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infer the parameters selecting the desired function. we can think of g as providing a nonlinear change of variables that transforms the distribution over into the desired distribution over. z x recall from equation that, for invertible, [UNK], continuous, 3. 47 g pz ( ) = z px ( ( ) ) g z det ( ∂g ∂z ). ( 20. 72 ) 694	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	709	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models this implicitly imposes a probability distribution over : x px ( ) = x pz ( g−1 ( ) ) x det ( ∂g ∂z ). ( 20. 73 ) of course, this formula may be [UNK] to evaluate, depending on the choice of g, so we often use indirect means of learning g, rather than trying to maximize log ( ) p x directly. in some cases, rather than using g to provide a sample of x directly, we use g to deﬁne a conditional distribution over x. for example, we could use a generator net whose ﬁnal layer consists of sigmoid outputs to provide the mean parameters of bernoulli distributions : p ( xi = 1 ) = ( ) \| z g z i. ( 20. 74 ) in this case, when we use g to deﬁne p ( x z \| ), we impose a distribution over x by marginalizing : z p ( ) = x ezp. ( ) x z \| ( 20. 75 ) both approaches deﬁne a distribution pg ( x ) and allow us to train various criteria of pg using the reparametrization trick of section. 20. 9 the two [UNK] approaches to formulating generator nets	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	710	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
x z \| ( 20. 75 ) both approaches deﬁne a distribution pg ( x ) and allow us to train various criteria of pg using the reparametrization trick of section. 20. 9 the two [UNK] approaches to formulating generator nets — emitting the parameters of a conditional distribution versus directly emitting samples — have complementary strengths and weaknesses. when the generator net deﬁnes a conditional distribution over x, it is capable of generating discrete data as well as continuous data. when the generator net provides samples directly, it is capable of generating only continuous data ( we could introduce discretization in the forward propagation, but doing so would mean the model could no longer be trained using back - propagation ). the advantage to direct sampling is that we are no longer forced to use conditional distributions whose form can be easily written down and algebraically manipulated by a human designer. approaches based on [UNK] generator networks are motivated by the success of gradient descent applied to [UNK] feedforward networks for classiﬁcation. in the context of supervised learning, deep feedforward networks trained with gradient - based learning seem practically guaranteed to succeed given enough hidden units and enough training data. can this same recipe for success transfer to generative modeling? generative modeling seems to be more [UNK] than class	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	710	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
supervised learning, deep feedforward networks trained with gradient - based learning seem practically guaranteed to succeed given enough hidden units and enough training data. can this same recipe for success transfer to generative modeling? generative modeling seems to be more [UNK] than classiﬁcation or regression because the learning process requires optimizing intractable criteria. in the context 695	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	710	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models of [UNK] generator nets, the criteria are intractable because the data does not specify both the inputs z and the outputs x of the generator net. in the case of supervised learning, both the inputs x and the outputs y were given, and the optimization procedure needs only to learn how to produce the speciﬁed mapping. in the case of generative modeling, the learning procedure needs to determine how to arrange space in a useful way and additionally how to map from to. z z x dosovitskiy 2015 et al. ( ) studied a simpliﬁed problem, where the correspondence between z and x is given. speciﬁcally, the training data is computer - rendered imagery of chairs. the latent variables z are parameters given to the rendering engine describing the choice of which chair model to use, the position of the chair, and other conﬁguration details that [UNK] the rendering of the image. using this synthetically generated data, a convolutional network is able to learn to map z descriptions of the content of an image to x approximations of rendered images. this suggests that contemporary [UNK] generator networks have [UNK] model capacity to be good generative models, and that contemporary optimization algorithms have the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	711	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##tional network is able to learn to map z descriptions of the content of an image to x approximations of rendered images. this suggests that contemporary [UNK] generator networks have [UNK] model capacity to be good generative models, and that contemporary optimization algorithms have the ability to ﬁt them. the [UNK] lies in determining how to train generator networks when the value of z for each x is not ﬁxed and known ahead of each time. the following sections describe several approaches to training [UNK] generator nets given only training samples of. x 20. 10. 3 variational autoencoders the variational autoencoder or vae (, ;, ) is a kingma 2013 rezende et al. 2014 directed model that uses learned approximate inference and can be trained purely with gradient - based methods. to generate a sample from the model, the vae ﬁrst draws a sample z from the code distribution pmodel ( z ). the sample is then run through a [UNK] generator network g ( z ). finally, x is sampled from a distribution pmodel ( x ; g ( z ) ) = pmodel ( x z \| ). however, during training, the approximate inference network ( or encoder ) q ( z x \| ) is used to obtain	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	711	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
sampled from a distribution pmodel ( x ; g ( z ) ) = pmodel ( x z \| ). however, during training, the approximate inference network ( or encoder ) q ( z x \| ) is used to obtain z and pmodel ( x z \| ) is then viewed as a decoder network. the key insight behind variational autoencoders is that they may be trained by maximizing the variational lower bound associated with data point : l ( ) q x l ( ) = q ez z x [UNK] ( \| ) log pmodel ( ) + ( ( ) ) z x, h q z \| x ( 20. 76 ) = ez z x [UNK] ( \| ) log pmodel ( ) x z \| −dkl ( ( ) q z \| x \| \| pmodel ( ) ) z ( 20. 77 ) ≤log pmodel ( ) x. ( 20. 78 ) 696	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	711	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models in equation, we recognize the ﬁrst term as the joint log - likelihood of the visible 20. 76 and hidden variables under the approximate posterior over the latent variables ( just like with em, except that we use an approximate rather than the exact posterior ). we recognize also a second term, the entropy of the approximate posterior. when q is chosen to be a gaussian distribution, with noise added to a predicted mean value, maximizing this entropy term encourages increasing the standard deviation of this noise. more generally, this entropy term encourages the variational posterior to place high probability mass on many z values that could have generated x, rather than collapsing to a single point estimate of the most likely value. in equation, we recognize the ﬁrst term as the reconstruction 20. 77 log - likelihood found in other autoencoders. the second term tries to make the approximate posterior distribution q ( z \| x ) and the model prior p model ( z ) approach each other. traditional approaches to variational inference and learning infer q via an opti - mization algorithm, typically iterated ﬁxed point equations ( section ). these 19. 4 approaches are slow and often require the ability to compute [UNK] log p model ( z x,	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	712	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
variational inference and learning infer q via an opti - mization algorithm, typically iterated ﬁxed point equations ( section ). these 19. 4 approaches are slow and often require the ability to compute [UNK] log p model ( z x, ) in closed form. the main idea behind the variational autoencoder is to train a parametric encoder ( also sometimes called an inference network or recognition model ) that produces the parameters of q. so long as z is a continuous variable, we can then back - propagate through samples of z drawn from q ( z x \| ) = q ( z ; f ( x ; θ ) ) in order to obtain a gradient with respect to θ. learning then consists solely of maximizing l with respect to the parameters of the encoder and decoder. all of the expectations in may be approximated by monte carlo sampling. l the variational autoencoder approach is elegant, theoretically pleasing, and simple to implement. it also obtains excellent results and is among the state of the art approaches to generative modeling. its main drawback is that samples from variational autoencoders trained on images tend to be somewhat blurry. the causes of this phenomenon are not yet known. one	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	712	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
results and is among the state of the art approaches to generative modeling. its main drawback is that samples from variational autoencoders trained on images tend to be somewhat blurry. the causes of this phenomenon are not yet known. one possibility is that the blurriness is an intrinsic [UNK] of maximum likelihood, which minimizes dkl ( pdatapmodel ). as illustrated in ﬁgure, this means that the model will assign high probability to 3. 6 points that occur in the training set, but may also assign high probability to other points. these other points may include blurry images. part of the reason that the model would choose to put probability mass on blurry images rather than some other part of the space is that the variational autoencoders used in practice usually have a gaussian distribution for pmodel ( x ; g ( z ) ). maximizing a lower bound on the likelihood of such a distribution is similar to training a traditional autoencoder with mean squared error, in the sense that it has a tendency to ignore features of the input that occupy few pixels or that cause only a small change in the brightness of the pixels that they occupy. this issue is not speciﬁc to vaes	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	712	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
mean squared error, in the sense that it has a tendency to ignore features of the input that occupy few pixels or that cause only a small change in the brightness of the pixels that they occupy. this issue is not speciﬁc to vaes and 697	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	712	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models is shared with generative models that optimize a log - likelihood, or equivalently, dkl ( pdatapmodel ), as argued by ( ) and by ( ). another theis et al. 2015 huszar 2015 troubling issue with contemporary vae models is that they tend to use only a small subset of the dimensions of z, as if the encoder was not able to transform enough of the local directions in input space to a space where the marginal distribution matches the factorized prior. the vae framework is very straightforward to extend to a wide range of model architectures. this is a key advantage over boltzmann machines, which require extremely careful model design to maintain tractability. vaes work very well with a diverse family of [UNK] operators. one particularly sophisticated vae is the deep recurrent attention writer or draw model (, ). gregor et al. 2015 draw uses a recurrent encoder and recurrent decoder combined with an attention mechanism. the generation process for the draw model consists of sequentially visiting [UNK] small image patches and drawing the values of the pixels at those points. vaes can also be extended to generate sequences by deﬁning variational rnns (	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	713	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
with an attention mechanism. the generation process for the draw model consists of sequentially visiting [UNK] small image patches and drawing the values of the pixels at those points. vaes can also be extended to generate sequences by deﬁning variational rnns (, ) by using a recurrent encoder and decoder within chung et al. 2015b the vae framework. generating a sample from a traditional rnn involves only non - deterministic operations at the output space. variational rnns also have random variability at the potentially more abstract level captured by the vae latent variables. the vae framework has been extended to maximize not just the traditional variational lower bound, but instead the importance weighted autoencoder (, ) objective : burda et al. 2015 lk ( ) = x, q ez ( 1 ),..., z ( ) k [UNK] \| q ( z x ) log 1 k k i = 1 pmodel ( x z, ( ) i ) q ( z ( ) i \| x ). ( 20. 79 ) this new objective is equivalent to the traditional lower bound l when k = 1. however, it may also be interpreted as forming an estimate of the true log pmodel ( x )	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	713	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
z ( ) i \| x ). ( 20. 79 ) this new objective is equivalent to the traditional lower bound l when k = 1. however, it may also be interpreted as forming an estimate of the true log pmodel ( x ) using importance sampling of z from proposal distribution q ( z x \| ). the importance weighted autoencoder objective is also a lower bound on log pmodel ( x ) and becomes tighter as increases. k variational autoencoders have some interesting connections to the mp - dbm and other approaches that involve back - propagation through the approximate inference graph ( goodfellow 2013b stoyanov 2011 brakel 2013 et al., ; et al., ; et al., ). these previous approaches required an inference procedure such as mean ﬁeld ﬁxed point equations to provide the computational graph. the variational autoencoder is deﬁned for arbitrary computational graphs, which makes it applicable to a wider range of probabilistic model families because there is no need to restrict the choice 698	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	713	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models of models to those with tractable mean ﬁeld ﬁxed point equations. the variational autoencoder also has the advantage that it increases a bound on the log - likelihood of the model, while the criteria for the mp - dbm and related models are more heuristic and have little probabilistic interpretation beyond making the results of approximate inference accurate. one disadvantage of the variational autoencoder is that it learns an inference network for only one problem, inferring z given x. the older methods are able to perform approximate inference over any subset of variables given any other subset of variables, because the mean ﬁeld ﬁxed point equations specify how to share parameters between the computational graphs for all of these [UNK] problems. one very nice property of the variational autoencoder is that simultaneously training a parametric encoder in combination with the generator network forces the model to learn a predictable coordinate system that the encoder can capture. this makes it an excellent manifold learning algorithm. see ﬁgure for examples of 20. 6 low - dimensional manifolds learned by the variational autoencoder. in one of the cases demonstrated in the ﬁgure, the algorithm discovered two independent factors of variation present	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	714	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
manifold learning algorithm. see ﬁgure for examples of 20. 6 low - dimensional manifolds learned by the variational autoencoder. in one of the cases demonstrated in the ﬁgure, the algorithm discovered two independent factors of variation present in images of faces : angle of rotation and emotional expression. 20. 10. 4 generative adversarial networks generative adversarial networks or gans (, ) are another goodfellow et al. 2014c generative modeling approach based on [UNK] generator networks. generative adversarial networks are based on a game theoretic scenario in which the generator network must compete against an adversary. the generator network directly produces samplesx = g ( z ; θ ( ) g ). its adversary, the discriminator network, attempts to distinguish between samples drawn from the training data and samples drawn from the generator. the discriminator emits a probability value given by d ( x ; θ ( ) d ), indicating the probability that x is a real training example rather than a fake sample drawn from the model. the simplest way to formulate learning in generative adversarial networks is as a zero - sum game, in which a function v ( θ ( ) g, θ ( )	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	714	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
real training example rather than a fake sample drawn from the model. the simplest way to formulate learning in generative adversarial networks is as a zero - sum game, in which a function v ( θ ( ) g, θ ( ) d ) determines the [UNK] the discriminator. the generator receives −v ( θ ( ) g, θ ( ) d ) as its own [UNK]. during learning, each player attempts to maximize its own [UNK], so that at convergence g∗ = arg min g max d v g, d. ( ) ( 20. 80 ) the default choice for is v v ( θ ( ) g, θ ( ) d ) = [UNK] ( ) + d x e [UNK] log ( 1 ( ) ) −d x. ( 20. 81 ) 699	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	714	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models figure 20. 6 : examples of two - dimensional coordinate systems for high - dimensional mani - folds, learned by a variational autoencoder ( kingma and welling 2014a, ). two dimensions may be plotted directly on the page for visualization, so we can gain an understanding of how the model works by training a model with a 2 - d latent code, even if we believe the intrinsic dimensionality of the data manifold is much higher. the images shown are not examples from the training set but images x actually generated by the model p ( x z \| ), simply by changing the 2 - d “ code ” z ( each image corresponds to a [UNK] choice of “ code ” z on a 2 - d uniform grid ). ( left ) the two - dimensional map of the frey faces manifold. one dimension that has been discovered ( horizontal ) mostly corresponds to a rotation of the face, while the other ( vertical ) corresponds to the emotional expression. the ( right ) two - dimensional map of the mnist manifold. this drives the discriminator to attempt to learn to correctly classify samples as real or fake. simultaneously, the generator attempts to fool the classiﬁer into believing its samples are real. at convergence	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	715	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
two - dimensional map of the mnist manifold. this drives the discriminator to attempt to learn to correctly classify samples as real or fake. simultaneously, the generator attempts to fool the classiﬁer into believing its samples are real. at convergence, the generator ’ s samples are indistinguishable from real data, and the discriminator outputs 1 2 everywhere. the discriminator may then be discarded. the main motivation for the design of gans is that the learning process requires neither approximate inference nor approximation of a partition function gradient. in the case where maxd v ( g, d ) is convex in θ ( ) g ( such as the case where optimization is performed directly in the space of probability density functions ) the procedure is guaranteed to converge and is asymptotically consistent. unfortunately, learning in gans can be [UNK] in practice when g and d are represented by neural networks and max d v ( g, d ) is not convex. goodfellow 700	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	715	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models ( ) identiﬁed non - convergence as an issue that may cause gans to underﬁt. 2014 in general, simultaneous gradient descent on two players ’ costs is not guaranteed to reach an equilibrium. consider for example the value function v ( a, b ) = ab, where one player controls a and incurs cost ab, while the other player controls b and receives a cost −ab. if we model each player as making inﬁnitesimally small gradient steps, each player reducing their own cost at the expense of the other player, then a and b go into a stable, circular orbit, rather than arriving at the equilibrium point at the origin. note that the equilibria for a minimax game are not local minima of v. instead, they are points that are simultaneously minima for both players ’ costs. this means that they are saddle points of v that are local minima with respect to the ﬁrst player ’ s parameters and local maxima with respect to the second player ’ s parameters. it is possible for the two players to take turns increasing then decreasing v forever, rather than landing exactly on the saddle point where neither player is capable of reducing its cost. it is not known to	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	716	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##a with respect to the second player ’ s parameters. it is possible for the two players to take turns increasing then decreasing v forever, rather than landing exactly on the saddle point where neither player is capable of reducing its cost. it is not known to what extent this non - convergence problem [UNK] gans. goodfellow 2014 ( ) identiﬁed an alternative formulation of the [UNK], in which the game is no longer zero - sum, that has the same expected gradient as maximum likelihood learning whenever the discriminator is optimal. because maximum likelihood training converges, this reformulation of the gan game should also converge, given enough samples. unfortunately, this alternative formulation does not seem to improve convergence in practice, possibly due to suboptimality of the discriminator, or possibly due to high variance around the expected gradient. in realistic experiments, the best - performing formulation of the gan game is a [UNK] formulation that is neither zero - sum nor equivalent to maximum likelihood, introduced by ( ) with a heuristic motivation. in goodfellow et al. 2014c this best - performing formulation, the generator aims to increase the log probability that the discriminator makes a mistake, rather than aiming to decrease the log probability that the discriminator makes the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	716	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##stic motivation. in goodfellow et al. 2014c this best - performing formulation, the generator aims to increase the log probability that the discriminator makes a mistake, rather than aiming to decrease the log probability that the discriminator makes the correct prediction. this reformulation is motivated solely by the observation that it causes the derivative of the generator ’ s cost function with respect to the discriminator ’ s logits to remain large even in the situation where the discriminator conﬁdently rejects all generator samples. stabilization of gan learning remains an open problem. fortunately, gan learning performs well when the model architecture and hyperparameters are care - fully selected. ( ) crafted a deep convolutional gan ( dcgan ) radford et al. 2015 that performs very well for image synthesis tasks, and showed that its latent repre - sentation space captures important factors of variation, as shown in ﬁgure. 15. 9 see ﬁgure for examples of images generated by a dcgan generator. 20. 7 the gan learning problem can also be simpliﬁed by breaking the generation 701	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	716	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models figure 20. 7 : images generated by gans trained on the lsun dataset. ( left ) images of bedrooms generated by a dcgan model, reproduced with permission from radford et al. ( ). images of churches generated by a lapgan model, reproduced with 2015 ( right ) permission from ( ). denton et al. 2015 process into many levels of detail. it is possible to train conditional gans ( mirza and osindero 2014, ) that learn to sample from a distribution p ( x y \| ) rather than simply sampling from a marginal distribution p ( x ). ( ) denton et al. 2015 showed that a series of conditional gans can be trained to ﬁrst generate a very low - resolution version of an image, then incrementally add details to the image. this technique is called the lapgan model, due to the use of a laplacian pyramid to generate the images containing varying levels of detail. lapgan generators are able to fool not only discriminator networks but also human observers, with experimental subjects identifying up to 40 % of the outputs of the network as being real data. see ﬁgure for examples of images generated by a lapgan 20. 7 generator. one unusual	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	717	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
not only discriminator networks but also human observers, with experimental subjects identifying up to 40 % of the outputs of the network as being real data. see ﬁgure for examples of images generated by a lapgan 20. 7 generator. one unusual capability of the gan training procedure is that it can ﬁt proba - bility distributions that assign zero probability to the training points. rather than maximizing the log probability of speciﬁc points, the generator net learns to trace out a manifold whose points resemble training points in some way. somewhat para - doxically, this means that the model may assign a log - likelihood of negative inﬁnity to the test set, while still representing a manifold that a human observer judges to capture the essence of the generation task. this is not clearly an advantage or a disadvantage, and one may also guarantee that the generator network assigns non - zero probability to all points simply by making the last layer of the generator network add gaussian noise to all of the generated values. generator networks that add gaussian noise in this manner sample from the same distribution that one obtains by using the generator network to parametrize the mean of a conditional 702	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	717	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models gaussian distribution. dropout seems to be important in the discriminator network. in particular, units should be stochastically dropped while computing the gradient for the generator network to follow. following the gradient of the deterministic version of the discriminator with its weights divided by two does not seem to be as [UNK]. likewise, never using dropout seems to yield poor results. while the gan framework is designed for [UNK] generator networks, similar principles can be used to train other kinds of models. for example, self - supervised boosting can be used to train an rbm generator to fool a logistic regression discriminator ( welling 2002 et al., ). 20. 10. 5 generative moment matching networks generative moment matching networks (, ;, li et al. 2015 dziugaite et al. 2015 ) are another form of generative model based on [UNK] generator networks. unlike vaes and gans, they do not need to pair the generator network with any other network — neither an inference network as used with vaes nor a discriminator network as used with gans. these networks are trained with a technique called moment matching. the basic idea behind moment matching is to train the generator	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	718	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
generator network with any other network — neither an inference network as used with vaes nor a discriminator network as used with gans. these networks are trained with a technique called moment matching. the basic idea behind moment matching is to train the generator in such a way that many of the statistics of samples generated by the model are as similar as possible to those of the statistics of the examples in the training set. in this context, a moment is an expectation of [UNK] powers of a random variable. for example, the ﬁrst moment is the mean, the second moment is the mean of the squared values, and so on. in multiple dimensions, each element of the random vector may be raised to [UNK] powers, so that a moment may be any quantity of the form exπixni i ( 20. 82 ) where n = [ n1, n2,..., nd ] is a vector of non - negative integers. upon ﬁrst examination, this approach seems to be computationally infeasible. for example, if we want to match all the moments of the form xix j, then we need to minimize the [UNK] between a number of values that is quadratic in the dimension of x. moreover, even matching all of the ﬁrst	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	718	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. for example, if we want to match all the moments of the form xix j, then we need to minimize the [UNK] between a number of values that is quadratic in the dimension of x. moreover, even matching all of the ﬁrst and second moments would only be [UNK] to ﬁt a multivariate gaussian distribution, which captures only linear relationships between values. our ambitions for neural networks are to capture complex nonlinear relationships, which would require far more moments. gans avoid this problem of exhaustively enumerating all moments by using a 703	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	718	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models dynamically updated discriminator that automatically focuses its attention on whichever statistic the generator network is matching the least [UNK]. instead, generative moment matching networks can be trained by minimizing a cost function called maximum mean discrepancy ( scholkopf and smola, 2002 gretton 2012 ; et al., ) or mmd. this cost function measures the error in the ﬁrst moments in an inﬁnite - dimensional space, using an implicit mapping to feature space deﬁned by a kernel function in order to make computations on inﬁnite - dimensional vectors tractable. the mmd cost is zero if and only if the two distributions being compared are equal. visually, the samples from generative moment matching networks are somewhat disappointing. fortunately, they can be improved by combining the generator network with an autoencoder. first, an autoencoder is trained to reconstruct the training set. next, the encoder of the autoencoder is used to transform the entire training set into code space. the generator network is then trained to generate code samples, which may be mapped to visually pleasing samples via the decoder. unlike gans, the cost function is deﬁ	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	719	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
autoencoder is used to transform the entire training set into code space. the generator network is then trained to generate code samples, which may be mapped to visually pleasing samples via the decoder. unlike gans, the cost function is deﬁned only with respect to a batch of examples from both the training set and the generator network. it is not possible to make a training update as a function of only one training example or only one sample from the generator network. this is because the moments must be computed as an empirical average across many samples. when the batch size is too small, mmd can underestimate the true amount of variation in the distributions being sampled. no ﬁnite batch size is [UNK] large to eliminate this problem entirely, but larger batches reduce the amount of underestimation. when the batch size is too large, the training procedure becomes infeasibly slow, because many examples must be processed in order to compute a single small gradient step. as with gans, it is possible to train a generator net using mmd even if that generator net assigns zero probability to the training points. 20. 10. 6 convolutional generative networks when generating images, it is often useful to use a generator network that includes a	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	719	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
to train a generator net using mmd even if that generator net assigns zero probability to the training points. 20. 10. 6 convolutional generative networks when generating images, it is often useful to use a generator network that includes a convolutional structure ( see for example goodfellow 2014c dosovitskiy et al. ( ) or et al. ( ) ). to do so, we use the “ transpose ” of the convolution operator, 2015 described in section. this approach often yields more realistic images and does 9. 5 so using fewer parameters than using fully connected layers without parameter sharing. convolutional networks for recognition tasks have information ﬂow from the image to some summarization layer at the top of the network, often a class label. 704	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	719	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models as this image ﬂows upward through the network, information is discarded as the representation of the image becomes more invariant to nuisance transformations. in a generator network, the opposite is true. rich details must be added as the representation of the image to be generated propagates through the network, culminating in the ﬁnal representation of the image, which is of course the image itself, in all of its detailed glory, with object positions and poses and textures and lighting. the primary mechanism for discarding information in a convolutional recognition network is the pooling layer. the generator network seems to need to add information. we cannot put the inverse of a pooling layer into the generator network because most pooling functions are not invertible. a simpler operation is to merely increase the spatial size of the representation. an approach that seems to perform acceptably is to use an “ un - pooling ” as introduced by dosovitskiy et al. ( ). this layer corresponds to the inverse of the max - pooling operation under 2015 certain simplifying conditions. first, the stride of the max - pooling operation is constrained to be equal to the width of the pooling region. second, the maximum input within each	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	720	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
to the inverse of the max - pooling operation under 2015 certain simplifying conditions. first, the stride of the max - pooling operation is constrained to be equal to the width of the pooling region. second, the maximum input within each pooling region is assumed to be the input in the upper - left corner. finally, all non - maximal inputs within each pooling region are assumed to be zero. these are very strong and unrealistic assumptions, but they do allow the max - pooling operator to be inverted. the inverse un - pooling operation allocates a tensor of zeros, then copies each value from spatial coordinate i of the input to spatial coordinate i k × of the output. the integer value k deﬁnes the size of the pooling region. even though the assumptions motivating the deﬁnition of the un - pooling operator are unrealistic, the subsequent layers are able to learn to compensate for its unusual output, so the samples generated by the model as a whole are visually pleasing. 20. 10. 7 auto - regressive networks auto - regressive networks are directed probabilistic models with no latent random variables. the conditional probability distributions in these models are represented by neural networks (	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	720	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
a whole are visually pleasing. 20. 10. 7 auto - regressive networks auto - regressive networks are directed probabilistic models with no latent random variables. the conditional probability distributions in these models are represented by neural networks ( sometimes extremely simple neural networks such as logistic regression ). the graph structure of these models is the complete graph. they decompose a joint probability over the observed variables using the chain rule of probability to obtain a product of conditionals of the form p ( xd \| xd−1,..., x1 ). such models have been called fully - visible bayes networks ( fvbns ) and used successfully in many forms, ﬁrst with logistic regression for each conditional distribution ( frey 1998, ) and then with neural networks with hidden units ( bengio and bengio 2000b larochelle and murray 2011, ;, ). in some forms of auto - regressive networks, such as nade (, ), described larochelle and murray 2011 705	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	720	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models in section below, we can introduce a form of parameter sharing that 20. 10. 10 brings both a statistical advantage ( fewer unique parameters ) and a computational advantage ( less computation ). this is one more instance of the recurring deep learning motif of reuse of features. x1 x1 x2 x2 x3 x3 x4 x4 p x ( 4 \| x1, x2, x3 ) p x ( 4 \| x1, x2, x3 ) p x ( 3 \| x1, x2 ) p x ( 3 \| x1, x2 ) p x ( 2 \| x1 ) p x ( 2 \| x1 ) p x ( 1 ) p x ( 1 ) x1 x1 x2 x2 x3 x3 x4 x4 figure 20. 8 : a fully visible belief network predicts the i - th variable from the i −1 previous ones. ( top ) ( bottom ) the directed graphical model for an fvbn. corresponding computational graph, in the case of the logistic fvbn, where each prediction is made by a linear predictor. 20. 10. 8 linear auto - regressive networks the simplest form of auto - regress	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	721	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
fvbn. corresponding computational graph, in the case of the logistic fvbn, where each prediction is made by a linear predictor. 20. 10. 8 linear auto - regressive networks the simplest form of auto - regressive network has no hidden units and no sharing of parameters or features. each p ( xi \| xi−1,..., x1 ) is parametrized as a linear model ( linear regression for real - valued data, logistic regression for binary data, softmax regression for discrete data ). this model was introduced by frey 1998 ( ) and has o ( d2 ) parameters when there are d variables to model. it is illustrated in ﬁgure. 20. 8 if the variables are continuous, a linear auto - regressive model is merely another way to formulate a multivariate gaussian distribution, capturing linear pairwise interactions between the observed variables. linear auto - regressive networks are essentially the generalization of linear classiﬁcation methods to generative modeling. they therefore have the same 706	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	721	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models advantages and disadvantages as linear classiﬁers. like linear classiﬁers, they may be trained with convex loss functions, and sometimes admit closed form solutions ( as in the gaussian case ). like linear classiﬁers, the model itself does not [UNK] a way of increasing its capacity, so capacity must be raised using techniques like basis expansions of the input or the kernel trick. x1 x1 x2 x2 x3 x3 x4 x4 h1 h1 h2 h2 h3 h3 p x ( 4 \| x1, x2, x3 ) p x ( 4 \| x1, x2, x3 ) p x ( 3 \| x1, x2 ) p x ( 3 \| x1, x2 ) p x ( 2 \| x1 ) p x ( 2 \| x1 ) p x ( 1 ) p x ( 1 ) figure 20. 9 : a neural auto - regressive network predicts thei - th variable xi from the i −1 previous ones, but is parametrized so that features ( groups of hidden units denotedhi ) that are functions of x1,..., xi can be reused in predicting all	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	722	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
predicts thei - th variable xi from the i −1 previous ones, but is parametrized so that features ( groups of hidden units denotedhi ) that are functions of x1,..., xi can be reused in predicting all of the subsequent variables xi + 1, xi + 2,..., xd. 20. 10. 9 neural auto - regressive networks neural auto - regressive networks (,, ) have the same bengio and bengio 2000a b left - to - right graphical model as logistic auto - regressive networks ( ﬁgure ) but 20. 8 employ a [UNK] parametrization of the conditional distributions within that graphical model structure. the new parametrization is more powerful in the sense that its capacity can be increased as much as needed, allowing approximation of any joint distribution. the new parametrization can also improve generalization by introducing a parameter sharing and feature sharing principle common to deep learning in general. the models were motivated by the objective of avoiding the curse of dimensionality arising out of traditional tabular graphical models, sharing the same structure as ﬁgure. in tabular discrete probabilistic models, each 20. 8 conditional distribution is represented by a table of probabilities	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	722	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
of avoiding the curse of dimensionality arising out of traditional tabular graphical models, sharing the same structure as ﬁgure. in tabular discrete probabilistic models, each 20. 8 conditional distribution is represented by a table of probabilities, with one entry and one parameter for each possible conﬁguration of the variables involved. by using a neural network instead, two advantages are obtained : 707	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	722	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models 1. the parametrization of each p ( xi \| xi−1,..., x1 ) by a neural network with ( i −1 ) × k inputs and k outputs ( if the variables are discrete and take k values, encoded one - hot ) allows one to estimate the conditional probability without requiring an exponential number of parameters ( and examples ), yet still is able to capture high - order dependencies between the random variables. 2. instead of having a [UNK] neural network for the prediction of each xi, a connectivity illustrated in ﬁgure allows one to merge all left - to - right 20. 9 the neural networks into one. equivalently, it means that the hidden layer features computed for predicting xi can be reused for predicting xi k + ( k > 0 ). the hidden units are thus organized in groups that have the particularity that all the units in the i - th group only depend on the input values x1,..., xi. the parameters used to compute these hidden units are jointly optimized to improve the prediction of all the variables in the sequence. this is an instance of the reuse principle that recurs throughout deep learning in scenarios ranging from recurrent and convolutional	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	723	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
used to compute these hidden units are jointly optimized to improve the prediction of all the variables in the sequence. this is an instance of the reuse principle that recurs throughout deep learning in scenarios ranging from recurrent and convolutional network architectures to multi - task and transfer learning. each p ( xi \| xi−1,..., x1 ) can represent a conditional distribution by having outputs of the neural network predict parameters of the conditional distribution of xi, as discussed in section. although the original neural auto - regressive 6. 2. 1. 1 networks were initially evaluated in the context of purely discrete multivariate data ( with a sigmoid output for a bernoulli variable or softmax output for a multinoulli variable ) it is natural to extend such models to continuous variables or joint distributions involving both discrete and continuous variables. 20. 10. 10 nade the neural autoregressive density estimator ( nade ) is a very successful recent form of neural auto - regressive network ( larochelle and murray 2011, ). the connectivity is the same as for the original neural auto - regressive network of bengio and bengio 2000b ( ) but nade introduces an additional	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	723	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)