Overparametrized deep models can memorize datasets with labels entirely randomized [ZBH16]
. It is consequently not entirely clear why such extremely flexible models are able to generalize well on unseen data and trained with algorithms as simple as stochastic gradient descent, although a lot of progress on these questions have recently been reported[DR17, JGH18, BM19, MMN18, RVE18, GML18].
The high-capacity of neural network models, and their ability to easily overfit complex datasets, makes them especially vulnerable to calibration issues. In many situations, standard deep-learning approaches are known to produce probabilistic forecasts that are over-confident[GPSW17]. In this text, we consider the regime where the size of the training sets are very small, which typically amplifies these issues. This can lead to problematic behaviours when deep neural networks are deployed in scenarios where a proper quantification of the uncertainty is necessary. Indeed, a host of methods [LPB17, MGI19, SHK14, GG16, Pre98] have been proposed to mitigate these calibration issues, even though no gold-standard has so far emerged. Many different forms of regularization techniques [PW17, ZBH16, ZH05] have been shown to reduce overfitting in deep neural networks. Importantly, practical implementations and approximations of Bayesian methodologies [MGI19, WHSX16, BCKW15, Gra11, LW16, RMW14, Mac92] have demonstrated their worth in several settings, although some of these techniques are not entirely straightforward to implement in practice. Ensembling approaches such as drop-outs [GG16] have been widely adopted, largely due to their ease of implementation. In this text, we investigate the practical use of Deep-Ensembles [LPB17, BC17, LPC15, SLJ15, FHL19, GPSW17], a straightforward approach that displays state-of-the-art performances in most regimes. Although deep-ensembles can be difficult to implement when training datasets are large (but calibration issues are less pronounced in this regime), the focus of this text is the data-scarce regime where the computational burden associated to deep-ensembles is not a significant problem.
Contributions: we study the interaction between three of the most simple and widely used methods for scaling deep-learning to the low-data regime: ensembling, temperature scaling, and mixup data-augmentation.
Despite the general belief that averaging models improves calibration properties, we show that, in general, standard ensembling practices do not lead to better-calibrated models. Instead, we show that averaging the predictions of a set of neural networks generally leads to less confident predictions: that is generally only beneficial in the oft-encountered regime when each network is overconfident. Although our results are based on Deep Ensembles, our empirical analysis extends to any class of model averaging, including sampling-based Bayesian Deep Learning.
We empirically demonstrate that networks trained with the mixup
data-augmentation scheme, a very common practice in computer vision, are typically under-confident. Consequently, subtle interactions between ensembling techniques and modern data-augmentation pipelines have to be taken into account for proper uncertainty quantification. The typical distributional-shift induced by the mixup data-augmentation strategy influences the calibration properties of the resulting trained neural networks.
Post-processing techniques such as temperature scaling can be successfully used in conjunction with deep-ensembling methods, but the order in which the aggregation and the calibration procedures are carried out does greatly influence the quality of the resulting uncertainty quantification. These findings lead us to formulate the straightforward Pool-Then-Calibrate strategy for post-processing deep-ensembles: (1) in a first stage, separately train deep models (2) in a second stage, fit a single temperature parameter by minimizing a proper scoring rule (eg. cross-entropy) on a validation set. In the low data-regime, this simple procedure can halve the Expected Calibration Error (ECE) on a range of benchmark classification problems when compared to standard deep-ensembles.
Consider a classification task with possible classes . For a sample , the quantity represents a probabilistic prediction, often obtained as for a neural network with weight and softmax function . We set and .
Augmentation: Consider a training dataset and denote by
the one-hot encoded version of the label. A stochastic augmentation process maps a pair to another augmented pair . In computer vision, standard augmentation strategies include rotations, translations, brightness and contrast manipulations. In this text, in addition to these standard agumentations, we also make use of the more recently proposed mixup augmentation strategy [ZCDL17] that has proven beneficial in several settings. For a pair , its mixup-augmented version is defined as
for a random coefficient drawn from a fixed mixing distribution often chosen as , and a random index drawn uniformly within .
Model averaging: Ensembling methods leverage a set of models by combining them into a aggregated model. In the context of deep learning, Bayesian averaging consists in weighting the predictions according to the Bayesian posterior
on the neural weights. Instead of finding an optimal set of weights by minimizing a loss function, predictions are averaged. Denoting bythe probabilistic prediction associated to sample and neural weight , the Bayesian approach advocates to consider
Designing sensible prior distributions is still an active area of research and data-augmentation schemes, crucial in practice, are not entirely straightforward to fit into this framework. Furthermore, the high-dimensional integral (2) is (extremely) intractable: the posterior distribution is multi-modal, high-dimensional, concentrated along low-dimensional structures, and any local exploration algorithm (eg. MCMC, Langevin dynamics and their variations) is bound to only explore a tiny fraction of the state space. Because of the typically large number of degrees of symmetries, many of these local modes correspond to essentially similar predictions, indicating that it is likely not necessary to explore all the modes in order to approximate (2). A detailed understanding of the geometric properties of the posterior distribution in Bayesian neural networks is still lacking, although a lot of recent progress have been made. Indeed, variational approximations have been reported to improve, in some settings, over standard empirical risk minimization procedures. Deep-ensembles can be understood as crude, but practical, approximations of the integral in Equation (2). The high-dimensional integral can be approximated by a simple non-weighted average over several modes of the posterior distribution found by minimizing the negative log-posterior, or some approximations of it, with standard optimization techniques:
We refer the interested reader to [Nea12, MMK03, WI20] for different perspectives on Bayesian neural network. Although simple and not very well understood, deep-ensembles have been shown to provide extremely robust uncertainty quantification when compared to more sophisticated approaches [LPB17, BC17, LPC15, SLJ15].
Post-processing Calibration Methods: The article [GPSW17] proposes a class of post-processing calibration methods that extend the more standard Platt Scaling approach [Pla99]. Temperature Scaling, the simplest of these methods, transforms the probabilistic outputs into a tempered version defined through the scaling function
for a temperature parameter .
The optimal parameter is usually found by minimizing a proper-scoring rules [GR07], often chosen as the negative log-likelihood, on a validation dataset.
Crucially, during this post-processing step, the parameters of the probabilistic model are kept fixed: the only parameter being optimized is the temperature .
In the low-data regime considered in this article, the validation set being also extremely small, we have empirically observed that the more sophisticated Vector and Matrix scaling post-processing calibration methods [GPSW17] do not offer any significant advantage over the simple and robust temperature scaling approach.
Calibration Metrics: The Expected Calibration Error (ECE) measures the discrepancy between prediction confidence and empirical accuracy. In this text, we also define the signed Expected Calibration Error (sECE) in order to differentiate under-confidence from over-confidence. For a partition of the unit interval and a labelled set , set and and . The quantities ECE and sECE are defined as
A model is calibrated if for all , i.e. . A large (resp. low) value of the sECE indicates over-confidence (resp. under-confidence). It is often instructive to display the associated reliability curve, i.e. the curve with on the x-axis and the difference on the y-axis. Figure 1 displays examples of such reliability curves. A perfectly calibrated model is flat (i.e. ), while the reliability curve associated to an under-confident (resp. over-confident) model prominantly lies above (resp. below) the flat line . In the sequel, we sometimes report the value of the Brier score [Bri50] defined as .
3 Empirical Observations
Linear pooling: It has been observed in several studies that averaging the probabilistic predictions of a set of independently trained neural networks, i.e. deep-ensembles, often leads to more accurate and better-calibrated forecasts [LPB17, BC17, LPC15, SLJ15, FHL19]. Figure 1 displays the reliability curves across three different datasets of a set of independently trained neural networks, as well as the reliability curves of the aggregated forecasts obtained by simply linear averaging the individual probabilistic predictions. These results suggest that deep-ensembles consistently lead to predictions that are less confident than the ones of its individual constituents. This can indeed be beneficial in the often encountered situation when each individual neural network is overconfident. Nevertheless, this phenomenon should not be mistaken with an intrinsic property of deep ensembles to lead to better-calibrated forecasts. For example, and as discussed further in Section 4, networks trained with the popular mixup data-augmentation are typically under-confident. Ensembling such a set of individual networks typically leads to predictions that are even more under-confident. In order to gain some insights into this phenomenon, recall the definition of the entropy functional ,
The entropy functional is concave on the probability simplex, i.e. for any
. Furthermore, tempering a probability distributionleads to increase in entropy if , as can be proved by examining the derivative of the function . The entropy functional is consequently a natural surrogate measure of (lack of) confidence. The concavity property of the entropy functional shows that ensembling a set of individual networks leads to predictions whose entropies are higher than the average of the entropies of the individual predictions. We have not been able to prove a similar property for the ECE functional.
In order to obtain a more quantitative understanding of this phenomenon, consider a binary classification framework. For a pair of random variables, with and , and a classification rule that approximates the conditional probability , define the Deviation from Calibration score as
The term is equivalent to the Brier score of the classification rule and the quantity is an entropic term (i.e. large for predictions close to uniform). Note that DC can take both positive and negative values and for a well-calibrated classification rule, i.e. for all . Furthermore, among a set of classification rules with the same Brier score, the ones with less confident predictions (i.e. larger entropy) have a lesser DC score. In summary, the DC score is a measure of confidence that vanishes for well-calibrated classification rules, and that is low (resp. high) for under-confident (resp.over-confident) classification rules. Contrarily to the entropy functional (6), the DC score is extremely tractable. Algebraic manipulations readily shows that, for a set of classification rules and non-negative weights , the linearly averaged classification rule satisfies
Equation (8) shows that averaging classifications rules decreases the DC score (i.e. the aggregated estimates are less confident). Furthermore, the more dissimilar the individual classification rules, the larger the decrease. Even if each individual model is well-calibrated, i.e. for , the averaged model is not well-calibrated as soon as at least two of them are not identical.
Distance to the training set: in order to gain some additional insights into the calibration properties of neural networks trained on small datasets, as well as the influence of the popular mixup augmentation strategy, we examine several metrics (i.e. signed ECE (sECE), Negative Log-likelihood (NLL), entropy) as a function of the distance to the (small) training set . We focus on the CIFAR10 dataset and train our networks on a balanced subset of training examples. Since there is no straightforward and semantically meaningful distance between images, we first use an unsupervised method (i.e. labels were not used) for learning a low-dimensional and semantically meaningful representation of dimension . For these experiments, we obtained a mapping , where denotes the unit sphere in , with the simCLR method of [CKNH20], although experiments with other metric learning approaches [HFW19, YZYC19] have led to essentially similar conclusions. We used the distance , which in this case is equivalent to the cosine distance between the -dimensional representations of the CIFAR10 images and . The distance of a test image to the training dataset is defined as . We computed the distances to the training set for each image contained in the standard CIFAR10 test set (last column of Figure 2). Not surprisingly, we note that the average Entropy, Negative Log-likelihood and Error Rate all increase as test samples are chosen further away from the training set.
Over-confidence: the predictions associated to samples chosen further away from the training set have a higher sECE. This indicates that the over-confidence of the predictions increases with the distance to the training set. In other words, even if the entropy increases as the distance increases (as it should), calibration issues do not vanish as the distance to the training set increases. This phenomenon is irrespective of the amount of mixup used for training the network.
Effect of mixup-augmentation: The first row of Figure 2 shows that increasing the amount of mixup augmentation consistently leads to an increase in entropy, decrease in over-confidence (i.e. sECE), as well as a more accurate predictions (lower NLL and higher accuracy). Additionally, the effect is less pronounced for . This is confirmed in Figure 3 that displays the more generally the effect of the mixup-augmentation on the reliability curves, over four different datasets.
Temperature Scaling: importantly, the second row of Figure 2 indicates that a post-processing temperature scaling for the individual models almost washes-out all the differences due to the mixup-augmentation scheme. For this experiment, an ensemble of networks is considered: before averaging the predictions, each network has been individually temperature scaled by fitting a temperature parameter (through negative likelihood minimization) on a validation set of size .
4 Calibrating Deep Ensembles
In order to calibrate deep ensembles, several methodologies can be considered:
Do nothing and hope that the averaging process intrinsically leads to better calibration
Calibrate each individual network before aggregating all the results
Simultaneously aggregate and calibrate the probabilistic forecasts of each individual model.
Aggregate first the estimates of each individual model before eventually calibrating the pooled estimate.
Pooling methods: as recognized in the operation research literature [JW08, WGCLJ19], simple pooling/aggregation rules that do not require a large number of tuning parameters are usually preferred, especially when training data is scarce. Simple aggregation rules are usually robust, conceptually easy to understand, and straightforward to implement and optimize. The standard average and median pooling of a set of probabilistic predictions are defined as
for a normalization constant , the median operation being executed component-wise over the components. Finally, , the trimmed mean [JW08] of real numbers , is obtained by first discarding the largest and smallest values before averaging the remaining elements. This means that where is a permutation such that . The trimmed mean pooling method is consequently defined as
for a normalization constant , with the trimmed-averaging being executed component-wise.
Pool-Then-Calibrate: any of the above-mentioned aggregation procedure can be used as a pooling strategy before fitting a temperature by a minimizing proper scoring rules on a validation set. In all our experiment, we minimized the negative log-likelihood (i.e. cross-entropy). In other words, given a set of probabilistic forecasts, the final prediction is defined as
Note that the aggregation procedure can be carried out entirely independently from the fitting of the optimal temperature .
Joint Pool-and-Calibrate: there are several situations when the so-called end-to-end training strategy consisting in jointly optimizing several component of a composite system leads to increased performances [MKS15, MPV16, GWR16]. In our setting, this means learning the optimal temperature concurrently with the aggregation procedure. The optimal temperature is found by minimizing a proper scoring rule on a validation set ,
where denotes the aggregated probabilistic prediction for sample .
In all our experiments, we have found it computationally more efficient and robust to use a simple grid search for finding the optimal temperature; we used temperatures equally spaced on a logarithmic scale in between and .
Importance of the Pooling and Calibration order: Figure 4 shows calibration curves when individual models are temperature scaled separately (i.e. group [B] of methods), as well as when the models are scaled with a common temperature parameter (i.e. group [C] of methods). Furthermore, the calibration curves of the pooled model (group [B] and [C] of methods) are also displayed. More formally, the group [B] of methods obtains for each individual model an optimal temperature as solution of the optimization procedure
where denotes the probabilistic output of the model for the example in validation dataset. The light blue calibration curves corresponds to the outputs for different models. The deep blue calibration curve corresponds the linear pooling of the individually scaled predictions. For the group [C] of methods, a single common temperature is obtained as solution of the optimization procedure
where denotes the aggregated probabilistic prediction for sample . The orange calibration curves are generated using the predictions and the red one corresponds to the prediction .
Notice that when scaled separately (by ) each of the individual models (light blue) is close to being calibrated, but the resulting pooled model (deep blue) is under-confident. However, when scaled by a common temperature, the optimization chooses a temperature that makes the individual models (orange) slightly over-confident, so that the resulting pooled model is nearly calibrated. This is in line with the findings discussed in section 3 and it also shows why the ordering of pooling and scaling is important.
Figure 5 compares the four methodologies A-B-C-D identified at the start of this section, with the three different pooling approaches and and . These methods are compared to the baseline approach (in dashed red line) consisting of fitting a single network trained with the same amount of mixup augmentation before being temperature scaled. All the experiments are executed times, on the same training set, but with different validation sets of size for CIFAR10, IMAGENETTE, IMAGEWOOF and for CIFAR100, and for the Diabetic Retinopathy dataset. The results indicate that on most metrics and datasets, the (naive) method consisting of simply averaging predictions is not competitive. Secondly, and as explained in the previous section, the method (B) consisting in first calibrating the individual networks before pooling the predictions is less efficient across metrics than the last two methods . Finally, the two methods perform comparably, the method (D) (i.e. pool-then-calibrate) being slightly more straightforward to implement. As regards the pooling methods, the intuitive robustness of the median and trimmed-averaging approaches does not seem to lead to any consistent gain across metrics and datasets. Note that ensembling a set of networks (without any form of post-processing) does lead to a very significant improvement in NLL and Brier score but lead to a serious deterioration of the ECE. The Pool-Then-Aggregate methodology allows to benefit from the gains in NLL/Brier score, without compromising any loss in ECE.
|CIFAR10 1000 samples|
|Metric||Group [A]||Group [B]||Group [C]||Group [D]|
|Linear Pool||Linear Pool||Linear Pool||Linear Pool|
|test ECE||13.9||11.1 3.6||4.8 2.7||4.9 2.9|
|test NLL||0.961||0.956 .031||0.915 .013||0.916 .015|
|test BRIER||0.431||0.431 .011||0.416 .004||0.417 .005|
|CIFAR100 5000 samples|
|test ECE||17.8||13.1 1.2||3.5 0.9||2.1 .5|
|test NLL||1.911||1.883 .016||1.799 .002||1.787 .002|
|test BRIER||0.623||0.616 0.004||0.594 .001||0.592 .0|
|Diabetic Retinopathy 5000 samples|
|test ECE||4.9||2.8 .7||2.8 .8||2.9 .8|
|test NLL||0.641||0.636 .001||0.637 .002||0.637 .002|
|test BRIER||0.450||0.445 .001||0.445 .001||0.446 .001|
|Imagewoof 1000 samples|
|test ECE||8.9||7.5 3.5||4.2 2.2||4.3 2.1|
|test NLL||1.044||1.065 0.26||1.044 .013||1.045 0.12|
|test BRIER||0.452||0.463 .008||0.456 .003||0.457 .003|
|Imagenette 1000 samples|
|test ECE||18.2||7.3 2.7||3.1 1.1||3.5 1.0|
|test NLL||0.753||0.659 .018||0.637 .006||0.638 .005|
|test BRIER||0.312||0.279 .005||0.272 .001||0.273 .001|
) and different datasets. The number of samples used for different setup are the same as mentioned in the main text. The mean and standard deviation is reported out of 50 different validation sets.
Importance of the validation set: it would be practically useful to be able to fit the temperature without relying on a validation set. We report that using the training set instead (obviously) does not lead to better calibrated models (i.e. the optimal temperature is close to ). We have tried to use a different amount of mixup-augmentation (and other types of augmentation) on the training set for fitting the temperature parameter, but have not been able to obtain satisfying results.
Size of the ensembles:
Figure 7 shows the performance of the different pooling methods (i.e. groups [B]-[D]) on the CIFAR10 dataset, as a function of the number of individual models in the ensemble. For clarity, the (non-calibrated) group [A] of methods are not reported. Recall that the group [A] pools the the predictions without any calibration procedure, the group [B] first calibrates each individual models separately before aggregating the results, the group [C] jointly calibrates and aggregates the prediction, and finally the group [D] first aggregates the results before calibrating the resulting prediction. Methods in group [C] and [D] performs similarly. For the CIFAR10 dataset, we observe that the performance under most metrics saturates for ensemble of sizes .
Table 1 reports the numerical results obtained when a linear averaging aggregation method is used within each group [A]–[D] of calibration procedures. Experiments are carried-out on different validations sets (and a single training set).
Role and effect of mixup-augmentation:
the mixup augmentation strategy is popular and straightforward to implement. As already empirically described in Section 3, increasing the amount of mixup-augmentation typically leads to a decrease in the confidence and increase in entropy of the predictions. This can be beneficial in some situations but also indicates that this approach should certainly be employed with care for producing calibrated probabilistic predictions. Contrarily to other geometric data-augmentation transformations such as image flipping, rotations, and dilatations, the mixup strategy produces non-realistic images that consequently lie outside the data-manifold of natural images: this typically leads to a large distributional shift. The mixup strategy relies on a subtle trade-off between the increase in training data diversity, which can help mitigate over-fitting problems, and the distributional shift that can be detrimental to the calibration properties of the resulting method. Figure 6 compares the performance of the Pool-Then-Calibrate approach when applied to a deep-ensemble of networks trained with different amount of mixup-augmentation. The results are compared to the same approach (i.e. Pool-then-Calibrate with networks) with no mixup-augmentation. The results indicate a clear benefit in using the mixup-augmentation in conjunction with temperature scaling.
Ablation study: For our ablation study, we focus on the CIFAR10 dataset with examples. As mentioned earlier, we reduce the training dataset by 50 training examples for steps involving the validation dataset. Similar to table 1 we evaluate methods requiring post-processing optimization on a random set of 50 different validation datasets. We provide the results of our ablation study in table 2. For setups involving training a single model, we report mean and standard deviations of the metric from a variety of 30 different trained models.
|Metric||(Ours) 30 models||30 models||single model||single model||single model|
|temp scaled||mixup||mixup||no mixup||no mixup|
|Augment + mixup||Augment||Augment||Augment||no Augment|
|test acc||69.92 .04||70.67||66.45 .61||63.73 .51||49.85 .66|
|test ECE||3.3 1.9||13.9||7.03 .7||20.7 .4||23.4 1.0|
|test NLL||0.910 .012||0.961||1.03 .13||1.509 .017||1.770 .045|
|test BRIER||0.414 .002||0.431||0.463 .005||0.556 .006||0.718 .009|
Cold posteriors: the article [WRV20] reports gains in several metrics when fitting Bayesian neural networks to a tempered posterior of type , where is the standard Bayesian posterior, for temperatures smaller than one. Although not identical to our setting, it should be noted that in all our experiments, the optimal temperature was consistently smaller than one. In our setting, this is because simply averaging predictions lead to under-confident results. We postulate that related mechanisms are responsible for the observations reported in [WRV20].
The problem of calibrating deep-ensembles has received surprisingly little attention in the literature. In this text, we examined the interaction between three of the most simple and widely used methods for scaling deep-learning to the low-data regime: ensembling, temperature scaling, and mixup data-augmentation. We highlight that ensembling in itself does not lead to better-calibrated predictions, that the mixup augmentation strategy is practically important and relies on non-trivial trade-offs, and that these methods subtly interact with each other. Crucially, we demonstrate that the order in which the pooling and temperature scaling procedures are executed is important to obtaining calibrated deep-ensembles. We advocate the Pool-Then-Calibrate approach consisting of first pooling the individual neural network predictions together before eventually post-processing the result with a simple and robust temperature scaling step. Furthermore, we note that this approach is insensitive to the choice of pooling method, the simple linear averaging procedure being essentially as robust as the median and trimmed averaging methods.
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