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understanding deep learning simon j.d. prince december 24, 2023 the most recent version of this document can be found at http://udlbook.com. copyright in this work has been licensed exclusively to the mit press, https://mitpress.mit.edu, which will be releasing the final version to the public in 2024. all inquiries reg... |
6 summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 draft: please send errata to udlbookmail@gmail.com.iv contents 5 loss functions 56 5.1 maximum likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.2 recipe for constructing loss functions . . . . . . . . . . . . . ... |
search through the family of possible equations (possible cyan curves) relating input to output to find the one that describes the training data most accurately. itfollowsthatthemodelsinfigure1.2requirelabeledinput/outputpairsfortraining. for example, the music classification model would require a large number of audio... |
. in practice, this takes the form of one sgd-like update within another. keskar this work is subject to a creative commons cc-by-nc-nd license. (c) mit press.notes 159 et al. (2017) showed that sgd finds wider minima as the batch size is reduced. this may be because of the batch variance term that results from implici... |
changingormanipulatingthecolorspace,noiseinjection,andapplyingspatialfilters. moreelaboratetechniquesincluderandomlymixingimages(inoue,2018;summers&dinneen, 2019), randomly erasing parts of the image (zhong et al., 2020), style transfer (jackson et al., 2019), and randomly swapping image patches (kang et al., 2017). in... |
l˜[ϕ]=l[ϕ]+ ϕ2, (9.23) 2α k k where ϕ are the parameters, and α is the learning rate. problem 9.6 consider a model with parameters ϕ = [ϕ ,ϕ ]t. draw the l0, l1, and l1 0 1 p2 regularizationtermsinasimilarformtofigure9.1b. thelp regularizationtermis d |ϕ |p. d=1 d this work is subject to a creative commons cc-by-nc-nd ... |
] . (10.2) in other words, f[x] is equivariant to the transformation t[x] if its output changes in the same way under the transformation as the input. networks for per-pixel image segmentation should be equivariant to transformations (figure 10.1c–f); if the image is translated, rotated, or flipped, the network f[x] sh... |
valid range (figure 10.2c). other possibilities include treating the input as circular or reflecting it at the boundaries. the second approach is to discard the output positions wherethe kernelexceeds the range of input positions. these valid convolutionshavethe advantage of introducing no extra information at the edge... |
a convolutional layer with kernel size three and stride two computes a weighted sum at every other position. f) this is also a special case of a fully connected network with a different sparse weight structure. figure10.5channels. typically,multipleconvolutionsareappliedtotheinputx and stored in channels. a) a convolut... |
convolutional net- workhas2,050parameters,andthefullyconnectednetworkhas150,185parameters. by the logic of figure 10.4, the convolutional network is a special case of the fully connected draft: please send errata to udlbookmail@gmail.com.168 10 convolutional networks figure 10.6 receptive fields for network with kernel... |
. the convolutional kernel is now a 2d object. a 3×3 kernel ω ∈ r3×3 applied to a 2d input comprising of elements x ij computes a single layer of hidden units h as: ij " # x3 x3 hij = a β+ ωmnxi+m−2,j+n−2 , (10.6) m=1n=1 where ω are the entries of the convolutional kernel. this is simply a weighted sum mn overasquare3×... |
, we apply downsampling separately to each channel, so the output has half the width and height but the same number of channels. 10.4.2 upsampling the simplest way to scale up a network layer to double the resolution is to duplicate all the channels at each spatial position four times (figure 10.12a). a second method i... |
ationwherethegoalistoassignalabel to each pixel according to which object is present. 10.5.1 image classification much of the pioneering work on deep learning in computer vision focused on image classificationusingtheimagenetdataset(figure10.15). thiscontains1,281,167training images, 50,000validationimages, and100,000t... |
considers unsupervised learning models. 1.2 unsupervised learning constructing a model from input data without corresponding output labels is termed unsupervised learning; theabsenceofoutputlabelsmeanstherecanbeno“supervision.” rather than learning a mapping from input to output, the goal is to describe or under- stand... |
% top-1errorrate. atthetime,thiswasanenormousleapforwardinperformanceatatask considered far beyond the capabilities of contemporary methods. this result revealed the potential of deep learning and kick-started the modern era of ai research. the vgg network was also targeted at classification in the imagenet task and ac... |
. an early network for semantic segmentation is depicted in figure 10.19. the input is a 224×224 rgb image, and the output is a 224×224×21 array that contains the probability of each of 21 possible classes at each position. thefirstpartofthenetworkisasmallerversionofvgg(figure10.17)thatcontains thirteenratherthansixtee... |
network depth indefinitely doesn’t continue to help; after a certain depth, the system becomes difficult to train. this is the motivation for residual connections, which are the topic of the next chapter. notes dumoulin&visin(2016)presentanoverviewofthemathematicsofconvolutionsthatexpands on the brief treatment in this ... |
used when inpainting missing pixels and account for the partial masking of the input. gated convolutions learn the mask from the previous layer (yu et al., 2019; chang et al., 2019b). hu et al. (2018b) propose squeeze-and-excitation networks which re-weight the channels using information pooled across all spatial posit... |
esteves et al. (2018) introduced polar transformer networks, which are invariant to translations and equivariant to rotation and scale. worrall et al. (2017) developed harmonic networks, the first example of a group cnn that was equivariant to continuous rotations. initialization and regularization: convolutional netwo... |
et al. (2021) and ulku & akagündüz (2022). visualizing convolutional networks: the dramatic success of convolutional networks led toaseriesofeffortstovisualizetheinformationtheyextractfromtheimage(seeqinetal.,2018, for a review). erhan et al. (2009) visualized the optimal stimulus that activated a hidden unit by starti... |
convolution with kernel size five, stride two, and a dilation rate of one. the second hidden layer h is computed using a convolution with kernelsize three, stride one, and 2 a dilation rate of one. the third hidden layer h is computed using a convolution with kernel 3 sizefive,strideone,andadilationrateoftwo. whatareth... |
11.1 sequential processing 187 figure 11.1 sequential processing. standard neural networks pass the output of each layer directly into the next layer. lineartransformation. inaconvolutionalnetwork,eachlayerconsistsofasetofconvolu- tions followed by an activation function, and the parameters comprise the convolutional k... |
). this work is subject to a creative commons cc-by-nc-nd license. (c) mit press.11.2 residual connections and residual blocks 189 shatteredgradientspresumablyarisebecausechangesinearlynetworklayersmodify theoutputinanincreasinglycomplexwayasthenetworkbecomesdeeper. thederivative of the output y with respect to the fir... |
id- ual blocks. a) the usual order of linear transformation or convolution followed byarelunonlinearitymeansthateach residualblockcanonlyaddnon-negative quantities. b) with the reverse order, bothpositiveandnegativequantitiescan beadded. however,wemustaddalinear transformation at the start of the net- workincasetheinpu... |
roomwassilentagain. istoodthereforafewmoments,tryingtomakesenseofwhathadjusthappened. thenirealizedthat thestudentswereallstaringatme,waitingformetosaysomething. itriedtothinkofsomethingwitty orclevertosay,butmymindwasblank. soijustsaid,“well,thatwasstrange,’andthenistartedmy lecture. figure 1.8 conditional text synthe... |
, we initialize the network parameters so that the expected variance of the activations(intheforwardpass)andgradients(inthebackwardpass)remainsthesame between layers. he initialization (section 7.5) achieves this for relu activations by initializing the biases β to zero and choosing normally distributed weights ω with ... |
atch normalization is applied independently to each hidden unit. in a standard neural network with k layers, each containing d hidden units, there would be kd problem11.6 learned offsets δ and kd learned scales γ. in a convolutional network, the normalizing statistics are computed over both the batch and the spatial po... |
information over a 3×3 pixel area using fewer parameters. problem11.8 the resnet-200 model (figure 11.8) contains 200 layers and was used for image clas- sification on the imagenet database (figure 10.15). the architecture resembles alexnet and vgg but uses bottleneck residual blocks instead of vanilla convolutional la... |
,thiscanonlybesustainedforafewlayersbecausethenumberofchannels (and hence the number of parameters required to process them) becomes increasingly large. this problem can be alleviated by applying a 1×1 convolution to reduce the number of channels before the next 3×3 convolution is applied. in a convolutional network,th... |
belonging to the cell if all five networks agree. adapted from falk et al. (2019). 11.6 why do nets with residual connections perform so well? residual networks allow much deeper networks to be trained; it’s possible to extend the resnet architecture to 1000 layers and still train effectively. the improvement in image ... |
net architecture, which concatenatesoutputsofallprior layerstofeedintothecurrentlayer, andu-nets, which incorporate residual connections into encoder-decoder models. notes residual connections: residualconnectionswereintroducedbyheetal.(2016a),whobuilt anetworkwith152layers,whichwaseighttimeslargerthanvgg(figure10.17),... |
by the biases. several regularization methods have been developed that are targeted specifically at residual architectures. resdrop (yamada et al., 2016), stochastic depth (huang et al., 2016), and randomdrop (yamada et al., 2019) all regularize residual networks by randomly dropping residualblocksduringthetrainingproc... |
batch normalization needs access to the whole batch. however, this may not be easily available when training is distributed across several machines. layernormalizationorlayernorm(baetal.,2016)avoidsusingbatchstatisticsbynormalizing eachdataexampleseparately,usingstatisticsgatheredacrossthechannelsandspatialposition (fi... |
has three advantages. first, we may need fewer text/image pairs to learn this mapping now that the inputs and outputs are lower dimensional. second, we are more likely to generate a plausible-looking image; any sensible values of the latent variables should produce something that looks like a plausible example. third, ... |
the potential benefits in healthcare, design, entertainment, transport, education, and almosteveryareaofcommerceareenormous. however,scientistsandengineersareoften unrealistically optimistic about the outcomes of their work, and the potential for harm is just as great. the following paragraphs highlight five concerns. ... |
and to reduce the potential for harm. we should consider what kind of organizations we are prepared to work for. how serious are they in their commitment to reducing the potential harms of ai? are they simply “ethics-washing” to reduce reputational risk, or do they actually implement mechanisms to halt ethically suspec... |
book contain a main body of text, a notes section, and asetofproblems. themainbodyofthetextisintendedtobeself-containedandcanbe readwithoutrecoursetotheotherpartsofthechapter. asmuchaspossible,background mathematics is incorporated into the main body of the text. however, for larger topics thatwouldbeadistractiontothem... |
2.1 supervised learning overview in supervised learning, we aim to build a model that takes an input x and outputs a prediction y. for simplicity, we assume that both the input x and output y are vectors ofapredeterminedandfixedsizeandthattheelementsofeachvectorarealwaysordered in the same way; in the prius example abo... |
(figure2.2a)consistsofi input/outputpairs{x ,y }. i i figures 2.2b–d show three lines defined by three sets of parameters. the green line in figure 2.2d describes the data more accurately than the other two since it is much closer to the data points. however, we need a principled approach for deciding which parameters ... |
values and visualize the loss function as a surface (figure 2.3). the “best” parameters are at the minimum of this surface. draft: please send errata to udlbookmail@gmail.com.22 2 supervised learning 2.2.3 training theprocessoffindingparametersthatminimizethelossistermedmodelfitting,training, or learning. the basic met... |
hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.6 summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 9 regularization 138 9.1 explicit regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.2 implicit regularization . . . . . . . . . . . . ... |
imized(i.e., the entire right-hand side of equation 2.5). a cost function can contain additional terms that are not associated with individual data points (see section 9.1). more generally, an objective function is any function that is to be maximized or minimized. generative vs. discriminative models: themodelsy=f[x,ϕ... |
). second, we pass the 10 11 20 21 30 31 three results through an activation function a[•]. finally, we weight the three resulting activations with ϕ ,ϕ , and ϕ , sum them, and add an offset ϕ . 1 2 3 0 to complete the description, we must define the activation function a[•]. there are many possibilities, but the most ... |
.3) is θ ϕ +θ ϕ , where the first term is the slope in 11 1 31 3 panel (g) and the second term is the slope in panel (i). each hidden unit contributes one “joint” to the function, so with three hidden units, notebook3.1 there can be four linear regions. however, only three of the slopes of these regions are shallownetw... |
subsetofthereallinetoarbitraryprecision. toseethis,considerthateverytimeweaddahiddenunit,weaddanotherlinearregionto the function. as these regions become more numerous, they represent smaller sections of the function, which are increasingly well approximated by a line (figure 3.5). the universal approximation theorem p... |
processing in network with two inputs x = [x ,x ]t, three hidden 1 2 units h ,h ,h , and one output y. a–c) the input to each hidden unit is a 1 2 3 linearfunctionofthetwoinputs,whichcorrespondstoanorientedplane. bright- ness indicates function output. for example, in panel (a), the brightness repre- sents θ +θ x +θ x ... |
= i {1,5,10,50,100}. the number of regions increases rapidly in high dimensions; with d = 500 units and input size d = 100, there can be greater than 10107 i regions(solidcircle). b)thesamedataareplottedasafunctionofthenumberof parameters. thesolidcirclerepresentsthesamemodelasinpanel(a)withd= 500 hidden units. this ne... |
. each layer is connected to the next by forward con- nections (arrows). for this reason, these models are referred to as feed-forward networks. when every variable in one layer connects to every variable in the next, we call this a fully connected network. each connection represents a slope parameter in the underlying... |
notes 37 figure 3.13 activation functions. a) logistic sigmoid and tanh functions. b) leaky relu and parametric relu with parameter 0.25. c) softplus, gaussian errorlinearunit,andsigmoidlinearunit. d)exponentiallinearunitwithparam- eters0.5and1.0,e)scaledexponentiallinearunit. f)swishwithparameters0.4, 1.0, and 1.4. hi... |
tasks. the optimal function was found to be a[x] = x/(1+exp[−βx]), where β is a learned parameter(figure3.13f). theytermedthisfunctionswish. interestingly,thiswasarediscovery of activation functions previously proposed by hendrycks & gimpel (2016) and elfwing et al. (2018). howardetal.(2019)approximatedswishbythehardsw... |
3.4 draw a version of figure 3.3 where the y-intercept and slope of the third hidden unit have changed as in figure 3.14c. assume that the remaining parameters remain the same. figure 3.14 processing in network with one input, three hidden units, and one outputforproblem3.4. a–c)theinputtoeachhiddenunitisalinearfunctio... |
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