  We propose a computationally efficient random walk on a convex body which
rapidly mixes and closely tracks a time-varying log-concave distribution. We
develop general theoretical guarantees on the required number of steps; this
number can be calculated on the fly according to the distance from and the
shape of the next distribution. We then illustrate the technique on several
examples. Within the context of exponential families, the proposed method
produces samples from a posterior distribution which is updated as data arrive
in a streaming fashion. The sampling technique can be used to track
time-varying truncated distributions, as well as to obtain samples from a
changing mixture model, fitted in a streaming fashion to data. In the setting
of linear optimization, the proposed method has oracle complexity with best
known dependence on the dimension for certain geometries. In the context of
online learning and repeated games, the algorithm is an efficient method for
implementing no-regret mixture forecasting strategies. Remarkably, in some of
these examples, only one step of the random walk is needed to track the next
distribution.
