Bayesian optimization is a technique for efficiently optimizing unknown functions in a black-box manner. To handle practical settings where gathering data requires use of finite resources, it is desirable to explicitly incorporate function evaluation costs into Bayesian optimization policies. To understand how to do so, we develop a previously-unexplored connection between cost-aware Bayesian optimization and the Pandora's Box problem, a decision problem from economics. The Pandora's Box problem admits a Bayesian-optimal solution based on an expression called the Gittins index, which can be reinterpreted as an acquisition function. We study the use of this acquisition function for cost-aware Bayesian optimization, and demonstrate empirically that it performs well, particularly in medium-high dimensions. We further show that this performance carries over to classical Bayesian optimization without explicit evaluation costs. Our work constitutes a first step towards integrating techniques from Gittins index theory into Bayesian optimization.

In many applications of online decision making, the environment is non-stationary and it is therefore crucial to use bandit algorithms that handle changes. Most existing approaches are designed to protect against non-smooth changes, constrained only by total variation or Lipschitzness over time, where they guarantee $T^{2/3}$ regret. However, in practice environments are often changing smoothly, so such algorithms may incur higher-than-necessary regret in these settings and do not leverage information on the rate of change. In this paper, we study a non-stationary two-arm bandit problem where we assume an arm’s mean reward is a $\beta$-Hölder function over (normalized) time, meaning it is $(\beta-1)$-times Lipschitz-continuously differentiable. We show the first separation between the smooth and non-smooth regimes by presenting a policy with $T^{3/5}$ regret for $\beta=2$. We complement this result by a $T^{\frac{\beta+1}{2\beta+1}}$ lower bound for any integer $\beta\ge 1$, which matches our upper bound for $\beta=2$.