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Table 5 Convex \(F\): Parallelization speedup factors for DU samplings. The factors below the line are special cases of the general expression. Maximum speedup is naturally obtained by the fully parallel sampling: \(\tfrac{n}{\omega }\)

From: Parallel coordinate descent methods for big data optimization

\(\hat{S}\) Parallelization speedup factor
Doubly uniform \(\frac{\mathbf {E}[|\hat{S}|]}{1 + \tfrac{(\omega -1)\left( (\mathbf {E}[|\hat{S}|^2]/\mathbf {E}[|\hat{S}|])-1\right) }{\max (1,n-1)}}\)
\((\tau ,p_b)\)-binomial \(\frac{\tau }{\tfrac{1}{p_b}+ \tfrac{(\omega -1)(\tau -1)}{\max (1,n-1)}}\)
\(\tau \)-nice \(\frac{\tau }{1+ \tfrac{(\omega -1)(\tau -1)}{\max (1,n-1)}}\)
Fully parallel \(\frac{n}{\omega }\)
Serial \(1\)
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