  We introduce a new class of adaptive importance samplers leveraging adaptive
optimisation tools, which we term AdaOAIS. We build on Optimised Adaptive
Importance Samplers (OAIS), a class of techniques that adapt proposals to
improve the mean-squared error of the importance sampling estimators by
parameterising the proposal and optimising the $\chi^2$-divergence between the
target and the proposal. We show that a naive implementation of OAIS using
stochastic gradient descent may lead to unstable estimators despite its
convergence guarantees. To remedy this shortcoming, we instead propose to use
adaptive optimisers (such as AdaGrad and Adam) to improve the stability of the
OAIS. We provide convergence results for AdaOAIS in a similar manner to OAIS.
We also provide empirical demonstration on a variety of examples and show that
AdaOAIS lead to stable importance sampling estimators in practice.
