51 citations · 202 across the 34 of their papers we have counts for
25 papers · 1 filter
One-step differentiation of iterative algorithms
Jérôme Bolte, Edouard Pauwels, Samuel Vaiter
In appropriate frameworks, automatic differentiation is transparent to the user at the cost of being a significant computational burden when the number of operations is large. For…
On the nature of Bregman functions
Edouard Pauwels
Let C be convex, compact, with nonempty interior and h be Legendre with domain C, continuous on C. We prove that h is Bregman if and only if it is strictly convex on C and C is a p…
The derivatives of Sinkhorn-Knopp converge
Edouard Pauwels, Samuel Vaiter
We show that the derivatives of the Sinkhorn-Knopp algorithm, or iterative proportional fitting procedure, converge towards the derivatives of the entropic regularization of the op…
Automatic differentiation of nonsmooth iterative algorithms
Jérôme Bolte, Edouard Pauwels, Samuel Vaiter
Differentiation along algorithms, i.e., piggyback propagation of derivatives, is now routinely used to differentiate iterative solvers in differentiable programming. Asymptotics is…
Subgradient sampling for nonsmooth nonconvex minimization
Jérôme Bolte, Tam Le, Edouard Pauwels
Risk minimization for nonsmooth nonconvex problems naturally leads to first-order sampling or, by an abuse of terminology, to stochastic subgradient descent. We establish the conve…
Semialgebraic Representation of Monotone Deep Equilibrium Models and Applications to Certification
Tong Chen, Jean-Bernard Lasserre, Victor Magron +1
Deep equilibrium models are based on implicitly defined functional relations and have shown competitive performance compared with the traditional deep networks. Monotone operator e…
Conservative parametric optimality and the ridge method for tame min-max problems
Edouard Pauwels
We study the ridge method for min-max problems, and investigate its convergence without any convexity, differentiability or qualification assumption. The central issue is to determ…
A Sublevel Moment-SOS Hierarchy for Polynomial Optimization
Tong Chen, Jean-Bernard Lasserre, Victor Magron +1
We introduce a sublevel Moment-SOS hierarchy where each SDP relaxation can be viewed as an intermediate (or interpolation) between the d-th and (d+1)-th order SDP relaxations of th…
Sequential convergence of AdaGrad algorithm for smooth convex optimization
Cheik Traoré, Edouard Pauwels
We prove that the iterates produced by, either the scalar step size variant, or the coordinatewise variant of AdaGrad algorithm, are convergent sequences when applied to convex obj…
Long term dynamics of the subgradient method for Lipschitz path differentiable functions
Jerome Bolte, Edouard Pauwels, Rodolfo Rios-Zertuche
We consider the long-term dynamics of the vanishing stepsize subgradient method in the case when the objective function is neither smooth nor convex. We assume that this function i…
Semialgebraic Optimization for Lipschitz Constants of ReLU Networks
Tong Chen, Jean-Bernard Lasserre, Victor Magron +1
The Lipschitz constant of a network plays an important role in many applications of deep learning, such as robustness certification and Wasserstein Generative Adversarial Network.…
Curiosities and counterexamples in smooth convex optimization
Jerome Bolte, Edouard Pauwels
Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. We show that block-coordinate, steepest descent with exact search or…
Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning
Jérôme Bolte, Edouard Pauwels
Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible calculus. We introduce generalized derivatives called conservative fields for which we d…
Semi-algebraic approximation using Christoffel-Darboux kernel
Swann Marx, Edouard Pauwels, Tillmann Weisser +2
We provide a new method to approximate a (possibly discontinuous) function using Christoffel-Darboux kernels. Our knowledge about the unknown multivariate function is in terms of f…
An unexpected connection between Bayes optimal designs and the Group Lasso
Guillaume Sagnol, Edouard Pauwels
We show that the -optimal design optimization problem over design points in is equivalent to minimizing a quadratic function plus a group lasso sparsity induc…
The multiproximal linearization method for convex composite problems
Jérôme Bolte, Zheng Chen, Edouard Pauwels
Composite minimization involves a collection of smooth functions which are aggregated in a nonsmooth manner. In the convex setting, we design an algorithm by linearizing each smoot…
Qualification Conditions in Semi-algebraic Programming
Jérôme Bolte, Antoine Hochart, Edouard Pauwels
For an arbitrary finite family of semi-algebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbation…
Extragradient Method in Optimization: Convergence and Complexity
Trong Phong Nguyen, Edouard Pauwels, Emile Richard +1
We consider the extragradient method to minimize the sum of two functions, the first one being smooth and the second being convex. Under the Kurdyka-Lojasiewicz assumption, we prov…
Positivity certificates in optimal control
Edouard Pauwels, Didier Henrion, Jean-Bernard Lasserre
We propose a tutorial on relaxations and weak formulations of optimal control with their semidefinite approximations. We present this approach solely through the prism of positivit…
The value function approach to convergence analysis in composite optimization
Edouard Pauwels
This works aims at understanding further convergence properties of first order local search methods with complex geometries. We focus on the composite optimization model which unif…
The Cyclic Block Conditional Gradient Method for Convex Optimization Problems
Amir Beck, Edouard Pauwels, Shoham Sabach
In this paper we study the convex problem of optimizing the sum of a smooth function and a compactly supported non-smooth term with a specific separable form. We analyze the block…
Linear conic optimization for inverse optimal control
Edouard Pauwels, Didier Henrion, Jean-Bernard Lasserre
We address the inverse problem of Lagrangian identification based on trajecto-ries in the context of nonlinear optimal control. We propose a general formulation of the inverse prob…
Majorization-minimization procedures and convergence of SQP methods for semi-algebraic and tame programs
Jérôme Bolte, Edouard Pauwels
In view of solving nonsmooth and nonconvex problems involving complex constraints (like standard NLP problems), we study general maximization-minimization procedures produced by fa…
Linear conic optimization for nonlinear optimal control
Didier Henrion, Edouard Pauwels
Infinite-dimensional linear conic formulations are described for nonlinear optimal control problems. The primal linear problem consists of finding occupation measures supported on…