  We prove local convergence of several notable gradient descent algorithms
used in machine learning, for which standard stochastic gradient descent theory
does not apply directly. This includes, first, online algorithms for recurrent
models and dynamical systems, such as \emph{Real-time recurrent learning}
(RTRL) and its computationally lighter approximations NoBackTrack and UORO;
second, several adaptive algorithms such as RMSProp, online natural gradient,
and Adam with $\beta^2\to 1$.Despite local convergence being a relatively weak
requirement for a new optimization algorithm, no local analysis was available
for these algorithms, as far as we knew. Analysis of these algorithms does not
immediately follow from standard stochastic gradient (SGD) theory. In fact,
Adam has been proved to lack local convergence in some simple situations
\citep{j.2018on}. For recurrent models, online algorithms modify the parameter
while the model is running, which further complicates the analysis with respect
to simple SGD.Local convergence for these various algorithms results from a
single, more general set of assumptions, in the setup of learning dynamical
systems online. Thus, these results can cover other variants of the algorithms
considered.We adopt an "ergodic" rather than probabilistic viewpoint, working
with empirical time averages instead of probability distributions. This is more
data-agnostic and creates differences with respect to standard SGD theory,
especially for the range of possible learning rates. For instance, with cycling
or per-epoch reshuffling over a finite dataset instead of pure i.i.d.\ sampling
with replacement, empirical averages of gradients converge at rate $1/T$
instead of $1/\sqrt{T}$ (cycling acts as a variance reduction method),
theoretically allowing for larger learning rates than in SGD.
