  In the online false discovery rate (FDR) problem, one observes a possibly
infinite sequence of $p$-values $P_1,P_2,\dots$, each testing a different null
hypothesis, and an algorithm must pick a sequence of rejection thresholds
$\alpha_1,\alpha_2,\dots$ in an online fashion, effectively rejecting the
$k$-th null hypothesis whenever $P_k \leq \alpha_k$. Importantly, $\alpha_k$
must be a function of the past, and cannot depend on $P_k$ or any of the later
unseen $p$-values, and must be chosen to guarantee that for any time $t$, the
FDR up to time $t$ is less than some pre-determined quantity $\alpha \in
(0,1)$. In this work, we present a powerful new framework for online FDR
control that we refer to as SAFFRON. Like older alpha-investing (AI)
algorithms, SAFFRON starts off with an error budget, called alpha-wealth, that
it intelligently allocates to different tests over time, earning back some
wealth on making a new discovery. However, unlike older methods, SAFFRON's
threshold sequence is based on a novel estimate of the alpha fraction that it
allocates to true null hypotheses. In the offline setting, algorithms that
employ an estimate of the proportion of true nulls are called adaptive methods,
and SAFFRON can be seen as an online analogue of the famous offline Storey-BH
adaptive procedure. Just as Storey-BH is typically more powerful than the
Benjamini-Hochberg (BH) procedure under independence, we demonstrate that
SAFFRON is also more powerful than its non-adaptive counterparts, such as LORD
and other generalized alpha-investing algorithms. Further, a monotone version
of the original AI algorithm is recovered as a special case of SAFFRON, that is
often more stable and powerful than the original. Lastly, the derivation of
SAFFRON provides a novel template for deriving new online FDR rules.
