MaNo: Exploiting Matrix Norm for Unsupervised Accuracy Estimation Under Distribution Shifts
Authors: RENCHUNZI XIE
Ambroise Odonnat
Vasilii Feofanov
Weijian Deng
Jianfeng Zhang
Bo An
Venue: NeurIPS 2024 poster (2024)
Keywords: Unsupervised Learning
Distribution Shifts
Unsupervised Accuracy Estimation
Generalization
Deep Learning
Primary area: safety_in_machine_learning
TL;DR: Exploiting Matrix Norm for Unsupervised Accuracy Estimation Under Distribution Shifts

Abstract: Leveraging the model’s outputs, specifically the logits, is a common approach to estimating the test accuracy of a pre-trained neural network on out-of-distribution (OOD) samples without requiring access to the corresponding ground-truth labels.
Despite their ease of implementation and computational efficiency, current logit-based methods are vulnerable to overconfidence issues, leading to prediction bias, especially under the natural shift. In this work, we first study the relationship between logits and generalization performance from the view of low-density separation assumption. Our findings motivate our proposed method \method{} that \textbf{(1)}~applies a data-dependent normalization on the logits to reduce prediction bias, and \textbf{(2)} takes the $L_p$ norm of the matrix of normalized logits as the estimation score. Our theoretical analysis highlights the connection between the provided score and the model's uncertainty. 
We conduct an extensive empirical study on common unsupervised accuracy estimation benchmarks and demonstrate that \method{} achieves state-of-the-art performance across various architectures in the presence of synthetic, natural, or subpopulation shifts. The code is available at https://github.com/Renchunzi-Xie/MaNo.

MANO: Exploiting Matrix Norm for Unsupervised Accuracy Estimation Under Distribution Shifts
Renchunzi Xie
Ambroise Odonnat"
Nanyang Technological University Singapore renchunzi xxeentu.eddi edu.
Huawei Noah's Ark Lab, Inria Paris, France
ambroise. owwnnattgmail..................m com
Vasilii Feofanov
Weijian Deng
Jianfeng Zhang
Huawei Noah's 's Ark Lab
Australian National University
Huawei Noah's Ark Lab Shenzhen, China
Paris, France
Canberra, Australia weijian deng@anu edu, au
vasilii, feo.iii.gaoaaaaonmaaaai com
banngiaanenn33 com
Bo An Skywork AI Nanyang Technological University Singapore boan@ntu. Coan@ntu.edu..g. edu.sg
Abstract
Leveraging the model's outputs, specifically the logits, is a common approach to estimating the test accuracy of pre-trained neural network out-of-distribution (OOD) samples without requiring access to the corresponding ground-truth la- bels. Despite their ease of implementation and computational efficiency, current logit-based methods are vulnerable to overconfidence issues, leading to prediction bias, especially under the natural shift. In this work, first study the relation- ship between logits and generalization performance from the view of low-density separation assumption. Our findings motivate our proposed method MANO that (1) applies a data-dependent normalization on the logits to reduce prediction bias, and (2) takes the Lp norm the matrix of normalized logits as the estimation score. Our theoretical analysis highlights the connection between the provided score and the model's uncertainty. We conduct an extensive empirical study on common unsupervised accuracy estimation benchmarks and demonstrate that MANO achieves ofte-offthe-art performance across various architectures the presence of synthetic, natural, or subpopulation shifts. The code is available at attps://gittee cow hemm/eenchunzii Kie/MaNo.
Introduction
The deployment of machine learning models in real-world scenarios frequently challenged by distribution shifts between the training and test data. These shifts can substantially deteriorate the model's 's performance during testing (Geirhos et al., 2018; Koh et al., 2021; Quionero-Candela et al., 2009) and introduce significant risks related to AI safety (Deng and Zheng, 2021; Hendrycks and Mazeika, 2022). To alleviate this issue, is common to monitor model performance by periodically collecting the ground truth labels for subset of the current test dataset (Lu et al., 2023). However,
"Equal contribution. Correspondence to: Bo boan@ntu. edu.sg sg.
Univ. Rennes 2, CNRS, IRISA
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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this approach is often resource- intersive and time-consuming, which motivates the importance of estimating the model's performance on ofu-s---iittittiion (OOD) data in an unsupervised manner, also known as Unsupervised Accuracy Estimation (Donmez et al., 2010).
Due to privacy constraints and computational efficiency, one of the most popular ways estimate accuracy without labels is to rely on the model's outputs, called logits, as source of confidence in the model's predictions (Deng et al., 2023; Garg et al., 2022; Guillory et al., 2021; Hendrycks and Gimpel, 2016). For instance, ConfScore (Hendrycks and Gimpel, 2016) leverages the average maximum sof tmax probability as the test accuracy estimator, while Deng et al. (2023) has recently proposed to estimate the accuracy via the nuclear norm of the tmax probability matrix. These approaches. however, tend underperform on the natural shift applications while the intuition behind the use of logits remains unclear. This motivates us to ask:
Question 1: What explains the correlation between logits and generalization performance?
In Section 3, we show that logits are connected to the model's margins, i.e,, the distances between the learned embeddings, and the decision boundaries. Inspired by the low-density separation (LDS) assumption (Chapelle and Zien, 2005; Feofanov et al., 2023) that optimal decision boundaries should lie in low-density regions, we propose MANO, an estimation score that aggregates the margins at dataset level by taking the Lp -orm of the normalized model's prediction matrix to evaluate the density around decision boundaries. Nevertheless, logit-based approaches are known to suffer from overconfidence (Odonnat al., 2024; Wei al., 2022a), resulting in high prediction bias, especially under poorly-calibrate scenarios. This leads us to another critical question:
Question 2: How to alleviate the overconfidence issues of logits- methods?
In Section 4, we reveal that this question is connected to the normalization of logits and show that the widely-used tmax normalization accumulates errors in the presence of prediction bias, which can lead to overconfidence and significantly degrade the performance of existing accuracy estimation methods in poorly-calibrated scenarios. To mitigate this issue, we propose novel normalization strategy called softrun that takes into account the empirical distribution of logits and aims to find a trade-off between information completeness of ground-trute logits and error accumulation.
Summary of our contributions. (1) We show that logits are informative of generalization per- formance through the lens of the low-density separation assumption by reflecting the distances to decision boundaries. (2) We identify the failure of the commonly-use softmax normalization that accumulates errors under poorly calibrated because of its overconfidence, leading to biased estimation. (3) We propose MANO, a training-free estimation method that quantifies the global distances to decision boundaries by taking the Lp norm of the logits matrix. MANo relies on softrun, novel normalization technique that makes a trade-off between information completeness and error accumu- lation and is robust to different calibration scenarios. In addition, demonstrate its connection to the model's uncertainty. (4) We demonstrate the superiority of MANO compared to competitors with large-scale empirical evaluation including 12 benchmarks across diverse distribution shifts. Results show that MANO consistently improves over the state-of-the-ar baselines, including on the challenging natural shift.
2 Problem Statement
Setting. Consider a classification task with input space X' R.D and label space y = (1, (1..... .......... K}. Let ps and pr be the source and target distributions on X' V. respectively, with # pr. We parameterize a neural network f: X - R as f = fw o fop where fp: X - RR9
feature
extractor and fw: Ra + R is a linear classifier with parameters W (wx) E R Further, we denote an input by x, its corresponding label by y. its representation by z = fx)(x) and logits by
q= f(x) (w) z)x K The accuracy of f on defined Acc(f f, D) D E(xDeD Lg=y
with predicted labels ğ. The probability simplex is denoted by Ax {p E [0, 1] 1 1 1}.

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Unsupervised aceuracy estimation. Given a model f pre-trained on a training set Durain with samples drawn i..d. from ps, the goal of unsupervised accuracy estimation assess generaliza- tion performance on given unlabeled test set Dust = {x: with N samples drawn i.i.d. from pr. More specifically, we assume (i) a source-free regime (no direct access to Draain), (ii) no access to test labels, and (iii) distribution shift, i.e. ps * pr. In this challenging setup, which often occurs in real-world scenarios when ground-truth labels are inaccessible at test time, we aim to design an estimation score Sf. Dustt that exhibits a linear correlation with the true OOD accuracy Afc D.est). Following the standard closed-se set setting, both pr and ps involve the same K classes. For an extended discussion of related work on unsupervised accuracy estimation, we refer the reader to Appendix B.
3 What Explains the Correlation between Logits and Test Accuracy?
Although existing literature has shown the feasibility of unsupervised accuracy prediction under distribution shift by utilizing the model's logits (Deng et al., 2023; Garg et al., 2022; Guillory et al., 2021), the reason behind this empirical success remains unclear. In this section, seek to understand when and why logits can be informative for analyzing generalization performance. Based on the derived understanding, we propose approach, MANO, for estimating generalization performance.
3.1 Motivation
Logits reflect the distances to decision boundaries. We analyze logits from linear classification perspective in the embedding space, where the decision boundary of class it is the hyperplane {z' E IR | z' = 0}. In Appendix D.2, we remind that the distance from point z to hyperplane wi is given by d(w,, 2) = w = || As the pre-trained model is fixed and W1: can be normalized, we derive that the logits absolute values are proportional to the distance from the learned embeddings to the decision boundaries, i.e,, |qx1 = |يا z| x d(w), 2), vk. This indicates that the magnitude of logits reflects how close the corresponding embedding is from each decision boundary.
Low-density separation assumption. When dealing with unlabeled data, it is required to make assumptions on the relationship between the dis- tance decision boundaries and gen- eralization performance. The low- density separation assumption (LDS, Chapelle and Zien, 2005) states that optimal decision boundaries should lie in low-density regions (Figure 1) so that unlabeled margin |w z| re- flects reliable confidence in predict- ing x to the class k. The assumption is often empirically supported as the misclassified samples tend to be sig- nificantly closer the decision bound- ary than the correctly classified ones (Mickisch et al., 2020). This might indicate that the absolute values of the logits are positively correlated to its generalization performance.
Seloon
20
becenn bandey
350
35
3.0
23
50
75
13.0
(a) High-density region
(b) Low-density region.
Figure Illustration of the LDS assumption. When the boundary passes through dense regions (a), margins have little predictive power and cannot be used without labels. On the contrary, margins are informative in sparse regions (b).
Assumptions on the prediction bias. It is important to note that the LDS assumption has been initially proposed for semi-supervised learning where labeled and unlabeled data are assumed to come from the same distribution, which is not the case in our setting. This leads to logits writing f(x) q" + in the general case", i.e,, subject to potentially non-negligible prediction bias E = (Ex)& with respect to the ground-truth logits q" E R The following proposition shows the impact of the prediction bias the divergence between the true class posterior probabilities, assumed modeled as p = softmax(q') E Ak. and the estimated ones s = softmax(f(xx) E Ak.
We write this decomposition without loss of generality as restrictions are imposed
3

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Proposition 3.1. Let ++ (maxi { - -. Then, the KL divergence between and verifies
0 (L(plls) S ET.p.
The proposition indicates that a large approximation error of the posterior may be caused by prediction bias that has a large norm and or bad alignment with the true probabilities. Thus, the logit- based methods assume that the magnitude of the bias is reasonably bounded while the direction of bias does not drastically harm the ranking of classes by probabilities. We elaborate on this discussion and present the proof of Proposition 3.1 in Appendix D.1.
3.2 MANO: Predicting Generalization Performance With Matrix Norm of Logits
We have shown a connection between the feature-to-boundary distances and generalization perfor- mance well the impact of the prediction bias. Based on the derived intuition, we introduce MANO that leverages the model margins at the dataset level performing two steps: normalization and aggregation. The pseudo-code of MANO is provided in Appendix A.
Step I: Normalization. Given that logits can exhibit significant variations in their scale depending on the imput x, it is crucial to normalize the logits within standardized range to prevent outliers from exerting disproportionate influence on the estimation. A natural range stems from the fact that most deep classifiers have outputs in Ak. which amounts applying normalization function a: IR + Ax on top of the pre-trained neural network (Mensch et al., 2019), where Ax refers probability simplex. This ensures having logits entries in [0, 1]. For each test sample x;, first extract its learned feature representation z; (x:). Then, logits corresponding to this representation are computed as qi = fw (z:) E RRK The normalization procedure results a prediction matrix IR Nx with each row Q; containing the normalized logits of input sample:
Q = ( E AA.
(1)
where a denotes the normalization function for the logits values. It is worth noting that not all normalization methods are appropriate candidates. The selection of suitable normalization function based on different calibration scenarios will be discussed in detail in Section 4.
Step 2: Aggregation. Once the logits are scaled, we aggregate the dataset-level information on distances by taking the entry-wise Lp norm of the prediction matrix Q, which can be expressed as:
N
K o (q)))
S(f, Drea) = VNK 1Q1p = NK Σ
)
(2)
As we have ||Q|pp S AKK max(Q.,) VNK (Qi E [0, the scaling by VNK leads to S(f. Drea) E J0. 1], providing a standardized metric regardless of variations in the size of the test dataset and the number of classes K. As p increases, MANO puts greater emphasis on high-margin terms, focusing on confident classification hyperplanes. In the extreme case where + oc, we have [Qll + max (Qiz). ). In practice, we choose p = in all experiments and provide an ablation study on p in Appendix G. 1. As the Lp norm is straightforward to compute, our approach is scalable and efficient compared to the current state-of-the-ar method Nuclear (Deng et al., 2023) that requires performing singular value decomposition.
3.3 Theoretical Analysis of MANO
In this section, we provide the theoretical support for the positive correlation between MANO and test accuracy. More specifically, we reveal that our proposed score is connected with the uncertainty of the neural network's ' predictions Theorem 3.3. Before presenting this result, we recall below the definition of Tsallis a-entropies introduced in Tsallis (1988).

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Offic no M6C5
Naceer
1.00
MCS Citccotome
Vahlo nt Softmax
10
Pañto of Tayler
probetniig
0.
6.75
0.70
0.4
0.55
080
D.6 Predicted probability
Df1 co-Tomm
RACS
0.0
(a) Calibration curves.
(b) Type of normalization.
(c) Order in Eq. (4).
Figure 2: Empirical evidence with Resnet18. The model is well-calibrated on Office-Home and miscalibrated on PACS. (b) sof is superior to the state-of-the-art Nuclear (Deng et al., 2023) in all scenarios while the sof tmax heavily fails on PACS. (c) Increasing the approximation order in Eq. (4) is detrimental on PACS and beneficial on Office-Homes The optimal trade-off in all calibration scenarios taking (2.3).
Definition 3.2 (Tsallis a-entropies (Tsallis, 1988)). Let a >> and > 0. The Tsallis o-entropy is defined
H
- 1)- ||
In this work, we choose k 1 following Blondel et al. (2019). The Tsallis entropies generalize the Shannon entropy (limit case + 1) and have been used in various upplications (Blondel et al., 2019, 2020; Muzellec et al., 2017). More details can be found in Appendix C. The following theorem, whose proof deferred to Appendix D.3, states that estimation score obtained with MANO is function of the average Tsallis entropy of the normalized neural network's logits.
Theorem 3.3 (Connection uncertainty). Let a= plp-1) and b Given test K Drest {x, /i=1 corresponding logits q, fxx,), a normalization function a: R AK
and p > 1, the estimation score S(f. Drest) provided by MANo (Algorithm 1) verifies
S(ʃ, Dusst
Σ H
H (a((a))) b.
(3)
N
2=1
As >> 0, Theorem 3.3 implies that the estimation score provided by MANO is negatively correlated with the average Tsallis- on the test In particular, the less certain the model on test data, the lower the test accuracy is and the higher the entropy term is in Eq. (3), resulting in lower score S(f,D Drest). ). As the converse sense holds, MANO provides a score positively correlated the test accuracy. This follows the findings of Guillory et al. (2021); Wang et al. (2021) and empirically confirmed Section 5 for various architectures, datasets, and types of shift.
How to Alleviate Overconfidence Issues of Logit-Baser Methods?
The most common normalization technique of existing logit-based approaches is the softmax normalization. In this section, we show that the widely used amax is sensitive to prediction bias, which hinders the quality of the estimation in poorly calibrated scenarios. To alleviate this issue, we propose a novel normalization strategy, softrun, which balances the information completeness and overconfidence accumulation based on calibration.
4.1 The Failure of Softmax Normalization Under Poorly-Calibrated Scenarios
It is widely known that the sof tmax normalization can suffer from overconfidence issues (Odonnat al., 2024; Wei al., 2022b) and saturation of its outputs (Chen al., 2017), with one entry close to one while the others close to zero.
5

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Analysis. To alleviate those issues, we first notice that the softmax can be decomposed as softmax(q) exp(a)/ Σk=1 exp(q=) (ø exp) where Φ: I& AK writes ((u) u/ Ei= in = 1/ While & has appealing property for normalization (see Proposition D.2), the exponential can accumulate prediction errors, leading to the tmat overconfidence and a biased accuracy estimation. In particular, assume that the k-th entry of the output of the neural network on a test sample x; writes Ai Ex, where 97 the ground-truth logits and EY is the prediction error.
Then, the th -order Taylor polynomial of the exponential writes
(9), + Ex)²
(qf + Ex)"
x R 1+ + (q +xx) +
(4)
2!
n!
Consequently, logit-based aceuracy estimation methods using sof tmax are sensitive to prediction bias, leading to low-quality estimations in poorly calibrated scenarios.
Empirical evidence. We illustrate this phenomenon in Figure 2(a) on two datasets, where a pre- trained 8 exhibits more pronounced calibration issues on PACS (Li et al., 2017) compared to Office Home (Venkateswara al., 2017). Figure 2(b) shows that using MAÑO with sof softmax normalization is appropriate on Office- Home where the model is well calibrated but not in mis- calibrated scenario on PACS. Conversely. using MANO with 2nd -odder Taylor approximation is appropriate under miscalibration on PACS but not on Office-Home. In both cases, we see that MANO can surpass the state-of-the-art method Nuclear (Deng et al., 2023) provided it uses the appropriate normalization order. We conclude that a trade-off is needed between information completeness on true logits and error accumulation depending the type of calibration scenario. Specifically, when the model is poorly calibrated on a given dataset (i.e., E/ large in absolute value), the normalization should focus on avoiding error accumulation, and when the model is well calibrated (i.e., Ey small in absolute value), the normalization should focus on information completeness.
Figure 2(c) illustrates the impact of truncating Eq. (4) up to the n-th
4.2 Softrun: The Proposed Normalization Strategy
The above analysis shows that different calibration scenarios emphasize different information during normalization. Therefore, we propose normalization strategy called sof trun that normalizes the model outputs based on the calibration scenario. Given logits qi E IR and reusing the function ø previously introduced, it takes the general form:
v(qi)
o (q))) (Ø (q:)) Σk= v(g)s.
E AAK-
(5)
where v: RRK + R is designed to avoid error accumulation under poorly-calibrated scenarios by truncating the exponential (Taylor n 2 in Eq. (4)) and using complete logits information under well-calibrated scenarios (softmax). As in practice, the calibration of the model on test data is unknown, softrun employs a simple yet effective strategy reminiscent of pseudo-labeling (Lee, 2013; Sohn et al., 2020). More specifically, given test dataset Dreet = {x N. and corresponding logits q, fx;),, criterion (Drest) is computed the dataset level and the normalized logits are defined as
e(q,) = 1 q 9 if (Drest) S 17 exp(c)),
(Taylor)
(6)
if (D vest) ) > v)
(sof
exp(q;)= We define (DDrest) = NK Ex= E log( Σ _x,, exp( which is equal, up to constant, to the average KI. divergence between the uniform distribution and the predicted sof tmax probabilities. follows from Tian et al. (2021) that showed that this KI. divergence was small when the uncertainty of the model was high and large for confident models. Hence, when the model is uncertain, i.e., & (Deest < 77, we truncate the exponential to reduce error accumulation, and when the model is certain, i.e., @ > 77, complete information is used with the exact exponential (and we recover the sof tmax). Thus, softrun is designed to treat the problem with an additional level of complexity often overlooked by previous methods. While this comes the cost of introducing the hyperparameter 77. we fix 7 = across all our experiments. This, along with the design of softrun, is justified both theoretically in Appendix E and experimentally in Appendix G.3.
Inn practice if Taylor is applied, we replace v(q:) by v(q.) min ((q:) to make sure the final outputs have nonnegative entries. This is especially needed for approximation orders n 2 3.

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Table I: Method comparison four benchmarks using ResNet18, ResNet50, and WRN-50-2 under the synthetic shift, where R refers to coefficients of determination, and p refers to the absolute value of Spearman correlation coefficients (higher is better). The best results for each metric are in bold. Overall, MANO achieves the highest Rt2 and p values across different datasets and architectures, indicating its superior performance.
Systheic Shilt
Rotazon
ContScurs
Exenay
Apuedione
Frödect
Dapanics
PrefNems
MDD
Nodca
MANo
0
0951
CIFAR
RoNes50
0.901
3.9
0.916
0.985
0.9
6/417 0.91
099
0.996
0.881
0.665
0,95
0.965
097
0.980
0.925
0.991
0.991
0.995
0MH
0.915
6.9%
CHEAR 100
WRN--002
0.000
09919
0.961 0:861
0,981 0.955 0.977
0997
0/922 0.667
01651
0,993
0.77
0,967
0972
0.965
0.991
0191
0,911
0.966
0.991
0.91
0.996
0,997
09N
01911
0.008
0,99
6.99
0.9%
0.907
0.983
0.988
0.906
0,907
0.991
0.993
6,997
R
0.795
0.900 0.670 0,807 0.912 0.542 0.561 0.853 0.751
0.945
0,235
0,900
0.005
0.996
0.970
0,991
0.041
0.993
8:985
0.094
0.983
0,994
0.001
8.0%
Thyhrogehi
RoNNe50
0.795
0.947 0.670 0.800 0:651 0,892
0.500 0.853 0.751
0.945
0:235
0.971
0.977
0.996
0.979
0,947
0.541
0.993
0.000
0,994
0.965
0.994
0.000
8.09
WRN-5002
0.778
0.667
0.257 0.051 0.714 0.85 0.64 0,004
0.655
0,997 0,864
0,664
0.065
0.990
0.665
0,643
0.061
0,596
0.585
0,997
0.662
0.885
0.59
8,597
Arstis
0,200
(3009
00/27 (1703) (1424 0.900 (,997
0,015
03722 0,577 037/
0,730
01587
0.9/2
0055
0.050
03713
0:344
03700
0.005
0370 0.00)
$359
Kosvett
0.999 (,MM) 0.965 0/M1
0.99 0.063
0,912
0.9/4
(149)
5.971
01775
0/M3
0,124
0.M9
0.5%
8.59
0/91
0/97
0/M2
6.977
InngeNi
0.000 0,094 0.007 0,992
0.500
0.003 0.855 0,074 0.008
0.005 0.996 0,903
0.805
0.094
0.593
0.995
0.983
0,997
0.991 0.098
WRN 29-2
0.461 0.9M 0.413 OWL
0.413
0/W1 0MM 0.4 0MW 6.9 09755 0/W1 0.009
0,90
0:M9
0.941
0.477
0.91 0,994 8.99
Average
0.061 0.90
3,90
0,92
0.96
6.007
0.855
0.999 0,579
0.972 3.981
0/91 0.097
0.915
0.915
0.995
0.955
0,991
0.99
5 Experiments
In this section, we conduct experiments with MANO that uses softrun to properly normalize logits.
5.1 Experimental Setup
Pre-training datasets. For pre-training the neural network, we use a diverse set of datasets includ- ing CIFAR-10, CIFAR- 100 (Krizhevsky and Hinton, 2009), TinyImageNet (Le and Yang, 2015), ImageNet (Deng et al., 2009), PACS (Li et al., 2017), Office-Home (Venkateswara et al., 2017), DomainNet (Peng et al., 2019) and RRI-WILDS (Taylor et al., 2019), and BREEDS (Santurkar et al., 2020) which leverages class hierarchy of ImageNet (Deng et al., 2009) to create 4 datasets including Living-17, Nonliving-26, Entity-13 and Entity-30.
Test datasets. In our comprehensive evaluation, we consider 12 datasets with 3 types of distribution shifts: the synthetic, the natural, and the subpopulation shifts. To verify the effectiveness of our method under the synthetic shift, we use CIFAR- 10C, CIFAR-10OCE and ImageNet-C (Hendrycks and Dietterich, 2019) that span 19 types of corruption across 5 severity levels. well as TinylmageNet-C (Hendrycks and Dietterich, 2019) with 15 types of corruption and severity levels. For the natural shift, we use the domains excluded from training from PACS, office-Home, DomainNet, and RR1- WILDS as the OOD datasets. For the novel subpopulation shift, we consider the BREEDS benchmark with Living- 17, Nonliving 26, Entity- 13, and Entity-30 which were constructed from ImageNet-C.
Training details. To show the versatility of our method across different architectures, we perform experiments with ResNet18, ResNet50 (He et al., 2016), and WRN-50-2 (Zagoruyko and Komodakis, 2016) models. We train them for 20 epochs for CIFAR- 10 (Krizhevsky and Hinton, 2009) and 50 epochs for the other datasets. In all cases, use SGD with learning rate of 10-3, cosine learning rate decay (Loshchilov and Hutter, 2016), a momentum of 0.9, and a batch size of 128.
Evaluation metries. We use the coefficient of determination R2 E [0. 1] (Nagelkerke et al., 1991) and the Spearman's 's rank correlation coefficient & E [-1. 1] (Kendall, 1948) to evaluate performance. The former measures the linearity and goodness of fit and indicates a perfect fit. / measures monotonicity and values close to {-1,1} 1} indicate strong correlation while 0 indicates no correlation.
Baselines. We consider 11 baselines commonly evaluated in the unsupervised accuracy estimation studies, including Rotation Prediction (Rotation) (Deng et al., 2021), Averaged Confidence (ConfS- core) (Hendryeks and Gimpel, 2016), Entropy (Guillory et al., 2021), Agreement Score (AgreeScore) (Jiang et al., 2021), Averuged Threshold Confulence (ATC) (Garg et al., 2022), AutoEval (Fréchet) (Deng and Zheng, 2021), ProjNorm (Yu et al., 2022), Dispersion Score (Dispersion) (Xie et al., 2023), MDE (Peng et al., 2024), COT (Lu al., 2024), and Nuclear Norm (Nuclear) (Deng et al., 2023).
7

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Table 2: Method comparison four benchmarks using ResNet18, ResNet50, and WRN-50-2 under subpopulation shift with R² and p metrics (the higher the better). The best results for each metric in bold. Overall, MANO surpasses all its competitors.
Sunpoputatne Stin
Earury
AgresSoree
Deperioe
PregNoem
11
Acrrage
Recirit8
09961
0.500
0:991
0:511
0,813
0.513
0611
01815
9.9%8
0.961
5/997
0:931
D494
D:571
(190%
(1900
0.991 8.99
ReaNe:00
W8SS5502
0.00
BeNSei88
0.66
0,973 0,013 0.733
0:891
0.970
0.515
0,736
0,540
0.913
0.718
0,000
0.000
0.058
3/999
(970
0:921
0995
0.972
0.964
0.975
0.067
0.500
0.09
Liitg 17
MaNNe33
11966 13.89 0.880 0/67
51811
(9961
01531
11902
US538
11916
01381
0.12)
2/112
19124
0/914
11985
11953
11915
11997
01906
0,575
WBN-302
0.87 0.624 0591
0/921
69977
16793 11566
0.184
03511
01067
0151
0,00
2013
E1314
11983
11560
(1550)
0.505
0.961
0.99
0.955
0.907
0/814
0.915 0:533
0:111
(1919
1.96)
0.995
ResNei8
0.900 0.905 0.781 0.925 0.738
0.909
0,543
0.510
0.554 0.079
0.914 0.000 0.958 0.981
0:939
(1978
0:929
0/989 0.963 0.992 0 0.970 0.009 0.978
0.991
Nontsing--2 26
WgN 0.917 0.977 0,082 0971 0012 0,599
Reckeet)
01916 0.900 0.807 0661 0.77 0,91 0.538
0.75 0,561 0.45 0.969 0.585 0,042 0,203 0,099
0.537 0.590 0.900 0.548 0.990 0 0.50
0.907
5924
0:873
0973 0.999 0.99) 0.562 0,579 0.560 0.568 0,678 0,592
(1917 0:967
01983 0.563 0.564 i 956 01985 0.995
0.095
Tostage 0.913 0.007 0,807 0040 1000
0333
0233 0.331 01157 0356
0352
0300 01113 0.74
1,89
(043 (0313
(091
10300 (1555 (1.002 01561
0.977
0.902
5.2 Main Takeaways
MANO improves over state-of-the-art. Tables and 2 present the numerical results of unsuper- vised accuracy estimation across 8 datasets using 3 different network architectures, evaluated under synthetic and subpopulation shifts. These shifts are quantified by R² and p. Empirical results demon- strate that these distribution shifts do not significantly impact calibration (i.e., (Duust) > 11). We observe that MANO consistently outperforms other baselines, achieving state-of-the-art performance. For instance, MANO achieves R² > 0.960 and p > 0.990 under subpopulation shift, whereas the performance of other baselines does not reach such consistently high levels.
MANO significantly boosts performance under the natural shift. Table 3 illustrates the results of accuracy estimation under the natural shift on four datasets. Under the challenging and natural shift that is more complex than the other distribution shifts, we empirically observe (DDeest) S n-- From Table 3, we observe a significant improvement compared with the other baselines. In particular, most methods have an R and & under 0.9 while MANO reaches higher values. In addition, our method achieves the best performance on average on all four datasets. To visualize the estimation performance, we provide the scatter plots for Dispersion Score, ProjNorm and MANO in Figure on Entity-18 18 with ResNet18. We find that MANO scores present a robust linear relationship with ground-truth OOD errors, while the other state-of-the-e baselines tend to exhibit a biased estimation of high test errors.
ConfScore v.s. Accuracy
ProjNorm v.s. Accuracy
MaNo v.S. Accuracy
Accuracy
70
Test Accuracy
Test Accuracy
60-
60-
50-
50
40
40
40
0.6
0.7
0.8
0.3
7.4
7.0
6.8
9.14
0.35
0.16
0:37
ContScore
Negative ProjNom
MaNo
Figure 3: OOD error prediction versus ground-truth error on Entity-13 with ResNet18. This scatter plot compares MANO with Dispersion Score and ProjNorm. Each point represents one dataset under specific type and severity of corruption. Different shapes indicate different types of corruption, while darker colors indicate higher severity levels. This indicates the qualitative superiority of MANo.
Improved robustness. Figure presents a box plot showing the estimation robustness across different distribution shifts on all datasets except ImageNet, using ResNet18. Results for ImageNet are excluded due to the lack of Rotation and AgreeScore data for this dataset, as these two methods require retraining the networks. We observe that the estimation performance of MANO is more stable than other baselines across three types of distribution shifts. Additionally, MANO achieves the highest median estimation performance.
8

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Table 3: Method comparison on four benchmarks using ResNet18, ResNet50 and WRN-50-2 under natural shift with Rº and metrics (the higher the better). The best results for each metric are in bold. Overall, MANO surpasses all the other baselines.
Natrral Stin
Cantfeere
Earngy
Agrodoome
Dapenios
PrefNven
RACS
Asarage
08832
0.795
0.90
0:26
0.811
0/51
01166
0.511
0.555
0.00%
0.795
0.651
0/61
019
0/017
0.311
DA50 01853
0.703
0,906 6.900
Office Hooee
W8NS550
Re38c:0
0.525
0.851
0.874
0.574
0.16
0.915
Reckenti
11.766 11,602 (1677)
0.776
0.121
0/003
0.135
11:25
11597
01900
0,312
0.516
0.219
0/511
0.358
D/415
11597
11910
11.75
11.709
0.90
DomazzNet
MesNez21
11338
11.100 (357))
0.544
0.821
0/433
M697
11345
14404
11872
0161
0.231
0.43)
0.719
(1:2)
11:000
11927
01.77
0,51 0.590
W-NN-30-2
(1540)
01.122 0.7714
0.874
0,711
0.943
0411
U1546
0.585
03551
0104
04554
0,963
0.405
0.529
00713
11,90
11915
01:50
0,911
0,500
0.778
Aurrage
(1195.
(0:61
0.091
0/683
0.639
01369
0.315
0.567
01919
0.9
ReSSc18
0.531 1.000 0.951
1.000
0.836
1,000
0:919
1,000
0.342
0.500
0,956
1.000
0.563
1000 0,859
1,000
0.927
1,000
0.569
1.000
0.555
1.000 0.583
1,00
BRI-WILDS
W8N-5502
RecSec0
0,081
(1740 1.000 0.918
0,500 0.041
1.00
100
0,946
0/819
1,000
1,000
0/45
0:931
1,000
1.000
0.55
11966
1.000
1.000
0,922
0.955
1,000
1000
0.524
0.737
1,000
1000
0.807
0,874
1,000
1,000
0:354
0.938
1,800
1,000
0:500
0.934
1.000
1.000
0.540
8.006
1.000
1.000
0.978
0,500
1,00
1,00
03330
01335
03007
1100)
1/000
0033
1,000
10.15
01315
00:51
1,00
00300]
0.233
0,303
12000
(134)
12000
00333
1,000
01317
1.00
0.977
1290
1.0
ResNets
ConvNext
ViT
1.0-
0.9
0.4
0.8
0.8
0.7-
koaaton
Confcore
Entragy
Agreet
ATC
Frecheet
Prafnarm
Disperston
Nucletr
MaNo
0.6-
Methods
ConfScore Nuclear
MaNo
Figure 4: R2 distribution with & on all distribution shifts. Overall, MANO leads to the best and most robust estimations.
Figure 5: R² distribution using ResNets (av- erage), ConvNext, or ViT on all distribution shifts. Again, MANO is the best method.
5.3 Additional Experiments
In this section, we discuss the results obtained with additional architectures, the ablation studies we conducted to validate our implementation choices, and the generalization capabilities of MANO.
Beyond ResNets. To demonstrate the efficiency and versatility of MANO, we conduct experiments on recent models such Vision Transformers (Dosovitskiy, 2020, ViT) and ConvNeXt (Liu et al., 2022). We compare MANO to its best competitors on 6 datasets for 3 distribution shifts in Figure 5. The full results are gathered in Table 5 of Appendix F. We note that ConfScore is particularly strong with ConvNexts while MANO works the best with ResNets and ViT. Again, we observe that MANO the best method overall.
Ablation study. To motivate our choices of implementation, we provide in Appendix G ablation studies on the Lp norm and the Taylor order n well as sensitive analysis the calibration threshold n7
Generalization capabilities of MANO. To verify the generalization capabilities of MANO, we utilize designed scores calculated from ImageNet-C and their corresponding accuracy to fit linear regression model. This model is then used to predict the test accuracy on ImageNet- V2-C., which is generated using the 10 new corruptions provided by (Mintun et al., 2021) on ImageNet-Vas V2 (Recht et al., 2019). These new corruptions are perceptually dissimilar from those in ImageNet-CC, including warps, blurs, color distortions, noise additions, and obscuring effects. Figure 6 shows that ConfScore and Dispersion give two distinct trends, while Nuclear exhibits some deviations for ImageNet-V2? V2- C. In comparison, our MANO exhibits consistent prediction pattern for both ImageNet-C and V22-C, aligning well with the linear regression model trained on ImageNet-C. Additionally,

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ConfScore v.S. Test Accuracy
Dispersion v.s. Test Accuracy
Nuclear v.S. Test Accuracy
MaNo V.S. Test Accuracy
80
ImngchetC
imagehebb122
60
70
Image Vak
70
ImagshitiC ImagelieVa
aee
60-
magalistV2C
60
50
10
MMA--2.38
HAB- 9.342
30
30
20
MA5-3.908
7.5
65
75
ConfScore
Dispersion
Nudear
MaNo
Figure 6: Comparison of generalization capability across four methods. Each subplot displays a linear regression model fitted on ImageNet-C, which is used to predict the accuracy on ImageNet-V2-C. The mean absolute error (MAE) is reported. All experiments are conducted using ResNet18.
experimental results on ImageNet-C and ImageNet-C, generated from the validation set ImageNet, are provided in Appendix G.5, further demonstrating the superiority of MANO.
5.4 Discussion
In this section, we discuss how MANO can be applied in practice, the benefit of combining sof trut with other estimation baselines, and the limitations of our approach.
Real-world applications. Appendix H, we discuss how MANO can be used in real-world applications. In particular, our additional results with the Mean Absolute Error (MAE) metric confirm the superiority of MANO.
Can softrun enhance other logit-based methods? To study this, we conducted an ablation study by equipping softrun with Nuclear (Deng et al., 2023), ConfS- core (Hendrycks and Gimpel, 2016), and our MANO. In Table 4, we observe that softrun significantly enhances the estimation perfor- mance R² of Nuclear. For example, Nuclear is improved from 0.692 to 0.826 poorly- calibrated Office-Home.
Table Impact of softrun on other logit-based methods. softrun significantly boosts the perfor- mance of Nuclear. The metric used is R².
ConfScore
MANO
Nuclear
Dataset
w/o
w/
w/o
w/
w/o
w/
PACS
0.594
0.574
0.541
0.827
0.609
0.851
Office-Home
0.795
0.829
0.929
0.926
0.692
0.826
Limitations. Despite its soundness and strong empirical performance, we acknowledge that our method has potential areas for improvement. One of these the dependence on of the selection criterion in Eq. (6). We elaborate this discussion in Appendix E.3. In future work, will explore smoother way to automatically select the optimal normalization function without requiring hyperparameters. Additionally, if multiple validation sets provided, as in (Deng et al., 2021; Deng and Zheng, 2021), we could select based on those sets.
Conclusion
In this paper, we introduce MANO, a simple yet effective training-free method to estimate test accuracy in an unsupervised manner using the Matrix Norm of neural network predictions on test data. Our approach is inspired by the LDS assumption that optimal decision boundaries should lie in low-density regions. To mitigate the negative impact of different distribution shifts on estimation performance, we first demonstrate the failure of softmax normalization under poor calibration, due to the accumulation of overconfident errors. We then propose normalization strategy based on Taylor polynomial approximation, balancing logits information and error accumulation. Extensive experiments show that MANO consistently outperforms previous methods across various distribution shifts. This work highlights that logits imply the feature-to-bounne impact of calibration on estimation performance. We hope our insights inspire future research to explore the relationship between model outputs and generalization.
distance and considers the
10

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Acknowledgements
Ambroise Odonnat would like thank Alexandre Ramé and Youssef Attia El Hili for the fruitful discussions that led to this work. The authors thank the anonymous reviewers and meta-reviewers for their time and constructive feedback. This work was enabled thanks open-source software such as Python (Van Rossum and Drake Jr, 1995), PyTorch (Paszke et al., 2019) and Matplotlib (Hunter, 2007). This research is supported by the National Research Foundation Singapore and DSO National Laboratories under the Al Singapore Programme (AISGAward No: AISG2-GG--003-009)
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Appendix
Roadmap. We provide the pseudo-code of MANO in Appendix A. We discuss related work Appendix and provide some background on Tsallis entropies Appendix C. Appendix contains detailed proofs of our theoretical results. Additional discussion and theoretical insights into Section are given in Appendix E. In Appendix F, we conduct experiments with ViT and ConvNext architectures and provide thorough ablation study and sensitivity analysis in Appendix G. Finally, we explain how MANO can used in practice in Appendix We display the corresponding table of contents below.
Table of Contents
Pseudo-Code of MANO
18
B Extended Related Work
18
Background on Tsallis Entropies
19
D Proofs
19
D.I Impact of Prediction Errors
19
D.2 Distance to the Hyperplane
D.3 Proof of Theorem 3.3
21
D.4 Properties of d
21
Theoretical Insights into Criterion @ (uest)
22
E.1 Choice of Criterion (DDret)
22
E.2 Choice of hyperparameter "
25
E.3 Potential Limitations
26
F
Beyond ResNets: Experiments with Vision Transformers and ConvNeXts
27
Sensitivity Analysis and Ablation Study
28
G.I Choice of Lp Norm
28
G.2 Choice of Taylor Approximation Order
28
G.3 Choice of Calibration Threshold ?)
28
G.4 Superiority of softrun...
28
G.5 Generalization Capabilities of MANO on ImageNet-C
29
How to Use MANO in World Applications?
29
17

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A Pseudo-Code of MANO
Algorithm 1 summarizes MANO introduced in Section 3.2, which is a lightweight, training- free method for unsupervised accuracy estimation using the neural network's outputs. We open-sourced the code of MAÑO at https: /ttpte////ttttttt/gg com/Renchunzi A/Moo.
Algorithm 1: Our proposed algorithm, MANO, for unsupervised accuracy estimation.
Input: Model f' pre-trained on Dursin test dataset Drest = (x, Ji
Parameters: Hyperparameter p > 1.
Initialization: Empty prediction matrix Q E RNNK
Criterion: compute criterion (Dtest) and select a following Eq. (5) and Eq. (6).
for i E N1 do
Inference: recover logits q, =
E E E
Normalization: obtain normalized logits a())) E AK.
Update: fill the prediction matrix Q: (q:) following Eq. (1).
end
Output Estimation score S(,, Drest)
|Qlp following Eq. (2).
VNK
B Extended Related Work
Unsupervised accuracy estimation. This task aims to estimate model generalization per- formance on unlabeled test sets. To achieve this, several directions have been proposed. (1) Utilizing model outputs: One popular research direction is to use the model outputs on distribution- shifted data to construct linear relationship with the test accuracy (Deng et al., 2023; Garg et al., 2022; Guillory et al., 2021; Hendrycks and Gimpel, 2016; Xie et al., 2024). Some of these approaches are limited by requiring access to the training set (Chen et al., 2021; Chuang et al., 2020). The most recent work (Deng et al., 2023) uses the nuclear norm of the softmax probability matrix as training-free accuracy estimator. However, it significantly suffers from the overconfidence is- sues (Wei et al., 2022a), leading to fluctuating estimation performance across natural distribution shifts. Our work focuses on addressing this issue by balancing logit-information completeness and peerconfienneeiiffrmmatti
accumulation. (2) Considering distribution discrepancy: another direction examines the negative relation between test accuracy and the distribution discrepancy between the training and test datasets (Deng and Zheng, 2021; Lu et al., 2023; Tu et al., 2023; Yu et al., 2022). However, commonly-used distribution distances do not guarantee stable accuracy estimation under different distribution shifts (Guillory et al., 2021; Xie et al., 2023), and some of these methods time-consuming on large- large-scale datasets due to the requirement of training data (Deng and Zheng, 2021). (3) Constructing unsupervised losses: methods such as data augmentation and multiple-classine agreement have also been introduced (Jiang et al., 2021; ;; Madani et al., 2004; Platanios et al., 2017, 2016). However, they usually require special model architectures, undermining their practical applicability.
Distance to decision boundaries. The idea of treating distance to the decision boundary as an indicator of confidence originates from classical support vector machines (Vapnik, 1998). Decision boundaries of deep neural networks have been studied in various contexts. For example, some works explore the geometric properties deep neural networks either in the input space (Fawzi et al., 2018; Karimi et al., 2019; Montufar et al., 2014; Poole al., 2016) or in the weight space (Chaudhari et al., 2019; Choromanska et al., 2015; Dauphin et al., 2014; Dinh et al., 2017; Freeman and Bruna, 2016). Some works apply the properties of decision boundaries to address practical questions, such as adversarial defense (Croce and Hein, 2020; He et al., 2018; Heo et al., 2019), OOD detection (Lee al., 2020), and domain generalization (Li al., 2022: Yousefzadeh, 2021). As we discuss in Section 3. those approaches that use the distance to the decision boundary as an unsupervised indicator of confidence, rely on the low-density separation assumption (Chapelle and Zien, 2005), which states that the classifier must mistake mostly in the low margin zone (Feofanov et al., 2019, 2024). In our work, under this assumption, similar to Li et al. (2018), we use the distance between the learned intermediate feature to each decision boundary in the last hidden space.
18

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Normalization in deep learning. Normalization is crucial technique extensively utilized across various fields in deep learning, including domain generalization (Fan al., 2021; Seo al., 2020; Wang et al., 2021), 1), metric learning (Oord et al., 2018: Sohn, 2016; Wu et al., 2018), face recognition (Deng al., 2019; Liu et al., 2017; Ranjan al., 2017; Wang et al., 2017: Zhang al., 2019) and self-supervised learning (Chen et al., 2020). For example, TENT (Wang et al., 2021) normalizes features of test data using the mean value and standard deviation estimated from the target data. Lz-constrained softmax (Ranjan et al., 2017) introduces L2 normalization on features. These normalization techniques are primarily employed to adapt new samples to familiar domains, calculate similarity, and speed up convergence. However, proposed normalization focuses reducing the negative implications of poorly calibrated scenarios.
C Background on Tsallis Entropies
The definition of Tsallis a-entropies (Tsallis, 1988) given below.
Definition C.1 (Tsallis a-entropies), Letp E Ak be probability distribution. Let >> 1 and 0. The Tsallis a-entropy is defined as:
H k(a - 1) (p
a following Blondel al. (2019). The Tsallis a-entropy generalizes the Boltzmann- theory of statistic mechanics nonextensive systems. It has been used a measure of disorder and uncertainty in many applications (Gell- Mann and Tsallis, 2004; Negrinho and Martins, 2014; Sneddon, 2007: Teimoori et al., 2024), including in Machine Learning (Blon- del et al., 2019, 2020; Muzellec al., 2017). Moreover, they general- ize two widely-known measures of uncertainty. Indeed, the limit case -> leads to the Shannon entropy H3 (see Peters et al., 2019, Ap- pendix A. L... lim (p) = Hs(p) = - -E Pj In(p)), while taking a = leads to the Gini index G, a popular impurity measure fo decision trees (Gini, 1912), i.e., H. (p)
It is common to take k
g
6.0
Figure 7: Tsallis a-entropies
of [t, 1 for E [0, 1].
(1-11p11))
Gpp). Tsallis entropies measure the uncertainty: the higher the entropy the greater the uncertainty. From a probabilistic perspective, the entropy will take high values for uncertain probability distri- butions, ie., close to the uniform distribution. We visualize the evolution the Tsallis entropy for varying parameters in Figure where the
corresponds the Shannon entropy.
D Proofs
In this section, we detail the proofs of our theoretical results.
Notations. Scalar values are denoted by regular letters (e.g. parameter A), vectors are represented in bold lowercase letters (e.g., vector x) and matrices are represented by bold capital letters (e.g.. matrix A). The i-th row of the matrix is denoted by A, its j-th column is denoted by A., and its elements are denoted by Aij- The trace of matrix A is denoted by (T(AA) and its transpose by A T_
The Tp norm of a vector x is denoted by ||x|pp and by abuse of notation we denote it by ||A] l|p for
matrix with || 1 = Al1. E A,gA Let AK {p [o, 1] Σ E P: 13 be the K K-dimensional probability simplex.
D.1 Impact of Prediction Errors
Let Drest be test sample with ground-truth label y E fl...., In multi-class classification, the softmax operator is used to approximate the posterior probability p(y[x) (see Bishop, 2006, chap.4, p. 198). Reusing the notations of Section 4, because the distribution shifts between source and target, logits are subject to a prediction bias (Ex); and write f(x) q" + where q" are ground-truth logits. this section, we study the impact of such bias on the approximation of the posterior x).
19

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Impact on the posterior approximation. Proposition 3.1 shows the impact of the prediction bias on the KL divergence between the true class posterior probabilities, assumed modeled as softmax(q") E Ax, and the estimated ones softmax([(x)) Ax. In particular, it states that
0 < (plp)l)) S sT..p.
where e+ = (max) (x1) - Ex)E E R The proof is given below.
We denote = [(x) E R the neural network's outputs on given test sample x. We first remark that
Proof.
[p|lss
Σ Pk
In
P&
Sk
exp(gx)
Σ/ exp(a] + Ej) expla + tx)
Σ P In
(8)
Σラ= exp(qj)
K
Σx expla) + +8/)
Σ Pa In exp(-ex)
Σ/= explax)
k
To obtain the upper-bound, we notice that
exp(q} +æz) = ppp(a)))pppe S exp(qş) mxx(cx((E))).
This leads
Σ j=1 exp EEy)
(exp(EE)(E Σe= exp(q;)
max(exp(<i)).
(9)
Σj=1 ===expp'))
Σj=1 exp(q;)
Using the fact that all the terms are positive and that In and exp are increasing functions, we obtain from Eq. (8) that
Σ/1 exp(a) +0₂)
Σ P& In exp(-ex)
Σ Pk In(exp(-ex) max(exp(ei)))
(g=rexp(''))
Ps (i(ll(exp(mmxx(=))) sx]
(10)
< Pk [mam((==) - Ex]
TTp.
Combining Eq. (8) and Eq. (10) gives the upper bound.
The quantities 54 a linear transformation of the prediction bias e RK and has nonnegative entries. which means each class is overestimated, representing an overconfident model. Proposition 3.1 shows that the discrepancy between the error approximation of the posterior probabilities controlled by the alignment between the posterior and this extreme prediction bias. In addition, by simple application of
Eq. (7) and using the fact that |p|| Ex- P S Ex- Ps KL(plls) < ||e|||2- In particular, in the perfect situation where & O, e. equal to 0 and the softmax probabilities perfectly approximate the posterior. summary, Proposition 3.1 indicates that not only the norm of the prediction bias but also its alignment to the posterior is responsible for the approximation error of the posterior. In our setting, it means that logits- methods need a low prediction bias on classes which the model is confident such that softmax probabilities can be reliably used to estimate accuracy, This follows our analysis and empirical verification from Section 4.
we have
A real-world example. Although we usually tend to think that high prediction bias shifts the predicted posterior towards the uniform distribution, in the general case, other situations may happen that hinder the quality of the accuracy estimation. For example, one may think of a letter recognition task with a neural network pre-trained on the Latin alphabet and tested on the Cyrillic one. this case, some prediction probabilities will be adversarial as the neural network will not be aware of the semantic differences between the Latin "B" and the Cyrillie "B", therefore predicting a wrong class with high probability.
20

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D.2 Distance to the Hyperplane
Lemma D.1 (Dohmatob (2020)), Let w ER" be non zero and b E R. The distance between any point E Rd and the hyperplane (x|w x b= 03 writes d(w,, 2) w x blf ||w||
Proof. The proof follows the geometric intuition from Dohmatob (2020). We recall it here for the sake of self-consistency. The distance between and the hyperplane H. {x E I 3
= 0} is equal to the distance between and its orthogonal projection on H. We consider the line L {z + twott e R that is orthogonal to H and passes through z. The desired orthogonal projection simply the point z + l'w such that L and H. intersects, i.e., such that
w1 + b + w z
+t ||
w 2 + b
t
[|W|] # 0)
||m|
follows that the distance between z and H writes
w + b
w T z + bl |
d(w z) 1| 2 + tw
x w
3
D.3 Proof of Theorem 3.3
Proof. Reusing the notations introduced in Section 3.2 and Algorithm 1, we have that
N
Sif, Drest)"
Q
Σ
Q
(Definition of [[]]pp
NK
NK
N
(Definition of Q; in Algorithm 1)
NK
1=1
Σ
-
(lo(g:)))))
NK
1=1
N
1
p(p
1)
(1 [lo(q))))
K
NK
p(p 1)
i=1
N
1
p(p
1)
H (a(qi))
(Definition of H
K
NK
1=1
b
H (o(a))
N i=1
where
2(p-1) 0 and b $. Rearranging the terms concludes the proof.
D.4 Properties of Φ
The sof tmax can be decomposed as softmax(q) = exp(q)/ Σ exp(q)x (ø exp) where 6: R > AR writes (uu) u/ Σ Exm1 u₁ u/||u//1- We extend the domain of & IR. by setting (0) k. The following proposition states the properties of Φ.
Proposition D.2 (Properties of Φ).
1. Generalized injectivity. Vu, v E R (0), Φ(u) 6(v) > Ea E R*, s.t. u ov
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roughly distinguish the five following cases illustrated in 2. Evaluation on constant inputs. Let alk with 2 0. Then, we have p(u) = K 1K
Proof. We start by proving the first part of Proposition D.2. Let u, R$ \ O. We have
V
| |
(uu) o(v)
xv.
|v]l
Then, we prove the second part of the proposition. Let 2 0 and consider al E R If 0, then and by definition, (du) x00) π K. Assuming 0, we have
u
u
a
1 K
x (
x x =
K
k= uk
Ek=1 a
The first part of the proposition is dubbed "generalized injectivity" the injectivity can be retrieved by fixing = 1 in Proposition D.2. It ensures that & only has equal outputs if the inputs are similar. To illustrate that, consider the logits q. E RK From Proposition D.2, having softmax(q) softmax( 6') is equivalent to having exp(q) = exp(o) for some a * 0. By positivity of both sides, implies a > 0. and taking the logarithm leads q = 8 + In a. means that q equals ô up to fixed constant. From learning perspective, those logits will thus have the same predicted label and normalized logits. Proposition D.2 shows that using Φ preserves the information from the neural network. In addition, if the neural network's output is not informative,iee, all entries are equal, then the link function gives equal probability to all classes.
E Theoretical Insights into Criterion 4 (Dtesst))
In Section 4, we explained the main drawbacks of using the softmax normalization in the presence of the prediction bias proposing new alternative normalization, softrun, that we recall
q. if (Drust))
((q)) =
q
(Taylor)
exp(q:),
if (Drrett
(softmax)
where (Duustt the selection criterion defined as
log( exp(q;)=
(Drest
NK
K Σj= exp(q:);)
In this section, we first would like to give more insights on the choice of 4 (Puest) and the selection rule. Then, we motivate our choice of the hyperparameter Finally, we discuss the potential limitations our approach.
E.1 Choice of Criterion & (D
Intuition. We first provide some high-level intuition behind the selection criterion of Section 4. As discussed before, the main idea of the logit- based approach is rely on the model's 's confidence whose reliability depends on the prediction bias induced by possible distribution shift. Depending on exact values of confidence and bias, Figure and described as follows:
High confidence, high bias. The model is self-confident but practically makes lot of mistakes. This corresponds to the case when assumptions are not met (Section 3.1), so logits are generally uninformative, and no normalization technique can really fix it. Thus, practice, we have to make sure that the low-density separation (LDS) assumption holds, which is generally the case for well-calibrated models trained on diverse training sets. However, a user has to be careful when applying test-time adaptation methods (Chen et al., 2022; Rusak et al., 2022; Wang et al., 2021) to original pre-trained model, since these approaches perform unsupervised confidence maximization making LDS not guaranteed anymore.
22

---

Use Taylor
X
LDS violated
Prediction Bias, &
f Conf
T Conf
1 Bias
Bias
Use Taylor
J Bias
1 Bias
f Conf
T Conf
X
Doesn't happen
Use softmax
Confidence
Figure 8: A schematic illustration of possible cases that may may not happen and what normaliza- tion technique we use depending on the model's ' confidence and the value of the prediction bias. The two anadmissible scenarios correspond to the red sub-squares.
Low confidence, low bias. The model tends to output confidence that low but overall precise. This situation unlikely happen mainly for the following two reasons. Firstly, the classification problem is poorly posed as the considered case implies that the true posterior probabilities are close to uniform ones. Secondly, is known that deep neural networks tend to be overconfident in their predictions (Wei et al., 2022a), which we also observed in our experiments. Thus, we do not consider this case and leave it as subject of future work.
3. High confidence, low bias. The model tends to be self-confident being overall precise. This is a favorable case as logits are very reliable, so we can use softmax normalization without being afraid to be too optimistic.
4. Low confidence, high bias. The model is not confident in their predictions and it indeed makes lot of mistakes due the high prediction bias. This also favorable scenario as low confidence correlates with low performance. In this case, want posterior probabilities to be close to uniform, and we use the Taylor normalization due to its smoother behavior.
mixed scenario when different examples may refer to different cases. As it is generally difficult to know what normalization technique would be the most relevant, we would opt for more conservative solution in this situation. This is where the Taylor normalizer becomes useful as it does not exacerbate prediction bias to the same degree as the softmax (see Section 4. for more details).
Grey zone. corresponds
Formulation of (Diestt As we discussed in the main body of the paper, the criterion {DD equal, up to constant, to the average KI. divergence between the uniform distribution and the predicted sof tmax probabilities (Tian et al., 2021). This implies that the criterion reflects the model's confidence being high when predicted probabilities are far away from the uniform ones. As we rely on the LDS assumption, high values of (D correspond to the 3rd case (high confidence, low bias), and the softmax normalization selected. Conversely, when the (D is lower than the threshold 77. we apply Taylor, which corresponds either the 4th case (low confidence, high bias) or the 5th case (grey zone).
Connection between criterion and misclassification error. The next proposition provides further insights into our selection process. Let u k 1k E A be the uniform probability. The test dataset N with corresponding logits qi f(x) and ground-truth touuhhtruth labels Deest
writes Drest
23

---

{us NN (unavailable in practice), We denote the softmax probabilities by s = softmax(q:) exp(q:) E exp(ς)): Ak. We introduce the entropy of probability vector as H(p)
K Dk=1 Pa In(p:). In particular, it a measure of uncertainty and takes high value when the model uncertain, Le,, outputs probabilities close the uniform. We establish in the following proposition the connection between the criterion ((Dwst), the miscalibration error, the model's confidence, and its entropy.
Proposition E.1 (Dett))) misclassification error, confidence and entropy). We have
{(Peett )rest)
testt + (Drost)
S (D
+In(e
R)
misclassificuttin
confidence
entropy
criterion
where E(Dtest Frest) H(Drest) = Σi H(st).
N E sy ((Drest)
N EE KL(U
|| s*), and
Proof. We first present the following lemma that introduces the change of measure incquality (Banerjee, 2006; Seldin and Tishby, 2010).
Lemma E.2 (Change of measure inequality (Seldin and Tishby, 2010)). Let Z be random variable on {1...., K} and μa = (μx)x E Ar and v = (Vk) E AK be two probability distributions. For any measurable function wi + R, the following inequality holds:
K
pkb((k) S KL(H || v) In Σ ik exp(v(()))
Proof. We have
K
(k)(()
Σ
Mb In
exp(v(())
Vs
k=1
K
K
Σ
+
Σ
Mh In (x)(()(()))
k=1
k=1
= KK( v) +
Σ
M& In (x)((<(k)))
(Definition of KL(- -))
M4
Am l
<KI(H || v) + In
Σ
Mi (x)()((k))
U/
(Jensen inequality)
K
= KL(H v) In
Σ
u/ exp((((()))
We now proceed to the proof of Proposition (E.1). For given test sample E Drost, we first notice that
In 1 1 In(s()
KL(u s²)
u
IIn(K)
In(st).
K
1=1
k=
24

---

Similarly, we obtain (s(s' [) - s In(s)) + In(K)
Combining those results leads to
K
K
KL(u s²) + KT(ss || = K
In(s
Sy
In(s),
k=1
K
- KL(s' u) -KL(uu s')
In(s))
Σ sy In(s)).
Consider the function (k) [l( * k) that takes the value when y: * k and 0 otherwise. Using Lemma E.2 with the measures = s', V = u and ub and the previous equation, we obtain
K Styl + k) (L(s')
u) + In Σ
u exp(I((i
# +b))
K
K
K +
K
( + k) < -L 8*)
1
In(s
Sy In((s)) + In
Σ
up exp # k))
K
1=1
k=1
K
1 sy s -KT((u ||
1
In(s))
H(s*)
In
}( (Ke+1)
k= 8 = 1)
k
K
k=1
v. KT + H(s)) S
In(s))
In
c K
Summing over all the test samples and dividing by N leads to
N
1 N
N
N NK 1=1 R=
K
In(st) + In c 1 K ).
E KL(U
|(s
Σ H(S))
N
N
N
=De(Deen))
which concludes the proof by using the notations introduced in Proposition E.I.
Interpretation. The term {(rest )teat), dubbed misclassification error, is the average error the test set between the optimal probability on the true label (i.e., 1) and the predicted probability sy. takes high values when the model makes lot of mistakes, assigning low confidence to the true class labels, and low values otherwise. (Deestt is the average KL divergence on the test set between the predicted probabilities and the uniform distribution and it measures the model's confidence (Tian et al., 2021). It takes high values when the predicted probabilities are far from the uniform (confidence) and low values when they are close the uniform (uncertain). H(Drest) is the average entropy on the test set of the predicted probabilities. It takes high values when predicted probabilities are close to the uniform and low values otherwise. Proposition E.1 implies that when the model makes few mistakes (E(Dtest: (esa)) is low) and is confident (M((tess) is high and H( is low), then the criterion (D ) takes high values. This matches the empirical evidence from from Tian et al. (2021). Proposition E. is harder to analyze in other scenarios, i.e, when the misclassification error, the confidence, or the entropy behaves differently, mostly because of the interplay between (DDeett and H(Duust. However, we experimentally show the benefits of (Deestt and trun Section 5 where MANO achieves superior performance against 11 commonly used baselines for various architectures and types of shifts on 12 datasets.
E.2 Choice of hyperparameter ?)
In all our experiments, we take n 5 for the selection criterion in Eq. (6). We motivate this choice in what follows. In our setting, we consider test samples x; E Drest drawn i.i.d. from the test distribution pr. As the model f pre-trained Persin is deterministic function, the logits qi are i.i.d. random variables and the decision threshold (Dtest) NK Σn Σk In (exx(())))/ Σ exp(q:), is a random variable with mean and variance v. Applying the Chebyshev's inequality leads to
1
(( - 14| > pn) <
(11)
25

---

The threshold ((Drest) is used to determine how calibrated the model is on given test dataset Drest. Figure 2(b) shows that our proposed normalization is optimal in poorly calibrated datasets and performs slightly below the sof in calibrated situations. Hence, we can afford to be conservative and we want to consider the model calibrated only for extreme values of + (Drest). From Eq. (11), taking 5 ensures that the probability that (Dtest) deviates from its mean by several standard deviations with probability smaller than 5% (23 < 0.05). It should be noted that we do not claim the optimality of this choice nor the optimality of our automatic selection in Eq. (6). However, it is particularly difficult to define decision rules in unsupervised and semi-supervised settings (Amini al., 2022). Moreover, using Eq. (6), MAÑO remains suitable even when test labels not available which is often the case in real-world applications, and we demonstrate state-of- oatte--f-hhe-eeee art performance for various architecture and types of shifts in Section 5. For the sake of ceff-consistency, also provide a sensitivity analysis on the values of in Appendix G.3.
E.3 Potential Limitations
It should be noted that the selection criterion of Eq. (6) remains somehow heuristic and might depend on the model, the data, or the threshold yj. As stated above, the chosen value of v) is motivated by a probabilistic argument and by experiments. However, as it can be seen in Eq. (11), the mean and standard deviation of & (Prest) can impact the validity of 17. In particular, this could be the case when applying MANO on other data modalities than images in other learning settings (e.g., classification with a huge number of classes, regression tasks, auto-regressive settings). We believe this is the subject of future work to improve the robustness and versatility of our method.
Impact of the number of classes. As a first research direction, we provide a motivating example with synthetic data the impact of the number of classes K on the values of Φ (Dtest). We uniformly draw random vectors of & in [-5, 5] to mimic the logits obtained from 100000 independent models. We compute the corresponding (Dttstt for
each model and recover the 0.5th and 99.5th
8
percentile to obtain 99% confidence interval.
Criterion (Drett)
We repeat this experiment for K [2, 100].
The evolution of the confidence interval is dis- played in Figure 9. We observe that soon K >> 3, the upper bound of the confidence inter- val has very slow increase. However, the lower bound increases quickly in the beginning until K 25 and then adopts the same increase pace as the upper bound. In summary, the range of values of (Ptest) becomes thinner and more concentrated on high values for K > 25. In particular, we observe that 99% of the models have an associated (Deest)) higher than 17 5, which means that in this situation, the softmax would always be selected as normalization While the conclusions from this experiment are not directly applicable to our real experimental setting (in particular, Taylor and sof tmax cases of Eq. (6) occur both for datasets with K > 25 and K < 25), we believe it motivates further work to make 7) more robust to the values of ((Prest). In particular, one could propose to compute 7) based on statistics of the data or function of the number of classes.
5-
0.5 percentile
2-
99.5th percentile
25
50
75
100
Number of classes K
Figure 9: Evolution of the 99% confidence interval on (Dtest) with the number of classes K. As K increases, (DDrest will likely be higher than 17
Dispersion of the softmax. In the previous paragraph, we showed that the number of classes K can have an impact the values taken by criterion (Dtest) As the criterion relies on the softmax probabilities of the model on test data, it is natural to investigate the impact of K on the sof smft function. The lemma below shows that the softmax must disperse all entries the number of classes K increases. It means that the total weights on the softmax entries (equal to 1) cannot be concentrated on few entries the number of classes K increases.
26

---

Lemma E.3. Let 0 E. RK be logits of a neural network f such that ||0||1 S c some 0. Then, as the number of classes grows, ie., K + 00, we have
softmax (0) = K
where the equality holds at the component level.
Proof. Following the proof of Zekri et al. (2024, Lemma D.7), we can show for all E
[K]
that
exp (-c) exp (0.)
exp (c)
a
I softmax(0))
b
ΣØ exp (c)
j= exp (0,)
2j=1 exp (-c)
K
K'
where a = exp (-2c), = exp (2c) are constant. This concludes the proof.
Lemma E.3 implies that, as the number of classes grows, the highest value an individual entry can have decreases. Hence, the number of classes impacts the distribution of the weights among the softmax entries (recalling that it must sum at 1 as it is a probability vector). As by definition, (Diest) depends on the soft probability distributions, this will impact its value. We believe that empirically and theoretically studying this phenomenon could be insightful in deriving more robust selection criteria and threshold values. We note that Lemma E.3 similar but more general than Veličković et al. (2024, Lemma 2.1) as our global bounding condition on O encompasses their entry-wise condition.
F Beyond ResNets: Experiments with Vision Transformers and ConyNeXts
To evaluate the efficiency of MANO across diverse model architectures, we conducted additional experiments with the Vision Transformer (Dosovitskiy, 2020, ViT) and the ConvNeXt (Liu et al., 2022) architectures. The numerical results are gathered in Table 5. Two methods stand out from the rest of the baselines: ConfScore and MANO. In particular, ConfScore is particularly strong with the ConvNeXt architecture while MANO is better with the Vision Transformer. Overall, MANO leads to a better accuracy estimation on average.
Table 5: Method comparison using Vision Transformer (ViT) and ConvNext under synthetic, sub- population and natural shifts with R² and P metrics (the higher the better). The best results for each metric are in bold. Overall, MANO surpasses its competitors while ConfScore appears to be stronger with ViT and ConvNext than with ResNets.
Dataset
Network
ConfScore
Entropy
ATC
MDE
COT
Nucker
MANo
R²
P
R²
R²
R"
D.885
0.936
(1980
0.996
0.997
0.873
0.950
0.997
0.9%6
CIFAR 10
ComNNeXi
0.936
0.996
0.924
0.995
0.911
0.994
0.002
0.534
0.978
0.994
0,991
0.994
0.916
0.995
Average
0.611
0.9%
0.952
0.996
0.951
0.966
0.435
0.704
0.964
0.996
0.964
0.991
0.954
0.9%
VIT
0.983
0.997
0.981
0.995
0.987
0.995
0.974
0.983
0.993
0.996
0.977
0.995
0.989
0.996
CIHAR 100
ComvNeXi
B.766
0.995
0.957
0.992
0.976
09933
0.617
0.399
D.881
0.996
0
0.994
0.954
0.94
Average
0.961
0.99
0.952
0.996
0.982
0.991
0.795
0.691
0.987
0.99
0.977
0.995
0.971
0.995
VIT
0.711
0.783
0.631
0.727
0.426
0.503
0.180
0.209
0.742
0.797
0,823
0,860
0.705
0.755
PACS
ConvNeXt
0.900
0.895
0.872
0.853
0.727
0.580
0.062
0.814
0.748
0.790
0.874
0.755
Average
10000
0.339
0.752
0.750
0.577
01541
03092
0.778
0.772
IL829
0.825
0.759
0.755
VIT
0.947
0.958
0.928
0.979
0.896
0.902
0.117
0.755
0.642
0.856
0.861
0.958
0.953
0.979
OfficeHHmmc
CanvNeXi
0.784
0,860
0.649
0.612
0.733
0.818
Average
0.865
0.909
0.788
0.902
0.832
0.912
0.128
0.615
0.642
0.856
0.687
0.736
0.843
0.898
VIT
D.930
0.950
0.925
0,950
0.950
0.971
0.816
0.884
0.923
0.958
0,873
0,882
0.958
0.971
CamoNeXt
0.943
0.970
0.931
0.902
0.805
0.937
0,942
0.930
0.963
Average
03377
0.360
0.928
0.555
0.926
0.37
0.892
0.841
0.90
0.948
0,907
0.912
0.444
0.967
VIT
0.950
0.972
0.937
0.968
0.948
0.970
0.819
0.908
0.939
0.971
0,905
0.927
0.959
0.975
Entity33
CanoNexi
0.968
0.988
0.955
0.981
0936
0.61
0.442
0.960
0.990
Avenige
B359
0.500
0.946
0975
0353
094
0341
0.75
0.45
0.960
0.982
27

---

3.00
Actrege Offo--tame
MCC Cansipon
1.0
10
0.81
0.97
0.39
5.25
0P58-15
OPAN
InvbrogeMec:
0.35
Enciyy 10
Borkwing
37
03
PACS
Doma Velee: Oftce-lloe WIL35-80
Catasets
(a) Impact the Ly norm
(b) Approximation order 71
(c) Benefits of softrun
Figure 10: Sensitivity analysis with Resnet18. (a) Effect of the L, norm types. (b) Impact of the Taylor approximation order, i.e., the number of terms in Eq 4. For instance, an order of 3 means that 3 terms are taken, which corresponds the default setting in Eq. (6) and is used in all our experiments. (c) Type of normalization function.
G Sensitivity Analysis and Ablation Study
G.1 Choice Tpp Norm
To reflect the impact of different Lp norms on estimation performance, we conduct a sensitivity study on 5 datasets with ResNet18, whose results are shown in Figure 10(a). The performance for = 1 is ignored as in this case, |QQ|| = because Q is right-stochastic. We can see that when we choose p [2, 5). the results fluctuate within satisfying range. This can be explained by the fact that within this range, we emphasize adequately the large positive feature-to-boundary distances without ignoring the other comparatively small distances.
G.2 Choice of Taylor Approximation Order
In Figure 10(b), verify the impact of Taylor formula approximation final accuracy estimation performance. should be noted that for orders higher than in the default setting in Eq. (6), the positivity is lost. To alleviate this issue, the obtained vector to each its entries to ensure having an output in R This extends de Brébisson and Vincent (2016) to orders higher than 2. From this figure, three terms the Taylor formula, the average estimation performance is optimal. For well-calibrated datasets such Office-Home and WILDS, there exists increased trend of estimation performance when we reserve more Taylor formula terms. As for suboptimal-calibrate datasets such as PACS and Office-31, their performance rises when fewer terms reserved. It empirically certifies that the normalization technique is a trade-off tool between the ground-truth logits' information and error accumulation. In addition, the optimal choice is to keep terms in Eq. (4) which motivates our default setting in Eq (6).
consistently remove for all orders the minimum value of
that when reserve the first
G.3 Choice of Calibration Threshold !)
In Table 6, we display the performance comparison for varying values threshold 7 on three datasets with ResNet18. It should be noted that taking 1) 0 corresponds to the case where the smat always taken, i.e., the common choice in the literature. This matches our theoretical insights in Appendix E.2 and confirms that taking 1) 5 is robust and effective choice for softrun.
G.4 Superiority of softrun
To verify the effectiveness of our proposed normalization technique, coftrun, we conduct an ablation study by replacing our normalization with softmax function under the natural shift. Figure 10(c), we observe that our proposed normalization significantly enhances the estimation performance of datasets from the natural shift. Especially, R² for poorly-calibranl datasets such PACS improved from 0.541 to 0.812.
28

---

Table Performance comparison for varying 7) (, on CIFAR-10, Office-Home, and PACS with ResNet18. The metric used is R² (the higher the better). The best results are in bold. The results motivate our choice of rj
Dataset
1) = 0
7)
1) =
3
1)
1)
7
1)
Cifar-10
0.995
0.995
0.995
0.995
0.995
0.995
Office-Home
0.926
0.926
0.926
0.926
0.777
0.777
PACS
0.541
0.541
0.541
0.827
0.827
0.827
Average
0.820
0.820
0.820
0.916
0.866
0.866
G.S Generalization Capabilities of MANO ImageNet-C
To further demonstrate the generalization capability of MANo, we provide a similar experiment with that in Section 5.4 ImageNet-C and ImageNet-C' (Mintun et al., 2021) in Figure 11. In particular, we fit a linear regression function on ImageNet-C and use the linear function to predict the accuracy of ImageNet-C. This figure shows that MANO has better estimation performance than the other baselines when meeting different corruption types.
ConfScore Accuracy
Dispersian v.s. Accuracy
Accuracy
Accuracy
80- Imagabie:
Men.
ImagelietC
70
60:
6b-
6000
to-
S0
60
Toxt Accurall
60
so-
MAE-5.592
MAE-1.842
300
MAE-2,704
13
20
19
01
022 ConfScore
0'3
7.75 8.00 8.25 Dispersion
8.50 6.75 9.50
6.0
65
7.0
7.
8.0
15
2.0
25
Nudear
MaNo
Figure 11: Comparison of generalization capability across four methods. Each subplot shows linear regression model fitted on ImageNet-C to predict accuracy on ImageNet-C. Mean absolute error (MAB) is calculated on ImageNet-C (Mintun et al., 2021). All experiments ResNet18.
H How to Use MANO in Real-World Applications?
This work demonstrates the strong correlation between ground-truth OOD accuracy and the designed score, which can be particularly useful for model deployment applications. this section, we provide two examples.
Find difficult (under-performed) test set. cases such as retraining on under-performed datasets or annotating hard datasets, we only need to know the rank of datasets by accuracy. Therefore, we can calculate the proposed score for each dataset directly and fulfill the task based on this score's 's ranking.
Deployement risk estimation. When deploying the model into production, it is important estimate its safety. If the cost of getting test labels is prohibitive, our method can help to estimate the model's 's accuracy on the product's 's test data. A practitioner can additionally look the variability of the score on multiple test sets. When multiple datasets are not available, we can alternatively construct adequate synthetic datasets via various visual transformations.
In Table we provide an example, using 90% of datasets to train a linear regression model and estimating the test accuracy of the rest 10% of datasets via the trained linear regression model. The results are measured by Mean Absolute Error (MAE). From this table, we observe the superiority of MANO for application in the real world.
29

---

Table 7: Method comparison on four benchmarks using ResNet18, ResNet50, and WRN-50-2 under natural shift with the MAE metric (the lower the better). The best results are highlighted in bold. MANO provides the best accuracy estimation overall.
Dataset
Network
Con/Score
Entropy
ATC
Fréchet
Dispersion
MDE
COT
Nuclear
MANo
ResNet18
5.131
5.265
3.968
1.964
3.842
1.763
0.952
1.357
0.394
ResNet50
1.955
1.891
1.706
2.477
3.842
0.928
0.670
1.024
0.450
CIFAR-10
WRN-50-2
2.956
3.160
3.086
3,547
2.671
0.846
0.355
0,843
0.406
Average
3.344
5,439
2.920
2.662
2,539
1.179
0.639
1.075
0,417
ResNet18
2.944
5,265
3.968
1.964
3.846
1.763
0.952
1.357
0.394
CIFAR-100
ResNet50
2.128
1.891
2.477
2.671
0.670
WRN-50
1.323
3.160
3.086
3.547
1.102
0.846
0.355
0.843
0.406
Average
3.344
3,439
2.920
2.662
2,339
1.179
0.639
1.075
0,417
ResNe118
3.822
3,559
3.752
5,998
2,822
1.926
1.297
1.165
0.612
TinylnageNet
ResNet50
3.376
3.696
3.435
5.616
1.667
2.687
1.653
1.226
1,005
WRN-50-2
2.712
2,854
5.011
4.862
1.054
1.703
1,286
1.053
1.145
Average
3.303
3,370
4,066
5,492
1.848
2.106
1.412
1.148
ResNer18
3.616
3.812
2.503
3,750
0.705
2.679
1.057
Imageeet
ResNet50
3.325
3.357
2.911
3.242
7,318
6.346
1.796
3,203
1.388
WRN-50-2
2,990
4.132
3.388
5.210
10.050
6.755
2.564
4.091
0.695
Average
3.310
3.767
2.934
4.067
6.656
5,973
1.688
3.325
1,047
ResNet18
1.987
2.878
10.014
1.404
7.976
8.228
0.602
0.885
0,880
ResNet50
0.855
L.874
7,509
1,254
7.285
9,374
0.730
2.676
3.995
Office-Home
WRN-50-2
2.607
3,933
11,272
0.924
7,591
7.974
2.382
2.764
Average
1.816
2.895
9.598
L.947
7.617
8.525
1.238
1,941
3.068
ResNet18
3,375
7.067
10.875
7,416
8.200
7.066
5.954
7,313
3,830
ResNet50
4.778
7.742
10.039
3,533
8.949
9.917
5.005
7.407
4.230
DomainNet
WRN-50-2
5,513
6.310
3.513
10.001
8.695
4,605
6.953
5.827
Average
4.555
7,039
8.142
6.983
8.615
8,738
5.188
7.224
4.629
ResNet18
7.448
7,416
7,391
5,544
2.343
3.798
2.205
2.182
0.790
Entity-13
ResNe150
5.367
WRN-50-2
2.748
2,893
3.128
1,817
5,431
3.159
2.276
1.297
0.674
Average
5.126
5.225
5.338
3.419
4.824
3.699
2.871
1.917
ResNet18
5.060
5.544
5.731
4.098
3.768
3.665
1.867
2.269
Entity-30
ResNet50
4.415
5.14
4,630
4,499
7.185
4.326
4.767
2,158
1.455
WRN-50-2
3.477
4,200
3.363
2.490
4.17
4.080
2.103
1.797
0.918
Average
4.317
3.695
ResNet18
4.095
4.098
3.699
3.767
4.262
3.737
1.373
2.569
1.975
ResNet50
2,802
2.757
1.574
9.687
4,361
3,535
3.260
1.860
1,573
living-17
WRN-50-2
4.059
4.250
2.535
3.889
4.925
3.833
2.922
2.974
3.281
Average
3.652
3.701
2.603
3.781
4.516
2.319
2.467
2.277
ResNet18
2.907
3.767
3.386
1.891
2.773
3.453
1.881
2.168
1.010
Nonliving-26
ResNet50
4.004
4.663
4.685
3,547
6.461
4.236
3.955
2.087
2.189
WRN-50-2
1.903
2.444
2.593
2.259
4.220
3.781
2.785
2.094
Average
2.938
3.625
3.555
2.565
4.485
3.823
2.874
2.116
1.534
30

---

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Reviews:
[{"review_id": "Fxra1ObZ2E", "reviewer": "Reviewer_6N1W", "summary": "The paper presents MANO, a method for unsupervised accuracy estimation under distribution shifts. The method addresses the challenge of estimating model performance on out-of-distribution (OOD) samples without access to ground-truth labels. Firstly, the authors investigate the correlation between logits and test accuracy, and propose a novel approach that involves normalizing logits and using the Lp norm of the normalized logits matrix as an estimation score, motivated by the low-density separation assumption. Then, the authors find that the commonly used softmax normalization method tends to harm the performance estimation when facing overconfidence issue on some particular dataset. The authors demonstrate that MANO achieves state-of-the-art performance across various benchmarks and architectures.", "questions": "Please refer to the weaknesses.", "limitations": "N/A.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. This paper introduces a novel method of using the norm of normalized logits for accuracy estimation, providing a fresh perspective on handling accuracy estimation problem under distribution shifts in unsupervised settings. This innovative approach builds on the low-density separation assumption, which is theoretically sound and practically relevant.\n\n2. The authors consider the common overconfidence issue faced by deep models and propose a new normalization method based softmax operator. The illustrated experiments show that the new SoftTrun method shows improvement in some cases.", "weaknesses": "1. Despite the authors designing a method based on the low-density assumption from theoretical perspective, this approach appears rather trivial and is quite similar to entropy-based methods like the variant of ATC. Moreover, it measures the overall output smoothness of the test set, making the connection with uncertainty somewhat trivial.\n\n2. The proposed SoftTrun method, which is an improvement over softmax, shows performance enhancements in some cases (as shown in Figure 2(b)), but can also degrade performance in others. For new tasks, SoftTrun may not always outperform the traditional softmax method. Additionally, the experimental validation for this aspect is insufficient, as Tables 1 and 2 lack comparative results with the softmax-based MANO baseline method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the positive support and constructive comments on this work! Please find our responses below.\n\n**1.1. Relation to entropy-based methods such as ATC.**\n\nWe thank the reviewer for this comment. The proposed MaNo and the ATC indeed share similarities as they both belong to the class of logit-based accuracy estimation methods. However, there are several important aspects that distinguish the two methods, not only in terms of methodology but also effectiveness.\n* *[ATC calculates entropy that MaNo never considers.]* ATC utilizes the logits to calculate the entropy and measures how many test samples have a confidence larger than a threshold, which is a hyperparameter needed to be chosen. They proposed to search it from the labeled source validation set, which might be not a good idea in cases when softmax tends to be overconfident and the prediction bias is noticeable as we discussed in Section 4.1. In contrast, MaNo directly calculates the $L_p$ norm of the normalized logit matrix, thereby avoiding the introduction of a confidence threshold and leading to a simpler and more efficient approach.\n* *[Stability.]* Just being based purely on the experimental results, we can see that the entropy-based score function does not seem to be stable in performance and is worse compared to the proposed MaNo (regardless if it is softmax or SoftTrun normalization scheme).\n* *[New normalization scheme]* To the best of our knowledge, the question of calibration and logit normalization has never been touched before in the unsupervised accuracy estimation domain, so the proposed SoftTrun normalization scheme makes our contribution conceptually different from other approaches including ATC. \n\n**1.2. Simplicity of the approach.**  \nWe would like to open a debate with respect to this point as we believe that the simplicity of our approach is one of its main strengths given its superiority compared to 11 competitors over 3 different shifts and 12 datasets with a very noticeable difference in natural shift setting.  In addition, we provide the following arguments.\n -  Our approach is source-free and training-free (in contrast to ATC and ProjNorm, resp.), which makes it relevant for model deployment applications. \n -  As the reviewer noticed, our method has emerged from the theoretical perspective of the low-density separation assumption, which makes it intuitive and easy to analyze. \n - The versatility of the approach allows us to painlessly test our method for different neural network architectures. Therefore, in order to strengthen this point, we have conducted additional experiments using two other commonly used vision models: ViT and Swin. In addition, we added a new evaluation metric (mean absolute error) based on the suggestion of Reviewer A5Zk. We will include all the experimental results in the revision, meanwhile displaying some of the results in the two tables below. Our approach has superior performance on these two architectures as well, which validates our versatility claim. Our experimental results also include an ablation baseline, which is Softmax $MaNo$ that uses softmax normalization function instead of the proposed SoftTrun.\n\n**Table A: $R^2$ on CIFAR-10C and Office-Home with ViT.**\n|ViT|Dispersion|Nuclear|COT|Softmax MaNo|MaNo (Ours)|\n| --- | - | - | -|-|-|\n|Cifar10|0.945|0.963|0.950|0.984|**0.984**|\n|Office-Home|0.216|0.531|0.732|0.805|**0.805**|\n\n**Table B: Mean absolute error on CIFAR-10.1, ImageNet-S and ImageNet-A with Swin**\n|Swin|Dispersion|Nuclear|COT|Softmax MaNo|MaNo (Ours)|\n| --- | - | - | -|-|-|\n|Cifar10.1|10.758|5.074|4.669|1.042|**1.042**|\n|ImageNet-S|7.352|4.054|5.379|3.336|**2.091**|\n|ImageNet-A|24.533|7.521|4.597|1.273|**1.001**|\n\n**2.1. Clarifications regarding SoftTrun.**\nWe apologize for the possible misunderstanding regarding Figure2(b) and the design of SoftTrun. Please find our comments below.\n\n* *[Typo in Figure 2(b)]* We have made a typo, and the bar named \"SoftTrun (first 2-order)\" should be called as \"Taylor approximation 2nd order\" instead. This bar refers to the case when we apply only Taylor normalization function (as if we would always choose the 1st scenario in Eq.(6)), while \"MaNo w/ Softmax\" refers to applying softmax only (as if we would always choose the 2nd scenario in Eq.(6)).\n* *[Purpose of Figure 2]* The goal of this figure is to motivate the proposed SoftTrun defined by Eq.(6), where we automatically choose one of the normalization functions depending on a calibration scenario. For well-calibrated scenarios (Office-Home), we should use the softmax normalization for information completeness, while for poorly-calibrated ones (PACS), Taylor normalization is preferred in order to be more robust to prediction errors.\n* *[SoftTrun]* Thus, the proposed SoftTrun selects one of these two normalization schemes depending on the uncertainty value, and the final experimental results are displayed in Table 1, 2, 3. \n\nWe will update the labeling of Figure 2(b) and the writing of Section 4.2 to clear out any potential misunderstanding. \n\n**2.2. Results of softmax-based MaNo in Tables 1 and 2.**\nThanks for suggesting this idea. We did not display this baseline for the synthetic and subpopulation shifts as we found that the prediction bias is quite low on these problems which leads to $\\Phi(D_{test})$ much greater than the proposed fixed $\\eta=5$, so SoftTrun always chooses softmax normalization for these problems. The *key difference* between SoftTrun and softmax-always normalization schemes comes when *natural shift* is considered. In the table below, please find the experimental results of comparing the two schemes on 4 natural shift benchmarks, confirming SoftTrun superiority.\n\n**Table C: Softmax MaNo v.s. the proposed MaNo measuring by $R^2$ under natural shifts with ResNet-18.**\n|Method|PACS|Office-Home|DomainNet|RR1|\n| --- | - | - | -|-|\n|Softmax MaNo| 0.541 | **0.929** |0.894|0.971| \n|SoftTrun MaNo| **0.827** | 0.926 |**0.902**|**0.983**|", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "N9rCH4tweJ", "reviewer": "Reviewer_5Gdr", "summary": "This paper proposes the OOD accuracy estimation method, named MANO, by leveraging the positive correlations between features to decision boundary distance and generalization performance. Along with Softtrunc for preventing error accumulation in overconfidence scenarios, the proposed method outperforms existing estimation baselines over various benchmark datasets.", "questions": "- Averaged confidence can be considered as particular case of MANO with softmax, i.e., using p as infinity. Still, in Table 4, even using Softtruc, AC is not competitive against MANO. So, what makes MANO, considering featuer-to-boundary for all classes, superior against AC, which corresponds to only considering the most confident (probable) class?", "limitations": "See weaknesses and questions.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "- This paper provides the theoretical insights that motivate the simple but effective OOD accuracy estimation method without supervision.\n- Analysis on extensive set of distribution shift benchmarks as well as ablation studies demonstrate that the proposed method shows promising results in various distribution shifts.", "weaknesses": "1. The method requires hyperparameters, such as $p$ in aggregation step and determining model’s calibration for softrunc, which requires the OOD labels for optimal hyperparameter search.\n2. As mentioned in Section D.2, the selection of criterion ($\\Phi (D_{test})$) higher than $\\eta$ could still include the high errors and high confidence (overconfindence), i.e., first scenario has high lower bound of $\\Phi (D_{test})$ as in third scenario. Therefore, more thorough analysis or ablation studies on how sensitive MANO is under such potential failures in correctly determining calibration is required. In addition, miscalibration includes the underconfidence, where models have underconfident predictions compared to their actual accuracy. \n2. Important estimation baseline is omitted, Agreement-on-the-Line, Baek et al. (2022).  \n\nBaek et al., Agreement-on-the-Line: Predicting the Performance of Neural Networks under Distribution Shift, NeurIPS 2022", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their precious comments which help us further improve the paper. We hope our answers below could precisely address the reviewer's concerns. Please let us know if any issues remain.\n\n**1. Hyperparameter tuning and OOD labels.**\n\nWe thank the reviewer for this comment. In order to avoid any confusion, we would like to clarify several important aspects of problem setup:\n\n* *[No access to OOD labels.]* We consider the unsupervised setting, so we do not have any validation set. This is eventually one of the main challenges of the framework as we are not able to tune hyperparameters in contrast to the supervised approaches.\n* *[Source-free.]* We have constrained ourselves to a setup where only the pre-trained model is available without direct access to source data.\n\nGiven these constraints, we have come up with the proposed method, where we have made the following model choices.\n* *[Fixed hyperparameters across all distribution shifts.]* We have set all the hyperparameters to fixed values ($p=4$ and $\\eta=5$) across all the datasets. The empirical results show the superiority of MaNo without searching the optimal hyperparameter values. We have also performed a sensitivity study (please see Figure 7(a) for $p$ and response to Reviewer A5Zk for $\\eta$).\n* *[Necessity of introducing $\\eta$]* The proposed normalization SoftTrun eventually reveals an additional level of complexity of the problem overlooked by previous methods. Table 4 shows that SoftTrun improves ConfScore and Nuclear, which justifies the importance of normalization, which, however, comes with the cost of introducing the hyperparameter $\\eta$. \n\nWhen there is access to the training data, a possible way to choose hyperparameters could be to generate multiple synthetic datasets from the training set via various visual transformations to determine hyperparameters' values. \n \nWe will revise the manuscript to better clarify our problem setup and the motivation behind the model choices we have made.\n\n**2. MaNo and SoftTrun in the case of the erroneous overconfidence and underconfidence.**\n\nThanks for this insightful comment and we reply to it in two parts.\n\n* *[high errors and high confidence]* This great remark concerns actual assumptions we must make in our setup (Section 3.1). One may notice that overconfidence and high errors imply that predicted probabilities are misaligned with the true ones in terms of class ranking, which is a big issue that may go beyond calibration. Considering unsupervised cases and no access to the training phase, logits are the only source of information we have, so we find it reasonable to assume that the model mistakes are mostly in low-confidence regions.\n\n* *[underconfidence]* We totally agree. That is why we do not impose any assumption on the prediction bias $\\mathbf{\\epsilon}$, so it can have positive/negative entries covering overconfidence and underconfidence. In practice, it has been shown that deep neural networks with softmax are overconfident [a], so the situation of underconfidence is unlikely to happen. As we did not observe this phenomenon in our experiments, it would be a good direction for future work to find benchmarks where underconfidence issues are present. We will include the above discussion in the revision.\n\n**3. Relation to Agreement-on-the-Line (ALine-D).**\nWe thank the reviewer for suggesting this interesting work. We would like to clarify the differences between ALine-D and MaNo.\n* *[Different settings.]* ALine-D operates under a model-centric setting, aiming to accurately estimate OOD accuracy across many different models. In contrast, MaNo is designed for a data-centric setting, focusing on estimating a single model’s OOD accuracy on various datasets.\n* *[Different assumptions.]* ALine-D assumes that (1) a set of diverse pre-trained models is available during evaluation; (2) agreement-on-the-line phenomenon holds consistently.\nIn practice, accessing a diverse set of models may be infeasible, and assumption (2) does not always hold [b, c]. On the other hand, MaNo and other data-centric methods do not face these limitations.\n\nBased on the above, ALine-D cannot be directly compared with MaNo, but we provide two indirect ways to do that.\n* In our experimental results, MaNo outperforms AgreeScore, which is based on a similar idea to ALine-D.\n* In Table D, we compare the results provided in [d], Section 5.2 with ours on CIFAR-10C to further illustrate the efficiency of MaNo even without the two assumptions. \n\n**Table D: MaNo v.s. Agreement-on-the-Line (ALine-D) on CIFAR-10C with ResNet-18.**\n|Metric|ALine-D|MaNo|\n| --- | - | - | \n| $R^2$| **0.995** | **0.995** | \n| $\\rho$| 0.974 | **0.997** |\n\n**4. What makes MaNo superior to AC?**\nThanks for raising this question. First, we would like to clarify a possible misunderstanding between $L_\\infty$ norm and AC. \n* *[ AC is not a specific case of $p=\\infty$]* When we consider $L_\\infty$-norm, it means that we compute the maximum value over the whole prediction matrix, while AC extracts the maximum value of each row in that matrix.\n\nThen, to answer the question:\n* *[AC ignores the distances to subconfident decision boundaries.]* Our analysis in Section 3.1 shows that distance to each class boundary is important, so by considering the confidence of the most probable class only,  AC may bring a loss of information.\n* *[AC considers each sample separatively.]* MaNo puts greater emphasis on high-margin terms globally when $p$ increases, while AC considers the high-margin values of each sample with the same weight. \n\n---\n\n[a] Leveraging ensemble diversity for robust self-training in the presence of sample selection bias.\n\n[b] ID and OOD performance are sometimes inversely correlated on real-world datasets.\n\n[c] Accuracy on the Line: On the Strong Correlation Between Out-of-Distribution and In-Distribution Generalization.\n\n[d] Agreement-on-the-Line: Predicting the Performance of Neural Networks under Distribution Shift.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qOkWySXFUe", "reviewer": "Reviewer_A5Zk", "summary": "​​This paper presents MANO, a straightforward and efficient training-free approach for estimating test accuracy in an unsupervised manner, leveraging the Matrix Norm of neural network predictions on test data. The method is inspired by the low-density separation assumption, which posits that optimal decision boundaries should reside in low-density regions. An extensive empirical study on standard unsupervised accuracy estimation benchmarks reveals that MANO consistently achieves state-of-the-art performance across diverse architectures, even in the presence of synthetic, natural, or subpopulation shifts.", "questions": "1. Could the authors also report the performance of the approaches under the absolute estimation error metric?\n\n2. Could the authors perform an ablation study on the effect of the hyper-parameter $\\eta$?\n\n3. Could the authors add the missing related works on unsupervised accuracy estimation?", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. It demonstrates that logits can effectively indicate generalization performance by reflecting distances to decision boundaries, in alignment with the low-density separation assumption.\n\n2. It introduces MANO, a training-free method for estimating test accuracy by computing the $L_p$ norm of the logits matrix, which quantifies global distances to decision boundaries. MANO employs a novel normalization technique that balances information completeness and error accumulation, and is resilient to various calibration scenarios. Additionally, it reveals a connection to the model’s uncertainty.\n\n3. It conducts a comprehensive empirical evaluation, encompassing 12 benchmarks across diverse distribution shifts, to showcase MANO’s superiority over 11 baseline methods. The results consistently show that MANO outperforms state-of-the-art baselines, even under challenging natural shifts.", "weaknesses": "1. It does not address how to translate the proposed MANO into a practical estimated accuracy, which is crucial for real-world applications. Additionally, it should report the performance of the methods using the absolute estimation error metric, defined as the absolute difference between the estimated accuracy and the actual accuracy, for unsupervised accuracy estimation.\n\n2. The justification for selecting $\\eta=5$ is not robust. An ablation study on the impact of varying $\\eta$ should be conducted to strengthen the argument.\n\n3. It lacks a discussion of some key related works on unsupervised accuracy estimation, such as [1][2].\n\n[1] Chen, Jiefeng, et al. \"Detecting errors and estimating accuracy on unlabeled data with self-training ensembles.\" Advances in Neural Information Processing Systems 34 (2021): 14980-14992.\n\n[2] Chuang, Ching-Yao, Antonio Torralba, and Stefanie Jegelka. \"Estimating Generalization under Distribution Shifts via Domain-Invariant Representations.\" International Conference on Machine Learning. PMLR, 2020.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive comments and very valuable suggestions to improve the quality of our paper. We address the reviewer's concerns below. Please let us know if any issues remain.\n\n**1. How to use $\\texttt{MaNo}$ in practice?**\nThis work demonstrates the strong correlation between ground-truth OOD accuracy and the designed score, which can be particularly useful for model deployment applications. In the following, we provide two examples.\n* *[Find difficult (under-performed) test set]* In cases such as retraining on under-performed datasets or annotating hard datasets, we only need to know the rank of datasets by accuracy. Therefore, we can calculate the proposed score for each dataset directly and fulfill the task based on this score's ranking. \n* *[Deployement risk estimation.]* When deploying the model into production, it is important to estimate its safety. If the cost of getting test labels is prohibitive, our method can help estimate the model's accuracy on the product's test data. A practitioner can additionally look at the variability of the score on multiple test sets. When multiple datasets are not available, we can alternatively construct adequate synthetic datasets via various visual transformations. \n\nWe thank the reviewer for raising this question, we will update our paper accordingly.\n\n**2. Numerical results by using the absolute estimation error metric.**\n\nThank you very much for introducing this metric to us, which will enhance the experiment quality significantly. In below Table E, we provide partial numerical results of our experiments by using the absolute estimation error. We will include all the results using this metric in our final version. \n\n**Table E: Mean absolute error on CIFAR-10 and Office-Home with ResNet-18.**\n|Dataset|ConfScore|Entropy|ATC|COT|Nuclear|MaNo|\n| --- | - | - | -|-|-|-|\n| CIFAR-10| 3.371 | 3.188 |2.677|1.381|1.358|**0.427**|\n| Office-Home| 4.212|4.568|6.523|3.330|3.886|**2.230**|\n\n\n**3. Ablation study on $\\eta$.**\nWe thank the reviewer for this suggestion. We provide the ablation study in Table F below, where we observed that $\\eta=5$ can effectively distinguish the calibration scenarios.\n\n**Table F: Performance on CIFAR-10, Office-31, and PACS with ResNet18 for varying value of $\\eta$ . The metric used in this table is the coefficient of determination $R^2$.**\n|Dataset|$\\eta$=0|$\\eta$=1|$\\eta$=3|$\\eta$=5|$\\eta$=7|$\\eta$=9|\n| --- | - | - | -|-|-|-|\n| CIFAR-10| 0.995 | 0.995 |0.995|0.995|0.995|0.995|\n| Office-Home| 0.926 |0.926|0.926|0.926|0.777|0.777|\n| PACS|0.541|0.541|0.541|0.827|0.827|0.827\n| Average| 0.820|0.820|0.820|**0.916**|0.866|0.866|\n\nIt is worth noting that we provide the theoretical analysis to demonstrate the general reason why we choose $\\eta$=5 in Section D.3. It indicates that $\\eta$=5 ensures that $\\Phi(D_{test})$ deviates from its mean by the variance with probability smaller than 0.05.\n\n**4. Discussion about the two related works.**\nWe thank the reviewer for suggesting these two significant works. Both approaches utilize the disagreement between a pre-trained model and a target-adapted model to predict accuracy on each test set. In contrast, our MANO leverages the model’s logits for accuracy estimation without any training on target datasets. We will include these works and clarify the differences in our revision.\n\nWe thank the reviewer for their questions which helped us improve our work. We remain open to further discussion in case some issues remain unaddressed.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "e0JPDqVpNd", "reviewer": "Reviewer_Uw9o", "summary": "The paper addresses the challenge of estimating the test accuracy of pre-trained neural networks on out-of-distribution (OOD) samples without access to ground-truth labels. Current logit-based methods often suffer from overconfidence, leading to prediction bias. The authors propose a new method called MANO, which applies data-dependent normalization on the logits and uses the Lp norm of the normalized logits matrix as the estimation score. This approach aims to reduce prediction bias and improve accuracy estimation. Theoretical and empirical analyses demonstrate that MANO outperforms existing methods across various types of distribution shifts and architectures.", "questions": "See weakness.", "limitations": "N/A.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper provides a solid theoretical analysis connecting logits to model uncertainty and generalization performance, supported by the low-density separation assumption.\n2. The paper is well-written and logically structured, making it easy to follow the authors' arguments and methodology.\n3. Extensive empirical studies on multiple benchmarks show that MANO outperforms existing state-of-the-art methods in different distribution shift scenarios.", "weaknesses": "1. How can you ensure the universality of the Low-Density Separation (LDS) assumption in practical applications? Are there specific scenarios or datasets where this assumption might not hold?\n2. What are the specific mathematical advantages of the SoftTrun normalization strategy compared to traditional softmax? To what extent do these advantages rely on particular data distributions?\n3. In Table 1 and Table 2, the performance improvements over existing methods appear to be relatively minor in many instances. Is it possible to calculate other uncertainty measures to assess the significance of these results more accurately?\n4. In most of the experimental results, the outcomes are close to the maximum value of the evaluation metrics. Does this imply that the research problem has already been solved?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the positive comments as well as the insightful suggestions. We will address the reviewer's concerns below. Please let us know if any issues remain, we would be happy to continue this discussion to address them.\n\n**1. The universality of the Low-Density Separation (LDS) assumption in practical applications.**\n\nThis is a very interesting and important question, so our answer will be a bit detailed. We apologize for the length. In case if the reviewer wants to know more details on the LDS assumption, please refer to Chapter 1 of Olivier Chapelle et al.’s book [e]. \n\nWhen dealing with classification problems, we always implicitly rely on the smoothness assumption that two data points close to each other are from the same class with high probability. Originally, the LDS assumption was proposed as an attempt to apply this reasoning to unlabeled data in order to train semi-supervised models, and it could be considered as a natural hypothesis we put forward on the problem at hand. In our case, we pose this assumption with respect to the *pre-trained model* in order to be able to analyze its behavior on unlabeled data. Hence, the LDS assumption does not hold when the pre-trained model makes lots of mistakes on examples that are far away from the decision boundary, which, for example, can happen in the presence of unreasonable distribution shifts that exclude deploying the model on the target domain. In the paragraph `Assumptions on the prediction bias` of Section 3.1, we elaborated further on the assumptions that we should make with respect to the pre-trained model and amount of distribution shift for the safe application of accuracy estimation methods (*because this reasoning applies, to some extent, to all our competitors*).\n\n**2. The specific mathematical advantages of the $\\texttt{SoftTrun}$ normalization strategy compared to traditional softmax.**\nThe softmax can be seen as a composition between the exponential and the scaling $\\phi \\colon u \\mapsto u/\\lVert u \\rVert_1$. We study the nice mathematical properties of $\\phi$ in Appendix D.6 and design $\\texttt{SoftTrunc}$ such that it preserves those properties. As shown in Equation (6), $\\texttt{SoftTrunc}$ recovers the softmax when the model is well-calibrated, and when it is not, it uses a Taylor approximation of the exponential (which is a $2^{nd}$ order polynomial) that reduces the influence of the prediction bias $\\mathbf{\\epsilon}$ as shown in Equation (4) and Figure 2(b). \n\n**3. To what extent do advantages of $\\texttt{SoftTrun}$ rely on particular data distributions?**\nThe advantages of $\\texttt{SoftTrun}$ do not rely on the data distributions, since we do not make any assumptions on the prediction bias $\\mathbf{\\epsilon}$ and the criterion $\\Phi(D_{test})$. It autonomously distinguishes calibration scenarios on data from different distribution shifts.\n\n**4. Enhancing the performance in Table 1 and Table 2 by selecting other uncertainty measures.**\n\nWe thank the reviewer for raising this question. We would distinguish the two following reasons why the performance enhancement in Table 1 and Table 2 seems minor.\n* *[The estimation performance under synthetic and subpopulation shifts are close to the optimal.]* In our experimental protocol, we followed other papers in this domain, but it is true that the performance for synthetic and subpopulation shift datasets is almost perfect, so the improvement with respect to the state-of-the-art is not very large. Thus, the community may need more challenging datasets (e.g., natural shifts in Table 3) for these types of domain shifts.\n* *[The selection of metrics.]* We utilize commonly-used metrics including $R^2$ and $\\rho$ to measure the linear relationship between designed scores and true test accuracy, and, from our experience, these metrics may not provide a clear insight into the degree of the improvement. Nevertheless, thanks to the suggestion of Reviewer A5Zk, we will add to the revised manuscript the results evaluated by the mean absolute error. In Table G below, we can see that with this metric the superiority of MaNo on CIFAR-10 is more evident now.\n\n**Table G: Mean absolute error on CIFAR-10 with ResNet-18.**\n|Dataset|ConfScore|Entropy|ATC|COT|Nuclear|MaNo|\n| --- | - | - | -|-|-|-|\n| CIFAR-10| 3.371 | 3.188 |2.677|1.381|1.358|**0.427**|\n\n**5. Does this imply that the research problem has already been solved?**\nWe believe that there exists still many open questions in this field. These are some of them:\n* *[Room for improvement under the natural shift]* The numerical results in Table 3 show that current test accuracy estimation under the natural shift needs to be further improved, which is a more complex and practically meaningful question.\n* *[Generalization of LLM / other data modalities.]* Unsupervised accuracy estimation papers have a focus today mostly on computer vision applications, but it would be important to explore also other data modalities and network architectures. In particular, a promising research direction would be to analyze the possibility of unsupervised performance estimation for large language models.\n* *[The underlying working mechanisim of generalization is still unclear.]* One of the essential goals of this field is to understand when and why the model is able to generalize to unseen domains. However, the generalization capability of models is still a mysterious desideratum. \n\n---\n[e] Chapelle, O., Scholkopf, B., & Zien, A. (2006). Semi-supervised learning. 2006.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We thank the reviewers for their time and valuable suggestions. We are deeply grateful to them for acknowledging the **novelty and quality** of our study (Reviewer 6N1W, A5Zk, Uw9o) while noting its **effectiveness and superiority on large-scale experiments** (Reviewers 6N1W, 5Gdr, A5Zk, Uw9o). We are also encouraged to know that all reviewers found that our proposed method is built on **solid theoretical support** (Reviewer 6N1W, 5Gdr, A5Zk, Uw9o). 

We provided all the additional experiments requested by the reviewers and remain open to continuing this constructive discussion for the length of the rebuttal period.

We believe that the paper strongly benefited from the reviews and hope that this, together with the multiple additional experiments provided in the individual answers, will allow the reviewers to reconsider their evaluation of our work if they think we addressed their concerns.

Meta-review: All the reviewers recommended acceptance. The paper proposes a novel method for accuracy estimation based on the matrix norm of logits, based on a solid theoretical foundation. The paper is clearly written and presents an extensive set of experiments with convincing results.
Decision: Accept (poster)