SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Then, we can show that, for an appropriate choice of  $\mathcal{C}$ , if  $f_{B}$  satisfies  $\alpha$ -multicalibration with respect to  $\mathcal{C}$ , then it satisfies  $\alpha$ -aligned calibration with respect to  $f_{H}$ . More specifically, we have the following result:
Theorem 8. If  $f_{B}$  satisfies  $(\alpha /2)$ -multicalibration with respect to  $\{S_h\}_{h\in \mathcal{H}}$ , with  $S_{h} = \{(x,H)\in \mathcal{Z}\mid H = h\}$ , then  $f_{B}$  satisfies  $\alpha$ -aligned calibration with respect to  $f_{H}$ .
The above theorem suggests that, given a classifier's confidence function  $f_{B}$ , we can multicalibrate  $f_{B}$  with respect to  $\{S_{h}\}_{h\in \mathcal{H}}$  to achieve  $\alpha$ -aligned calibration with respect to  $f_{H}$ . To achieve multicalibration guarantees using finite amounts of (calibration) data, multicalibration algorithms need to discretize the range of  $f_{B}$  [9, 11, 12]. In what follows, we briefly revisit two algorithms, which carry out this discretization differently, and discuss their complexity and data requirements with respect to achieving  $\alpha$ -aligned calibration.
Multicalibration algorithm via  $\lambda$ -discretization. This algorithm, which was introduced by Hébert-Johnson et al. [11], discretizes the range of  $f_{B}$ , i.e., the interval  $[0, 1]$ , into bins of fixed size  $\lambda > 0$  with values  $\Lambda[0, 1] = \{\frac{\lambda}{2}, \frac{3\lambda}{2}, \dots, 1 - \frac{\lambda}{2}\}$ .
Let  $\lambda(b) = [b - \lambda/2, b + \lambda/2)$ . The algorithm partitions each subspace  $S_h$  into  $1/\lambda$  groups  $S_{h,\lambda(b)} = \{(x,h) \in S_h \mid f_B(x,h) \in \lambda(b)\}$ , with  $b \in \Lambda[0,1]$ . It iteratively updates the confidence values of function  $f_B$  for these groups until  $f_B$  satisfies a discretized notion of  $\alpha'$ -multicalibration over these groups. The algorithm then returns a discretized confidence function  $f_{B,\lambda}(x,h) = \mathbb{E}[f_B(X,H) \mid f_B(X,H) \in \lambda(b)]$ , with  $b \in \Lambda[0,1]$  such that  $f_B(x,h) \in \lambda(b)$ , which is guaranteed to satisfy  $(\alpha' + \lambda)$ -multicalibration. Refer to Algorithm 1 in Appendix B for a pseudocode of the algorithm.
Then, as a direct consequence of Theorem 8, we can obtain a (discretized) confidence function  $f_{B,\lambda}$  that satisfies  $\alpha$ -aligned calibration by setting  $\alpha' = \lambda = \alpha / 4$ . However, the following proposition shows that, to satisfy just  $\alpha$ -alignment, it is enough to set  $\alpha' = \frac{3}{8}\alpha > \alpha / 4$  and  $\lambda = \alpha / 4$ :
Proposition 1. The discretized confidence function  $f_{B,\lambda}$  returned by Algorithm 1 satisfies  $(2\alpha' + \lambda)$ -alignment with respect to  $f_{H}$ .
Finally, it is worth noting that, to implement Algorithm 1, we need to compute empirical estimates of the expectations and probabilities above using a calibration set  $\mathcal{D}$ . In this context, Theorem 2 in Hébert-Johnson et al. [11] shows that, if we use a calibration set of size  $O(\log (|\mathcal{H}| / (\alpha \gamma \xi)) / \alpha^{11 / 2}\gamma^{3 / 2})$ , with  $P((X,H)\in S_h) > \gamma$  for all  $h\in \mathcal{H}$ , then  $f_{B,\lambda}$  is guaranteed to satisfy  $\alpha$ -multicalibration with probability at least  $1 - \xi$  in time  $O(|\mathcal{H}|\cdot \mathrm{poly}(1 / \alpha ,1 / \gamma))$ .
Multicalibration algorithm via uniform mass binning. Uniform mass binning (UMD) [9, 12] has been originally designed to calibrate  $f_{B}$  with respect to  $\mathcal{Z}$  using a calibration set  $\mathcal{D}$ . However, since the subspaces  $\{S_h\}_{h \in \mathcal{H}}$  are disjoint, i.e.,  $S_{h} \cap S_{h^{\prime}} = \emptyset$  for every  $h \neq h^{\prime}$ , we can multicalibrate  $f_{B}$  with respect to  $\{S_h\}_{h \in \mathcal{H}}$  by just running  $|\mathcal{H}|$  instances of UMD, each using the subset of samples  $\mathcal{D} \cap S_{h}$ . Here, we would like to emphasize that we can use UMD to achieve multicalibration because, in our setting, the subspaces  $\{S_h\}_{h \in \mathcal{H}}$  are disjoint.
Each instance of UMD discretizes the range of  $f_{B}$ , i.e., the interval [0, 1], into  $N = 1 / \lambda$  bins with values  $\Lambda_h[0,1] = \{\hat{P}(Y = 1 \mid f_B(X,h) \in [0,\hat{q}_1]), \dots, \hat{P}(Y = 1 \mid f_B(X,h) \in [\hat{q}_{N-1},\hat{q}_N])\}$ , where  $\hat{q}_i$  denotes the  $(i/N)$ -th empirical quantile of the confidence values  $f_{B}(x,h)$  of the samples  $(x,h) \in \mathcal{D} \cap \mathcal{S}_h$  and  $\hat{P}$  denotes an empirical estimate of the probability using samples from  $\mathcal{D} \cap \mathcal{S}_h$ , as well. Here, note that, by construction, the bins have similar probability mass. Then, for each  $(x,h) \in \mathcal{Z}$ , the corresponding instance of UMD provides the value of the discretized confidence function  $f_{B,\lambda}(x,h) = b$ , where  $b \in \Lambda_h[0,1]$  denotes the value of the bin whose corresponding defining interval includes  $f_{B}(x,h)$ . Finally, we have the following theorem, which guarantees that, as
(a) Art
(b) Sarcasm
(c) Cities
(d) Census
Figure 2: Empirical estimate of the probabilities  $P(Y = 1 \mid (X,Y) \in S_{h,\lambda(b)})$ , where  $b \in \Lambda[0,1]$  and  $h \in \{\text{low}, \text{mid}, \text{high}\}$  are the discretized confidence values for the classifiers and human participants, respectively. Error bars represent  $90\%$  confidence intervals and hatched bars mark alignment violations between confidence pairs  $(h,b)$  with  $|S_{h,\lambda(b)}| \geq 30$ .
long as the calibration set is large enough, the discretized confidence function  $f_{B,\lambda}$  satisfies  $\alpha$ -aligned calibration with respect to  $f_{H}$  with high probability:
Theorem 9. The discretized confidence function  $f_{B,\lambda}$  returned by  $|\mathcal{H}|$  instances of UMD, one per  $S_h$ , satisfies  $\alpha$ -aligned calibration with respect to  $f_H$  with probability at least  $1 - \xi$  as long as the size of the calibration set  $|\mathcal{D}| = O\left(\frac{|\mathcal{H}|}{\alpha^2\lambda\gamma}\log \left(\frac{|\mathcal{H}|}{\lambda\xi}\right)\right)$ , with  $P((X,H)\in S_h)\geq \gamma$ .
In this section, we validate our theoretical results using a dataset with real expert predictions in an AI-assisted decision making scenario comprising four different binary classification tasks<sup>7</sup>.
Data description. We experiment with the publicly available Human-AI Interactions dataset [35]. The dataset comprises 34,783 unique predictions from 1,088 different human participants on four different binary prediction tasks ("Art", "Sarcasm", "Cities" and "Census"). Overall, there are approximately 32 different instances per task. In the "Art" task, participants need to determine the art period of a painting given two choices and, overall, there are paintings from four art periods. In the "Sarcasm" task, participants need to detect if sarcasm is present in text snippets from the Reddit sarcasm dataset [36]. In the "Cities" task, participants need to determine which large US city is depicted in an image given two choices and, overall, there are images of four different US cities. Finally, in the "Census" task, participants need to determine if an individual earns more than  $50\mathrm{k}$  a year based on certain demographic information in tabular form. For "Sarcasm",  $x$  is a representation of the text snippets and we set  $y = 1$  if sarcasm is present, for "Art" and "Cities",  $x$  is a representation of the images and we set  $y = 1$  and  $y = 0$  at random for each different instance and, for "Census",  $x$  summarizes demographic information and we set  $y = 1$  if an individual earns more than  $50\mathrm{k}$  a year. In each of the tasks, human participants provide confidence values about their predictions before  $(h)$  and after  $(h_{+AI})$  receiving AI advice from a classifier in form of the classifier's confidence values  $b$ . The original dataset contains predictions by participants from different, but overlapping, sets
<sup>7</sup>We release the code to reproduce our analysis at https://github.com/Networks-Learning/human-aligned-calibration.
Refer to Appendix C for more details on the dataset.
Table 1: Misalignment, miscalibration and AUC.
Task	Misalignment	Miscalibration	AUC
EAE	MAE	ECE	MCE	πB	πH	πH+AI
Art	4.5·10-4	0.058	0.084	0.186	86.7%	72.7%	82.0%
Sarcasm	3.8·10-3	0.224	0.085	0.310	89.9%	82.5%	86.5%
Cities	6.2·10-5	0.013	0.066	0.158	84.4%	79.0%	84.7%
Census	9.0·10-3	0.298	0.109	0.270	80.0%	77.3%	79.9%
of countries across tasks, who were told the AI advice had different values of accuracy. $^{9}$  In our experiments, to control for these confounding factors, we focus on participants from the US who were told the AI advice was  $80\%$  accurate, resulting in 15,063 unique predictions from 471 different human participants.
Experimental setup and evaluation metrics. For each of the tasks, we first measure (i) the degree of misalignment between the classifiers' confidence values  $b$  and the participants' confidence values  $h$  before receiving AI advice  $b$  and (ii) the difference  $(h_{+AI} - h)$  between the human participant's confidence values before and after receiving AI advice  $b$ . Then, we compare the utility achieved by a AI-assisted decision policy  $\pi_{H + AI}$  that predicts the value of  $y$  by thresholding the humans' confidence values  $h_{+AI}$  after observing the classifier's confidence values against two baselines: (i) a decision policy  $\pi_B$  that predicts the value of  $y$  by thresholding the classifier's confidence values  $b$  and (ii) a decision policy  $\pi_H$  that predicts the value of  $y$  by thresholding the humans' confidence values  $h$  before observing the classifier's confidence values.
To measure the degree of misalignment, we discretize the confidence values  $b$  and  $h$  into bins. For the classifiers' confidence  $b$ , we use 8 uniform sized bins per task with (centered) values  $\Lambda[0,1]$ , where  $\lambda = 1/8$ . For the human participants' confidence  $h$  before receiving AI advice  $b$ , we use three bins per task ('low', 'mid' and 'high'), where we set the bin boundaries so that each bin contains approximately the same probability mass and set the bin values to the average confidence value within each bin. In what follows, we refer to the pairs of discretized confidence values  $(h, b)$  as cells, where samples  $(x, y) \in \mathcal{Z}$  whose confidence values lie in the cell  $(h, b)$  define the group  $S_{h, \lambda(b)}$ , and note that we choose a rather low number of bins for both  $b$  and  $h$  so that most cells have sufficient data samples to reliable estimate several misalignment metrics, which we describe next.
We use three different misalignment metrics: (i) the number of alignment violations between cell pairs, (ii) the expected alignment error (EAE) and (iii) the maximum alignment error (MAE). There is an alignment violation between cells pairs  $(h,b)$  and  $(h^{\prime},b^{\prime})$ , with  $h\leq h^{\prime}$  and  $b\leq b^{\prime}$ , if
$$
P (Y = 1 | (X, Y) \in S _ {h, \lambda (b)}) > P (Y = 1 | (X, Y) \in S _ {h ^ {\prime}, \lambda \left(b ^ {\prime}\right)}).
$$
Moreover, we have that:
$$
\mathrm {E A E} = \frac {1}{N} \cdot \sum_ {h \leq h ^ {\prime}, b \leq b ^ {\prime}} \left[ P (Y = 1 \mid (X, Y) \in \mathcal {S} _ {h, \lambda (b)}) - P (Y = 1 \mid (X, Y) \in \mathcal {S} _ {h ^ {\prime}, \lambda (b ^ {\prime}))} \right] _ {+},
$$
$$
\mathrm {M A E} = \max _ {h \leq h ^ {\prime}, b \leq b ^ {\prime}} P (Y = 1 \mid (X, Y) \in \mathcal {S} _ {h, \lambda (b)}) - P (Y = 1 \mid (X, Y) \in \mathcal {S} _ {h ^ {\prime}, \lambda (b ^ {\prime}))},
$$
where  $N = |\{h \leq h', b \leq b'\}|$ . Here, note that the number of alignment violations tells us how frequently is the left hand side of Eq. 6 positive across cell pairs given  $\tilde{S}_h = S_h$  and the EAE and MAE quantify the average and maximum value of the left hand side of Eq. 6 across cells violating alignment. To obtain reliable estimates of the above metrics, we only consider cells  $(h,b)$  with  $|S_{h,\lambda(b)}| \geq 30$  samples. Moreover, we also report the expected calibration error (ECE) and maximum calibration error (MCE) [12, 37], which are natural counterparts to EAE and MAE, respectively.
As a measure of utility, we estimate the true positive rate (TPR) and false positive rate (FPR) of the decision policies  $\pi_B$ ,  $\pi_H$  and  $\pi_{H + \mathrm{AI}}$  for all possible choices of threshold values, which we summarize using the area under the ROC curve (AUC) and, in Appendix C, we also report ROC curves.
Participants were also either told that the advice is from a "Human" or from an "AI" based on a random assignment of participants to a treatment or control group. Since the actual advice received in both groups was identical for the same instance and the "perceived advice source" is randomized, we use data from both treatment and control groups in the experiments.
(d) Census
Figure 3: Empirical estimate of the average difference  $\mathbb{E}[h_{+\mathrm{AI}} - h\mid (X,Y)\in S_{h,\lambda (b)}]$ , where  $b\in \Lambda [0,1]$  and  $h\in \{\mathrm{low},\mathrm{mid},\mathrm{high}\}$  are the discretized confidence values for the classifier and human participants, respectively. Error bars represent  $90\%$  confidence intervals and hatched bars mark alignment violations between confidence pairs  $(h,b)$  with  $|S_{h,\lambda (b)}|\geq 30$
Results. We start by looking at the empirical estimates of the probabilities  $P(Y = 1 \mid (X,Y) \in S_{h,\lambda(b)})$  and of our measures of misalignment (EAE, MAE) and miscalibration (ECE, MCE) in Figure 2 and Table 1 (left and middle columns). The results show that, for "Cities", the probabilities  $P(Y = 1 \mid (X,Y) \in S_{h,\lambda(b)})$  are (approximately) monotonically increasing with respect to the classifier's confidence values  $b$ . More specifically, as shown in Figure 2, there is only one alignment violation between cell pairs and, hence, our metrics of misalignment acquire also very low values. In contrast, for "Art", "Sarcasm" and especially "Census", there is an increasing number of alignment violations and our misalignment metrics acquire higher values, up to several orders of magnitude higher for "Census". These results also show that misalignment and miscalibration go hand in hand, however, in terms of miscalibration, "Census" does not stand up so strongly.
Next, we look at the difference  $h_{+ \mathrm{AI}} - h$  between the human participant's recorded confidence values before and after receiving AI advice  $b$  across samples in each of the subsets  $S_{h,\lambda(b)}$  induced by the discretized confidence values used above. Figure 3 summarizes the results, which reveal that the difference  $h_{+ \mathrm{AI}} - h$  increases monotonically with respect to the classifier's confidence  $b$ . This suggests that participants always expect  $b$  to reflect the probability of a positive outcome irrespective of their confidence value  $h$  before receiving AI advice, providing support for our hypothesis that (rational) decision makers implement monotone AI-assisted decisions policies. Further, this finding also implies that, for "Art", "Sarcasm" and "Census", any policy  $\pi_{H_{+ \mathrm{AI}}}$  that predicts the value of the label  $y$  by thresholding the confidence value  $h_{+ \mathrm{AI}}$  will be necessarily suboptimal because the probabilities  $P(Y = 1 \mid (X,Y) \in S_{h,\lambda(b)})$  are not monotone increasing with  $b$ .
Finally, we look at the AUC achieved by decision policies  $\pi_B$ ,  $\pi_H$  and  $\pi_{H + \mathrm{AI}}$ . Table 1 (right columns) summarizes the results, which shows that  $\pi_{H + \mathrm{AI}}$  outperforms  $\pi_H$  consistently across all tasks but it only outperforms  $\pi_B$  in a single task ("Cities") out of four. These findings provide empirical support for Theorem 3, which predicts that, in the presence of human-alignment violations as those observed in "Art", "Sarcasm" and "Census", any monotone AI-assisted decision policy will be suboptimal, and they also provide support for Theorem 5, which predicts that, under human-alignment, there exist near-optimal AI-assisted decision policies satisfying monotonicity.
In this section, we discuss the intended scope of our work and identify several limitations of our theoretical and experimental results, which may serve as starting points for future work.
Decision making setting. We have focused on decision making settings where both decisions and outcomes are binary. However, we think that it may be feasible to extend our theoretical analysis to settings with multi-categorical (or real-valued) outcome variables and decisions. One of the main challenges would be to identify which natural conditions utility functions may satisfy in such settings. Further, we also think that it would be significantly more challenging to extend our theoretical analysis to sequential settings—multicalibration in sequential settings is an open area of research—but our ideas may still be a useful starting point. In addition, our theoretical analysis assumes that the decision makers aim to maximize the average utility of their decisions. However, whenever human decisions are consequential to individuals, the decision maker may have fairness desiderata.
Confidence values. In our causal model of AI-assisted decision making, we allow the classifier's confidence values to depend on the decision maker's confidence values because this is necessary to achieve human-alignment via multicalibration as described in Section 5. However, we would like to clarify that both Theorems 3 and 5 still hold if the classifier's confidence values do not depend on the decision maker's confidence, as it is typically the status quo today. Looking into the future, our work questions this status quo by showing that, by allowing the classifier's confidence values to depend on the decision maker's confidence values, a decision maker may end up taking decisions with higher utility. Moreover, we would also like to clarify that, while the motivation behind our work is AI-assisted human decision making, our theoretical results do not depend on who—be it a classifier or another human—gives advice. As long as the advice comes in the form of confidence values, our results are valid. Finally, while we have shown that human-alignment can be achieved via multicalibration, we hypothesize that algorithms specifically designed to achieve human-alignment may have lower data and computational requirements than multicalibration algorithms.
Experimental results. Our experimental results demonstrate that, across tasks, the average utility achieved by decision makers is relatively higher if the classifier they use satisfies human-alignment. However, they do not empirically demonstrate that, for a fixed task, there is an improvement in average utility achieved by decision makers if the classifier they use satisfies human-alignment. The reason why we could not demonstrate the latter is because, in our experiments, we used an observational dataset gathered by others [35]. Looking into the future, it would be very important to run a human subject study to empirically demonstrate the latter and, for now, treat our conclusions with caution.
We have introduced a theoretical framework to investigate what properties confidence values should have to help decision makers take better decisions. We have shown that there exists data distribution for which a rational decision maker using calibrated confidence values will always take suboptimal decisions. However, we have further shown that, if the confidence values satisfy a natural alignment property, which can be achieved via multicalibration, then a rational decision maker using these confidence values can take optimal decisions. Finally, we have illustrated our theoretical results using real human predictions on four AI-assisted decision making tasks.
Acknowledgements. We would like to thank Nastaran Okati for fruitful discussions at an early stage of the project. Gomez-Rodriguez acknowledges support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 945719).
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A.1 Additional Lemmas
Lemma 1 (Monotonicity). If a utility function  $u$  satisfies Eq. 1, then  $u$  is monotone with respect to the probability that  $Y = 1$ , i.e., for any  $P, P' \in \mathcal{P}(\{0,1\})$  such that  $P(Y = 1) \leq P'(Y = 1)$ , it holds that  $\mathbb{E}_{Y \sim P}[u(1,Y)] \leq \mathbb{E}_{Y \sim P'}[u(1,Y)]$ .
Proof. We readily have that
$$
\begin{array}{l} \mathbb {E} _ {Y \sim P} [ u (1, Y) ] = P (Y = 1) \cdot u (1, 1) + (1 - P (Y = 1)) \cdot u (1, 0) \\ \leq P ^ {\prime} (Y = 1) \cdot u (1, 1) + (1 - P ^ {\prime} (Y = 1)) \cdot u (1, 0) \\ = \mathbb {E} _ {Y \sim P ^ {\prime}} [ u (1, Y) ], \\ \end{array}
$$
where, in the above inequality, we use that  $u(1,1) > u(1,0)$  and  $P(Y = 1) \leq P'(Y = 1)$ .
Lemma 2 (Trivial policies are not always optimal). If a utility function  $u$  satisfies Eq. 1, then there exist  $P, P' \in \mathcal{P}(\{0,1\})$  such that the trivial policies  $\pi$  that either always decide  $T = 1$  or always decide  $T = 0$  are suboptimal. In particular, for any  $P, P' \in \mathcal{P}(\{0,1\})$  such that  $P(Y = 1) < c$  and  $P'(Y = 1) > c$ , where
$$
c = \frac {u (0 , 0) - u (1 , 0)}{u (1 , 1) - u (1 , 0) + u (0 , 0) - u (0 , 1)} \in (0, 1), \tag {8}
$$
$$
\mathbb {E} _ {Y \sim P} [ u (1, Y) ] <   \mathbb {E} _ {Y \sim P} [ u (0, Y) ] \quad a n d \quad \mathbb {E} _ {Y \sim P ^ {\prime}} [ u (1, Y) ] > \mathbb {E} _ {Y \sim P ^ {\prime}} [ u (0, Y) ]. \tag {9}
$$
Proof. Let  $P$  be any distribution such that
$$
P (Y = 1) <   c = \frac {u (0 , 0) - u (1 , 0)}{u (1 , 1) - u (1 , 0) + u (0 , 0) - u (0 , 1)},
$$
where  $c \in (0,1)$  because, by assumption,  $u$  satisfies Eq. 1. Now, by rearranging the above inequality, we have that
$P(Y = 1)\cdot u(1,1) + (1 - P(Y = 1))\cdot u(1,0) <   P(Y = 1)\cdot u(0,1) + (1 - P(Y = 1))\cdot u(0,0),$  and, using the definition of the expectation, it immediately follows that
$$
\mathbb {E} _ {Y \sim P} [ u (1, Y) ] <   \mathbb {E} _ {Y \sim P} [ u (0, Y) ].
$$
The same argument can be used to show that, for any distribution  $P'$  such that  $P'(Y = 1) > c$ , it holds that  $\mathbb{E}_{Y \sim P'}[u(1,Y)] > \mathbb{E}_{Y \sim P'}[u(0,Y)]$ . Finally, note that, since  $c \in (0,1)$ , we know that such distributions  $P$  and  $P'$  exist.
A.2 Proof of Theorem 3
Before proving Theorem 3, we rewrite the expected utility with respect to the probability distribution  $P^{\mathcal{M}}$  in terms of confidence  $H$  and  $B$  by using the law of total expectation,
$$
\mathbb {E} _ {\pi} [ u (T, Y) ] = \mathbb {E} _ {H, B \sim P ^ {\mathcal {M}} (H, B)} \left[ \mathbb {E} _ {\pi} [ u (T, Y) | H, B ] \right].
$$
Here, to simplify notation, we will write
$$
\mathbb {E} _ {H, B} \left[ \mathbb {E} _ {\pi} [ u (T, Y) \mid H, B ] \right],
$$
where note that, using the law of total expectation, we can write the inner expectation in the above expression in terms of the utilities of the trivial policies, i.e.,
$$
\begin{array}{l} \mathbb {E} _ {\pi} [ u (T, Y) \mid H, B ] = \mathbb {E} [ u (1, Y) \mid H, B ] \cdot P _ {\pi} (T = 1 \mid H, B) \\ + \mathbb {E} [ u (0, Y) \mid H, B ] \cdot P _ {\pi} (T = 0 \mid H, B), \tag {10} \\ \end{array}
$$
and we will use  $P$  to refer to probabilities induced by  $\operatorname{SCM} \mathcal{M}$ , e.g.,  $P(H, B)$  to denote  $P^{\mathcal{M}}(H, B)$ . Now, we restate and prove Theorem 3.