10 papers
Computing Kurdyka-Łojasiewicz exponents via composition and symmetry
Cédric Josz, Wenqing Ouyang
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…
Implicit regularization of normalized gradient descent
Cédric Josz
How to find flat minima? We propose running normalized gradient descent, usually reserved for nonsmooth optimization, with sufficiently slowly diminishing step sizes. This induces…
Subdifferentiation with symmetry
Cédric Josz
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 satisf…
Reachability of gradient descent
Cédric Josz, Wenqing Ouyang
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 r…
Lyapunov stability of the Euler method
Cédric Josz
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 discret…
On the geometry of flat minima
Cédric Josz
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 d…
Global convergence of gradient descent for phase retrieval
Théodore Fougereux, Cédric Josz, Xiaopeng Li
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…
Nonsmooth rank-one symmetric matrix factorization landscape
Cédric Josz, Lexiao Lai
We consider nonsmooth rank-one symmetric matrix factorization. It has no spurious second-order stationary points.
Proximal random reshuffling under local Lipschitz continuity
Cédric Josz, Lexiao Lai, Xiaopeng Li
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 assumin…
Singular perturbation in heavy ball dynamics
Cedric Josz, Xiaopeng Li
Given a lower bounded function definable in an o-minimal structure on the real field, we show that the singular pertur…