SlowGuess/NeurIPS_2023_OCR huggingface dataset:
7 in Appendix K.2. More specifically, with more noise, the best number of estimators, as chosen based on cross-validation, decreases. We describe the setup in detail in Appendix K.2.
Figure 1: Practitioner's validation process in the presence of noise for CART. For a fixed tree depth, as we add noise, the gap between training and validation accuracy increases (Subfigure a). As we use cross validation to select tree depth, the best tree depth decreases with noise (Subfigure b).
4.4 Step 4. Rashomon Ratio is Larger for Simpler Spaces
For simpler hypothesis spaces, it may not be immediately obvious that the Rashomon ratio is larger, i.e., a larger fraction of a simpler model class consists of "good" models. Intuitively, this is because the denominator of the ratio (the total number of models) increases faster than the numerator (the number of good models) as the complexity of the model class increases. As we will see, this is because the good models in the simpler model class tend to give rise to more bad models in the more complex class, and the bad models do not tend to give rise to good models as much. We explore two popular model classes: decision trees and linear models. The results thus extend to forests (collections of trees) and generalized additive models (which are linear models in enhanced feature space).
Consider data that live on a hypercube (e.g., the data have been binarized, which is a common pre-processing step [29, 3, 52, 47]) and a hypothesis space of fully grown trees (complete trees with the last level also filled) of a given depth  $d$ . Denote this hypothesis space as  $\mathcal{F}_d$ . For example, a depth 1 tree has 1 node and 2 leaves. Under natural assumptions on the quality of features of classifiers and the purity of the leaves of trees in the Rashomon set, we show that the Rashomon ratio is larger for hypothesis spaces of smaller-depth trees.
Proposition 6 (Rashomon ratio is larger for decision trees of smaller depth). For a dataset  $S = X \times Y$  with binary feature matrix  $X \in \{0,1\}^{n \times m}$ , consider a hypothesis space  $\mathcal{F}_d$  of fully grown trees of depth  $d$ . Let the number of dimensions  $m < 2^{2^d}$ . Assume: (Leaves are correct) all leaves in all trees in the Rashomon set have at least  $\lceil \theta n \rceil$  more correctly classified points than incorrectly classified points; (Bad features) there is a set of  $m_{bad} \geq d$  "bad" features such that the empirical risk minimizer of models using only the bad features is not in the Rashomon set. Then  $\hat{R}_{ratio}(\mathcal{F}_{d+1}, \theta) < \hat{R}_{ratio}(\mathcal{F}_d, \theta)$ .
The proof of Proposition 6 is in Appendix E. Both assumptions are typically satisfied in practice.
To demonstrate our point, we computed the Rashomon ratio and pattern Rashomon ratio for 19 different datasets for hypothesis spaces of decision trees and linear models of different complexity (see Figure 2). As the complexity of the hypothesis space increases, we see an obvious decrease in the Rashomon ratio and pattern Rashomon ratio.
To compute the Rashomon ratio, for trees, we used TreeFARMS [52], which allows us to enumerate the whole Rashomon set for sparse trees. For linear models, we designed a two-step approach that allows us to compute all patterns in the Rashomon set. First, for every sample  $z_{i}$  we solved a simple optimization problem that checks if there exists a model in the Rashomon set that misclassifies this sample. If there is no such model, changing the label of  $z_{i}$  will not add more patterns to the Rashomon set, therefore, we can ignore  $z_{i}$ . In the second step, we grew a search tree and bounded all paths that lead to patterns outside of the Rashomon set. We describe our approach for pattern computation in detail in Appendix K.3 and discuss the experimental setup in Appendix K.4.
Completion of the Path. After reducing the complexity of the hypothesis space in Step 3, the practitioner has already arrived at a simpler hypothesis space, which is the goal. Continuing on the
(b)
Figure 2: Calculation showing that the Rashomon ratio (a) and pattern Rashomon ratio (b) decrease for the hypothesis space of decision trees of fixed depth from 1 to 7 for 14 different datasets (a) and for the hypothesis space of linear models of sparsity from 1 to 4 for 5 different datasets (b). Each line represents a different dataset, each dot represents the log of the Rashomon ratio or pattern Rashomon ratio. Both ratios decrease as we move to a more complex hypothesis space.
path, they can reduce complexity further. Specifically, they find a larger Rashomon ratio for the newly chosen lower complexity hypothesis space according to Step 4. As a reminder of the thesis of Semenova et al. [39], with large Rashomon ratios, there are many good models, among which may exist even simpler models that perform well. Thus, the path, starting from noise, is a powerful way to explain what we see in practice, which is that simple models often perform well [17].
Note that to follow the path, the machine learning practitioner does not need to know the exact amount of noise. As long as they suspect some noise present in the dataset, the results of this paper apply, and the practitioner would expect a good performance from simpler models.
5 Rashomon Ratio for Ridge Regression Increases under Additive Attribute Noise
For linear regression, adding multiplicative or additive noise to the training data is known to be equivalent to regularizing the model parameters [5]. Moreover, the more noise is added, the stronger this regularization is. Thus, noise leads directly to Step 3 and a choice of a smaller (simpler) hypothesis space. Also, for ridge regression, the Rashomon volume (the numerator of the Rashomon ratio) can be computed directly [39] and depends on the regularization parameter. Building upon these results, we prove that noise leads to an increase of the Rashomon ratio (as in Step 4 of the path).
Given dataset  $S$ , the ridge regression model is learned by minimizing the penalized sum of squared errors:  $\hat{L} (\omega) = \hat{L}_{LS}(\omega) + C\omega^{T}\omega$ , where  $\hat{L}_{LS}(\omega) = \frac{1}{n}\sum_{i = 1}^{n}\left(x_i^T\omega -y_i\right)^2$  is the least squares loss,  $C$  is a regularization parameter, and  $\omega \in \mathbb{R}^m$  is a parameter vector for a linear model  $f = \omega^T x$ .
We will assume that there is a maximum loss value  $\hat{L}_{\mathrm{max}}$ , such that any linear model that has higher regularized loss than  $\hat{L}_{\mathrm{max}}$  is not being considered within the hypothesis space. For instance, an upper bound for  $\hat{L}_{\mathrm{max}}$  is the value of the loss at the model that is identically  $\bar{0}$ , namely  $\hat{L}(\bar{0}) = \frac{1}{n} \sum_{i} y_i^2$ . Thus, for every reasonable model  $f = \omega^T x$ ,  $f \in \mathcal{F} \equiv \hat{L}(\omega) \leq \hat{L}_{\mathrm{max}}$ . On the other hand, the best possible value of the least squares loss is  $\hat{L}_{LS} = 0$ , and therefore we get that  $C w^T w \leq \hat{L}_{\mathrm{max}}$ , or alternatively,  $w^T w \leq \hat{L}_{\mathrm{max}} / C$ . This defines the hypothesis space as an  $\ell_2$ -norm ball in  $m$ -dimensional space, the volume of which we can compute.
To measure the numerator of the Rashomon ratio we will use the Rashomon volume  $\mathcal{V}(\hat{R}_{set}(\mathcal{F},\theta))$  as defined in Semenova et al. [39]. In the case of ridge regression, the Rashomon set is an ellipsoid in  $m$ -dimensions, thus the Rashomon volume can be computed directly. Therefore, we have the Rashomon ratio as the ratio of the Rashomon volume to the volume of the  $\ell_2$ -norm ball that defines the hypothesis space.
Next, we show that under additive attribute noise, the Rashomon ratio increases:
Theorem 7 (Rashomon ratio increases with noise for ridge regression). Consider dataset  $S = X \times Y$ ,  $X$  is a non-zero matrix, and a hypothesis space of linear models  $\mathcal{F} = \{f = \omega^T x, \omega \in \mathbb{R}^m, \omega^T\omega \leq$
$\hat{L}_{\max} / C\}$ . Let  $\epsilon_{i}$ , such that  $\epsilon_{i} \sim \mathcal{N}(\bar{0}, \lambda I)$  ( $\lambda > 0$ ,  $I$  is identity matrix), be i.i.d. noise vectors added to every sample:  $x_{i}' = x_{i} + \epsilon_{i}$ . Consider options  $\lambda_{1} > 0$  and  $\lambda_{2} > 0$  that control how much noise we add to the dataset. For ridge regression, if  $\lambda_{1} < \lambda_{2}$ , then the Rashomon ratios obey  $\hat{R}_{ratio_{\lambda_1}}(\mathcal{F}, \theta) < \hat{R}_{ratio_{\lambda_2}}(\mathcal{F}, \theta)$ .
The proof of Theorem 7 is in Appendix F. Note that while in Theorem 7 we directly show that adding noise to the training data is equivalent to stronger regularization leading us directly to Step 3, Steps 1 and 2 of the path identified in the previous section are automatically satisfied. We formally prove that additive noise still leads to an increase of the variance of losses for least squares loss (similar to Theorems 2, 12, and 15) in Theorem 19 in Appendix F, and we show that an increase in the maximum variance of losses leads to worse generalization bound (similar to Theorem 4) for the squared loss in Theorem 20 in Appendix F.
Returning to the Path. For ridge regression, we have now built a direct noise-to-Rashomon-ratio argument showing that, in the presence of noise, the Rashomon ratios are larger. As before, for larger Rashomon ratios, there are multiple good models, including simpler ones that are easier to find.
6 Rashomon Set Characteristics in the Presence of Noise
Now we discuss a different mechanism for obtaining larger Rashomon sets. Suppose the practitioner knows the data are noisy. They would then expect a large Rashomon set, which we speculate in this section is explained by noise in the data. To show this, we define pattern diversity as a characteristic of the Rashomon set and show that it is likely to increase with label noise. Pattern diversity is an empirical measure of differences in patterns on the dataset. It computes the average distance between the patterns, which allows us not only to assess how large the Rashomon set is but also how diverse it is. Once the practitioner knows they have a large Rashomon set, they could hypothesize from the reasoning in Semenova et al. [39] that a simple model might perform well for their dataset.
6.1 Pattern Diversity: Definition, Properties and Upper Bound
Recall that  $\pi (\mathcal{F},\theta)$  is the set of unique classification patterns produced by the Rashomon set of  $\mathcal{F}$  with the Rashomon parameter  $\theta$
Definition 8 (Pattern diversity). For Rashomon set  $\hat{R}_{set}(\mathcal{F},\theta)$ , the pattern diversity  $\operatorname{div}(\hat{R}_{set}(\mathcal{F},\theta))$  is defined as:
$$
div(\hat{R}_{set}(\mathcal{F},\theta)) = \frac{1}{n  \Pi  \Pi}\sum_{\substack{j\leq \Pi \\ p_{j}\sim \pi (\mathcal{F},\theta)}}\sum_{\substack{k\leq \Pi \\ p_{k}\sim \pi (\mathcal{F},\theta)}}H(p_{j},p_{k}),
$$
where  $n = |S|$ ,  $\Pi = |\pi(\mathcal{F},\theta)|$ , and  $H(p_j,p_k) = \sum_{i=1}^{n}\mathbb{1}_{[p_j^i\neq p_k^i]}$  is the Hamming distance (in our case it computes the number of samples at which predictions are different), and  $|\cdot|$  denotes cardinality.
Pattern diversity measures pairwise differences between patterns of functions in the pattern Rashomon set. Pattern diversity is in the range  $[0,1)$ , where it is 0 if the pattern set contains one pattern or no patterns. Among different measures of the Rashomon set, the pattern diversity is the closest to the pattern Rashomon ratio [38] and expected pairwise disagreement (as in [6]). In Appendix G, we discuss similarities and differences between pattern diversity and these measures.
Given a sample  $z_{i}$ , let  $a_{i} = \frac{1}{\Pi}\sum_{k=1}^{\Pi}\mathbb{1}_{[p_{k}^{i}=y_{i}]}$ , where  $p_{k}^{i}$  is  $i^{\text{th}}$  index of the  $k^{\text{th}}$  pattern, denote the probability with which patterns from the pattern Rashomon set classify  $z_{i}$  correctly. We will call  $a_{i}$  sample agreement over the pattern Rashomon set. When  $a_{i} = 1$ , then all patterns agreed and correctly classified  $z_{i}$ . If  $a_{i} = \frac{1}{2}$ , only half of the models were able to correctly predict the label. As we will show, when more samples have sample agreement near  $\frac{1}{2}$ , we have higher pattern diversity. We can compute pattern diversity using average sample agreements instead of the Hamming distance according to the theorem below.
Theorem 9 (Pattern diversity via sample agreement). For 0-1 loss, dataset  $S$ , and pattern Rashomon set  $\pi(\mathcal{F}, \theta)$ , pattern diversity can be computed as  $\text{div}(\hat{R}_{\text{set}}(\mathcal{F}, \theta)) = \frac{2}{n} \sum_{i=1}^{n} a_i(1 - a_i)$ , where  $a_i = \frac{1}{\Pi} \sum_{k=1}^{\Pi} \mathbb{1}_{[p_k^i = y_i]}$  is sample agreement over the pattern Rashomon set.
The proof of Theorem 9 is in Appendix H. In Theorem 23 in Appendix I, we show that average sample agreement (over all samples  $z_{i}$ ) is inversely proportional to the average loss of the patterns from the pattern Rashomon set. Using this intuition, we can upper bound the pattern diversity by the empirical risk of the empirical risk minimizer and the Rashomon parameter  $\theta$ .
Theorem 10 (Upper bound on pattern diversity). Consider hypothesis space  $\mathcal{F}$ , 0-1 loss, and empirical risk minimizer  $\hat{f}$ . For any  $\theta \geq 0$ , pattern diversity can be upper bounded by
$$
d i v \left(\hat {R} _ {s e t} (\mathcal {F}, \theta)\right) \leq 2 \left(\hat {L} (\hat {f}) + \theta\right) \left(1 - \left(\hat {L} (\hat {f}) + \theta\right)\right) + 2 \theta . \tag {2}
$$
The proof of Theorem 10 is in Appendix I. The bound (2) emphasizes how important the performance of the empirical risk minimizer is for understanding pattern diversity. If dataset is well separated so that the empirical risk is small, then pattern diversity will also be small, as there are not many different ways to misclassify points and stay within the Rashomon set. As dataset becomes noisier, on average, we expect the empirical risk to increase and thus pattern diversity as well. We will show theoretically and experimentally that pattern diversity is likely to increase under label noise.
6.2 Label Noise is Likely to Increase Pattern Diversity
Let  $S_{\rho}$  be a version of  $S$  with uniformly random label noise, creating randomly perturbed labels  $\tilde{y}$  with probability  $0 < \rho < \frac{1}{2}$ :  $P(\tilde{y}_i \neq y_i) = \rho$ . Let  $\Omega(S_{\rho})$  be the uniform distribution over all  $S_{\rho}$ . From Theorem 10 denote the upper bound on pattern diversity as  $U_{div}(\hat{R}_{set}(\mathcal{F}, \theta)) = 2(\hat{L}(\hat{f}) + \theta)(1 - (\hat{L}(\hat{f}) + \theta)) + 2\theta$ . We show that it increases with uniform label noise.
Theorem 11 (Upper bound on pattern diversity increases with label noise). Consider a hypothesis space  $\mathcal{F}$ , 0-1 loss, and a dataset  $S$ . Let  $\rho \in (0, \frac{1}{2})$  be the probability with which each label  $y_i$  is flipped independently, and let  $S_{\rho} \sim \Omega(S_{\rho})$  denote a noisy version of  $S$ . For the Rashomon parameter  $\theta \geq 0$ , if  $\inf_{f \in \mathcal{F}} \mathbb{E}_{S_{\rho}} \hat{L}_{S_{\rho}}(f) < \frac{1}{2} - \theta$  and  $\hat{L}_S(\hat{f}_S) < \frac{1}{2}$ , then adding noise to the dataset increases the upper bound on pattern diversity of the expected Rashomon set:
$$
U _ {d i v} (\hat {R} _ {s e t _ {S}} (\mathcal {F}, \theta)) <   U _ {d i v} (\hat {R} _ {s e t _ {\mathbb {E} _ {S _ {\rho} \sim \Omega (S _ {\rho})} S _ {\rho}}} (\mathcal {F}, \theta)).
$$
The proof of Theorem 11 is in Appendix J. In the general case, it is challenging to find a closed-form formula for pattern diversity or design a lower bound without strong assumptions about the data distribution or the hypothesis space. Therefore, we empirically examine the behavior of pattern diversity alongside other characteristics of the Rashomon set for different datasets and show these characteristics tend to increase with more noise.
6.3 Experiment for Pattern Diversity and Label Noise
We expect many different datasets to have larger Rashomon set measurements as data become more noisy. As before, we consider uniform label noise, where each label is flipped independently with probability  $\rho$ . For different noise levels, we computed diversity, and the number of patterns in the Rashomon set for 12 different datasets for the hypothesis space of sparse decision trees of depth 3 (note that we underfitted for some of the datasets); see Figure 3 (a)-(b). For every dataset, we introduced up to  $25\%$  label noise ( $\rho \in [0,0.25]$ ), or Accuracy( $\hat{f}$  -  $50\%$ , whichever is smaller. This means that data with lower accuracy (before adding noise) will have shorter plots, since noise is along the horizontal axis of our plots in Figure 3. For each noise level  $\rho$ , we performed 50 draws of  $S_{\rho}$  and for each draw  $S_{\rho}$  we recomputed the Rashomon set. For decision trees we used TreeFARMS [52], which allows us to compute the number of trees in the Rashomon set. We set the Rashomon parameter to  $5\%$ . We discuss results for the hypothesis space of linear models in Appendix K.5.
Some key observations from Figure 3: First, the number of trees and patterns in the Rashomon set on average increases with noise. This means that the Rashomon ratio and pattern Rashomon ratio increase with noise as well since the hypothesis space stays the same. Second, datasets, that initially had higher empirical risk (e.g., COMPAS, Coffee House, FICO) tend to have more models in the Rashomon set (and thus higher Rashomon ratios) as compared to datasets with lower empirical risk (Car Evaluation, Monks1, Monks3). Finally, pattern diversity, on average, tends to increase with noise for the majority of the datasets.
(c)
Figure 3: Rashomon set characteristics such as the number of trees in the Rashomon set (Subfigure a), the number of patterns in the Rashomon set (Subfigure b), and pattern diversity (Subfigure c) tend to increase with uniform label noise for hypothesis spaces of sparse decision trees. For readability, the top row of the figure shows datasets with lower empirical risk and the bottom row shows datasets with higher empirical risk.
Returning to the Path. If the practitioner observes more noise in the data, it could be already the case that the Rashomon set is large. Then, there are many good models, among which simpler or interpretable model are likely to exist [39].
While we believe our results have illuminated that the Rashomon effect often exists when data are noisy, the connection between noise and increased Rashomon metrics is mostly supported by experiments, rather than a tight set of theoretical bounds. Specifically, we do not have a lower bound on diversity nor a correlation between diversity and the pattern Rashomon ratio (or Rashomon ratio) yet. This connection deserves further exploration and strengthening.
In previous sections, we showed that we expect an increase in pattern diversity while adding more label noise. In Section 6.3, we also observe an increase in the number of trees and patterns in the Rashomon set under label noise. In both cases, we used 0-1 loss. While in Section 5 we studied ridge regression (and thus used squared loss), ways to extend our results (both experimental and theoretical) to different distance-based classification loss functions are not yet fully clear. In the case of classification with distance-based losses, such as exponential loss, the effect of label noise can be difficult to model without taking properties of the data distribution into account. For example, regardless of the amount of noise in the dataset, a single misclassified outlier could essentially define the Rashomon set. The exponential loss could be very sensitive to the position of this outlier, leading to a small Rashomon set. This does not apply to the analysis in this paper, since the 0-1 loss we used is robust to outliers. Perhaps techniques that visualize the loss landscape [26] can be helpful in characterizing the Rashomon set for distance-based classification losses.
Our results have profound policy implications, as they underscore the critical need to prioritize interpretable models in high-stakes decision-making. In a world where black box models are often used for high-stakes decisions, and yet the data generation processes are known to be noisy, our work sheds light on the false premise of this dangerous practice – that black box models are likely to be more accurate. Our findings have particular relevance for critical domains such as criminal justice, healthcare, and loan decisions, where individuals are subjected to the outputs of these models. The use of interpretable models in these areas can safeguard the rights and well-being of these individuals and ensure that decision-making processes are transparent, fair, and accountable.
We gratefully acknowledge support from grants DOE DE-SC0023194, NIH/NIDA R01 DA054994, and NSF IIS-2130250.
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