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Online Resource Allocation with Replenishable Budgets
Eleonora Fidelia Chiefari, Francesco Emanuele Stradi, Alberto Marchesi
TL;DR
Classical online resource allocation assumes budgets only decrease, leaving replenishable resources outside its standard model. The paper develops a dual-based algorithm for non-monotonic budgets and reports sublinear regret guarantees with strict budget satisfaction. Active replenishment tightens the bounds and improves the adversarial competitive ratio.
Problem
Classical ORA assumes actions deplete resources, excluding settings where actions can replenish budgets and complicating dual updates when costs are negative.
Source: §1 · p. 3, §1.2 · p. 4Method
A dual-based framework handles non-monotonic resource use, with dual updates paused during replenishment to avoid artificially decreasing multipliers.
Source: §3 · p. 8, §5 · p. 11Results
The algorithm achieves sublinear expected regret in stochastic environments and sublinear fractional regret in adversarial ones while strictly satisfying budget constraints.
Source: §1 · p. 3, §5 · p. 11Takeaways & Limitations
An active replenishment margin tightens regret bounds and improves the adversarial competitive ratio without requiring prior knowledge of β.
Source: §1.1 · p. 3, §5 · p. 11The model assumes one void action that uniformly replenishes every resource by at least β, rather than resource-specific replenishing actions.
Source: §5 · p. 11
Abstract
from arXiv · showhide
Online Resource Allocation (ORA) is a fundamental framework for sequential decision-making problems under budget constraints. Classical ORA models typically assume that resources are monotonic, meaning that selecting actions can only decrease the available budget. In this work, we study a more general setting with replenishable budgets, in which actions may either consume or replenish resources over time. This extension is necessary to capture scenarios such as inventory systems or energy markets in which capacity can be actively recovered. We develop a dual-based algorithm that recovers best-of-both-worlds guarantees for standard ORA when the replenishment factor $\beta = 0$, and improves them when $\beta > 0$. In particular, our algorithm attains $\widetilde{\mathcal O}(\sqrt{T})$ regret in the stochastic setting and $\widetilde{\mathcal O}(\sqrt{T})$ $\alpha$-regret in the adversarial setting, where $\alpha$ depends on the per-round budget and on the replenishment factor of the void action. Moreover, the algorithm ensures strict satisfaction of the budget constraints.
1 Introduction
Online resource allocation assumes actions deplete budgets, but replenishable resources make allocation non-monotonic and complicate dual updates. The paper develops a dual-based framework with stochastic and adversarial guarantees, distinguishes its full-information setting from prior bandit work, and exploits replenishment to improve bounds.
1 Introduction
- Classical ORA assumes monotonic budget depletion, excluding settings such as inventory or cloud computing where actions can restore capacity.ORA reveals each request’s reward and resource use for every possible action before the decision, leaving uncertainty about future requests as the main challenge.Source: §1 · p. 3
1.1 Original Contribution
- The framework recovers state-of-the-art guarantees when β = 0, improves bounds when β > 0, and strictly satisfies hard budgets in both regimes.Its bounds depend on the budget and replenishment margins; the algorithm does not require prior knowledge of β.Source: §1.1 · pp. 3–4, Table 1 · p. 4
1.2 Technical Challenges
- Negative costs can decrease dual multipliers, allowing an action that restores one resource to mask severe depletion of another and risk premature exhaustion.Standard primal-dual updates can therefore encourage over-consumption when replenishment breaks monotonicity.Source: §1 · p. 3, §1.2 · p. 4
- The proposed dual-based framework prevents artificial deflation of Lagrangian multipliers to handle non-monotonic consumption and unaligned resource constraints.This addresses cross-constraint compensation, in which replenishing one resource can mask depletion of another.Source: §1.2 · p. 4
1.3 Related works
- Prior stochastic ORA methods include two-phase approaches and periodically re-solved linear programs, while later gradient-based schemes avoid repeated LP solvers but require specific assumptions or structures.The related-work discussion also identifies stochastic, non-stationary, and adversarial lines of research.Source: §1 · p. 3, §1.3 · p. 5
- Earlier best-of-both-worlds ORA work addresses stochastic and adversarial regimes, while replenishable-budget methods cited here remain confined to bandit feedback.This work instead studies active replenishment in the full-information ORA setting.Source: §1 · p. 3, §1.3 · p. 5
2 Preliminaries
ORA with replenishable budgets lets actions consume or restore resources, while decisions are made with full information about each round’s possible actions. The paper formulates the budget constraints through a Lagrangian and evaluates performance separately under stochastic and adversarial inputs.
2 Preliminaries
- Actions may consume or replenish resources, with negative consumption representing replenishment; each round’s reward and resource functions are fully revealed before the decision.The horizon is T rounds, and the void action guarantees at least β replenishment per resource while yielding zero reward.Source: §2 · p. 6
2.1 Lagrangian Formulation
- The instantaneous Lagrangian incorporates resource constraints into the objective, with dual multipliers penalizing consumption above the per-round budget.Replenishing actions can produce negative resource terms in the Lagrangian.Source: §2.1 · p. 6
2.2 Stochastic Setting
- In the stochastic setting, inputs are i.i.d. from a fixed unknown distribution, and performance is measured against an expected constrained optimum.The benchmark averages the optimal reward over input realizations.Source: §2.2 · p. 7
2.3 Adversarial Setting
- In the adversarial setting, inputs may be chosen arbitrarily each round, so performance is measured by α-regret against a fraction of the optimum.The passage motivates α-regret by the impossibility of simultaneously guaranteeing sublinear regret and constraint satisfaction under fully adversarial inputs.Source: §2.3 · p. 7
3 Algorithm
Replenishing Dual Gradient Descent chooses actions by maximizing an instantaneous Lagrangian when budgets are safe, and switches to the void action when a resource falls below the safety threshold. It pauses dual updates during forced replenishments and maintains nonnegative budgets.
3 Algorithm
- When any available resource budget falls below 1, the algorithm plays the void action; otherwise, it greedily maximizes the instantaneous Lagrangian.A forced round replenishes capacity, while a safe active round updates the overall budget based on the chosen action.Source: §3 · pp. 7–8
- During forced replenishment rounds, dual variables remain unchanged, preventing artificial decreases that could lead to over-consumption after recovery.The algorithm pauses the dual updates as a deliberate design choice.Source: §3 · p. 8
- The budget remains non-negative at every round because active rounds start with at least 1 unit per resource and consume at most 1, while forced rounds only replenish.The active-round consumption bound follows from the resource limits stated in the algorithm discussion.Source: §3 · pp. 7–8
- After each active decision, projected OGD adjusts dual multipliers to penalize above-budget consumption or encourage future use when resources are underutilized.The update is projected onto the positive orthant.Source: §3 · p. 8
4 Theoretical results
The theoretical analysis bounds the dual variables despite non-monotonic resource dynamics, then establishes regret guarantees in stochastic and adversarial settings. Replenishment increases the effective margin used in the multiplier bound, and the algorithm’s guarantees do not require prior knowledge of β.
4.1 Bounding the Lagrangian Multipliers
- The dual variables remain bounded in both settings, with the effective margin ρ + β arising from guaranteed replenishment and tightening the multiplier bound.Lemma 4.1 states the bound under η ≤ 1/(2m); the analysis controls the squared Euclidean norm.Source: §4.1 · pp. 8–9
4.2 Stochastic Setting
- The stochastic and adversarial analyses establish cumulative regret guarantees for Algorithm 1, with the stochastic guarantee described as optimal for stochastic ORA.The stochastic proof bounds the benchmark using weak duality and an expected Lagrangian, then controls performance over active rounds.Source: §4.2 · pp. 9–10
- The algorithm preserves sublinearity without knowing β by tuning its learning rate using the known per-round budget ρ, although constants may worsen.The stochastic-setting remark says an accurate β estimate can tighten the bound.Source: §4.2 · p. 10
4.3 Adversarial Setting
- The adversarial analysis establishes an α-regret bound by controlling dual penalties on active rounds and accounting for forced replenishment rounds separately.The algorithm plays the void action during forced rounds, whose regret is bounded by the number of such rounds.Source: §4.3 · pp. 10–11
5 Discussion and Open Problems
The paper introduces replenishable budgets and a dual-based algorithm for non-monotonic resource use. It reports sublinear regret guarantees in stochastic and adversarial settings, while identifying assumptions that future work could relax.
5 Discussion and Open Problems
- The dual-based framework handles non-monotonic resource use and achieves sublinear expected regret in stochastic settings and sublinear fractional regret in adversarial settings, with deterministic budget satisfaction.The paper also reports that the regret bounds tighten when active replenishment is available.Source: §5 · p. 11
- The current model assumes the void action replenishes every resource by at least β for every input tuple.The authors identify replenishment only in conditional expectation as a possible relaxation and raise whether the algorithm could estimate β.Source: §5 · p. 11
- The current model uses one void action that uniformly replenishes all m resources by at least β, rather than resource-specific actions.The authors note that multiple independent replenishing actions with different margins would better represent some inventory settings.Source: §5 · p. 11
A Additional Related Works
The related work distinguishes replenishable ORA from general long-term-constraint models and situates it alongside work on constraints that can receive negative per-round contributions. It also traces related results on fixed, time-varying, and budget constraints.
A Additional Related Works
- Unlike replenishable ORA, general long-term-constraint models have soft constraints and no single action guaranteed to offset violations across all constraints.In replenishable ORA, a known void action contributes at most −β for every input tuple.Source: A Additional Related Works · p. 14
- Prior work established simultaneous sublinear regret and constraint-violation guarantees first for fixed constraints, then for time-varying constraints.The passage names Mahdavi et al. (2012) and Jenatton et al. (2015) for fixed constraints, and Mannor et al. (2009), Yu et al. (2017), and Sun et al. (2022) for time-varying constraints.Source: A Additional Related Works · p. 14
- Bandits with Knapsacks was introduced for budget constraints, with later work extending results to stochastic, adversarial, and non-stationary environments.The passage identifies generalizations beyond budgets to arbitrary long-term constraints as especially relevant.Source: A Additional Related Works · p. 14
B.1 Omitted Proofs and Lemmas of Section 4
The omitted proofs analyze the dual updates and regret by separating active rounds from forced replenishments. They establish a dual-variable bound and use learning-rate choices in the stochastic and adversarial analyses.
B.1 Omitted Proofs and Lemmas of Section 4
- The proof separates active decision rounds from forced replenishment rounds, when the algorithm is forced to play the void action.The appendix uses this partition to bound cumulative dual penalties and regret.Source: §B.1 · pp. 14–15
- In both stochastic and adversarial settings, Algorithm 1 with 0 < η ≤ 1/2m guarantees that the ℓ1-norm of the dual variables is bounded for all t ∈ [T+1].Source: §B.1 · p. 17
- The stochastic proof bounds regret using the expected Lagrangian and tunes the learning rate according to whether β is known.The proof first bounds the hindsight stochastic optimum and then combines that bound with the dual analysis.Source: §B.1 · pp. 18–19