Title: TRAINING-FREE UNCERTAINTY ESTIMATION FOR NEURAL NETWORKS

Abstract: Uncertainty estimation is an essential step in the evaluation of the robustness for deep learning models in computer vision, especially when applied in risk-sensitive areas. However, most state-of-the-art deep learning models either fail to obtain uncertainty estimation or need significant modification (e.g., formulating a proper Bayesian treatment) to obtain it. None of the previous methods are able to take an arbitrary model off the shelf and generate uncertainty estimation without retraining or redesigning it. To address this gap, we perform the first systematic exploration into training-free uncertainty estimation. We propose three simple and scalable methods to analyze the variance of output from a trained network under tolerable perturbations: infer-transformation, infernoise, and infer-dropout. They operate solely during inference, without the need to re-train, re-design, or fine-tune the model, as typically required by other stateof-the-art uncertainty estimation methods. Surprisingly, even without involving such perturbations in training, our methods produce comparable or even better uncertainty estimation when compared to other training-required state-of-the-art methods. Last but not least, we demonstrate that the uncertainty from our proposed methods can be used to improve the neural network training.

1. INTRODUCTION:

Deep learning is already able to achieve excellent or even super-human performance in many tasks (Krizhevsky et al., 2012; He et al., 2015; Silver et al., 2016) . While most previous work in the field has focused on improving accuracy in various tasks, in several risk-sensitive areas such as autonomous driving (Chen et al., 2015) and healthcare (Zhang et al., 2019) , reliability and robustness are arguably more important and interesting than accuracy.

Recently, several novel approaches have been proposed to take into account an estimation of uncertainty during training and inference. Some use probabilistic formulations for neural networks (Graves, 2011; Hernández-Lobato & Adams, 2015; Wang et al., 2016; Shekhovtsov & Flach, 2018) and model the distribution over the parameters (weights) and/or the neurons. Such formulations naturally produce distributions over the possible outputs. Others utilize the randomness induced during training and inference (e.g., dropout and ensembling) to obtain an uncertainty estimation (Gal & Ghahramani, 2015; Lakshminarayanan et al., 2017) .

All methods above require specific designs or a special training pipeline in order to involve the uncertainty estimation during training. Unfortunately, there are many cases where such premeditated designs or pipelines cannot be implemented. For example, if one wants to study the uncertainty of trained models released online, retraining is never an option, especially when only a black-box model is provided or the training data is not available. Moreover, most models are deterministic and do not have stochasticity. A straightforward solution is to add dropout layers into proper locations and finetune the model (Gal & Ghahramani, 2016) . However, this is impractical for many state-ofthe-art and published models, especially those trained on large datasets (e.g. ImageNet (Deng et al., 2009) ) with a vast amount of industrial computing resources. In addition, models that have already been distilled, pruned, or binarized fall short of fitting re-training (Han et al., 2015a; Hou et al., 2016) .

To fill this gap, we first propose and define the problem of training-free uncertainty estimation: how to obtain an uncertainty estimation of any given model without re-designing, re-training, or fine-tuning it. We focus on two scenarios: black-box uncertainty estimation (BBUE) , where one has access to the model only as a black box, and gray-box uncertainty estimation (GBUE) , where one has access to intermediate-layer neurons of the model (but not the parameters). To the best of our knowledge, ours is the first systematic exploration into the problem.

We propose a set of simple and scalable training-free methods to analyze the variance of output from a trained network. Our main idea is to add a tolerable perturbation into inputs or feature maps during inference. Different from an adversarial perturbation aiming to change the outputs during inference (Madry et al., 2017) , a tolerable perturbation does not dramatically alter the original distribution while allowing generation of multiple diverse outputs that could later be used for uncertainty estimation. The first method, which we call infer-transformation, is to apply transformation that exploits the natural characteristics of a CNN: that it is variant to input transformation such as rotation (Cohen & Welling, 2016) . Transformations have been frequently used for augmentation but rarely evaluated for uncertainty estimation. The second method, infer-noise, is to inject Gaussian noise with a zero-mean and a small standard deviation into intermediate-layer neurons. The third one, which we call infer-dropout is to perform inference-time dropout in a chosen layer. Although at first blush infer-dropout is similar to MC-dropout, where dropout is performed during both training and inference in the same layers, they are different in several aspects: (1) Infer-dropout is involved only during inference. (2) Infer-dropout can be applied to arbitrary layers, even those without dropout training. Surprisingly, we find that even without involving dropout during training, infer-dropout is still comparable to, or even better than, MC-dropout for the purpose of uncertainty estimation.

For classification, we note that the softmax output in classification models is naturally a distribution, the entropy of which could be directly used for training-free uncertainty estimation. Hence, using entropy for uncertainty estimation qualifies as a training-free method. We evaluate this method in two classification tasks (see details in Appendix A.1) and find that it already yields satisfactory uncertainty estimation (even more correlated with error compared with MC dropout). Therefore in this paper we focus on regression tasks where output distributions are not readily available. Following the previous work, we evaluate our proposed methods on two regression tasks, monocular depth estimation and single image super resolution, shown in Figure 1 . Our major contributions are thus:

1. We perform, to the best of our knowledge, the first systematic exploration of training-free uncertainty estimation during inference with a post-hoc analysis, given any trained models. 2. We propose simple and scalable methods for regression models, using a tolerable perturbation such as infer-transformation or infer-noise injection to effectively and efficiently estimate uncertainty. 3. Surprisingly, we find that our methods are able to generate a comparable or higher correlation between variance and error than baseline methods, MC dropout and deep ensemble, in both large-scale regression tasks. 4. We demonstrate that the uncertainty from the proposed methods can be used to improve neural network training.

2. RELATED WORK:

Probabilistic Neural Networks for Uncertainty Estimation: Probabilistic neural networks consider the input and model parameters as random variables which take effect as the source of stochasticity (Graves, 2011; Hernández-Lobato & Adams, 2015; Wang et al., 2016) . Traditional Bayesian neural networks model the distribution over the parameters (weights) (MacKay, 1992; Hinton & Van Camp, 1993; Graves, 2011) and obtain the output distribution by marginalizing out the parameters. Even with recent improvement (Balan et al., 2015; Hernández-Lobato & Adams, 2015) , one major limitation is that the size of network at least doubles under this assumption, and the propagation with a distribution is usually computationally expensive. Another set of popular and efficient methods (Gal & Ghahramani, 2015; Teye et al., 2018) formulate dropout (Srivastava et al., 2014) or batch normalization (Ioffe & Szegedy, 2015) as approximations to Bayesian neural networks. For example, MC dropout (Gal & Ghahramani, 2015) injects noise into some layers during both training and inference. Unlike most models that disable dropout during inference, MC-dropout feed-forwards the same example multiple times with dropout enabled, in order to form a distribution on the output. Meanwhile, other works (Wang et al., 2016; Shekhovtsov & Flach, 2018) propose sampling-free probabilistic neural networks as a lightweight Bayesian treatment for neural networks. Non-probabilistic Neural Networks for Uncertainty Estimation: Other scalable strategies such as deep ensemble (Lakshminarayanan et al., 2017) train an ensemble of neural networks from scratch, where some randomness is induced during the training process, i.e. the initial weight is randomly sampled from a distribution. During inference, these networks will generate a distribution of the output. Though simple and effective, training multiple networks costs even more time and memory than Bayesian neural networks. Another more efficient method is to train the network to have both original outputs and uncertainty predictions, jointly optimize for both (Zhang et al., 2019) . However, such a design requires re-training, introduces heavy implementation overhead, and sometimes makes the optimization process more challenging.

3. METHODOLOGY:

The idea at the core of our approach is to impose a tolerable perturbation on the model's input or intermediate representations (feature maps). We distinguish between three cases: 1. Black-box case: the model is given as a trained black box without any access to its internal structure. 2. Gray-box case: the internal representations (feature maps) of the model is accessible (while the parameters are not) and can be modified during inference. 3. White-box case: the model is available for all modifications (e.g. its weights can be modified). In this paper we focus on the black-box and graybox cases, for which we offer, correspondingly, two classes of methods.

As shown in Figure 1 , for the same image, they first (1) perform different tolerable perturbations on the input or intermediate-layer neurons, (2) generate multiple outputs under different perturbations, and (3) use the variance of the outputs as uncertainty estimation. Concretely, for the black-box case, we propose infer-transformation, which exploits the model's dependence on input transformations, e.g. rotation/flip. For the grey-box case, we propose infer-noise and infer-dropout, which introduce an internal embedding/representation manipulation -injecting a noise layer or a dropout layer. These three methods are illustrated in Figure 1 and are described below.

3.1. BLACK-BOX UNCERTAINTY ESTIMATION: INFER-TRANSFORMATION:

Given a black-box model, we explore the behavior of the outputs for different transformed versions of the input. We focus on transformations that preserve pertinent characteristics of the input, such as rotations, flips, etc. This method draws partial inspiration from adversarial attacks (Madry et al., 2017) . Instead of causing 'large' changes in the output from small perturbations in the input, such as adversarial attack, we introduce tolerable-perturbation transformations of the input, exploiting the model's dependence on input transformations in practice, shown in Equation 1, where T ∈ T is a transformation, and T is the inverse operation. In our method, T • T is cancelled out since the transformation is invertible, and F is the function represented by black-box neural networks. If input x is an H × W matrix, the variance t estimated through transformation is also an

H × W matrix. Assuming E T ∈T [T • F • T (x)] = F • T • T (x) = F (x)

, by the law of large number we have,

T • F • T (x) → N (F • T • T (x), t ) = N (F (x), t )

(1) For the sake of simplicity and with little loss of generality, we apply rotations and flips to the input in our implementation. While, in principle, arbitrary rotation angles could be chosen for the creation of different outputs, in order not to introduce interpolation noise, we limit this discussion to the simpler case where the rotations are pure multiples of 90 degrees. Combining both rotation and flip, at most 8 samples could be generated during inference. Note that more outputs could be easily generated by similarly exploiting other transformations incurring zero-interpolation noise. Meanwhile, if such transformations are also applied during training, one might expect a reduction in the variance computed from multiple outputs for transformed inputs during inference. In our experiment, we evaluate the state-of-the-art models in super resolution and depth estimation, trained without this imposed augmentation. We find the strong correlation with error is still comparable with the value from baselines such as MC dropout.

3.2. GRAY-BOX UNCERTAINTY ESTIMATION: INFER-NOISE AND INFER-DROPOUT:

To manipulate the intermediate representations (feature maps) of the model, we consider another class of methods for generating multiple outputs from a distribution: sampling from different latent codes. The main added benefit of this method is that it allows us to modulate the perturbation strength to ensure its tolerability.

Specifically we propose infer-noise, which introduces Gaussian noise at an intermediate representation within the trained model, and infer-dropout, which uses dropout instead. For infer-noise, the noise will be evenly distributed at the feature maps of a certain layer. This noise source is randomly sampled during inference allowing the corresponding resultant diverse outputs generation. Inferdropout is motivated by observations on network compression and pruning (Han et al., 2015b) : the intermediate layer has the most redundant information and therefore is the most effective location to prune (He et al., 2014) . Naturally, dropping features in such locations incurs more tolerable perturbations. During each inference, the dropout is performed randomly to generate output samples, the variance of which are then used as uncertainty estimation.

Our method could be interpreted as Equation 2, where F 1 represents the function of network layers before the perturbation, denoted as P ∈ P (i.e., Gaussian noise or dropout), and F 2 represents the function of network layers come after. F 2 • F 1 (x) is the gray-box network F (x), and we could use this perturbation P to an output distribution centered at F 2 • F 1 (x). Similarly, if input x is an H × W matrix, the variance n estimated through infer-noise or infer-dropout is also an H × W matrix. After performing inference of the model for multiple times, the computation of the variance n can naturally ensue over these output samples. We find the tuning of a proper location takes effect for uncertainty performance, which is not explored in MC-dropout. Note that conventionally in MC dropout, the location of noise layer is added before the fully connected (FC) layer, while infer-dropout allows for noise injection at arbitrary layers. Assuming

E P ∈P [F 2 • P • F 1 (x)] = F 2 • F 1 (x) = F (x)

, by the law of large number we have,

EQUATION

4. EXPERIMENTS:

In this section, we evaluate our three proposed approaches in two large scale regression tasks, superresolution and depth estimation, and compare our methods with two state-of-the-art methods, MC dropout and deep ensemble.

4.1. SINGLE IMAGE SUPER RESOLUTION:

The task of Single Image Super Resolution is to restore a high-resolution (HR) image from a lowresolution (LR) input. This problem is undetermined by definition, that is, given any low resolution input, there exist diverse plausible solutions. Here we focus on analyzing the state-of-the-art SRGAN model (Ledig et al., 2017) , which is able to restore photo-realistic high-quality images. SRGAN always outputs deterministic restorations since the conditional GAN (Mirza & Osindero, 2014) used in this model involves no latent variable sampling. However, we can still evaluate its uncertainty with our proposed methods. We apply our proposed methods to estimate uncertainty in one open-source version of this work (Dong et al., 2017) . The package provides two models trained with different loss functions: 1) SRresnet model with L 2 loss and 2) SRGAN model with a combination of L 2 loss and adversarial loss (details in Appendix A.2). We evaluate our methods on both models, and conduct experiments corresponding to the black-box/gray-box methodologies previously introduced:

Infer-Transformation: For infer-transformation, we apply rotation of K × 90 degrees (K = 0, 1, 2, 3) as well as horizontal flip to the LR input, feed it into the trained model during the inference, and apply the inverse transformation to its output. We could generate at most 8 samples using this strategy, and then calculate the pixel-wise variance.

Infer-Noise: In infer-noise, we take the trained model and add a Gaussian-noise layer, which has mean 0 and standard deviation σ ∈ {0.01, 0.02, 0.05, 0.1, 0.2, 0.5}, in different locations (layers). We choose 5 different locations for noise injection, as shown in Figure 2 , including the layers right after the input and right before the output, as well as some intermediate layers. For each experiment, we only add the layer into one location with a specific σ value. During the inference, the noise will be randomly sampled from the Gaussian distribution, and we could get one sample from each inference pass. Sample numbers of 8 and 32 are evaluated.

Infer-Dropout: In infer-dropout, we take the trained model and add a dropout layer with varied dropout rates in different locations. We choose dropout rates ρ from the set {0.01, 0.02, 0.05, 0.1, 0.2, 0.5} use the same of locations as the infer-noise. Similarly, for each experiment, we only add the layer into one location with one specific dropout rate. Sample numbers of 8 and 32 are evaluated.

Baselines: The first baseline we test is MC-dropout (Gal & Ghahramani, 2015) , instead of training the model with a dropout layer from scratch, we take the trained model released from (Dong et al., 2017) as a pre-trained model. We then add the dropout layer with varied dropout rate ρ ∈ {0.01, 0.02, 0.05, 0.1, 0.2, 0.5}. For each experiment, we add dropout layer only into one location with one dropout rate. After adding dropout layer, we train the model for more epochs until it converges. The same dropout rate is used for sampling during inference. We try different sample numbers of 8 and 32. The second baseline we test is deep ensemble (Lakshminarayanan et al., 2017) : we train the model from scratch for multiple times (specified as 4 and 8 in our experiments). We add the randomness into the initial weight, which is sampling from one Gaussian distribution, with the mean of zero and standard deviation σ of 0.02. We train these networks with the same number of epochs until they converge. During inference, each of them generates a single deterministic output, with 4 or 8 samples generated in total.

4.2. MONOCULAR DEPTH ESTIMATION:

Uncertainty estimation for the task of monocular depth estimation has been studied by many prior works (Postels et al., 2019; Kendall & Gal, 2017) . Here we use one of the state-of-the-art models for depth estimation using fully convolutional residual network (FCRN) (Laina et al., 2016) . It's trained with a Huber loss (Zwald & Lambert-Lacroix, 2012) . We directly used the trained model released by the original author; this is consistent with the scenarios of black-box and gray-box cases, since the code for training is not released. Note that the model has a dropout layer before the final fully connected (FC) layer during training; this dropout layer could directly be used during inference as an experiment of MC dropout. We compare MC dropout, where dropout is applied in the same layer Correlation values between uncertainty map and error map are also presented.

during training and inference, and infer-dropout, where dropout can be applied in arbitrary layers during inference.

We evaluate the model on NYU Depth Dataset V2. For infer-transformation, we avoid applying 90-degree rotation to input, since the orientation is a strong prior to predict depth. Therefore we only apply horizontal flip and generate 2 samples for uncertainty estimation. For infer-dropout, we choose two locations to add dropout layer and conduct similar experiments as described in the SR task. For the baseline MC dropout, we directly perform sampling from the existing dropout layer. Sample numbers of 2 or 8 are evaluated for both cases adding dropout. Experiment details are shown in Appendix A.3.

4.3. EXPERIMENT RESULTS:

Evaluation Metrics Here we define three evaluation metrics: pixel-wise correlation, mean correlation, and block-wise correlation. Take L 1 loss as an example, the uncertainty (variance) V ij estimated in these methods and the regression error L 1,ij are both pixel-wise values; naturally the pixel-wise L 1 correlation is defined as corr({V ij }, {L 1,ij }). The mean L 1 correlation is defined as corr({V z }, {L 1,z }), where V z and L 1,z are the average variance and average error of a single image z, respectively. This metric has been used in (Zhang et al., 2019) . The third evaluation metric is block-wise correlation -a new metric we propose in this work. To compute block-wise correlation, we first run a local segmentation algorithm on the output of the trained model to generate each cluster C i (block) with similar low-level context. Here we use the local center of mass approach (Aganj et al., 2018) . The local segmentation results with clusters are shown in Figure 3 . The block-wise

L 1 correlation is defined as corr({ V i }, { L 1,i })

, where V i and L 1,i are the average pixel-wise variance and error inside each cluster. The intuition is that in many situations it is more instructive and meaningful when uncertainty is visualized in each sub-region (e.g., in a sub-region with a possible tumor). Detailed descriptions of these metrics are in Appendix A.4.

Figure 3 shows some qualitative results for an example image in the SR task. We can see that the variance maps generated in our task are consistent to the level of ambiguity. Specifically, in our methods, high variance occurs in areas with high randomness and high-frequency spectral contents. For a depth estimation task however, high variance usually occurs in the area with high spatial resolution and large depth. More examples are available in Appendix A.6.2.

The Role of Tolerable Perturbations Tolerable perturbations play a crucial role in obtaining effective uncertainty estimation. Here tolerability can be measured by the disturbance of accuracy or loss. In general we observe that better tolerability corresponds to be better uncertainty estimation.

Figure 4 shows the correlation for different amount of perturbations (noise or dropout) in different locations. As we can see, the optimal cases to generate uncertainty maps with high correlation require that the loss should remain small after perturbation (high tolerability). Interestingly, different methods have different ways of achieving high tolerability: (1) For MC dropout, involving dropout Various dropout rates and noise levels have been evaluated. We compare our infer-noise and infer-dropout with baseline MC-dropout (Gal & Ghahramani, 2016) . Location 0 is right after the input; location 4 is right before the last convolutional layer; location 1, 2, 3 are intermediate layers.

Right: Intermediate layers are optimal locations to add dropout layers for uncertainty estimation, validated in both infer-dropout and MC-dropout methods. The corresponding correlation is the optimal value of each location, after tuning with dropout rates.

during training increases the robustness of model against perturbations, keeping the loss relatively small after adding dropout layer in most locations during inference;

(2) for infer-dropout, adding dropout layer in intermediate locations (i.e., location 2 and location 3) where the information is the most redundant, can effectively alleviate disturbance;

(3) for infer-noise, adding noise with small standard deviation effectively limits the perturbation level. More interestingly, we further find that for both MC-dropout and infer-dropout, adding perturbation in intermediate layers are usually the optimal choices for uncertainty estimation; it is different from convention where dropout is applied before the final FC layers. Applying infer-dropout in these intermediate layers, we could achieve comparable or even better correlation values compared to training-required approaches. For infernoise, locations do not have similar effect, and therefore further tuning of locations and σ is also required. The conclusion above is also consistent with the evaluation of other models (see more details in Appendix A.6.1).

Comparable Performance with Training-required Baselines We summarize the correlation value using the optimal hyper-parameters in different methods in Table 1 and Table 2 . As depicted in both tasks, our methods infer-transformation and infer-dropout are able to provide comparable results with the training-required state-of-the-art method, MC-dropout, and are better than deep ensemble. For the super-resolution task, we find that SRGAN has a higher correlation than the SRresnet model. One explanation is that the model trained with adversarial loss is more sensitive to perturbation, and will cause various artifacts in the area of high uncertainty during each inference.

For depth estimation, we find that using infer-dropout in the intermediate layers outperforms other methods. MC dropout only allows perturbation before the last convolutional layer, the location added with dropout layer during training for the original trained model, and therefore produces a highly localized variance map with poor correlation (more details available in Appendix A.6.2). If we are allowed to perform MC-dropout in intermediate layers and re-train the model, a correlation value comparable to that of infer-dropout should be expected. Interestingly, even with only 2 samples, infer-transformation can still outperform all other methods.

5. APPLICATIONS:

We demonstrate two applications of uncertainty maps estimated by our methods, in the context of super resolution. In general, estimated uncertainty can be adopted to improve the training process.

For more details see Sec. A.5. The first application is to take the pixel-wise uncertainty map as a weight term for regression loss, and provide a higher penalty in highly uncertain regions. As expected, loss weighted by uncertainty can provide a more photo-realistic SR output with finer structures and sharper edges, as shown in Figure 5 . (Gal & Ghahramani, 2016) . Models evaluated: FCRN model for depth destimation.

Another natural application is active learning (Gal et al., 2017) . In our experiment, we firstly use the original model trained on DIV2K training data to super-resolve the DIV2K test data (as our validation data here). Then the model using active learning strategy is fine-tuned with the 25% validation data with the highest uncertainty. Two baseline models are the original trained model and the model fine-tuned with 25% of validation data which is randomly chosen. Three models are evaluated in Set 14, image quality is quantified by standard metrics. We find that active learning based on our generated uncertainty estimation can improve the performance, shown in Figure 5 . We propose three simple, scalable, and effective methods, namely infer-transformation, infer-noise, and infer-dropout, for uncertainty estimation in both black-box and gray-box cases. Surprisingly, our training-free methods achieve comparable or even better results compared to training-required state-of-the-art methods. Furthermore, we demonstrate adding tolerable perturbations is the key to generating uncertainty maps with high correlation to error maps for all methods we studied. Our future work includes evaluating our methods on distilled, pruned and binarized models, as well as generalizing our methods for more complicated noise/transformation (e.g., non-Gaussian noise arbitrary-angle rotation). 

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Figures and Tables:

Figure 1: Left: Method description of training-free uncertainty estimation: Apply infertransformation T (top) and infer-noise or infer-dropout P (bottom) to a trained neural network F during inference for N times. Right: Examples of uncertainty maps generated using our proposed method in two different tasks, single image super resolution and monocular depth estimation.

Figure 2: Different locations for infer-noise and infer-dropout in SRGAN and SRresnet for super resolution. For each experiment, the noise is injected at a single location with one perturbation level.

Figure 3: Block-wise and pixel-wise uncertainty (variance) maps generated using infertransformation, infer-dropout, MC-dropout (Gal & Ghahramani, 2016) compared with L 1 loss map. Correlation values between uncertainty map and error map are also presented.

Figure 4: Left: Correlation changes with different locations where dropout and noise layers are inserted.Various dropout rates and noise levels have been evaluated. We compare our infer-noise and infer-dropout with baseline MC-dropout(Gal & Ghahramani, 2016). Location 0 is right after the input; location 4 is right before the last convolutional layer; location 1, 2, 3 are intermediate layers.Right: Intermediate layers are optimal locations to add dropout layers for uncertainty estimation, validated in both infer-dropout and MC-dropout methods. The corresponding correlation is the optimal value of each location, after tuning with dropout rates.

Figure 5: Left: We compare SR results that use loss reweighted by variance map and that do not. HR represents high resolution image. Right: in active learning setting, choosing samples with high uncertainty yields better results than choosing randomly.6 DISCUSSION AND CONCLUSIONIn the work, we perform the first systematic exploration into training-free uncertainty estimation. We propose three simple, scalable, and effective methods, namely infer-transformation, infer-noise, and infer-dropout, for uncertainty estimation in both black-box and gray-box cases. Surprisingly, our training-free methods achieve comparable or even better results compared to training-required state-of-the-art methods. Furthermore, we demonstrate adding tolerable perturbations is the key to generating uncertainty maps with high correlation to error maps for all methods we studied. Our future work includes evaluating our methods on distilled, pruned and binarized models, as well as generalizing our methods for more complicated noise/transformation (e.g., non-Gaussian noise arbitrary-angle rotation).

Figure 9: Visualization of uncertainty maps and error map from infer-transformation, infer-dropout compared with baseline MC-dropout, evaluated on the SRGAN model on Set14 dataset for superresolution task.

Figure 10: Visualization of uncertainty maps and error map from infer-transformation, infer-dropout compared with baseline MC-dropout, evaluated on the FCRN model on NYU depth dataset V2 for depth estimation task.

Mean/block-wise/pixel-wise correlation of L 1 /L 2 loss and uncertainty on SR benchmark dataset set 14. Our infer-transformation and infer-noise injection are compared with MC dropout(Gal & Ghahramani, 2016) and deep ensemble(Lakshminarayanan et al., 2017). Models evaluated: SRresnet trained with L 2 loss and SRGAN trained with L 2 loss and adversarial loss.

Table content: <table><tr><td/><td/><td colspan="6">SRGAN model: trained with adversarial loss and L2 loss</td><td/><td/><td/></tr><tr><td>Condition</td><td/><td/><td colspan="2">Training Free (Ours)</td><td/><td/><td/><td colspan="2">Training Required</td><td/></tr><tr><td>Method</td><td colspan="2">Infer-transformation</td><td colspan="2">Infer-dropout</td><td colspan="2">Infer-noise</td><td colspan="2">MC-dropout</td><td colspan="2">Deep Ensemble</td></tr><tr><td>Samples</td><td>4</td><td>8</td><td>8</td><td>32</td><td>8</td><td>32</td><td>8</td><td>32</td><td>4</td><td>8</td></tr><tr><td>mean L1</td><td>0.9380</td><td>0.9366</td><td>0.8932</td><td>0.8918</td><td>0.7858</td><td>0.7867</td><td>0.9515</td><td>0.9532</td><td>0.6521</td><td>0.6803</td></tr><tr><td>mean L2</td><td>0.8993</td><td>0.8990</td><td>0.8878</td><td>0.8808</td><td>0.8575</td><td>0.8559</td><td>0.9131</td><td>0.9115</td><td>0.6056</td><td>0.6307</td></tr><tr><td>block-wise L1</td><td>0.7654</td><td>0.7697</td><td>0.7222</td><td>0.7313</td><td>0.5899</td><td>0.5984</td><td>0.7477</td><td>0.7551</td><td>0.6362</td><td>0.6536</td></tr><tr><td>block-wise L2</td><td>0.6841</td><td>0.6905</td><td>0.6515</td><td>0.6615</td><td>0.6251</td><td>0.6334</td><td>0.6856</td><td>0.6920</td><td>0.5921</td><td>0.6120</td></tr><tr><td>pixel-wise L1</td><td>0.3671</td><td>0.3945</td><td>0.3233</td><td>0.3762</td><td>0.2450</td><td>0.2877</td><td>0.3388</td><td>0.3898</td><td>0.2537</td><td>0.2962</td></tr><tr><td>pixel-wise L2</td><td>0.2681</td><td>0.2956</td><td>0.2676</td><td>0.3105</td><td>0.2186</td><td>0.2566</td><td>0.2779</td><td>0.3198</td><td>0.2108</td><td>0.2474</td></tr><tr><td/><td/><td/><td colspan="4">SRResnet model: trained with L2 loss</td><td/><td/><td/><td/></tr><tr><td>Condition</td><td/><td/><td colspan="2">Training Free (Ours)</td><td/><td/><td/><td colspan="2">Training Required</td><td/></tr><tr><td>Method</td><td colspan="2">Infer-transformation</td><td colspan="2">Infer-dropout</td><td colspan="2">Infer-noise</td><td colspan="2">MC-dropout</td><td colspan="2">Deep Ensemble</td></tr><tr><td>Samples</td><td>4</td><td>8</td><td>8</td><td>32</td><td>8</td><td>32</td><td>8</td><td>32</td><td>4</td><td>8</td></tr><tr><td>mean L1</td><td>0.3654</td><td>0.3843</td><td>0.5239</td><td>0.5314</td><td>0.1044</td><td>0.0909</td><td>0.4813</td><td>0.4628</td><td>-0.1595</td><td>-0.2974</td></tr><tr><td>mean L2</td><td>0.2661</td><td>0.2804</td><td>0.3628</td><td>0.3770</td><td>0.3021</td><td>0.2933</td><td>0.3281</td><td>0.2866</td><td>-0.2440</td><td>-0.2935</td></tr><tr><td>block-wise L1</td><td>0.5014</td><td>0.5197</td><td>0.5079</td><td>0.5178</td><td>0.3714</td><td>0.3854</td><td>0.5346</td><td>0.5447</td><td>0.2771</td><td>0.2726</td></tr><tr><td>block-wise L2</td><td>0.4288</td><td>0.4468</td><td>0.4190</td><td>0.4286</td><td>0.3574</td><td>0.3688</td><td>0.4607</td><td>0.4702</td><td>0.2461</td><td>0.2275</td></tr><tr><td>pixel-wise L1</td><td>0.2372</td><td>0.2690</td><td>0.2584</td><td>0.3025</td><td>0.1721</td><td>0.2156</td><td>0.2638</td><td>0.3087</td><td>0.1708</td><td>0.1740</td></tr><tr><td>pixel-wise L2</td><td>0.1845</td><td>0.2097</td><td>0.2003</td><td>0.2339</td><td>0.1440</td><td>0.1796</td><td>0.2055</td><td>0.2414</td><td>0.1374</td><td>0.1390</td></tr><tr><td/><td/><td/><td colspan="4">FCRN model: Depth Estimation</td><td/><td/><td/><td/></tr><tr><td/><td colspan="2">Condition</td><td/><td colspan="2">Training Free (Ours)</td><td/><td colspan="2">Training Required</td><td/><td/></tr><tr><td/><td/><td>Method</td><td colspan="2">Infer-transformation</td><td colspan="2">Infer-dropout</td><td colspan="2">MC-dropout</td><td/><td/></tr><tr><td/><td/><td>Samples</td><td>2</td><td/><td>2</td><td>8</td><td>2</td><td>8</td><td/><td/></tr><tr><td/><td colspan="2">mean L1</td><td colspan="2">0.5960</td><td>0.6302</td><td>0.6770</td><td>0.4711</td><td>0.4706</td><td/><td/></tr><tr><td/><td colspan="2">mean L2</td><td colspan="2">0.5995</td><td>0.5886</td><td>0.6380</td><td>0.4060</td><td>0.4039</td><td/><td/></tr><tr><td/><td colspan="2">block-wise L1</td><td colspan="2">0.3544</td><td>0.3541</td><td>0.4487</td><td>0.2117</td><td>0.2186</td><td/><td/></tr><tr><td/><td colspan="2">block-wise L2</td><td colspan="2">0.3748</td><td>0.3696</td><td>0.4712</td><td>0.1837</td><td>0.1891</td><td/><td/></tr><tr><td/><td colspan="2">pixel-wise L1</td><td colspan="2">0.2082</td><td>0.1815</td><td>0.2836</td><td>0.0745</td><td>0.1335</td><td/><td/></tr><tr><td/><td colspan="2">pixel-wise L2</td><td colspan="2">0.2047</td><td>0.1750</td><td>0.2724</td><td>0.0541</td><td>0.0956</td><td/><td/></tr></table>



Table content: <table/>

Peer review (iclr2020):
This paper addresses the problem of deriving better predictive uncertainty estimates from pre-trained models. As the paper notes in the introduction, this is an important problem for machine learning models embedded in real-world decision-making systems, where the consequences of inaccurate predictions may be high, and better insight into model uncertainty can help avoid bad outcomes due to misplaced trust in model predictions. The paper focuses on training-free uncertainty estimation, in which uncertainty estimates are driven by input perturbations and modifications to ‘grey-box’ models (such as application of dropout to a trained model). These methods avoid the computational burden of applying uncertainty estimation during the training process. The paper evaluates the methods on two regression tasks: image super-resolution and monocular depth estimation. Evaluation metrics are based on the correlation between predictive variance and loss. The paper concludes that the training-free methods examined give comparable or better performance for uncertainty estimation, compared with state-of-the-art methods applied during training. The paper addresses an important topic, and is generally well-executed and well-written. However, it focuses on a narrow set regression tasks (with little indication that results generalize further), and uses evaluation metrics that are nonstandard and not well-substantiated. For these reasons, I recommend that the paper be rejected. A systematic study of training-free uncertainty estimation, which is the goal of the paper, would be a significant contribution. The paper chooses to focus on regression tasks, which is a valid narrowing of scope (although the implication that uncertainty estimation for classification is a solved problem, via entropy methods, I believe is incorrect). Image super-resolution and monocular depth estimation, the two regression tasks that the paper examines, are important but do not represent a broad enough set of tasks to justify claims about the general performance of the methods (though they enable the observations in equations (1) and (2)). The performance of the methods on these two tasks alone I don’t believe is a significant enough finding to warrant acceptance to ICLR. My second major concern with the paper is in the choice of evaluation metrics, which are based on the correlation between variance of predictions (over a sample of perturbations) and the value of the loss. While I understand how these metrics provide intuition on the quality of uncertainty estimates (i.e. when a model is “more wrong”, ideally it should be more uncertain), the paper does not give a good justification for the validity of the metrics beyond intuition (nor does the cited paper, Zhang et al. (2019)). The correlation metrics obscure a lot of information on the relationship between the predictive distribution and the true values, and without further justification, it is difficult to imagine trusting the resulting uncertainty estimates in a consequential downstream decision-making task. Proper scoring rules (Gneiting and Rafferty (2007), Lakshminarayanan et al. (2017)) could provide a more principled and useful basis for assessing the quality of uncertainty estimates. Smaller points: - The paper makes the implicit assumption that training/test sets are IID. While I understand that prediction on out-of-distribution (OOD) samples is beyond the paper’s scope, high uncertainty on OOD samples is a desirable property for uncertainty estimation methods, and some discussion of this or indication of the methods’ performance on OOD samples would strengthen the paper. - The paper implies that uncertainty estimation via input perturbation is a novel contribution; a similar method was used by Liang et al. (2018; https://arxiv.org/pdf/1706.02690.pdf) for the different but related task of out-of-distribution detection. - Is there a way to derive error bars for the results in Tables 1 and 2? Some of the correlation coefficients, reported to 4 decimal places, are very close and it’s unclear whether the results are significantly different.