Title: MaNo: Exploiting Matrix Norm for Unsupervised Accuracy Estimation Under Distribution Shifts
Venue: NeurIPS 2024
Accepted: yes
Keywords: Unsupervised Learning

Distribution Shifts

Unsupervised Accuracy Estimation

Generalization

Deep Learning
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

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 MANO that (1) applies a data-dependent normalization on the logits to reduce prediction bias, and (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 MANO 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 .

1 Introduction

The deployment of machine learning models in real-world scenarios is frequently challenged by distribution shifts between the training and test data. These shifts can substantially deteriorate the model'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, it is common to monitor model performance by periodically collecting the ground truth labels for a subset of the current test dataset (Lu et al., 2023). However,

this approach is often resource-intensive and time-consuming, which motivates the importance of estimating the model's performance on out-of-distribution (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 to estimate accuracy without labels is to rely on the model's outputs, called logits, as a 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 softmax probability as the test accuracy estimator, while Deng et al. (2023) has recently proposed to estimate the accuracy via the nuclear norm of the softmax probability matrix. These approaches, however, tend to 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 a dataset level by taking the L p -norm 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 et al., 2024; Wei et al., 2022a), resulting in high prediction bias, especially under poorly-calibrated scenarios. This leads us to another critical question:

Question 2: How to alleviate the overconfidence issues of logits-based methods?

In Section 4, we reveal that this question is connected to the normalization of logits and show that the widely-used softmax 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 a 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-truth logits and error accumulation.

Summary of our contributions. (1) We show that logits are informative of generalization performance through the lens of the low-density separation assumption by reflecting the distances to decision boundaries. (2) We identify the failure of the commonly-used 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 L p norm of the logits matrix. MANO relies on softrun , a novel normalization technique that makes a trade-off between information completeness and error accumulation and is robust to different calibration scenarios. In addition, we demonstrate its connection to the model's uncertainty. (4) We demonstrate the superiority of MANO compared to 11 competitors with a large-scale empirical evaluation including 12 benchmarks across diverse distribution shifts. Results show that MANO consistently improves over the state-of-the-art 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 , . . . , K } . Let p S and p T be the source and target distributions on X × Y , respectively, with p S = p T . We parameterize a neural network f : X → R K as f = f W ◦ f φ , where f φ : X → R q is a feature extractor and f W : R q → R K is a linear classifier with parameters W = ( ω k ) K k =1 ∈ R q × K . Further, we denote an input by x , its corresponding label by y , its representation by z = f φ ( x ) and logits by q = f ( x ) = ( ω ⊤ k z ) k ∈ R K . The accuracy of f on D is defined as Acc( f, D ) := 1 |D| ∑ ( x ,y ) ∈D ✶ ˆ y = y with predicted labels ˆ y . The probability simplex is denoted by ∆ K = { p ∈ [0 , 1] K | ✶ ⊤ K p = 1 } .

̸

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Unsupervised accuracy estimation. Given a model f pre-trained on a training set D train with samples drawn i.i.d. from p S , the goal of unsupervised accuracy estimation is to assess its generalization performance on a given unlabeled test set D test = { x i } N i =1 with N samples drawn i.i.d. from p T . More specifically, we assume (i) a source-free regime (no direct access to D train ), (ii) no access to test labels, and (iii) a distribution shift, i.e. p S = p T , . In this challenging setup, which often occurs in real-world scenarios when ground-truth labels are inaccessible at a test time, we aim to design an estimation score S ( f, D test ) that exhibits a linear correlation with the true OOD accuracy Acc( f, D test ) . Following the standard closed-set setting, both p T and p S 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, we seek to understand when and why logits can be informative for analyzing generalization performance. Based on the derived understanding, we propose our approach, MANO, for estimating generalization performance.

3.1 Motivation

Logits reflect the distances to decision boundaries. We analyze logits from a linear classification perspective in the embedding space, where the decision boundary of class k is the hyperplane { z ′ ∈ R q | ω ⊤ k z ′ = 0 } . In Appendix D.2, we remind that the distance from a point z to hyperplane ω k is given by d( ω k , z ) = | ω ⊤ k z | / ∥ ω k ∥ . As the pre-trained model is fixed and ω k can be normalized, we derive that the logits in absolute values are proportional to the distance from the learned embeddings to the decision boundaries, i.e. , | q k | = | ω ⊤ k z | ∝ d ( ω k , z ) , ∀ k . 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 distance to decision boundaries and generalization performance. The lowdensity separation assumption (LDS, Chapelle and Zien, 2005) states that optimal decision boundaries should lie in low-density regions (Figure 1) so that unlabeled margin | ω ⊤ k z | reflects reliable confidence in predicting x to the class k . The assumption is often empirically supported as the misclassified samples tend to be significantly closer to the decision boundary than the correctly classified ones

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Class 1

Class 2

Decision Boundary

(a) High-density region

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5.0

7.5

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Class 2

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(b) Low-density region.

(Mickisch et al., 2020). This might indicate that the absolute values of the logits are positively correlated to its generalization performance .

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 ∗ + ε in the general case ⋄ , i.e. , subject to a potentially non-negligible prediction bias ε = ( ε k ) k with respect to the ground-truth logits q ∗ ∈ R K . The following proposition shows the impact of the prediction bias on the divergence between the true class posterior probabilities, assumed modeled as p = softmax( q ∗ ) ∈ ∆ K , and the estimated ones s = softmax( f ( x )) ∈ ∆ K .

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 performance as well as 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 1: Normalization. Given that logits can exhibit significant variations in their scale depending on the input x , it is crucial to normalize the logits within a 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 ∆ K , which amounts to applying a normalization function σ : R K → ∆ K on top of the pre-trained neural network (Mensch et al., 2019), where ∆ K refers to probability simplex. This ensures having logits entries in [0 , 1] . For each test sample x i , we first extract its learned feature representation z i = f φ ( x i ) . Then, logits corresponding to this representation are computed as q i = f W ( z i ) ∈ R K . The normalization procedure results in a prediction matrix Q ∈ R N × K with each row Q i containing the normalized logits of an input sample:

where σ denotes the normalization function for the logits values. It is worth noting that not all normalization methods are appropriate candidates. The selection of a suitable normalization function σ 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 feature-to-boundary distances by taking the entry-wise L p norm of the prediction matrix Q , which can be expressed as:

As we have || Q || p ≤ p √ NK max( Q ij ) = p √ NK ( Q ij ∈ [0 , 1] ), the scaling by p √ NK leads to S ( f, D test ) ∈ [0 , 1] , providing a standardized metric regardless of variations in the size of the test dataset N 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 p →∞ , we have | Q || p → max( Q ij ) . In practice, we choose p = 4 in all experiments and provide an ablation study on p in Appendix G.1. As the L p norm is straightforward to compute, our approach is scalable and efficient compared to the current state-of-the-art method Nuclear (Deng et al., 2023) that requires performing a 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 in Theorem 3.3. Before presenting this result, we recall below the definition of Tsallis α -entropies introduced in Tsallis (1988).

0.0

0.2

0.4

0.6

0.8

1.0

Predicted probability

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0.2

0.4

0.6

0.8

1.0

True probability

Office-Home

PACS

(a) Calibration curves.

(b) Type of normalization.

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0.60

0.65

0.70

0.75

0.80

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PACS

Office-Home

(c) Order

n

in Eq. (4).

Definition 3.2 (Tsallis α -entropies (Tsallis, 1988)) . Let α > 1 and k > 0 . The Tsallis α -entropy is defined as:

In this work, we choose k = 1 α following Blondel et al. (2019). The Tsallis entropies generalize the Shannon entropy (limit case α → 1 ) and have been used in various applications (Blondel et al., 2019, 2020; Muzellec et al., 2017). More details can be found in Appendix C. The following theorem, whose proof is deferred to Appendix D.3, states that the estimation score obtained with MANO is a function of the average Tsallis entropy of the normalized neural network's logits.

Theorem 3.3 (Connection to uncertainty) . Let p > 1 , a = p ( p -1) K and b = 1 K . Given a test set D test = { x i } N i =1 , corresponding logits q i = f ( x i ) , a normalization function σ : R K → ∆ K and p > 1 , the estimation score S ( f, D test ) provided by MANO (Algorithm 1) verifies

As a > 0 , Theorem 3.3 implies that the estimation score provided by MANO is negatively correlated with the average Tsallis-entropy on the test set. In particular, the less certain the model is on test data, the lower the test accuracy is and the higher the entropy term is in Eq. (3), resulting in a lower score S ( f, D test ) . As the converse sense holds, MANO provides a score positively correlated to the test accuracy. This follows the findings of Guillory et al. (2021); Wang et al. (2021) and empirically confirmed in Section 5 for various architectures, datasets, and types of shift.

4 How to Alleviate Overconfidence Issues of Logit-Based Methods?

The most common normalization technique of existing logit-based approaches is the softmax normalization. In this section, we show that the widely used softmax 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 softmax normalization can suffer from overconfidence issues (Odonnat et al., 2024; Wei et al., 2022b) and saturation of its outputs (Chen et al., 2017), with one entry close to one while the others are close to zero.

Analysis. To alleviate those issues, we first notice that the softmax can be decomposed as softmax( q ) = exp( q ) / ∑ K k =1 exp( q k ) = ( ϕ ◦ exp)( q ) , where ϕ : R K + → ∆ K writes ϕ ( u ) = u / ∑ K k =1 u k = u / ∥ u ∥ 1 . While ϕ has appealing property for normalization (see Proposition D.2), the exponential can accumulate prediction errors, leading to the softmax 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 i writes q ∗ i,k + ε k , where q ∗ i,k are the ground-truth logits and ε k is the prediction error. Then, the n th -order Taylor polynomial of the exponential writes

Consequently, logit-based accuracy estimation methods using softmax 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 pretrained ResNet18 exhibits more pronounced calibration issues on PACS (Li et al., 2017) compared to Office Home (Venkateswara et al., 2017). Figure 2(b) shows that using MANO with softmax normalization is appropriate on Office-Home where the model is well calibrated but not in a miscalibrated scenario on PACS. Conversely, using MANO with 2 nd -order 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 σ . Figure 2(c) illustrates the impact of truncating Eq. (4) up to the n -th order. We conclude that a trade-off is needed between information completeness on true logits and error accumulation depending on the type of calibration scenario. Specifically, when the model is poorly calibrated on a given dataset ( i.e. , ε k large in absolute value), the normalization should focus on avoiding error accumulation, and when the model is well calibrated ( i.e. , ε k small in absolute value), the normalization should focus on information completeness.

4.2 Softrun: The Proposed Normalization Strategy

The above analysis shows that different calibration scenarios emphasize different information during normalization. Therefore, we propose a normalization strategy called softrun that normalizes the model outputs based on the calibration scenario. Given logits q i ∈ R K and reusing the function ϕ previously introduced, it takes the general form:

where v : R K → R K + 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 a test dataset D test = { x i } N i =1 and corresponding logits q i = f ( x i ) , a criterion Φ( D test ) is computed at the dataset level and the normalized logits are defined as 1

We define Φ( D test ) = -1 NK ∑ N j =1 ∑ K k =1 log( exp( q j ) k ∑ K j =1 exp( q i ) j ) ) , which is equal, up to a constant, to the average KL divergence between the uniform distribution and the predicted softmax probabilities. It follows from Tian et al. (2021) that showed that this KL divergence was small when the uncertainty of the model was high and large for confident models. Hence, when the model is uncertain, i.e. , Φ( D test ) ≤ η , we truncate the exponential to reduce error accumulation, and when the model is certain, i.e. , Φ( D test ) > η , complete information is used with the exact exponential (and we recover the softmax ). Thus, softrun is designed to treat the problem with an additional level of complexity often overlooked by previous methods. While this comes at the cost of introducing the hyperparameter η , we fix η = 5 across all our experiments. This, along with the design of softrun , is justified both theoretically in Appendix E and experimentally in Appendix G.3.

Synthetic Shift

Dataset

Model

Rotation

ConfScore

Entropy

AgreeScore

ATC

Fr´ echet

Dispersion

ProjNorm

MDE

COT

Nuclear

MANO

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2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

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ρ

R

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ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

CIFAR 10

ResNet18

0.822

0.951

0.869

0.985

0.899

0.987

0.663

0.929

0.884

0.985

0.950

0.971

0.968

0.990

0.936

0.982

0.957

0.987

0.989

0.995

0.995

0.997

0.995

0.997

ResNet50

0.835

0.961

0.935

0.993

0.945

0.994

0.835

0.985

0.946

0.994

0.858

0.964

0.987

0.990

0.944

0.989

0.978

0.963

0.984

0.996

0.994

0.996

0.996

0.997

WRN-50-2

0.862

0.976

0.943

0.994

0.942

0.994

0.856

0.986

0.947

0.994

0.814

0.973

0.962

0.988

0.961

0.989

0.930

0.809

0.988

0.994

0.994

0.995

0.996

0.992

Average

0.840

0.963

0.916

0.991

0.930

0.992

0.785

0.967

0.926

0.991

0.874

0.970

0.972

0.990

0.947

0.987

0.955

0.920

0.987

0.995

0.995

0.996

0.996

0.995

CIFAR 100

ResNet18

0.860

0.936

0.916

0.985

0.891

0.979

0.902

0.973

0.938

0.986

0.888

0.968

0.952

0.988

0.979

0.980

0.975

0.994

0.991

0.995

0.989

0.995

0.996

0.996

ResNet50

0.908

0.962

0.919

0.984

0.884

0.977

0.922

0.982

0.921

0.984

0.837

0.972

0.951

0.985

0.988

0.991

0.988

0.995

0.985

0.996

0.979

0.994

0.995

0.997

WRN-50-2

0.924

0.970

0.971

0.984

0.968

0.981

0.955

0.977

0.978

0.993

0.865

0.987

0.980

0.991

0.990

0.991

0.995

0.994

0.987

0.997

0.962

0.988

0.996

0.998

Average

0.898

0.956

0.936

0.987

0.915

0.983

0.927

0.982

0.946

0.988

0.864

0.976

0.962

0.988

0.985

0.987

0.986

0.994

0.988

0.996

0.977

0.993

0.996

0.997

TinyImageNet

ResNet18

0.786

0.946

0.670

0.869

0.592

0.842

0.561

0.853

0.751

0.945

0.826

0.970

0.966

0.986

0.970

0.981

0.941

0.993

0.985

0.994

0.983

0.994

0.981

0.996

ResNet50

0.786

0.947

0.670

0.869

0.651

0.892

0.560

0.853

0.751

0.945

0.826

0.971

0.977

0.986

0.979

0.987

0.941

0.993

0.980

0.994

0.965

0.994

0.980

0.996

WRN-50-2

0.878

0.967

0.757

0.951

0.704

0.935

0.654

0.904

0.635

0.897

0.884

0.984

0.968

0.986

0.965

0.983

0.961

0.996

0.985

0.997

0.962

0.988

0.979

0.997

Average

0.805

0.959

0.727

0.920

0.650

0.890

0.599

0.878

0.693

0.921

0.847

0.976

0.970

0.987

0.972

0.984

0.950

0.995

0.984

0.995

0.968

0.993

0.980

0.996

ImageNet

ResNet18

-

-

0.979

0.991

0.963

0.991

-

-

0.974

0.983

0.802

0.974

0.940

0.971

0.975

0.993

0.924

0.994

0.996

0.998

0.992

0.997

0.992

0.997

ResNet50

-

-

0.980

0.994

0.967

0.992

-

-

0.970

0.983

0.855

0.974

0.938

0.968

0.986

0.993

0.886

0.994

0.993

0.996

0.985

0.997

0.991

0.998

WRN-50-2

-

-

0.983

0.991

0.963

0.991

-

-

0.983

0.993

0.909

0.988

0.939

0.976

0.978

0.993

0.880

0.997

0.989

0.994

0.987

0.998

0.996

0.998

Average

-

-

0.981

0.993

0.969

0.992

-

-

0.976

0.987

0.855

0.979

0.939

0.972

0.980

0.993

0.897

0.995

0.993

0.996

0.988

0.998

0.993

0.998

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 including 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 RR1-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-100C, and ImageNet-C (Hendrycks and Dietterich, 2019) that span 19 types of corruption across 5 severity levels, as well as TinyImageNet-C (Hendrycks and Dietterich, 2019) with 15 types of corruption and 5 severity levels. For the natural shift, we use the domains excluded from training from PACS, Office-Home, DomainNet, and RR1WILDS 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, we use SGD with a 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 metrics. We use the coefficient of determination R 2 ∈ [0 , 1] (Nagelkerke et al., 1991) and the Spearman's rank correlation coefficient ρ ∈ [ -1 , 1] (Kendall, 1948) to evaluate performance. The former measures the linearity and goodness of fit and 1 indicates a perfect fit. ρ measures monotonicity and values close to {-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 (ConfScore) (Hendrycks and Gimpel, 2016), Entropy (Guillory et al., 2021), Agreement Score (AgreeScore) (Jiang et al., 2021), Averaged Threshold Confidence (ATC) (Garg et al., 2022), AutoEval (Fr´ echet) (Deng and Zheng, 2021), ProjNorm (Yu et al., 2022), Dispersion Score (Dispersion) (Xie et al., 2023), MDE (Peng et al., 2024), COT (Lu et al., 2024), and Nuclear Norm (Nuclear) (Deng et al., 2023).

Subpopulation Shift

Dataset

Model

Rotation

ConfScore

Entropy

AgreeScore

ATC

Fr´ echet

Dispersion

ProjNorm

MDE

COT

Nuclear

MANO

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

Entity-13

ResNet18

0.927

0.961

0.795

0.940

0.794

0.935

0.543

0.919

0.823

0.945

0.950

0.981

0.937

0.968

0.952

0.981

0.927

0.995

0.960

0.985

0.978

0.991

0.992

0.996

ResNet50

0.932

0.976

0.728

0.941

0.698

0.928

0.901

0.964

0.783

0.950

0.903

0.959

0.764

0.892

0.944

0.974

0.912

0.993

0.935

0.971

0.989

0.996

0.993

0.998

WRN-50-2

0.939

0.983

0.930

0.977

0.919

0.973

0.871

0.935

0.936

0.980

0.906

0.958

0.815

0.905

0.950

0.977

0.925

0.995

0.944

0.979

0.989

0.995

0.992

0.996

Average

0.933

0.973

0.817

0.953

0.804

0.945

0.772

0.939

0.847

0.958

0.920

0.966

0.948

0.977

0.839

0.922

0.921

0.995

0.947

0.979

0.985

0.994

0.993

0.996

Entity-30

ResNet18

0.964

0.979

0.570

0.836

0.553

0.832

0.542

0.935

0.611

0.845

0.849

0.978

0.929

0.968

0.952

0.987

0.931

0.994

0.971

0.993

0.980

0.993

0.991

0.996

ResNet50

0.961

0.980

0.878

0.969

0.838

0.956

0.914

0.975

0.924

0.973

0.835

0.956

0.783

0.914

0.937

0.986

0.918

0.995

0.958

0.982

0.978

0.994

0.988

0.997

WRN-50-2

0.940

0.978

0.897

0.974

0.878

0.970

0.826

0.955

0.936

0.984

0.927

0.973

0.927

0.973

0.959

0.986

0.925

0.995

0.944

0.979

0.985

0.996

0.988

0.997

Average

0.955

0.978

0.781

0.926

0.756

0.919

0.728

0.956

0.823

0.934

0.871

0.969

0.880

0.952

0.949

0.987

0.925

0.995

0.970

0.988

0.981

0.994

0.989

0.996

Living-17

ResNet18

0.876

0.973

0.913

0.973

0.898

0.970

0.586

0.736

0.940

0.973

0.768

0.950

0.900

0.958

0.923

0.970

0.927

0.985

0.972

0.984

0.975

0.987

0.980

0.991

ResNet50

0.906

0.956

0.880

0.967

0.853

0.961

0.633

0.802

0.938

0.976

0.771

0.926

0.851

0.929

0.903

0.924

0.914

0.985

0.953

0.973

0.967

0.976

0.975

0.997

WRN-50-2

0.909

0.957

0.928

0.980

0.921

0.977

0.652

0.793

0.966

0.984

0.931

0.967

0.931

0.966

0.915

0.970

0.914

0.983

0.965

0.990

0.951

0.978

0.961

0.996

Average

0.933

0.974

0.907

0.973

0.814

0.969

0.623

0.777

0.948

0.978

0.817

0.949

0.894

0.951

0.913

0.969

0.918

0.984

0.963

0.982

0.964

0.980

0.972

0.995

Nonliving-26

ResNet18

0.906

0.955

0.781

0.925

0.739

0.909

0.543

0.810

0.854

0.939

0.914

0.980

0.958

0.981

0.939

0.978

0.929

0.989

0.982

0.992

0.970

0.989

0.978

0.991

ResNet50

0.916

0.970

0.832

0.942

0.776

0.918

0.638

0.837

0.893

0.960

0.848

0.950

0.805

0.907

0.873

0.972

0.907

0.993

0.962

0.984

0.956

0.985

0.975

0.995

WRN-50-2

0.917

0.977

0.932

0.971

0.912

0.959

0.676

0.861

0.945

0.969

0.885

0.942

0.893

0.939

0.924

0.973

0.909

0.991

0.962

0.979

0.960

0.988

0.978

0.992

Average

0.913

0.967

0.849

0.946

0.809

0.929

0.618

0.836

0.897

0.956

0.882

0.957

0.913

0.974

0.886

0.943

0.915

0.991

0.969

0.985

0.962

0.987

0.977

0.992

Subpopulation Shift

Dataset

Model

Rotation

ConfScore

Entropy

AgreeScore

ATC

Fr´ echet

Dispersion

ProjNorm

MDE

COT

Nuclear

MANO

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

Entity-13

ResNet18

0.927

0.961

0.795

0.940

0.794

0.935

0.543

0.919

0.823

0.945

0.950

0.981

0.937

0.968

0.952

0.981

0.927

0.995

0.960

0.985

0.978

0.991

0.992

0.996

ResNet50

0.932

0.976

0.728

0.941

0.698

0.928

0.901

0.964

0.783

0.950

0.903

0.959

0.764

0.892

0.944

0.974

0.912

0.993

0.935

0.971

0.989

0.996

0.993

0.998

WRN-50-2

0.939

0.983

0.930

0.977

0.919

0.973

0.871

0.935

0.936

0.980

0.906

0.958

0.815

0.905

0.950

0.977

0.925

0.995

0.944

0.979

0.989

0.995

0.992

0.996

Average

0.933

0.973

0.817

0.953

0.804

0.945

0.772

0.939

0.847

0.958

0.920

0.966

0.948

0.977

0.839

0.922

0.921

0.995

0.947

0.979

0.985

0.994

0.993

0.996

Entity-30

ResNet18

0.964

0.979

0.570

0.836

0.553

0.832

0.542

0.935

0.611

0.845

0.849

0.978

0.929

0.968

0.952

0.987

0.931

0.994

0.971

0.993

0.980

0.993

0.991

0.996

ResNet50

0.961

0.980

0.878

0.969

0.838

0.956

0.914

0.975

0.924

0.973

0.835

0.956

0.783

0.914

0.937

0.986

0.918

0.995

0.958

0.982

0.978

0.994

0.988

0.997

WRN-50-2

0.940

0.978

0.897

0.974

0.878

0.970

0.826

0.955

0.936

0.984

0.927

0.973

0.927

0.973

0.959

0.986

0.925

0.995

0.944

0.979

0.985

0.996

0.988

0.997

Average

0.955

0.978

0.781

0.926

0.756

0.919

0.728

0.956

0.823

0.934

0.871

0.969

0.880

0.952

0.949

0.987

0.925

0.995

0.970

0.988

0.981

0.994

0.989

0.996

Living-17

ResNet18

0.876

0.973

0.913

0.973

0.898

0.970

0.586

0.736

0.940

0.973

0.768

0.950

0.900

0.958

0.923

0.970

0.927

0.985

0.972

0.984

0.975

0.987

0.980

0.991

ResNet50

0.906

0.956

0.880

0.967

0.853

0.961

0.633

0.802

0.938

0.976

0.771

0.926

0.851

0.929

0.903

0.924

0.914

0.985

0.953

0.973

0.967

0.976

0.975

0.997

WRN-50-2

0.909

0.957

0.928

0.980

0.921

0.977

0.652

0.793

0.966

0.984

0.931

0.967

0.931

0.966

0.915

0.970

0.914

0.983

0.965

0.990

0.951

0.978

0.961

0.996

Average

0.933

0.974

0.907

0.973

0.814

0.969

0.623

0.777

0.948

0.978

0.817

0.949

0.894

0.951

0.913

0.969

0.918

0.984

0.963

0.982

0.964

0.980

0.972

0.995

Nonliving-26

ResNet18

0.906

0.955

0.781

0.925

0.739

0.909

0.543

0.810

0.854

0.939

0.914

0.980

0.958

0.981

0.939

0.978

0.929

0.989

0.982

0.992

0.970

0.989

0.978

0.991

ResNet50

0.916

0.970

0.832

0.942

0.776

0.918

0.638

0.837

0.893

0.960

0.848

0.950

0.805

0.907

0.873

0.972

0.907

0.993

0.962

0.984

0.956

0.985

0.975

0.995

WRN-50-2

0.917

0.977

0.932

0.971

0.912

0.959

0.676

0.861

0.945

0.969

0.885

0.942

0.893

0.939

0.924

0.973

0.909

0.991

0.962

0.979

0.960

0.988

0.978

0.992

Average

0.913

0.967

0.849

0.946

0.809

0.929

0.618

0.836

0.897

0.956

0.882

0.957

0.913

0.974

0.886

0.943

0.915

0.991

0.969

0.985

0.962

0.987

0.977

0.992

5.2 Main Takeaways

MANO improves over state-of-the-art. Tables 1 and 2 present the numerical results of unsupervised accuracy estimation across 8 datasets using 3 different network architectures, evaluated under synthetic and subpopulation shifts. These shifts are quantified by R 2 and ρ . Empirical results demonstrate that these distribution shifts do not significantly impact calibration ( i.e. , Φ( D test ) > η ). We observe that MANO consistently outperforms other baselines, achieving state-of-the-art performance. For instance, MANO achieves R 2 > 0 . 960 and ρ > 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 Φ( D test ) ≤ η . From Table 3, we observe a significant improvement compared with the other baselines. In particular, most methods have an R 2 and a ρ 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 3 on Entity-18 with ResNet18. We find that MANO scores present a robust linear relationship with ground-truth OOD errors, while the other state-of-the-art baselines tend to exhibit a biased estimation of high test errors.

0.6

0.7

0.8

0.9

ConfScore

40

50

60

70

80

Test Accuracy

ConfScore v.s. T est Accuracy

7.4

7.2

7.0

6.8

Negative ProjNorm

40

50

60

70

80

Test Accuracy

ProjNorm v.s. T est Accuracy

0.14

0.15

0.16

0.17

40

50

60

70

80

Test Accuracy

MaNo v.s. Test Accuracy

severity

1

2

3

4

5

corruption

frost

impulse_noise

snow

zoom_blur

brightness

elastic_transform

gaussian_blur

jpeg_compression

pixelate

spatter

contrast

gaussian_noise

motion_blur

saturate

speckle_noise

defocus_blur

fog

glass_blur

shot_noise

Improved robustness. Figure 4 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.

Natural Shift

Dataset

Model

Rotation

ConfScore

Entropy

AgreeScore

ATC

Fr´ echet

Dispersion

ProjNorm

MDE

COT

Nuclear

MANO

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

R

2

ρ

PACS

ResNet18

0.822

0.895

0.594

0.755

0.624

0.755

0.613

0.832

0.514

0.650

0.624

0.804

0.832

0.825

0.161

0.419

0.003

0.153

0.790

0.783

0.609

0.874

0.827

0.909

ResNet50

0.860

0.923

0.070

0.069

0.061

0.055

0.463

0.622

0.192

0.265

0.463

0.622

0.073

0.167

0.244

0.587

0.059

0.104

0.891

0.790

0.611

0.888

0.923

0.958

WRN-50-2

0.865

0.902

0.646

0.678

0.629

0.671

0.377

0.858

0.752

0.832

0.558

0.832

0.111

0.167

0.474

0.650

0.072

0.244

0.890

0.888

0.607

0.867

0.924

0.972

Average

0.849

0.906

0.437

0.501

0.438

0.494

0.488

0.770

0.486

0.582

0.548

0.337

0.338

0.275

0.293

0.552

0.045

0.065

0.857

0.820

0.609

0.876

0.891

0.946

Office-Home

ResNet18

0.822

0.930

0.795

0.909

0.761

0.881

0.054

0.146

0.571

0.615

0.605

0.755

0.453

0.664

0.064

0.202

0.331

0.650

0.863

0.874

0.692

0.783

0.926

0.930

ResNet50

0.851

0.944

0.769

0.895

0.742

0.853

0.026

0.216

0.487

0.734

0.607

0.685

0.383

0.727

0.169

0.475

0.342

0.622

0.762

0.846

0.731

0.895

0.838

0.916

WRN-50-2

0.823

0.958

0.741

0.874

0.696

0.846

0.132

0.405

0.383

0.643

0.589

0.706

0.456

0.713

0.172

0.531

0.342

0.650

0.863

0.874

0.766

0.874

0.800

0.895

Average

0.832

0.944

0.768

0.892

0.733

0.860

0.071

0.256

0.480

0.664

0.601

0.715

0.431

0.702

0.135

0.403

0.339

0.650

0.781

0.855

0.730

0.850

0.854

0.913

DomainNet

ResNet18

0.568

0.692

0.670

0.736

0.423

0.609

0.326

0.668

0.429

0.597

0.704

0.903

0.202

0.516

0.219

0.443

0.358

0.445

0.897

0.910

0.758

0.789

0.902

0.937

ResNet50

0.588

0.703

0.570

0.706

0.344

0.573

0.455

0.697

0.245

0.404

0.746

0.872

0.002

0.041

0.220

0.430

0.379

0.527

0.903

0.927

0.809

0.879

0.910

0.950

WRN-50-2

0.609

0.712

0.774

0.874

0.711

0.845

0.437

0.698

0.846

0.918

0.585

0.831

0.003

0.034

0.363

0.466

0.520

0.713

0.885

0.935

0.850

0.911

0.893

0.978

Average

0.588

0.702

0.671

0.722

0.493

0.676

0.406

0.688

0.507

0.639

0.678

0.869

0.069

0.197

0.234

0.446

0.419

0.562

0.894

0.919

0.805

0.895

0.899

0.949

RR1-WILDS

ResNet18

0.821

1.000

0.951

1.000

0.836

1.000

0.929

1.000

0.342

0.500

0.936

1.000

0.843

1.000

0.859

1.000

0.927

1.000

0.969

1.000

0.885

1.000

0.983

1.000

ResNet50

0.740

1.000

0.918

1.000

0.819

1.000

0.938

1.000

0.986

1.000

0.935

1.000

0.737

1.000

0.867

1.000

0.938

1.000

0.960

1.000

0.906

1.000

0.978

1.000

WRN-50-2

0.031

0.500

0.941

1.000

0.846

1.000

0.946

1.000

0.988

1.000

0.922

1.000

0.824

1.000

0.878

1.000

0.954

1.000

0.934

1.000

0.840

1.000

0.969

1.000

Average

0.530

0.833

0.937

1.000

0.833

1.000

0.938

1.000

0.779

0.833

0.931

1.000

0.801

0.833

0.868

1.000

0.940

1.000

0.953

1.000

0.877

1.000

0.977

1.000

1.0

0.8

0.6

0.4

0.2

0.0

Rotation

Confscore

Entropy

Agree

ATC

Projnorm

Nuclear

Methods

Dispersion

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 as 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 is the best method overall.

Ablation study. To motivate our choices of implementation, we provide in Appendix G ablation studies on the L p norm and the Taylor order n as well as a sensitive analysis on the calibration threshold η .

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 a 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-V2 (Recht et al., 2019). These new corruptions are perceptually dissimilar from those in ImageNet-C, 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-¯ C . In comparison, our MANO exhibits a consistent prediction pattern for both ImageNet-C and ImageNet-V2-¯ C , aligning well with the linear regression model trained on ImageNet-C. Additionally,

ResNets

1.0

0.9

0.8

0.7

0.6

ConfScore Nuclear mean

ConvNext

ViT

2

R

2

R

0.1

0.2

0.3

ConfScore

20

40

60

80

Test Accuracy

MAE=9.342

ConfScore v.s. T est Accuracy

ImageNet-C

ImageNet-V2-

C

7.0

7.5

8.0

8.5

9.0

Dispersion

10

20

30

40

50

60

70

Test Accuracy

MAE=55.032

Dispersion v.s. T est Accuracy

ImageNet-C

ImageNet-V2-

C

6.0

6.5

7.0

7.5

8.0

Nuclear

10

20

30

40

50

60

70

Test Accuracy

MAE=3.908

Nuclear v.s. T est Accuracy

ImageNet-C

ImageNet-V2-

C

1.5

2.0

2.5

0

10

20

30

40

50

60

70

Test Accuracy

MAE=2.387

MaNo v.s. Test Accuracy

ImageNet-C

ImageNet-V2-

C

experimental results on ImageNet-C and ImageNet-¯ C , generated from the validation set of 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 softrun with other estimation baselines, and the limitations of our approach.

Real-world applications. In 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), ConfScore (Hendrycks and Gimpel, 2016), and our MANO. In Table 4, we observe that softrun significantly enhances the estimation performance R 2 of Nuclear. For example, Nuclear is improved from 0 . 692 to 0 . 826 on poorlycalibrated Office-Home.

Limitations. Despite its soundness and strong empirical performance, we acknowledge that our method has potential areas for improvement. One of these is the dependence on η of the selection criterion in Eq. (6). We elaborate on this discussion in Appendix E.3. In future work, we will explore a smoother way to automatically select the optimal normalization function without requiring hyperparameters. Additionally, if multiple validation sets are provided, as in (Deng et al., 2021; Deng and Zheng, 2021), we could select η based on those sets.

6 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 a 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-boundary distance and considers the impact of calibration on estimation performance. We hope our insights inspire future research to explore the relationship between model outputs and generalization.

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Appendix

Roadmap. We provide the pseudo-code of MANO in Appendix A. We discuss related work in Appendix B and provide some background on Tsallis entropies in Appendix C. Appendix D contains detailed proofs of our theoretical results. Additional discussion and theoretical insights into Section 4 are given in Appendix E. In Appendix F, we conduct experiments with ViT and ConvNext architectures and provide a thorough ablation study and sensitivity analysis in Appendix G. Finally, we explain how MANO can be used in practice in Appendix H. We display the corresponding table of contents below.

Table of Contents

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 MANO at https://github.com/Renchunzi-Xie/MaNo .

B Extended Related Work

Unsupervised accuracy estimation. This task aims to estimate model generalization performance 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 distributionshifted data to construct a 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 a training-free accuracy estimator. However, it significantly suffers from the overconfidence issues (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 overconfidence-information 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 are time-consuming on 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-classifier 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 of deep neural networks either in the input space (Fawzi et al., 2018; Karimi et al., 2019; Montufar et al., 2014; Poole et 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 et al., 2020), and domain generalization (Li et al., 2022; Yousefzadeh, 2021). As we discuss in Section 3.1, 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.

Normalization in deep learning. Normalization is a crucial technique extensively utilized across various fields in deep learning, including domain generalization (Fan et al., 2021; Seo et al., 2020; Wang et al., 2021), metric learning (Oord et al., 2018; Sohn, 2016; Wu et al., 2018), face recognition (Deng et al., 2019; Liu et al., 2017; Ranjan et al., 2017; Wang et al., 2017; Zhang et 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. L 2 -constrained softmax (Ranjan et al., 2017) introduces L 2 normalization on features. These normalization techniques are primarily employed to adapt new samples to familiar domains, calculate similarity, and speed up convergence. However, our proposed normalization focuses on reducing the negative implications of poorly calibrated scenarios.

C Background on Tsallis Entropies

The definition of Tsallis α -entropies (Tsallis, 1988) is given below.

Definition C.1 (Tsallis α -entropies) . Let p ∈ ∆ K be a probability distribution. Let α > 1 and k ≥ 0 . The Tsallis α -entropy is defined as:

It is common to take k = 1 or k = 1 α following Blondel et al. (2019). The Tsallis α -entropy generalizes the Boltzmann-Gibbs theory of statistic mechanics to nonextensive systems. It has been used as a measure of disorder and uncertainty in many applications (GellMann and Tsallis, 2004; Negrinho and Martins, 2014; Sneddon, 2007; Teimoori et al., 2024), including in Machine Learning (Blondel et al., 2019, 2020; Muzellec et al., 2017). Moreover, they generalize two widely-known measures of uncertainty. Indeed, the limit case α → 1 leads to the Shannon entropy H S (see Peters et al., 2019, Appendix A.1), i.e. , lim α → 1 H T α ( p ) = H S ( p ) = -∑ K j =1 p j ln( p j ) , while taking α = 2 leads to the Gini index G , a popular impurity measure fo decision trees (Gini, 1912), i.e. , H T 2 ( p ) = 1 2 (1 -∥ p ∥ 2 2 ) =

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([t, 1

t])

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G ( p ) . 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 distributions, i.e. , close to the uniform distribution. We visualize the evolution of the Tsallis entropy for varying parameters α in Figure 7, where the case α = 1 corresponds to 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 λ ), 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 A is denoted by A i , its j -th column is denoted by A · ,j and its elements are denoted by A ij . The trace of a matrix A is denoted by Tr( A ) and its transpose by A ⊤ . The L p norm of a vector x is denoted by ∥ x ∥ p , and by abuse of notation we denote it by ∥ A ∥ p for a matrix A with ∥ A ∥ p p = ∑ i ∥ A i ∥ p p = ∑ ij | A ij | p . Let ∆ K := { p ∈ [0 , 1] K | ∑ K i =1 p i = 1 } be the K -dimensional probability simplex.

D.1 Impact of Prediction Errors

Let x ∈ D test be a test sample with ground-truth label y ∈ { 1 , . . . , K } . 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 of the distribution shifts between source and target, logits are subject to a prediction bias ε = ( ε k ) k and write f ( x ) = q ∗ + ε where q ∗ are ground-truth logits. In this section, we study the impact of such bias on the approximation of the posterior p ( y | x ) .

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 p = softmax( q ∗ ) ∈ ∆ K , and the estimated ones s = softmax( f ( x )) ∈ ∆ K . In particular, it states that

where ε + = (max l { ε l } -ε k ) k ∈ R K + . The proof is given below.

Proof. We denote q = f ( x ) ∈ R K the neural network's outputs on a given test sample x . We first remark that

To obtain the upper-bound, we notice that

This leads to

Using the fact that all the terms are positive and that ln and exp are increasing functions, we obtain from Eq. (8) that

Combining Eq. (8) and Eq. (10) gives the upper bound.

The quantities ε + is a linear transformation of the prediction bias ε ∈ R K 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 is controlled by the alignment between the posterior and this extreme prediction bias. In addition, by a simple application of Cauchy-Schwartz in Eq. (7) and using the fact that | p ∥ = ∑ k =1 p 2 k ≤ ∑ k =1 p k = 1 , we have KL( p || s ) ≤ ∥ ε + ∥ 2 . In particular, in the perfect situation where ε = 0 , ε + is equal to 0 and the softmax probabilities perfectly approximate the posterior. In 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 on 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.

A real-world example. Although we usually tend to think that a 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. In 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 Cyrillic 'B', therefore predicting a wrong class with high probability.

D.2 Distance to the Hyperplane

Lemma D.1 (Dohmatob (2020)) . Let ω ∈ R n be non zero and b ∈ R . The distance between any point z ∈ R n and the hyperplane { x | ω ⊤ x + b = 0 } writes d( ω , z ) = | ω ⊤ x + b | / ∥ ω ∥ .

Proof. The proof follows the geometric intuition from Dohmatob (2020). We recall it here for the sake of self-consistency. The distance between z and the hyperplane H = { x ∈ R n | ω ⊤ x + b = 0 } is equal to the distance between z and its orthogonal projection on H . We consider the line L = { z + t ω | t ∈ R } that is orthogonal to H and passes through z . The desired orthogonal projection is simply the point z + t ∗ ω such that L and H intersects, i.e. , such that

It follows that the distance between z and H writes

D.3 Proof of Theorem 3.3

Proof. Reusing the notations introduced in Section 3.2 and Algorithm 1, we have that

where a = p ( p -1) K > 0 and b = 1 K . Rearranging the terms concludes the proof.

D.4 Properties of ϕ

The softmax can be decomposed as softmax( q ) = exp( q ) / ∑ K k =1 exp( q ) k = ( ϕ ◦ exp)( q ) , where ϕ : R K + → ∆ K writes ϕ ( u ) = u / ∑ K k =1 u k = u / ∥ u ∥ 1 . We extend the domain of ϕ to R K + by setting ϕ ( 0 ) = 1 K ✶ K . The following proposition states the properties of ϕ .

Proposition D.2 (Properties of ϕ ) .

2. Evaluation on constant inputs. Let u = α ✶ K with α ≥ 0 . Then, we have ϕ ( u ) = 1 K ✶ K . Proof. We start by proving the first part of Proposition D.2. Let u , v ∈ R K + \ 0 . We have

Then, we prove the second part of the proposition. Let α ≥ 0 and consider u = α ✶ K ∈ R K + . If α = 0 , then u = 0 and by definition, ϕ ( u ) = ϕ ( 0 ) = 1 K K . Assuming α > 0 , we have

̸

The first part of the proposition is dubbed 'generalized injectivity' as the injectivity can be retrieved by fixing α = 1 in Proposition D.2. It ensures that ϕ only has equal outputs if the inputs are similar . To illustrate that, consider the logits q , δ ∈ R K . From Proposition D.2, having softmax( q ) = softmax( δ ) is equivalent to having exp( q ) = α exp( δ ) for some α = 0 . By positivity of both sides, it implies α > 0 , and taking the logarithm leads to q = δ +ln α . It means that q equals δ up to a fixed constant. From a 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, i.e. , all entries are equal, then the link function gives equal probability to all classes.

E Theoretical Insights into Criterion Φ( D test )

In Section 4, we explained the main drawbacks of using the softmax normalization in the presence of the prediction bias proposing a new alternative normalization, softrun , that we recall is

where Φ( D test ) is the selection criterion defined as

In this section, we first would like to give more insights on the choice of Φ( D test ) and the selection rule. Then, we motivate our choice of the hyperparameter η . Finally, we discuss the potential limitations of our approach.

E.1 Choice of Criterion Φ( D test )

Intuition. We first provide some high-level intuition behind the selection criterion of Section 4. As we discussed before, the main idea of the logit-based approach is to rely on the model's confidence whose reliability depends on the prediction bias induced by possible distribution shift. Depending on exact values of confidence and bias, we can roughly distinguish the five following cases illustrated in Figure 8 and described as follows:

High confidence, high bias. The model is self-confident but practically makes a 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, in 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 an original pre-trained model, since these approaches perform unsupervised confidence maximization making LDS not guaranteed anymore.

Confidence

Doesn't happen

Use softmax

LDS violated

Conf

Conf

Conf

Conf

Low confidence, low bias. The model tends to output confidence that is low but overall precise. This situation is unlikely to 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, it 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 a subject of future work.

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.

Low confidence, high bias. The model is not confident in their predictions and it indeed makes a lot of mistakes due to the high prediction bias. This is also a favorable scenario as low confidence correlates with low performance. In this case, we want posterior probabilities to be close to uniform, and we use the Taylor normalization due to its smoother behavior.

Grey zone. It corresponds to a 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 a 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 4.1 for more details).

Formulation of Φ( D test ) . As we discussed in the main body of the paper, the criterion Φ( D test ) is equal, up to a constant, to the average KL divergence between the uniform distribution and the predicted softmax 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 Φ( D test ) correspond to the 3 rd case ( high confidence, low bias ), and the softmax normalization is selected. Conversely, when the Φ( D test ) is lower than the threshold η , we apply Taylor, which corresponds to either the 4 th case ( low confidence, high bias ) or the 5 th case ( grey zone ).

Connection between criterion and misclassification error. The next proposition provides further insights into our selection process. Let u = 1 K ✶ K ∈ ∆ K be the uniform probability. The test dataset writes D test = { x i } N i =1 with corresponding logits q i = f ( x i ) and ground-truth labels Y test =

{ y i } N i =1 (unavailable in practice). We denote the softmax probabilities by s i = softmax( q i ) = exp( q i ) / ∑ K k =1 exp( q i ) k ∈ ∆ K . We introduce the entropy of a probability vector as H( p ) = -1 K ∑ K k =1 p k ln( p k ) . In particular, it is a measure of uncertainty and takes a high value when the model is uncertain, i.e. , outputs probabilities close to the uniform. We establish in the following proposition the connection between the criterion Φ( D test ) , the miscalibration error, the model's confidence, and its entropy.

Proposition E.1 ( Φ( D test ) , misclassification error, confidence and entropy) . We have

where ξ ( D test , Y test ) = 1 N ∑ N i =1 ( 1 -s i y i ) , U ( D test ) = 1 N ∑ N i =1 KL( u ∥ s i ) , and H ( D test ) = 1 N ∑ N i =1 H( s i ) .

Proof. We first present the following lemma that introduces the change of measure inequality (Banerjee, 2006; Seldin and Tishby, 2010).

Lemma E.2 (Change of measure inequality (Seldin and Tishby, 2010)) . Let Z be a random variable on { 1 , . . . , K } and µ = ( µ k ) k ∈ ∆ K and ν = ( ν k ) k ∈ ∆ K be two probability distributions. For any measurable function ψ : Z → R , the following inequality holds:

Proof. We have

We now proceed to the proof of Proposition (E.1). For a given test sample x i ∈ D test , we first notice that

Similarly, we obtain KL( s i ∥ u ) = ∑ K k =1 s i k ln( s i k ) + ln( K ) . Combining those results leads to

̸

Consider the function ψ ( k ) = I ( y i = k ) that takes the value 1 when y i = k and 0 otherwise. Using Lemma E.2 with the measures µ = s i , ν = u and ψ , and the previous equation, we obtain

̸

̸

̸

̸

Summing over all the test samples and dividing by N leads to

which concludes the proof by using the notations introduced in Proposition E.1.

Interpretation. The term ξ ( D test , Y test ) , dubbed misclassification error, is the average error on the test set between the optimal probability on the true label ( i.e. , 1 ) and the predicted probability s i y . It takes high values when the model makes a lot of mistakes, assigning low confidence to the true class labels, and low values otherwise. U ( D test ) 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 to the uniform (uncertain). H ( D test ) 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 ( ξ ( D test , Y test ) is low) and is confident ( U ( D test ) is high and H ( D test ) is low), then the criterion Φ( D test ) takes high values. This matches the empirical evidence from from Tian et al. (2021). Proposition E.1 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 U ( D test ) and H ( D test ) . However, we experimentally show the benefits of Φ( D test ) and softrun in 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 η

In all our experiments, we take η = 5 for the selection criterion in Eq. (6). We motivate this choice in what follows. In our setting, we consider test samples x i ∈ D test drawn i.i.d. from the test distribution p T . As the model f pre-trained on D train is a deterministic function, the logits q i are i.i.d. random variables and the decision threshold Φ( D test ) = -1 NK ∑ N i =1 ∑ K k =1 ln ( exp( q i ) k / ∑ K j =1 exp( q i ) j ) is a random variable with mean µ and variance ν . Applying the Chebyshev's inequality leads to

̸

The threshold Φ( D test ) is used to determine how calibrated the model is on a given test dataset D test . Figure 2(b) shows that our proposed normalization is optimal in poorly calibrated datasets and performs slightly below the softmax in calibrated situations. Hence, we can afford to be conservative and we want to consider the model calibrated only for extreme values of Φ( D test ) . From Eq. (11), taking η = 5 ensures that the probability that Φ( D test ) deviates from its mean by several standard deviations with probability smaller than 5% ( 1 25 < 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 et al., 2022). Moreover, using Eq. (6), MANO remains suitable even when test labels are not available which is often the case in real-world applications, and we demonstrate state-of-the-art performance for various architecture and types of shifts in Section 5. For the sake of self-consistency, we also provide a sensitivity analysis on the values of η 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 η . As stated above, the chosen value of η is motivated by a probabilistic argument and by our experiments. However, as it can be seen in Eq. (11), the mean and standard deviation of Φ( D test ) can impact the validity of η . In particular, this could be the case when applying MANO on other data modalities than images or 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 on the impact of the number of classes K on the values of Φ( D test ) . We uniformly draw random vectors of R K in [ -5 , 5] to mimic the logits obtained from 100000 independent models.

We compute the corresponding Φ( D test ) for each model and recover the 0 . 5 th and 99 . 5 th percentile to obtain a 99% confidence interval. We repeat this experiment for K ∈ J 2 , 100 K . The evolution of the confidence interval is displayed in Figure 9. We observe that as soon as K > 3 , the upper bound of the confidence interval has a 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 Φ( D test ) becomes thinner and more concentrated on high values for K > 25 . In particular, we observe that 99% of the models have an associated Φ( D test ) , higher than η = 5 , which means that in this situation, the softmax would always be selected as a normalization σ . While the conclusions from this experiment are not directly applicable to our real experimental setting (in particular, Taylor and softmax cases

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of Eq. (6) occur both for datasets with K > 25 and K ≤ 25 ), we believe it motivates further work to make η more robust to the values of Φ( D test ) . In particular, one could propose to compute η based on statistics of the data or as a function of the number of classes.

Dispersion of the softmax. In the previous paragraph, we showed that the number of classes K can have an impact on the values taken by criterion Φ( D test ) . 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 softmax function. The lemma below shows that the softmax must disperse on all entries as the number of classes K increases. It means that the total weights on the softmax entries (equal to 1 ) cannot be concentrated on a few entries as the number of classes K increases.

Lemma E.3. Let θ ∈ R K be logits of a neural network f such that ∥ θ ∥ 1 ≤ c for some c > 0 . Then, as the number of classes grows, i.e., K →∞ , we have

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 i ∈ [ K ] that

where a = exp( -2 c ) , b = exp(2 c ) 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, Φ( D test ) depends on the softmax 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 is similar but more general than Veliˇ ckovi´ c et al. (2024, Lemma 2.1) as our global bounding condition on θ encompasses their entry-wise condition.

F Beyond ResNets: Experiments with Vision Transformers and ConvNeXts

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.

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G Sensitivity Analysis and Ablation Study

G.1 Choice of L p Norm

To reflect the impact of different L p 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 p = 1 is ignored as in this case, ∥ Q ∥ 1 = 1 because Q is right-stochastic. We can see that when we choose p ∈ [2 , 5] , the results fluctuate within a 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), we verify the impact of Taylor formula approximation on final accuracy estimation performance. It should be noted that for orders higher than in the default setting in Eq. (6), the positivity is lost. To alleviate this issue, we consistently remove for all orders the minimum value of the obtained vector to each of its entries to ensure having an output in R K + . This extends de Br´ ebisson and Vincent (2016) to orders higher than 2 . From this figure, we can see that when we reserve the first three terms in the Taylor formula, the average estimation performance is optimal. For well-calibrated datasets such as Office-Home and WILDS, there exists an increased trend of estimation performance when we reserve more Taylor formula terms. As for suboptimal-calibrated datasets such as PACS and Office-31, their performance rises when fewer terms are 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 3 terms in Eq. (4) which motivates our default setting in Eq (6).

G.3 Choice of Calibration Threshold η

In Table 6, we display the performance comparison for varying values of threshold η on three datasets with ResNet18. It should be noted that taking η = 0 corresponds to the case where the softmax is always taken, i.e., the common choice in the literature. This matches our theoretical insights in Appendix E.2 and confirms that taking η = 5 is a robust and effective choice for softrun .

G.4 Superiority of softrun

To verify the effectiveness of our proposed normalization technique, softrun , we conduct an ablation study by replacing our normalization with the softmax function under the natural shift. In Figure 10(c), we observe that our proposed normalization significantly enhances the estimation performance of datasets from the natural shift. Especially, R 2 for poorly-calibrated datasets such as PACS is improved from 0.541 to 0.812.

G.5 Generalization Capabilities of MANO on ImageNet-¯ C

To further demonstrate the generalization capability of MANO, we provide a similar experiment with that in Section 5.4 on 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.

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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. In this section, we provide two examples.

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.

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 to 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.

In Table 7, 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.

NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n1. 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.\n\nWEAKNESSES:\n1. 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?\n\nQUESTIONS:\nSee weakness.\n\nLIMITATIONS:\nN/A.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-25"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\n​​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.\n\nSTRENGTHS:\n1. 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.\n\nWEAKNESSES:\n1. 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.\n\nQUESTIONS:\n1. 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?\n\nLIMITATIONS:\nThe authors have adequately addressed the limitations and potential negative societal impact of their work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n- 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.\n\nWEAKNESSES:\n1. 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\n\nQUESTIONS:\n- 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?\n\nLIMITATIONS:\nSee weaknesses and questions.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\n1. 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.\n\nWEAKNESSES:\n1. 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.\n\nQUESTIONS:\nPlease refer to the weaknesses.\n\nLIMITATIONS:\nN/A.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-20"}