We revisit the central online problem of ski rental in the "algorithms with predictions" framework from the point of view of distributional predictions. Ski rental was one of the first problems to be studied with predictions, where a natural prediction is simply the number of ski days. But it is both more natural and potentially more powerful to think of a prediction as a distribution p-hat over the ski days. If the true number of ski days is drawn from some true (but unknown) distribution p, then we show as our main result that there is an algorithm with expected cost at most OPT + O(min(max({eta}, 1) * sqrt(b), b log b)), where OPT is the expected cost of the optimal policy for the true distribution p, b is the cost of buying, and {eta} is the Earth Mover's (Wasserstein-1) distance between p and p-hat. Note that when {eta} < o(sqrt(b)) this gives additive loss less than b (the trivial bound), and when {eta} is arbitrarily large (corresponding to an extremely inaccurate prediction) we still do not pay more than O(b log b) additive loss. An implication of these bounds is that our algorithm has consistency O(sqrt(b)) (additive loss when the prediction error is 0) and robustness O(b log b) (additive loss when the prediction error is arbitrarily large). Moreover, we do not need to assume that we know (or have any bound on) the prediction error {eta}, in contrast with previous work in robust optimization which assumes that we know this error. We complement this upper bound with a variety of lower bounds showing that it is essentially tight: not only can the consistency/robustness tradeoff not be improved, but our particular loss function cannot be meaningfully improved.

翻译：本文从分布预测的角度重新审视"带预测算法"框架下的核心在线问题——滑雪租赁问题。滑雪租赁是最早被研究带预测的问题之一，其自然预测形式是滑雪天数。然而，将预测视为滑雪天数上的分布 p-hat 更为自然且可能更具表现力。如果真实滑雪天数服从某个真实（但未知）的分布 p，我们的主要结果表明：存在一种算法，其期望成本至多为 OPT + O(min(max({η}, 1) * sqrt(b), b log b))，其中 OPT 是针对真实分布 p 的最优策略期望成本，b 为购买成本，{η} 是 p 与 p-hat 之间的推土机（Wasserstein-1）距离。值得注意的是，当 {η} < o(sqrt(b)) 时，该算法的附加损失小于 b（平凡上界）；当 {η} 任意大（对应极不准确的预测）时，附加损失仍不超过 O(b log b)。这些界限的推论是：我们的算法具有 O(sqrt(b)) 的一致性（预测误差为 0 时的附加损失）和 O(b log b) 的鲁棒性（预测误差任意大时的附加损失）。与以往需要已知预测误差 {η}（或其界限）的鲁棒优化研究不同，我们的算法无需此假设。我们通过多种下界证明该上界本质上是紧的：不仅一致性/鲁棒性权衡无法改进，我们特定的损失函数也无法获得实质性优化。