Estimating Performance for Safe Deployment of Machine Learning Models

Zeju Li; Konstantinos Kamnitsas; Mobarakol Islam; Melanie Roschewitz; Cheng Ouyang; Chen Chen; Ben Glocker

cs.LGOct 21, 2026

Estimating Performance for Safe Deployment of Machine Learning Models

Authors: Zeju Li, Konstantinos Kamnitsas, Mobarakol Islam, Melanie Roschewitz, Cheng Ouyang, Chen Chen, Ben Glocker

Organizations: College of Biomedical Engineering, Fudan University, Shanghai, China · Oxford Centre for Integrative Neuroimaging, University of Oxford, Oxford, UK · Department of Engineering Scienc, University of Oxford, Oxford, UK · Wellcome/EPSRC Centre for Interventional and Surgical Sciences (WEISS), University College London, London, UK ·

Abstract

The main challenge in applying deep neural networks to real-world scenarios lies in distributional discrepancies between training and test datasets, often caused by variations in data acquisition conditions. Assessing the performance of pre-trained models without labels is crucial for safety, but this is complicated by deep models' overly optimistic probabilistic outputs, often exacerbated by class imbalances, leading to unreliable predictions. To address these issues, we introduce MOVAL, a novel MOdel eVALuation tool that estimates performance using unlabeled test data. MOVAL incorporates class-wise confidence calibration to adjust model outputs, allowing for the estimation of metrics beyond accuracy and mitigating the detrimental effects of class imbalance on confidence calibration. Additionally, we propose an ensemble approach that combines multiple calibrated confidence scores for a more robust solution. We validate MOVAL on natural and medical image datasets with various acquisition shifts, covering tasks from classification to segmentation. It achieves state-of-the-art results across evaluation metrics such as accuracy, F1-Score, AUC, and more. Furthermore, we demonstrate that MOVAL provides accurate performance estimations at the class, instance, and pixel levels, ensuring the transparent and safe deployment of machine learning models. We provide MOVAL as a deployment tool for monitoring and estimating performance under a range of domain shifts.

Figures & tables

Figure 1: Detailed descriptions of the confidence calibration process. Here

p_{ij}

denotes the raw (uncalibrated) softmax probability that unit

i

belongs to class

j

, where

i

indexes a sample for classification and a pixel/voxel for segmentation, assuming a single volume is under consideration; after calibration,

p_{ij}

is replaced by the calibrated probability

\hat{p}_{ij}

used to compute all reported metric estimates. (a) : Based on the model probabilities

\boldsymbol{p}_{i}

, multiple performance metrics can be estimated. As we illustrate with the example of accuracy, vanilla model probabilities are well known to be overconfident. Therefore, the estimation is prone to being too optimistic and not accurate. (b) : The calibration process adjusts the probabilities such that the estimation aligns with the real metrics on the validation dataset. (c) : The same principle can be applied to estimating other group-wise metrics. (d) : The calibrated probabilities are expected to produce more accurate estimations when exposed to domain shifts.

Figure 2: Class-wise calibration and multi-metric estimation with MOVAL. We show an estimation example on CIFAR10-LT, where sample counts decrease from class 0 to class 9. (a) : MOVAL facilitates class-wise calibration with class-specific parameters. Class-wise calibration allows for the alignment of estimated class-wise metrics and the true underlying counterparts, which can mitigate model biases caused by class imbalance. (b) : Overall accuracy estimation across different methods. By aggregating multiple confidence scores, MOVAL Ensemble achieves the most accurate results. (c) : In addition to accuracy estimation, which is provided by existing methods, class-wise alignment enables MOVAL to estimate class-wise metrics, thus providing a comprehensive evaluation.

Figure 3: We extend existing model calibration methods to class-specific variants by introducing parameters for different classes. This enables class-wise calibration and improves estimation accuracy, especially when the model is trained with class-imbalanced data.

Figure 4: Class-wise calibration improves the estimation results for all metrics in classification tasks. (a) : The mean absolute error (MAE) between real metrics and estimated results based on different estimation methods for classification tasks. The size of the circles represents the MAE, with smaller circles indicating better performance. We report the average MAE for various tasks here. (b) : MAE comparison between the original estimation methods and our proposed class-specific variants for classification tasks. Paired t-tests were used for statistical analysis.

Figure 5: Class-wise calibration is even more crucial for segmentation tasks. (a, c) : MAE between real metrics and estimated results from different methods for 2D and 3D segmentation tasks. (b, d) : MAE comparison between the original estimation methods and proposed class-specific variants for 2D and 3D segmentation tasks. Statistical tests are conducted using paired t-tests.

Figure 6: Agreement between predicted and real metrics for different tasks. For each subfigure: Top: Predicted and real label distribution. We benchmark MOVAL ensemble (blue) against MCP baselines (light gray) and the best-performing existing method (dark gray). Middle: Predicted versus real metrics. Each point represents the model’s performance on unseen target data under specific conditions. Bottom: Histogram of MAE for different estimation methods. (a) : Classification tasks. (b) : 2D segmentation tasks. (c) : 3D segmentation tasks.

Figure 7: MOVAL Ensemble achieves state-of-the-art performance. (a, c) : MAE comparison between proposed algorithms and existing standards (Best-performing existing method versus MOVAL Ensemble, paired t-test,

p=1.05\times 10^{-7}

for accuracy estimation of classification tasks,

p=3.54\times 10^{-4}

for F1-Score estimation of 2D segmentation tasks,

p=4.28\times 10^{-27}

for F1-Score estimation of 3D segmentation tasks, see the results of other metrics in Extended Fig. 16 and the results of individual datasets metrics in Extended Fig. 17 and Fig. 18 ). (b) : MAE comparison between estimation algorithms based on MCP and other confidence scores for classification tasks (see Extended Fig. 19 for other tasks).

Figure 8: MOVAL Ensemble maintains stability in realistic settings. (a) : Classification and segmentation results when models are trained with advanced training strategies designed for class imbalance or domain generalization (Classification, Balanced Training I: label-distribution-aware margin (LDAM) loss ( Cao et al., 2019 ) , Balanced Training II: deferred re-weighting (DRW) training schedule ( Cao et al., 2019 ) , Robust Training I: cutout ( DeVries and Taylor, 2017 ) , Robust Training II: RandAugment ( Cubuk et al., 2020 ) ; Segmentation, Balanced Training I: asymmetric large margin loss ( Li et al., 2020 ) , Balanced Training II: asymmetric focal loss ( Li et al., 2020 ) , Robust Training I: global intensity non-linear augmentation (GIN) technique ( Ouyang et al., 2022a ) , Robust Training II: mixup ( Zhang et al., 2017 ) ). (b) : Top: MAE comparison between the original estimation methods and our proposed class-specific variants with various training strategies. Bottom: MAE comparison between best-performing existing method and MOVAL Ensemble with various training strategies. (c) : MAE achieved by MOVAL Ensemble when optimized with different amounts of validation data.

Figure 9: Fine-grained performance indicators at class and instance levels. (a) : Alignment of confidence scores and class-wise correctness for skin lesion classification. (b) : Top: Alignment of instance-wise DSC estimation and real DSC for 2D cardiac segmentation. Bottom: Alignment for 3D brain lesion segmentation.

Figure 10: Fine-grained performance indicators at pixel level. Left: Visualization of calibrated confidence scores for 2D cardiac segmentation. Right: 3D scatter plots of segmentation error and uncertainty regions from calibrated confidence scores.

Appendix figures & tables30 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset and Training Strategy				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
CIFAR10	Synthetic-95	Vanilla Training	Vanilla	32.6 $\,\pm$ 6.2	29.4 $\,\pm$ 5.2	30.5 $\,\pm$ 6.1	36.3 $\,\pm$ 6.5	—	—	—	—	2.8 $\,\pm$ 2.1
			TS	6.5 $\,\pm$ 5.0	13.5 $\,\pm$ 6.7	5.5 $\,\pm$ 5.0	4.9 $\,\pm$ 4.0	7.1 $\,\pm$ 5.4	12.4 $\,\pm$ 7.6	5.9 $\,\pm$ 4.9	4.0 $\,\pm$ 3.7
			DoC	11.7 $\,\pm$ 6.2	8.5 $\,\pm$ 5.3	10.9 $\,\pm$ 6.1	12.6 $\,\pm$ 6.5	10.9 $\,\pm$ 6.5	8.5 $\,\pm$ 5.9	10.3 $\,\pm$ 6.1	11.6 $\,\pm$ 6.7
			ATC	4.1 $\,\pm$ 3.7	6.2 $\,\pm$ 3.7	4.2 $\,\pm$ 3.7	4.1 $\,\pm$ 3.7	3.5 $\,\pm$ 3.5	3.2 $\,\pm$ 1.8	3.3 $\,\pm$ 3.2	3.4 $\,\pm$ 3.4

Appendix

Table 1: Related to Fig. 4 . Evaluation of accuracy estimation for different classification tasks under various types of domain shifts based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Dataset			Confidence Score
		Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
		Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
Cardiac-2D	Synthetic-65	Vanilla	17.8 $\,\pm$ 7.7	52.5 $\,\pm$ 9.0	16.3 $\,\pm$ 7.2	17.5 $\,\pm$ 7.5	—	—	—	—	4.3 $\,\pm$ 5.0
		TS	14.8 $\,\pm$ 7.1	19.8 $\,\pm$ 8.4	13.3 $\,\pm$ 6.7	13.7 $\,\pm$ 6.7	5.1 $\,\pm$ 5.0	6.3 $\,\pm$ 7.0	4.7 $\,\pm$ 4.2	4.5 $\,\pm$ 3.9
		DoC	17.4 $\,\pm$ 7.7	17.0 $\,\pm$ 8.0	15.9 $\,\pm$ 7.2	17.1 $\,\pm$ 7.5	5.6 $\,\pm$ 6.0	5.9 $\,\pm$ 6.1	5.2 $\,\pm$ 5.5	5.5 $\,\pm$ 5.9
		ATC	6.5 $\,\pm$ 4.5	19.8 $\,\pm$ 8.4	6.4 $\,\pm$ 4.4	6.5 $\,\pm$ 4.5	3.6 $\,\pm$ 4.2	4.5 $\,\pm$ 7.7	3.6 $\,\pm$ 4.3	3.6 $\,\pm$ 4.2

Appendix

Table 2: Related to Fig. 5 . Evaluation of DSC (F1-Score) estimation for different segmentation tasks under various types of domain shifts based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Figure 11: Comparison of MOVAL with existing standards. (a) : Within a unified framework, MOVAL can be readily applied to classification, 2D, and 3D segmentation tasks. (b) : After extending existing methods in many dimensions, we find that an ensemble of multiple confidence scores leads to more robust and state-of-the-art results. (c) : MOVAL explores performance estimation based on confidence scores rather than maximum class probability (MCP), which is the basis of all existing methods.

Figure 12: Summary of the datasets used in this study. (a) : Classification task I, Long-tailed version of the CIFAR10 dataset. (b) : Classification task II, Long-tailed version of the CIFAR10 dataset. (c) : Classification task III, skin lesion classification from 8 sources of data. (d) : 2D segmentation task I, cardiac MRI segmentation from 5 different scanners. (e) : 3D segmentation task I, prostate MRI segmentation from 6 different scanners. (f) : 3D segmentation task II, brain lesion segmentation from 13 different scanners.

Figure 13: In real-world scenarios, significant discrepancies often exist between training and testing environments, affecting the distribution of acquired images. For instance, in natural images, such discrepancies can be caused by image corruptions like blur or noise ( Hendrycks and Dietterich, 2019a ) . In medical images, these can arise from scanner effects or variations in imaging protocols ( Castro et al., 2020 ) . For segmentation tasks, regions of interest are highlighted with different colors.

Figure 14: Examples of brain lesion segmentation datasets. (a) : Visualization of Brain MR images of In-Domain and Out-Domain Examples. Top: Brain MR images used for training. Brain lesion regions are highlighted with red color. Bottom: Brain MR images with natural and synthesized domain shifts. (b) : Intensity histograms of training and test data for 12 test conditions.

Figure 15: Accuracy and AUC are less useful for segmentation tasks. (a, b) : MAE between real metrics and estimated results based on different estimation methods for 2D and 3D segmentation tasks. We also report the average MAE for natural and synthetic domain shifts here. The smaller the MAE, the better. (c, d) : Top: Predicted and real label distribution. We benchmark MOVAL ensemble (blue) against MCP baselines (light gray) and the best-performing existing method (dark gray). Middle: Predicted versus real metrics for 2D and 3D segmentation tasks. Bottom: Histogram of MAE for different estimation methods. As the class ratios are extremely imbalanced for segmentation tasks, accuracy and AUC are less informative as they are close to 100 $\%$ .

Figure 16: MOVAL Ensemble achieves significant improvements on major evaluation metrics. (a, b, c) : MAE comparison between proposed algorithms and existing methods (Existing best method versus MOVAL Ensemble, paired t-test, $T(196)=9.54,p=6.04\times 10^{-18}$ for F1-Score estimation of classification tasks, $T(196)=4.97,p=1.43\times 10^{-6}$ for precision estimation of classification tasks, $T(196)=7.80,p=3.54\times 10^{13}$ for sensitivity estimation of classification tasks, $T(196)=5.10,p=8.18\times 10^{-7}$ for AUC estimation of classification tasks, $T(68)=-0.26,p=0.80$ for sensitivity estimation of 2D segmentation tasks, $T(68)=5.42,p=8.54\times 10^{-7}$ for precision estimation of 2D segmentation tasks, $T(182)=3.96,p=1.09\times 10^{-4}$ for sensitivity estimation for 3D segmentation tasks and $T(182)=6.79,p=1.53\times 10^{-10}$ for precision estimation for 3D segmentation tasks.

Figure 17: MOVAL Ensemble achieves state-of-the-art performance for several classification tasks. (a) : MAE comparison between proposed algorithms and existing methods for different classification tasks. (b) : MAE comparison between the original estimation methods and our proposed class-specific variants for different classification tasks.

Figure 18: MOVAL Ensemble achieves state-of-the-art performance for several segmentation tasks. (a) : MAE comparison between proposed algorithms and existing methods for different segmentation tasks. (b) : MAE comparison between the original estimation methods and our proposed class-specific variants for different segmentation tasks.

Figure 19: MOVAL Ensemble combines advanced calibration and confidence scores for better performance. (a) : MAE comparison between estimation algorithms based on different confidence calibration algorithms. (b) : MAE comparison between estimation algorithms based on MCP and other confidence scores for segmentation tasks.

Figure 20: Group-wise calibration enhances the robustness of the MOVAL Ensemble against a reduction in validation samples. The proposed group-wise calibration method generally requires fewer validation samples because it has fewer learnable parameters. However, it may negatively impact the estimation of false positive predictions, which require class-specific adjustments. This could potentially affect the estimation of AUC and precision when data are sufficient. The standard error is shown alongside the bar plots.

Figure 21: Confidence score distributions of different calibration methods on CIFAR10-LT. For each class and calibration method, we show the estimated accuracy (gray gradient bar), the true accuracy (black tick), and the gap between them (red), together with the underlying confidence-score distribution (blue). Confidence distributions are shown as kernel density estimates (KDE) of the per-sample confidence values, for visualization only, to make the shift in confidence across classes easier to see. MOVAL Ensemble, which combines multiple calibrated confidence scores, shows a distinctly bimodal distribution concentrated near 0 and 1, providing a more distinct indicator of prediction correctness than the single-score methods.

	Accuracy	F1-Score	Sensitivity	Precision	AUC	Advanced Training	Reduced Data	Total
Classification	197	197	197	197	197	380	2375	3740
2D Segmentation	69	69	69	69	69	0	276	621
3D Segmentation	183	183	183	183	183	216	1080	2211
Total	449	449	449	449	449	596	3731	6572

Appendix

Table 3: Summary of experimental results covered in the main body.

	nv (%)	mel (%)	bcc (%)	df (%)	bkl (%)	vasc (%)	akiec (%)	Total
HAM	6705 (67)	1113 (11)	514 (5)	115 (1)	1099 (11)	142 (1)	327 (3)	10015
BCN	4206 (34)	2857 (23)	2809 (23)	124 (1)	1138 (9)	111 (1)	1168 (9)	12413
VIE	4331 (99)	34 (1)	0	0	0	0	0	4365
MSK	2202 (62)	826 (23)	30 (1)	5 ( $<$ 1)	470 (13)	0	7 ( $<$ 1)	3540
UDA	408 (67)	193 (31)	3 ( $<$ 1)	2 ( $<$ 1)	7 (1)	0	0	613
OTH	4523 (55)	1669 (20)	513 (6)	124 (2)	889 (11)	95 (1)	388 (5)	8201

Appendix

Table 4: Dataset statistics of skin lesion classification datasets used in this study. nv: Melanocytic nevi, mel: Melanoma, bcc: Basal cell carcinoma, df: Dermatofibroma, bkl: Benign keratosis, vasc: Vascular skin lesions, akiec: Actinic Keratoses (Solar Keratoses) and Intraepithelial Carcinoma (Bowen’s disease).

Evaluation Metric and Dataset				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
Sensitivity	CIFAR10	Synthetic-95	Vanilla	32.3 $\,\pm$ 5.2	31.4 $\,\pm$ 3.2	30.6 $\,\pm$ 5.1	35.9 $\,\pm$ 5.7	—	—	—	—	9.5 $\,\pm$ 3.6
			TS	20.8 $\,\pm$ 3.7	24.6 $\,\pm$ 5.5	21.3 $\,\pm$ 3.7	21.6 $\,\pm$ 3.5	12.2 $\,\pm$ 5.5	15.4 $\,\pm$ 6.9	11.8 $\,\pm$ 5.1	10.9 $\,\pm$ 4.4
			DoC	21.9 $\,\pm$ 4.2	22.9 $\,\pm$ 4.6	21.7 $\,\pm$ 4.2	22.6 $\,\pm$ 4.5	13.8 $\,\pm$ 6.1	13.1 $\,\pm$ 5.8	13.3 $\,\pm$ 5.8	14.5 $\,\pm$ 6.3
			ATC	18.9 $\,\pm$ 3.0	25.0 $\,\pm$ 3.9	19.1 $\,\pm$ 3.1	18.9 $\,\pm$ 3.0	9.5 $\,\pm$ 3.8	10.8 $\,\pm$ 4.2	9.4 $\,\pm$ 3.7	9.4 $\,\pm$ 3.7

Appendix

Table 5: Evaluation of sensitivity and precision estimation for different classification tasks under various types of domain shifts based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Evaluation Metric and Dataset				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
F1-Score	CIFAR10	Synthetic-95	Vanilla	40.2 $\,\pm$ 7.4	16.0 $\,\pm$ 2.9	38.9 $\,\pm$ 7.3	42.4 $\,\pm$ 7.7	—	—	—	—	8.5 $\,\pm$ 3.1
			TS	22.3 $\,\pm$ 6.0	28.7 $\,\pm$ 8.1	21.6 $\,\pm$ 6.0	19.4 $\,\pm$ 5.9	11.8 $\,\pm$ 5.5	15.1 $\,\pm$ 6.8	11.3 $\,\pm$ 5.0	10.2 $\,\pm$ 4.1
			DoC	26.4 $\,\pm$ 7.1	24.9 $\,\pm$ 6.8	25.9 $\,\pm$ 6.9	27.3 $\,\pm$ 7.5	14.0 $\,\pm$ 6.4	13.1 $\,\pm$ 5.8	13.5 $\,\pm$ 6.2	14.9 $\,\pm$ 6.9
			ATC	17.1 $\,\pm$ 4.4	17.2 $\,\pm$ 6.1	17.3 $\,\pm$ 4.5	17.1 $\,\pm$ 4.4	8.5 $\,\pm$ 3.1	11.6 $\,\pm$ 4.8	8.7 $\,\pm$ 3.2	8.5 $\,\pm$ 3.1

Appendix

Table 6: Evaluation of F1-Score and AUC estimation for different classification tasks under various types of domain shifts based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Dataset and Training Strategy				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
CIFAR10	Synthetic-95	Vanilla Training	Vanilla	57.0 $\,\pm$ 10.4	8.6 $\,\pm$ 5.3	54.5 $\,\pm$ 10.4	61.1 $\,\pm$ 10.5	—	—	—	—	—
			TS	28.8 $\,\pm$ 9.6	39.4 $\,\pm$ 9.5	27.2 $\,\pm$ 9.9	24.2 $\,\pm$ 11.1	27.4 $\,\pm$ 4.4	35.9 $\,\pm$ 6.1	24.3 $\,\pm$ 3.7	18.9 $\,\pm$ 3.4
			DoC	36.1 $\,\pm$ 10.4	32.4 $\,\pm$ 8.5	35.0 $\,\pm$ 10.4	37.4 $\,\pm$ 10.5	34.1 $\,\pm$ 5.5	28.9 $\,\pm$ 4.5	32.9 $\,\pm$ 5.3	35.2 $\,\pm$ 5.6
			ATC	23.1 $\,\pm$ 11.3	20.2 $\,\pm$ 10.1	22.9 $\,\pm$ 11.4	23.2 $\,\pm$ 11.3	15.7 $\,\pm$ 4.1	11.2 $\,\pm$ 5.2	14.7 $\,\pm$ 4.2	15.2 $\,\pm$ 4.1

Appendix

Table 7: Evaluation of accuracy estimation for different classification tasks under various types of domain shifts based on MAE. The test datasets include only samples from the 2 most minority classes. The best and second-best results within the same category are in bold and underlined for each configuration.

Evaluation Metric and Dataset				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
Sensitivity	Cardiac-2D	Synthetic-65	Vanilla	15.1 $\,\pm$ 6.5	62.3 $\,\pm$ 9.2	12.4 $\,\pm$ 5.9	14.5 $\,\pm$ 6.3	—	—	—	—	3.3 $\,\pm$ 2.5
			TS	9.8 $\,\pm$ 5.8	18.8 $\,\pm$ 7.8	7.4 $\,\pm$ 5.3	8.1 $\,\pm$ 5.3	4.4 $\,\pm$ 3.7	6.0 $\,\pm$ 6.3	4.2 $\,\pm$ 3.2	4.1 $\,\pm$ 3.1
			DoC	14.4 $\,\pm$ 6.5	13.5 $\,\pm$ 7.0	11.8 $\,\pm$ 5.9	13.8 $\,\pm$ 6.3	5.1 $\,\pm$ 5.1	5.5 $\,\pm$ 5.1	4.6 $\,\pm$ 4.5	4.9 $\,\pm$ 4.9
			ATC	3.2 $\,\pm$ 2.4	18.8 $\,\pm$ 7.8	3.3 $\,\pm$ 2.5	3.2 $\,\pm$ 2.4	3.1 $\,\pm$ 2.5	4.5 $\,\pm$ 5.9	3.1 $\,\pm$ 2.5	3.1 $\,\pm$ 2.5

Appendix

Table 8: Evaluation of sensitivity estimation for different segmentation tasks under various types of domain shifts based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Evaluation Metric and Dataset				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
Precision	Cardiac-2D	Synthetic-65	Vanilla	9.5 $\,\pm$ 1.5	55.0 $\,\pm$ 3.7	8.2 $\,\pm$ 0.9	9.3 $\,\pm$ 1.3	—	—	—	—	8.4 $\,\pm$ 0.8
			TS	10.8 $\,\pm$ 1.5	17.7 $\,\pm$ 7.7	11.0 $\,\pm$ 2.0	11.0 $\,\pm$ 1.9	10.1 $\,\pm$ 0.6	52.1 $\,\pm$ 3.6	10.1 $\,\pm$ 1.9	9.3 $\,\pm$ 1.9
			DoC	10.8 $\,\pm$ 1.6	10.3 $\,\pm$ 1.4	11.0 $\,\pm$ 1.3	10.9 $\,\pm$ 1.5	9.2 $\,\pm$ 0.6	52.1 $\,\pm$ 3.6	9.2 $\,\pm$ 1.0	9.0 $\,\pm$ 0.7
			ATC	—	—	—	—	—	—	—	—

Appendix

Table 9: Evaluation of precision estimation for different segmentation tasks under various types of domain shifts based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Evaluation Metric and Dataset				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
Accuracy	Cardiac-2D	Synthetic-65	Vanilla	0.8 $\,\pm$ 0.3	80.9 $\,\pm$ 0.2	0.7 $\,\pm$ 0.3	0.8 $\,\pm$ 0.3	—	—	—	—	0.3 $\,\pm$ 0.5
			TS	0.2 $\,\pm$ 0.4	1.2 $\,\pm$ 0.5	0.3 $\,\pm$ 0.6	0.3 $\,\pm$ 0.6	0.2 $\,\pm$ 0.4	0.7 $\,\pm$ 0.5	0.3 $\,\pm$ 0.6	0.3 $\,\pm$ 0.6
			DoC	0.2 $\,\pm$ 0.3	0.1 $\,\pm$ 0.2	0.3 $\,\pm$ 0.5	0.3 $\,\pm$ 0.5	0.2 $\,\pm$ 0.3	0.2 $\,\pm$ 0.2	0.3 $\,\pm$ 0.5	0.3 $\,\pm$ 0.5
			ATC	0.4 $\,\pm$ 0.7	1.0 $\,\pm$ 0.3	0.4 $\,\pm$ 0.8	0.4 $\,\pm$ 0.7	0.4 $\,\pm$ 0.8	0.6 $\,\pm$ 0.6	0.4 $\,\pm$ 0.8	0.4 $\,\pm$ 0.8

Appendix

Table 10: Evaluation of accuracy estimation for different segmentation tasks under various types of domain shifts based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Evaluation Metric and Dataset				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
AUC	Cardiac-2D	Synthetic-65	Vanilla	2.9 $\,\pm$ 0.6	45.6 $\,\pm$ 0.8	3.0 $\,\pm$ 0.7	2.7 $\,\pm$ 0.6	—	—	—	—	2.3 $\,\pm$ 0.4
			TS	4.3 $\,\pm$ 1.0	3.1 $\,\pm$ 0.8	4.3 $\,\pm$ 1.1	4.1 $\,\pm$ 1.0	2.5 $\,\pm$ 0.5	33.1 $\,\pm$ 0.6	2.4 $\,\pm$ 0.4	2.3 $\,\pm$ 0.4
			DoC	3.4 $\,\pm$ 0.6	5.0 $\,\pm$ 1.3	3.5 $\,\pm$ 0.7	3.2 $\,\pm$ 0.6	2.5 $\,\pm$ 0.5	19.2 $\,\pm$ 0.3	2.4 $\,\pm$ 0.4	2.3 $\,\pm$ 0.4
			ATC	—	—	—	—	—	—	—	—

Appendix

Table 11: Evaluation of AUC estimation for different segmentation tasks under various types of domain shifts based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Dataset and Training Strategy				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
CIFAR10	Synthetic-95	Balanced Training I	Vanilla	25.2 $\,\pm$ 8.2	21.8 $\,\pm$ 8.5	49.8 $\,\pm$ 10.3	13.0 $\,\pm$ 2.7	—	—	—	—	11.5 $\,\pm$ 6.4
			TS	27.9 $\,\pm$ 8.0	32.9 $\,\pm$ 11.0	27.3 $\,\pm$ 7.8	27.7 $\,\pm$ 8.2	15.7 $\,\pm$ 7.7	23.5 $\,\pm$ 11.9	14.3 $\,\pm$ 6.9	14.4 $\,\pm$ 7.4
			DoC	31.5 $\,\pm$ 10.2	32.8 $\,\pm$ 11.0	31.9 $\,\pm$ 10.3	29.6 $\,\pm$ 9.1	19.9 $\,\pm$ 9.9	23.5 $\,\pm$ 11.9	20.4 $\,\pm$ 10.2	16.0 $\,\pm$ 7.8
			ATC	22.0 $\,\pm$ 5.6	36.5 $\,\pm$ 10.3	28.3 $\,\pm$ 8.8	26.7 $\,\pm$ 8.1	9.4 $\,\pm$ 6.2	29.0 $\,\pm$ 14.3	13.3 $\,\pm$ 7.1	12.0 $\,\pm$ 6.6

Appendix

Table 12: Evaluation of F1-Score estimation for CIFAR10 classification under different training strategies based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Dataset and Training Strategy				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
CIFAR10	Synthetic-95	Balanced Training I	Vanilla	37.3 $\,\pm$ 9.3	8.3 $\,\pm$ 7.2	52.4 $\,\pm$ 9.4	20.2 $\,\pm$ 8.9	—	—	—	—	12.4 $\,\pm$ 11.7
			TS	12.4 $\,\pm$ 7.7	22.5 $\,\pm$ 10.9	11.2 $\,\pm$ 7.7	10.5 $\,\pm$ 9.4	13.6 $\,\pm$ 10.3	20.2 $\,\pm$ 13.6	12.7 $\,\pm$ 10.3	11.1 $\,\pm$ 10.7
			DoC	18.4 $\,\pm$ 9.3	22.4 $\,\pm$ 10.9	19.0 $\,\pm$ 9.4	14.0 $\,\pm$ 9.3	18.2 $\,\pm$ 11.9	20.2 $\,\pm$ 13.6	18.8 $\,\pm$ 12.2	14.3 $\,\pm$ 11.0
			ATC	4.5 $\,\pm$ 6.0	26.4 $\,\pm$ 12.9	8.1 $\,\pm$ 9.8	8.0 $\,\pm$ 8.7	6.4 $\,\pm$ 8.8	23.1 $\,\pm$ 17.1	8.3 $\,\pm$ 9.8	8.2 $\,\pm$ 9.5

Appendix

Table 13: Evaluation of accuracy estimation for CIFAR10 classification under different training strategies based on MAE. The proposed CS variants underperform in accuracy estimation compared to their counterparts when models are trained using advanced balanced training strategies. We observe that balanced training strategies largely affect the model’s predicted probabilities, resulting in a more significant difference in the confusion matrix when exposed to domain shifts. These changes, in particular, create difficulties for CS variants, as they rely on predictions to choose class parameters. In such cases, we recommend performing class-wise calibration with group-wise parameters (c.f. Table 14 ) or alternatively tracking the F1-Score. The best and second-best results within the same category are in bold and underlined for each configuration.

Dataset and Training Strategy				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
CIFAR10	Synthetic-95	Balanced Training I	Vanilla	37.3 $\,\pm$ 9.3	8.3 $\,\pm$ 7.2	52.4 $\,\pm$ 9.4	20.2 $\,\pm$ 8.9	—	—	—	—	11.5 $\,\pm$ 6.9
			TS	12.4 $\,\pm$ 7.7	22.5 $\,\pm$ 10.9	11.2 $\,\pm$ 7.7	10.5 $\,\pm$ 9.4	12.2 $\,\pm$ 7.4	20.7 $\,\pm$ 9.8	11.1 $\,\pm$ 7.5	10.6 $\,\pm$ 9.3
			DoC	18.4 $\,\pm$ 9.3	22.4 $\,\pm$ 10.9	19.0 $\,\pm$ 9.4	14.0 $\,\pm$ 9.3	17.6 $\,\pm$ 8.7	20.7 $\,\pm$ 9.8	18.2 $\,\pm$ 8.9	13.8 $\,\pm$ 9.1
			ATC	4.5 $\,\pm$ 6.0	26.4 $\,\pm$ 12.9	8.1 $\,\pm$ 9.8	8.0 $\,\pm$ 8.7	4.7 $\,\pm$ 6.0	24.2 $\,\pm$ 11.2	8.3 $\,\pm$ 10.2	8.1 $\,\pm$ 9.1

Appendix

Table 14: Related to Extended Table 13 . Evaluation of accuracy estimation for CIFAR10 classification under different training strategies based on MAE. To achieve stable estimation in complex scenarios, we employ a group-wise optimization strategy. Specifically, we estimate two parameters for the CS variants: one shared by the 8 classes with the highest number of predicted samples, and another shared by the remaining 2 classes. Compared to CS variants applied to individual classes, this technique reduces the number of required validation samples and mitigates the complexity of the confusion matrix (e.g., misclassifications within a group have less impact on the estimation results). In this ways, the CS variants can effectively generalize with advanced balanced training strategies. The best and second-best results within the same category are in bold and underlined for each configuration.

Dataset and Training Strategy				Confidence Score
			Estimation Algorithm	Class-Agnostic				Class-Specific				MOVAL Ensemble
			Estimation Algorithm	MCP	Energy	Entropy	Doctor	MCP	Energy	Entropy	Doctor	MOVAL Ensemble
Brain Lesion-3D	Synthetic-42	Balanced Training I	Vanilla	24.8 $\,\pm$ 9.7	70.2 $\,\pm$ 10.0	21.6 $\,\pm$ 7.5	23.3 $\,\pm$ 8.5	—	—	—	—	2.6 $\,\pm$ 4.2
			TS	22.5 $\,\pm$ 8.9	25.9 $\,\pm$ 10.5	19.6 $\,\pm$ 6.8	20.2 $\,\pm$ 7.1	5.8 $\,\pm$ 9.0	6.6 $\,\pm$ 10.4	2.8 $\,\pm$ 4.7	2.9 $\,\pm$ 4.8
			DoC	24.8 $\,\pm$ 9.7	25.9 $\,\pm$ 10.5	21.5 $\,\pm$ 7.5	23.2 $\,\pm$ 8.5	5.8 $\,\pm$ 9.2	6.5 $\,\pm$ 10.2	4.0 $\,\pm$ 6.8	4.8 $\,\pm$ 7.9
			ATC	17.1 $\,\pm$ 5.2	25.9 $\,\pm$ 10.5	17.1 $\,\pm$ 5.2	17.1 $\,\pm$ 5.2	2.7 $\,\pm$ 4.0	2.7 $\,\pm$ 4.0	2.7 $\,\pm$ 4.0	2.7 $\,\pm$ 4.0

Appendix

Table 15: Evaluation of F1-Score estimation for brain lesion segmentation under different training strategies based on MAE. The best and second-best results within the same category are in bold and underlined for each configuration.

Estimating Metric	Group-Wise Calibration	Samples Used for Estimating
Estimating Metric	Group-Wise Calibration	3000 (100 $\%$ )	1500 (50 $\%$ )	600 (20 $\%$ )	300 (10 $\%$ )	150 (5 $\%$ )	90 (3 $\%$ )
Accuracy	No	2.8 $\,\pm$ 2.1	2.8 $\,\pm$ 2.3	2.9 $\,\pm$ 2.3	3.4 $\,\pm$ 2.1	3.9 $\,\pm$ 3.0	4.6 $\,\pm$ 3.6
Sensitivity	No	9.5 $\,\pm$ 3.6	9.7 $\,\pm$ 3.5	11.0 $\,\pm$ 3.6	12.2 $\,\pm$ 4.0	16.4 $\,\pm$ 4.4	25.2 $\,\pm$ 8.1
Precision	No	31.6 $\,\pm$ 7.0	32.0 $\,\pm$ 7.2	31.7 $\,\pm$ 6.8	32.0 $\,\pm$ 7.0	31.9 $\,\pm$ 6.8	32.8 $\,\pm$ 7.2
F1-Score	No	8.5 $\,\pm$ 3.1	8.7 $\,\pm$ 3.0	10.5 $\,\pm$ 3.8	11.4 $\,\pm$ 4.2	15.4 $\,\pm$ 4.4	20.4 $\,\pm$ 6.2
AUC	No	3.0 $\,\pm$ 1.1	3.0 $\,\pm$ 1.1	3.3 $\,\pm$ 1.4	3.6 $\,\pm$ 1.6	3.8 $\,\pm$ 1.8	4.0 $\,\pm$ 2.0

Appendix

Table 16: MAE of estimation results for CIFAR10 classification when MOVAL Ensemble is optimized with different amounts of validation data. For group-wise calibration, we split the 10 cases into 5 groups, with each group containing 2 classes.

Dataset	Samples Used for Estimating
Dataset	19 (100 $\%$ )	12 (63 $\%$ )	6 (32 $\%$ )	3 (16 $\%$ )	2 (11 $\%$ )	1 (5 $\%$ )
Synthetic-65	4.3 $\,\pm$ 5.0	4.6 $\,\pm$ 4.4	4.7 $\,\pm$ 4.6	5.4 $\,\pm$ 4.2	6.2 $\,\pm$ 3.3	8.3 $\,\pm$ 2.4
Natural-4	2.6 $\,\pm$ 0.4	2.6 $\,\pm$ 0.3	3.3 $\,\pm$ 0.3	3.6 $\,\pm$ 0.3	4.2 $\,\pm$ 0.2	6.5 $\,\pm$ 0.4
Average	3.5	3.6	4.0	4.5	5.2	7.4

Appendix

Table 17: MAE of F1-Score estimation results for cardiac segmentation when MOVAL Ensemble is optimized with different amounts of validation data.

Dataset	Training Strategy	Samples Used for Estimating
Dataset	Training Strategy	9 (100 $\%$ )	6 (66 $\%$ )	3 (33 $\%$ )	2 (22 $\%$ )	1 (11 $\%$ )
Synthetic-42	Vanilla	0.5 $\,\pm$ 0.6	2.9 $\,\pm$ 0.3	4.0 $\,\pm$ 0.4	5.8 $\,\pm$ 0.1	13.5 $\,\pm$ 0.3
	Balanced Training I	0.5 $\,\pm$ 0.5	3.4 $\,\pm$ 0.1	3.8 $\,\pm$ 0.1	6.5 $\,\pm$ 0.2	14.3 $\,\pm$ 0.4
	Balanced Training II	0.2 $\,\pm$ 0.3	1.5 $\,\pm$ 0.1	3.5 $\,\pm$ 0.4	5.4 $\,\pm$ 0.2	8.4 $\,\pm$ 0.2
	Robust Training I	0.9 $\,\pm$ 0.7	4.4 $\,\pm$ 0.3	4.3 $\,\pm$ 0.3	8.8 $\,\pm$ 0.4	21.0 $\,\pm$ 0.8
	Robust Training II	0.6 $\,\pm$ 0.6	3.8 $\,\pm$ 0.4	3.6 $\,\pm$ 0.2	6.8 $\,\pm$ 0.1	15.0 $\,\pm$ 0.2

Appendix

Table 18: MAE of F1-Score estimation results for brain lesion segmentation under different training strategies when MOVAL Ensemble is optimized with different amounts of validation data.

Estimating Metric	Classification
Estimating Metric	Choice 1	Choice 2	Choice 3
Accuracy	MCP $\|$ CS TS-ATC	ENT $\|$ CS TS-ATC	ENE $\|$ CS ATC
Sensitivity	DOC $\|$ CS TS-ATC	ENT $\|$ CS ATC	MCP $\|$ CS ATC
Precision	DOC $\|$ CS TS	MCP $\|$ CS DoC	DOC $\|$ CS DoC
F1-Score	DOC $\|$ CS ATC	ENT $\|$ CS ATC	MCP $\|$ CS ATC
AUC	ENT $\|$ CS TS	DOC $\|$ CS TS	DOC $\|$ CS DoC

Appendix

Table 19: The confidence scores chosen for creating MOVAL Ensemble.

Distribution Shift

Distribution shift studies how changes between training and deployment data affect model reliability, and how to detect, measure, and address them.

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Inside-Out: Measuring Generalization in Vision Transformers Through Inner Workings
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School of Computer Science and Engineering, Sun Yat-sen University, China · 2Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education, China · 3Shenzhen Loop Area Institute, China +1

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Inside-Out: Measuring Generalization in Vision Transformers Through Inner Workings

Yunxiang Peng, Mengmeng Ma, Ziyu Yao, Xi Peng

University of Delaware · George Mason University

OOD Generalization Vision Transformer

Open paper AskPDF arXiv

Reliable generalization metrics are fundamental to the evaluation of machine learning models. Especially in high-stakes applications where labeled target data are scarce, evaluation of models' generalization performance under distribution shift is a pressing need. We focus on two practical scenarios: (1) Before deployment, how to select the best model for unlabeled target data? (2) After deployment, how to monitor model performance under distribution shift? The central need in both cases is a reliable and label-free proxy metric. Yet existing proxy metrics, such as model confidence or accuracy-on-the-line, are often unreliable as they only assess model output while ignoring the internal mechanisms that produce them. We address this limitation by introducing a new perspective: using the inner workings of a model, i.e., circuits, as a predictive metric of generalization performance. Leveraging circuit discovery, we extract the causal interactions between internal representations as a circuit, from which we derive two metrics tailored to the two practical scenarios. (1) Before deployment, we introduce Dependency Depth Bias, which measures different models' generalization capability on target data. (2) After deployment, we propose Circuit Shift Score, which predicts a model's generalization under different distribution shifts. Across various tasks, both metrics demonstrate significantly improved correlation with generalization performance, outperforming existing proxies by an average of 13.4% and 34.1%, respectively. Our code is available at https://github.com/deep-real/GenCircuit.

Estimating Metric	Classification
Estimating Metric	Choice 1	Choice 2	Choice 3
Accuracy	MCP $\|$ CS TS-ATC	ENT $\|$ CS TS-ATC	ENE $\|$ CS ATC
Sensitivity	DOC $\|$ CS TS-ATC	ENT $\|$ CS ATC	MCP $\|$ CS ATC
Precision	DOC $\|$ CS TS	MCP $\|$ CS DoC	DOC $\|$ CS DoC
F1-Score	DOC $\|$ CS ATC	ENT $\|$ CS ATC	MCP $\|$ CS ATC
AUC	ENT $\|$ CS TS	DOC $\|$ CS TS	DOC $\|$ CS DoC

Abstract

Distribution Shift

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