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exponentially with the depth of the forward propagation graph. this large cost would be incurred because the same computation for ∂u ( ) i ∂u ( ) j would be redone many times. to avoid such recomputation, we can think of back - propagation as a table - filling algorithm that takes advantage of storing intermediate resul... | /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 |
chapter 6. deep feedforward networks back - propagation avoids repeating many common subexpressions. this table - filling strategy is sometimes called. dynamic programming 6. 5. 7 example : back - propagation for mlp training as an example, we walk through the back - propagation algorithm as it is used to train a multil... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 234 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
operation that can compute max { 0, z } element - wise. the predictions of the unnormalized log probabilities over classes are then given by hw ( 2 ). we assume that our graph language includes a cross _ entropy operation that computes the cross - entropy between the targets y and the probability distribution defined by... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 234 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
that it would be tedious to draw or to read. this demonstrates one of the benefits of the back - propagation algorithm, which is that it can automatically generate gradients that would be straightforward but tedious for a software engineer to derive manually. we can roughly trace out the behavior of the back - propagati... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 234 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks x w ( 1 ) w ( 1 ) u ( 1 ) u ( 1 ) matmul h relu u ( 3 ) u ( 3 ) sqr u ( 4 ) u ( 4 ) sum λ u ( 7 ) u ( 7 ) w ( 2 ) w ( 2 ) u ( 2 ) u ( 2 ) matmul y j mle j mle cross _ entropy u ( 5 ) u ( 5 ) sqr u ( 6 ) u ( 6 ) sum u ( 8 ) u ( 8 ) j + × + figure 6. 11 : the computational graph used ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 235 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
branches. on the shorter branch, it adds h g to the gradient on w ( 2 ), using the back - propagation rule for the second argument to the matrix multiplication operation. the other branch corresponds to the longer chain descending further along the network. first, the back - propagation algorithm computes ∇hj = gw ( 2 ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 235 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks matrix, resulting in o ( w ) multiply - adds, where w is the number of weights. during the backward propagation stage, we multiply by the transpose of each weight matrix, which has the same computational cost. the main memory cost of the algorithm is that we need to store the input ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 236 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
wish to compute both the maximum value in a tensor and the index of that value, it is best to compute both in a single pass through memory, so it is most [UNK] to implement this procedure as a single operation with two outputs. we have not described how to control the memory consumption of back - propagation. back - pr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 236 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the gradient requested by the user is undefined. various other technicalities make real - world [UNK] more complicated. these technicalities are not insurmountable, and this chapter has described the key intellectual tools needed to compute derivatives, but it is important to be aware that many more subtleties exist. 6.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 236 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks concerning how to perform [UNK]. more generally, the field of automatic [UNK] is concerned with how to compute derivatives algorithmically. the back - propagation algorithm described here is only one approach to automatic [UNK]. it is a special case of a broader class of techniques c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 237 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
), ( 6. 57 ) where we build the softmax function out of exponentiation, summation and division operations, and construct a cross - entropy loss j = − i pilog qi. a human mathematician can observe that the derivative of j with respect to zi takes a very simple form : qi −pi. the back - propagation algorithm is not capab... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 237 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
is of the same order as the number of computations for the forward computation : this can be seen in algorithm because each local partial derivative 6. 2 ∂u ( ) i ∂u ( ) j needs to be computed only once along with an associated multiplication and addition for the recursive chain - rule formulation ( equation ). the ove... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 237 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks each scalar internal node in the original forward graph, the naive implementation computes k gradients instead of a single gradient. when the number of outputs of the graph is larger than the number of inputs, it is sometimes preferable to use another form of automatic [UNK] called ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 238 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the end and going backwards only requires matrix - vector products. this corresponds to the backward mode. instead, starting to multiply from the left would involve a series of matrix - matrix products, which makes the whole computation much more expensive. however, if a has fewer rows than d has columns, it is cheaper... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 238 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
- propagation rules to be developed for each operation, allowing the developer to improve speed or stability in non - obvious ways that an automatic procedure would presumably be unable to replicate. back - propagation is therefore not the only way or the optimal way of computing the gradient, but it is a very practica... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 238 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks 6. 5. 10 higher - order derivatives some software frameworks support the use of higher - order derivatives. among the deep learning software frameworks, this includes at least theano and tensorflow. these libraries use the same kind of data structure to describe the expressions for ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 239 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. krylov methods are a set of iterative techniques for performing various operations like approximately inverting a matrix or finding approximations to its eigenvectors or eigenvalues, without using any operation other than matrix - vector products. in order to use krylov methods on the hessian, we only need to be able ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 239 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, it is possible to do with hessian vector products. one simply computes he ( ) i for all i = 1,..., n, where e ( ) i is the one - hot vector with e ( ) i i = 1 and all other entries equal to 0. 6. 6 historical notes feedforward networks can be seen as [UNK] nonlinear function approximators based on using gradient desc... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 239 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks from this point of view, the modern feedforward network is the culmination of centuries of progress on the general function approximation task. the chain rule that underlies the back - propagation algorithm was invented in the 17th century (, ;, ). calculus and algebra have leibniz ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 240 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
per - ceptron and a means of computing the gradient through such a model. [UNK] applications of the chain rule based on dynamic programming began to appear in the 1960s and 1970s, mostly for control applications (, ; kelley 1960 bryson and denham 1961 dreyfus 1962 bryson and ho 1969 dreyfus 1973, ;, ;, ;, ) but also fo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 240 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the ideas put forward by the authors of that book and in particular by rumelhart and hinton go much beyond back - propagation. they include crucial ideas about the possible computational implementation of several central aspects of cognition and learning, which came under the name of “ connectionism ” because of the im... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 240 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks approaches to gradient descent are still in use. most of the improvement in neural network performance from 1986 to 2015 can be attributed to two factors. first, larger datasets have reduced the degree to which statistical generalization is a challenge for neural networks. second, n... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 241 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
greatly improved the performance of feedforward networks was the replacement of sigmoid hidden units with piecewise linear hidden units, such as rectified linear units. rectification using the max { 0, z } function was introduced in early neural network models and dates back at least as far as the cognitron and neocognit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 241 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
important factor in improving the performance of a recognition system ” among several [UNK] factors of neural network architecture design. for small datasets, ( ) observed that using rectifying non - jarrett et al. 2009 linearities is even more important than learning the weights of the hidden layers. random weights ar... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 241 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 6. deep feedforward networks rectified linear units are also of historical interest because they show that neuroscience has continued to have an influence on the development of deep learning algorithms. ( ) motivate rectified linear units from glorot et al. 2011a biological considerations. the half - rectifying no... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 242 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
known that with the right resources and engineering practices, feedforward networks perform very well. today, gradient - based learning in feedforward networks is used as a tool to develop probabilistic models, such as the variational autoencoder and generative adversarial networks, described in chapter. rather than be... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 242 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7 regularization for deep learning a central problem in machine learning is how to make an algorithm that will perform well not just on the training data, but also on new inputs. many strategies used in machine learning are explicitly designed to reduce the test error, possibly at the expense of increased train... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 243 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
learning. if you are already familiar with these concepts, feel free to skip the relevant sections. however, most of this chapter is concerned with the extension of these basic concepts to the particular case of neural networks. in section, we defined regularization as “ any modification we make to 5. 2. 2 a learning alg... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 243 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning on the test set. sometimes these constraints and penalties are designed to encode specific kinds of prior knowledge. other times, these constraints and penalties are designed to express a generic preference for a simpler model class in order to promote generalization. sometime... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 244 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
or ( 2 ) matched the true data generating process, or ( 3 ) included the generating process but also many other possible generating processes — the overfitting regime where variance rather than bias dominates the estimation error. the goal of regularization is to take a model from the third regime into the second regime... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 244 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
model family ). what this means is that controlling the complexity of the model is not a simple matter of finding the model of the right size, with the right number of parameters. instead, we might find — and indeed in practical deep learning scenarios, we almost always do find — that the best fitting model ( in the sense ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 244 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning 7. 1 parameter norm penalties regularization has been used for decades prior to the advent of deep learning. linear models such as linear regression and logistic regression allow simple, straightforward, and [UNK] regularization strategies. many regularization approaches are ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 245 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
objective j on the training data and some measure of the size of the parameters θ ( or some subset of the parameters ). [UNK] choices for the parameter norm can result in [UNK] solutions being preferred. ω in this section, we discuss the [UNK] of the various norms when used as penalties on the model parameters. before ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 245 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
use the vector w to indicate all of the weights that should be [UNK] by a norm penalty, while the vector θ denotes all of the parameters, including both w and the unregularized parameters. in the context of neural networks, it is sometimes desirable to use a separate penalty with a [UNK] α [UNK] for each layer of the n... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 245 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning 7. 1. 1 l2 parameter regularization we have already seen, in section, one of the simplest and most common kinds 5. 2. 2 of parameter norm penalty : the l2 parameter norm penalty commonly known as weight decay. this regularization strategy drives the weights closer to the orig... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 246 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
y ) = w + ∇wj,. ( ; w x y ) ( 7. 3 ) to take a single gradient step to update the weights, we perform this update : w w w ← −α ( + ∇wj,. ( ; w x y ) ) ( 7. 4 ) written another way, the update is : w w ← − ( 1 α ) −∇ wj,. ( ; w x y ) ( 7. 5 ) we can see that the addition of the weight decay term has modified the learning... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 246 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
). if the objective function is truly quadratic, as in the case of fitting a linear regression model with 1more generally, we could regularize the parameters to be near any specific point in space and, surprisingly, still get a regularization [UNK], but better results will be obtained for a value closer to the true one, ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 246 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning mean squared error, then the approximation is perfect. the approximation [UNK] is given by [UNK] j ( ) = θ ( w∗ ) + 1 2 ( w w − ∗ ) h w w ( − ∗ ), ( 7. 6 ) where h is the hessian matrix of j with respect to w evaluated at w∗. there is no first - order term in this quadratic ap... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 247 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to represent the location of the minimum. α [UNK] h + ( [UNK] w − ∗ ) = 0 ( 7. 8 ) ( + ) h αi [UNK] hw = ∗ ( 7. 9 ) [UNK] h i = ( + α ) −1hw∗. ( 7. 10 ) as α approaches 0, the regularized solution [UNK] approaches w∗. but what happens as α grows? because h is real and symmetric, we can decompose it into a diagonal matr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 247 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
is to rescale w ∗along the axes defined by the eigenvectors of h. specifically, the component of w ∗that is aligned with the i - th eigenvector of h is rescaled by a factor of λi λi + α. ( you may wish to review how this kind of scaling works, first explained in figure ). 2. 3 along the directions where the eigenvalues of ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 247 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning w1 w2 w∗ [UNK] w figure 7. 1 : an illustration of the [UNK] ofl2 ( or weight decay ) regularization on the value of the optimal w. the solid ellipses represent contours of equal value of the unregularized objective. the dotted circles represent contours of equal value of thel... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 248 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##w2 relatively little. only directions along which the parameters contribute significantly to reducing the objective function are preserved relatively intact. in directions that do not contribute to reducing the objective function, a small eigenvalue of the hessian tells us that movement in this direction will not sign... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 248 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning the sum of squared errors : ( ) xw y − ( ) xw y −. ( 7. 14 ) when we add l2 regularization, the objective function changes to ( ) xw y − ( ) + xw y − 1 2αww. ( 7. 15 ) this changes the normal equations for the solution from w x = ( x ) −1x y ( 7. 16 ) to w x = ( x i + α ) −1x... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 249 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
input x as having higher variance, which makes it shrink the weights on features whose covariance with the output target is low compared to this added variance. 7. 1. 2 l1 regularization while l2 weight decay is the most common form of weight decay, there are other ways to penalize the size of the model parameters. ano... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 249 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
zero, but instead towards some parameter value w ( ) o. in that case the l1 regularization would introduce the term ω ( ) = θ | | − w w ( ) o | | 1 = i | wi −w ( ) o i |. 234 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 249 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning of regularization. as with l2 weight decay, l1 weight decay controls the strength of the regularization by scaling the penalty using a positive hyperparameter ω α. thus, the regularized objective function [UNK], ( ; w x y ) is given by [UNK], α ( ; w x y ) = | | | | w 1 + ( ;... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 250 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
wi ; instead it is a constant factor with a sign equal to sign ( wi ). one consequence of this form of the gradient is that we will not necessarily see clean algebraic solutions to quadratic approximations of j ( x y, ; w ) as we did for l2 regularization. our simple linear model has a quadratic cost function that we c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 250 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
n ] ), where each hi, i > 0. this assumption holds if the data for the linear regression problem has been preprocessed to remove all correlation between the input features, which may be accomplished using pca. our quadratic approximation of the l1 regularized objective function decom - poses into a sum over the paramet... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 250 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning consider the situation where w∗ i > i 0 for all. there are two possible outcomes : 1. the case where w∗ i ≤ α hi, i. here the optimal value of wi under the regularized objective is simply wi = 0. this occurs because the contribution of j ( w ; x y, ) to the regularized object... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 251 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
sparse. sparsity in this context refers to the fact that some parameters have an optimal value of zero. the sparsity of l1 regularization is a qualitatively [UNK] behavior than arises with l2 regularization. equation gave the 7. 13 solution [UNK] for l2 regularization. if we revisit that equation using the assumption o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 251 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
well known lasso (, ) ( least absolute shrinkage and tibshirani 1995 selection operator ) model integrates an l1 penalty with a linear model and a least squares cost function. the l1 penalty causes a subset of the weights to become zero, suggesting that the corresponding features may safely be discarded. in section, we... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 251 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning from the point of view of learning via maximization with respect to w, we can ignore the terms because they do not depend on. log log 2 α − w 7. 2 norm penalties as constrained optimization consider the cost function regularized by a parameter norm penalty : [UNK], j, α. ( ; ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 252 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( ) θ, α x y, j θ x y, α θ k. ) ( 7. 26 ) the solution to the constrained problem is given by θ∗ = arg min θ max α, α≥0l ( ) θ, α. ( 7. 27 ) as described in section, solving this problem requires modifying both 4. 4 θ and α. section provides a worked example of linear regression with an 4. 5 l2 constraint. many [UNK] p... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 252 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
function of : θ θ∗ = arg min θ l ( θ, α∗ ) = arg min θ j, α ( ; θ x y ) + ∗ω ( ) θ. ( 7. 28 ) this is exactly the same as the regularized training problem of minimizing [UNK]. we can thus think of a parameter norm penalty as imposing a constraint on the weights. if is the ω l2 norm, then the weights are constrained to ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 252 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning limited l1 norm. usually we do not know the size of the constraint region that we impose by using weight decay with [UNK] α∗because the value of α∗does not directly tell us the value of k. in principle, one can solve for k, but the relationship between k and α∗depends on the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 253 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
value of k is appropriate and do not want to spend time searching for the value of α that corresponds to this. k another reason to use explicit constraints and reprojection rather than enforcing constraints with penalties is that penalties can cause non - convex optimization procedures to get stuck in local minima corr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 253 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. finally, explicit constraints with reprojection can be useful because they impose some stability on the optimization procedure. when using high learning rates, it is possible to encounter a positive feedback loop in which large weights induce large gradients which then induce a large update to the weights. if these u... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 253 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning of a neural net layer, rather than constraining the frobenius norm of the entire weight matrix. constraining the norm of each column separately prevents any one hidden unit from having very large weights. if we converted this constraint into a penalty in a lagrange function, ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 254 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
generating distri - bution truly has no variance in some direction, or when no variance is observed in some direction because there are fewer examples ( rows of x ) than input features ( columns of x ). in this case, many forms of regularization correspond to inverting xx i + α instead. this regularized matrix is guara... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 254 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##ow, at which point its behavior will depend on how the programmer has decided to handle values that are not real numbers. most forms of regularization are able to guarantee the convergence of iterative methods applied to underdetermined problems. for example, weight decay will cause gradient descent to quit increasin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 254 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning the moore - penrose pseudoinverse. recall that one definition of the pseudoinverse x + of a matrix is x x + = lim α0 ( xx i + α ) −1x. ( 7. 29 ) we can now recognize equation as performing linear regression with weight 7. 29 decay. specifically, equation is the limit of equatio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 255 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
classification. a classifier needs to take a compli - cated, high dimensional input x and summarize it with a single category identity y. this means that the main task facing a classifier is to be invariant to a wide variety of transformations. we can generate new ( x, y ) pairs easily just by transforming the inputs in o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 255 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##tion and pooling techniques described in chapter. many other operations such as rotating the image or scaling 9 the image have also proven quite [UNK]. one must be careful not to apply transformations that would change the correct class. for example, optical character recognition tasks require recognizing the [UNK] b... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 255 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning there are also transformations that we would like our classifiers to be invariant to, but which are not easy to perform. for example, out - of - plane rotation can not be implemented as a simple geometric operation on the input pixels. dataset augmentation is [UNK] for speech ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 256 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##oising autoencoder ( vincent 2008 et al., ). noise injection also works when the noise is applied to the hidden units, which can be seen as doing dataset augmentation at multiple levels of abstraction. poole 2014 et al. ( ) recently showed that this approach can be highly [UNK] provided that the magnitude of the nois... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 256 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##set augmentation schemes. suppose that algorithm a performs poorly with no dataset augmentation and algorithm b performs well when combined with numerous synthetic transformations of the input. in such a case it is likely the synthetic transformations caused the improved performance, rather than the use of machine le... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 256 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning 7. 5 noise robustness section has motivated the use of noise applied to the inputs as a dataset 7. 4 augmentation strategy. for some models, the addition of noise with infinitesimal variance at the input of the model is equivalent to imposing a penalty on the norm of the weigh... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 257 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a stochastic implementation of bayesian inference over the weights. the bayesian treatment of learning would consider the model weights to be uncertain and representable via a probability distribution that reflects this uncertainty. adding noise to the weights is a practical, stochastic way to reflect this uncertainty. n... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 257 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
now assume that with each input presentation we also include a random perturbation w [UNK] ( ; 0, ηi ) of the network weights. let us imagine that we have a standard l - layer mlp. we denote the perturbed model as [UNK] ( x ). despite the injection of noise, we are still interested in minimizing the squared error of th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 257 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning ηep, y ( x ) ∇w [UNK] ( ) x 2. this form of regularization encourages the parameters to go to regions of parameter space where small perturbations of the weights have a relatively small influence on the output. in other words, it pushes the model into regions where the model i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 258 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
labels. it can be harmful to maximize log p ( y | x ) when y is a mistake. one way to prevent this is to explicitly model the noise on the labels. for example, we can assume that for some small constant, the training set label y is correct with probability 1−, and otherwise any of the other possible labels might be cor... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 258 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
scenario using other regularization strategies like weight decay. label smoothing has the advantage of preventing the pursuit of hard probabilities without discouraging correct classification. this strategy has been used since the 1980s and continues to be featured prominently in modern neural networks ( szegedy et al.,... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 258 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning that examples from the same class have similar representations. unsupervised learning can provide useful cues for how to group examples in representation space. examples that cluster tightly in the input space should be mapped to similar representations. a linear classifier in... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 259 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
one ( such as −log p ( x ) or −log p ( x y, ) ). the generative criterion then expresses a particular form of prior belief about the solution to the supervised learning problem (, ), namely that the structure of lasserre et al. 2006 p ( x ) is connected to the structure of p ( y x | ) in a way that is captured by the s... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 259 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
information about semi - supervised learning. chapelle et al. 2006 7. 7 multi - task learning multi - task learning (, ) is a way to improve generalization by pooling caruana 1993 the examples ( which can be seen as soft constraints imposed on the parameters ) arising out of several tasks. in the same way that addition... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 259 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning factors. the model can generally be divided into two kinds of parts and associated parameters : 1. task - specific parameters ( which only benefit from the examples of their task to achieve good generalization ). these are the upper layers of the neural network in figure. 7. 2 2... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 260 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. the lower layers of a deep network ( whether it is supervised and feedforward or includes a generative component with downward arrows ) can be shared across such tasks, while task - specific parameters ( associated respectively with the weights into and from h ( 1 ) and h ( 2 ) ) can be learned on top of those yieldin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 260 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the factors that explain some of the input variations but are not relevant for predicting y ( 1 ) or y ( 2 ). improved generalization and generalization error bounds (, ) can be baxter 1995 achieved because of the shared parameters, for which statistical strength can be 245 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 260 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning 0 50 100 150 200 250 time ( epochs ) 0 00. 0 05. 0 10. 0 15. 0 20. loss ( negative log - likelihood ) training set loss validation set loss figure 7. 3 : learning curves showing how the negative log - likelihood loss changes over time ( indicated as number of training iterati... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 261 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the factors that explain the variations observed in the data associated with the [UNK] tasks, some are shared across two or more tasks. 7. 8 early stopping when training large models with [UNK] representational capacity to overfit the task, we often observe that training error decreases steadily over time, but validatio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 261 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning algorithm terminates when no parameters have improved over the best recorded validation error for some pre - specified number of iterations. this procedure is specified more formally in algorithm. 7. 1 algorithm 7. 1 the early stopping meta - algorithm for determining the best ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 262 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##←i v v ← else j j ← + 1 end if end while best parameters are θ∗, best number of training steps is i∗ this strategy is known as early stopping. it is probably the most commonly used form of regularization in deep learning. its popularity is due both to its [UNK] and its simplicity. one way to think of early stopping i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 262 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning performance curve. most hyperparameters that control model capacity have such a u - shaped validation set performance curve, as illustrated in figure. in the case of 5. 3 early stopping, we are controlling the [UNK] capacity of the model by determining how many steps it can ta... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 263 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
available, then the cost of these periodic evaluations may be reduced by using a validation set that is small compared to the training set or by evaluating the validation set error less frequently and obtaining a lower resolution estimate of the optimal training time. an additional cost to early stopping is the need to... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 263 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
weight decay, where one must be careful not to use too much weight decay and trap the network in a bad local minimum corresponding to a solution with pathologically small weights. early stopping may be used either alone or in conjunction with other regulariza - tion strategies. even when using regularization strategies... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 263 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning of the data. in this second training pass, we train for the same number of steps as the early stopping procedure determined was optimal in the first pass. there are some subtleties associated with this procedure. for example, there is not a good way of knowing whether to retra... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 264 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
subtrain and y ( ) subtrain for training data and x ( valid ) and y ( valid ) for validation data. this returns i∗, the optimal number of steps. set to random values again. θ train on x ( ) train and y ( ) train for i∗steps. another strategy for using all of the data is to keep the parameters obtained from the first rou... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 264 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. this procedure is presented more formally in algorithm. 7. 3 early stopping is also useful because it reduces the computational cost of the training procedure. besides the obvious reduction in cost due to limiting the number of training iterations, it also has the benefit of providing regularization without requiring ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 264 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning algorithm 7. 3 meta - algorithm using early stopping to determine at what objec - tive value we start to overfit, then continue training until that value is reached. let x ( ) train and y ( ) train be the training set. split x ( ) train and y ( ) train into ( x ( ) subtrain, x... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 265 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
model? bishop ( ) and ( ) argued that early stopping has the [UNK] of 1995a sjoberg and ljung 1995 restricting the optimization procedure to a relatively small volume of parameter space in the neighborhood of the initial parameter value θo, as illustrated in figure. more specifically, imagine taking 7. 4 τ optimization s... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 265 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( θ = w ). we can model the cost function j with a quadratic approximation in the neighborhood of the empirically optimal value of the weights w∗ : [UNK] j ( ) = θ ( w∗ ) + 1 2 ( w w − ∗ ) h w w ( − ∗ ), ( 7. 33 ) where h is the hessian matrix of j with respect to w evaluated at w∗. given the assumption that w∗is a min... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 265 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning w1 w2 w∗ [UNK] w w1 w2 w∗ [UNK] w figure 7. 4 : an illustration of the [UNK] of early stopping. ( left ) the solid contour lines indicate the contours of the negative log - likelihood. the dashed line indicates the trajectory taken by sgd beginning from the origin. rather tha... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 266 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
by analyzing gradient descent on [UNK] : w ( ) τ = w ( 1 ) τ− −∇ w [UNK] ( w ( 1 ) τ− ) ( 7. 35 ) = w ( 1 ) τ− −h w ( ( 1 ) τ− −w ∗ ) ( 7. 36 ) w ( ) τ −w∗ = ( ) ( i h − w ( 1 ) τ− −w∗ ). ( 7. 37 ) let us now rewrite this expression in the space of the eigenvectors ofh, exploiting the eigendecomposition of h : h = q q ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 266 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
) ( 7. 39 ) 3for neural networks, to obtain symmetry breaking between hidden units, we cannot initialize all the parameters to 0, as discussed in section. however, the argument holds for any other 6. 2 initial value w ( 0 ). 251 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 266 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning assuming that w ( 0 ) = 0 and that is chosen to be small enough to guarantee | 1−λi | < 1, the parameter trajectory during training after τ parameter updates is as follows : qw ( ) τ = [ ( ) i −i −λ τ ] qw∗. ( 7. 40 ) now, the expression for [UNK] in equation for 7. 13 l2 reg... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 267 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
under the quadratic approximation of the objective function ). going even further, by taking logarithms and using the series expansion for log ( 1 + x ), we can conclude that if all λi are small ( that is, λi 1 and λi / α 1 ) then τ ≈1 α, ( 7. 44 ) α ≈1 τ. ( 7. 45 ) that is, under these assumptions, the number of train... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 267 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
point that corresponds to a minimum of the l2 - regularized objective. early stopping is of course more than the mere restriction of the trajectory length ; instead, early stopping typically involves monitoring the validation set error in order to stop the trajectory at a particularly good point in space. early stoppin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 267 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning 7. 9 parameter tying and parameter sharing thus far, in this chapter, when we have discussed adding constraints or penalties to the parameters, we have always done so with respect to a fixed region or point. for example, l2 regularization ( or weight decay ) penalizes model pa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 268 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
w ( ) b. the two models map the input to two [UNK], but related outputs : [UNK] ( ) a = f ( w ( ) a, x ) and [UNK] ( ) b = ( g w ( ) b, x ). let us imagine that the tasks are similar enough ( perhaps with similar input and output distributions ) that we believe the model parameters should be close to each other : [UNK]... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 268 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of another model, trained in an unsupervised paradigm ( to capture the distribution of the observed input data ). the architectures were constructed such that many of the parameters in the classifier model could be paired to corresponding parameters in the unsupervised model. while a parameter norm penalty is one way to... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 268 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
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