  This paper focuses on $\alpha$-divergence minimisation methods for
Variational Inference. More precisely, we are interested in algorithms
optimising the mixture weights of any given mixture model, without any
information on the underlying distribution of its mixture components
parameters. The Power Descent, defined for all $\alpha \neq 1$, is one such
algorithm and we establish in our work the full proof of its convergence
towards the optimal mixture weights when $\alpha <1$. Since the
$\alpha$-divergence recovers the widely-used forward Kullback-Leibler when
$\alpha \to 1$, we then extend the Power Descent to the case $\alpha = 1$ and
show that we obtain an Entropic Mirror Descent. This leads us to investigate
the link between Power Descent and Entropic Mirror Descent: first-order
approximations allow us to introduce the Renyi Descent, a novel algorithm for
which we prove an $O(1/N)$ convergence rate. Lastly, we compare numerically the
behavior of the unbiased Power Descent and of the biased Renyi Descent and we
discuss the potential advantages of one algorithm over the other.
