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we can write a kernel function k ( x, x ( ) i ) = min ( x, x ( ) i ) that is exactly equivalent to the corresponding infinite - dimensional dot product. the most commonly used kernel is the gaussian kernel k,, σ ( u v u v ) = ( n − ; 0 2i ) ( 5. 84 ) where n ( x ; µ, σ ) is the standard normal density. this kernel is al... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 157 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to euclidean distance, the gaussian kernel has a large response, indicating that xis very similar to the x template. the model then puts a large weight on the associated training label y. overall, the prediction will combine many such training labels weighted by the similarity of the corresponding training examples. su... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 157 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics as support vectors. kernel machines also [UNK] from a high computational cost of training when the dataset is large. we will revisit this idea in section. kernel machines with 5. 9 generic kernels struggle to generalize well. we will explain why in section. the 5. 11 modern incarnatio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 158 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
algorithm as not having any parameters, but rather implementing a simple function of the training data. in fact, there is not even really a training stage or learning process. instead, at test time, when we want to produce an output y for a new test input x, we find the k - nearest neighbors to x in the training data x.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 158 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
- nearest neighbor converges to double the bayes error as the 1 number of training examples approaches infinity. the error in excess of the bayes error results from choosing a single neighbor by breaking ties between equally distant neighbors randomly. when there is infinite training data, all test points x will have infi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 158 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics distribution, but only a single variable x1 is relevant to the output. suppose further that this feature simply encodes the output directly, i. e. that y = x1 in all cases. nearest neighbor regression will not be able to detect this simple pattern. the nearest neighbor of most points ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 159 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics 0 1 01 111 0 1 011 1111 1110 110 10 010 00 1110 1111 110 10 01 00 010 011 11 111 11 figure 5. 7 : diagrams describing how a decision tree works. ( top ) each node of the tree chooses to send the input example to the child node on the left ( 0 ) or or the child node on the right ( 1 ).... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 160 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the center of the region of examples they receive. the result is a piecewise - constant function, with one piece per leaf. each leaf requires at least one training example to define, so it is not possible for the decision tree to learn a function that has more local maxima than the number of training examples. 145 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 160 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics another type of learning algorithm that also breaks the input space into regions and has separate parameters for each region is the decision tree (, breiman et al. 1984 ) and its many variants. as shown in figure, each node of the decision 5. 7 tree is associated with a region in the i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 161 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
used, with axis - aligned splits and constant outputs within each node, struggle to solve some problems that are easy even for logistic regression. for example, if we have a two - class problem and the positive class occurs wherever x2 > x1, the decision boundary is not axis - aligned. the decision tree will thus need ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 161 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##vised algorithms are those that experience 5. 1. 3 only “ features ” but not a supervision signal. the distinction between supervised and unsupervised algorithms is not formally and rigidly defined because there is no objective test for distinguishing whether a value is a feature or a target provided by a supervisor. ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 161 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics related examples. a classic unsupervised learning task is to find the “ best ” representation of the data. by ‘ best ’ we can mean [UNK] things, but generally speaking we are looking for a representation that preserves as much information about x as possible while obeying some penalty ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 162 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
overall structure of the representation that tends to distribute data along the axes of the representation space. independent representations attempt to disentangle the sources of variation underlying the data distribution such that the dimensions of the representation are statistically independent. of course these thr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 162 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. 1 principal components analysis in section, we saw that the principal components analysis algorithm provides 2. 12 a means of compressing data. we can also view pca as an unsupervised learning algorithm that learns a representation of data. this representation is based on two of the criteria for a simple representati... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 162 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics − − 20 10 0 10 20 x1 −20 −10 0 10 20 x2 − − 20 10 0 10 20 z1 −20 −10 0 10 20 z2 figure 5. 8 : pca learns a linear projection that aligns the direction of greatest variance with the axes of the new space. ( left ) the original data consists of samples ofx. in this space, the variance m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 163 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
input x to a representation z as shown in figure. in section, we saw that 5. 8 2. 12 we could learn a one - dimensional representation that best reconstructs the original data ( in the sense of mean squared error ) and that this representation actually corresponds to the first principal component of the data. thus we can... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 163 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics pca finds a representation ( through linear transformation ) z = xw where var [ ] z is diagonal. in section, we saw that the principal components of a design matrix 2. 12 x are given by the eigenvectors of xx. from this view, xx w w = λ. ( 5. 86 ) in this section, we exploit an alterna... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 164 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
: x x var [ ] = x 1 m −1xx ( 5. 88 ) = 1 m −1 ( u w σ ) u w σ ( 5. 89 ) = 1 m −1wσu u w σ ( 5. 90 ) = 1 m −1wσ2 w, ( 5. 91 ) where we use the fact that uu = i because the u matrix of the singular value decomposition is defined to be orthogonal. this shows that if we take z = xw, we can ensure that the covariance of is d... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 164 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics the above analysis shows that when we project the data x to z, via the linear transformation w, the resulting representation has a diagonal covariance matrix ( as given by σ2 ) which immediately implies that the individual elements of z are mutually uncorrelated. this ability of pca t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 165 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a simple linear transformation. 5. 8. 2 - means clustering k another example of a simple representation learning algorithm isk - means clustering. the k - means clustering algorithm divides the training set intok [UNK] clusters of examples that are near each other. we can thus think of the algorithm as providing a k - ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 165 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
all examples in the same cluster are similar to each other ) and it confers the computational advantage that the entire representation may be captured by a single integer. the k - means algorithm works by initializingk [UNK] centroids { µ ( 1 ),..., µ ( ) k } to [UNK] values, then alternating between two [UNK] steps un... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 165 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics one [UNK] pertaining to clustering is that the clustering problem is inherently ill - posed, in the sense that there is no single criterion that measures how well a clustering of the data corresponds to the real world. we can measure properties of the clustering such as the average eu... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 166 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
ask each clustering algorithm to find two clusters, one algorithm may find a cluster of cars and a cluster of trucks, while another may find a cluster of red vehicles and a cluster of gray vehicles. suppose we also run a third clustering algorithm, which is allowed to determine the number of clusters. this may assign the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 166 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
and one representing whether it is a car or a truck. it is still not entirely clear what the optimal distributed representation is ( how can the learning algorithm know whether the two attributes we are interested in are color and car - versus - truck rather than manufacturer and age? ) but having many attributes reduc... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 166 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics gradient descent algorithm introduced in section. 4. 3 a recurring problem in machine learning is that large training sets are necessary for good generalization, but large training sets are also more computationally expensive. the cost function used by a machine learning algorithm oft... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 167 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of this operation is o ( m ). as the training set size grows to billions of examples, the time to take a single gradient step becomes prohibitively long. the insight of stochastic gradient descent is that the gradient is an expectation. the expectation may be approximately estimated using a small set of samples. specifi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 167 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics gradient descent in general has often been regarded as slow or unreliable. in the past, the application of gradient descent to non - convex optimization problems was regarded as foolhardy or unprincipled. today, we know that the machine learning models described in part work very well... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 168 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, as m approaches infinity, the model will eventually converge to its best possible test error before sgd has sampled every example in the training set. increasing m further will not extend the amount of training time needed to reach the model ’ s best possible test error. from this point of view, one can argue that the... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 168 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
with tens of thousands of examples. soon after, deep learning garnered additional interest in industry, because it provided a scalable way of training nonlinear models on large datasets. stochastic gradient descent and many enhancements to it are described further in chapter. 8 5. 10 building a machine learning algorit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 168 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics x y and, the cost function j, b ( w ) = −ex, [UNK] log pmodel ( ) y | x, ( 5. 100 ) the model specification pmodel ( y | x ) = n ( y ; xw + b, 1 ), and, in most cases, the optimization algorithm defined by solving for where the gradient of the cost is zero using the normal equations. by... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 169 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
| w 2 2 −ex, [UNK] log pmodel ( ) y | x. ( 5. 101 ) this still allows closed - form optimization. if we change the model to be nonlinear, then most cost functions can no longer be optimized in closed form. this requires us to choose an iterative numerical optimization procedure, such as gradient descent. the recipe for... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 169 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
) = x wxw. in some cases, the cost function may be a function that we cannot actually evaluate, for computational reasons. in these cases, we can still approximately minimize it using iterative numerical optimization so long as we have some way of approximating its gradients. most machine learning algorithms make use o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 169 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics hand - designed, it can usually be understood as using a special - case optimizer. some models such as decision trees or k - means require special - case optimizers because their cost functions have flat regions that make them inappropriate for minimization by gradient - based optimize... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 170 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
examples becomes exponentially more [UNK] when working with high - dimensional data, and how the mechanisms used to achieve generalization in traditional machine learning are [UNK] to learn complicated functions in high - dimensional spaces. such spaces also often impose high computational costs. deep learning was desi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 170 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics figure 5. 9 : as the number of relevant dimensions of the data increases ( from left to right ), the number of configurations of interest may grow exponentially. ( left ) in this one - dimensional example, we have one variable for which we only care to distinguish 10 regions of interes... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 171 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, we seem to need o ( v d ) regions and examples. this is an instance of the curse of dimensionality. figure graciously provided by nicolas chapados. the curse of dimensionality arises in many places in computer science, and especially so in machine learning. one challenge posed by the curse of dimensionality is a stat... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 171 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
training examples in the same unit volume cell as x, divided by the total number of training examples. if we wish to classify an example, we can return the most common class of training examples in the same cell. if we are doing regression we can average the target values observed over the examples in that cell. but wh... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 171 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics meaningful about these new configurations? many traditional machine learning algorithms simply assume that the output at a new point should be approximately the same as the output at the nearest training point. 5. 11. 2 local constancy and smoothness regularization in order to generali... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 172 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
“ priors ” is the smoothness prior or local constancy prior. this prior states that the function we learn should not change very much within a small region. many simpler algorithms rely exclusively on this prior to generalize well, and as a result they fail to scale to the statistical challenges involved in solving ai ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 172 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
x ( for example, if x is a labeled training example ) then that answer is probably good in the neighborhood of x. if we have several good answers in some neighborhood we would combine them ( by some form of averaging or interpolation ) to produce an answer that agrees with as many of them as much as possible. an extrem... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 172 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics region containing all the points x that have the same set of k nearest neighbors in the training set. for k = 1, the number of distinguishable regions cannot be more than the number of training examples. while the k - nearest neighbors algorithm copies the output from nearby training ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 173 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
- based learning because they break the input space into as many regions as there are leaves and use a separate parameter ( or sometimes many parameters for extensions of decision trees ) in each region. if the target function requires a tree with at least n leaves to be represented accurately, then at least n training... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 173 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
underlying function will not allow a learner to do that. for example, imagine that the target function is a kind of checkerboard. a checkerboard contains many variations but there is a simple structure to them. imagine what happens when the number of training examples is substantially smaller than the number of black a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 173 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics figure 5. 10 : illustration of how the nearest neighbor algorithm breaks up the input space into regions. an example ( represented here by a circle ) within each region defines the region boundary ( represented here by the lines ). they value associated with each example defines what th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 174 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics entire checkerboard right is to cover each of its cells with at least one example. the smoothness assumption and the associated non - parametric learning algo - rithms work extremely well so long as there are enough examples for the learning algorithm to observe high points on most pe... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 175 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##ize well to new inputs — is yes. the key insight is that a very large number of regions, e. g., o ( 2k ), can be defined with o ( k ) examples, so long as we introduce some dependencies between the regions via additional assumptions about the underlying data generating distribution. in this way, we can actually genera... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 175 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
much too complex to be limited to simple, manually specified properties such as periodicity, so we want learning algorithms that embody more general - purpose assumptions. the core idea in deep learning is that we assume that the data was generated by the composition of factors or features, potentially at multiple level... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 175 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics 5. 11. 3 manifold learning an important concept underlying many ideas in machine learning is that of a manifold. a manifold is a connected region. mathematically, it is a set of points, associated with a neighborhood around each point. from any given point, the manifold locally appear... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 176 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
or dimensions, embedded in a higher - dimensional space. each dimension corresponds to a local direction of variation. see figure for an 5. 11 example of training data lying near a one - dimensional manifold embedded in two - dimensional space. in the context of machine learning, we allow the dimensionality of the manif... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 176 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics many machine learning problems seem hopeless if we expect the machine learning algorithm to learn functions with interesting variations across all of rn. manifold learning algorithms surmount this obstacle by assuming that most of rn consists of invalid inputs, and that interesting in... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 177 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
evidence in favor of this assumption consists of two categories of observations. the first observation in favor of the manifold hypothesis is that the proba - bility distribution over images, text strings, and sounds that occur in real life is highly concentrated. uniform noise essentially never resembles structured inp... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 177 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics figure 5. 12 : sampling images uniformly at random ( by randomly picking each pixel according to a uniform distribution ) gives rise to noisy images. although there is a non - zero probability to generate an image of a face or any other object frequently encountered in ai applications... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 178 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics examples, with each example surrounded by other highly similar examples that may be reached by applying transformations to traverse the manifold. the second argument in favor of the manifold hypothesis is that we can also imagine such neighborhoods and transformations, at least inform... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 179 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
al., 2000 brand 2003 belkin and niyogi 2003 donoho and grimes 2003 weinberger ;, ;, ;, ; and saul 2004, ) clearly support the hypothesis for a large class of datasets of interest in ai. when the data lies on a low - dimensional manifold, it can be most natural for machine learning algorithms to represent the data in te... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 179 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a manifold structure. in figure, we will see 20. 6 how a machine learning algorithm can successfully accomplish this goal. this concludes part, which has provided the basic concepts in mathematics i and machine learning which are employed throughout the remaining parts of the book. you are now prepared to embark upon yo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 179 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics figure 5. 13 : training examples from the qmul multiview face dataset (, ) gong et al. 2000 for which the subjects were asked to move in such a way as to cover the two - dimensional manifold corresponding to two angles of rotation. we would like learning algorithms to be able to disco... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 180 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
part ii deep networks : modern practices 166 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 181 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
this part of the book summarizes the state of modern deep learning as it is used to solve practical applications. deep learning has a long history and many aspirations. several approaches have been proposed that have yet to entirely bear fruit. several ambitious goals have yet to be realized. these less - developed bra... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 182 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
and reflect in order to accomplish the task, remain beyond the scope of deep learning for now. this part of the book describes the core parametric function approximation technology that is behind nearly all modern practical applications of deep learning. we begin by describing the feedforward deep network model that is ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 182 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6 deep feedforward networks deep feedforward networks, also often called feedforward neural networks, or multilayer perceptrons ( mlps ), are the quintessential deep learning models. the goal of a feedforward network is to approximate some functionf ∗. for example, for a classifier, y = f ∗ ( x ) maps an input x... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 183 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
are of extreme importance to machine learning practi - tioners. they form the basis of many important commercial applications. for example, the convolutional networks used for object recognition from photos are a specialized kind of feedforward network. feedforward networks are a conceptual stepping stone on the path t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 183 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks length of the chain gives the depth of the model. it is from this terminology that the name “ deep learning ” arises. the final layer of a feedforward network is called the output layer. during neural network training, we drive f ( x ) to match f∗ ( x ). the training data provides us... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 184 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
does not show the desired output for each of these layers, these layers are called hidden layers. finally, these networks are called neural because they are loosely inspired by neuroscience. each hidden layer of the network is typically vector - valued. the dimensionality of these hidden layers determines the width of ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 184 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
network research is guided by many mathematical and engineering disciplines, and the goal of neural networks is not to perfectly model the brain. it is best to think of feedforward networks as function approximation machines that are designed to achieve statistical generalization, occasionally drawing some insights fro... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 184 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks nonlinear transformation. equivalently, we can apply the kernel trick described in section, to obtain a nonlinear learning algorithm based on implicitly applying 5. 7. 2 the φ mapping. we can think of φ as providing a set of features describing x, or as providing a new representatio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 185 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of human [UNK] for each separate task, with practitioners specializing in [UNK] domains such as speech recognition or computer vision, and with little transfer between domains. 3. the strategy of deep learning is to learn φ. in this approach, we have a model y = f ( x ; θ w, ) = φ ( x ; θ ) w. we now have parameters θ ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 185 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##rst approach by being highly generic — we do so by using a very broad family φ ( x ; θ ). this approach can also capture the benefit of the second approach. human practitioners can encode their knowledge to help generalization by designing families φ ( x ; θ ) that they expect will perform well. the advantage is that ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 185 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks mappings from x to y that lack feedback connections. other models presented later will apply these principles to learning stochastic mappings, learning functions with feedback, and learning probability distributions over a single vector. we begin this chapter with a simple example o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 186 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
layers should be connected to each other, and how many units should be in each layer. learning in deep neural networks requires computing the gradients of complicated functions. we present the back - propagation algorithm and its modern generalizations, which can be used to [UNK] compute these gradients. finally, we cl... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 186 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
as similar as possible to f∗. in this simple example, we will not be concerned with statistical generalization. we want our network to perform correctly on the four points x = { [ 0, 0 ], [ 0, 1 ], [ 1, 0 ], and [ 1, 1 ] }. we will train the network on all four of these points. the only challenge is to fit the training ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 186 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks appropriate cost function for modeling binary data. more appropriate approaches are described in section. 6. 2. 2. 2 evaluated on our whole training set, the mse loss function is j ( ) = θ 1 4 x∈x ( f∗ ( ) ( ; ) ) x −f x θ 2. ( 6. 1 ) now we must choose the form of our model, f ( x ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 187 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. one way to solve this problem is to use a model that learns a [UNK] feature space in which a linear model is able to represent the solution. specifically, we will introduce a very simple feedforward network with one hidden layer containing two hidden units. see figure for an illustration of 6. 2 this model. this feedfo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 187 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( 2 ) ( f ( 1 ) ( ) ) x. what function should f ( 1 ) compute? linear models have served us well so far, and it may be tempting to make f ( 1 ) be linear as well. unfortunately, if f ( 1 ) were linear, then the feedforward network as a whole would remain a linear function of its input. ignoring the intercept terms for ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 187 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 0 1 x1 0 1 x2 original space x 0 1 2 h1 0 1 h2 learned space h figure 6. 1 : solving the xor problem by learning a representation. the bold numbers printed on the plot indicate the value that the learned function must output at each point. ( left ) a linear model applied directly to... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 188 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
in feature space. in other words, the nonlinear features have mapped both x = [ 1, 0 ] and x = [ 0, 1 ] to a single point in feature space, h = [ 1, 0 ]. the linear model can now describe the function as increasing in h1 and decreasing in h2. in this example, the motivation for learning the feature space is only to mak... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 188 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks y h x w w y h1 h1 x1 x1 h2 h2 x2 x2 figure 6. 2 : an example of a feedforward network, drawn in two [UNK] styles. specifically, this is the feedforward network we use to solve the xor example. it has a single hidden layer containing two units. ( left ) in this style, we draw every un... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 189 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to y. we typically omit the intercept parameters associated with each layer when labeling this kind of drawing. model, we used a vector of weights and a scalar bias parameter to describe an [UNK] transformation from an input vector to an output scalar. now, we describe an [UNK] transformation from a vector x to a vecto... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 189 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
{, w x c + } + b. ( 6. 3 ) we can now specify a solution to the xor problem. let w = 1 1 1 1, ( 6. 4 ) c = 0 −1, ( 6. 5 ) 174 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 189 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 0 z 0 g z ( ) = max 0 {, z } figure 6. 3 : the rectified linear activation function. this activation function is the default activation function recommended for use with most feedforward neural networks. applying this function to the output of a linear transformation yields a nonline... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 190 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks w = 1 −2, ( 6. 6 ) and. b = 0 we can now walk through the way that the model processes a batch of inputs. let x be the design matrix containing all four points in the binary input space, with one example per row : x = 0 0 0 1 1 0 1 1. ( 6. 7 ) the first step in the neural network is ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 191 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
each example, we apply the rectified linear transformation : h 0 0 1 0 1 0 2 1. ( 6. 10 ) this transformation has changed the relationship between the examples. they no longer lie on a single line. as shown in figure, they now lie in a space where a 6. 1 linear model can solve the problem. we finish by multiplying by the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 191 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks the neural network has obtained the correct answer for every example in the batch. in this example, we simply specified the solution, then showed that it obtained zero error. in a real situation, there might be billions of model parameters and billions of training examples, so one ca... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 192 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. in section, we described 5. 10 how to build a machine learning algorithm by specifying an optimization procedure, a cost function, and a model family. the largest [UNK] between the linear models we have seen so far and neural networks is that the nonlinearity of a neural network causes most interesting loss functions... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 192 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
random values. the biases may be initialized to zero or to small positive values. the iterative gradient - based opti - mization algorithms used to train feedforward networks and almost all other deep models will be described in detail in chapter, with parameter initialization in 8 particular discussed in section. for ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 192 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks more specifically, are most often improvements of the stochastic gradient descent algorithm, introduced in section. 5. 9 we can of course, train models such as linear regression and support vector machines with gradient descent too, and in fact this is common when the training set is... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 193 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
is the choice of the cost function. fortunately, the cost functions for neural networks are more or less the same as those for other parametric models, such as linear models. in most cases, our parametric model defines a distribution p ( y x | ; θ ) and we simply use the principle of maximum likelihood. this means we us... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 193 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##ization strategies. more advanced regularization strategies for neural networks will be described in chapter. 7 6. 2. 1. 1 learning conditional distributions with maximum likelihood most modern neural networks are trained using maximum likelihood. this means that the cost function is simply the negative log - likelih... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 193 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks as the cross - entropy between the training data and the model distribution. this cost function is given by j ( ) = θ −ex y, [UNK] log pmodel ( ) y x |. ( 6. 12 ) the specific form of the cost function changes from model to model, depending on the specific form of log pmodel. the expa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 194 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. the discarded θ constant is based on the variance of the gaussian distribution, which in this case we chose not to parametrize. previously, we saw that the equivalence between maximum likelihood estimation with an output distribution and minimization of mean squared error holds for a linear model, but in fact, the eq... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 194 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the hidden units or the output units saturate. the negative log - likelihood helps to avoid this problem for many models. many output units involve an exp function that can saturate when its argument is very negative. the log function in the negative log - likelihood cost function undoes the exp of some output units. w... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 194 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks can control the density of the output distribution ( for example, by learning the variance parameter of a gaussian output distribution ) then it becomes possible to assign extremely high density to the correct training set outputs, resulting in cross - entropy approaching negative i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 195 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##fic parametric form. from this point of view, we can view the cost function as being a functional rather than just a function. a functional is a mapping from functions to real numbers. we can thus think of learning as choosing a function rather than merely choosing a set of parameters. we can design our cost functiona... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 195 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
[UNK] | | − | | y f ( ) x 2 ( 6. 14 ) yields f∗ ( ) = x [UNK] ( ) y x | [ ] y, ( 6. 15 ) so long as this function lies within the class we optimize over. in other words, if we could train on infinitely many samples from the true data generating distribution, minimizing the mean squared error cost function gives a functi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 195 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks [UNK] cost functions give [UNK] statistics. a second result derived using calculus of variations is that f ∗ = arg min f ex y, [UNK] | | − | | y f ( ) x 1 ( 6. 16 ) yields a function that predicts the median value of y for each x, so long as such a function may be described by the f... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 196 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
we simply use the cross - entropy between the data distribution and the model distribution. the choice of how to represent the output then determines the form of the cross - entropy function. any kind of neural network unit that may be used as an output can also be used as a hidden unit. here, we focus on the use of th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 196 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
features h, a layer of linear output units produces a vector [UNK] = wh + b. linear output layers are often used to produce the mean of a conditional gaussian distribution : p ( ) = ( ; y x | n y [UNK] i, ). ( 6. 17 ) 181 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 196 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks maximizing the log - likelihood is then equivalent to minimizing the mean squared error. the maximum likelihood framework makes it straightforward to learn the covariance of the gaussian too, or to make the covariance of the gaussian be a function of the input. however, the covarian... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 197 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
be cast in this form. the maximum - likelihood approach is to define a bernoulli distribution over y conditioned on. x a bernoulli distribution is defined by just a single number. the neural net needs to predict only p ( y = 1 | x ). for this number to be a valid probability, it must lie in the interval [ 0, 1 ]. satisfy... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 197 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
how to improve the corresponding parameters. instead, it is better to use a [UNK] approach that ensures there is always a strong gradient whenever the model has the wrong answer. this approach is based on using sigmoid output units combined with maximum likelihood. a sigmoid output unit is defined by [UNK] σ = wh + b ( ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 197 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks where is the logistic sigmoid function described in section. σ 3. 10 we can think of the sigmoid output unit as having two components. first, it uses a linear layer to compute z = wh + b. next, it uses the sigmoid activation function to convert into a probability. z we omit the depe... | /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 |
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