Training-Free Uncertainty Estimation for Embedding Models

Young-Jin Park; Addison Kristanto Julistiono; Hao Wang; Shervin Ardeshir; Navid Azizan

cs.LGMay 31, 2023

Training-Free Uncertainty Estimation for Embedding Models

Authors: Young-Jin Park, Addison Kristanto Julistiono, Hao Wang, Shervin Ardeshir, Navid Azizan

Organizations: Massachusetts Institute of Technology · MIT-IBM Watson AI Lab

Abstract

Embedding models, often obtained via self-supervised learning, extract general-purpose representations from data. Quantifying the reliability of these representations is crucial, as many downstream models rely on them as input for their own tasks. To this end, we introduce a formal definition of representation reliability: the representation for a given test point is considered to be reliable if the downstream models built on top of that representation can, on average, consistently generate accurate predictions for that test point across various downstream tasks. However, accessing the downstream data to quantify the representation reliability is often limited or restricted for various reasons. We propose training-free methods for estimating the representation reliability without access to the downstream data. Our method is based on the concept of neighborhood consistency (NC) across distinct pre-trained representation spaces. The key insight is to find shared neighboring points as anchors to align these representation spaces before comparing them. We provide theoretical justifications for NC and develop two practical approaches: (1) directly computing NC when multiple pre-trained models are available, and (2) a perturbation-based NC (PNC), which creates synthetic ensembles from a single model through isotropic Gaussian noise, avoiding the computational cost of training deep ensembles. We further propose PNC-spread tuning, which systematically determines the perturbation magnitude by maximizing the spread of the PNC scores on a reference set. We demonstrate through comprehensive numerical experiments that our methods effectively capture the representation reliability with a high degree of correlation, achieving robust and favorable performance compared with baseline methods.

Figures & tables

Figure 1 : Illustration of representation reliability ( $\mathsf{Reli}$ ) and neighborhood consistency ( $\mathsf{NC}$ ). For a test point $x^{*}$ and a collection of pre-trained backbone models $\mathcal{H}=\{h_{1},\cdots,h_{M}\}$ , the representation reliability is defined as the average performance of downstream models when using the representations of $x^{*}$ provided by the backbones in $\mathcal{H}$ . Our $\mathsf{NC}$ estimates $\mathsf{Reli}$ without requiring any prior knowledge of the downstream tasks. It operates by measuring the number of consistent neighbors of $x^{*}$ among reference points across different representation spaces.

Pretraining Algorithms	Method	ResNet-18				ResNet-50
		CIFAR-10		CIFAR-100		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier	Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{NC}_{100}$	0.3865	0.3262	0.3130	0.2590	0.3786	0.3206	0.3070	0.2516
	$\mathsf{Dist}_{1}$	0.3021	0.2544	0.0757	0.0635	0.2750	0.2376	0.1100	0.0919
	$\mathsf{Norm}$	0.2907	0.2429	0.1198	0.0649	0.2789	0.2385	0.1311	0.0710
	$\mathsf{FV}$	-0.0476	-0.0458	-0.1891	-0.1710	-0.0402	-0.0393	-0.1984	-0.1870

Table 1 : Comparison of our neighborhood consistency (

\mathsf{NC}_{100}

) with baseline methods in terms of their correlation with the representation reliability. We use Kendall’s

\tau

coefficient to measure this correlation. Downstream tasks are in-distribution tasks where the model is pre-trained and fine-tuned on the same dataset. Performance on these tasks is evaluated using either negative predictive entropy or negative Brier score. The highest and second-highest scores are highlighted in bold and underlined, respectively. As shown, our method consistently receives a favorable score compared with baselines.

Table 2 : Comparison of our neighborhood consistency (

\mathsf{NC}_{100}

) with baseline methods in terms of their correlation with the representation reliability. Here the downstream tasks are transfer learning tasks, where embedding functions are pre-trained on TinyImagenet and fine-tuned on CIFAR-10 , CIFAR-100 , and STL-10 .

Panel of Figure 2 — Figure 2 : Ablation studies on the (a)–(b) ensemble size ( $M$ ), (c)–(d) the number of neighbors ( $k$ ) for $\mathsf{NC}_{k}$ (ours) and baselines, and (e)–(f) the robustness to the choice of distance metric for $\mathsf{NC}_{k}$ . Brier score is used for the downstream performance metric. The comprehensive results can be found in Appendix D .

Pretraining Algorithms	Method	ResNet-18				ResNet-50
		CIFAR-10		CIFAR-100		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier	Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{PNC}_{100}^{s}$	0.3262	0.2779	0.2175	0.1724	0.2961	0.2525	0.2396	0.1954
	$\mathsf{Dist}_{1}$	0.2383	0.2058	0.0580	0.0572	0.2351	0.2049	0.0973	0.0838
	$\mathsf{Norm}$	0.2668	0.2157	0.1242	0.0620	0.2629	0.2156	0.1370	0.0690
BYOL	$\mathsf{PNC}_{100}^{s}$	0.2306	0.1361	0.2184	0.1364	0.3473	0.1764	0.3130	0.1231

Table 3 : Quantifying the reliability of individual embedding functions without retraining (single model only). Kendall’s

\tau

coefficient correlation between each method and the downstream performance of individual models on in-distribution binary classification tasks. The baselines (

\mathsf{Dist}_{1}

\mathsf{Norm}

) are computed from the single model without ensemble averaging. The highest and second-highest scores are highlighted in bold and underlined, respectively.

Table 4 : Quantifying the reliability of individual embedding functions without retraining (single model only). Kendall’s

\tau

coefficient correlation between each method and the downstream performance on transfer learning binary classification tasks. Embedding functions are pre-trained on TinyImagenet and fine-tuned on CIFAR-10 , CIFAR-100 , and STL-10 .

Panel of Figure 3 — Figure 3 : Validation of $\mathsf{PNC}$ -spread tuning for SimCLR on in-distribution binary classification tasks. Blue: Kendall’s $\tau$ coefficient between $\mathsf{PNC}_{100}$ scores and downstream Brier scores as a function of $\sigma_{\text{pert}}$ . Red: standard deviation of $\mathsf{PNC}_{100}$ scores on $X_{\text{ref}}$ . The peak correlation aligns with the peak spread, supporting the spread-based tuning criterion. Dashed gray line: largest considered $\sigma_{pert}$ that did not cause a floating-point overflow.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4 : Graphical visualization of the sketch for the proof of Theorem 2 . Let $\mathcal{Z}i$ and $\mathcal{Z}j$ denote the representation spaces defined by the embedding functions $h_{i}$ and $h_{j}$ , respectively. Suppose that there is a reliable neighboring point $x^{r}$ that is located close to the test point $x^{*}$ in each representation space. For any downstream task $t$ , a reliable neighboring point $x^{r}$ serves as an anchor for comparing different representations $\bm{z}_{i}^{*}=h_{i}(x^{*})$ and $\bm{z}_{j}^{*}=h_{j}(x^{*})$ of the test point $x^{*}$ . The key idea is that $y_{i,t}^{*}$ and $y_{j,t}^{*}$ — the downstream predictions on the test point using the two different embedding functions — should be similar because the predictions $y_{i,t}^{*}$ and $y_{i,t}^{r}$ as well as $y_{j,t}^{r}$ and $y_{j,t}^{*}$ are similar due to the Lipschitz continuity of the downstream predictors. Additionally, since $x^{r}$ is a reliable point, the predictions $y_{i,t}^{r}$ and $y_{j,t}^{r}$ are similar. Thus, it follows that $y_{i,t}^{*}$ and $y_{j,t}^{*}$ are similar as well.

Pretraining Algorithms	Method	ResNet-18
		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{NC}_{100}$	0.3865 $\pm$ 0.0024	0.3262 $\pm$ 0.0022	0.3130 $\pm$ 0.0031	0.2590 $\pm$ 0.0028
	$\mathsf{Dist}_{1}$	0.3021 $\pm$ 0.0063	0.2544 $\pm$ 0.0051	0.0757 $\pm$ 0.0084	0.0635 $\pm$ 0.0075
	$\mathsf{Norm}$	0.2907	0.2429	0.1198	0.0649
	$\mathsf{FV}$	-0.0476	-0.0458	-0.1891	-0.1710

Appendix

Table 5 : Comparison on the correlation between our neighborhood consistency (

\mathsf{NC}_{100}

) and baseline methods in relation to performance on in-distribution downstream tasks.

Pretraining Algorithms	Method	ResNet-18
		$\texttt{TinyImagenet}\rightarrow$
		CIFAR-10		CIFAR-100		STL-10
		Entropy	Brier	Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{NC}_{100}$	0.1729 $\pm$ 0.0090	0.1292 $\pm$ 0.0089	0.1680 $\pm$ 0.0032	0.1343 $\pm$ 0.0032	0.2176 $\pm$ 0.0112	0.1513 $\pm$ 0.0106
	$\mathsf{Dist}_{1}$	0.1311 $\pm$ 0.0127	0.0933 $\pm$ 0.0115	0.0138 $\pm$ 0.0071	-0.0251 $\pm$ 0.0049	0.1098 $\pm$ 0.0151	0.0520 $\pm$ 0.0116
	$\mathsf{Norm}$	0.2124	0.1642	0.1220	0.0507	0.1641	0.0852

Appendix

Table 6 : Comparison on the correlation between our neighborhood consistency (

\mathsf{NC}_{100}

) and baseline methods in relation to performance on transfer learning tasks from TinyImagenet to CIFAR-10 , CIFAR-100 , and STL-10 .

Pretraining Algorithms	Method	ResNet-18
		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{NC}_{100}$	0.3848 $\pm$ 0.0016	0.3202 $\pm$ 0.0019	0.2998 $\pm$ 0.0051	0.2166 $\pm$ 0.0030
	$\mathsf{Dist}_{1}$	0.3025 $\pm$ 0.0079	0.2533 $\pm$ 0.0055	0.0641 $\pm$ 0.0070	0.0698 $\pm$ 0.0072
	$\mathsf{Norm}$	0.3098	0.2462	0.1725	0.0467
	$\mathsf{FV}$	-0.0485	-0.0357	-0.1755	-0.1358

Appendix

Table 7 : Comparison on the correlation between our neighborhood consistency (

\mathsf{NC}_{100}

) and baseline methods in relation to performance on in-distribution downstream tasks, for the multi-class downstream classification tasks. Our approach consistently demonstrates strong performance when compared to baseline methods, similar to the results observed when using the OVO binary classification tasks, as shown in Table 5 .

Pretraining Algorithms	Method	ResNet-18
		$\texttt{TinyImagenet}\rightarrow$
		CIFAR-10		CIFAR-100		STL-10
		Entropy	Brier	Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{NC}_{100}$	0.1673 $\pm$ 0.0080	0.1230 $\pm$ 0.0073	0.1486 $\pm$ 0.0046	0.1031 $\pm$ 0.0018	0.2012 $\pm$ 0.0106	0.1271 $\pm$ 0.0100
	$\mathsf{Dist}_{1}$	0.1331 $\pm$ 0.0122	0.0982 $\pm$ 0.0116	0.0082 $\pm$ 0.0064	-0.0162 $\pm$ 0.0051	0.0633 $\pm$ 0.0122	0.0382 $\pm$ 0.0107
	$\mathsf{Norm}$	0.2294	0.1654	0.1527	0.0348	0.1431	0.0624

Appendix

Table 8 : Comparison on the correlation between our neighborhood consistency (

\mathsf{NC}_{100}

) and baseline methods in relation to performance on transfer learning tasks from TinyImagenet to CIFAR-10 , CIFAR-100 , and STL-10 , for the multi-class downstream classification tasks.

Pretraining Algorithms	Method	ResNet-18
		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{NC}_{100}$	0.3237 $\pm$ 0.0023	0.2727 $\pm$ 0.0021	0.2868 $\pm$ 0.0014	0.2408 $\pm$ 0.0021
	$\mathsf{Dist}_{1}$	-0.0590 $\pm$ 0.0071	-0.0460 $\pm$ 0.0065	-0.0317 $\pm$ 0.0061	0.0027 $\pm$ 0.0064
	$\mathsf{Norm}$	0.2907	0.2429	0.1198	0.0649
	$\mathsf{FV}$	-0.3224	-0.2704	-0.2370	-0.1723

Appendix

Table 9 : Comparison on the correlation between our neighborhood consistency (

\mathsf{NC}_{100}

) and baseline methods in relation to performance on in-distribution downstream tasks, for the unnormalized representation. The overall correlation is weaker compared to the one observed when using normalized representation as presented in Table 5 . Nevertheless, our approach consistently shows a robust performance compared to baseline methods.

Pretraining Algorithms	Method	ResNet-18
		$\texttt{TinyImagenet}\rightarrow$
		CIFAR-10		CIFAR-100		STL-10
		Entropy	Brier	Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{NC}_{100}$	0.1392 $\pm$ 0.0072	0.1025 $\pm$ 0.0067	0.1520 $\pm$ 0.0052	0.1218 $\pm$ 0.0057	0.1540 $\pm$ 0.0103	0.1054 $\pm$ 0.0080
	$\mathsf{Dist}_{1}$	-0.1318 $\pm$ 0.0092	-0.1101 $\pm$ 0.0090	-0.1067 $\pm$ 0.0079	-0.0671 $\pm$ 0.0053	-0.1211 $\pm$ 0.0120	-0.0838 $\pm$ 0.0086
	$\mathsf{Norm}$	0.2124	0.1642	0.1220	0.0507	0.1641	0.0852

Appendix

Table 10 : Comparison on the correlation between our neighborhood consistency (

\mathsf{NC}_{100}

) and baseline methods in relation to performance on transfer learning tasks from TinyImagenet to CIFAR-10 , CIFAR-100 , and STL-10 , for the unnormalized representation.

Panel of Figure 5 — Figure 5 : Ablation over the ensemble size ( $M$ ) for $\mathsf{NC}_{k}$ (ours) and baselines. Brier score is used for the downstream performance metric.

Panel of Figure 6 — Figure 6 : Ablation over the number of neighbors ( $k$ ) for $\mathsf{NC}_{k}$ (ours) and baselines. Brier score is used for the downstream performance metric.

Pretraining Algorithms	Method	ResNet-18
		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{PNC}_{100}^{s}$	0.3300 $\pm$ 0.0325	0.2716 $\pm$ 0.0218	0.2183 $\pm$ 0.0278	0.1427 $\pm$ 0.0159
	$\mathsf{Dist}_{1}$	0.2407 $\pm$ 0.0251	0.2053 $\pm$ 0.0176	0.0520 $\pm$ 0.0315	0.0640 $\pm$ 0.0134
	$\mathsf{Norm}$	0.2903 $\pm$ 0.0421	0.2146 $\pm$ 0.0319	0.1808 $\pm$ 0.0308	0.0363 $\pm$ 0.0260
BYOL	$\mathsf{PNC}_{100}^{s}$	0.3198 $\pm$ 0.0740	0.0863 $\pm$ 0.0161	0.2793 $\pm$ 0.0884	0.0494 $\pm$ 0.0161

Appendix

Table 11 : Quantifying the reliability of individual embedding functions without retraining. Kendall’s

\tau

coefficient correlation between each method and the downstream performance of individual models on in-distribution multi-class classification tasks. The baselines (

\mathsf{Dist}_{1}

and

\mathsf{Norm}

) are computed from the single model without ensemble averaging. The highest and second-highest scores are highlighted in bold and underlined, respectively.

Pretraining Algorithms	Method	ResNet-18
		$\texttt{TinyImagenet}\rightarrow$
		CIFAR-10		CIFAR-100		STL-10
		Entropy	Brier	Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{PNC}_{100}^{s}$	0.1769 $\pm$ 0.0154	0.1301 $\pm$ 0.0117	0.1747 $\pm$ 0.0080	0.0947 $\pm$ 0.0048	0.2251 $\pm$ 0.0198	0.1479 $\pm$ 0.0149
	$\mathsf{Dist}_{1}$	0.1210 $\pm$ 0.0159	0.0882 $\pm$ 0.0127	-0.0228 $\pm$ 0.0100	-0.0188 $\pm$ 0.0058	0.0589 $\pm$ 0.0191	0.0357 $\pm$ 0.0150
	$\mathsf{Norm}$	0.2214 $\pm$ 0.0121	0.1534 $\pm$ 0.0091	0.1363 $\pm$ 0.0101	0.0166 $\pm$ 0.0071	0.1435 $\pm$ 0.0214	0.0617 $\pm$ 0.0168

Appendix

Table 12 : Quantifying the reliability of individual embedding functions without retraining. Kendall’s

\tau

coefficient correlation between each method and the downstream performance of individual models on transfer learning multi-class classification tasks. The baselines (

\mathsf{Dist}_{1}

and

\mathsf{Norm}

) are computed from the single model without ensemble averaging. Embedding functions are pre-trained on TinyImagenet and fine-tuned on CIFAR-10 , CIFAR-100 , and STL-10 . The highest and second-highest scores are highlighted in bold and underlined, respectively.

Pretraining Algorithms	Method	ResNet-18				ResNet-50
		CIFAR-10		CIFAR-100		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier	Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{PNC}_{100}^{o}$	0.3361	0.2863	0.2405	0.1929	0.3116	0.2658	0.2475	0.2004
SimCLR	$\mathsf{PNC}_{100}^{s}$	0.3262	0.2779	0.2175	0.1724	0.2961	0.2525	0.2396	0.1954
BYOL	$\mathsf{PNC}_{100}^{o}$	0.2458	0.1509	0.2624	0.1656	0.3530	0.1911	0.3157	0.1405
BYOL	$\mathsf{PNC}_{100}^{s}$	0.2306	0.1361	0.2184	0.1364	0.3473	0.1764	0.3130	0.1231

Appendix

Table 13 : Oracle (

\mathsf{PNC}_{100}^{o}

) vs. spread-tuned (

\mathsf{PNC}_{100}^{s}

) Kendall’s

\tau

coefficient correlation with downstream performance on in-distribution binary classification tasks.

Table 14 : Oracle (

\mathsf{PNC}_{100}^{o}

) vs. spread-tuned (

\mathsf{PNC}_{100}^{s}

) Kendall’s

\tau

coefficient correlation with downstream performance on transfer learning binary classification tasks.

Pretraining Algorithms	Method	ResNet-18				ResNet-50
		CIFAR-10		CIFAR-100		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier	Entropy	Brier	Entropy	Brier
SimCLR	$\mathsf{PNC}_{100}^{o}$	0.3397	0.2806	0.2367	0.1594	0.3107	0.2613	0.2412	0.1641
SimCLR	$\mathsf{PNC}_{100}^{s}$	0.3300	0.2716	0.2183	0.1427	0.2946	0.2474	0.2318	0.1605
BYOL	$\mathsf{PNC}_{100}^{o}$	0.3365	0.1013	0.3468	0.0736	0.4493	0.1082	0.5505	0.0516
BYOL	$\mathsf{PNC}_{100}^{s}$	0.3198	0.0863	0.2793	0.0494	0.4408	0.0932	0.5472	0.0172

Appendix

Table 15 : Oracle (

\mathsf{PNC}_{100}^{o}

) vs. spread-tuned (

\mathsf{PNC}_{100}^{s}

) Kendall’s

\tau

coefficient correlation with downstream performance on in-distribution multi-class classification tasks.

Table 16 : Oracle (

\mathsf{PNC}_{100}^{o}

) vs. spread-tuned (

\mathsf{PNC}_{100}^{s}

) Kendall’s

\tau

coefficient correlation with downstream performance on transfer learning multi-class classification tasks.

Panel of Figure 7 — Figure 7 : Validation of $\mathsf{PNC}$ -spread tuning for BYOL and MoCo on in-distribution binary classification tasks. Blue: Kendall’s $\tau$ coefficient between $\mathsf{PNC}_{100}$ scores and downstream Brier scores as a function of $\sigma_{\text{pert}}$ . Red: standard deviation of $\mathsf{PNC}_{100}$ scores on $X_{\text{ref}}$ . The peak correlation aligns with the peak spread, supporting the spread-based tuning criterion. Dashed gray line: largest considered $\sigma_{pert}$ that did not cause a floating-point overflow.

Downstream Task Type	Method	ResNet-18				ResNet-50
		CIFAR-10		CIFAR-100		CIFAR-10		CIFAR-100
		Entropy	Brier	Entropy	Brier	Entropy	Brier	Entropy	Brier
Binary	$\mathsf{PNC}_{100}^{\mathbin{\\|},o}$	0.3622	0.3062	0.2490	0.1968	0.3636	0.3107	0.2546	0.2016
	$\mathsf{PNC}_{100}^{\mathbin{\\|},s}$	0.3574	0.3025	0.2479	0.1961	0.3618	0.3087	0.2546	0.2016
	$\mathsf{PNC}_{100}^{\perp,o}$	0.3267	0.2787	0.2349	0.1901	0.2873	0.2446	0.2458	0.2018
	$\mathsf{PNC}_{100}^{\perp,s}$	0.3193	0.2719	0.2119	0.1677	0.2637	0.2246	0.2113	0.1759

Appendix

Table 17 : Quantifying the reliability of individual embedding functions without retraining for different choices of

\mathcal{P}

. The following are Kendall’s

\tau

coefficient correlation between each method and the downstream performance of individual models on in-distribution classification tasks.

Table 18 : Quantifying the reliability of individual embedding functions without retraining for different choices of

\mathcal{P}

. The following are Kendall’s

\tau

coefficient correlation between each method and the downstream performance of individual models on transfer learning classification tasks. Embedding functions are pre-trained on TinyImagenet and fine-tuned on CIFAR-10 , CIFAR-100 , and STL-10 .

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Decodable but Not Accessible: Auditing Distance-Based Reliability Estimation on Disentangled Skin-Lesion Representations

Duc-Vinh Tran

School of Information and Communication Technology, Hanoi University of Science and Technology, 1 Dai Co Viet Road, Hai Ba Trung, Hanoi 100000, Vietnam

Representation Space Orthogonality Model Auditing Dose Decoding

Open paper AskPDF arXiv

Distance-based reliability estimation assumes that a representation's geometry reflects its trustworthiness, yet this assumption is rarely tested under training interventions that reshape geometry directly. We audit this assumption under domain-adversarial representation learning using a disentanglement dose-response ladder. Three checkpoint families share the same architecture and a 16-dimensional representation, differing only in orthogonality strength (lambda = 0, 1, 5). Representation geometry changed substantially with disentanglement strength: the condition number shifted by two orders of magnitude (Kendall tau = 0.84, exact p = 2.8e-5). This change was not accompanied by improved reliability estimation: Mahalanobis-distance AUROC (ISIC-test vs. PAD-UFES) remained flat and below chance (about 0.40) at every level, with no significant association with any of five geometry metrics tested. The same failure was observed for cosine-to-centroid and pooled k-nearest-neighbor scorers, plus three non-distance-based scorers: an energy-based confidence score, Virtual-Logit Matching, and a kernel density estimator. Seven of eight scorers converged on the same result; the energy-based score showed an isolated upward trend that we report but do not treat as evidence against the overall pattern. A supervised probe with no access to the training objective recovered domain membership from the identical embeddings at 0.72-0.81 AUROC across every level, showing that the relevant information was not absent from the representation. These findings indicate that classification performance alone can overlook whether information in a learned representation is organized in a form that downstream reliability estimators can use. Information can remain decodable while becoming largely inaccessible to non-probing reliability estimators.

Abstract

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