We investigate further improvement of boosting
in the case that 
the target concept belongs to the class of r-of-k threshold Boolean functions,
which answers ``$+1$''
if at least $r$ of $k$ relevant variables are positive,
and answers ``$-1$'' otherwise.

Given $m$ examples of a r-of-k function and literals as base hypotheses,
popular boosting algorithms (e.g., AdaBoost)
construct a consistent final hypothesis in $O(k^2 \log m)$ iterations.
While this convergence speed is tight in general,
we show that
a modification of AdaBoost(confidence-rated AdaBoostor InfoBoost)
can make use of the property of r-of-k functions that make less error on one-side
to find a consistent final hypothesis in $O(kr \log m)$ iterations.

Our result extends the previous investigation
by Hatano and Warmuth
and gives more general examples
where confidence-rated AdaBoost or InfoBoost has an advantage over AdaBoost.