Stochastic-approximation step-size conditions (${\sum }_n{\alpha }_n=\infty ,{\sum }_n{\alpha }_n^2<\infty$)

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.