  Richardson extrapolation is a classical technique from numerical analysis
that can improve the approximation error of an estimation method by combining
linearly several estimates obtained from different values of one of its
hyperparameters, without the need to know in details the inner structure of the
original estimation method. The main goal of this paper is to study when
Richardson extrapolation can be used within machine learning, beyond the
existing applications to step-size adaptations in stochastic gradient descent.
We identify two situations where Richardson interpolation can be useful: (1)
when the hyperparameter is the number of iterations of an existing iterative
optimization algorithm, with applications to averaged gradient descent and
Frank-Wolfe algorithms (where we obtain asymptotically rates of $O(1/k^2)$ on
polytopes, where $k$ is the number of iterations), and (2) when it is a
regularization parameter, with applications to Nesterov smoothing techniques
for minimizing non-smooth functions (where we obtain asymptotically rates close
to $O(1/k^2)$ for non-smooth functions), and ridge regression. In all these
cases, we show that extrapolation techniques come with no significant loss in
performance, but with sometimes strong gains, and we provide theoretical
justifications based on asymptotic developments for such gains, as well as
empirical illustrations on classical problems from machine learning.
