Uncertainty Quantification and Deep Ensembles

1 Introduction . Overparametrized deep models can memorize datasets with labels entirely randomized [ 48 ] . 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 [ 8 , 19 , 2 , 31 , 39 , 10 ] . 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 [ 16 ] . In this text , we consider the regime where the size of the training sets is very small , which typically amplifies these issues . This can lead to problematic behaviors when deep neural networks are deployed in scenarios where a proper quantification of the uncertainty is necessary . Indeed , a host of methods [ 22 , 30 , 40 , 12 , 37 ] have been proposed to mitigate these calibration issues , even though no gold standard has so far emerged . Many different forms of regularization techniques [ 35 , 48 , 50 ] have been shown to reduce overfitting in deep neural networks . Importantly , practical implementations and 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . approximations of Bayesian methodologies [ 30 , 44 , 3 , 14 , 27 , 38 , 28 ] have demonstrated their worth in several settings . However , some of these techniques are not entirely straightforward to implement in practice . Ensembling approaches such as drop-outs [ 12 ] have been widely adopted , largely due to their ease of implementation . Recently , [ 1 ] provides a study on different ensembling techniques and describes pitfalls of certain metric for in-domain uncertainty quantification . Also subsequent to our work , several articles also studied the interaction between data-augmentation and calibration issues . Importantly , the CAMixup approach is proposed as a promising solution in [ 42 ] . Furthermore , [ 47 ] analyzes the under-confidence of ensembles due to augmentations from a theoretical perspective . In this text , we investigate the practical use of Deep-Ensembles [ 22 , 4 , 25 , 41 , 9 , 16 ] , a straightforward approach that leads to 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 setting where the computational burden associated with deep-ensembles is not a significant problem . Contributions : We study the interaction between three of the most simple and widely used methods for adopting deep-learning to the low-data regime : ensembling , temperature scaling , and mixup data augmentation . • Despite the widely-held belief that model averaging 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 methods . • We empirically demonstrate that networks trained with the mixup data-augmentation scheme , a widespread practice in computer vision , are typically under-confident . Consequently , subtle interactions between ensembling techniques and modern data-augmentation pipelines have to be considered 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 . In these settings , a standard ensembling approach typically worsens the calibration issues . • Post-processing techniques such as temperature scaling are sometimes regarded as competing methods when comparing the performance of many modern model-averaging techniques . Instead , to mitigate the under-confidence of model averaging , temperature scaling should be used in conjunction with deep-ensembling methods . More importantly , the order in which the aggregation and the calibration procedures are carried out greatly influences 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 . Although straightforward to implement , to the best of our knowledge this strategy has not been investigated in the literature prior to our work . 2 Background . Consider a classification task with C ≥ 2 possible classes Y ≡ { 1 , . . . , C } . For a sample x ∈ X , the quantity p ( x ) ∈ ∆C = { p ∈ RC+ : p1 + . . . + pC = 1 } represents a probabilistic prediction , often obtained as p ( x ) = σSM [ fw ( x ) ] for a neural network fw : X → RC with weight w ∈ RD and softmax function σSM : RC → ∆C . We set ŷ ( x ) ≡ arg maxc pc ( x ) and p̂ ( x ) = maxp ( x ) . Augmentation : Consider a training dataset D ≡ { xi , yi } Ni=1 and denote by y ∈ ∆C the one-hot encoded version of the label y ∈ Y . A stochastic augmentation process Aug : X ×∆C → X ×∆C maps a pair ( x , y ) ∈ X × ∆C to another augmented pair ( x ? , y ? ) . 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 [ 49 ] that has proven beneficial in several settings . For a pair ( x , y ) ∈ X ×∆C , its mixup-augmented version ( x ? , y ? ) is defined as x ? = γ x+ ( 1− γ ) xJ and y ? = γ y + ( 1− γ ) yJ for a random coefficient γ ∈ ( 0 , 1 ) drawn from a fixed mixing distribution often chosen as Beta ( α , α ) , and a random index J drawn uniformly within { 1 , . . . , N } . Model averaging : Ensembling methods leverage a set of models by combining them into an aggregated model . In the context of deep learning , Bayesian averaging consists of weighting the predictions according to the Bayesian posterior π ( dw | Dtrain ) on the neural weights . Instead of finding an optimal set of weights by minimizing a loss function , predictions are averaged . Denoting by pw ( x ) ∈ ∆C the probabilistic prediction associated to sample x ∈ X and neural weight w , the Bayesian approach advocates to consider ( prediction ) ≡ ∫ pw ( x ) π ( dw | Dtrain ) ∈ ∆C . ( 1 ) 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 ( 1 ) is ( extremely ) intractable : the posterior distribution π ( dw|Dtrain ) 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 ( 1 ) well . A detailed understanding of the geometric properties of the posterior distribution in Bayesian neural networks is still lacking , although a lot of recent progress has 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 ( 1 ) . The high-dimensional integral can be approximated by a simple non-weighted average over several modes w1 , . . . , wK of the posterior distribution found by minimizing the negative log-posterior , or some approximations of it , with standard optimization techniques : ( prediction ) ≡ 1 K { pw1 ( x ) + . . .+ pwK ( x ) } ∈ ∆C . We refer the interested reader to [ 34 , 29 , 45 , 3 ] for different perspectives on Bayesian neural networks . Although simple and not well understood , deep-ensembles have been shown to provide highly robust uncertainty quantification when compared to more sophisticated approaches [ 22 , 4 , 25 , 41 ] . Post-processing Calibration Methods : The article [ 16 ] proposes a class of post-processing calibration methods that extend the more standard Platt Scaling approach [ 36 ] . Temperature Scaling , the simplest of these methods , transforms the probabilistic outputs p ( x ) ∈ ∆C into a tempered version Scale [ p ( x ) , τ ] ∈ ∆C defined through the scaling function Scale ( p , τ ) ≡ σSM ( logp/τ ) = 1 Z ( p 1/τ 1 , . . . , p 1/τ C ) ∈ ∆C , ( 2 ) for a temperature parameter τ > 0 and normalization Z > 0 . The optimal parameter τ ? > 0 is usually found by minimizing proper-scoring rules [ 13 ] , 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 τ > 0 . In the low-data regime , the validation set being also extremely small , we have empirically observed that the more sophisticated Vector and Matrix scaling post-processing calibration methods [ 16 ] do not offer any significant advantage over temperature scaling approach and in fact overfit the extremely small validation dataset as chosen by our setup . Calibration Metrics : The Expected Calibration Error ( ECE ) measures the discrepancy between prediction confidence and empirical accuracy . For a partition 0 = c0 < . . . < cM = 1 of the unit interval and a labelled set { xi , yi } Ni=1 , set Bm = { i : cm−1 < p̂ ( xi ) ≤ cm } . The quantity ECE is then defined as ECE = M∑ m=1 |Bm| N ∣∣ confm− accm ∣∣ ( 3 ) where accm = 1 |Bm| ∑ i∈Bm 1 ( ŷ ( xi ) = yi ) and confm = 1 |Bm| ∑ i∈Bm p̂ ( xi ) . ( 4 ) A model is calibrated if accm ≈ confm for all 1 ≤ m ≤ M . It is often instructive to display the associated reliability curve , i.e . the curve with confm on the x-axis and the difference ( accm− confm ) on the y-axis . Figure 1 displays examples of such reliability curves . A perfectly calibrated model is flat ( i.e . accm− confm = 0 ) , while the reliability curve associated to an under-confident ( resp . over-confident ) model prominently lies above ( resp . below ) the flat line accm− confm = 0 . We sometimes also report the value of the Brier score [ 5 ] defined as 1N ∑N i=1 ‖p ( xi ) − yi‖22 . Setup and implementation details : For our experiments , we use standard neural architectures . For CIFAR10/100 [ 21 ] we use ResNet18 , ResNet34 [ 17 ] for Imagenette/Imagewoof [ 18 ] , and for the Diabetic Retinopathy [ 7 ] , similar to [ 26 ] we use the architecture ( not containing any residual connection ) from the 5th place solution of the associated Kaggle challenge . We also include the results for LeNet [ 23 ] trained on the MNIST [ 24 ] dataset in the supplementary . A very low number of training examples ( CIFAR10 : 1000 , CIFAR100 : 5000 , Image { nette , woof } : 5000 , MNIST : 500 ) was used for all the datasets . However , we also show that our observations extend to full-data setups in 4 . The validation dataset is chosen from the leftover training dataset . The test dataset is kept as the original and is hidden during both training and validation step .

Summary: This paper empirically studies the calibration of deep ensembles in low-data settings, as well as the interaction between ensembling, post-hoc calibration (i.e. temperature scaling) and mixup data augmentation (as a form of regularization). The main experimental findings include that in the low-data regime, 1) ensembling neural networks (i.e. averaging their predictions) _reduces_ their confidence, 2) mixup regularization generally makes neural networks _underconfident_, 3) as a result, ensembling mixup-regularized models does _not_ improve calibration but instead exacerbates their underconfidence, and 4) temperature scaling can help improve the calibration of deep ensembles, but only if it is applied _after_ averaging the individual model predictions.