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chapter 16. structured probabilistic models for deep learning the model until it is just barely possible to train or use. we often use models whose marginal distributions cannot be computed, and are satisfied simply to draw approximate samples from these models. we often train models with an intractable objective functi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 602 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
single layer of latent variables that may be used to learn a representation for the input. in chapter, we will 20 see how rbms can be used to build many deeper models. here, we show how the rbm exemplifies many of the practices used in a wide variety of deep graphical models : its units are organized into large groups c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 602 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##ined, real - valued, learnable parameters. we can see that the model is divided into two groups of units : v and h, and the interaction between them is described by a matrix w. the model is depicted graphically in figure. as this figure makes clear, an important aspect of this model is 16. 14 that there are no direct i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 602 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 16. structured probabilistic models for deep learning h1h1 h2h2 h3h3 v1 v1 v2 v2 v3 v3 h4h4 figure 16. 14 : an rbm drawn as a markov network. and p ( ) = π v h | ip ( vi | h ). ( 16. 12 ) the individual conditionals are simple to compute as well. for the binary rbm we obtain : p ( hi = 1 ) = | v σ vw :, i + bi,... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 603 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
function of the parameters, it is easy to take its derivatives. for example, ∂ ∂wi, j e, ( v h ) = −vihj. ( 16. 15 ) these two properties — [UNK] gibbs sampling and [UNK] derivatives — make training convenient. in chapter, we will see that undirected models may be 18 trained by computing such derivatives applied to sam... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 603 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 16. structured probabilistic models for deep learning figure 16. 15 : samples from a trained rbm, and its weights. image reproduced with permission from ( ). lisa 2008 ( left ) samples from a model trained on mnist, drawn using gibbs sampling. each column is a separate gibbs sampling process. each row represent... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 604 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17 monte carlo methods randomized algorithms fall into two rough categories : las vegas algorithms and monte carlo algorithms. las vegas algorithms always return precisely the correct answer ( or report that they failed ). these algorithms consume a random amount of resources, usually memory or time. in contras... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 605 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods integrals at reduced cost. sometimes we use this to provide a significant speedup to a costly but tractable sum, as in the case when we subsample the full training cost with minibatches. in other cases, our learning algorithm requires us to approximate an intractable sum or integral, such... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 606 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
x ( ) = x p [ ( ) ] f x ( 17. 1 ) or s = p f d e ( ) x ( ) x x = p [ ( ) ] f x ( 17. 2 ) be the sum or integral to estimate, rewritten as an expectation, with the constraint that p is a probability distribution ( for the sum ) or a probability density ( for the integral ) over random variable. x we can approximate s by... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 606 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods provided that the variance of the individual terms, var [ f ( x ( ) i ) ], is bounded. to see this more clearly, consider the variance of [UNK] as n increases. the variance var [ [UNK] ] decreases and converges to 0, so long as var [ ( f x ( ) i ) ] < ∞ : var [ [UNK] ] = 1 n2 n i = 1 var... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 607 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
normal distribution with mean s and variance var [ ( ) ] f x n. this allows us to estimate confidence intervals around the estimate [UNK], using the cumulative distribution of the normal density. however, all this relies on our ability to easily sample from the base distribution p ( x ), but doing so is not always possi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 607 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
under that probability distribution ) is to be estimated. there is no unique decomposition because p ( x ) f ( x ) can always be rewritten as p f q ( ) x ( ) = x ( ) x p f ( ) x ( ) x q ( ) x, ( 17. 8 ) where we now sample from q and average pf q. in many cases, we wish to compute an expectation for a given p and an f,... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 607 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods from the start as an expectation suggests that this p and f would be a natural choice of decomposition. however, the original specification of the problem may not be the the optimal choice in terms of the number of samples required to obtain a given level of accuracy. fortunately, the for... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 608 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
[UNK] ] = eq [ [UNK] ] = s. ( 17. 11 ) however, the variance of an importance sampling estimator can be greatly sensitive to the choice of. the variance is given by q var [ [UNK] ] = var [ p f ( ) x ( ) x q ( ) x ] / n. ( 17. 12 ) the minimum variance occurs when is q q∗ ( ) = x p f ( ) x | ( ) x | z, ( 17. 13 ) where ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 608 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
single sample from the optimal distribution. any choice of sampling distribution q is valid ( in the sense of yielding the correct expected value ) and q∗is the optimal one ( in the sense of yielding minimum variance ). sampling from q∗is usually infeasible, but other choices of q can be feasible while still reducing t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 608 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods another approach is to use biased importance sampling, which has the advantage of not requiring normalized p or q. in the case of discrete variables, the biased importance sampling estimator is given by [UNK] = n i = 1 p ( x ( ) i ) q ( x ( ) i ) f ( x ( ) i ) n i = 1 p ( x ( ) i ) q ( x... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 609 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
p and q and the x ( ) i are the samples from q. this estimator is biased because e [ [UNK] bis ] = s, except asymptotically when n →∞and the denominator of equation converges to 1. hence this estimator 17. 14 is called asymptotically unbiased. although a good choice of q can greatly improve the [UNK] of monte carlo est... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 609 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
in q causes it to match p or p f | | poorly. when q ( x ( ) i ) p ( x ( ) i ) | f ( x ( ) i ) |, importance sampling collects useless samples ( summing tiny numbers or zeros ). on the other hand, when q ( x ( ) i ) p ( x ( ) i ) | f ( x ( ) i ) |, which will happen more rarely, the ratio can be huge. because these latt... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 609 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods distribution ) in section, and to estimate the log - likelihood in deep directed 18. 7 models such as the variational autoencoder, in section. importance sampling 20. 10. 3 may also be used to improve the estimate of the gradient of the cost function used to train model parameters with s... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 610 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
undirected model. in these cases, we introduce a mathematical tool called a markov chain to approximately sample from pmodel ( x ). the family of algorithms that use markov chains to perform monte carlo estimates is called markov chain monte carlo methods ( mcmc ). markov chain monte carlo methods for machine learning ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 610 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
on a case - by - case basis for [UNK] families of such distributions. in the context of deep learning, it is most common to rely on the most general theoretical guarantees that naturally apply to all energy - based models. to understand why drawing samples from an energy - based model is [UNK], consider an ebm over jus... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 610 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods of obtaining a sample. in an ebm, we can avoid this chicken and egg problem by sampling using a markov chain. the core idea of a markov chain is to have a state x that begins as an arbitrary value. over time, we randomly update x repeatedly. eventually x becomes ( very nearly ) a fair sa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 611 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
values of x map back to [UNK] states in the original problem. x consider what happens when we run infinitely many markov chains in parallel. all of the states of the [UNK] markov chains are drawn from some distribution q ( ) t ( x ), where t indicates the number of time steps that have elapsed. at the beginning, q ( 0 )... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 611 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the probability of a single state landing in state xis given by q ( + 1 ) t ( x ) = x q ( ) t ( ) ( x t x | x. ) ( 17. 18 ) using our integer parametrization, we can represent the [UNK] of the transition operator using a matrix. we define so that t a a ai, j = ( t x = = ) i | x j. ( 17. 19 ) using this definition, we can... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 611 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods applying the markov chain update repeatedly corresponds to multiplying by the matrix a repeatedly. in other words, we can think of the process as exponentiating the matrix : a v ( ) t = atv ( 0 ). ( 17. 21 ) the matrix a has special structure because each of its columns represents a prob... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 612 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the eigenvalues that are not equal to to decay to zero. 1 under some additional mild conditions, a is guaranteed to have only one eigenvector with eigenvalue. the process thus converges to a 1 stationary distribution, sometimes also called the. at convergence, equilibrium distribution v = = av v, ( 17. 23 ) and this sa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 612 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
with countable states can be generalized to continuous variables. in this situation, some authors call the markov chain a harris chain but we use the term markov chain to describe both conditions. in general, a markov chain with transition operator t will converge, under mild conditions, to a fixed point described by th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 612 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods regardless of whether the state is continuous or discrete, all markov chain methods consist of repeatedly applying stochastic updates until eventually the state begins to yield samples from the equilibrium distribution. running the markov chain until it reaches its equilibrium distributi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 613 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
sample after reaching equilibrium. if one desires truly independent samples, one can run multiple markov chains in parallel. this approach uses extra parallel computation to eliminate latency. the strategy of using only a single markov chain to generate all samples and the strategy of using one markov chain for each de... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 613 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
point of view of a matrix a acting on a vector of probabilities v, then we know that the chain mixes when at has [UNK] lost all of the eigenvalues from a besides the unique eigenvalue of. 1 this means that the magnitude of the second largest eigenvalue will determine the mixing time. however, in practice, we cannot act... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 613 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods successive samples. 17. 4 gibbs sampling so far we have described how to draw samples from a distribution q ( x ) by repeatedly updating x x ← [UNK] ( x | x ). however, we have not described how to ensure that q ( x ) is a useful distribution. two basic approaches are considered in this ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 614 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. to obtain the desired q ( ) x, we must choose an appropriate t ( x | x ). a conceptually simple and [UNK] approach to building a markov chain that samples from pmodel ( x ) is to use gibbs sampling, in which sampling from t ( x | x ) is accomplished by selecting one variable xi and sampling it from pmodel conditioned... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 614 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
called block gibbs sampling. alternate approaches to designing markov chains to sample from pmodel are possible. for example, the metropolis - hastings algorithm is widely used in other disciplines. in the context of the deep learning approach to undirected modeling, it is rare to use any approach other than gibbs samp... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 614 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods from p ( x ) would be completely independent from each other and would visit many [UNK] regions in x space proportional to their probability. instead, especially in high dimensional cases, mcmc samples become very correlated. we refer to such behavior as slow mixing or even failure to mi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 615 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##guration ( higher energy than the typical ones from p ( x ) ), the chain tends to gradually reduce the energy of the state and only occasionally move to another mode. once the chain has found a region of low energy ( for example, if the variables are pixels in an image, a region of low energy might be a connected man... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 615 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
that probability is the shape of the “ energy barrier ” between these modes. transitions between two modes that are separated by a high energy barrier ( a region of low probability ) are exponentially less likely ( in terms of the height of the energy barrier ). this is illustrated in figure. the problem arises when the... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 615 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
= σ ( w ). if w is large, the sigmoid saturates, and the probability of also assigning b to be 1 is close to 1. likewise, if a = −1, the probability of assigning b to be −1 is close to 1. according to pmodel ( a b, ), both signs of both variables are equally likely. 600 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 615 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods figure 17. 1 : paths followed by gibbs sampling for three distributions, with the markov chain initialized at the mode in both cases. ( left ) a multivariate normal distribution with two independent variables. gibbs sampling mixes well because the variables are independent. a multivariat... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 616 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
greater because we care not only about making transitions between two modes but more generally between all the many modes that a real model might contain. if several such transitions are [UNK] because of the [UNK] of mixing between modes, then it becomes very expensive to obtain a reliable set of samples covering most ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 616 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods figure 17. 2 : an illustration of the slow mixing problem in deep probabilistic models. each panel should be read left to right, top to bottom. ( left ) consecutive samples from gibbs sampling applied to a deep boltzmann machine trained on the mnist dataset. consecutive samples are simil... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 617 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
it well, which implies that h and x should have very high mutual information. these two goals are at odds with each other. we often learn generative models that very precisely encode x into h but are not able to mix very well. this situation arises frequently with boltzmann machines — the sharper the distribution a bol... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 617 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods 17. 5. 1 tempering to mix between modes when a distribution has sharp peaks of high probability surrounded by regions of low probability, it is [UNK] to mix between the [UNK] modes of the distribution. several techniques for faster mixing are based on constructing alternative versions of... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 618 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
and rises to infinity, the energy - based model becomes β deterministic. when the temperature rises to infinity and β falls to zero, the distribution ( for discrete ) becomes uniform. x typically, a model is trained to be evaluated at β = 1. however, we can make use of other temperatures, particularly those where β < 1. ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 618 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the model. the transition operator includes stochastically swapping states between two [UNK] temperature levels, so that a [UNK] high - probability sample from a high - temperature slot can jump into a lower temperature slot. this approach has also been applied to rbms ( desjardins et al. et al., ; 2010 cho, ). althoug... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 618 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 17. monte carlo methods 17. 5. 2 depth may help mixing when drawing samples from a latent variable model p ( h x, ), we have seen that if p ( h x | ) encodes x too well, then sampling from p ( x h | ) will not change x very much and mixing will be poor. one way to resolve this problem is to make h be a deep rep... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 619 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
bengio 2013a et al. ( ) observed that deeper stacks of regularized autoencoders or rbms yield marginal distributions in the top - level h - space that appeared more spread out and more uniform, with less of a gap between the regions corresponding to [UNK] modes ( categories, in the experiments ). training an rbm in tha... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 619 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18 confronting the partition function in section we saw that many probabilistic models ( commonly known as undi - 16. 2. 2 rected graphical models ) are defined by an unnormalized probability distribution [UNK] ( x ; θ ). we must normalize [UNK] by dividing by a partition function z ( θ ) in order to obtain a va... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 620 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function 18. 1 the log - likelihood gradient what makes learning undirected models by maximum likelihood particularly [UNK] is that the partition function depends on the parameters. the gradient of the log - likelihood with respect to the parameters has a term corresponding to the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 621 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
primarily covered in chapter. this chapter focuses 19 on the [UNK] of the negative phase. let us look more closely at the gradient of : log z ∇θ log z ( 18. 5 ) = ∇θz z ( 18. 6 ) = ∇θ x [UNK] ( ) x z ( 18. 7 ) = x [UNK] ( ) x z. ( 18. 8 ) for models that guarantee p ( x ) > 0 for all x, we can substitute exp ( log [UNK... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 621 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function = ex x [UNK] ( ) ∇θ log [UNK]. ( ) x ( 18. 13 ) this derivation made use of summation over discrete x, but a similar result applies using integration over continuous x. in the continuous version of the derivation, we use leibniz ’ s rule for [UNK] under the integral sign t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 622 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( x ) | ≤r ( x ) for all θ and almost all x. fortunately, most machine learning models of interest have these properties. this identity ∇θ log = z ex x [UNK] ( ) ∇θ log [UNK] ( ) x ( 18. 15 ) is the basis for a variety of monte carlo methods for approximately maximizing the likelihood of models with intractable partiti... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 622 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
samples drawn from the model, as illustrated in figure. 18. 1 18. 2 stochastic maximum likelihood and contrastive divergence the naive way of implementing equation is to compute it by burning in 18. 15 a set of markov chains from a random initialization every time the gradient is needed. when learning is performed using... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 622 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function training procedure presented in algorithm. the high cost of burning in the 18. 1 markov chains in the inner loop makes this procedure computationally infeasible, but this procedure is the starting point that other more practical algorithms aim to approximate. algorithm 18.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 623 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
., from a uniform or normal distribution, or possibly a distribution with marginals matched to the model ’ s marginals ). for do i k = 1 to for do j m = 1 to [UNK] ( ) j ←gibbs _ update ( [UNK] ( ) j ). end for end for g g ← − 1 m m i = 1 ∇θ log [UNK] ( [UNK] ( ) i ; ) θ. θ θ ← +. g end while we can view the mcmc appro... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 623 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
model ’ s distri - bution, we can think of it as finding points that the model believes in strongly. because the negative phase acts to reduce the probability of those points, they are generally considered to represent the model ’ s incorrect beliefs about the world. they are frequently referred to in the literature as ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 623 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function x p ( x ) the positive phase pmodel ( ) x pdata ( ) x x p ( x ) the negative phase pmodel ( ) x pdata ( ) x figure 18. 1 : the view of algorithm as having a “ positive phase ” and “ negative phase. ” 18. 1 ( left ) in the positive phase, we sample points from the data dist... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 624 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
gradient ( in expectation ) and training must terminate. for dreaming in humans and other animals ( crick and mitchison 1983, ), the idea being that the brain maintains a probabilistic model of the world and follows the gradient of log [UNK] while experiencing real events while awake and follows the negative gradient o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 624 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a less expensive alternative to algorithm. 18. 1 the main cost of the naive mcmc algorithm is the cost of burning in the markov chains from a random initialization at each step. a natural solution is to initialize the markov chains from a distribution that is very close to the model distribution, 609 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 624 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function so that the burn in operation does not take as many steps. the contrastive divergence ( cd, or cd - k to indicate cd with k gibbs steps ) algorithm initializes the markov chain at each step with samples from the data distribution ( hinton 2000 2010,, ). this approach is pr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 625 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to allow a markov chain sampling from p ( x ; θ ) to mix when initialized from pdata. perhaps 1 - 20 to train an rbm on a small image patch. while not converged do sample a minibatch of examples m { x ( 1 ),..., x ( ) m } from the training set. g ← 1 m m i = 1 ∇θ log [UNK] ( x ( ) i ; ) θ. for do i m = 1 to [UNK] ( ) i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 625 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
is that it fails to suppress regions of high probability that are far from actual training examples. these regions that have high probability under the model but low probability under the data generating distribution are called spurious modes. figure illustrates why this happens. essentially, it is because modes in the... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 625 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function x p ( x ) pmodel ( ) x pdata ( ) x figure 18. 2 : an illustration of how the negative phase of contrastive divergence ( algo - rithm ) can fail to suppress spurious modes. a spurious mode is a mode that is 18. 2 present in the model distribution but absent in the data dist... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 626 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
mode along the number line in r. this corresponds to a markov chain based on making local moves with a single x variable in r. for most deep probabilistic models, the markov chains are based on gibbs sampling and can make non - local moves of individual variables but cannot move all of the variables simultaneously. for... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 626 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##mc methods. bengio and delalleau 2009 ( ) showed that cd can be interpreted as discarding the smallest terms of the correct mcmc update gradient, which explains the bias. cd is useful for training shallow models like rbms. these can in turn be stacked to initialize deeper models like dbns or dbms. however, cd does no... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 626 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function to obtain samples of the hidden units given samples of the visible units. since the hidden units are not included in the data, initializing from training points cannot solve the problem. even if we initialize the visible units from the data, we will still need to burn in a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 627 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##d design advantage. sutskever and tieleman 2010 ( ) showed that the cd update direction is not the gradient of any function. this allows for situations where cd could cycle forever, but in practice this is not a serious problem. a [UNK] strategy that resolves many of the problems with cd is to initial - ize the marko... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 627 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
from the current step. it follows that the samples from the previous model ’ s distribution will be very close to being fair samples from the current model ’ s distribution, so a markov chain initialized with these samples will not require much time to mix. because each markov chain is continually updated throughout th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 627 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function marlin 2010 et al. ( ) compared sml to many of the other criteria presented in this chapter. they found that sml results in the best test set log - likelihood for an rbm, and that if the rbm ’ s hidden units are used as features for an svm classifier, sml results in the bes... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 628 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
model trained on mnist might sample exclusively 7s on one step. the learning process will then push down strongly on the mode corresponding to 7s, and the model might sample exclusively 9s on the next step. algorithm 18. 3 the stochastic maximum likelihood / persistent contrastive divergence algorithm using gradient as... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 628 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
do sample a minibatch of examples m { x ( 1 ),..., x ( ) m } from the training set. g ← 1 m m i = 1 ∇θ log [UNK] ( x ( ) i ; ) θ. for do i k = 1 to for do j m = 1 to [UNK] ( ) j ←gibbs _ update ( [UNK] ( ) j ). end for end for g g ← − 1 m m i = 1 ∇θ log [UNK] ( [UNK] ( ) i ; ) θ. θ θ ← +. g end while care must be taken... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 628 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function initialized from a random starting point after the model is done training. the samples present in the persistent negative chains used for training have been influenced by several recent versions of the model, and thus can make the model appear to have greater capacity than ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 629 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of the enhanced mcmc techniques described in chapter, such as parallel tempering (, 17 desjardins et al. 2010 cho 2010 ; et al., ). one approach to accelerating mixing during learning relies not on changing the monte carlo sampling technology but rather on changing the parametrization of the model and the cost function... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 629 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to change. typically one also applies significant weight decay to the fast weights, encouraging them to converge to small values, after only transiently taking on large values long enough to encourage the markov chain to change modes. one key benefit to the mcmc - based methods described in this section is that they prov... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 629 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function incompatible with bound - based positive phase methods. 18. 3 pseudolikelihood monte carlo approximations to the partition function and its gradient directly confront the partition function. other approaches sidestep the issue, by training the model without computing the p... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 630 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##nd the conditional distribution over, b contains the variables we want to condition on, and c contains the variables that are not part of our query. p ( ) = a b | p, ( a b ) p ( ) b = p, ( a b ) a c, p,, ( a b c ) = [UNK], ( a b ) a c, [UNK],, ( a b c ). ( 18. 18 ) this quantity requires marginalizing out a, which ca... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 630 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( p x p x1 ) + log ( p x2 | x1 ) + + ( · · · p xn | x1 : 1 n− ). ( 18. 19 ) in this case, we have made a maximally small, but c can be as large as x2 : n. what if we simply move c into b to reduce the computational cost? this yields the pseudolikelihood (, ) objective function, based on predicting the value besag 1975 ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 630 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function if each random variable has k [UNK] values, this requires onlyk n × evaluations of [UNK] to compute, as opposed to the kn evaluations needed to compute the partition function. this may look like an unprincipled hack, but it can be proven that estimation by maximizing the p... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 631 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
1 ) = 1,..., n the generalized pseudolikelihood recovers the log - likelihood. in the extreme case of m = n and s ( ) i = { } i, the generalized pseudolikelihood recovers the pseudolikelihood. the generalized pseudolikelihood objective function is given by m i = 1 log ( p xs ( ) i | x−s ( ) i ). ( 18. 21 ) the performa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 631 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
while leaving out groups of variables that only have negligible correlation. for example, in natural images, pixels that are widely separated in space also have weak correlation, so the generalized pseudolikelihood can be applied with each set being a small, spatially localized window. s one weakness of the pseudolikel... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 631 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function that interact with each other. however, pseudolikelihood is still useful for deep learning, because it can be used to train single layer models, or deep models using approximate inference methods that are not based on lower bounds. pseudolikelihood has a much greater cost ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 632 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the asymptotic [UNK] of pseudolikelihood. 18. 4 score matching and ratio matching score matching (, ) provides another consistent means of training a hyvarinen 2005 model without estimating z or its derivatives. the name score matching comes from terminology in which the derivatives of a log density with respect to its... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 632 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the [UNK] associated with [UNK] the partition function z because z is not a function of x and therefore ∇xz = 0. initially, score matching appears to have a new [UNK] : computing the score of the data distribution requires knowledge of the true distribution generating the training data, pdata. fortunately, minimizing t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 632 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function equivalent to minimizing the expected value of [UNK], ( x θ ) = n j = 1 ∂2 ∂x2 j log pmodel ( ; ) + x θ 1 2 ∂ ∂xj log pmodel ( ; ) x θ 2 ( 18. 25 ) where is the dimensionality of. n x because score matching requires taking derivatives with respect to x, it is not applicabl... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 633 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
deep boltzmann machines. while score matching can be used to pretrain the first hidden layer of a larger model, it has not been applied as a pretraining strategy for the deeper layers of a larger model. this is probably because the hidden layers of such models usually contain some discrete variables. while score matchin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 633 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
not work in high dimensional discrete spaces where the observed probability of many events is 0. a more successful approach to extending the basic ideas of score matching to discrete data is ratio matching (, ). ratio matching applies hyvarinen 2007b specifically to binary data. ratio matching consists of minimizing the... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 633 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function where returns with the bit at position flipped. ratio matching avoids f, j ( x ) x j the partition function using the same trick as the pseudolikelihood estimator : in a ratio of two probabilities, the partition function cancels out. ( ) marlin et al. 2010 found that ratio ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 634 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the data. ratio matching can also be useful as the basis for dealing with high - dimensional sparse data, such as word count vectors. this kind of data poses a challenge for mcmc - based methods because the data is extremely expensive to represent in dense format, yet the mcmc sampler does not yield sparse values until... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 634 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
27 ) rather than the true pdata. the distribution q ( x y | ) is a corruption process, usually one that forms by adding a small amount of noise to. x y denoising score matching is especially useful because in practice we usually do not have access to the true pdata but rather only an empirical distribution defined by sa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 634 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function described in section. ( ) introduced a procedure for 5. 4. 5 kingma and lecun 2010 performing regularized score matching with the smoothing distribution q being normally distributed noise. recall from section that several autoencoder training algorithms are 14. 5. 1 equiva... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 635 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
x pmodel ( ; ) + x θ c, ( 18. 28 ) where c is explicitly introduced as an approximation of −log z ( θ ). rather than estimating only θ, the noise contrastive estimation procedure treats c as just another parameter and estimates θ and c simultaneously, using the same algorithm for both. the resulting log pmodel ( x ) th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 635 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
way that maximum likelihood estimation in this supervised 1nce is also applicable to problems with a tractable partition function, where there is no need to introduce the extra parameter c. however, it has generated the most interest as a means of estimating models with [UNK] partition functions. 620 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 635 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function learning problem defines an asymptotically consistent estimator of the original problem. specifically, we introduce a second distribution, the noise distribution pnoise ( x ). the noise distribution should be tractable to evaluate and to sample from. we can now construct a m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 636 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
from the noise distribution. formally, ptrain ( y = 1 ) = 1 2, ptrain ( x | y = 1 ) = pdata ( x ), and ptrain ( = 0 ) = x | y pnoise ( ) x. we can now just use standard maximum likelihood learning on the supervised learning problem of fitting pjoint to ptrain : θ, c = arg max θ, c ex, p [UNK] log pjoint ( ) y | x. ( 18.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 636 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
) x ( 18. 35 ) = σ −log pnoise ( ) x pmodel ( ) x ( 18. 36 ) = ( log σ pmodel ( ) log x − pnoise ( ) ) x. ( 18. 37 ) 621 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 636 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function nce is thus simple to apply so long as log [UNK] is easy to back - propagate through, and, as specified above, pnoise is easy to evaluate ( in order to evaluate pjoint ) and sample from ( in order to generate the training data ). nce is most successful when applied to probl... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 637 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the basic marginal statistics. imagine learning a model of images of faces, using unstructured gaussian noise as pnoise. if pmodel learns about eyes, it can reject almost all unstructured noise samples without having learned anything about other facial features, such as mouths. the constraint that pnoise must be easy t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 637 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
an upper bound on pjoint ( = 1 ) y | x. when the model distribution is copied to define a new noise distribution before each gradient step, nce defines a procedure called self - contrastive estimation, whose expected gradient is equivalent to the expected gradient of maximum likelihood (, ). the special case of nce where... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 637 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function noise contrastive estimation is based on the idea that a good generative model should be able to distinguish data from noise. a closely related idea is that a good generative model should be able to generate samples that no classifier can distinguish from data. this idea yi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 638 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
) = 1 za [UNK] ( x ; θa ) and model mb defining a probability distribution pb ( x ; θb ) = 1 zb [UNK] ( x ; θb ). a common way to compare the models is to evaluate and compare the likelihood that both models assign to an i. i. d. test dataset. suppose the test set consists of m examples { x ( 1 ),..., x ( ) m }. if i pa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 638 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
knowledge of the partition function. unfortunately, equation seems to require evaluating the log probability that 18. 38 the model assigns to each point, which in turn requires evaluating the partition function. we can simplify the situation slightly by re - arranging equation 18. 38 into a form where we need to know o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 638 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
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