Smooth Non-Stationary Bandits

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 {\it smoothly}, so such algorithms may incur higher-than-necessary regret in these settings and do not leverage information on the {\it 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 $\beta$-H\"older function over (normalized) time, meaning it is $(\beta-1)$-times Lipschitz-continuously differentiable. We show the first {\it 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 $\beta=2$.