Title: Fairness-Regularized Online Optimization with Switching Costs
Venue: NeurIPS 2025
Accepted: yes
Keywords: Online optimization

long-term cost

health-informed decision making
Abstract: Fairness and action smoothness are two crucial considerations in many online optimization problems, but they have yet to be addressed simultaneously. In this paper, we study a new and challenging setting of fairness-regularized smoothed online convex optimization with switching costs. First, to highlight the fundamental challenges introduced by the long-term fairness regularizer evaluated based on the entire sequence of actions, we prove that even without switching costs, no online algorithms can possibly achieve a sublinear regret or finite competitive ratio compared to the offline optimal algorithm as the problem episode length $T$ increases. Then, we propose **FairOBD** (Fairness-regularized Online Balanced Descent), which reconciles the tension between minimizing the hitting cost, switching cost, and fairness cost. Concretely, **FairOBD** decomposes the long-term fairness cost into a sequence of online costs by introducing an auxiliary variable and then leverages the auxiliary variable to regularize the online actions for fair outcomes. Based on a new approach to account for switching costs, we prove that **FairOBD** offers a worst-case asymptotic competitive ratio against a novel benchmark---the optimal offline algorithm with parameterized constraints---by considering $T\to\infty$. Finally, we run trace-driven experiments of dynamic computing resource provisioning for socially responsible AI inference to empirically evaluate **FairOBD**, showing that **FairOBD** can effectively reduce the total fairness-regularized cost and better promote fair outcomes compared to existing baseline solutions.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper studies an online optimization problem, where an algorithm must make an irrevocable decision in every round\nwithout knowing future costs. The authors study an objective that is composed of three parts: the cost of the chosen action\nin each round (hitting cost), the cost of switching the action from the previous round (switching cost), and a cost that measure\nthe fairness of the solution. Importantly, the fairness regularization can only be evaluated at the end of the instance.\nThis smooth fairness regularized setting thus generalizes previously studied smoothed online optimization without fairness and fairness-regularized online optimization without switching cost (smoothness). \n\nThe authors present an online algorithm and prove regret and competitive ratio bounds, which recover the results of the previous settings.\nA crucial component of the author's approach is to decompose the fairness cost into online costs via a new regularizer and auxiliary variables, and to maintain approximate fairness through a Lagrangian‐multiplier mirror‐descent update, while balancing hitting cost, switching cost, and fairness cost in each round.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- Formulation and motivation of the problem\n- Asymptotically tight guarantees; good model of resource augmentation (restricted optimum)\n- The algorith is neat; it uses mirror descent with an additional auxiliary variable. \n- Empirical evaluation that confirms the theoretical findings.\n\nWeaknesses:\n- Requires strong assumptions on fairness cost: convex and Lipschitz.\n- The result is somewhat incremental (given that different subsets of the problem have been studied before)\n\nQUESTIONS:\n- What can be done if the fairness cost is non-convex and Lipschitz?   \n- How should the different hyperparameters (e.g. \\lambda) be chosen for a good performance in practice?\n\nLIMITATIONS:\nAll limitations have been addressed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-24"}

comments: SUMMARY:
The paper works on a smoothed online optimization problem with a long-term horizon fairness regularizer. In the problem, the agent obtains a cost function at each time slot and a matrix. Then the agent will select an action, incurring three costs: hitting cost by the cost function, switching cost induced from changing to a different action from the last time slot, and horizon fairness cost. The problem aims to minimize the total cost. In this paper, the authors first analyze the worst-case regret and the competitive ratio of the problem, using the offline setting as a benchmark. Then the authors provide an algorithm and analyze its performance.

STRENGTHS AND WEAKNESSES:
Strength:

The paper studies an interesting and challenging problem, which is well-motivated by several real world applications.

The paper is well organized.
 
Weakness

The lower bound and upper bound does not match. That is, the lower bound is the comparison of the optimal online algorithm and the optimal offline algorithm, while the upper bound is the comparison of the proposed algorithm and so-called (R,\delta)-optimal algorithm. These are two different stories.

The technique is standard.

QUESTIONS:
In Theorem 5.1 and 5.2, the (R,\delta)-optimal algorithm is online or offline? Or they are the same? Please clarify.

LIMITATIONS:
NA
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this work, the authors model an online learning setting. A decision-maker is tasked with selecting an $N$-dimensional decision at each step. They observe a complete cost function that maps this chosen decision to a non-negative cost. Furthermore, an $M\\times N$ weighting matrix transforms the decision into an $M$-dimensional utility vector, upon which a long-term fairness objective is imposed, yielding a scalar fairness value. The decision-maker also incurs a switching cost, which captures the expense associated with changing decisions over time. The core objective is to devise a policy that guarantees a worst-case asymptotic competitive ratio compared to the offline optimum as the time horizon tends towards infinity.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths.**\n\nThe paper is well-written, and its results are clearly presented. The authors' contributions are distinct, technically sound, and easily understood. They begin by establishing significant negative results in Theorems 4.1 and 4.2, demonstrating the impossibility of deriving meaningful guarantees for both regret and competitive ratio against an unrestricted adversary, like the one typically assumed in Online Convex Optimization (OCO). The authors then address this by relaxing their objective, introducing the concept of an $(R,\\delta)$-optimal algorithm, and deriving guarantees against this new benchmark. Their main result is presented as a competitive ratio guarantee for the proposed algorithm.\n\n**Weaknesses.**\n\n* The *significance of the results and the problem statement's motivation remain unclear*. It is not evident how an additive combination of a fairness criterion with a cost objective is meaningful. Fairness goals, such as min-max or proportional fairness (as exemplified by the authors), typically incorporate notions of efficiency, like Pareto optimality. Therefore, the system designer's secondary objective is ambiguous, especially given that efficiency is already implicitly accounted for in the fairness metric's definition. For instance, providing zero allocation to involved parties, while potentially fair, may not represent an efficient solution.\n\n* The *algorithmic complexity of Algorithm 1 is not transparent*. The cost function is defined only as continuous and non-negative over the action set, providing no inherent structure. Consequently, there is no clear justification that the crucial step in Equation (5) can be solved in polynomial time, rendering the algorithm impractical in real-world applications. If solving Equation (5) requires approximation for efficiency, this could potentially compromise the theoretical guarantees proposed by the authors.\n\n* The authors claim to use *mirror descent* as an integral part of their algorithm's design. However, the specific advantages or efficiencies that mirror descent, particularly the choice of its associated mirror map (regularizer), offers are unclear, as no apparent link to the obtained results is provided. It is not evident why standard gradient descent would not suffice.\n\nQUESTIONS:\n* Given that fairness objectives often inherently account for efficiency (e.g., Pareto optimality), could you clarify the system designer's intended secondary goal when combining these objectives additively?\n* The algorithmic complexity of Algorithm 1 remains unclear. Since the cost function has no specified structure beyond being continuous and non-negative, could you justify how the crucial step in Equation (5) can be solved in polynomial time? If approximations are necessary for practical efficiency, how might this impact the theoretical guarantees presented in the paper?\n* You state that mirror descent is integral to the algorithm's design. Could you please elaborate on the specific advantages or efficiency gains it provides over simpler optimization methods like standard gradient descent?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces FairOBD, an online algorithm designed to address the challenge of incorporating both long-term fairness and smoothness (via switching costs) in online convex optimization (OCO). The work formalizes a new problem setting that jointly penalizes hitting costs, action changes, and fairness violations over time. The approach decomposes the long-term fairness regularizer using auxiliary variables and dynamically updates dual multipliers through mirror descent. Theoretical analysis demonstrates an asymptotic competitive ratio guarantee against a constrained offline oracle, while empirical experiments on AI inference workload balancing across real data center traces show meaningful improvements in both fairness and overall cost compared to established baselines.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. The paper tackles an important gap in the literature by studying online optimization with both switching (smoothness) costs and long-term fairness, providing a model relevant for a variety of impactful applications, such as energy-efficient and socially equitable resource allocation.\n\n2. The introduction of FairOBD represents a meaningful algorithmic contribution. It reformulates long-term fairness regularizer as long-term constraint, which is then addressed through the use of auxiliary variables and dual updates via mirror descent.  The theoretical analysis is also carefully done, clearly establishing the competitive ratio and explicitly accounting for the complexities introduced by switching costs and fairness costs. \n\n3. The paper validates FairOBD on a realistic setting: trace-driven experiments for AI inference load balancing across several Microsoft data centers using real-world traces. The results demonstrate that FairOBD achieves improved fairness and overall cost compared to five baselines, including the offline optimum and established algorithms.\n\nWeaknesses:\n\n\n1. The transformation from long-term fairness regularizer to long-term fairness constraints in equation (3) is interesting. However, the subsequent treatment largely follows standard techniques, combining a conventional primal-dual framework with Online Balanced Descent [10,16].\n\n2. The model assumes a convexity assumption on fairness cost and squared l2-norm switching costs. While justified by the need for theoretical tractability, this restricts applicability to a lot of problems. Although the limitations section briefly mentions that relaxing the convexity assumption or generalizing the form of switching costs is left for future work, a more concrete discussion of how the algorithm might be adapted under such cases would significantly strengthen the contribution.\n\n3. Some relevant works [Li et al., 2025, Chen et al., 2024, Ma et al., 20204] are missing and should be cited.\n\n[Li et al., 2025] Learning for Sustainable Online Scheduling with Competitive Fairness Guarantees.\n[Chen et al., 2024] Network Revenue Management with Demand Learning and Fair Resource-Consumption Balancing.\n[Ma et al., 20204] Optimal Regularized Online Allocation by Adaptive Re-Solving.\n\nQUESTIONS:\nHave you considered your problem in the stochastic setting where { (f_t, A_t) }_t are i.i.d.? Would it be possible to achieve sublinear regret in this case?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-04"}