这篇论文分析了连续随机消耗下在线资源分配的遗憾理论，给出了具体的下界和上界，还拿两个常见分布举例说明p值怎么算，挺实用。
论文研究了一个在线资源分配模型，其中奖励和消耗大小均连续分布，请求顺序到达且必须不可逆地接受或拒绝。该模型允许确定性流体松弛退化，并引入active weighted-mass exponent p来刻画遗憾。当p>1时，所有在线策略的遗憾下界为Ω(T^(1/2 - 1/(2p)))；当p=1时，边际策略达到O((log T)^2)遗憾。例如，size与value-to-size比率独立均匀分布时p=1，size与reward独立均匀时p=2。该策略在无流体非退化假设下实现了o(√T)遗憾。
Online Resource Allocation with Continuous Random Consumption: Regret under Degeneracy
We study online resource allocation when both rewards and consumption sizes may be continuously distributed. Requests arrive sequentially and must be accepted or rejected irrevocably under fixed resource capacities. Each request belongs to one of finitely many observable types; conditional on an observable request type, both the reward and the scalar size are random, and the realized size scales a fixed type-specific resource-consumption vector. The model allows the deterministic fluid relaxation to be degenerate. We show that additive regret is governed by the size-weighted mass of requests whose value-to-size ratios lie near the active acceptance cutoffs. We formalize this quantity through an active weighted-mass exponent p. When p > 1, this cutoff mass is thin, and the problem is genuinely hard: every online policy must incur regret of order at least $T^{1/2 - 1/(2p)}$, and this holds for every p > 1. A sample-path marginal policy matches this lower bound up to polylogarithmic factors; and when p = 1, so that the mass grows linearly near the cutoff, it attains $O((\log T)^2)$ regret. For example, if the size and the value-to-size ratio are independent and uniformly distributed, then p = 1; if instead the size and the reward are independent and uniformly distributed, then p = 2. Thus the policy achieves $o(\sqrt{T})$ regret throughout this regularity class without any fluid non-degeneracy assumption, allowing both primal degeneracy and dual non-uniqueness.