Online Resource Allocation with Continuous Random Consumption: Regret under Degeneracy

2026-07-02

AI Summary

This study addresses an online resource allocation problem where requests arrive sequentially, the total resource budget is fixed, and each request type yields a reward and incurs a scalar consumption drawn from continuous random variables; the actual resource consumption is scaled by a type-specific vector, allowing for degenerate fluid relaxations. The authors introduce an active index \( p \), defined via the weighted quality around the truncation point of the value-to-consumption ratio, to characterize instance difficulty without requiring non-degeneracy of the fluid limit. They propose a sample-path marginal policy that integrates continuous distribution modeling with fluid analysis to handle cases of non-unique dual solutions and primal degeneracy. Theoretically, they establish a regret lower bound of \( T^{1/2 - 1/(2p)} \) for any policy when \( p > 1 \), while their proposed policy achieves \( O((\log T)^2) \) regret when \( p = 1 \) and attains \( o(\sqrt{T}) \) performance under typical uniform distributions.

Abstract

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.

Problem

• online resource allocation
• continuous random consumption
• regret analysis
• degeneracy
• value-to-size ratio

Innovation

• online resource allocation
• continuous random consumption
• regret analysis
• degenerate fluid relaxation
• sample-path marginal policy