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think of each of these regions as a category or symbol : by having a separate degree of freedom for each symbol ( or region ), we can learn an arbitrary decoder mapping from symbol to value. however, this does not allow us to generalize to new symbols for new regions. if we are lucky, there may be some regularity in the target function, besides being smooth. for example, a convolutional network with max - pooling can recognize an object regardless of its location in the image, even though spatial translation of the object may not correspond to smooth transformations in the input space. let us examine a special case of a distributed representation learning algorithm, that extracts binary features by thresholding linear functions of the input. each binary feature in this representation divides rd into a pair of half - spaces, as illustrated in ﬁgure. the exponentially large number of intersections of 15. 7 n of the corresponding half - spaces determines how many regions this distributed representation learner can distinguish. how many regions are generated by an arrangement of n hyperplanes in rd? by applying a general result concerning the intersection of hyperplanes (, ), one can show ( zaslavsky 1975 pascanu 2014b et al., ) that the number of regions this	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	565	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
by an arrangement of n hyperplanes in rd? by applying a general result concerning the intersection of hyperplanes (, ), one can show ( zaslavsky 1975 pascanu 2014b et al., ) that the number of regions this binary feature representation can distinguish is d j = 0 n j = ( o nd ). ( 15. 4 ) therefore, we see a growth that is exponential in the input size and polynomial in the number of hidden units. 1potentially, we may want to learn a function whose behavior is distinct in exponentially many regions : in a d - dimensional space with at least 2 [UNK] values to distinguish per dimension, we might want to [UNK] in f 2d [UNK] regions, requiring o ( 2 d ) training examples. 550	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	565	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15. representation learning this provides a geometric argument to explain the generalization power of distributed representation : with o ( nd ) parameters ( for n linear - threshold features in rd ) we can distinctly represent o ( nd ) regions in input space. if instead we made no assumption at all about the data, and used a representation with one unique symbol for each region, and separate parameters for each symbol to recognize its corresponding portion of rd, then specifying o ( nd ) regions would require o ( nd ) examples. more generally, the argument in favor of the distributed representation could be extended to the case where instead of using linear threshold units we use nonlinear, possibly continuous, feature extractors for each of the attributes in the distributed representation. the argument in this case is that if a parametric transformation with k parameters can learn about r regions in input space, with k r, and if obtaining such a representation was useful to the task of interest, then we could potentially generalize much better in this way than in a non - distributed setting where we would need o ( r ) examples to obtain the same features and associated partitioning of the input space into r regions. using fewer parameters to represent the model means that we have fewer parameters to ﬁt, and thus	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	566	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
a non - distributed setting where we would need o ( r ) examples to obtain the same features and associated partitioning of the input space into r regions. using fewer parameters to represent the model means that we have fewer parameters to ﬁt, and thus require far fewer training examples to generalize well. a further part of the argument for why models based on distributed represen - tations generalize well is that their capacity remains limited despite being able to distinctly encode so many [UNK] regions. for example, the vc dimension of a neural network of linear threshold units is only o ( w w log ), where w is the number of weights ( sontag 1998, ). this limitation arises because, while we can assign very many unique codes to representation space, we cannot use absolutely all of the code space, nor can we learn arbitrary functions mapping from the representation space h to the output y using a linear classiﬁer. the use of a distributed representation combined with a linear classiﬁer thus expresses a prior belief that the classes to be recognized are linearly separable as a function of the underlying causal factors captured by h. we will typically want to learn categories such as the set of all images of all green objects or the set of all images of cars,	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	566	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
that the classes to be recognized are linearly separable as a function of the underlying causal factors captured by h. we will typically want to learn categories such as the set of all images of all green objects or the set of all images of cars, but not categories that require nonlinear, xor logic. for example, we typically do not want to partition the data into the set of all red cars and green trucks as one class and the set of all green cars and red trucks as another class. the ideas discussed so far have been abstract, but they may be experimentally validated. ( ) ﬁnd that hidden units in a deep convolutional network zhou et al. 2015 trained on the imagenet and places benchmark datasets learn features that are very often interpretable, corresponding to a label that humans would naturally assign. in practice it is certainly not always the case that hidden units learn something that has a simple linguistic name, but it is interesting to see this emerge near the top levels of the best computer vision deep networks. what such features have in 551	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	566	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15. representation learning - + = figure 15. 9 : a generative model has learned a distributed representation that disentangles the concept of gender from the concept of wearing glasses. if we begin with the repre - sentation of the concept of a man with glasses, then subtract the vector representing the concept of a man without glasses, and ﬁnally add the vector representing the concept of a woman without glasses, we obtain the vector representing the concept of a woman with glasses. the generative model correctly decodes all of these representation vectors to images that may be recognized as belonging to the correct class. images reproduced with permission from ( ). radford et al. 2015 common is that one could imagine learning about each of them without having to see all the conﬁgurations of all the others. ( ) demonstrated that radford et al. 2015 a generative model can learn a representation of images of faces, with separate directions in representation space capturing [UNK] underlying factors of variation. figure demonstrates that one direction in representation space corresponds 15. 9 to whether the person is male or female, while another corresponds to whether the person is wearing glasses. these features were discovered automatically, not ﬁxed a priori. there is no need to have labels for	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	567	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
one direction in representation space corresponds 15. 9 to whether the person is male or female, while another corresponds to whether the person is wearing glasses. these features were discovered automatically, not ﬁxed a priori. there is no need to have labels for the hidden unit classiﬁers : gradient descent on an objective function of interest naturally learns semantically interesting features, so long as the task requires such features. we can learn about the distinction between male and female, or about the presence or absence of glasses, without having to characterize all of the conﬁgurations of the n −1 other features by examples covering all of these combinations of values. this form of statistical separability is what allows one to generalize to new conﬁgurations of a person ’ s features that have never been seen during training. 552	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	567	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15. representation learning 15. 5 exponential gains from depth we have seen in section that multilayer perceptrons are universal approxima - 6. 4. 1 tors, and that some functions can be represented by exponentially smaller deep networks compared to shallow networks. this decrease in model size leads to improved statistical [UNK]. in this section, we describe how similar results apply more generally to other kinds of models with distributed hidden representations. in section, we saw an example of a generative model that learned about 15. 4 the explanatory factors underlying images of faces, including the person ’ s gender and whether they are wearing glasses. the generative model that accomplished this task was based on a deep neural network. it would not be reasonable to expect a shallow network, such as a linear network, to learn the complicated relationship between these abstract explanatory factors and the pixels in the image. in this and other ai tasks, the factors that can be chosen almost independently from each other yet still correspond to meaningful inputs are more likely to be very high - level and related in highly nonlinear ways to the input. we argue that this demands deep distributed representations, where the higher level features ( seen as functions of the input ) or factors ( seen as generative causes	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	568	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
inputs are more likely to be very high - level and related in highly nonlinear ways to the input. we argue that this demands deep distributed representations, where the higher level features ( seen as functions of the input ) or factors ( seen as generative causes ) are obtained through the composition of many nonlinearities. it has been proven in many [UNK] settings that organizing computation through the composition of many nonlinearities and a hierarchy of reused features can give an exponential boost to statistical [UNK], on top of the exponential boost given by using a distributed representation. many kinds of networks ( e. g., with saturating nonlinearities, boolean gates, sum / products, or rbf units ) with a single hidden layer can be shown to be universal approximators. a model family that is a universal approximator can approximate a large class of functions ( including all continuous functions ) up to any non - zero tolerance level, given enough hidden units. however, the required number of hidden units may be very large. theoretical results concerning the expressive power of deep architectures state that there are families of functions that can be represented [UNK] by an architecture of depth k, but would require an exponential number of hidden units ( with respect to the input size ) with [UNK] depth ( depth 2 or depth	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	568	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
expressive power of deep architectures state that there are families of functions that can be represented [UNK] by an architecture of depth k, but would require an exponential number of hidden units ( with respect to the input size ) with [UNK] depth ( depth 2 or depth ). k −1 in section, we saw that deterministic feedforward networks are universal 6. 4. 1 approximators of functions. many structured probabilistic models with a single hidden layer of latent variables, including restricted boltzmann machines and deep belief networks, are universal approximators of probability distributions ( le roux and bengio 2008 2010 montufar and ay 2011 montufar 2014 krause,, ;, ;, ; et al., 2013 ). 553	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	568	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15. representation learning in section, we saw that a [UNK] deep feedforward network can have 6. 4. 1 an exponential advantage over a network that is too shallow. such results can also be obtained for other models such as probabilistic models. one such probabilistic model is the sum - product network or spn ( poon and domingos 2011, ). these models use polynomial circuits to compute the probability distribution over a set of random variables. ( ) showed that there exist delalleau and bengio 2011 probability distributions for which a minimum depth of spn is required to avoid needing an exponentially large model. later, ( ) martens and medabalimi 2014 showed that there are signiﬁcant [UNK] between every two ﬁnite depths of spn, and that some of the constraints used to make spns tractable may limit their representational power. another interesting development is a set of theoretical results for the expressive power of families of deep circuits related to convolutional nets, highlighting an exponential advantage for the deep circuit even when the shallow circuit is allowed to only approximate the function computed by the deep circuit (, cohen et al. 2015 ). by comparison, previous theoretical work made claims regarding only the case	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	569	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##tional nets, highlighting an exponential advantage for the deep circuit even when the shallow circuit is allowed to only approximate the function computed by the deep circuit (, cohen et al. 2015 ). by comparison, previous theoretical work made claims regarding only the case where the shallow circuit must exactly replicate particular functions. 15. 6 providing clues to discover underlying causes to close this chapter, we come back to one of our original questions : what makes one representation better than another? one answer, ﬁrst introduced in section, is 15. 3 that an ideal representation is one that disentangles the underlying causal factors of variation that generated the data, especially those factors that are relevant to our applications. most strategies for representation learning are based on introducing clues that help the learning to ﬁnd these underlying factors of variations. the clues can help the learner separate these observed factors from the others. supervised learning provides a very strong clue : a label y, presented with each x, that usually speciﬁes the value of at least one of the factors of variation directly. more generally, to make use of abundant unlabeled data, representation learning makes use of other, less direct, hints about the underlying factors. these hints take the form of implicit prior beliefs that we, the designers of	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	569	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
factors of variation directly. more generally, to make use of abundant unlabeled data, representation learning makes use of other, less direct, hints about the underlying factors. these hints take the form of implicit prior beliefs that we, the designers of the learning algorithm, impose in order to guide the learner. results such as the no free lunch theorem show that regularization strategies are necessary to obtain good generalization. while it is impossible to ﬁnd a universally superior regularization strategy, one goal of deep learning is to ﬁnd a set of fairly generic regularization strategies that are applicable to a wide variety of ai tasks, similar to the tasks that people and animals are able to solve. 554	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	569	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15. representation learning we provide here a list of these generic regularization strategies. the list is clearly not exhaustive, but gives some concrete examples of ways that learning algorithms can be encouraged to discover features that correspond to underlying factors. this list was introduced in section 3. 1 of ( ) and has bengio et al. 2013d been partially expanded here. • smoothness : this is the assumption that f ( x + d ) ≈f ( x ) for unit d and small. this assumption allows the learner to generalize from training examples to nearby points in input space. many machine learning algorithms leverage this idea, but it is [UNK] to overcome the curse of dimensionality. • linearity : many learning algorithms assume that relationships between some variables are linear. this allows the algorithm to make predictions even very far from the observed data, but can sometimes lead to overly extreme predictions. most simple machine learning algorithms that do not make the smoothness assumption instead make the linearity assumption. these are in fact [UNK] assumptions — linear functions with large weights applied to high - dimensional spaces may not be very smooth. see goodfellow et al. ( ) for a further discussion of the limitations of the linearity assumption. 2014b • multiple explanatory factors : many representation	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	570	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
functions with large weights applied to high - dimensional spaces may not be very smooth. see goodfellow et al. ( ) for a further discussion of the limitations of the linearity assumption. 2014b • multiple explanatory factors : many representation learning algorithms are motivated by the assumption that the data is generated by multiple underlying explanatory factors, and that most tasks can be solved easily given the state of each of these factors. section describes how this view motivates semi - 15. 3 supervised learning via representation learning. learning the structure of p ( x ) requires learning some of the same features that are useful for modeling p ( y \| x ) because both refer to the same underlying explanatory factors. section 15. 4 describes how this view motivates the use of distributed representations, with separate directions in representation space corresponding to separate factors of variation. • causal factors : the model is constructed in such a way that it treats the factors of variation described by the learned representation h as the causes of the observed data x, and not vice - versa. as discussed in section, this 15. 3 is advantageous for semi - supervised learning and makes the learned model more robust when the distribution over the underlying causes changes or when we use the model for a	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	570	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
observed data x, and not vice - versa. as discussed in section, this 15. 3 is advantageous for semi - supervised learning and makes the learned model more robust when the distribution over the underlying causes changes or when we use the model for a new task. • depth a hierarchical organization of explanatory factors, or : high - level, abstract concepts can be deﬁned in terms of simple concepts, forming a hierarchy. from another point of view, the use of a deep architecture 555	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	570	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15. representation learning expresses our belief that the task should be accomplished via a multi - step program, with each step referring back to the output of the processing accomplished via previous steps. • shared factors across tasks : in the context where we have many tasks, corresponding to [UNK] yi variables sharing the same input x or where each task is associated with a subset or a function f ( ) i ( x ) of a global input x, the assumption is that each yi is associated with a [UNK] subset from a common pool of relevant factors h. because these subsets overlap, learning all the p ( yi \| x ) via a shared intermediate representation p ( h x \| ) allows sharing of statistical strength between the tasks. • manifolds : probability mass concentrates, and the regions in which it con - centrates are locally connected and occupy a tiny volume. in the continuous case, these regions can be approximated by low - dimensional manifolds with a much smaller dimensionality than the original space where the data lives. many machine learning algorithms behave sensibly only on this manifold (, ). some machine learning algorithms, especially goodfellow et al. 2014b autoencoders, attempt to explicitly learn the structure of the manifold. • natural clustering : many machine learning	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	571	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
algorithms behave sensibly only on this manifold (, ). some machine learning algorithms, especially goodfellow et al. 2014b autoencoders, attempt to explicitly learn the structure of the manifold. • natural clustering : many machine learning algorithms assume that each connected manifold in the input space may be assigned to a single class. the data may lie on many disconnected manifolds, but the class remains constant within each one of these. this assumption motivates a variety of learning algorithms, including tangent propagation, double backprop, the manifold tangent classiﬁer and adversarial training. • temporal and spatial coherence : slow feature analysis and related algorithms make the assumption that the most important explanatory factors change slowly over time, or at least that it is easier to predict the true underlying explanatory factors than to predict raw observations such as pixel values. see section for further description of this approach. 13. 3 • sparsity : most features should presumably not be relevant to describing most inputs — there is no need to use a feature that detects elephant trunks when representing an image of a cat. it is therefore reasonable to impose a prior that any feature that can be interpreted as “ present ” or “ absent ” should be	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	571	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
describing most inputs — there is no need to use a feature that detects elephant trunks when representing an image of a cat. it is therefore reasonable to impose a prior that any feature that can be interpreted as “ present ” or “ absent ” should be absent most of the time. • simplicity of factor dependencies : in good high - level representations, the factors are related to each other through simple dependencies. the simplest 556	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	571	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 15. representation learning possible is marginal independence, p ( h ) = ip ( hi ), but linear dependencies or those captured by a shallow autoencoder are also reasonable assumptions. this can be seen in many laws of physics, and is assumed when plugging a linear predictor or a factorized prior on top of a learned representation. the concept of representation learning ties together all of the many forms of deep learning. feedforward and recurrent networks, autoencoders and deep probabilistic models all learn and exploit representations. learning the best possible representation remains an exciting avenue of research. 557	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	572	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16 structured probabilistic models for deep learning deep learning draws upon many modeling formalisms that researchers can use to guide their design [UNK] and describe their algorithms. one of these formalisms is the idea of structured probabilistic models. we have already discussed structured probabilistic models brieﬂy in section. that brief presentation was 3. 14 [UNK] to understand how to use structured probabilistic models as a language to describe some of the algorithms in part. now, in part, structured probabilistic ii iii models are a key ingredient of many of the most important research topics in deep learning. in order to prepare to discuss these research ideas, this chapter describes structured probabilistic models in much greater detail. this chapter is intended to be self - contained ; the reader does not need to review the earlier introduction before continuing with this chapter. a structured probabilistic model is a way of describing a probability distribution, using a graph to describe which random variables in the probability distribution interact with each other directly. here we use “ graph ” in the graph theory sense — a set of vertices connected to one another by a set of edges. because the structure of the model is deﬁned by a graph, these models are often also referred	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	573	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
each other directly. here we use “ graph ” in the graph theory sense — a set of vertices connected to one another by a set of edges. because the structure of the model is deﬁned by a graph, these models are often also referred to as graphical models. the graphical models research community is large and has developed many [UNK] models, training algorithms, and inference algorithms. in this chapter, we provide basic background on some of the most central ideas of graphical models, with an emphasis on the concepts that have proven most useful to the deep learning research community. if you already have a strong background in graphical models, you may wish to skip most of this chapter. however, even a graphical model expert 558	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	573	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning may beneﬁt from reading the ﬁnal section of this chapter, section, in which we 16. 7 highlight some of the unique ways that graphical models are used for deep learning algorithms. deep learning practitioners tend to use very [UNK] model structures, learning algorithms and inference procedures than are commonly used by the rest of the graphical models research community. in this chapter, we identify these [UNK] in preferences and explain the reasons for them. in this chapter we ﬁrst describe the challenges of building large - scale proba - bilistic models. next, we describe how to use a graph to describe the structure of a probability distribution. while this approach allows us to overcome many challenges, it is not without its own complications. one of the major [UNK] in graphical modeling is understanding which variables need to be able to interact directly, i. e., which graph structures are most suitable for a given problem. we outline two approaches to resolving this [UNK] by learning about the dependen - cies in section. finally, we close with a discussion of the unique emphasis that 16. 5 deep learning practitioners place on speciﬁc approaches to graphical modeling in section. 16. 7 16. 1 the challenge of unstructured modeling the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	574	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
cies in section. finally, we close with a discussion of the unique emphasis that 16. 5 deep learning practitioners place on speciﬁc approaches to graphical modeling in section. 16. 7 16. 1 the challenge of unstructured modeling the goal of deep learning is to scale machine learning to the kinds of challenges needed to solve artiﬁcial intelligence. this means being able to understand high - dimensional data with rich structure. for example, we would like ai algorithms to be able to understand natural images, 1 audio waveforms representing speech, and documents containing multiple words and punctuation characters. classiﬁcation algorithms can take an input from such a rich high - dimensional distribution and summarize it with a categorical label — what object is in a photo, what word is spoken in a recording, what topic a document is about. the process of classiﬁcation discards most of the information in the input and produces a single output ( or a probability distribution over values of that single output ). the classiﬁer is also often able to ignore many parts of the input. for example, when recognizing an object in a photo, it is usually possible to ignore the background of the photo. it is possible to ask probabilistic models to	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	574	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
classiﬁer is also often able to ignore many parts of the input. for example, when recognizing an object in a photo, it is usually possible to ignore the background of the photo. it is possible to ask probabilistic models to do many other tasks. these tasks are often more expensive than classiﬁcation. some of them require producing multiple output values. most require a complete understanding of the entire structure of 1 a natural image is an image that might be captured by a camera in a reasonably ordinary environment, as opposed to a synthetically rendered image, a screenshot of a web page, etc. 559	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	574	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning the input, with no option to ignore sections of it. these tasks include the following : • density estimation : given an input x, the machine learning system returns an estimate of the true density p ( x ) under the data generating distribution. this requires only a single output, but it does require a complete understand - ing of the entire input. if even one element of the vector is unusual, the system must assign it a low probability. • denoising : given a damaged or incorrectly observed input [UNK], the machine learning system returns an estimate of the original or correct x. for example, the machine learning system might be asked to remove dust or scratches from an old photograph. this requires multiple outputs ( every element of the estimated clean example x ) and an understanding of the entire input ( since even one damaged area will still reveal the ﬁnal estimate as being damaged ). • missing value imputation : given the observations of some elements of x, the model is asked to return estimates of or a probability distribution over some or all of the unobserved elements of x. this requires multiple outputs. because the model could be asked to restore any of the elements of x, it must understand the entire input. • sampling	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	575	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
to return estimates of or a probability distribution over some or all of the unobserved elements of x. this requires multiple outputs. because the model could be asked to restore any of the elements of x, it must understand the entire input. • sampling : the model generates new samples from the distribution p ( x ). applications include speech synthesis, i. e. producing new waveforms that sound like natural human speech. this requires multiple output values and a good model of the entire input. if the samples have even one element drawn from the wrong distribution, then the sampling process is wrong. for an example of a sampling task using small natural images, see ﬁgure. 16. 1 modeling a rich distribution over thousands or millions of random variables is a challenging task, both computationally and statistically. suppose we only wanted to model binary variables. this is the simplest possible case, and yet already it seems overwhelming. for a small, 32 ×32 2 pixel color ( rgb ) image, there are 3072 possible binary images of this form. this number is over 10800 times larger than the estimated number of atoms in the universe. in general, if we wish to model a distribution over a random vector x containing n discrete variables capable of taking on k values each,	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	575	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
images of this form. this number is over 10800 times larger than the estimated number of atoms in the universe. in general, if we wish to model a distribution over a random vector x containing n discrete variables capable of taking on k values each, then the naive approach of representing p ( x ) by storing a lookup table with one probability value per possible outcome requires kn parameters! this is not feasible for several reasons : 560	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	575	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning figure 16. 1 : probabilistic modeling of natural images. ( top ) example 32× 32 pixel color images from the cifar - 10 dataset (, ). samples krizhevsky and hinton 2009 ( bottom ) drawn from a structured probabilistic model trained on this dataset. each sample appears at the same position in the grid as the training example that is closest to it in euclidean space. this comparison allows us to see that the model is truly synthesizing new images, rather than memorizing the training data. contrast of both sets of images has been adjusted for display. figure reproduced with permission from ( ). courville et al. 2011 561	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	576	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning • memory : the cost of storing the representation : for all but very small values of n and k, representing the distribution as a table will require too many values to store. • statistical [UNK] : as the number of parameters in a model increases, so does the amount of training data needed to choose the values of those parameters using a statistical estimator. because the table - based model has an astronomical number of parameters, it will require an astronomically large training set to ﬁt accurately. any such model will overﬁt the training set very badly unless additional assumptions are made linking the [UNK] entries in the table ( for example, like in back - [UNK] smoothed n - gram models, section ). 12. 4. 1 • runtime : the cost of inference : suppose we want to perform an inference task where we use our model of the joint distribution p ( x ) to compute some other distribution, such as the marginal distribution p ( x1 ) or the conditional distribution p ( x2 \| x1 ). computing these distributions will require summing across the entire table, so the runtime of these operations is as high as the intractable memory cost of storing the model. • runtime : the cost of	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	577	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
distribution p ( x2 \| x1 ). computing these distributions will require summing across the entire table, so the runtime of these operations is as high as the intractable memory cost of storing the model. • runtime : the cost of sampling : likewise, suppose we want to draw a sample from the model. the naive way to do this is to sample some value u [UNK] ( 0, 1 ), then iterate through the table, adding up the probability values until they exceed u and return the outcome corresponding to that position in the table. this requires reading through the whole table in the worst case, so it has the same exponential cost as the other operations. the problem with the table - based approach is that we are explicitly modeling every possible kind of interaction between every possible subset of variables. the probability distributions we encounter in real tasks are much simpler than this. usually, most variables inﬂuence each other only indirectly. for example, consider modeling the ﬁnishing times of a team in a relay race. suppose the team consists of three runners : alice, bob and carol. at the start of the race, alice carries a baton and begins running around a track. after completing her lap around the track, she hands the baton to bob. bob then	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	577	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. suppose the team consists of three runners : alice, bob and carol. at the start of the race, alice carries a baton and begins running around a track. after completing her lap around the track, she hands the baton to bob. bob then runs his own lap and hands the baton to carol, who runs the ﬁnal lap. we can model each of their ﬁnishing times as a continuous random variable. alice ’ s ﬁnishing time does not depend on anyone else ’ s, since she goes ﬁrst. bob ’ s ﬁnishing time depends on alice ’ s, because bob does not have the opportunity to start his lap until alice has completed hers. if alice ﬁnishes faster, bob will ﬁnish faster, all else being 562	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	577	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning equal. finally, carol ’ s ﬁnishing time depends on both her teammates. if alice is slow, bob will probably ﬁnish late too. as a consequence, carol will have quite a late starting time and thus is likely to have a late ﬁnishing time as well. however, carol ’ s ﬁnishing time depends only indirectly on alice ’ s ﬁnishing time via bob ’ s. if we already know bob ’ s ﬁnishing time, we will not be able to estimate carol ’ s ﬁnishing time better by ﬁnding out what alice ’ s ﬁnishing time was. this means we can model the relay race using only two interactions : alice ’ s [UNK] on bob and bob ’ s [UNK] on carol. we can omit the third, indirect interaction between alice and carol from our model. structured probabilistic models provide a formal framework for modeling only direct interactions between random variables. this allows the models to have signiﬁcantly fewer parameters and therefore be estimated reliably from less data. these smaller models also have dramatically reduced computational cost in terms of storing the model, performing inference in the model, and drawing samples from the model. 16. 2 using graphs	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	578	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##ﬁcantly fewer parameters and therefore be estimated reliably from less data. these smaller models also have dramatically reduced computational cost in terms of storing the model, performing inference in the model, and drawing samples from the model. 16. 2 using graphs to describe model structure structured probabilistic models use graphs ( in the graph theory sense of “ nodes ” or “ vertices ” connected by edges ) to represent interactions between random variables. each node represents a random variable. each edge represents a direct interaction. these direct interactions imply other, indirect interactions, but only the direct interactions need to be explicitly modeled. there is more than one way to describe the interactions in a probability distribution using a graph. in the following sections we describe some of the most popular and useful approaches. graphical models can be largely divided into two categories : models based on directed acyclic graphs, and models based on undirected graphs. 16. 2. 1 directed models one kind of structured probabilistic model is the directed graphical model, otherwise known as the belief network bayesian network or 2 ( pearl 1985, ). directed graphical models are called “ directed ” because their edges are directed, 2 judea pearl suggested using the term “ bayesian network ” when one wishes to “ emphasize the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	578	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
known as the belief network bayesian network or 2 ( pearl 1985, ). directed graphical models are called “ directed ” because their edges are directed, 2 judea pearl suggested using the term “ bayesian network ” when one wishes to “ emphasize the judgmental ” nature of the values computed by the network, i. e. to highlight that they usually represent degrees of belief rather than frequencies of events. 563	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	578	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning t0t0 t1t1 t2t2 alice bob carol figure 16. 2 : a directed graphical model depicting the relay race example. alice ’ s ﬁnishing time t0 inﬂuences bob ’ s ﬁnishing time t1, because bob does not get to start running until alice ﬁnishes. likewise, carol only gets to start running after bob ﬁnishes, so bob ’ s ﬁnishing time t1 directly inﬂuences carol ’ s ﬁnishing time t2. that is, they point from one vertex to another. this direction is represented in the drawing with an arrow. the direction of the arrow indicates which variable ’ s probability distribution is deﬁned in terms of the other ’ s. drawing an arrow from a to b means that we deﬁne the probability distribution over b via a conditional distribution, with a as one of the variables on the right side of the conditioning bar. in other words, the distribution over b depends on the value of a. continuing with the relay race example from section, suppose we name 16. 1 alice ’ s ﬁnishing time t0, bob ’ s ﬁnishing time t1, and carol ’	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	579	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
, the distribution over b depends on the value of a. continuing with the relay race example from section, suppose we name 16. 1 alice ’ s ﬁnishing time t0, bob ’ s ﬁnishing time t1, and carol ’ s ﬁnishing time t2. as we saw earlier, our estimate of t1 depends on t0. our estimate of t2 depends directly on t1 but only indirectly on t0. we can draw this relationship in a directed graphical model, illustrated in ﬁgure. 16. 2 formally, a directed graphical model deﬁned on variables x is deﬁned by a directed acyclic graph g whose vertices are the random variables in the model, and a set of local conditional probability distributions p ( xi \| pag ( xi ) ) where pag ( xi ) gives the parents of xi in g. the probability distribution over x is given by p ( ) = π x ip ( xi \| pag ( xi ) ). ( 16. 1 ) in our relay race example, this means that, using the graph drawn in ﬁgure, 16. 2 p ( t0, t1, t2 ) = ( p t0 ) ( p t 1 \| t0 ) ( p	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	579	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
) in our relay race example, this means that, using the graph drawn in ﬁgure, 16. 2 p ( t0, t1, t2 ) = ( p t0 ) ( p t 1 \| t0 ) ( p t2 \| t1 ). ( 16. 2 ) this is our ﬁrst time seeing a structured probabilistic model in action. we can examine the cost of using it, in order to observe how structured modeling has many advantages relative to unstructured modeling. suppose we represented time by discretizing time ranging from minute 0 to minute 10 into 6 second chunks. this would make t0, t1 and t2 each be a discrete variable with 100 possible values. if we attempted to represent p ( t0, t1, t2 ) with a table, it would need to store 999, 999 values ( 100 values of t0 × 100 values of t1 × 100 values of t2, minus 1, since the probability of one of the conﬁgurations is made 564	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	579	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning redundant by the constraint that the sum of the probabilities be 1 ). if instead, we only make a table for each of the conditional probability distributions, then the distribution over t0 requires 99 values, the table deﬁning t1 given t0 requires 9900 values, and so does the table deﬁning t2 given t1. this comes to a total of 19, 899 values. this means that using the directed graphical model reduced our number of parameters by a factor of more than 50! in general, to model n discrete variables each having k values, the cost of the single table approach scales like o ( k n ), as we have observed before. now suppose we build a directed graphical model over these variables. if m is the maximum number of variables appearing ( on either side of the conditioning bar ) in a single conditional probability distribution, then the cost of the tables for the directed model scales like o ( km ). as long as we can design a model such that m < < n, we get very dramatic savings. in other words, so long as each variable has few parents in the graph, the distribution can be represented with very few parameters. some restrictions on the graph	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	580	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
as we can design a model such that m < < n, we get very dramatic savings. in other words, so long as each variable has few parents in the graph, the distribution can be represented with very few parameters. some restrictions on the graph structure, such as requiring it to be a tree, can also guarantee that operations like computing marginal or conditional distributions over subsets of variables are [UNK]. it is important to realize what kinds of information can and cannot be encoded in the graph. the graph encodes only simplifying assumptions about which variables are conditionally independent from each other. it is also possible to make other kinds of simplifying assumptions. for example, suppose we assume bob always runs the same regardless of how alice performed. ( in reality, alice ’ s performance probably inﬂuences bob ’ s performance — depending on bob ’ s personality, if alice runs especially fast in a given race, this might encourage bob to push hard and match her exceptional performance, or it might make him overconﬁdent and lazy ). then the only [UNK] alice has on bob ’ s ﬁnishing time is that we must add alice ’ s ﬁnishing time to the total amount of time we think bob needs to run. this observation allows us to deﬁ	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	580	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)