Title: Generalizing while preserving monotonicity in comparison-based preference learning models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Monotonicity

Preference Learning

Alignement

Bradley-Terry.
Abstract: If you tell a learning model that you prefer an alternative $a$ over another alternative $b$, then you probably expect the model to be *monotone*, that is, the valuation of $a$ increases, and that of $b$ decreases. Yet, perhaps surprisingly, many widely deployed comparison-based preference learning models, including large language models, fail to have this guarantee. Until now, the only comparison-based preference learning algorithms that were proved to be monotone are the Generalized Bradley-Terry models. Yet, these models are unable to generalize to uncompared data. In this paper, we advance the understanding of the set of models with generalization ability that are *monotone*. Namely, we propose a new class of Linear Generalized Bradley-Terry models with Diffusion Priors, and identify sufficient conditions on alternatives' embeddings that guarantee monotonicity. Our experiments show that this monotonicity is far from being a general guarantee, and that our new class of generalizing models improves accuracy, especially when the dataset is limited.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses to key problems in preference-learning models, monotonicity and score-generalization. \nIn particular, the authors tackle both by extending a well-established model (GBT) to include embeddings of the alternatives and a more general regularization term.\nThe authors then prove the conditions necessary to make their extended model monotonic and show how every diffusion embedding as defined in def5 satisfies these conditions.\nFinally, the authors perform a limited experiment to showcase their model against GBT.\n\n\n\n\n\n\n\n\n\n\nThe paper addresses the problem of monotonicity, or lack thereof, in preference learning models. \nIn particular, the authors propose a new class of preference learning algorithms, based on the Generalized Bradly-Terry model, which also considers embeddings of the alternatives. \nWith appropriate conditions on the abovementioned embeddings the authors show that they can guarantee monotonicity.\n\nSTRENGTHS AND WEAKNESSES:\n## S1 - Interesting model\nExtending GBT to allow for score-generalization is potentially impactful for the community, as it addresses a strong motivation.\nAlso, the fact that every diffusion embedding can be used within the model potentially makes it flexible in tackling different scenarios.\n\n## W1 - Notation\nThe mathematical notation is often inconsistent and confusing, making it quite hard to follow the text and, especially, the proofs.\nNotable issues include:\n1. The indiscriminate usage of the same letter $\\theta$ for both a matrix and a vector; both $\\theta_{ab}$ and $\\theta_a$ indicate a real number in different parts of the text.\n2. An inconsistent usage of typefacing. Matrices are sometimes indicated with lowercase letters ($x$), greek letters ($\\theta$), and uppercase letters ($L, H$). On the contrary, the same typeface is used for multiple different kinds of objects; examples are $A, D, N \\in \\mathbb N$ and $L$ being a matrix, as well as $\\beta$ being a matrix and $\\sigma$ being a real number. \n3. Undefined symbols, such as $e_{ab}$ in Lemma 1; I am guessing $e$ is the identity matrix of some dimension, but in Definition 3 the symbol $I$ was introduced for that.\n4. Inconsistencies in Lemma 1 and its proof, where both $\\frac{d\\theta^\\*(\\lambda)}{d\\lambda}$ (eq9) and $\\frac{d\\theta^*\\_\\lambda}{d\\lambda}(\\mathbf D, o)$ (eq30) are used to (potentially?) indicate the same object.\n\n## W2 - Experiments\nThe experimental evaluation is insufficient to showcase the benefits and potential drawbacks of the proposed model over other approaches.\nIn my opinion, both of the following issues with the experimental setup stem from W3. \n\n### W2.1 Competitors\nThe experiments omit comparisons with other potential competitors such as the ones described in L79--87.\n\n### W2.2 Data-generating process\nTo my understanding, the data-generating process described in Section 5.2 aligns perfectly with the proposed model's assumptions, making it a very unfair comparison ground for standard GBT. \n\n## W2.3 - Error bars\nThe authors report in the checklist that they are purposely not reporting error bars in their figures, as they are considering an arguably large number of repetitions (100). I believe error bars (preferably, standard deviation) should instead be included, as it helps discerning how significant the differences between the models' performances are. \nIdeally, some statistical analysis of the results would also help.\n\n## W3 - Assumptions of the model\nAs described in Section 3, the model assumes access to a known embedding matrix $x$, which is a strong and unrealistic assumption in most practical scenarios. \nThe sensitivity of the method to this assumption is evident in Figure 2a, where not choosing the “correct” embedding lead to significant performance degradation. The \"correct\" embedding is the one aligning with the data-generating process.\nWithout a mechanism to learn or robustly adapt this embedding, the practical utility of the method remains limited.\nI also wonder, if learning the embedding is possible, whether that would make the optimization problem intractable or very hard to optimizer, given the increased number of variables in the model.\n\nQUESTIONS:\nMajor questions:\nSee the weaknesses above.\n\nClarifications:\n- L117--120: There might be a contradiction here. If EXCHANGE replaces $(a_n, b_n, r_n)$ with its opposite $(a_n, b_n, -r_n)$, I don't see why preference algorithms should interpret the two comparison samples identically. At the limit, one can exchange all of the comparison samples with their opposites, most certainly affecting training.\n- L140: The authors should provide proof that $\\preceq_a$ is indeed a preorder; in particular, I am skeptical about reflexivity, as I cannot see a way to define an operation (infinitely) favors $a$ while leaving a dataset $\\mathbf D$ unchanged.\n- L163: In what sense is proposition 1 a generalization of [9]?\n- L274: How does $H$ depend on $L$? The sentence build like the two are indeed related, but I cannot see how.\n- L284: Shouldn't $x$ be $(L + \\tilde H)$-good?\n- L467: Based on eq7, shouldn't it be $\\mathcal L_\\lambda(\\beta|\\mathbf D, o) - \\mathcal L(\\beta|\\mathbf D) = \\lambda \\cdot (\\Phi_f(\\theta_{ab}) - r\\theta_{ab})$?\n\nSpelling and notation: \n- L56: \"monotony\" should be \"monotonicity; this may have happened in other parts of the manuscript.\n- L107, eq3: $\\mathcal R$ is used with different meanings.\n- L116: consider reserving $\\mathcal O$ for its usual meaning in asymptotic analysis.\n- L130, eq6, eq7, L262: the notation for UPDATE is inconsistent; this might be the case for the other operations as well. \n- L139 (and others): Given the definition of $\\Omega$, it's equivalent (and formally correct) to replace $\\geq$ with $\\in$. \n- L149: \",.\" at the end.\n- eq3: $\\theta_b(\\beta)$ is not defined; is it $\\theta(\\beta)_b$? The first symbol is not defined.\n- eq4: $x_{ab}^T\\beta$ is confusing for two reasons: (1) if it means $(x^T)\\_{ab}$  why not use $x_{ba}$? (2) $\\beta$ was defined on L181 as a vector and $x^T_{ab}$ looks like a scalar, meaning that $x_{ab}^T\\beta \\in \\mathbb  R^D$, which is in conflict with the rest of the terms in the equation. \n- L255, 226: proposes, provides.\n- L250: $o_\\lambda$ is not defined.\n- eq9: What is $e$? L305 defines $I$ as the identity matrix. \n- The figures are not saved in a vector format, making the labels hard to read when the paper is printed out.\n\nLIMITATIONS:\nThe authors discuss some limitations of their model, especially regarding the extension to non-diffusion embeddings.\nWhat is missing here, in my opinion, is a discussion on the drawbacks stemming from trying to use the model in practice (W3).", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors consider the problem of generalizable preference learning from pairwise comparisons. They focus on a desirable monotonicity property, which holds for a preference learning algorithm if observing data that favors an item $a$ only causes the learned value of $a$ to increase. Surprisingly, Bradley-Terry with linear utilities violates monotonicity. However, the authors show that if item embeddings satisfy certain properties, then a generalized Bradley-Terry model (GBT) does satisfy monotonicity. In simulation experiments, the authors examine the probability that random Gaussian embeddings satisfy their \"goodness\" property (unfortunately, this is small when there are few embedding dimensions relative to items). In experiments on synthetic data, they demonstrate how generalization across items is achieved through the model's Laplacian regularization and item embeddings.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The contribution is clear and significant\n2. The writing and notation are good\n3. The problem is timely and interesting\n\nWeaknesses:\n1. A few minor points that could be made clearer (see questions)\n\nQuality: Good quality, well situated in the literature, good mix of theory and experiments, well-structured. \n\nClarity: The clarity is high overall. A few points could be better explained (see questions) \n\nSignificance: Important and relevant to preference learning.\n\nOriginality: The model itself uses standard techniques (utility = linear function of features, Laplacian regularization), but the properties of embeddings required for monotonicity and the associated analysis are original to my knowledge. I'm not sure I'd call GBT with diffusion priors a \"new model\"; e.g., see (1) https://doi.org/10.1017/nws.2023.20 and (2) https://arxiv.org/abs/1703.07520 for other work applying item features and Laplacian regularization to learning preference models. The only difference between GBT with diffusion priors and the Laplacian-regularized conditional logit model in (1) is the use of the comparison score $r$ rather than a fixed choice of preferred item (and the focus here on pairwise comparisons). I don't think this is an issue; the major contribution of this work is the analysis of monotonicity. I would just tone down the claimed novelty of the model.\n\nQUESTIONS:\n1. If I'm understanding the definition of monotonicity correctly, any dataset operation that favors $a$ should increase the learned score score $\\theta_a$ of $a$. Is this sufficient to cover the \"buggy\" example in the introduction? Suppose adding a comparison where $a$ beats $b$ does increase the score of $a$, but it also increases the score of $c$ by even more, causing $a$ to now perform worse against $c$. (I know this is a weird learning algorithm, but perhaps $c$'s embedding indicates it has all the good features of $a$ and none of the bad features of $b$.) How is this avoided by monotonicity? \n\n2. Is there some intuition for what embeddings are diffusion embeddings? Are the embeddings in the introduction example not diffusion embeddings, and is this possible to tell without computing inverses?\n\n3. What parts of the model are required for monotonicity? Can we get rid of the $L_2$ regularization? The Laplacian regularization?\n\nMinor:\n- line 56: \"monotony\"\n- Line 68: another relevant choice citation is McFadden's multinomial/conditional logit\n- Line 149: has an extra comma\n- Line 162: this should be referencing Proposition 1, right?\n- $\\mathcal R$ is used both for the set of comparison scores and the regularization term\n- Smoothed rather than smoothened?\n\nLIMITATIONS:\nNice concise limitations section. I don't see this work having negative societal impacts.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a Generalised Linear Bradley-Terry model with a specific set of embeddings they name - diffusion embeddings. The paper proves that these embeddings result in monotonic behaviour during preference learning.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n- The setting studied by the authors is novel to the best of my knowledge.\n- The paper lays out rigorous theoretical proofs of its claims and supports these with experimental results.\n- The work is mostly clear and easy to follow.\n\n**Weaknesses**\n\nI recommend a weak reject at the moment because I do not believe the motivation for the work aligns with the method proposed.\n\n- The authors motivate the work through DPO, GPO and other RLHF works that focus on alignment within the LLM literature, however I am uncertain if the results presented in this paper align with that motivation. The authors do not present any experiments on an LLM setting and the class of problems considered in this work is not easily extended to the class of LLM models, as to the best of my knowledge the setting is akin to tabular with an embedding for each alternative A. I may well have misunderstood this aspect of the work though - thus my low confidence score.\n- The final paragraph of the introduction states that this work finds direct applications within the collaborative scoring of social media content - I am unfamiliar with this setting but would be keen to understand more as the work may present better when motivated in this context.\n- The experiments lack baselines against other preference optimization work - particularly other works from the collaborative content social media content literature would enrich the work and provide further comparisons to evaluate the method against.\n- Some clarity issues in the paper - in line 185 the regularisation $\\mathcal{R}$ overrides the notation for the comparison value defined in line 107.\n\nQUESTIONS:\nSee Weaknesses above\n\nLIMITATIONS:\nThe authors highlight the constraint that the proposed method is dependent upon diffusion embeddings.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the overlooked issue of monotonicity in preference learning, where models should assign higher value to preferred alternatives. While existing models like LLMs often violate this, the paper introduces a novel class of Linear Generalized Bradley-Terry models with Diffusion Priors, which ensure monotonicity under specific embedding conditions and can generalize to unseen comparisons. Experiments confirm improved accuracy, especially in low-data regimes.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. This paper focuses on a neglected but important data-driven issue in LLM fine-tuning, that is, the generalization of reward evaluations.\n2. Theoretical contributions are valid and identify conditions under which monotonicity can be guaranteed in linear comparison-based models with diffusion priors.\n\nWeakness:\n1. The experiments are not organized at an industrial level. The numerical evidence is on a miniature scale, insufficient for technical audiences.\n\nQUESTIONS:\nQ1. To provide more convincing real-world numerical evidence, can the experiments be designed and operated at an industrial level, e.g., using UltraFeedback or Anthropic-HH datasets?\n\nQ2. Generalisation of BT model has been discussed in various formats and contexts.\n\nIn many real-world datasets, binary reward models, such as the Bradley-Terry (BT) model, have been known to be vulnerable to 'intransitivity' risk because they rely on scalar variables, assuming all preferences are transitive. \n- The literature below studies representative preference datasets in the real world, where the 'transitive' relationship between preference annotations may not always hold, and presents a generalized version of the BT model that considers the intransitivity of human preferences.\n- https://arxiv.org/abs/2409.19325 (Duan et al, 2017) : Quantitative investigation into the importance of generalization and a generalized model that focuses on explainable representation learning.\n- 10.1109/ICPR48806.2021.9412462 (Gu et al, 2021): A low-rank matrix approach for the generalization.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-06"}