Title: Fair Continuous Resource Allocation with Equality of Impact
Venue: NeurIPS 2025
Accepted: yes
Keywords: resource allocation

fairness

machine learning

fair machine learning

equality of impact

group fairness

fair resource allocation

fairness regret

constant regret

online learning

marginal returns

constraint learning

censored feedback

online convex optimization
Abstract: Recent works have studied fair resource allocation in social settings, where fairness is judged by the impact of allocation decisions rather than more traditional minimum or maximum thresholds on the allocations themselves. Our work significantly adds to this literature by developing continuous resource allocation strategies that adhere to *equality of impact*, a generalization of equality of opportunity. We derive methods to maximize total welfare across groups subject to minimal violation of equality of impact, in settings where the outcomes of allocations are unknown but have a diminishing marginal effect. While focused on a two-group setting, our study addresses a broader class of welfare dynamics than explored in prior work. Our contributions are threefold. First, we introduce *Equality of Impact (EoI)*, a fairness criterion defined via group-level impact functions. Second, we design an online algorithm for non-noisy settings that leverages the problem’s geometric structure and achieves constant cumulative fairness regret. Third, we extend this approach to noisy environments with a meta-algorithm and empirically demonstrate that our methods find fair allocations and perform competitively relative to representative baselines.


Peer reviews:
comments: SUMMARY:
This paper studies the following problem. There is a learning agent that receives q units of a continuous resource in every round, for T rounds. The agent distributes the resource across two groups, A and B, according to an allocation rule. Each group has a reward function and an impact function (the latter quantifies societal benefit/harm); both functions are unknown. Fairness is defined as “equality of impact,” that is, intuitively, the difference in impact between the allocation of A and B, is bounded. The goal of the agent is to maximize her utility, which is the sum of rewards.

The authors give a characterization of the optimal allocation (the interest is “no-regret” here); under assumptions (nice behavior of reward/impact + diminishing marginal returns) the authors give an algorithm with regret O(1/T) in the noise-free setting (exact values are observed after allocation). In the noisy setting, the authors give a similar algorithm, but do not formally prove guarantees.

STRENGTHS AND WEAKNESSES:
The paper studies a natural problem. The writing is, for the most part, clear.
On the other hand, the overall contribution feels incremental. Specifically, it is not clear to me what a nice take-home message of this work is, either from a technical view (are there any new techniques or algorithmic ideas of interest?) or from a conceptual view.

QUESTIONS:
N/A

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-20

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes the study of fair resource allocation in an online setting when fairness is judged by the impact of allocation decisions instead of focusing on individuals. This is termed as Equality of Impact, and shown as a generalization of Allocative Equality of Opportunity. The paper derives properties of this problem setting in the presence of concave, continuous and non-decreasing utility and impact functions. The authors derive properties in the two-group case, defining Equality of Impact (EoI) in a manner similar to $\\epsilon$-demographic parity. They then cast the optimization problem as a search for the most utilitarian allocation among the set of all allocations satisfying EoI, and present an algorithm to learn this allocation online during repeated allocations with minimal prior knowledge. This is studied for both the noise-free case and the noisy case, and algorithms are provided for both, showing a sub-linear regret property in the noise-free case. The paper ends with a brief empirical analysis using three environments, providing evidence of the regret bound and comparing to two baselines for the cumulative regret.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper explores a new kind of fairness in resource allocation with 2 groups, considering the impact of allocations instead of just the benefit. This is important and has real-world significance.\n2. The theoretical properties presented characterize the problem studied in detail, which is useful for future researchers.\n\n\nWeaknesses:\n1. The paper is not well-written and is poorly organized. It is unclear when going through the paper why certain sections are included, and this makes it hard for readers to appreciate the intent. E.g. in section 2.1, there is no comment or exposition for why the two examples are needed. What are the authors trying to highlight? \n2. There is improper or incomplete notation. For example, EoI should be a function of G (Eq.2), and the optimization objective (4) should also be stated in terms of G. It is not clear what threshold is being optimized for otherwise. Similarly, $C^i$ is not defined (line 98). \n3. The paper seems to consider resources as continuous, but in some places, the exposition and examples seem to hint otherwise. E.g. in the proof for Lemma 3.1, the authors say 'all resources are allocated to group B', which suggests resources are countable and discrete, and the authors use funding (money) as an example of resources to be allocated. This may be a minor nitpick, but in this context it can matter. The authors use the Intermediate Value Theorem (IVT) to prove Lemma 3.1, and rely on the functions to be continuous, but in cases where there is discretization (like money), IVT is not valid. While it does not invalidate the paper's results, I feel it is worth pointing out.\n4. The property of sublinear regret seems trivial for any learning algorithm to achieve. Even a grid-search based method will achieve sublinear regret once it lands on the optimal point, given enough time. It would be helpful to contextualize how strong the proposed algorithm's sublinearity is when compared to naive baselines like a line search.\n5. Adding to the above point, the used baselines are very old. Why have the authors not used any other recent methods to solve the problem? They mention links to multi-armed bandits and RL, and these can be applied to solve the given problem.\n\nQUESTIONS:\nQ1. How does the proposed algorithm compare to a naive baseline like line search? It seems unreasonable that the baselines have increasing cumulative regret over long horizons. They are also not compared in the non-noisy setting.\n\nQ2. Why did the authors not consider RL or MAB-based techniques?\n\nQ3. What would it take to extend the approach to more than 2 groups?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nOn an abstract level, the authors consider an online constrained optimization problem. There are two groups where each group has a reward function and an impact function. Initially these functions are unknown. Then in each round, the agent selects an action (resource allocation) for both groups subject to a capacity constraint. The objective is to maximize the sum of the rewards. There is an additional “equality of impact” constraint: requiring that for each round, the difference in impact of the two groups is no larger than a constant G. To evaluate the effectiveness of an algorithm, the authors consider both the regret in reward compared to the optimal feasible allocation, and the fairness regret, which is the total violation in the equality constraint.\n\nUnder concavity assumptions, the authors show that the optimal action for each period should use all resources, turning the original two-dimensional decision into one-dimensional. Using this property and the concavity assumptions, this paper constructs confidence bounds for the reward and impact functions, using which it gives an algorithm with constant fairness regret. It then discusses heuristics to extend the algorithm to a noisy setting and offers several numerical results.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\n1.\tThe model in this paper generalizes previously studied fair allocation problems such as allocative equality of opportunity and water allocation.\n\n2.\tThe way to construct confidence bounds is pretty intuitive and the algorithm appears to be efficient in shrinking the set of optimal solutions.\n\n3.\tNumerical results support the benefit of the proposed algorithm.\n\nWeakness:\n\n1.\tThe studied model seems to be a special case of online convex optimization with long-term constraints. It is typical that one can achieve square-root T regret guarantee and constant fairness regret (constraint violation); see for example, https://www.jmlr.org/papers/v21/16-494.html. The authors should better compare their results with this literature and be clear why this work presents new results. \n\n2.\tMoreover, the model is rather restrictive in the sense that only two groups are considered. It is not explained why the setting of two groups is sufficiently interesting and captures real-world applications (such as the listed application of water allocation).\n\n3.\tIn addition, although the paper bounds the fairness regret in Theorem 5.2, there is no guarantee on the regret. There is also no guarantee for the noisy setting.\n\nQUESTIONS:\n1.\tWhy the existing approaches from online convex optimization with constraints do not work for your setting? \n\n2.\tTo what extent the proposed algorithm may extend to multiple groups?\n\n3.\tWhat guarantee for the regret on reward does the proposed algorithm have? Is it still a constant independent of the time horizon like the fairness regret result?\n\n4.\tWould Assumption 6.1 help give regret bound for the noisy setting?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses fair continuous resource allocation in settings where fairness should be measured by comparing the impact of allocations on groups rather than by comparing the impact of allocations on individuals between groups. This is relevant because some important resources, such as clean water or air, are not allocated directly to individuals but are nevertheless inequitably available to different groups.  The authors introduce Equality of Impact, a generalization of Equality of Opportunity, as a measure of the fairness property they introduce and develop algorithms for both noise-free and noisy settings that achieve sublinear regret in optimizing for Equality of Impact.\n\nSTRENGTHS AND WEAKNESSES:\nSTRENGTHS: The paper introduces a fairness characterization and metric, which is in itself an original contribution. The paper motivates why this metric is needed in a compelling way and provides a rigorous characterization.  The main theoretical result for the non-noisy setting algorithm (Algorithm 1), the demonstration that the algorithm provides a sublinear regret rate and therefore approaches optimality in the long term, is clearly described.  \n\nWEAKNESSES: The scope of applicability and scalability to more complex scenarios is not yet explored, leaving it somewhat unclear how useful the metric could be in a real-world scenario. The fact that the experiments appear to use synthetic data (or are so briefly described that it is hard to tell what kind of data they use) does not help with the projection to potential use cases. I think both of these concerns can be addressed, however.\n\nQUESTIONS:\nSCALE: \n(a) It would be helpful to say more about generalization to the >2 group setting. Many real world settings have more than two relevant groups and have potentially complex group structures, such as overlapping groups or subgroups formed by intersectional identities.  While the paper need not address all these cases, it would be helpful to have pointers to how the results are expected to generalize. \n\n(b) I also wonder how the exhaustive search method scales to higher dimensional problems or more complex fairness complaints. Mentioning this aspect of scale might also be helpful in determining the real-world utility of the techniques. \n\nEXPERIMENTS: While the experiments are described primarily in the appendix, they provide important support for the claims of the paper. If I am understanding correctly that both experiments are on synthetic data, I am curious about this choice and would like to understand why neither used empirical real-world data. In my view, using real data would strengthen the evidentiary value of the experimental results.   \n\nFAIRNESS TRADEOFFs: It would be great to have more detail about the tradeoff between fairness and welfare maximization. In particular, how sensitive is the welfare maximization performance to the choice of the fairness metric?  Do some fairness metrics force greater welfare losses than others, or does the severity of the tradeoff depend on the underlying structure of the data?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-13"}