Choosing $\alpha_{n}(s)$

Since :

$\vert r_{n}(s)-r_{0}(s)\vert \leq \sum_{i} \vert r_{i+1}(s)-r_{i}(s)\vert \leq ... ...s) \vert(Hr_{i})(s)-r_{i}(s)+W_{i}(s)\vert \leq A \sum_{i=1}^{n} \alpha_{i}(s)$
(where A is a constant that bounds |(Hr_i)(s)-r_i(s)+W_i(s)| ).

We need to require that $\sum_{i} \alpha_{i}(s) = \infty$ : $\forall s\in S$. Otherwise, we have to assume that the distance ||r^* - r₀|| is bounded. The above condition insures that any starting point converges to the optimal value.

We add a condition to insure that the converge rate is fast enough :

$\forall S\in S$ : $\sum_{i} \alpha_{i}^{2} < \infty$

One simple choice could be : $\alpha_{i}(s) = \frac {1}{i}$