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chapter 18. confronting the partition function we can thus determine whether ma is a better model than mb without knowing the partition function of either model but only their ratio. as we will see shortly, we can estimate this ratio using importance sampling, provided that the two models are similar. if, however, we w... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 639 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
continuous variables using integrals, but it can be readily applied to discrete variables by replacing the integrals with summation. we use a proposal distribution p0 ( x ) = 1 z 0 [UNK] ( x ) which supports tractable sampling and tractable evaluation of both the partition function z0 and the unnormalized distribution ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 639 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##normalized [UNK] and the proposal p0. we see also that this approach allows us to estimate the ratio between the partition functions as 1 k k k = 1 [UNK] ( x ( ) k ) [UNK] ( x ( ) k ) s t :.. x ( ) k [UNK]. ( 18. 45 ) this value can then be used directly to compare two models as described in equation. 18. 39 624 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 639 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function if the distribution p0 is close to p1, equation can be an [UNK] way of 18. 44 estimating the partition function ( minka 2005, ). unfortunately, most of the time p1 is both complicated ( usually multimodal ) and defined over a high dimensional space. it is [UNK] to find a tra... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 640 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
) [UNK] ( x ( ) k ) [UNK] 2. ( 18. 46 ) this quantity is largest when there is significant deviation in the values of the importance weights [UNK] ( x ( ) k ) [UNK] ( x ( ) k ). we now turn to two related strategies developed to cope with the challeng - ing task of estimating partition functions for complex distribution... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 640 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##zynski 1997 neal 2001 ). consider a sequence of distributions pη0,..., pηn, with 0 = η0 < η1 < < · · · ηn−1 < ηn = 1 so that the first and last distributions in the sequence are p0 and p1 respectively. this approach allows us to estimate the partition function of a multimodal distribution defined over a high - dimensio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 640 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function we can now write the ratio z1 z0 as z1 z0 = z1 z0 zη 1 zη 1 · · · zηn−1 zηn−1 ( 18. 47 ) = zη 1 z0 zη 2 zη 1 · · · zηn−1 zηn−2 z1 zηn−1 ( 18. 48 ) = n−1 j = 0 zηj + 1 zηj ( 18. 49 ) provided the distributions pηj and pηj + 1, for all 0 ≤ ≤ − j n 1, are [UNK] close, we can ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 641 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, it can be specifically constructed to suit the problem domain. one general - purpose and popular choice for the intermediate distributions is to use the weighted geometric average of the target distribution p1 and the starting proposal distribution ( for which the partition function is known ) p0 : pη j [UNK] 1 p1−ηj ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 641 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
passes through all of the random variables or other kinds of iterations. the ais sampling strategy is then to generate samples from p0 and then use the transition operators to sequentially generate samples from the intermediate distributions until we arrive at samples from the target distribution p1 : • for k... k = 1 ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 641 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function – sample x ( ) k η2 [UNK] ( x ( ) k η2 | x ( ) k η1 ) –... – sample x ( ) k ηn−1 [UNK] ( x ( ) k ηn−1 | x ( ) k ηn−2 ) – sample x ( ) k ηn [UNK] ( x ( ) k ηn | x ( ) k ηn−1 ) • end for sample k, we can derive the importance weight by chaining together the importance weight... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 642 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
compute log w ( ) k by adding and subtracting log probabilities, rather than computing w ( ) k by multiplying and dividing probabilities. with the sampling procedure thus defined and the importance weights given in equation, the estimate of the ratio of partition functions is given by : 18. 52 z1 z0 ≈1 k k k = 1 w ( ) k... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 642 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
xηn−2 | xηn−1 )... [UNK] 1 ( xη1 | xη2 ), ( 18. 55 ) where [UNK] is the reverse of the transition operator defined by ta ( via an application of bayes ’ rule ) : [UNK] ( x | x ) = pa ( x ) pa ( ) x ta ( x x | ) = [UNK] ( x ) [UNK] ( ) x ta ( x x | ). ( 18. 56 ) plugging the above into the expression for the joint distri... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 642 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function = [UNK] ( x1 ) [UNK] n−1 ( xηn−1 ) [UNK] n−1 ( x1 ) tηn−1 ( x 1 | xηn−1 ) n−2 i = 1 [UNK] ( xηi ) [UNK] ( xηi + 1 ) tη i ( xηi + 1 | xηi ) ( 18. 58 ) = [UNK] ( x1 ) [UNK] ( x1 ) tηn−1 ( x1 | xηn−1 ) [UNK] ( xη 1 ) n−2 i = 1 [UNK] + 1 ( xηi + 1 ) [UNK] ( xηi + 1 ) tη i ( xη... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 643 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##0 ( xη1 ) tη1 ( xη2 | xη1 )... tηn−1 ( x1 | x ηn−1 ). ( 18. 60 ) we have a joint distribution on the extended space given by equation. taking 18. 59 q ( xη1,..., xη n−1, x1 ) as the proposal distribution on the extended state space from which we will draw samples, it remains to determine the importance weights : w ( ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 643 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
are the same as proposed for ais. thus we can interpret ais as simple importance sampling applied to an extended state and its validity follows immediately from the validity of importance sampling. annealed importance sampling ( ais ) was first discovered by ( ) jarzynski 1997 and then again, independently, by ( ). it i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 643 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function intermediate distributions, bridge sampling relies on a single distribution p∗, known as the bridge, to interpolate between a distribution with known partition function, p0, and a distribution p1 for which we are trying to estimate the partition function z1. bridge samplin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 644 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##idging distribution is given by p ( ) opt ∗ ( x ) [UNK] [UNK] ( ) [UNK] x p1 ( ) x [UNK] ( ) + [UNK] x p1 ( ) x where r = z1 / z0. at first, this appears to be an unworkable solution as it would seem to require the very quantity we are trying to estimate, z1 / z0. however, it is possible to start with a coarse estimat... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 644 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
far apart for a single distribution p∗to bridge the gap then one can at least use ais with potentially many intermediate distributions to span the distance between p0 and p1. neal ( ) showed how his linked importance sampling method leveraged the power of 2005 the bridge sampling strategy to bridge the intermediate dis... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 644 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 18. confronting the partition function rbm throughout the training process. the strategy is based on the maintenance of independent estimates of the partition functions of the rbm at every temperature operating in the parallel tempering scheme. the authors combined bridge sampling estimates of the ratios of par... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 645 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19 approximate inference many probabilistic models are [UNK] to train because it is [UNK] to perform inference in them. in the context of deep learning, we usually have a set of visible variables v and a set of latent variables h. the challenge of inference usually refers to the [UNK] problem of computing p ( h... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 646 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. later, in chapter, we will describe how to use 20 these techniques to train probabilistic models that would otherwise be intractable, such as deep belief networks and deep boltzmann machines. intractable inference problems in deep learning usually arise from interactions between latent variables in a structured graph... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 646 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference figure 19. 1 : intractable inference problems in deep learning are usually the result of interactions between latent variables in a structured graphical model. these can be due to edges directly connecting one latent variable to another, or due to longer paths that are activated when t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 647 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
structures depicted above. this is possible if the conditional probability distributions are chosen to introduce additional independences beyond those described by the graph. for example, probabilistic pca has the graph structure shown in the right, yet still has simple inference due to special properties of the specifi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 647 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference 19. 1 inference as optimization many approaches to confronting the problem of [UNK] inference make use of the observation that exact inference can be described as an optimization problem. approximate inference algorithms may then be derived by approximating the underlying optimization ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 648 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
p v θ dkl ( ( ) ( ; ) ) q h v | p h v | θ ( 19. 1 ) where is an arbitrary probability distribution over. q h because the [UNK] between log p ( v ) and l ( v θ,, q ) is given by the kl divergence and because the kl divergence is always non - negative, we can see that l always has at most the same value as the desired lo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 648 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
h v | ( 19. 3 ) = log ( ; ) p v θ [UNK] log q ( ) h v | p, ( h v θ ; ) p ( ; ) v θ ( 19. 4 ) = log ( ; ) p v θ [UNK] [ log ( ) log ( ; ) + log ( ; ) ] q h v | − p h v, θ p v θ ( 19. 5 ) = [UNK] [ log ( ) log ( ; ) ] q h v | − p h v, θ. ( 19. 6 ) this yields the more canonical definition of the evidence lower bound, l ( ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 648 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference approximations of p ( h v | ), the lower bound l will be tighter, in other words, closer to log p ( v ). when q ( h v | ) = p ( h v | ), the approximation is perfect, and l ( ) = log ( ; ) v θ,, q p v θ. we can thus think of inference as the procedure for finding the q that maximizes l.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 649 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
get tighter or looser bounds that are cheaper or more expensive to compute depending on how we choose to approach this optimization problem. we can obtain a poorly matched q but reduce the computational cost by using an imperfect optimization procedure, or by using a perfect optimization procedure over a restricted fam... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 649 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
) i | v ( ) i ; θ ( 0 ) ) for all indices i of the training examples v ( ) i we want to train on ( both batch and minibatch variants are valid ). by this we mean q is defined in terms of the current parameter value of θ ( 0 ) ; if we vary θ then p ( h v | ; θ ) will change but q p ( ) h v | will remain equal to ( ; h v ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 649 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference with respect to using your optimization algorithm of choice. θ this can be viewed as a coordinate ascent algorithm to maximize l. on one step, we maximize l with respect to q, and on the other, we maximize l with respect to. θ stochastic gradient ascent on latent variable models can be... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 650 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the value θ ( 0 ) used in the e - step. fortunately, the e - step reduces the gap to zero again as we enter the loop for the next time. the em algorithm contains a few [UNK] insights. first, there is the basic structure of the learning process, in which we update the model parameters to improve the likelihood of a comp... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 650 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
optimal large m - step update, so this second insight which is more unique to the em algorithm is rarely used. 19. 3 map inference and sparse coding we usually use the term inference to refer to computing the probability distribution over one set of variables given another. when training probabilistic models with laten... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 650 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference of latent variable models, this means computing h∗ = arg max h p. ( ) h v | ( 19. 9 ) this is known as maximum a posteriori inference, abbreviated map inference. map inference is usually not thought of as approximate inference — it does compute the exact most likely value of h∗. howeve... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 651 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
inference by restricting the family of distributions q may be drawn from. specifically, we require to take on a dirac distribution : q q δ. ( ) = h v | ( ) h µ − ( 19. 11 ) this means that we can now control q entirely via µ. dropping terms of l that do not vary with, we are left with the optimization problem µ µ∗ = arg... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 651 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
respect to θ. the procedure as a whole can be justified by the fact that l is a lower bound on log p ( v ). in the case of map inference, this justification is rather vacuous, because the bound is infinitely loose, due to the dirac distribution ’ s [UNK] entropy of negative infinity. however, adding noise to would make the... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 651 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference map inference is commonly used in deep learning as both a feature extractor and a learning mechanism. it is primarily used for sparse coding models. recall from section that sparse coding is a linear factor model that imposes 13. 4 a sparsity - inducing prior on its hidden units. a com... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 652 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
model were gaussian then these interactions could be modeled [UNK] via the covariance matrix, but the sparse prior makes these interactions non - gaussian. because p ( h v | ) is intractable, so is the computation of the log - likelihood and its gradient. we thus cannot use exact maximum likelihood learning. instead, w... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 652 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
large. h w we can minimize j by alternating between minimization with respect to h and minimization with respect to w. both sub - problems are convex. in fact, the minimization with respect to w is just a linear regression problem. however, minimization of j with respect to both arguments is usually not a convex proble... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 652 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference 19. 4 variational inference and learning we have seen how the evidence lower bound l ( v θ,, q ) is a lower bound on log p ( v ; θ ), how inference can be viewed as maximizing l with respect to q, and how learning can be viewed as maximizing l with respect to θ. we have seen that the e... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 653 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
q is a factorial distribution : q ( ) = h v | i q h ( i | v ). ( 19. 17 ) this is called the mean field approach. more generally, we can impose any graphi - cal model structure we choose on q, to flexibly determine how many interactions we want our approximation to capture. this fully general graphical model approach is ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 653 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
used to represent q. calculus of variations is the origin of the names “ variational learning ” and “ variational inference, ” though these names apply even when the latent variables are discrete and calculus of variations is not needed. in the case of continuous latent variables, calculus of variations is a powerful t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 653 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference in this sense, we are fitting q to p. however, we are doing so with the opposite direction of the kl divergence than we are used to using for fitting an approximation. when we use maximum likelihood learning to fit a model to data, we minimize dkl ( pdatapmodel ). as illustrated in figure,... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 654 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
h v | ) p ( h v | ) ) involves evaluating expectations with respect to q, so by designing q to be simple, we can simplify the required expectations. the opposite direction of the kl divergence would require computing expectations with respect to the true posterior. because the form of the true posterior is determined b... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 654 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. after determining how to represent q, we simply optimize its parameters. in the case of discrete latent variables, this is just a standard optimization problem. in principle the selection of q could be done with any optimization algorithm, such as gradient descent. because this optimization must occur in the inner lo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 654 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference criterion. to make this more concrete, we show how to apply variational inference to the binary sparse coding model ( we present here the model developed by henniges et al. ( ) but demonstrate traditional, generic mean field applied to the model, 2010 while they introduce a specialized ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 655 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
adding gaussian noise to the sum of m [UNK] components which can each be present or absent. each component is switched on or [UNK] the corresponding hidden unit in h ∈ { } 0 1, m : p h ( i = 1 ) = ( σ bi ) ( 19. 19 ) p, ( ) = ( ; v h | n v w h β−1 ) ( 19. 20 ) where b is a learnable set of biases, w is a learnable weig... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 655 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference h1 h1 h2 h2 h3 h3 v1 v1 v2 v2 v3 v3 h4 h4 h1 h1 h2 h2 h3 h3 h4 h4 figure 19. 2 : the graph structure of a binary sparse coding model with four hidden units. ( left ) the graph structure of p ( h v, ). note that the edges are directed, and that every two hidden units are co - parents of... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 656 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
p h. ( 19. 27 ) this requires computing expectations with respect to p ( h v | ). unfortunately, p ( h v | ) is a complicated distribution. see figure for the graph structure of 19. 2 p ( h v, ) and p ( h v | ). the posterior distribution corresponds to the complete graph over the hidden units, so variable elimination a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 656 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
v ) = [UNK]. we impose a restriction that [UNK] is never equal to 0 or to 1, in order to avoid errors when computing, for example, log [UNK]. we will see that the variational inference equations never assign or to 0 1 [UNK] 641 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 656 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference analytically. however, in a software implementation, machine rounding error could result in or values. in software, we may wish to implement binary sparse 0 1 coding using an unrestricted vector of variational parameters z and obtain [UNK] via the relation [UNK] = σ ( z ). we can thus ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 657 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
log ( p hi ) + n i = 1 log ( p vi | − h ) m i = 1 log ( q hi | v ) ( 19. 32 ) = m i = 1 [UNK] ( log ( σ bi ) log − [UNK] ) + ( 1 [UNK] ) ( log ( σ −bi ) log ( 1 − [UNK] ) ) ( 19. 33 ) + [UNK] n i = 1 log βi 2π exp −βi 2 ( vi −wi, : h ) 2 ( 19. 34 ) = m i = 1 [UNK] ( log ( σ bi ) log − [UNK] ) + ( 1 [UNK] ) ( log ( σ −b... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 657 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
show that l can be expressed in a small number of simple arithmetic operations. the evidence lower bound l is therefore tractable. we can use l as a replacement for the intractable log - likelihood. in principle, we could simply run gradient ascent on both v and h and this would make a perfectly acceptable combined inf... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 657 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference second, we would like to be able to extract the features [UNK] very quickly, in order to recognize the content of v. in a realistic deployed setting, we would need to be able to compute [UNK] in real time. for both these reasons, we typically do not use gradient descent to compute the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 658 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
by more than some tolerance amount, or when the cycle does not change [UNK] by more than some amount. iterating mean field fixed point equations is a general technique that can provide fast variational inference in a broad variety of models. to make this more concrete, we show how to derive the updates for the binary spa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 658 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, kwj, [UNK] ( 19. 40 ) = log ( σ bi ) log − [UNK] − − 1 + log ( 1 [UNK] i ) + 1 log ( − σ −bi ) ( 19. 41 ) + n j = 1 βj vjwj, i −1 2w 2 j, i − k i = wj, kwj, i [UNK] ( 19. 42 ) 643 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 658 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference = bi −log [UNK] + log ( 1 [UNK] ) + v βw :, i −1 2w :, iβw :, i − j i = w :, j βw :, [UNK]. ( 19. 43 ) to apply the fixed point update inference rule, we solve for the [UNK] that sets equation to 0 : 19. 43 [UNK] = σ bi + vβw :, i −1 2w :, iβw :, i − j i = w :, jβw :, [UNK]. ( 19. 44 ) ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 659 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
this theme are described in chapter. 20 in the case of binary sparse coding, we can see that the recurrent network connection specified by equation consists of repeatedly updating the hidden 19. 44 units based on the changing values of the neighboring hidden units. the input always sends a fixed message of vβw to the hid... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 659 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
have both active. unfortunately, explaining away interactions cannot be modeled by the factorial q used for mean field, so the mean field approximation is forced to choose one mode to model. this is an instance of the behavior illustrated in figure. 3. 6 we can rewrite equation into an equivalent form that reveals some fu... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 659 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference error in v given the code of the other units. we can thus think of sparse coding as an iterative autoencoder, that repeatedly encodes and decodes its input, attempting to fix mistakes in the reconstruction after each iteration. in this example, we have derived an update rule that update... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 660 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
2009 ( ) for more information about choosing the degree of synchrony and damping strategies in message passing algorithms. 19. 4. 2 calculus of variations before continuing with our presentation of variational learning, we must briefly introduce an important set of mathematical tools used in variational learning : calcu... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 660 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
valued argument, we can take functional derivatives, also known as variational derivatives, of a functional j [ f ] with respect to individual values of the function f ( x ) at any specific value of x. the functional derivative of the functional j with respect to the value of the function at point is denoted f x δ δf x ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 660 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference to gain some intuition for this identity, one can think of f ( x ) as being a vector with uncountably many elements, indexed by a real vector x. in this ( somewhat incomplete view ), the identity providing the functional derivatives is the same as we would obtain for a vector θ ∈rn ind... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 661 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
by solving for the function where the functional derivative at every point is equal to zero. as an example of how this process works, consider the problem of finding the probability distribution function over x ∈r that has maximal [UNK] entropy. recall that the entropy of a probability distribution is defined as p x ( ) ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 661 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
fixed variance σ2. finally, the problem is underdetermined because the distribution can be shifted arbitrarily without changing the entropy. to impose a unique solution, we add a constraint that the mean of the distribution be µ. the lagrangian functional for this optimization problem is l [ ] = p λ1 p x dx ( ) −1 + λ2 ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 661 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference = λ1p x λ ( ) + 2p x x λ ( ) + 3p x x µ ( ) ( − ) 2 −p x p x ( ) log ( ) dx λ − 1 −µλ2 −σ2λ3. ( 19. 51 ) to minimize the lagrangian with respect to p, we set the functional derivatives equal to 0 : [UNK], δ δp x ( ) l = λ1 + λ2x λ + 3 ( ) x µ − 2 − − 1 log ( ) = 0 p x. ( 19. 52 ) this ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 662 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a functional. to finish the minimization problem, we must choose the λ values to ensure that all of our constraints are satisfied. we are free to choose any λ values, because the gradient of the lagrangian with respect to the λ variables is zero so long as the constraints are satisfied. to satisfy all of the constraints, ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 662 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
minimizes the entropy? why did we not find a second critical point corresponding to the minimum? the reason is that there is no specific function that achieves minimal entropy. as functions place more probability density on the two points x = µ + σ and x = µ σ −, and place less probability density on all other values of ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 662 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference described as a mixture of dirac distributions. because dirac distributions are not described by a single probability distribution function, no dirac or mixture of dirac distribution corresponds to a single specific point in function space. these distributions are thus invisible to our m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 663 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
) = h v | i q h ( i | v ), ( 19. 55 ) and fix q ( hj | v ) for all j = i, then the optimal q ( h i | v ) may be obtained by normalizing the unnormalized distribution [UNK] h ( i | v ) = exp [UNK] ( h−i | v ) log [UNK], ( v h ) ( 19. 56 ) so long as p does not assign probability to any joint configuration of variables. 0 ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 663 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
more than that. it tells us the functional form that the optimal solution will take, whether we arrive there by fixed point equations or not. this means we can take the functional form from that equation but regard some of the values that appear in it as parameters, that we can optimize with any optimization algorithm w... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 663 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference interesting ; we have constructed it only to provide a simple demonstration of how calculus of variations may be applied to probabilistic modeling. the true posterior is given, up to a normalizing constant, by p ( ) h v | ( 19. 57 ) [UNK], ( h v ) ( 19. 58 ) = ( p h1 ) ( p h2 ) ( ) p v... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 664 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
not factorize over h1 and h2. applying equation, we find that 19. 56 [UNK] h ( 1 | v ) ( 19. 62 ) = exp [UNK] ( h2 | v ) log [UNK], ( v h ) ( 19. 63 ) = exp −1 [UNK] ( h2 | v ) h2 1 + h2 2 + v2 + h2 1w2 1 + h2 2w2 2 ( 19. 64 ) −2vh1w1 −2vh2w2 + 2h 1w1h2w2 ]. ( 19. 65 ) from this, we can see that there are [UNK] only two... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 664 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
+ h2 2 + v2 + h2 1w2 1 + h2 2w2 2 ( 19. 66 ) −2vh1w1 − 2v h2w2 + 2h 1w1h2w2 ]. ( 19. 67 ) from this, we can see that [UNK] has the functional form of a gaussian. we can thus conclude q ( h v | ) = n ( h ; µ β, −1 ) where µ and diagonal β are variational parameters that we can optimize using any technique we choose. it ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 664 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference respect to l. using the same approach on a [UNK] model could yield a [UNK] functional form of. q this was of course, just a small case constructed for demonstration purposes. for examples of real applications of variational learning with continuous variables in the context of deep lear... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 665 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
have low probability under. q ( ) h v | this behavior causes our approximating assumptions to become self - fulfilling prophecies. if we train the model with a unimodal approximate posterior, we will obtain a model with a true posterior that is far closer to unimodal than we would have obtained by training the model wit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 665 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
variational approximation, we would need to know θ∗ = maxθ log p ( v ; θ ). it is possible for l ( v θ,, q ) ≈log p ( v ; θ ) and log p ( v ; θ ) log p ( v ; θ∗ ) to hold simultaneously. if maxql ( v θ, ∗, q ) log p ( v ; θ∗ ), because θ∗induces too complicated of a posterior distribution for our q family to capture, t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 665 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference 19. 5 learned approximate inference we have seen that inference can be thought of as an optimization procedure that increases the value of a function l. explicitly performing optimization via iterative procedures such as fixed point equations or gradient - based optimization is often ve... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 666 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
from v to h depends on the choice of model family, and evolves throughout the learning process as θ changes. the wake - sleep algorithm ( hinton 1995b frey 1996 et al., ; et al., ) resolves this problem by drawing samples of both h and v from the model distribution. for example, in a directed model, this can be done ch... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 666 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
algorithms use to approximate the negative gradient of the log partition function of undirected models. another possible explanation for biological dreaming is that it is providing samples from p ( h v, ) which can be used to train an inference network to predict h given v. in some senses, this explanation is more sati... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 666 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference not readily apparent how this schedule could support monte carlo training of an undirected model. learning algorithms based on maximizing l can be run with prolonged periods of improving q and prolonged periods of improving θ, however. if the role of biological dreaming is to train net... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 667 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
learning community. 19. 5. 2 other forms of learned inference this strategy of learned approximate inference has also been applied to other models. salakhutdinov and larochelle 2010 ( ) showed that a single pass in a learned inference network could yield faster inference than iterating the mean field fixed point equation... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 667 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to implement the kind of competition between units that we have seen in mean field inference. however, that problem can be remedied by training a deep encoder to perform learned approximate inference, as in the ista technique (, ). gregor and lecun 2010b learned approximate inference has recently become one of the domin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 667 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 19. approximate inference using approximate inference, it is possible to train and use a wide variety of models. many of these models are described in the next chapter. 653 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 668 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 20 deep generative models in this chapter, we present several of the specific kinds of generative models that can be built and trained using the techniques presented in chapters –. all of 16 19 these models represent probability distributions over multiple variables in some way. some allow the probability distri... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 669 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
., ;, ). variants of the boltzmann machine that include other kinds of variables have long ago surpassed the popularity of the original. in this section we briefly introduce the binary boltzmann machine and discuss the issues that come up when trying to train and perform inference in the model. we define the boltzmann ma... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 669 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 20. deep generative models meaning we define the joint probability distribution using an energy function : p ( ) = x exp ( ( ) ) −e x z, ( 20. 1 ) where e ( x ) is the energy function and z is the partition function that ensures that x p ( ) = 1 x. the energy function of the boltzmann machine is given by e ( ) =... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 670 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
model ( logistic regression ) from the values of the other units. the boltzmann machine becomes more powerful when not all the variables are observed. in this case, the latent variables, can act similarly to hidden units in a multi - layer perceptron and model higher - order interactions among the visible units. just a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 670 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
3 ) boltzmann machine learning learning algorithms for boltzmann machines are usually based on maximum likelihood. all boltzmann machines have an intractable partition function, so the maximum likelihood gradient must be ap - proximated using the techniques described in chapter. 18 one interesting property of boltzmann... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 670 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 20. deep generative models network participates in shaping those statistics, but the weight can be updated without knowing anything about the rest of the network or how those statistics were produced. this means that the learning rule is “ local, ” which makes boltzmann machine learning somewhat biologically pl... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 671 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
local statistics seem to require us to hypothesize the existence of more machinery than this. for example, for the brain to implement back - propagation in a multilayer perceptron, it seems necessary for the brain to maintain a secondary communication network for transmitting gradient information backwards through the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 671 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
1986 machines are some of the most common building blocks of deep probabilistic models. we have briefly described rbms previously, in section. here we review the 16. 7. 1 previous information and go into more detail. rbms are undirected probabilistic graphical models containing a layer of observable variables and a sing... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 671 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 20. deep generative models h1 h1 h2 h2 h3 h3 v1 v1 v2 v2 v3 v3 h4 h4 h ( 1 ) 1 h ( 1 ) 1 h ( 1 ) 2 h ( 1 ) 2 h ( 1 ) 3 h ( 1 ) 3 v1 v1 v2 v2 v3 v3 h ( 2 ) 1 h ( 2 ) 1 h ( 2 ) 2 h ( 2 ) 2 h ( 2 ) 3 h ( 2 ) 3 h ( 1 ) 4 h ( 1 ) 4 ( a ) ( b ) h ( 1 ) 1 h ( 1 ) 1 h ( 1 ) 2 h ( 1 ) 2 h ( 1 ) 3 h ( 1 ) 3 v1 v1 v2 v2 v... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 672 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
c ) figure 20. 1 : examples of models that may be built with restricted boltzmann machines. ( a ) the restricted boltzmann machine itself is an undirected graphical model based on a bipartite graph, with visible units in one part of the graph and hidden units in the other part. there are no connections among the visibl... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 672 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a completely undirected graph, but it would need intralayer connections to capture the dependencies between parents. a deep boltzmann machine is an undirected graphical ( c ) model with several layers of latent variables. like rbms and dbns, dbms lack intralayer connections. dbms are less closely tied to rbms than dbns... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 672 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 20. deep generative models we begin with the binary version of the restricted boltzmann machine, but as we see later there are extensions to other types of visible and hidden units. more formally, let the observed layer consist of a set of n v binary random variables which we refer to collectively with the vect... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 673 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
}. ( 20. 6 ) it is apparent from the definition of the partition function z that the naive method of computing z ( exhaustively summing over all states ) could be computationally intractable, unless a cleverly designed algorithm could exploit regularities in the probability distribution to compute z faster. in the case ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 673 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 20. deep generative models = 1 z exp nh j = 1 cjhj + nh j = 1 vw :, jhj ( 20. 10 ) = 1 z nh j = 1 exp cjhj + vw :, jhj ( 20. 11 ) since we are conditioning on the visible units v, we can treat these as constant with respect to the distribution p ( h v | ). the factorial nature of the conditional p ( h v | ) fol... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 674 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##w :, j exp 0 + exp { } { cj + vw :, j } ( 20. 13 ) = σ cj + vw :, j. ( 20. 14 ) we can now express the full conditional over the hidden layer as the factorial distribution : p ( ) = h v | nh j = 1 σ ( 2 1 ) ( + h − c wv ) j. ( 20. 15 ) a similar derivation will show that the other condition of interest to us, p ( v h... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 674 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
models 18 that have intractable partition functions. this includes cd, sml ( pcd ), ratio matching and so on. compared to other undirected models used in deep learning, the rbm is relatively straightforward to train because we can compute p ( h | v ) 659 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 674 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 20. deep generative models exactly in closed form. some other deep models, such as the deep boltzmann machine, combine both the [UNK] of an intractable partition function and the [UNK] of intractable inference. 20. 3 deep belief networks deep belief networks ( dbns ) were one of the first non - convolutional mod... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 675 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
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