  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.
