SlowGuess/NeurIPS_2023_OCR huggingface dataset:
**Construction.** Let  $\mathcal{U}$  be the set of all discrete unimodal distributions on  $\mathcal{C}$ . For any  $\mathbf{x}$  with  $\hat{p}(\mathbf{x}) = \hat{P}_{Y|X}(Y|X = \mathbf{x}) \in \mathcal{U}$ , the class probabilities  $[\hat{p}_k(\mathbf{x})]_{k=1}^K$  would be unimodal, i.e., there exists a mode  $c_{\hat{m}} \in \mathcal{C}$  such that the sequence is monotonically non-decreasing before  $c_{\hat{m}}$  and non-increasing post  $c_{\hat{m}}$ . The corresponding log probabilities denoted by  $z_k = \log (\hat{p}_k(\mathbf{x})) \in \mathbb{R}^-$ ,  $[k]_1^K$  also form a unimodal sequence with the same mode and are all negative valued.
Let  $\psi : \mathbb{R}^+ \to \mathbb{R}^-$  be any strictly monotonically decreasing bijective function, Then, for any  $z \in \mathbb{R}^-$ , there is a unique  $r \in \mathbb{R}^+$  such that  $\psi(r) = z$ . Let  $r_k = \psi^{-1}(z_k)$ ,  $[k]_m^K$  and  $r_k = -\psi^{-1}(z_k)$ ,  $[k]_1^{m-1}$  then given  $[z_k]_{k=1}^K$ , we obtain a unique^2 non-decreasing sequence  $[r_k]_{k=1}^K$ . Let  $\psi^E: R \to R^-$  be the "even" extension of  $\psi(\cdot)$ , i.e.,  $\psi^E(r) = \psi(r)$  for  $r \in R^+$  and  $\psi^E(r) = \psi(-r)$  for  $r \in R^-$ , then we have  $\psi^E(r_k) = z_k$ . Possible choices include  $\psi^E(x) = -|x|^d$  for any real  $d$ . The monotonically increasing  $[r_k]_{k=1}^K$  can be generated from the output vector  $\mathbf{v}$  of a DNN constrained such that  $v_k \geq 0$ ,  $[k]_2^K$  using a non-negative transformation  $\phi: R \mapsto R^+$  followed by a cumulative summation. Fig. 3 depicts our proposed construction for any DNN  $\mathbf{f}(\cdot)$  along with equations below.
$$
\begin{array}{l} \eta (\mathbf {x}) = \mathbf {f} (\mathbf {x}, \theta); v _ {1} = \eta_ {1} (\mathbf {x}); v _ {k} = \phi (\eta_ {k} (\mathbf {x})), [ k ] _ {2} ^ {K}, \\ r _ {1} = v _ {1}; r _ {k} = r _ {k - 1} + v _ {k}, [ k ] _ {2} ^ {K}; z _ {k} = \psi^ {E} (r _ {k}); \hat {p} _ {k} = \frac {\exp (z _ {k})}{\sum_ {k = 1} ^ {K} \exp (z _ {k})}, [ k ] _ {1} ^ {K}. \tag {5} \\ \end{array}
$$
This network can be trained with standard CE loss. From Thm. 2, any distribution in  $\mathcal{U}$  maps to an output vector  $\eta (\mathbf{x})\in R^d$ , which can be approximated using DNNs with appropriate capacity [26]. Fig. 8 in Appendix C.4 presents some illustrative examples of Unimodal model fit of COPOC on a public dataset.
Theorem 2. Let  $\eta : \mathcal{X} \to R^K$ ,  $\phi : R \to R^+$  and  $\psi^E : R \to R^-$  such that  $\psi^E(r) = \psi^E(-r)$ ,  $\forall r \in R$  and its restriction to  $R^+$  is a strictly monotonically decreasing bijective function. (a) Then, the model output constructed as per Eqn. 5 is always unimodal, i.e.,  $\hat{p}(\mathbf{x}) \in \mathcal{U}$ ,  $\forall \mathbf{x} \in \mathcal{X}$ . (b) Further, given any  $\hat{p}(\mathbf{x}) \in \mathcal{U}$  for  $\mathbf{x} \in \mathcal{X}$ , there exists a well defined function  $\eta(\mathbf{x}) \in R^K$  that satisfies Eqn. 5 if  $\phi(\cdot)$  is surjective on  $R^+$ .
Proof Sketch: Part (a) follows by observing that the non-decreasing sequence  $[r_k]_{k=1}^K$  will result in a unimodal  $[z_k]_{k=1}^K$  and unimodal  $\hat{p}(\mathbf{x}) \in \mathcal{U}$  with mode corresponding to sign change in  $r_k$  sequence. Part (b) follows by constructing a unique inverse for  $\psi^E(\cdot)$  by pivoting on the largest indexed mode  $\hat{m}$  of  $\hat{p}(\mathbf{x})$ . For  $k < \hat{m}$ , we choose  $(\psi^E)^{-1}(z_k) \in R^{-}$  and for  $k \geq \hat{m}$ , we choose  $(\psi^E)^{-1}(z_k) \in R^{+}$ .
We evaluate the utility of our approach (COPOC) for generating minimal contiguous PS for a desired coverage for ordinal classification on multiple datasets. We investigate the following questions:
2Note for uniqueness,  $\hat{m}$  has to be defined as the largest value of mode in case of multiple modes.
Table 1: Results on HCI: Mean & std. error is reported for 5 trials. Best results bolded.
MAE	Acc@1	Acc@2	Acc@3	|PSI|	CV%
HCI	CE	0.68 ± 0.03	54.3 ± 2.6	75.3 ± 3.1	88.9 ± 1.6	3.28 ± 0.14	24.4 ± 1.2
POE	0.66 ± 0.05	56.5 ± 1.8	76.5 ± 2.5	89.0 ± 2.1	3.1 ± 0.18	9.8 ± 1.2
SORD	0.65 ± 0.06	56.2 ± 2.8	77.1 ± 2.9	89.8 ± 2.6	2.96 ± 0.19	2.7 ± 1.1
AVDL	0.64 ± 0.08	56.8 ± 1.5	77.9 ± 2.4	89.8 ± 1.05	2.98 ± 0.11	2.1 ± 1.4
Binomial	0.68 ± 0.05	54.5 ± 1.2	75.8 ± 2.6	88.8 ± 1.8	3.01 ± 0.16	0
Binomial-temp	0.66 ± 0.04	55.5 ± 1.8	78 ± 2.2	90.1 ± 2.1	2.90 ± 0.11	0
Uni-loss	0.67 ± 0.09	54.5 ± 3.1	74.8 ± 2.5	88.1 ± 2.5	3.05 ± 0.38	5.1 ± 1.9
COPOC	0.65 ± 0.04	56.5 ± 2.0	79.8 ± 1.6	91.7 ± 2.8	2.66 ± 0.13	0
- RQ1: How does COPOC perform relative to SOTA methods on OC tasks?
- RQ2: What benefits does COPOC offer relative to parametric modeling on synthetic datasets?
- RQ3: Under what conditions does COPOC outperform vanilla model trained with CE loss (V-CE)? Is COPOC an approximately consistent estimator of the underlying distribution?
- RQ4: How does COPOC performance vary with a different choice of conformal inference such as LAC [36] instead of APS?
- RQ5: How does Ordinal-APS [25] which also outputs a contiguous prediction set (PS) over ordinal labels with regular model training fares against COPOC with it's unimodal training?
Algorithms: We compare our proposed unimodal construction (COPOC) against six SOTA methods as well as modeling with vanilla cross-entropy loss  $V-CE$  using APS for conformal inference for all methods for a fair comparison. For COPOC, we experimented with various choices of  $\phi$  and  $\psi$  and chose  $\phi(x) = |x|$  and  $\psi^E(x) = -|x|$  based on performance (see Appendix C.3). The SOTA methods include approaches that (a) utilize soft labels generated from linear exponentially decaying distributions  $SORD$  [[12], and linear adaptive Gaussians with variance learning  $AVDL$  [42], (b) specifically model unimodal distribution through parametric assumptions  $Binomial$  [4] and  $Binomial-Temp$ , and (c) non-parametric methods based on unimodality aware loss function  $Uni-Loss$  [22] or ordinality imposed in embedding space  $POE[23]$ . In Sec. 5.5, we also evaluate an alternative conformal procedure LAC [36] with our unimodal construction.
Metrics: Similar to [35, 9] we compute PS Size (|PS|) @90% coverage i.e.,  $\alpha = 0.1$ . Since APS does not guarantee contiguity for models that do not output unimodal distribution, we consider a minimal contiguous interval that covers the output PS and report the size. We also report the fraction of instances which resulted in non-contiguous PS for which we needed to expand the original non-contiguous set ( $CV\%$ ). To evaluate performance on OC tasks, we report MAE - mean absolute difference between predicted and ground-truth and Accuracy@K (Acc@K) that captures if the ground truth was in the top  $k$  predictions. This can benefit unimodal constrained models where classes adjacent to the mode will receive the next greatest probability mass. For comparing different conformal prediction methods in Sec. 5.5, we use size-stratified coverage violation (SSCV) [1] that measures violations of the conditional coverage property and is suited for high dimensional data.
Datasets: We evaluate COPOC on four public image datasets: age-detection (Adience [13]), historical image dating (HCI [30]), image aesthetics estimation (Aesthetic [37]) and biomedical classification (Retina-MNIST [44]). These datasets contain 8, 5, 5, and 5 ordered classes respectively and the number of instances is 26580, 1325, 15680, and 1600. More details are included in Appendix C.1. We also experimented with synthetic data with the generation process and result discussed in Sec 5.3.
5.2 RQ1: Performance on ordinal classification on real world datasets
Table 1 presents comparison of COPOC with other methods on the HCI dataset. Results on other datasets included in the Appendix C.1 due to space constraints. COPOC performs at par with other baselines in terms of MAE & Acc@1 and significantly outperforms them on Acc@2, Acc@3 & |PS| (8% smaller compared to the next best one). We observe that V-CE and POE have the highest
Table 2: Results on Synthetic Data: We report KL-Div, PS Size [Ground truth Oracle size],  $CV\%$  and MAE. Mean and std. error is reported across 10 random trials. Best mean results are bolded.
V-CE	SORD	AVDL	Binomial	Binomial-temp	Uni-Loss	COPOC
D1	KL Div.	0.04 ± 0.01	0.03 ± 0.01	0.07 ± 0.01	0.1 ± 0.02	0.09 ± 0.01	0.12 ± 0.10	0.04 ± 0.01
|PSI [3.03]	3.09 ± 0.06	3.09 ± 0.04	3.12 ± 0.05	3.16 ± 0.04	3.16 ± 0.04	3.61 ± 0.15	3.09 ± 0.05
CV%	0.3 ± 0.01	0	0	0	0	1.7 ± 0.4	0
MAE	0.65 ± 0.02	0.65 ± 0.01	0.67 ± 0.02	0.68 ± 0.02	0.69 ± 0.01	0.68 ± 0.03	0.65 ± 0.02
D2	KL Div.	0.06 ± 0.01	0.17 ± 0.02	0.18 ± 0.04	0.19 ± 0.03	0.17 ± 0.02	0.26 ± 0.11	0.04 ± 0.01
|PSI [2.56]	2.65 ± 0.04	2.72 ± 0.04	2.78 ± 0.04	2.85 ± 0.06	2.81 ± 0.02	3.12 ± 0.09	2.59 ± 0.01
CV%	2.3 ± 0.05	0.3 ± 0.1	0.6 ± 0.1	0	0	2.9 ± 0.7	0
MAE	0.56 ± 0.01	0.59 ± 0.01	0.60 ± 0.01	0.61 ± 0.02	0.61 ± 0.01	0.63 ± 0.04	0.57 ± 0.02
D3	KL Div.	0.13 ± 0.01	0.38 ± 0.02	0.17 ± 0.01	0.49 ± 0.03	0.44 ± 0.02	0.4 ± 0.09	0.09 ± 0.01
|PSI [1.58]	1.73 ± 0.02	1.96 ± 0.06	1.85 ± 0.04	2.39 ± 0.04	2.38 ± 0.02	2.35 ± 0.1	1.66 ± 0.02
CV%	2.9 ± 0.1	1.1 ± 0.2	0.8 ± 0.2	0	0	2.7 ± 0.4	0
MAE	0.24 ± 0.02	0.25 ± 0.02	0.23 ± 0.03	0.28 ± 0.01	0.26 ± 0.02	0.27 ± 0.04	0.23 ± 0.02
D4	KL Div.	0.14 ± 0.01	0.33 ± 0.01	0.31 ± 0.02	0.44 ± 0.04	0.35 ± 0.02	0.35 ± 0.1	0.08 ± 0.01
|PSI [4.40]	4.67 ± 0.03	4.73 ± 0.04	4.78 ± 0.05	4.83 ± 0.04	4.82 ± 0.06	5.04 ± 0.2	4.50 ± 0.02
CV%	4.7 ± 0.8	2.7 ± 0.4	2.8 ± 0.3	0	0	5.6 ± 1.1	0
MAE	1.26 ± 0.02	1.27 ± 0.03	1.27 ± 0.02	1.31 ± 0.04	1.29 ± 0.02	1.30 ± 0.03	1.24 ± 0.01
contiguity violations  $CV\%$ . Though SORD, AVDL, and Uni-loss enforce unimodality in training, it does not necessarily translate to unimodality in test samples (indicated by high  $CV\%$ ) resulting in poor performance on Acc@2 and Acc@3 and |PS|, which is critical in OC tasks. Since Binomial-temp enforces unimodality by construction, it performs better than the above methods on Acc@k. However, unimodality constraint on DNN in COPOC results in even better model fit for OC tasks which translates to higher Acc@K & shorter  $|PS|$ .
5.3 RQ2: Ablative Study of Unimodal Modeling Methods on Synthetic Data
To generate synthetic data, we consider the label set  $\mathcal{C}$  of size 10, i.e.,  $Y \in \{c_1, \ldots, c_{10}\}$  and input space  $\mathcal{X} \subseteq R^{10}$ , where  $X_1 \sim \text{Uniform}(1, 10)$  and  $X_2, \ldots, X_{10}$  follow independent normal distribution. Further, we define  $h(X) = 100 * (\sin(0.2 * X_1) + \cos(0.4 * X_1))$ . To generate the labels, we consider a conditional distribution function  $Q_{Z|X}$  that generates  $Z$  with  $h(X)$  as distributional parameters and a mapping function  $g: R \to \mathcal{C}$  that maps  $Z$  to the output label  $Y \in \mathcal{C}$ . Thus, for each  $x$ , we generate  $z \sim Q_{Z|X}(x)$  and  $y = g(z)$ . We consider two choices of mapping functions  $g(\cdot)$ . The first choice  $g_{equi}$  partitions the range of  $Z$  into 10 equi-bins which can be mapped to the label set in a linear order while the second choice  $g_{non - equi}$  partitions the range of  $Z$  into 10 bins of arbitrary width to assign labels. From the construction, one can see that  $g_{equi}$  considers classes to be equi-spaced categories while  $g_{non - equi}$  does not. We consider the following four synthetic datasets in increasing order of complexity and train models (details in Appendix C.2)
-  $D1$ :  $Q_{Z|X}(x) = \text{Double-Exponential with mean at } h(x) \text{ and constant variance}$ ;  $g = g_{\text{equi}}$ .
-  $D2$ :  $Q_{Z|X}(x) = \text{Double-Exponential with mean at } h(x) \text{ and constant variance}$ ;  $g = g_{\text{non - equi}}$ .
-  $D3$ :  $Q_{Z|X}(x) = \text{Gaussian with mean and variance varying with } h(x); g = g_{\text{non - equi}}$ .
-  $D4: Q_{Z|X}(x) = \text{Gaussian, Poisson or double Exponential with mean and variance varying with } h(x) \text{ chosen at random for each } x; g = g_{\text{non - equi}}$ .
Since we have the true distribution  $P_{Y|X}$ , we compute the Oracle PS size and compare it with that of other methods. We report KL Divergence (KL) between the true and predicted distributions with lower KL indicating better model fit. Table 2 shows that for  $D1$ , SORD fits the data well with lowest KL and MAE as it explicitly models exponential distribution assuming all classes to be equi-spaced. However, on  $D2$  and  $D3$ , COPOC outperforms all the other methods on the model fit and also results in shorter  $|PS|$  closer to Oracle. On  $D2$  and  $D3$ , SORD performs poorly because it assumes both constant variance across input spaces and equi-distant classes which are not valid for these datasets. Since  $D3$  draws samples from Gaussian distribution with heteroskedastic noise, AVDL outperforms SORD but is inferior to COPOC. Since  $D4$  is drawn from a more complex distribution, COPOC outperforms the rest with an even larger margin. The Binomial variants underperform throughout due to distributional
Figure 4: Comparison of KL Div. indicates superior model fit of COPOC vs. V-CE for smaller train size with convergence as data size increases.
Figure 5: Comparison of |PSI and  $CV\%$  indicates that COPOC is superior against V-CE with smaller train size.
Figure 6: LAC vs APS comparison by varying  $\alpha$ . (Top) |PSI and (Bottom) SSCV.
mismatch while Uni-Loss exhibits high variance indicating convergence issues. As with the real benchmarks, enforcing unimodality via loss or soft-labels does not guarantee unimodailty on test data, indicated by high  $CV\%$ . Interestingly V-CE performs exceedingly well throughout in terms of KL beating more sophisticated methods. The overall observation is that the performance of the methods depends largely on the validity of the assumptions and relatively unconstrained nature of COPOC makes it more versatile.
5.4 RQ3: COPOC vs Vanilla-CE and Consistency Properties
We compare COPOC against  $V-CE$  by varying train-data size ( $= N$ ) on synthetic data  $D4$ . Fraction of samples used to train each model is  $3/4$ , and rest is used for calibration and test. All experiments are repeated 10 times with random realization of train set. Comparing the  $KL$  loss on train set in Fig. 4, we observe that COPOC has lower  $KL$  in early stages with lesser data and performance for both converges when  $N$  reaches to  $\sim 80K$  and  $KL$  asymptotically becomes negligible. Fig. 5 compares |PSI and CV% by varying  $N$  on  $D4$ . |PSI| of COPOC approaches those of Oracle with far lesser samples compared to VC-E. Thus, COPOC with its inductive bias fits the data better with lesser samples compared to unbiased CE yielding shorter sets with finite samples. Above results also suggest that COPOC might be approximately consistent. Note that with increased  $N$ ,  $V-CE$  results in a better model fit, lower CV% and shorter sets, indicating that these are correlated.
5.5 RQ4: Choice of Conformal Inference: LAC vs. APS
Table 3 provides a comparison of PS size and size-stratified coverage violation(SSCV) of APS against LAC when applied on the output of our proposed unimodal model for  $\alpha = 0.1$ . Both methods generate contiguous prediction sets. Although LAC consistently produces shorter sets, it also has slightly worse SSCV metrics across datasets. In Fig. 6, we plot |PSI| and SSCV across different  $\alpha$  for synthetic data D4 and results are similar. We observe that LAC achieves the smallest prediction set size but sacrifices adaptiveness (conditional coverage) in the process.
Table 3: LAC vs APS with COPOC on image datasets. Mean is reported after 100 trials.
LAC	APS
|PSI|	SSCV	|PSI|	SSCV
HCI	2.61	0.09	2.66	0.09
Adience	2.42	0.13	2.52	0.10
Aesthetic	1.58	0.19	1.65	0.07
RetinaMnist	3.08	0.24	3.14	0.09
5.6 RQ5: COPOC vs Ordinal-APS [25]
Table 4 shows an empirical comparison of COPOC against APS and Ordinal-APS applied over a Vanilla DNN trained with Cross-entropy loss (V-CE) on public datasets and synthetic data  $D4$  described in Sec. 5.2 and 5.3. For V-CE with APS we consider a minimal contiguous interval that covers the output PS and report its size. We observe that Ordinal-APS produces significantly shorter sets compared to V-CE with APS. However, COPOC significantly outperforms Ordinal-APS because of better unimodal data fit. Note that Ordinal-APS outputs a contiguous PS over ordinal labels
Table 4: Ordinal-APS [25] vs COPOC on Synthetic data  $D4$  and public image datasets. Mean and std. error is reported after 10 trials.
APS	Ordinal-APS	COPOC
Synthetic D4	4.67 ± 0.03	4.59 ± 0.03	4.50 ± 0.02
HCI	3.28 ± 0.14	3.03 ± 0.15	2.66 ± 0.13
Adience	4.82 ± 0.24	2.67 ± 0.12	2.26 ± 0.06
Aesthetic	1.96 ± 0.2	1.77 ± 0.05	1.70 ± 0.06
Retina MNIST	3.6 ± 0.08	3.28 ± 0.02	3.03 ± 0.01
irrespective of whether the posterior distribution generated by the model is unimodal or not. COPOC with unimodal regularization results in better model fit and hence shorter PS sizes.
6 Validity of the Unimodality Assumption
Unimodality assumption might not be universally applicable for all OC scenarios (eg., prediction of preference ratings, event-hour-of-day etc.). However, there do exist a large number of OC applications where it is beneficial to assume unimodality as validated by multiple notable works in computer vision domain [12, 20, 42, 4]. Table 5 shows the negative log-likelihood (NLL) of vanilla DNN (V-CE) fitted with CE loss (unconstrained fit) and COPOC (unimodal fit) on four public datasets described in Sec. 5.2. The
Table 5: Comparison of COPOC against V-CE in terms of NLL. Mean and std. error is reported after 10 trials.
V-CE	COPOC
HCI	1.73 ± 0.13	1.59 ± 0.15
Adience	2.33 ± 0.18	1.66 ± 0.21
Aesthetic	1.49 ± 0.01	0.71 ± 0.02
Retina MNIST	1.24 ± 0.04	1.23 ± 0.04
superior fit of COPOC indicated by lower NLL justifies the unimodality assumption for these datasets. COPOC makes an assumption on the underlying distribution being unimodal and one could potentially check the validity of the assumption by comparing the likelihood of the unimodal and unconstrained fits. Note that even if the unimodality assumption is not true, the theoretical coverage guarantees of the prediction set produced by COPOC would still hold but the cardinality bounds would be weaker since the fitted distribution would deviate significantly from the true underlying distribution.
7 Conclusion, Broader Impact, and Limitations
We proposed an approach to construct minimal contiguous prediction sets for ordinal classification with guarantees on coverage and cardinality along with empirical validation of efficacy on both real and synthetic datasets. Our solution employs a novel architecture for non-parametric modelling of arbitrary unimodal class distributions without sacrificing representation ability. With ML-driven systems playing an increasingly important role in our society, it is critical to provide practical guarantees on the reliability and uncertainty of the ML model output. While existing CP methods [35] address this concern for the general classification setting, there is a need for specialized solutions for application scenarios with inherent ordinality and unimodality in class labels, e.g., predicting cancer stage of biopsy signals. Our work makes an important contribution in this direction and also provides a way to effectively communicate the limitations of the model adaptively across the input space with bigger prediction sets for harder instances. The proposed methods have numerous practical applications in improving recommendations based on an ordered attribute (e.g., budget), and reducing human supervision effort for ordinal classification tasks. Future directions include extensions to tasks where classes are partially ordered or hierarchically grouped (e.g., topic hierarchies). Consistency of our proposed unimodal construction is also worth investigating.
Limitations: COPOC makes an assumption on the underlying distribution being unimodal and might lead to a sub-optimal fit if the assumption does not hold. Furthermore, it can be paired only with CP methods such as LAC and APS where the prediction sets can be viewed as upper-level sets of the predicted class probabilities, i.e., set of classes with probability higher than a threshold. Due to the reliance on the CP methods, the coverage guarantees are also valid only when the assumption on the exchangeability of the data points holds true.
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Our current work on contiguous conformal predictions for ordinal classification is foundational in nature and has multiple real-world applications.