhibit univariate lower bounds (that thus hold in the multivariate case), and analyze the performance of these methods with respect to the same criteria.

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arXiv2024

Interpolation Conditions for Linear Operators and Applications to Performance Estimation Problems

The Performance Estimation Problem methodology makes it possible to determine the exact worst-case performance of an optimization method. In this work, we generalize this framework to first-order methods involving linear operators. This extension requires an explicit formulation of interpolation conditions for those linear operators. We consider the class of linear operators $\mathcal{M}:x \mapsto Mx$ where matrix $M$ has bounded singular values, and the class of linear operators where $M$ is symmetric and has bounded eigenvalues. We describe interpolation conditions for these classes, i.e. necessary and sufficient conditions that, given a list of pairs $\{(x_i,y_i)\}$, characterize the existence of a linear operator mapping $x_i$ to $y_i$ for all $i$. Using these conditions, we first identify the exact worst-case behavior of the gradient method applied to the composed objective $h\circ \mathcal{M}$, and observe that it always corresponds to $\mathcal{M}$ being a scaling operator. We then investigate the Chambolle-Pock method applied to $f+g\circ \mathcal{M}$, and improve the existing analysis to obtain a proof of the exact convergence rate of the primal-dual gap. In addition, we study how this method behaves on Lipschitz convex functions, and obtain a numerical convergence rate for the primal accuracy of the last iterate. We also show numerically that averaging iterates is beneficial in this setting.

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arXiv2024

Comparison of Proximal First-Order Primal and Primal-Dual algorithms via Performance Estimation

Selecting the fastest algorithm for a specific signal/image processing task is a challenging question. We propose an approach based on the Performance Estimation Problem framework that numerically and automatically computes the worst-case performance of a given optimization method on a class of functions. We first propose a computer-assisted analysis and comparison of several first-order primal optimization methods, namely, the gradient method, the forward-backward, Peaceman-Rachford, and Douglas-Rachford splittings. We tighten the existing convergence results of these algorithms and extend them to new classes of functions. Our analysis is then extended and evaluated in the context of the primal-dual Chambolle-Pock and Condat-Vũ methods.

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arXiv2023

Interpolation Constraints for Computing Worst-Case Bounds in Performance Estimation Problems

The Performance Estimation Problem (PEP) approach consists in computing worst-case performance bounds on optimization algorithms by solving an optimization problem: one maximizes an error criterion over all initial conditions allowed and all functions in a given class of interest. The maximal value is then a worst-case bound, and the maximizer provides an example reaching that worst case. This approach was introduced for optimization algorithms but could in principle be applied to many other contexts involving worst-case bounds. The key challenge is the representation of infinite-dimensional objects involved in these optimization problems such as functions, and complex or non-convex objects as linear operators and their powers, networks in decentralized optimization etc. This challenge can be resolved by interpolation constraints, which allow representing the effect of these objects on vectors of interest, rather than the whole object, leading to tractable finite dimensional problems. We review several recent interpolation results and their implications in obtaining of worst-case bounds via PEP.

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