A Unifying Perspective on Multi-Calibration: Game Dynamics for Multi-Objective Learning

Summary

This paper presents a novel approach to multicalibration by leveraging game dynamics and no-regret learning algorithms. The central idea is that multicalibration can be modeled as a multi-objective learning problem where an adversary and a learner play against each other, guided by no-regret dynamics. The authors propose three types of dynamics: no-regret no-regret (NRNR), no-regret best-response (NRBR), and best-response no-regret (BRNR), each serving different purposes. These dynamics are used to establish the multicalibration algorithms that match or improve the fastest known sample complexity rates for multicalibration, provide deterministic multicalibrated predictors, and offer online multicalibration. This work also extends the application of these dynamics to other fairness notions, specifically multi-group learning and multi-distribution learning. For the multi-group learning problem, the authors devise an optimal multi-group learning algorithm that relies on NRNR dynamics and is simpler than the existing approaches. The authors validate their theoretical claims with empirical results on a few standard datasetst. The results highlight the importance of effective no-regret algorithms for better multicalibration, with the Optimistic Hedge outperforming others in the no-regret no-regret dynamics. Overall, this paper contributes to the field by providing a unified, game-theoretic framework for multicalibration that unites disparate existing results and offers more efficient solutions, both in theory and practice. Moreover, it highlights the broader applicability of game dynamics and no-regret algorithms to other fairness notions in machine learning.

Strengths

The paper introduces a novel perspective by approaching multicalibration and other fairness notions using game dynamics and no-regret learning algorithms. It presents multicalibration as a multi-objective learning problem within a game-theoretic framework. However, while the application to multicalibration is new, the use of no-regret algorithms and game-theoretic models is well-established in other areas of calibration literature. The paper offers a technically sound approach with rigorous mathematical derivations. The theorems are well-proved, and the proposed algorithms are clearly detailed. However, the empirical section could be expanded upon to strengthen the validation of the theoretical claims with real-world data. The authors have done a reasonable job in articulating complex concepts and methodologies, with the paper being generally well-structured. The use of tables aids understanding, though more insights can be provided for the technical results. The work has the potential to unify and extend prior work on multicalibration and other fairness notions. However, the impact of the work may depend heavily on how effectively these theoretical insights can be translated into practical applications. Furthermore, while the findings from the empirical results could guide the selection of no-regret algorithms in multicalibration tasks, the experimental validation is relatively limited and does not fully exploit the range of situations where these algorithms could be applied.

Weaknesses

The paper provides a theoretically strong development of multicalibration algorithms using game dynamics. However, there seems to be a gap between the theoretical development and the empirical results. The authors could improve this aspect by designing more comprehensive experiments that test a wide range of scenarios to validate their theoretical claims. These could include different types of datasets, varying levels of complexity, and possibly real-world use cases. The experimental evaluation is currently limited to only a few standard but simple dataset. This may not sufficiently test the robustness of the proposed algorithms under different conditions. More experiments with diverse datasets would provide a better understanding of the algorithms' performance and potential limitations. Additionally, it would be beneficial to compare the proposed methods with more baseline or state-of-the-art algorithms for multicalibration to understand the relative performance. Also, the paper could be improved by providing more detailed descriptions of the proposed algorithms. While the authors do discuss the high-level ideas behind using game dynamics for multicalibration, it may be beneficial for readers to have more specific details about how these algorithms are implemented. Although the authors discuss several previous works, it is not entirely clear how their contributions improve upon or differ from these existing methods. Besides theoretical comparison, a clearer empirical comparison would make the authors' contributions more evident. The paper could benefit from a more explicit discussion of the limitations of the proposed methods.

Questions

The paper provides an improved theoretical framework for multicalibration using game dynamics. However, it seems there's a disconnect between the theory and empirical results presented. Could the authors elaborate on why specific experiments were chosen and how they directly validate the theoretical developments? The empirical evaluation mainly focuses on the standard datasets like UCI. How do the authors envision their proposed algorithms performing on other types of datasets, especially those with different characteristics, such as image datasets or high-dimensional problem? Could they provide any insights or theoretical guarantees on their algorithms' robustness across varied datasets? The high-level descriptions of the proposed algorithms are appreciated, but having more concrete, step-by-step details of the algorithm implementations would be very helpful. Could the authors provide pseudocode or a more in-depth explanation of their proposed algorithms? The paper could benefit from a more detailed comparison with previous works. How do the authors' contributions improve upon or differentiate from existing methods, specifically in terms of performance and applicability? The paper presents a novel framework that generalizes previous works. How far can this generalization go? Are there any theoretical or practical constraints that limit its applications? Are there any lower bounds? Any insight into this would be beneficial for the understanding and application of the framework.

Rating

4: Borderline reject: Technically solid paper where reasons to reject, e.g., limited evaluation, outweigh reasons to accept, e.g., good evaluation. Please use sparingly.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

2 fair

Presentation

3 good

Contribution

3 good

Limitations

The authors do not provide a clear discussion of the limitations of their work or potential negative societal impacts, which is a critical part of any comprehensive research paper. This is an area where the paper could be improved. Here are a few suggestions for potential limitations and societal impact considerations: Discuss the computational complexity and scalability of the proposed methods. This is crucial for real-world applications, particularly when dealing with large and complex datasets. Given that the paper deals with fairness notions in learning, the authors could consider discussing the implications of misapplication of these methods. While the goal is to improve fairness, any shortcomings or misuse could unintentionally lead to bias or unfair outcomes.

Summary

The paper provides a two-player dynamics framework that seeks to unify many strands of recent work on multicalibration and multiobjective optimization. With three possible setups considered: No regret against No regret, Best response against No regret, and Best response against Best response, efficient algorithms are provided to demonstrate that these generic setups apply in particular to (multi-)calibration-like constraints. The framework is then applied to several existing and new settings in the literature with the goal of improving some convergence bounds or, respectively, simplifying and streamlining the analysis of specific algorithms. Some experiments are given that show what happens when proposed dynamics are run with specific well-known no regret algorithms in the driver's seat.

Strengths

To me, the main strength of this paper is the rather clean nature of the setup, which promises that by combining, in any combination, best-responding and no-regret agents, it is possible (though not without further work) to recover existing multigroup fairness results. Previous guarantees are (with a couple exceptions) improved somewhat marginally, or greater simplicity of the framework is claimed in some other cases, but overall it is the generality of this framework --- showcased by its adaptability and ability to recover various insights about calibration (not just regret bounds, but also questions of the simplicity of the output calibrated model, amount of randomness required, etc) --- that is the paper's main forte. Additionally, the conditional multicalibration setup presented in the paper (and generalizing previous conditional results in the quantile setting) looks like an interesting addition to the literature.

Weaknesses

No particular weaknesses, other than the relatively well-studied nature of, and relatively small gains in, some of the applications (but no big deal), as well as the somewhat terse and cramped presentation. For instance, there are a few "intuitive-sounding" claims made about where NRNR, BRNR, BRBR may be applied to the greatest utility, but the actual applications are then scattered throughout the paper rather than recalled immediately (there is also Table 1, but I would appreciate a more intuitive dive into which dynamic was used where and why, at the point where these dynamics were actually introduced). For another example, I would be interested in further comments on potential alternative definitions of conditional multicalibration and a slightly expanded treatment of the comparison to the existing 1/sqrt(# people in group) type of guarantee. Still, the paper is overall solid and well-done and I enjoyed reading it.

Questions

See the above section.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

4: You are confident in your assessment, but not absolutely certain. It is unlikely, but not impossible, that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work.

Soundness

3 good

Presentation

2 fair

Contribution

3 good

Limitations

N/A

Summary

This work exploits connections to game dynamics to propose a unifying algorithmic framework to address the multicalibration problem which has been recently used for tackling fairness concerns in machine learning. More precisely, based on the classic game dynamics approach used in learning problems, it is shown that multicalibration results can be seen as learning dynamics for two-player zero-sum games relying on no-regret algorithms or best response dynamics. Using this unifying framework, the paper recovers some guarantees for existing multicalibration algorithms with simplified proofs and results. Multi-objective learning guarantees are shown and new guarantees are also established for several multicalibration settings including in particular an exponential reduction in the complexity of k-class multicalibration over prior work. Experiments were conducted to evaluate the empirical performance of multicalibration algorithms on some real-world datasets.

Strengths

- The paper proposes a unifying approach using game dynamics which recovers prior results with simpler proofs and establishes novel guarantees improving over prior work in multiple settings. - This paper is well-written, the presentation is very clear. - While I am not familiar with the calibration/multicalibration literature and I did not go through the long appendix supporting the main part in details, the results are sound, quite rigorously exposed and the proofs which leverage for instance prior online learning results seem solid to the best of my knowledge. - Several additional experimental results are also provided in the appendix to support the theoretical findings.

Weaknesses

Regarding novelty, the related work section mentions in l. 55-56 that ‘no work has established a broad connection between no-regret learning and multicalibration’. No-regret learning has been priorly used for calibrated forecasting as acknowledged by the paper. While it is mentioned that multicalibration ‘has very different challenges than calibrated forecasting’, the discussion regarding these is reduced to a single line l. 57-58 in the related work section. Given that one of the main contributions of this work is the unifying framework based on the connection with game dynamics, I would expect a more detailed discussion to further clarify the novelty with respect to prior work and calibrated forecasting regarding this particular aspect. For instance, some proofs such as the one of Theorem 3.8 follow similar lines to the proof of calibration as also mentioned in l. 246-247. Moreover, it seems that this connection to game dynamics was known for multi-objective learning and multicalibration is linked to it via the simple facts 2.5 and 2.6.

Questions

- The game dynamics approach to multi-objective learning is a ‘common approach’ as mentioned in l. 138. The paper connects multicalibration to multi-objective learning in Facts 2.5 and 2.6 which leads to a connection between multicalibration and the game dynamics approach. Are Facts 2.5 and 2.6 novel results? Is there a particular reason for using ‘Facts’ for these results instead of ‘Lemma’ for instance? Is this related to the simplicity of the proofs in Appendix B once the losses are defined as in the facts? - How do you justify the equality in l. 230-231? - Section 3.1 is about multicalibration but section 3 is about multi-objective learning. Section 4 is 'Multicalibration with Game Dynamics'. Would it be more appropriate to move section 3.1 to section 4? **Minor questions and suggestions:** - Why is the baseline of (2) not also used in (1) so that the formulations are unified? - The notations max and min are used for minimization over a hypothesis class or maximization over a class of loss functions and a set of data distributions. How are these guaranteed to be reached without further assumptions? Are the classes supposed to be finite or are the more general notations sup and inf more appropriate? For instance, cardinalities are used in Lemma 3.5 while the preliminaries do not seem to specify the nature of the classes. - Minor suggestion: in Definition 2.2., you could keep the same notation as in Definition 2.1 and only change the dependence on $t$ for $D$ and $p$ to ease the reading (with the same order for the quantities and a single indicator function). - l. 82: $\Delta(\mathcal{Y})^{\mathcal{X}}$ with parenthesis.

Rating

6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.

Confidence

3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

4 excellent

Contribution

3 good

Limitations

Limitations of the results do not seem to be clearly discussed throughout the paper to the best of my knowledge.

Summary

The authors proposed a unified framework for multicalibration learning by exploiting its connection to the game dynamics in multi-objective learning. Strong theoretical guarantees were given and its extension to address group fairness was discussed.

Strengths

1. The analysis of the game dynamics in multi-objective learning was novel and strong. 2. Connection between multicalibration learning and game dynamics in multi-objective learning was well exploited, unified framework was given. 2. The author gave a clear presentation of the key ideas of the work despite substantial material.

Weaknesses

1. The experiment parts seem to compare with algorithms within the proposed framework. Is there any comparison with existing baseline algorithms? 2. How does the multi-objective learning discussed in the paper related to the Pareto optimal one? 3. Given different learner choices shown in the empirical section, is there any learner who fits the proposed framework better?

Questions

see weakness.

Rating

7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.

Confidence

1: Your assessment is an educated guess. The submission is not in your area or the submission was difficult to understand. Math/other details were not carefully checked.

Soundness

4 excellent

Presentation

3 good

Contribution

3 good

Limitations

NA

Summary

This paper provides a unifying framework for the algorithm design and performance analysis of multicalibrated predictors. In this paper, the multicalibraion problems is placed in the setting of multi-objective learning. Under this interpretation, approaches based on game dynamics is proposed and analyzed. It is shown that this approach yields improved performance guarantees.

Strengths

- Bounds are improved for a variety of problems. - Experimental results support the effectiveness of the proposed method.

Weaknesses

- The approach based on game dynamics does not seem very surprising as it is a common approach to transform min-max optimization into a problem of finding an equilibrium solution by interpreting it as a zero-sum game. - I have concerns about whether the definition is consistent with that of existing studies. (please refer to "Questions") Minor comments: - When citing existing results in Table 1, etc., it would be better to indicate the theorem number or the relevant section. I had a hard time checking the corresponding part.

Questions

- In previous studies (e.g., [17,21,34]), multicalibrated predictors appear to be defined in terms of conditional expectation. On the other hand, in Definition 2.1 of this paper, it appears to be defined in some sense by joint probabilities. I believe this means that the errors bounded in 2.1 are smaller than those adopted as definitions in previous studies. This raises concerns about whether comparisons with existing research bounds are valid. I would appreciate an answer to this concern to see if there is a problem. - Is there lower bounds that can be compared to the results obtained?

Rating

5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.

Confidence

2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.

Soundness

3 good

Presentation

3 good

Contribution

3 good

Limitations

I have no concerns about the limitations and potential negative societal impact.

Acknowledgement

I thank the authors for their clarifications regarding related work and concerning the challenges raised by multicalibration compared to calibrated forecasting. Most of my questions have been answered. I maintain my positive score.

Acknowledgment

Thank you for the reply. With the expectation that the flow of the paper will be improved --- especially with regards to the uses of and intuition on various combinations of no regret (NR) and best responding (BR) agents --- I will keep my current score.