= Stochastic-approximation step-size conditions
{title2=$\sum_n\alpha_n=\infty,\quad\sum_n\alpha_n^2<\infty$}
A common condition along each repeatedly sampled component is $\sum_n\alpha_n=\infty$ and $\sum_n\alpha_n^2<\infty$. The first permits continuing correction; the second limits accumulated noise under the relevant stochastic-approximation hypotheses. The example $\alpha_n=1/n$ has a divergent harmonic sum and a convergent sum of squares. These conditions are not sufficient without suitable assumptions about the update dynamics and sampling.
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