We consider the sequential Bayesian optimization problem with bandit
feedback, adopting a formulation that allows for the reward function to vary
with time. We model the reward function using a Gaussian process whose
evolution obeys a simple Markov model. We introduce two natural extensions of
the classical Gaussian process upper confidence bound (GP-UCB) algorithm. The
first, R-GP-UCB, resets GP-UCB at regular intervals. The second, TV-GP-UCB,
instead forgets about old data in a smooth fashion. Our main contribution
comprises of novel regret bounds for these algorithms, providing an explicit
characterization of the trade-off between the time horizon and the rate at
which the function varies. We illustrate the performance of the algorithms on
both synthetic and real data, and we find the gradual forgetting of TV-GP-UCB
to perform favorably compared to the sharp resetting of R-GP-UCB. Moreover,
both algorithms significantly outperform classical GP-UCB, since it treats
stale and fresh data equally.Comment: To appear in AISTATS 201