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Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks

Shiyu LiangYixuan LiR. Srikant

article2018ICLR2,606 citations

Proposes ODIN, a post-hoc method that significantly lowers false positive rates in out-of-distribution image detection using only temperature scaling and small input perturbations without retraining the neural network.

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Modern neural networks generalize effectively on data drawn from the same distribution as their training set but often assign high-confidence predictions to unrecognizable or irrelevant inputs. This creates safety risks in real-world visual recognition tasks where testing distributions cannot be fully controlled.

The article sets out to develop and evaluate a practical method for detecting out-of-distribution images on already-trained networks without any retraining.

The authors introduce ODIN, which combines temperature scaling in the softmax function with small, controlled input perturbations. They test the approach on DenseNet and Wide ResNet models trained on CIFAR-10 and CIFAR-100, measuring performance against multiple out-of-distribution sets including cropped and resized TinyImageNet and LSUN images plus synthetic noise, using metrics such as false-positive rate at 95 percent true-positive rate, AUROC, and AUPR.

ODIN consistently lowers the false-positive rate by large margins relative to the prior baseline; for example, it reduces the rate from 34.7 percent to 4.3 percent on a DenseNet trained on CIFAR-10 when distinguishing TinyImageNet crop images at 95 percent true-positive rate. Similar gains appear across architectures and dataset pairs, with performance remaining strong even when hyperparameters are transferred across validation sets. Larger distributional distances between in- and out-of-distribution data correlate with easier detection.

These results indicate that simple post-processing can substantially improve a network’s ability to flag unfamiliar inputs, lowering the chance that erroneous high-confidence predictions reach downstream decisions. The approach adds negligible computational cost at inference time and preserves accuracy on in-distribution examples.

The method can be applied immediately to existing models in safety-critical settings. Further validation on additional domains such as speech and natural-language processing would strengthen confidence before broad deployment. Performance degrades when out-of-distribution images closely resemble in-distribution data, and optimal hyperparameters require a modest validation set; results are therefore most reliable when such a set is available.

Cornell UniversityUniversity of Illinois Urbana-ChampaignUniversity of Wisconsin Madisonnetwork architecturedetection algorithmrecurrent neural networksODINOut‑of‑distribution detectionTemperature scaling
arXiv: 1706.02690facebookresearch/odin
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Abstract

We consider the problem of detecting out-of-distribution images in neural networks. We propose ODIN, a simple and effective method that does not require any change to a pre-trained neural network. Our method is based on the observation that using temperature scaling and adding small perturbations to the input can separate the softmax score distributions between in- and out-of-distribution images, allowing for more effective detection. We show in a series of experiments that ODIN is compatible with diverse network architectures and datasets. It consistently outperforms the baseline approach by a large margin, establishing a new state-of-the-art performance on this task. For example, ODIN reduces the false positive rate from the baseline 34.7% to 4.3% on the DenseNet (applied to CIFAR-10) when the true positive rate is 95%.

Topics

network architecturedetection algorithmrecurrent neural networkshigh resolution imagefeedforward neural networks

Keywords

ODINOut‑of‑distribution detectionTemperature scalingInput preprocessingSoftmax scoreFalse positive rateCIFAR‑10DenseNetWide ResNetMaximum mean discrepancy

Table of Contents

  • 1 Introduction
  • 2 Problem Statement
  • 3 ODIN: Out-of-distribution Detector
  • 4 Experiments
  • 4.1 Training Setup
  • 4.2 Out-of-distribution Datasets
  • 4.3 Evaluation metrics
  • 4.4 Experimental Results
  • 5 Discussions
  • 5.1 Analysis on Temperature Scaling
  • 5.2 Analysis on Input Preprocessing
  • 6 Related Works and Future Directions
  • 7 Conclusions
  • References
  • A Supplementary Results in Section and
  • B Taylor Expansion
  • C Proposition
  • D Analysis of Temperature
  • E Additional Results on Distance Measurement
  • F Additional Discussions

Knowls

  1. Knowl 1 — ODIN Out-of-Distribution Detection Method

    model/method

    ODIN (Out-of-DIstribution detector for Neural networks) is a post-processing method for detecting whether a test input xx originates from an in-distribution PXP_X or an out-of-distribution QXQ_X using a pre-trained classifier f = (f_1, arsc, f_N) without retraining or modifying the network architecture.

    The detector integrates two mechanisms:

    1. Input Preprocessing: A small, label-free perturbation is added to the input along the sign of the log-softmax gradient w.r.t. the input: x~=x−εsign⁡(−∇xlog⁡Sy^(x;T))\tilde{x} = x - \varepsilon \operatorname{sign}(-\nabla_x \log S_{\hat{y}}(x; T)), where y^=arg⁡max⁡ifi(x)\hat{y} = \arg\max_i f_i(x) is the predicted class and ε\varepsilon is the perturbation magnitude.
    2. Temperature Scaling: The model's logits fi(x~)f_i(\tilde{x}) are divided by a temperature parameter T>1T > 1 before computing the softmax score Si(x~;T)=exp⁡(fi(x~)/T)∑j=1Nexp⁡(fj(x~)/T)S_i(\tilde{x}; T) = \frac{\exp(f_i(\tilde{x})/T)}{\sum_{j=1}^N \exp(f_j(\tilde{x})/T)}.

    The maximum calibrated score max⁡iSi(x~;T)\max_i S_i(\tilde{x}; T) is compared against a decision threshold δ\delta. The sample is classified as in-distribution if max⁡iSi(x~;T)>δ\max_i S_i(\tilde{x}; T) > \delta, and out-of-distribution if ≤δ\le \delta. For samples classified as in-distribution, original unmodified inputs xx are processed through ff to obtain the final classification label.

  2. Knowl 2 — Input Perturbation via Log-Softmax Gradient Preprocessing

    equation

    To enhance the separation between in-distribution and out-of-distribution inputs, ODIN preprocesses each input image xx by adding a directional perturbation:

    x~=x−εsign⁡(−∇xlog⁡Sy^(x;T))\tilde{x} = x - \varepsilon \operatorname{sign}\left(-\nabla_x \log S_{\hat{y}}(x; T)\right)

    where:

    • x∈Xx \in \mathcal{X} is the original input image.
    • ε≥0\varepsilon \ge 0 is the perturbation magnitude hyperparameter.
    • y^=arg⁡max⁡ifi(x)=arg⁡max⁡iSi(x;T)\hat{y} = \arg\max_i f_i(x) = \arg\max_i S_i(x; T) is the index of the predicted class label with the highest logit output.
    • Sy^(x;T)=exp⁡(fy^(x)/T)∑j=1Nexp⁡(fj(x)/T)S_{\hat{y}}(x; T) = \frac{\exp(f_{\hat{y}}(x)/T)}{\sum_{j=1}^N \exp(f_j(x)/T)} is the temperature-scaled softmax score for class y^\hat{y}.
    • ∇xlog⁡Sy^(x;T)\nabla_x \log S_{\hat{y}}(x; T) is the gradient of the log-softmax score with respect to xx, efficiently calculated via standard back-propagation of the cross-entropy loss.

  3. Knowl 3 — ODIN Out-of-Distribution Detector Decision Rule

    algorithm

    The algorithm executes the ODIN test-time detection procedure for a single test input xx given pre-tuned hyperparameters (T,ε,δ)(T, \varepsilon, \delta):

    Input: Input image xx, pre-trained neural network f=(f1,…,fN)f = (f_1, \dots, f_N), temperature T>0T > 0, perturbation magnitude ε≥0\varepsilon \ge 0, detection threshold δ\delta
    Output: Decision g∈{0,1}g \in \{0, 1\} where 0 denotes in-distribution and 1 denotes out-of-distribution
    Compute unnormalized logits (f1(x),…,fN(x))(f_1(x), \dots, f_N(x))
    Determine predicted label y^=arg⁡max⁡i∈{1,…,N}fi(x)\hat{y} = \arg\max_{i \in \{1, \dots, N\}} f_i(x)
    Compute gradient gx=∇xlog⁡Sy^(x;T)g_x = \nabla_x \log S_{\hat{y}}(x; T)
    Apply input preprocessing perturbation x~=x−εsign⁡(−gx)\tilde{x} = x - \varepsilon \operatorname{sign}(-g_x)
    Compute perturbed logits (f1(x~),…,fN(x~))(f_1(\tilde{x}), \dots, f_N(\tilde{x}))
    Compute calibrated maximum softmax score S(x~;T)=exp⁡(fy^(x~)/T)∑j=1Nexp⁡(fj(x~)/T)S(\tilde{x}; T) = \frac{\exp(f_{\hat{y}}(\tilde{x})/T)}{\sum_{j=1}^N \exp(f_j(\tilde{x})/T)}
    if S(x~;T)≤δS(\tilde{x}; T) \le \delta:
        return 1
    else:
        return 0

  4. Knowl 4 — Asymptotic Detection Error Under Large Temperature Scaling Without Preprocessing

    theoretical result

    Let PXP_X be the in-distribution, QXQ_X the out-of-distribution, and PZP_Z the prior distribution over in-distribution (Z=0Z=0) versus out-of-distribution (Z=1Z=1) mixture components. When no input preprocessing is applied (ε=0\varepsilon = 0), the detection error Pe(T,0)P_e(T, 0) as temperature T→∞T \to \infty converges to a constant cc that depends only on the first-order logit difference function U1(X)=1N−1∑i≠y^[fy^(X)−fi(X)]U_1(X) = \frac{1}{N-1} \sum_{i \neq \hat{y}} [f_{\hat{y}}(X) - f_i(X)], PXP_X, QXQ_X, and PZP_Z:

    lim⁡T→∞Pe(T,0)=0.05P(Z=0)+P(Z=1)QX((N−1)U1(X)>α∗)=c\lim_{T \to \infty} P_e(T, 0) = 0.05 P(Z = 0) + P(Z = 1) Q_X\left((N - 1)U_1(X) > \alpha^*\right) = c

    where α∗\alpha^* is chosen such that PX((N−1)U1(X)>α∗)=0.95P_X((N - 1)U_1(X) > \alpha^*) = 0.95, ensuring a true positive rate (TPR) of 95%. Consequently, arbitrarily increasing temperature TT without input perturbation causes the detection error to plateau at a fixed floor rather than decreasing indefinitely.

  5. Knowl 5 — Taylor Expansion and Separation Mechanism of Temperature Scaling

    theoretical result

    For a sufficiently large temperature TT, the Taylor expansion of the maximum softmax score Sy^(x;T)S_{\hat{y}}(x; T) truncated to second order is:

    Sy^(x;T)≈1N−N−1TU1(x)+N−12T2U2(x)S_{\hat{y}}(x; T) \approx \frac{1}{N - \frac{N-1}{T} U_1(x) + \frac{N-1}{2T^2} U_2(x)}

    where:

    • U1(x)=1N−1∑i≠y^[fy^(x)−fi(x)]=Mean⁡(Δ)U_1(x) = \frac{1}{N-1}\sum_{i \neq \hat{y}} [f_{\hat{y}}(x) - f_i(x)] = \operatorname{Mean}(\Delta) measures the extent to which the largest unnormalized logit exceeds the average of the remaining logits (with Δi=fy^(x)−fi(x)\Delta_i = f_{\hat{y}}(x) - f_i(x)).
    • U2(x)=1N−1∑i≠y^[fy^(x)−fi(x)]2=Variance⁡(Δ)+Mean⁡(Δ)2U_2(x) = \frac{1}{N-1}\sum_{i \neq \hat{y}} [f_{\hat{y}}(x) - f_i(x)]^2 = \operatorname{Variance}(\Delta) + \operatorname{Mean}(\Delta)^2 measures the dispersion among non-maximal logits.

    Because Sy^(x;T)S_{\hat{y}}(x; T) is proportional to (U1−U2/2T)/T(U_1 - U_2/2T)/T, U1U_1 increases the score for in-distribution data (where neural networks are more confident), whereas U2U_2 decreases it. Selecting a large TT (e.g., T=1000T=1000) suppresses the disruptive variance penalty term U2/(2T)U_2/(2T), allowing U1U_1 to dominate and significantly widening the score gap between in-distribution and out-of-distribution data.

  6. Knowl 6 — First-Order Log-Softmax Expansion and Gradient-Norm Separation Mechanism

    theoretical result

    The first-order Taylor expansion of the log-softmax score of the perturbed input x~=x−εsign⁡(−∇xlog⁡Sy^(x;T))\tilde{x} = x - \varepsilon \operatorname{sign}(-\nabla_x \log S_{\hat{y}}(x; T)) with respect to perturbation magnitude ε\varepsilon is:

    log⁡Sy^(x~;T)=log⁡Sy^(x;T)+ε∥∇xlog⁡Sy^(x;T)∥1+o(ε)\log S_{\hat{y}}(\tilde{x}; T) = \log S_{\hat{y}}(x; T) + \varepsilon \|\nabla_x \log S_{\hat{y}}(x; T)\|_1 + o(\varepsilon)

    In-distribution images empirically produce significantly larger 1-norms of the log-softmax input gradient ∥∇xlog⁡Sy^(x;T)∥1\|\nabla_x \log S_{\hat{y}}(x; T)\|_1 than out-of-distribution images, including when conditioned on identical pre-perturbation softmax scores S(x;T)S(x; T). As a result, adding the gradient-directed perturbation ε\varepsilon induces a much larger increase in confidence for in-distribution images than for out-of-distribution images, increasing their statistical separability.

  7. Knowl 7 — Out-of-Distribution Detection Benchmark Performance on DenseNet-BC

    data/table

    The table below compares the out-of-distribution detection performance of the baseline maximum softmax probability method against ODIN across multiple test distributions on DenseNet-BC (L=100,k=12L=100, k=12) trained on CIFAR-10 and CIFAR-100. ODIN uses T=1000T = 1000 and perturbation magnitude ε\varepsilon tuned on the independent iSUN validation set (ε=0.0014\varepsilon = 0.0014 for CIFAR-10; ε=0.002\varepsilon = 0.002 for CIFAR-100). All values are percentages.

    In-Dist Out-of-Dist Dataset FPR (95% TPR) ↓\downarrow Detection Error ↓\downarrow AUROC ↑\uparrow AUPR In ↑\uparrow AUPR Out ↑\uparrow
    Baseline / ODIN
    CIFAR-10 TinyImageNet (crop) 34.7 / 4.3 10.0 / 4.7 95.3 / 99.1 96.4 / 99.1 93.8 / 99.1
    TinyImageNet (resize) 40.8 / 7.5 11.5 / 6.1 94.1 / 98.5 95.1 / 98.6 92.4 / 98.5
    LSUN (crop) 39.3 / 11.4 10.2 / 7.2 94.8 / 97.9 96.0 / 98.0 93.1 / 97.9
    LSUN (resize) 33.6 / 3.8 9.8 / 4.4 95.4 / 99.2 96.4 / 99.3 94.0 / 99.2
    Uniform Noise 23.5 / 0.0 5.3 / 0.5 96.5 / 99.0 97.8 / 100.0 93.0 / 99.0
    Gaussian Noise 12.3 / 0.0 4.7 / 0.2 97.5 / 100.0 98.3 / 100.0 95.9 / 100.0
    CIFAR-100 TinyImageNet (crop) 67.8 / 26.9 36.4 / 12.9 83.0 / 94.5 85.3 / 94.7 80.8 / 94.5
    TinyImageNet (resize) 82.2 / 57.0 43.6 / 22.7 70.4 / 85.5 71.4 / 86.0 68.6 / 84.8
    LSUN (crop) 69.4 / 18.6 37.2 / 9.7 83.7 / 96.6 86.2 / 96.8 80.9 / 96.5
    LSUN (resize) 83.3 / 58.0 44.1 / 22.3 70.6 / 86.0 72.5 / 87.1 68.0 / 84.8
    Uniform Noise 100.0 / 100.0 35.9 / 17.9 43.1 / 99.5 63.2 / 87.5 41.9 / 65.1
    Gaussian Noise 100.0 / 100.0 41.2 / 38.0 30.6 / 40.5 53.4 / 60.5 37.6 / 40.9

    ODIN dramatically reduces False Positive Rate (FPR at 95% TPR) across both natural and synthetic noise datasets, reducing FPR by up to a factor of 8 on CIFAR-10 pairs and elevating AUROC beyond 98% across natural test sets.

  8. Knowl 8 — Transferability of ODIN Hyperparameters Across Validation Sets

    empirical result

    Hyperparameters (T,ε)(T, \varepsilon) tuned on one out-of-distribution validation dataset transfer across other distinct out-of-distribution test datasets with negligible sensitivity.

    On DenseNet-BC pre-trained on CIFAR-10, tuning (T,ε)(T, \varepsilon) using any individual validation dataset (TinyImageNet crop, TinyImageNet resize, LSUN crop, LSUN resize, iSUN, Gaussian noise, or Uniform noise) yields closely matching FPR (at 95% TPR) when evaluated on test distributions: FPR on TinyImageNet (crop) test set evaluates to 4.3% when tuned on TinyImageNet (resize), LSUN (resize), iSUN, Gaussian, or Uniform datasets, and 6.6% when tuned on LSUN (crop). Similarly, testing on Gaussian noise yields 0.0% FPR across all natural and synthetic tuning sets except LSUN crop (0.4%).

  9. Knowl 9 — Relationship Between Dataset Distributional Distance and Out-of-Distribution Detection Difficulty

    empirical result

    Statistical distances between in-distribution and out-of-distribution datasets measured by Maximum Mean Discrepancy (MMD) with a Gaussian RBF kernel or Energy distance correlate negatively with OOD detection performance:

    • Cropped image datasets (TinyImageNet crop, LSUN crop) exhibit larger distributional distances to CIFAR-100 (MMD ≈0.41−0.43\approx 0.41 - 0.43, Energy distance ≈2.25−2.31\approx 2.25 - 2.31) because random patches contain localized context distinct from full CIFAR images.
    • Resized image datasets (TinyImageNet resize, LSUN resize) preserve global object structure and are much closer in distance (MMD ≈0.088−0.12\approx 0.088 - 0.12, Energy distance ≈0.54−0.63\approx 0.54 - 0.63).

    Both FPR at 95% TPR and detection error increase systematically as the statistical distance decreases, demonstrating that out-of-distribution detection becomes harder when out-of-distribution images are distributionally closer to the in-distribution data.

  10. Knowl 10 — Correlation Between Classification Accuracy and Out-of-Distribution Detection Scores

    empirical result

    In-distribution test samples with lower ODIN softmax scores are disproportionately more difficult for the neural network to classify correctly.

    When evaluating a DenseNet-BC model trained on CIFAR-10:

    • Inputs with calibrated softmax scores above the threshold corresponding to 80% True Positive Rate achieve up to 98.5% classification accuracy on CIFAR-10 test data.
    • Inputs with calibrated softmax scores below the same 80% TPR threshold achieve only ~82% classification accuracy.

    This shows that in-distribution samples that are difficult to distinguish from out-of-distribution data are strongly correlated with samples that the classifier is prone to misclassifying.

Coverage note — No substantial contributed material was omitted; supplementary experimental plots for Wide-ResNet-28-10 and Wide-ResNet-40-4 architectures were integrated into the primary empirical and analytical knowls.

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Citation

MLA
Liang, S., et al. “Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks.”. ICLR, 2018, https://dblp.org/rec/conf/iclr/LiangLS18.
APA
Liang, S., Li, Y., & Srikant, R. (2018). Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks. ICLR. https://dblp.org/rec/conf/iclr/LiangLS18
Chicago
Liang, S., Y. Li, and R. Srikant. 2018. “Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks.”. ICLR. https://dblp.org/rec/conf/iclr/LiangLS18.
Harvard
Liang, S., Li, Y. and Srikant, R. (2018) “Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks.”, ICLR. Available at: https://dblp.org/rec/conf/iclr/LiangLS18.
Vancouver
1. Liang S, Li Y, Srikant R (2018) Enhancing The Reliability of Out-of-distribution Image Detection in Neural Networks. ICLR

BibTeX

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