Stochastic Online Optimization using Kalman Recursion

Joseph de Vilmarest (1) , Olivier Wintenberger (1)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

We study the Extended Kalman Filter in constant dynamics, offering a bayesian perspective of stochastic optimization. We obtain high probability bounds on the cumulative excess risk in an unconstrained setting. In order to avoid any projection step we propose a two-phase analysis. First, for linear and logistic regressions, we prove that the algorithm enters a local phase where the estimate stays in a small region around the optimum. We provide explicit bounds with high probability on this convergence time. Second, for generalized linear regressions, we provide a martingale analysis of the excess risk in the local phase, improving existing ones in bounded stochastic optimization. The EKF appears as a parameter-free online algorithm with O(d^2) cost per iteration that optimally solves some unconstrained optimization problems.

Mots clés: Logistic Regression; Kalman filter; stochastic optimization; online learning; extended kalman filter; Online Optimization

Domaines: Apprentissage [cs.LG]; Intelligence artificielle [cs.AI]; Statistiques [math.ST]

Joseph de Vilmarest, Olivier Wintenberger. Stochastic Online Optimization using Kalman Recursion. Journal of Machine Learning Research, 2021, 22, pp.1-55. ⟨hal-02468701v2⟩