Smooth Non-Stationary Bandits
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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 β-Hölder function over (normalized) time, meaning it is (β-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 β=2. We complement this result by a T^β+1/2β+1 lower bound for any integer β≥ 1, which matches our upper bound for β=2.
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