  Adaptive importance samplers are adaptive Monte Carlo algorithms to estimate
expectations with respect to some target distribution which \textit{adapt}
themselves to obtain better estimators over a sequence of iterations. Although
it is straightforward to show that they have the same $\mathcal{O}(1/\sqrt{N})$
convergence rate as standard importance samplers, where $N$ is the number of
Monte Carlo samples, the behaviour of adaptive importance samplers over the
number of iterations has been left relatively unexplored. In this work, we
investigate an adaptation strategy based on convex optimisation which leads to
a class of adaptive importance samplers termed \textit{optimised adaptive
importance samplers} (OAIS). These samplers rely on the iterative minimisation
of the $\chi^2$-divergence between an exponential-family proposal and the
target. The analysed algorithms are closely related to the class of adaptive
importance samplers which minimise the variance of the weight function. We
first prove non-asymptotic error bounds for the mean squared errors (MSEs) of
these algorithms, which explicitly depend on the number of iterations and the
number of samples together. The non-asymptotic bounds derived in this paper
imply that when the target belongs to the exponential family, the $L_2$ errors
of the optimised samplers converge to the optimal rate of
$\mathcal{O}(1/\sqrt{N})$ and the rate of convergence in the number of
iterations are explicitly provided. When the target does not belong to the
exponential family, the rate of convergence is the same but the asymptotic
$L_2$ error increases by a factor $\sqrt{\rho^\star} > 1$, where $\rho^\star -
1$ is the minimum $\chi^2$-divergence between the target and an
exponential-family proposal.
