  Valid online inference is an important problem in contemporary multiple
testing research,to which various solutions have been proposed recently. It is
well-known that these existing methods can suffer from a significant loss of
power if the null $p$-values are conservative. In this work, we extend the
previously introduced methodology to obtain more powerful procedures for the
case of super-uniformly distributed $p$-values. These types of $p$-values arise
in important settings, e.g. when discrete hypothesis tests are performed or
when the $p$-values are weighted. To this end, we introduce the method of
super-uniformity reward (SUR) that incorporates information about the
individual null cumulative distribution functions. Our approach yields several
new 'rewarded' procedures that offer uniform power improvements over known
procedures and come with mathematical guarantees for controlling online error
criteria based either on the family-wise error rate (FWER) or the marginal
false discovery rate (mFDR). We illustrate the benefit of super-uniform
rewarding in real-data analyses and simulation studies. While discrete tests
serve as our leading example, we also show how our method can be applied to
weighted $p$-values.
