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Cédric Josz

Publications and source records attributed to Cédric Josz.

At least 19 recordsLinked to original sources
arXiv2026

Proximal random reshuffling under local Lipschitz continuity

We study proximal random reshuffling (PRR) for minimizing the sum of locally Lipschitz or locally smooth functions and a proper lower semicontinuous convex function without assuming coercivity or the existence of limit points. The algorithmic guarantees pertaining to near approximate stationarity rely on a new tracking lemma linking the iterates to trajectories of conservative fields. One of the novelties in the analysis consists in handling set-valued mappings with unbounded values. In the locally smooth case, it improves the known convergence rate from nearly $O(k^{-1/4})$ to nearly $o(k^{-1/2})$.

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arXiv2026

Computing Kurdyka-Łojasiewicz exponents via composition and symmetry

We devise calculus rules for the Kurdyka-Łojasiewicz exponent using the rank theorem and Lie group actions. They apply to a wide class of composite and invariant functions, and are particularly suitable for handling nonisolated local minima. Notably, smoothness plays no role, eschewing gradient and Hessian computations. This provides a unified framework for establishing linear convergence of various algorithms in matrix factorization, $\ell_1$-matrix factorization, matrix sensing, and linear neural networks.

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arXiv2026

Reachability of gradient descent

We show that gradient descent can converge to any local minimum of a smooth semi-algebraic function. This holds if the step sizes are nonsummable and sufficiently small. The same results hold for the subgradient method on locally Lipschitz semi-algebraic functions if the step size is constant.

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arXiv2026

Implicit regularization of normalized gradient descent

How to find flat minima? We propose running normalized gradient descent, usually reserved for nonsmooth optimization, with sufficiently slowly diminishing step sizes. This induces implicit regularization towards flat minima if an appropriate Lyapunov functions exists in the gradient dynamics. Our analysis shows that implicit regularization is intrinsically a question of nonsmooth analysis, for which we deploy the full power of variational analysis and stratification theory.

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arXiv2026

On the geometry of flat minima

What does it mean to be flat? We propose to define it by measuring the maximal variation around a point, or from a dual perspective, the distance to neighboring level sets. After developing some calculus rules, we show how flat minima, conservation laws, and symmetries are intertwined. Gradient flows of conserved quantities are of particular interest, due to their flattening properties.

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arXiv2026

Lyapunov stability of the Euler method

We extend the Lyapunov stability criterion to Euler discretizations of differential inclusions. It relies on a pair of Lyapunov functions, one in continuous time and one in discrete time. In the context of optimization, this yields sufficient conditions for the stability of nonisolated local minima when using the Bouligand subgradient method.

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arXiv2025

Subdifferentiation with symmetry

Given an objective function that is invariant under an action of a Lie group, we study how its subgradients relate to the orbits of the action. Our main finding is that they satisfy projection formulae analogous to those stemming from the Whitney and Verdier stratifications. If the function is definable in an o-minimal structure on the real field, then we also obtain an invariant variational stratification. On the application side, we derive a conservation law for subgradient dynamics under minimal assumptions. It can be used to detect instability in discrete subgradient dynamics.

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arXiv2024

Global convergence of the gradient method for functions definable in o-minimal structures

We consider the gradient method with variable step size for minimizing functions that are definable in o-minimal structures on the real field and differentiable with locally Lipschitz gradients. We prove that global convergence holds if continuous gradient trajectories are bounded, with the minimum gradient norm vanishing at the rate $o(1/k)$ if the step sizes are greater than a positive constant. If additionally the gradient is continuously differentiable, all saddle points are strict, and the step sizes are constant, then convergence to a local minimum holds almost surely over any bounded set of initial points.

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arXiv2024

Global convergence of gradient descent for phase retrieval

We propose a tensor-based criterion for benign landscape in phase retrieval and establish boundedness of gradient trajectories. This implies that gradient descent will converge to a global minimum for almost every initial point.

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arXiv2024

Convergence of the momentum method for semialgebraic functions with locally Lipschitz gradients

We propose a new length formula that governs the iterates of the momentum method when minimizing differentiable semialgebraic functions with locally Lipschitz gradients. It enables us to establish local convergence, global convergence, and convergence to local minimizers without assuming global Lipschitz continuity of the gradient, coercivity, and a global growth condition, as is done in the literature. As a result, we provide the first convergence guarantee of the momentum method starting from arbitrary initial points when applied to principal component analysis, matrix sensing, and linear neural networks.

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arXiv2023

Global stability of first-order methods for coercive tame functions

We consider first-order methods with constant step size for minimizing locally Lipschitz coercive functions that are tame in an o-minimal structure on the real field. We prove that if the method is approximated by subgradient trajectories, then the iterates eventually remain in a neighborhood of a connected component of the set of critical points. Under suitable method-dependent regularity assumptions, this result applies to the subgradient method with momentum, the stochastic subgradient method with random reshuffling and momentum, and the random-permutations cyclic coordinate descent method.

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arXiv2023

Certifying the absence of spurious local minima at infinity

When searching for global optima of nonconvex unconstrained optimization problems, it is desirable that every local minimum be a global minimum. This property of having no spurious local minima is true in various problems of interest nowadays, including principal component analysis, matrix sensing, and linear neural networks. However, since these problems are non-coercive, they may yet have spurious local minima at infinity. The classical tools used to analyze the optimization landscape, namely the gradient and the Hessian, are incapable of detecting spurious local minima at infinity. In this paper, we identify conditions that certify the absence of spurious local minima at infinity, one of which is having bounded subgradient trajectories. We check that they hold in several applications of interest.

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arXiv2023

Lyapunov stability of the subgradient method with constant step size

We consider the subgradient method with constant step size for minimizing locally Lipschitz semi-algebraic functions. In order to analyze the behavior of its iterates in the vicinity of a local minimum, we introduce a notion of discrete Lyapunov stability and propose necessary and sufficient conditions for stability.

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arXiv2022

Nonsmooth rank-one matrix factorization landscape

We provide the first positive result on the nonsmooth optimization landscape of robust principal component analysis, to the best of our knowledge. It is the object of several conjectures and remains mostly uncharted territory. We identify a necessary and sufficient condition for the absence of spurious local minima in the rank-one case. Our proof exploits the subdifferential regularity of the objective function in order to eliminate the existence quantifier from the first-order optimality condition known as Fermat's rule.

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arXiv2021

Verifying Global Optimality of Candidate Solutions to Polynomial Optimization Problems using a Determinant Relaxation Hierarchy

We propose a method for verifying that a given feasible point for a polynomial optimization problem is globally optimal. The approach relies on the Lasserre hierarchy and the result of Lasserre regarding the importance of the convexity of the feasible set as opposed to that of the individual constraints. By focusing solely on certifying global optimality and relaxing the Lasserre hierarchy using necessary conditions for positive semidefiniteness based on matrix determinants, the proposed method is implementable as a computationally tractable linear program. We demonstrate this method via application to several instances of polynomial optimization, including the optimal power flow problem used to operate electric power systems.

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arXiv2018

Sparse polynomial interpolation: sparse recovery, super resolution, or Prony?

We show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the $n$-dimensional torus. Therefore the semidefinite programming approach initiated by Candès \\& Fernandez-Granda \cite{candes\_towards\_2014} in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that exact recovery is guaranteed provided that a geometric spacing condition on the supports holds and the number of evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of $\ell\_1$-norm minimization is also guaranteed to provide exact recovery {\it provided that} theevaluations are made in a certain manner and even though the Restricted Isometry Property for exact recovery is not satisfied. (A naive sparse recovery LP-approach does not offer such a guarantee.) Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments, one in which the super-resolution technique and Prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e. a semidefinite optimization).

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