  Bayesian Optimization (BO) has proven to be very successful at optimizing a
static, noisy, costly-to-evaluate black-box function $f : \mathcal{S} \to
\mathbb{R}$. However, optimizing a black-box which is also a function of time
(i.e., a dynamic function) $f : \mathcal{S} \times \mathcal{T} \to \mathbb{R}$
remains a challenge, since a dynamic Bayesian Optimization (DBO) algorithm has
to keep track of the optimum over time. This changes the nature of the
optimization problem in at least three aspects: (i) querying an arbitrary point
in $\mathcal{S} \times \mathcal{T}$ is impossible, (ii) past observations
become less and less relevant for keeping track of the optimum as time goes by
and (iii) the DBO algorithm must have a high sampling frequency so it can
collect enough relevant observations to keep track of the optimum through time.
In this paper, we design a Wasserstein distance-based criterion able to
quantify the relevancy of an observation with respect to future predictions.
Then, we leverage this criterion to build W-DBO, a DBO algorithm able to remove
irrelevant observations from its dataset on the fly, thus maintaining
simultaneously a good predictive performance and a high sampling frequency,
even in continuous-time optimization tasks with unknown horizon. Numerical
experiments establish the superiority of W-DBO, which outperforms
state-of-the-art methods by a comfortable margin.
