[Paper Review] Online Learning Demands in Max-min Fairness

Kirthevasan Kandasamy, Gur-Eyal Sela+3 more|arXiv (Cornell University)|Dec 15, 2020

Advanced Bandit Algorithms Research62 references4 citations

TL;DR

This paper proposes an online learning mechanism for max-min fair resource allocation in dynamic environments where users do not know their true resource demands. By leveraging feedback from past rounds, the mechanism learns user demands while ensuring efficiency, fairness, and strategy-proofness, achieving sublinear regret and asymptotic fairness under stochastic and nonparametric feedback models.

ABSTRACT

We describe mechanisms for the allocation of a scarce resource among multiple users in a way that is efficient, fair, and strategy-proof, but when users do not know their resource requirements. The mechanism is repeated for multiple rounds and a user's requirements can change on each round. At the end of each round, users provide feedback about the allocation they received, enabling the mechanism to learn user preferences over time. Such situations are common in the shared usage of a compute cluster among many users in an organisation, where all teams may not precisely know the amount of resources needed to execute their jobs. By understating their requirements, users will receive less than they need and consequently not achieve their goals. By overstating them, they may siphon away precious resources that could be useful to others in the organisation. We formalise this task of online learning in fair division via notions of efficiency, fairness, and strategy-proofness applicable to this setting, and study this problem under three types of feedback: when the users' observations are deterministic, when they are stochastic and follow a parametric model, and when they are stochastic and nonparametric. We derive mechanisms inspired by the classical max-min fairness procedure that achieve these requisites, and quantify the extent to which they are achieved via asymptotic rates. We corroborate these insights with an experimental evaluation on synthetic problems and a web-serving task.

Motivation & Objective

• To address the challenge of fair resource allocation when users cannot accurately specify their resource demands in shared computing environments.
• To design a mechanism that learns user demands over time through feedback while maintaining efficiency, fairness, and strategy-proofness.
• To formalize online learning in fair division under three feedback models: deterministic, parametric stochastic, and nonparametric stochastic.
• To extend classical max-min fairness to dynamic, learning-based settings where user demands are initially unknown.
• To quantify the performance of the proposed mechanism via asymptotic regret and fairness bounds.

Proposed method

• The mechanism uses a hierarchical partitioning of the allocation space to estimate user demands, with feedback collected after each round to refine estimates.
• It applies a variant of max-min fairness that dynamically adjusts allocations based on learned demand estimates, ensuring no agent is worse off than their fair share.
• The method employs Lipschitz continuity assumptions on utility functions and uses concentration inequalities to bound estimation errors across feedback types.
• For stochastic feedback, the mechanism uses confidence bounds and exploration-exploitation trade-offs to minimize regret, with bounds derived via martingale concentration and covering arguments.
• It decomposes regret into components based on under-allocation, misclassification, and estimation error, each bounded using recursive inequalities and tail probability bounds.
• Theoretical analysis derives asymptotic regret bounds of order $ O(T^{1/2}/G^{1/2}) $, where $ G $ controls granularity and $ T $ is the number of rounds.

Experimental results

Research questions

• RQ1 Can a fair and efficient resource allocation mechanism be designed when users do not know their true resource demands?
• RQ2 How can feedback from past allocations be used to learn user demands in a way that preserves fairness and strategy-proofness?
• RQ3 What are the asymptotic performance bounds (regret and fairness) of such a learning-based fair allocation mechanism?
• RQ4 How do different feedback models—deterministic, parametric stochastic, and nonparametric stochastic—affect the convergence and performance of the mechanism?
• RQ5 Can the mechanism maintain fairness while learning user demands, even when users attempt to manipulate the system by misreporting?

Key findings

• The proposed mechanism achieves sublinear regret in the number of rounds $ T $, with an asymptotic regret bound of $ O(T^{1/2}/G^{1/2}) $, where $ G $ controls the resolution of the allocation grid.
• The mechanism ensures asymptotic fairness, with the difference between the optimal and actual utility converging to zero as $ T $ increases, under the given feedback models.
• For parametric and nonparametric stochastic feedback, the mechanism achieves fairness and efficiency by balancing exploration and exploitation using confidence bounds and hierarchical partitioning.
• Theoretical analysis shows that the loss due to misallocation is bounded by $ O(L^2/G) $, where $ L $ is the utility Lipschitz constant and $ G $ is a granularity parameter.
• Experimental evaluation on synthetic and web-serving tasks confirms the mechanism's robustness and convergence to fair allocations under varying feedback conditions.
• The mechanism maintains strategy-proofness by ensuring that truthful reporting of load (not demand) minimizes regret and maximizes utility, discouraging manipulation.