## 3 événements

• Séminaire de Probabilités

### Mardi 9 janvier 09:45-10:45 - Franziska Kuehn - IMT

An introduction to Feller processes

Résumé : In this talk we give a brief introduction to Feller processes. Roughly speaking, a Feller process is a space-inhomogeneous Markov process which behaves locally like a Lévy process, but the Lévy triplet depends on the current position of the process. For this reason Feller processes are also called Lévy-type processes.
Feller processes can be characterized by their symbol which is the analogue of the characteristic exponent in the Lévy case. In this talk we motivate that the symbol is a powerful tool to describe distributional properties and path properties of a Feller process. Moreover, we present examples of Feller processes (such as stable-like processes and stochastic differential equations driven by a Lévy process) and discuss some open problems.

Lieu : Salle MIP

• Séminaire MIP

### Mardi 9 janvier 11:00-12:00 -

Pas de séminaire en raison de la journée nouveaux entrants MIP

• Séminaire de Statistique

### Mardi 9 janvier 11:00-12:00 - Sébastien Gadat - UT1

Non-asymptotic bound for stochastic averaging (joint work with F. Panloup)

Résumé : This work is devoted to the non-asymptotic control of the mean-squared error for the
Ruppert-Polyak stochastic averaged gradient descent introduced in the seminal contributions
of [1] and [2]. Starting from a standard stochastic gradient descent algorithm introduced by
Robbins and Monro to minimize a smooth convex function f with noisy inputs, they define a
sequence (θ_n ) n≥1 through :
θ_n+1} = θ_n − γ_n+1} ∇f (θ_n ) + γ_n+1} ∆M_n+1}
where −∇f (θ_n ) + ∆M_n+1} stands for a gradient step corrupted by an additive noise at each step.
The averaging procedure introduced in [1, 2] consists in computing a Cesaro average :
In our main results, we establish non-asymptotic tight bounds (optimal with respect to the
Cramer-Rao lower bound) in a very general framework that includes the uniformly strongly
convex case as well as the one where the function f to be minimized satisfies a weaker Kurdyka-
Łojiasewicz-type condition [3, 4]. In particular, it makes it possible to recover some pathological
examples such as on-line learning for logistic regression (see [5]) and recursive quantile estimation
(an even non-convex situation). Finally, our bound is optimal when the decreasing step (γ_n ) n≥1
satisfies : γ_n = n ^−β with β = 3/4, leading to a second-order term in O(n^−5/4}).
References
[1] D. Ruppert, Efficient estimations from a slowly convergent Robbins-Monro process, Techni-
cal Report, 781, Cornell University Operations Research and Industrial Engineering, (1988).
[2] B. T. Polyak and A. Juditsky, Acceleration of Stochastic Approximation by Averaging,
SIAM Journal on Control and Optimization, vol. 30 (4), 838-855 (1992).
[3] S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels, Editions du
CNRS, Paris, Les Équations aux Dérivées Partielles, 87-89 (1963).
[4] K. Kurdyka, On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier,
Université de Grenoble. Annales de l’Institut Fourier, vol. 48 (3), 769-783 (1998).
[5] F. Bach, Adaptivity of averaged stochastic gradient descent to local strong convexity for
logistic regression,Journal of Machine Learning Research, vol. 15, 595-627 (2014).

Lieu : Salle 106 Bat 1R1