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MethodDependenceTransformationspaceStochasticity(0 sampling)Phase
Gaussian noise on input(Bishop,1995a; An,1996)p(）InputRandomTraining
Gaussian noise on hidden units(DeVries and Taylor, 2017)p(0)Hidden featuresRandomTraining
Dropout (Hinton et al., 2012; Srivastavaet al., 2014)p(）Input andhidden featuresRandomTraining
Random dropout probability(Bouthillier et al., 2015, Sec. 4)p()Input andhidden featuresRandomTraining
Curriculum dropout(Morerio et al., 2017)p(0|time)Input andhidden featuresRandomTraining
Bayesian dropout(Maeda,2014)p(0|π)Input andhidden featuresRandomTraining
Standout (adaptive dropout)(Ba and Frey, 2013)p(0|x,π)Input andhidden featuresRandomTraining
"Projection” of dropout noise into inputspace (Bouthillier et al., 2015, Sec. 3)p(0lx,f,w)InputUses auxiliary Tin hidden-featurespace.RandomTraining
Approximation of Gaussian process bytest-time dropout(Gal and Ghahramani, 2016)p(0)Input andhidden featuresRandomTest
Stochastic depth (Huang et al., 2016b)p（）Hidden featuresRandomTraining
Noisy activation functions(Nair and Hinton, 2010; Xu et al., 2015;Gülcehre et al., 2016a)p(0x）Hidden featuresRandomTraining
Training with adversarial examples(Szegedy et al., 2014)p(0|x,t')InputAdaptiveConstrainedTraining
Network fooling (adversarial examples)(Szegedy et al., 2014)(Not for regularization)p(0lx,t）InputAdaptiveConstrainedTest
Synthetic minority oversampling inhidden-feature space (Wong et al., 2016)p(0x,t,D)Hidden featuresRandomTraining
Inter-and extrapolation in hidden-featurespace (DeVries and Taylor, 2017)p（0|x,t,D）Hidden featuresRandomTraining
Batch normalization (Ioffe and Szegedy,2015),Ghost batch normalization (Hofferet al., 2017)p(0|x)Hidden featuresDeterministicTrainingand test
Layer normalization(Ba et al., 2016)p(0|x)Hidden featuresDeterministicTrainingand test
Annealed noise on targets(Wang and Principe, 1999)p(0|time)TargetRandomTraining
Label smoothing (Szegedy et al., 2016,Sec. 7; Goodfellow et al.,2016, Chap.7)p()TargetDeterministicTraining
Model compression (mimic models,distilled models) (Bucila et al., 2006; Baand Caruana,2014; Hinton et al., 2015)p（0|x,D)TargetDeterministicTraining
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MethodDependenceTransformationspaceStochasticity(0 sampling)Phase
Rigid and elastic image transformation(Baird,1990; Yaegger et al., 1996; Simardet al.,2003; Ciresan et al., 2010)p(0）InputRandomTraining
Test-time image transformations(Simonyan and Zisserman, 2015;Dielemanet al., 2015)p(0)InputRandomTest
Sound transformations(Salamon and Bello, 2017)p(）InputRandomTraining
Error-maximizing rigid imagetransformations(Loosli et al., 2007; Fawzi et al., 2016)p(）InputAdaptivestochastic&constrained,respectivelyTraining
Learning class-specific elasticimage-deformation fields(Hauberg et al., 2016)p(0|t,D)InputRandomTraining
Any handcrafted data preprocessing, forexample scale-invariant feature transform(SIFT) for images (Lowe,1999)p(）InputDeterministicTrainingand test
Overfeat (Sermanet et al., 2013)p(0)InputDeterministicTrainingand test
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MethodMethod classAssumptions about an appropriate learnable input-output mapping
Any chosen (not overlycomplex） architecture*Mapping can be well approximated by functions from the chosen familywhich are easily accessible by optimization.
Small network*Mapping is simple (complexity of the mapping depends on the number ofnetwork units and layers).
Deep network*The mapping is complex,but can be decomposed into a composition (orgenerally into a directed acyclic graph) of simple nonlinear transformations,e.g.affine transformation followed by simple nonlinearity(fully-connectedlayer),“multi-channel convolution”followed by simple nonlinearity (convo-lutional layer),etc.
Hard bottleneck (layer withfew neurons)；soft bottleneck(e.g.Jacobian penalty (Rifaiet al.,201lc),see Section 6)Layer operationData concentrates around a lower-dimensional manifold; has few factors ofvariation.
Convolutional networks(Fukushima and Miyake,1982;Rumelhart et al.,1986,Pp.348-352; LeCun et al.,1989;Simard et al., 2003)Layer operationSpatially local and shift-equivariant feature extraction is all we need.
Dilated convolutions(Yu and Koltun,2015)Layer operationLike convolutional networks.Additionally:Sparse sampling of wide localneighborhoods provides relevant information,and better preserves rele-vant high-resolution information than architectures with downscaling andupsampling.
Strided convolutions (seeDumoulin and Visin,2016)Layer operationThe mapping is reliable at reacting to features that do not vary tooabruptly in space,i.e.which are present in several neighboring pixels andcan be detected even if the filter center skips some of the pixels.The out-put is robust towards slight changes of the location of features,and changesof strength/presence of spatially strongly varying features.
PoolingLayer operationThe output is invariant to slight spatial distortions of the input (slightchanges of the location of (deep) features).Features that are sensitive tosuch distortions can be discarded.
Stochastic pooling(Zeiler and Fergus,2013)Layer operationThe output is robust towards slight changes of the location (like pooling)but also of the strength/presence of (deep) features.
Training with different kindsof noise (including Dropout;see Section 3)NoiseThe mapping is robust to noise:the given class of perturbations of theinput or deep features should not affect the output too much.
Dropout (Hinton et al., 2012;Srivastava etal.,2014),DropConnect (Wan et al.,2013),and related methodsNoiseExtracting complementary (non-coadapted） features is helpful.Non-coadapted features are more informative,better disentangle factors of vari-ation.(We want to disentangle factors of variation because they are en-tangled in different ways in inputs vs.in outputs.)When interpreted as ensemble learning:usual assumptions of ensemblelearning (predictions of weak learners have complementary info and can becombined to strong prediction).
Maxout units(Goodfellow et al., 2013)Layer operationAssumptions similar to Dropout,with more accurate approximation ofmodel averaging (when interpreted as ensemble learning)
Skip-connections (Long et al.,2015;Huang et al.,2016a)Connections be-tween layersCertain lower-level features can directly be reused in a meaningful way at(several) higher levels of abstraction
Linearly augmentedfeed-forward network (van derSmagt and Hirzinger,1998)Connections be-tween layersSkip-connections that share weights with the non-skip-connections.Helpsagainst vanishing gradients. Rather changes the learning algorithm thanthe network mapping.
Residual learning(He et al., 2016)Connections be-tween layersLearning additive difference of a mapping f (or its compositional parts)from the identity mapping is easier than learning f itself. Meaningful deepfeatures can be composed asa sum of lower-level and intermediate-levelfeatures.
Stochastic depth(Huang et al.,2016b),DropIn(Smith et al., 2015)Connectionsbetween layers;noiseSimilar to Dropout:extracting complementary (non-coadapted) featuresacross different levels of abstraction is helpful; implicit model ensem-ble.Similar to Residual learning: meaningful deep features can be com-posed as a sum of lower-level and intermediate-level features,with theintermediate-level ones being optional,and leaving them out beingmeaningful data augmentation. Similar to Mollifying networks: simpli-fying random parts of the mapping improves training.
Mollifying networks(Gülcehre et al., 2016b)Connectionsbetweenlayers;noiseThe mapping can be easier approximated by estimating its decreasinglylinear simplified version
Network information criterion(Murata et al.,1994),Networkgrowing and network pruning(see Bishop,1995a, Sec. 9.5)Model selectionOptimal generalization is reached by a network that has the right numberof units (not too few,not too many)
Multi-task learning (seeCaruana,1998;Ruder,2017)*Several tasks can help each other to learn mutually useful feature extrac-tors,as long as the tasks do not compete for resources (network capacity)
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MethodDescriptionDependencyEquivalence
wy
Weight decay (see Plaut et al.,1986;Lang and Hinton,1990;Goodfellow et al.,2016,Chap.7)L² norm on network weights (notbiases). Favors smaller weights,thus for usual architectures tendsto make the mapping less“extreme",more robust to noise in the input.×Early stopping (seeCollobert and Bengio,2004;Goodfellow et al., 2016,Chap.7)
Weight smoothing(Lang and Hinton,1990)Penalizes L²norm of gradientsof learned filters,making themsmooth. Not beneficial in practice.×
Weight elimination(Weigend et al.,1991)Similar to weight decay but favorsfew stronger connections over manyweak ones.×Goal similar to Narrow andbroad Gaussians
Soft weight-sharing(Nowlan and Hinton,1992)Mixture-of-Gaussians prior onweights.Generalization of weightdecay.Weights are pushed to forma predefined number of groups withsimilar values.X
Narrow and broad Gaussians(Nowlan and Hinton,1992;Blundell et al., 2015)Weights come from two Gaussians,a narrow and a broad one.Specialcase of Soft weight-sharing.XGoal similar to Weightelimination
Fast dropout approximation(Wang and Manning,2013)Approximates the loss that dropoutminimizes. Weighted L2weightpenalty. Only for shallow networks.XxDropout
Mutual exclusivity(Sajjadi et al., 2016)Unlabeled samples push decisionboundaries to low-density regions ininput space,promoting sharp (con-fident） predictions.X
Segmentation with binarypotentials (BenTaieb andHamarneh,2016)Penalty on anatomically implausi-ble image segmentations.X
Flat minima search(Hochreiter and Schmidhuber,1995)Penalty for sharp minima,i.e. forweight configurations where smallweight perturbation leads to higherror increase.Flat minima havelow Minimum description length(i.e.exhibit ideal balance betweentraining error and model complex-ity)and thus should generalize bet-ter (Rissanen,1986).×X
Tangent prop(Simard et al.,1992)L² penalty on directional derivativeof mapping in the predefined tan-gent directions that correspond toknown input-space transformations.XSimple data augmentation
Jacobian penalty(Rifai et al., 2011c)L2penalty on the Jacobian of(partsof） the network mapping-smoothness prior.XNoise on inputs injection(not exact (see An,1996))
Manifold tangent classifier(Rifai et al., 2011a)Like tangent prop,but the input“tangent” directions are extractedfrom manifold learned by a stack ofcontractive autoencoders and thenperforming SVD of the Jacobian ateach input sample.×
Hessian penalty(Rifai et al.,2011b)Fast waytoapproximateL²penalty of thepenalizingJacobianwithnoisyinput.X
Tikhonov regularizers(Bishop,1995b)L² penalty on (up to) n-th deriva-tive of the learned mapping w.r.t.input.×Forpenalty on firstderivative: noise on inputsinjection (not exact (see An,1996)）
Loss-invariant backpropagation(Demyanov et al., 2015, Sec. 3.1;Lyu et al., 2015)(L²)norm of gradient of loss w.r.t.input.Changes the mapping suchthat the loss becomes rather invari-ant to changes of the input.×Adversarial training
Prediction-invariantbackpropagation(Demyanov et al.,2015, Sec. 3.2)(L2） norm of directional derivativeof mapping w.r.t.input in the di-rection of x causing the largest in-crease in loss.××Adversarial training
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