  Consider the online testing of a stream of hypotheses where a real--time
decision must be made before the next data point arrives. The error rate is
required to be controlled at {all} decision points. Conventional
\emph{simultaneous testing rules} are no longer applicable due to the more
stringent error constraints and absence of future data. Moreover, the online
decision--making process may come to a halt when the total error budget, or
alpha--wealth, is exhausted. This work develops a new class of
structure--adaptive sequential testing (SAST) rules for online false discover
rate (FDR) control. A key element in our proposal is a new alpha--investment
algorithm that precisely characterizes the gains and losses in sequential
decision making. SAST captures time varying structures of the data stream,
learns the optimal threshold adaptively in an ongoing manner and optimizes the
alpha-wealth allocation across different time periods. We present theory and
numerical results to show that the proposed method is valid for online FDR
control and achieves substantial power gain over existing online testing rules.
