Abstract: In the online false discovery rate (FDR) problem, one observes apossibly infinite sequence of $p$-values $P_1,P_2,\dots$, each testinga different null hypothesis, and an algorithm must pick a sequence ofrejection 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, andcannot depend on $P_k$ or any of the later unseen $p$-values, and mustbe 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 thiswork, we present a powerful new framework for online FDR control thatwe refer to as ``SAFFRON''. Like older alpha-investing algorithms,SAFFRON starts off with an error budget (called alpha-wealth) that itintelligently allocates to different tests over time, earning backsome alpha-wealth whenever it makes a new discovery. However, unlikeolder methods, SAFFRON's threshold sequence is based on a novelestimate of the alpha fraction that it allocates to true nullhypotheses. In the offline setting, algorithms that employ an estimateof the proportion of true nulls are called ``adaptive'', henceSAFFRON can be seen as an online analogue of the offlineStorey-BH adaptive procedure. Just as Storey-BH is typically morepowerful than the Benjamini-Hochberg (BH) procedure underindependence, we demonstrate that SAFFRON is also more powerful thanits non-adaptive counterparts such as LORD.