Reconciling Model Multiplicity for Downstream Decision Making

We would like to thank the reviewer for their time and insightful comments. Please find our responses below.

Regarding related work: We have added the following paragraph to our additional related work section in Appendix A:

Within the literature on multigroup fairness, Kleinberg et al. (2016) highlighted the trade-offs in fairness by proving the incompatibility of three key fairness conditions—calibration, positive class balance, and negative class balance—emphasizing the need for prioritization in fairness criteria. Building on these foundational insights, Kim et al. (2019) introduced multiaccuracy. This post-processing technique ensures consistent accuracy across subpopulations without access to group labels, laying the baseline for future work on predictive fairness. Following such framework, Rothblum & Yona (2021) extended the PAC learning paradigm to multi-group scenarios. Shen et al. (2023) proposed a framework for fair representations across multiple fairness notions and unknown tasks. More recently, Hu et al. (2024) explored robustness in multigroup fairness and proposed algorithms tailored for subpopulations under data corruption. Liu & Molinari (2024) proposed a method to estimate the fairness-accuracy frontier for testing fairness in decision-making. Another line of work (Haghtalab et al., 2023) connects fairness and optimization with a unified game theoretic framework. However, our work does not focus on the fairness aspect of multicalibration but instead borrows its techniques.

Regarding the hyperparameters in Algorithms 1 and 2: The loss margin $\alpha$ and decision-calibration tolerance $\beta$ are the targets that we are optimizing for in Algorithm 2 and Algorithm 1, respectively. For a larger value of $\beta$ (Definition 2.5), a predictor $f$ is further away from the true label on the set of best-response event $\mathcal{E}$ compared to a smaller value of $\beta$. That is, a smaller value of $\beta$ requires Algorithm 1 to run for more iterations before it can converge, as seen in the first result of Theorem 3.4.

Similarly, the loss margin $\alpha$ is how much we can tolerate the difference in decision-making losses when the two predictors induce different best-response actions in the downstream task. A higher value of $\alpha$ leads to a shorter convergence time for Algorithm 2 (first result) and a worse guarantee of the improvement of the Brier score (second result) in Theorems 3.2 and 3.4.

Regarding the third result in Theorem 3.2: The theorem establishes that the decision loss will increase only minimally, if at all. In the experimental results shown in Figures 2 and 3, the decision loss strictly decreases when running ReDCal, which is ideal and confirms the theory.

Regarding our empirical findings: In Figures 4 and 7 (Appendix F), we show that our proposed algorithms can reduce the Brier score of the input models (second result in Theorems 3.3 and 3.5). In Figures 2 and 3, we show that our proposed algorithm can decrease the decision loss (third result) by more than the baseline Reconcile algorithm and the Decision Calibration algorithm (when our algorithm is used as post-processing). Also, the final predictor1 and predictor2 have similar loss values, which corresponds to the fourth result. Both the Imagenet dataset and the HAM10000 dataset are real-world datasets.

Regarding the computational complexity: with a larger dataset and large disagreement region for the base predictors, our algorithms still converge in the same number of time steps as it does not depend on the number of samples. The additional error in Theorem 3.4 compared to Theorem 3.2 goes to zero as we have more data. At each time step, Algorithms 1 and 2 compute a single patching operation (line 5 of Algorithm 1, line 6 of Algorithm 2) to update the current predictor. With a larger disagreement event, this patching operation will update on more individual samples, which should scale with the size of the disagreement event.

We would like to thank the reviewer for their time and insightful comments. Please find our responses below.

Regarding the experiment: The loss function in Zhao et al (2021) assigns a loss value to each of the 7 possible skin diseases and actions (treatment or no treatment). The loss function is described in their Figure 1. Blue indicates low loss, and red indicates high loss. Since the authors did not provide an exact value for this loss function, we map the color to the closest value (as shown in the table below) and add Gaussian noise to emulate their setup.

In Figure 3, the Reconcile algorithm only seeks to reconcile individual probability predictions and does not take into account the downstream decision-making loss. For both predictors f_1 and f_2, we observe that the decision loss gap increases as we run this algorithm for longer. On the other hand, our algorithm reduces the loss gap for both predictors. This indicates that we should not disregard decision loss, aligning with our argument in Theorem 2.7.

Reconciling more than 2 models: We thank the reviewer for their comment on the write-up in Section 3.3. We have rephrased the text.

We would like to thank the reviewer for their time and insightful comments. Please find our responses below:

Regarding the differences between the two sets of experimental results: The HAM10000 dataset only contains 10000 samples in total. We split the dataset into train/validation/test sets, with 20% of the data used for validation and 20% of the data used for testing. In comparison, the Imagenet dataset has 40000 samples used for calibration and 10000 samples for testing. Moreover, the downstream decision-making loss function for the HAM10000 experiment is motivated by medical domain knowledge in Zhao et al. (2021) with added Gaussian noise, while the loss function for the Imagenet experiment is randomly generated. The exact details of these differences can be found in the experiment setup in Section 4.1 and Appendix F.2.

Regarding comparison to other related work in model multiplicity: We focus on comparing our result with that of Roth et al. 2023 due to the similarity in our approaches to resolving model multiplicity. Moreover, we also compare our empirical results with the baseline algorithm of Decision-Calibration (Algorithm 1).

Regarding the robustness of our proposed algorithm: In Section 3.2, we provide a set of results when the decision-makers only have access to a finite dataset instead of the underlying probability distribution. To the best of our knowledge, we are not aware of any work in multicalibration that explicitly discusses robustness toward noise in the samples.

We would like to thank the reviewer for their time and insightful comments. Please find our responses below.

Regarding Theorem 3.4: We apologize for the typo in item 1, we have fixed it in our revision. For the questions:

1. The trade-off between prediction accuracy and decision loss: In the setting where the distribution $\mathcal{D}$ is given to us (Remark 3.3), we can set $\beta = \alpha / (T \sqrt{d} A)$ to ensure the decision-loss does not increase by more than $\alpha$. There are no trade-offs in this case. In a more practical setting where a dataset $D$ is given, as shown in the experiment results in Fig. 2 and Fig. 3, the decision loss consistently decreases with each iteration of ReDCal. This implies that for larger $T_i$​, both prediction accuracy and decision loss generally improve, reducing the need to balance these two factors.
2. Decision gap and numerical results: The third item in Theorem 3.4 indicates that the decision loss of the final predictors $f_i^T$ is no larger than the decision loss of the initial predictors $f_i^0$. This does not imply that the decision loss of $f_i^T$ is necessarily close to that of $f_i^0$; rather, a decrease in decision loss is the desired outcome. The numerical results confirm this behavior, showing a reduction in decision loss that aligns with our theoretical findings.

Regarding the two models at the end of ReDCal: The decision-maker can choose either of the final models after running ReDCal. Even when the decision-loss gap is controlled, it is important to ensure that the two output models agree on the best-response actions almost everywhere. In our motivating example, a doctor may feel more confident providing treatment to a patient when both of the final models suggest that treatment is the best-response action, compared to a scenario where each model induces a different best-response action.

Regarding the decision loss: If the goal is to avoid any changes to the decision loss, the "agreement" between the two predictors becomes irrelevant. In this case, we can set the disagreement region mass $\eta$ to a large value, such as 1. This choice of $\eta$ causes ReDCal to terminate immediately without updating the predictors.

However, in our model multiplicity setting, the objective is to minimize the size of the disagreement region. A smaller disagreement region is desirable because, during a disagreement event, at least one predictor either fails to provide the correct best-response action or yields an inaccurate expected loss. By reducing the disagreement region, we enhance the reliability of both predictors and the decisions they inform.

Does this clarification address your concern about changes in the decision loss?

Regarding the convergence rate in experiments: The upper bounds in Theorem 3.2 and 3.4 are worst-case upper bounds. Hence, it is reasonable to have the algorithm converge at a faster rate in numerical experiments.

Regarding the presentation of the paper: We thank the reviewer for their suggestion. Below we clarify each of the points and highlight the changes we made to our file.

1. Background on calibration: We have added a sentence explaining calibration in the introduction.
2. $A$ and $\mu$: For $A$, we have defined in line 210 that it is the number of actions and added it in the theorem statement. We also added the definition of $\mu$ right before section 3.1.
3. Distribution in Theorem 2.7: We have already provided the distribution in the first sentence of the proof (around line 284) by saying that $Pr[x=1]=Pr[x=2]=0.5$.
4. $\ell_a$: $\ell_a$ is introduced in the first paragraph of Section 2.2 (around line 212). We assume that the decision loss is linear in the label $y$. This assumption implies that there exists a vector $\ell_a$​, associated with action $a$, such that the loss can be expressed as $\ell(y, a) = \langle y, \ell_a \rangle$. Note that, in the case of classification (i.e. $y$ is one-hot vector indicating the class), $\ell_a$ can be explicitly defined as $\ell_a = (\ell(1, a), \ell(2,a), \ldots, \ell(d, a)]$.
5. Decision-Calibration and Decision-Calibration+ReDCal: In the second paragraph of Section 4.1, we explained that Decision-Calibration corresponds to Algorithm 1, while Decision-Calibration+ReDCal refers to the process of applying ReDCal as a post-processing step after running Decision-Calibration. Including the baseline of Algorithm 1 highlights that using the subroutine alone is suboptimal. By running ReDCal on top of it, we demonstrate that the predictors can be further improved, showcasing the added value of the combined approach.
6. In Theorem 3.2, we do not count the number of time steps for Decision-Calibration. In Theorem 3.4, we do count the number of time steps for Decision-Calibration.
7. Fixed typo in Definition 2.4, Theorem 3.4 item 1, and $h$ in Algorithm 3. Added a brief description of Reconcile in the paragraph before Theorem 2.7