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z log [UNK] y yz ( ) = ( 6. 20 ) [UNK] y yz ( ) = exp ( ) ( 6. 21 ) p y ( ) = exp ( ) yz 1 y = 0 exp ( yz ) ( 6. 22 ) p y σ y z. ( ) = ( ( 2 −1 ) ) ( 6. 23 ) probability distributions based on exponentiation and normalization are common throughout the statistical modeling literature. the z variable defining such a distr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 198 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##rized by a sigmoid is j p y ( ) = log θ − ( | x ) ( 6. 24 ) = log ( ( 2 1 ) ) − σ y − z ( 6. 25 ) = ( ( 1 2 ) ) ζ −y z. ( 6. 26 ) this derivation makes use of some properties from section. by rewriting 3. 10 the loss in terms of the softplus function, we can see that it saturates only when ( 1 −2y ) z is very negativ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 198 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks ( 1 −2y ) z, may be simplified to | | z. as | | z becomes large while z has the wrong sign, the softplus function asymptotes toward simply returning its argument | | z. the derivative with respect to z asymptotes to sign ( z ), so, in the limit of extremely incorrect z, the softplus ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 199 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
this reason, maximum likelihood is almost always the preferred approach to training sigmoid output units. analytically, the logarithm of the sigmoid is always defined and finite, because the sigmoid returns values restricted to the open interval ( 0, 1 ), rather than using the entire closed interval of valid probabilitie... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 199 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
probability distribution over a binary variable. softmax functions are most often used as the output of a classifier, to represent the probability distribution over n [UNK] classes. more rarely, softmax functions can be used inside the model itself, if we wish the model to choose between one of n [UNK] options for some ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 199 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks to generalize to the case of a discrete variable with n values, we now need to produce a vector [UNK], with [UNK] = p ( y = i | x ). we require not only that each element of [UNK] be between and, but also that the entire vector sums to so that 0 1 1 it represents a valid probability... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 200 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
exp function works very well when training the softmax to output a target value y using maximum log - likelihood. in this case, we wish to maximize log p ( y = i ; z ) = log softmax ( z ) i. defining the softmax in terms of exp is natural because the log in the log - likelihood can undo the of the softmax : exp log soft... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 200 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, log j exp ( zj ), observe that this term can be roughly approximated by maxj zj. this approximation is based on the idea that exp ( zk ) is insignificant for any zk that is noticeably less than maxj zj. the intuition we can gain from this approximation is that the negative log - likelihood cost function always strongl... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 200 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks the fraction of counts of each outcome observed in the training set : softmax ( ( ; ) ) z x θ i ≈ m j = 1 1y ( ) j = i, x ( ) j = x m j = 1 1x ( ) j = x. ( 6. 31 ) because maximum likelihood is a consistent estimator, this is guaranteed to happen so long as the model family is capab... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 201 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
change its output, even when the model makes highly confident incorrect predictions (, bridle 1990 ). to understand why these other loss functions can fail, we need to examine the softmax function itself. like the sigmoid, the softmax activation can saturate. the sigmoid function has a single output that saturates when ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 201 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
can derive a numerically stable variant of the softmax : softmax ( ) = softmax ( max z z − i zi ). ( 6. 33 ) the reformulated version allows us to evaluate softmax with only small numerical errors even when z contains extremely large or extremely negative numbers. ex - amining the numerically stable variant, we see tha... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 201 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks can cause similar [UNK] for learning if the loss function is not designed to compensate for it. the argument z to the softmax function can be produced in two [UNK] ways. the most common is simply to have an earlier layer of the neural network output every element of z, as described ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 202 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
= softmax ( z ) 1 with a two - dimensional z and z1 = 0. both the n −1 argument and the n argument approaches to the softmax can describe the same set of probability distributions, but have [UNK] learning dynamics. in practice, there is rarely much [UNK] between using the overparametrized version or the restricted vers... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 202 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
and the others are nearly 0 ). the name “ softmax ” can be somewhat confusing. the function is more closely related to the arg max function than the max function. the term “ soft ” derives from the fact that the softmax function is continuous and [UNK]. the arg max function, with its result represented as a one - hot v... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 202 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks a good cost function for nearly any kind of output layer. in general, if we define a conditional distribution p ( y x | ; θ ), the principle of maximum likelihood suggests we use as our cost function. − | log ( p y x θ ; ) in general, we can think of the neural network as representin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 203 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
computationally more expensive approach that does not require writing special - case code is to simply include the variance as one of the properties of the distribution p ( y | x ) that is controlled by ω = f ( x ; θ ). the negative log - likelihood −log p ( y ; ω ( x ) ) will then provide a cost function with the appr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 203 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
variance be one of the values output by f ( x ; θ ). a typical way to do this is to formulate the gaussian distribution using precision, rather than variance, as described in equation. 3. 22 in the multivariate case it is most common to use a diagonal precision matrix diag ( 6. 34 ) ( ) β. this formulation works well w... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 203 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks regardless of whether we use standard deviation, variance, or precision, we must ensure that the covariance matrix of the gaussian is positive definite. because the eigenvalues of the precision matrix are the reciprocals of the eigenvalues of the covariance matrix, this is equivalent... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 204 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
if the covariance is full and conditional, then a parametrization must be chosen that guarantees positive - definiteness of the predicted covariance matrix. this can be achieved by writing σ ( ) = ( ) x b x b ( ) x, whereb is an unconstrained square matrix. one practical issue if the matrix is full rank is that computin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 204 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
1994 neural networks with gaussian mixtures as their output are often called mixture density networks. a gaussian mixture output with n components is defined by the conditional probability distribution p ( ) = y x | n i = 1 p i ( = c | n x ) ( ; y µ ( ) i ( ) x, σ ( ) i ( ) ) x. ( 6. 35 ) the neural network must have th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 204 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks typically be obtained by a softmax over an n - dimensional vector, to guarantee that these outputs are positive and sum to 1. 2. means µ ( ) i ( x ) : these indicate the center or mean associated with the i - th gaussian component, and are unconstrained ( typically with no nonlinear... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 205 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
: these specify the covariance matrix for each component i. as when learning a single gaussian component, we typically use a diagonal matrix to avoid needing to compute determinants. as with learning the means of the mixture, maximum likelihood is complicated by needing to assign partial responsibility for each point t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 205 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##tive models of speech ( schuster 1999, ) or movements of physical objects ( graves 2013, ). the mixture density strategy gives a way for the network to represent multiple output modes and to control the variance of its output, which is crucial for obtaining a high degree of quality in these real - valued domains. an ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 205 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks x y figure 6. 4 : samples drawn from a neural network with a mixture density output layer. the input x is sampled from a uniform distribution and the output y is sampled from pmodel ( y x | ). the neural network is able to learn nonlinear mappings from the input to the parameters of... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 206 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
are common to most parametric machine learning models trained with gradient - based optimization. now we turn to an issue that is unique to feedforward neural networks : how to choose the type of hidden unit to use in the hidden layers of the model. the design of hidden units is an extremely active area of research and... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 206 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks describe here some of the basic intuitions motivating each type of hidden units. these intuitions can help decide when to try out each of these units. it is usually impossible to predict in advance which will work best. the design process consists of trial and error, intuiting that ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 207 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
its value significantly, as shown in figure. these ideas will be described further in chapter. because we do not 4. 3 8 expect training to actually reach a point where the gradient is 0, it is acceptable for the minima of the cost function to correspond to points with undefined gradient. hidden units that are not [UNK] ar... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 207 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
0 is and the right derivative is. software implementations of neural network training usually return one of 1 the one - sided derivatives rather than reporting that the derivative is undefined or raising an error. this may be heuristically justified by observing that gradient - based optimization on a digital computer is... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 207 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 6. 3. 1 rectified linear units and their generalizations rectified linear units use the activation function. g z, z ( ) = max 0 { } rectified linear units are easy to optimize because they are so similar to linear units. the only [UNK] between a linear unit and a rectified linear unit i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 208 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
g x b + ). ( 6. 36 ) when initializing the parameters of the [UNK] transformation, it can be a good practice to set all elements of b to a small, positive value, such as 0. 1. this makes it very likely that the rectified linear units will be initially active for most inputs in the training set and allow the derivatives ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 208 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##i ) + αi min ( 0, zi ). absolute value rectification fixes αi = −1 to obtain g ( z ) = | | z. it is used for object recognition from images (, ), where it makes sense to seek features that are jarrett et al. 2009 invariant under a polarity reversal of the input illumination. other generalizations of rectified linear uni... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 208 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks one of these groups : g ( ) z i = max j∈g ( ) i zj ( 6. 37 ) where g ( ) i is the set of indices into the inputs for group i, { ( i −1 ) k + 1,..., ik }. this provides a way of learning a piecewise linear function that responds to multiple directions in the input space. x a maxout u... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 209 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
learn to implement a totally [UNK] function altogether. the maxout layer will of course be parametrized [UNK] from any of these other layer types, so the learning dynamics will be [UNK] even in the cases where maxout learns to implement the same function of as one of the other layer types. x each maxout unit is now par... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 209 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
because each unit is driven by multiple filters, maxout units have some redun - dancy that helps them to resist a phenomenon called catastrophic forgetting in which neural networks forget how to perform tasks that they were trained on in the past (, ). goodfellow et al. 2014a rectified linear units and all of these gener... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 209 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks architectures, the lstm, propagates information through time via summation — a particular straightforward kind of such linear activation. this is discussed further in section. 10. 10 6. 3. 2 logistic sigmoid and hyperbolic tangent prior to the introduction of rectified linear units, ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 210 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
value when z is very negative, and are only strongly sensitive to their input when z is near 0. the widespread saturation of sigmoidal units can make gradient - based learning very [UNK]. for this reason, their use as hidden units in feedforward networks is now discouraged. their use as output units is compatible with ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 210 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
training the network easier. tanh sigmoidal activation functions are more common in settings other than feed - forward networks. recurrent networks, many probabilistic models, and some autoencoders have additional requirements that rule out the use of piecewise linear activation functions and make sigmoidal units more ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 210 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 6. 3. 3 other hidden units many other types of hidden units are possible, but are used less frequently. in general, a wide variety of [UNK] functions perform perfectly well. many unpublished activation functions perform just as well as the popular ones. to provide a concrete example... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 211 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
that have appeared in the literature. we highlight a few especially useful and distinctive ones. one possibility is to not have an activation g ( z ) at all. one can also think of this as using the identity function as the activation function. we have already seen that a linear unit can be useful as the output of a neu... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 211 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. if u produces q outputs, then u and v together contain only ( n + p ) q parameters, while w contains np parameters. for small q, this can be a considerable saving in parameters. it comes at the cost of constraining the linear transformation to be low - rank, but these low - rank relationships are often [UNK]. linear ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 211 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks a few other reasonably common hidden unit types include : • radial basis function or rbf unit : hi = exp −1 σ 2 i | | w :, i − | | x 2. this function becomes more active as x approaches a template w :, i. because it saturates to for most, it can be [UNK] to optimize. 0 x • softplus ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 212 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
very counterintuitive — one might expect it to have an advantage over the rectifier due to being [UNK] everywhere or due to saturating less completely, but empirically it does not. • hard tanh : this is shaped similarly to the tanh and the rectifier but unlike the latter, it is bounded, g ( a ) = max ( −1, min ( 1, a ) )... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 212 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the first layer is given by h ( 1 ) = g ( 1 ) w ( 1 ) x b + ( 1 ), ( 6. 40 ) the second layer is given by h ( 2 ) = g ( 2 ) w ( 2 ) h ( 1 ) + b ( 2 ), ( 6. 41 ) and so on. 197 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 212 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks in these chain - based architectures, the main architectural considerations are to choose the depth of the network and the width of each layer. as we will see, a network with even one hidden layer is [UNK] to fit the training set. deeper networks often are able to use far fewer units... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 213 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
want to learn. fortunately, feedforward networks with hidden layers provide a universal approxi - mation framework. specifically, the universal approximation theorem ( hornik et al., ;, ) states that a feedforward network with a linear output 1989 cybenko 1989 layer and at least one hidden layer with any “ squashing ” a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 213 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. a neural network may also approximate any function mapping from any finite dimensional discrete space to another. while the original theorems were first stated in terms of units with activation functions that saturate both for very negative and for very positive arguments, universal approximation theorems have also bee... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 213 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks may not be able to find the value of the parameters that corresponds to the desired function. second, the training algorithm might choose the wrong function due to overfitting. recall from section that the “ no free lunch ” theorem shows that 5. 2. 1 there is no universally superior m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 214 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
hidden units ( possibly with one hidden unit corresponding to each input configuration that needs to be distinguished ) may be required. this is easiest to see in the binary case : the number of possible binary functions on vectors v ∈ { 0, 1 } n is 22 n and selecting one such function requires 2n bits, which will in ge... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 214 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. such results were first proved for models that do not resemble the continuous, [UNK] neural networks used for machine learning, but have since been extended to these models. the first results were for circuits of logic gates (, ). later hastad 1986 work extended these results to linear threshold units with non - negati... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 214 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks ( ) showed that functions representable with a deep rectifier net can require 2014 an exponential number of hidden units with a shallow ( one hidden layer ) network. more precisely, they showed that piecewise linear networks ( which can be obtained from rectifier nonlinearities or max... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 215 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of deeper rectifier networks formally by ( ). montufar et al. 2014 ( left ) an absolute value rectification unit has the same output for every pair of mirror points in its input. the mirror axis of symmetry is given by the hyperplane defined by the weights and bias of the unit. a function computed on top of that unit ( th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 215 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, and units per hidden layer, is l n o n d d l ( −1 ) nd, ( 6. 42 ) i. e., exponential in the depth. in the case of maxout networks with filters per l k unit, the number of linear regions is o k ( 1 ) + l− d. ( 6. 43 ) 200 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 215 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks of course, there is no guarantee that the kinds of functions we want to learn in applications of machine learning ( and in particular for ai ) share such a property. we may also want to choose a deep model for statistical reasons. any time we choose a specific machine learning algori... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 216 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
outputs are not necessarily factors of variation, but can instead be analogous to counters or pointers that the network uses to organize its internal processing. empirically, greater depth does seem to result in better generalization for a wide variety of tasks (, ;, ;, ; bengio et al. 2007 erhan et al. 2009 bengio 200... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 216 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
architectural considerations so far we have described neural networks as being simple chains of layers, with the main considerations being the depth of the network and the width of each layer. in practice, neural networks show considerably more diversity. many neural network architectures have been developed for specifi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 216 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 3 4 5 6 7 8 9 10 11 92 0. 92 5. 93 0. 93 5. 94 0. 94 5. 95 0. 95 5. 96 0. 96 5. test accuracy ( percent ) figure 6. 6 : empirical results showing that deeper networks generalize better when used to transcribe multi - digit numbers from photographs of addresses. data from goodfellow ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 217 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
layer. these strategies for reducing the number of connections reduce the number of parameters and the amount of computation required to evaluate the network, but are often highly problem - dependent. for example, convolutional networks, described in chapter, use specialized patterns of sparse connections 9 that are ve... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 217 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 0 0 0 2 0 4 0 6 0 8 1 0...... number of parameters ×108 91 92 93 94 95 96 97 test accuracy ( percent ) 3, convolutional 3, fully connected 11, convolutional figure 6. 7 : deeper models tend to perform better. this is not merely because the model is larger. this experiment from goodf... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 218 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of functions the model can learn. specifically, it expresses a belief that the function should consist of many simpler functions composed together. this could result either in learning a representation that is composed in turn of simpler representations ( e. g., corners defined in terms of edges ) or in learning a progra... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 218 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 6. 5 back - propagation and other [UNK] algo - rithms when we use a feedforward neural network to accept an input x and produce an output [UNK], information flows forward through the network. the inputs x provide the initial information that then propagates up to the hidden units at ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 219 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. actually, back - propagation refers only to the method for computing the gradient, while another algorithm, such as stochastic gradient descent, is used to perform learning using this gradient. furthermore, back - propagation is often misunderstood as being specific to multi - layer neural networks, but in principle i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 219 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. the back - propagation algorithm can be applied to these tasks as well, and is not restricted to computing the gradient of the cost function with respect to the parameters. the idea of computing derivatives by propagating information through a network is very general, and can be used to compute values such as the jac... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 219 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 6. 5. 1 computational graphs so far we have discussed neural networks with a relatively informal graph language. to describe the back - propagation algorithm more precisely, it is helpful to have a more precise language. computational graph many ways of formalizing computation as gr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 220 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
operations with multiple outputs, but we avoid this case in our description because it introduces many extra details that are not important to conceptual understanding. if a variable y is computed by applying an operation to a variable x, then we draw a directed edge from x to y. we sometimes annotate the output node w... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 220 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( g ( x ) ) = f ( y ). then the chain rule states that dz dx = dz dy dy dx. ( 6. 44 ) we can generalize this beyond the scalar case. suppose that x ∈rm, y ∈rn, 205 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 220 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks z x y ( a ) × x w ( b ) u ( 1 ) u ( 1 ) dot b u ( 2 ) u ( 2 ) + [UNK] σ ( c ) x w u ( 1 ) u ( 1 ) matmul b u ( 2 ) u ( 2 ) + h relu x w ( d ) [UNK] dot λ u ( 1 ) u ( 1 ) sqr u ( 2 ) u ( 2 ) sum u ( 3 ) u ( 3 ) × figure 6. 8 : examples of computational graphs. the graph using the ( a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 221 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
which computes a design matrix of rectified linear unit activations h given a design matrix containing a minibatch of inputs x. examples a – c applied at most one operation to each variable, but it ( d ) is possible to apply more than one operation. here we show a computation graph that applies more than one operation t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 221 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks g maps from rm to rn, and f maps from rn to r. if y = g ( x ) and z = f ( y ), then ∂z ∂xi = j ∂z ∂yj ∂yj ∂xi. ( 6. 45 ) in vector notation, this may be equivalently written as ∇xz = ∂y ∂x ∇y z, ( 6. 46 ) where ∂y ∂x is the jacobian matrix of. n m × g from this we see that the gradi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 222 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
are arranged in a grid to form a tensor. we could imagine flattening each tensor into a vector before we run back - propagation, computing a vector - valued gradient, and then reshaping the gradient back into a tensor. in this rearranged view, back - propagation is still just multiplying jacobians by gradients. to denot... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 222 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. if and, then y x = ( g ) z f = ( ) y ∇x z = j ( ∇xyj ) ∂z ∂yj. ( 6. 47 ) 6. 5. 3 recursively applying the chain rule to obtain backprop using the chain rule, it is straightforward to write down an algebraic expression for the gradient of a scalar with respect to any node in the computational graph that produced that ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 222 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks will need to choose whether to store these subexpressions or to recompute them several times. an example of how these repeated subexpressions arise is given in figure. in some cases, computing the same subexpression twice would simply 6. 9 be wasteful. for complicated graphs, there c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 223 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##ification of the computational graph for computing the back - propagation. however, this formulation does not make explicit the manipulation and the construction of the symbolic graph that performs the gradient computation. such a formulation is presented below in section, with algorithm, where we also generalize to 6... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 223 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to the parameters of the model. we will assume that the nodes of the graph have been ordered in such a way that we can compute their output one after the other, starting at u ( ni + 1 ) and going up to u ( ) n. as defined in algorithm, each node 6. 1 u ( ) i is associated with an operation f ( ) i and is computed by eva... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 223 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks algorithm 6. 1 a procedure that performs the computations mapping ni inputs u ( 1 ) to u ( ni ) to an output u ( ) n. this defines a computational graph where each node computes numerical value u ( ) i by applying a function f ( ) i to the set of arguments a ( ) i that comprises the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 224 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( ) i ←f ( ) i ( a ( ) i ) end for return u ( ) n using the chain rule with respect to scalar output u ( ) n : ∂u ( ) n ∂u ( ) j = i j p a u : ∈ ( ( ) i ) ∂u ( ) n ∂u ( ) i ∂u ( ) i ∂u ( ) j ( 6. 49 ) as specified by algorithm. the subgraph 6. 2 b contains exactly one edge for each edge from node u ( ) j to node u ( ) i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 224 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( ) i. to summarize, the amount of computation required for performing the back - propagation scales linearly with the number of edges in g, where the computation for each edge corresponds to computing a partial derivative ( of one node with respect to one of its parents ) as well as performing one multiplication and o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 224 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks algorithm 6. 2 simplified version of the back - propagation algorithm for computing the derivatives of u ( ) n with respect to the variables in the graph. this example is intended to further understanding by showing a simplified case where all variables are scalars, and we wish to com... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 225 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
this example ) to obtain the activa - 6. 1 tions of the network initialize grad _ table, a data structure that will store the derivatives that have been computed. the entry grad table _ [ u ( ) i ] will store the computed value of ∂u ( ) n ∂u ( ) i. grad table _ [ u ( ) n ] 1 ← for do j n = −1 down to 1 the next line c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 225 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
ni } back - propagation thus avoids the exponential explosion in repeated subexpressions. however, other algorithms may be able to avoid more subexpressions by performing simplifications on the computational graph, or may be able to conserve memory by recomputing rather than storing some subexpressions. we will revisit ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 225 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks z x y w f f f figure 6. 9 : a computational graph that results in repeated subexpressions when computing the gradient. let w ∈r be the input to the graph. we use the same function f : r r → as the operation that we apply at every step of a chain : x = f ( w ), y = f ( x ), z = f ( y... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 226 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. this is the approach taken by the back - propagation algorithm. an alternative approach is suggested by equation, where the subexpression 6. 53 f ( w ) appears more than once. in the alternative approach, f ( w ) is recomputed each time it is needed. when the memory required to store the value of these expressions is... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 226 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks applying the back - propagation algorithm to this graph. algorithms and are demonstrations that are chosen to be simple and 6. 3 6. 4 straightforward to understand. however, they are specialized to one specific problem. modern software implementations are based on the generalized for... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 227 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
for simplicity, this demonstration uses only a single input example x. practical applications should use a minibatch. see section for a more realistic demonstration. 6. 5. 7 require : network depth, l require : w ( ) i, i,..., l, ∈ { 1 } the weight matrices of the model require : b ( ) i, i,..., l, ∈ { 1 } the bias par... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 227 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##fic values. these algebraic and graph - based representations are called symbolic representations. when we actually use or train a neural network, we must assign specific values to these symbols. we replace a symbolic input to the network x with a specific numeric value, such as [ 1 2 3 765 1 8 ].,., −.. 212 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 227 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks algorithm 6. 4 backward computation for the deep neural network of algo - rithm, which uses in addition to the input 6. 3 x a target y. this computation yields the gradients on the activations a ( ) k for each layer k, starting from the output layer and going backwards to the first h... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 228 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
into a gradient into the pre - nonlinearity activation ( element - wise multiplication if is element - wise ) : f g ←∇a ( ) k j f = g ( a ( ) k ) compute gradients on weights and biases ( including the regularization term, where needed ) : ∇b ( ) k j λ = + g ∇b ( ) k ω ( ) θ ∇w ( ) k j = g h ( 1 ) k− + λ∇w ( ) k ω ( ) ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 228 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks z x y w f f f z x y w f f f dz dy dz dy f dy dx dy dx f dz dx dz dx × dx dw dx dw f dz dw dz dw × figure 6. 10 : an example of the symbol - to - symbol approach to computing derivatives. in this approach, the back - propagation algorithm does not need to ever access any actual speci... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 229 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
do not explain how the back - propagation algorithm works. the purpose is only to illustrate what the desired result is : a computational graph with a symbolic description of the derivative. some approaches to back - propagation take a computational graph and a set of numerical values for the inputs to the graph, then ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 229 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
described in the same language as the original expression. because the derivatives are just another computational graph, it is possible to run back - propagation again, [UNK] the derivatives in order to obtain higher derivatives. computation of higher - order derivatives is described in section. 6. 5. 10 we will use th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 229 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks terms of constructing a computational graph for the derivatives. any subset of the graph may then be evaluated using specific numerical values at a later time. this allows us to avoid specifying exactly when each operation should be computed. instead, a generic graph evaluation engin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 230 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
1. we can then compute the gradient with respect to each parent of z in the graph by multiplying the current gradient by the jacobian of the operation that produced z. we continue multiplying by jacobians traveling backwards through the graph in this way until we reach x. for any node that may be reached by going backw... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 230 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
operation, and the get _ operation function. suppose we have a variable that is created by matrix multiplication, c = ab. then get operation _ ( v ) returns a pointer to an instance of the corresponding c + + class. • get consumers _ ( v, g ) : this returns the list of variables that are children of v in the computatio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 230 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks each operation op is also associated with a bprop operation. this bprop operation can compute a jacobian - vector product as described by equation. 6. 47 this is how the back - propagation algorithm is able to achieve great generality. each operation is responsible for knowing how t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 231 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, then the matrix operation is responsible for implementing the bprop method and specifying that the desired gradient is given by ag. the back - propagation algorithm itself does not need to know any [UNK] rules. it only needs to call each operation ’ s bprop rules with the right arguments. formally, op bprop inputs. (... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 231 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
bprop method should still return x as the derivative with respect to both inputs. the back - propagation algorithm will later add both of these arguments together to obtain 2x, which is the correct total derivative on. x software implementations of back - propagation usually provide both the opera - tions and their bpr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 231 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks algorithm 6. 5 the outermost skeleton of the back - propagation algorithm. this portion does simple setup and cleanup work. most of the important work happens in the subroutine of algorithm build _ grad 6. 6. require : t, the target set of variables whose gradients must be computed.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 232 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
have exponential runtime due to these repeated subexpressions. now that we have specified the back - propagation algorithm, we can understand its computational cost. if we assume that each operation evaluation has roughly the same cost, then we may analyze the computational cost in terms of the number of operations exec... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 232 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
can see that computing the gradient requires as most o ( n2 ) operations because the forward propagation stage will at worst execute all n nodes in the original graph ( depending on which values we want to compute, we may not need to execute the entire graph ). the back - propagation algorithm adds one jacobian - vecto... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 232 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks algorithm 6. 6 the inner loop subroutine build grad _ ( v,, g g, grad table _ ) of the back - propagation algorithm, called by the back - propagation algorithm defined in algorithm. 6. 5 require : v, the variable whose gradient should be added to and. g grad _ table require : g, the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 233 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
1 end for g ← ig ( ) i grad table _ [ ] = v g insert and the operations creating it into g g return g roughly chain - structured, causing back - propagation to have o ( n ) cost. this is far better than the naive approach, which might need to execute exponentially many nodes. this potentially exponential cost can be se... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 233 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
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