  This paper describes a methodology for detecting anomalies from sequentially
observed and potentially noisy data. The proposed approach consists of two main
elements: (1) {\em filtering}, or assigning a belief or likelihood to each
successive measurement based upon our ability to predict it from previous noisy
observations, and (2) {\em hedging}, or flagging potential anomalies by
comparing the current belief against a time-varying and data-adaptive
threshold. The threshold is adjusted based on the available feedback from an
end user. Our algorithms, which combine universal prediction with recent work
on online convex programming, do not require computing posterior distributions
given all current observations and involve simple primal-dual parameter
updates. At the heart of the proposed approach lie exponential-family models
which can be used in a wide variety of contexts and applications, and which
yield methods that achieve sublinear per-round regret against both static and
slowly varying product distributions with marginals drawn from the same
exponential family. Moreover, the regret against static distributions coincides
with the minimax value of the corresponding online strongly convex game. We
also prove bounds on the number of mistakes made during the hedging step
relative to the best offline choice of the threshold with access to all
estimated beliefs and feedback signals. We validate the theory on synthetic
data drawn from a time-varying distribution over binary vectors of high
dimensionality, as well as on the Enron email dataset.
