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chapter 7. regularization for deep learning convolutional neural networks by far the most popular and extensive use of parameter sharing occurs in convolutional neural networks ( cnns ) applied to computer vision. natural images have many statistical properties that are invariant to translation. for example, a photo of... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 269 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
detail in chapter. 9 7. 10 sparse representations weight decay acts by placing a penalty directly on the model parameters. another strategy is to place a penalty on the activations of the units in a neural network, encouraging their activations to be sparse. this indirectly imposes a complicated penalty on the model pa... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 269 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning −14 1 19 2 23 = 3 1 2 5 4 1 − − 4 2 3 1 1 3 − − − − − 1 5 4 2 3 2 3 1 2 3 0 3 − − − − − − 5 4 2 2 5 1 0 2 0 0 −3 0 y ∈rm b ∈rm n × h ∈rn ( 7. 47 ) in the first expression, we have an example of a sparsely parametrized linear regression model. in the second, we have linear regr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 270 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
before, we denote the regularized loss function by [UNK] : [UNK], j, α ( ; θ x y ) = ( ; θ x y ) + ω ( ) h ( 7. 48 ) where α ∈ [ 0, ∞ ) weights the relative contribution of the norm penalty term, with larger values of corresponding to more regularization. α just as an l1 penalty on the parameters induces parameter spar... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 270 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##fellow 2009 et al. ( ) and et al. ( ) both provide examples of strategies based on regularizing the average activation across several examples, 1 m i h ( ) i, to be near some target value, such as a vector with. 01 for each entry. other approaches obtain representational sparsity with a hard constraint on the activat... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 270 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning omp - k with the value of k specified to indicate the number of non - zero features allowed. ( ) demonstrated that omp - can be a very [UNK] coates and ng 2011 1 feature extractor for deep architectures. essentially any model that has hidden units can be made sparse. throughou... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 271 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. consider for example a set of k regression models. suppose that each model makes an error i on each example, with the errors drawn from a zero - mean multivariate normal distribution with variances e [ 2 i ] = v and covariances e [ ij ] = c. then the error made by the average prediction of all the ensemble models is ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 271 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
size. in other words, on average, the ensemble will perform at least as well as any of its members, and if the members make independent errors, the ensemble will perform significantly better than its members. [UNK] ensemble methods construct the ensemble of models in [UNK] ways. for example, each member of the ensemble ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 271 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning 8 8 first ensemble member second ensemble member original dataset first resampled dataset second resampled dataset figure 7. 5 : a cartoon depiction of how bagging works. suppose we train an 8 detector on the dataset depicted above, containing an 8, a 6 and a 9. suppose we ma... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 272 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the 8 are present. [UNK] kind of model using a [UNK] algorithm or objective function. bagging is a method that allows the same kind of model, training algorithm and objective function to be reused several times. specifically, bagging involves constructing k [UNK] datasets. each dataset has the same number of examples as... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 272 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##fit from model averaging even if all of the models are trained on the same dataset. [UNK] in random initialization, random selection of minibatches, [UNK] in hyperparameters, or [UNK] outcomes of non - deterministic imple - mentations of neural networks are often enough to cause [UNK] members of the 257 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 272 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning ensemble to make partially independent errors. model averaging is an extremely powerful and reliable method for reducing generalization error. its use is usually discouraged when benchmarking algorithms for scientific papers, because any machine learning algorithm can benefit s... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 273 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##entally adding neural networks to the ensemble. boosting has also been applied interpreting an individual neural network as an ensemble (, ), incrementally adding hidden bengio et al. 2006a units to the neural network. 7. 12 dropout dropout ( srivastava 2014 et al., ) provides a computationally inexpensive but powerf... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 273 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
inexpensive approximation to training and evaluating a bagged ensemble of exponentially many neural networks. specifically, dropout trains the ensemble consisting of all sub - networks that can be formed by removing non - output units from an underlying base network, as illustrated in figure. in most modern neural networ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 273 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning the [UNK] between the unit ’ s state and some reference value. here, we present the dropout algorithm in terms of multiplication by zero for simplicity, but it can be trivially modified to work with other operations that remove a unit from the network. recall that to learn wit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 274 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
sampling a mask value of one ( causing a unit to be included ) is a hyperparameter fixed before training begins. it is not a function of the current value of the model parameters or the input example. typically, an input unit is included with probability 0. 8 and a hidden unit is included with probability 0. 5. we then ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 274 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
are all independent. in the case of dropout, the models share parameters, with each model inheriting a [UNK] subset of parameters from the parent neural network. this parameter sharing makes it possible to represent an exponential number of models with a tractable amount of memory. in the case of bagging, each model is... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 274 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning y h1 h1 h2 h2 x1 x1 x2 x2 y h1 h1 h2 h2 x1 x1 x2 x2 y h1 h1 h2 h2 x2 x2 y h1 h1 h2 h2 x1 x1 y h2 h2 x1 x1 x2 x2 y h1 h1 x1 x1 x2 x2 y h1 h1 h2 h2 y x1 x1 x2 x2 y h2 h2 x2 x2 y h1 h1 x1 x1 y h1 h1 x2 x2 y h2 h2 x1 x1 y x1 x1 y x2 x2 y h2 h2 y h1 h1 y base network ensemble of s... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 275 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
network. here, we begin with a base network with two visible units and two hidden units. there are sixteen possible subsets of these four units. we show all sixteen subnetworks that may be formed by dropping out [UNK] subsets of units from the original network. in this small example, a large proportion of the resulting... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 275 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning [UNK] µx1 µx1 x1 x1 [UNK] x2 x2 µx2 µx2 h1 h1 h2 h2 µh1 µh1 µh2 µh2 [UNK] [UNK] h1 [UNK] y y h1 h1 h2 h2 x1 x1 x2 x2 figure 7. 7 : an example of forward propagation through a feedforward network using dropout. ( top ) in this example, we use a feedforward network with two inp... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 276 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning to make a prediction, a bagged ensemble must accumulate votes from all of its members. we refer to this process as inference in this context. so far, our description of bagging and dropout has not required that the model be explicitly probabilistic. now, we assume that the mo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 277 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
p ( µ ) is the probability distribution that was used to sample µ at training time. because this sum includes an exponential number of terms, it is intractable to evaluate except in cases where the structure of the model permits some form of simplification. so far, deep neural nets are not known to permit any tractable ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 277 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
be a probability distribution. to guarantee that the result is a probability distribution, we impose the requirement that none of the sub - models assigns probability 0 to any event, and we renormalize the resulting distribution. the unnormalized probability distribution defined directly by the geometric mean is given b... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 277 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning also possible. to make predictions we must re - normalize the ensemble : pensemble ( ) = y | x [UNK] ( ) y | x y [UNK] ( y | x ). ( 7. 55 ) a key insight (, ) involved in dropout is that we can approxi - hinton et al. 2012c mate pensemble by evaluating p ( y | x ) in one mode... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 278 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
2 the model as usual. another way to achieve the same result is to multiply the states of the units by during training. either way, the goal is to make sure that 2 the expected total input to a unit at test time is roughly the same as the expected total input to that unit at train time, even though half the units at tr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 278 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##izing the geometric mean over all ensemble members ’ predictions : pensemble ( = ) = y y | v [UNK] ( = ) y y | v [UNK] ( = y y | v ) ( 7. 58 ) where [UNK] ( = ) = y y | v 2n d∈ { } 0 1, n p y. ( = y | v ; ) d ( 7. 59 ) 263 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 278 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning to see that the weight scaling rule is exact, we can simplify [UNK] : [UNK] ( = ) = y y | v 2n d∈ { } 0 1, n p y ( = y | v ; ) d ( 7. 60 ) = 2n d∈ { } 0 1, n softmax ( w ( ) + ) d v b y ( 7. 61 ) = 2n d∈ { } 0 1, n exp w y, : ( ) + d v by yexp w y, : ( ) + d v by ( 7. 62 ) = ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 279 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
y | v [UNK] 2n d∈ { } 0 1, n exp w y, : ( ) + d v by ( 7. 64 ) = exp 1 2 n d∈ { } 0 1, n w y, : ( ) + d v by ( 7. 65 ) = exp 1 2w y, : v + by. ( 7. 66 ) substituting this back into equation we obtain a softmax classifier with weights 7. 58 1 2w. the weight scaling rule is also exact in other settings, including regressi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 279 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
approximations to the ensemble predictor. this held true even when the monte carlo approximation was allowed to sample up to 1, 000 sub - networks. ( ) found gal and ghahramani 2015 that some models obtain better classification accuracy using twenty samples and 264 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 279 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning the monte carlo approximation. it appears that the optimal choice of inference approximation is problem - dependent. srivastava 2014 et al. ( ) showed that dropout is more [UNK] than other standard computationally inexpensive regularizers, such as weight decay, filter norm con... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 280 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
advantage of dropout is that it does not significantly limit the type of model or training procedure that can be used. it works well with nearly any model that uses a distributed representation and can be trained with stochastic gradient descent. this includes feedforward neural networks, probabilistic models such as re... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 280 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
much lower when using dropout, but this comes at the cost of a much larger model and many more iterations of the training algorithm. for very large datasets, regularization confers little reduction in generalization error. in these cases, the computational cost of using dropout and larger models may outweigh the benefit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 280 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning each input feature. the magnitude of each feature ’ s weight decay [UNK] is determined by its variance. similar results hold for other linear models. for deep models, dropout is not equivalent to weight decay. the stochasticity used while training with dropout is not necessar... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 281 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
problem. just as stochasticity is not necessary to achieve the regularizing [UNK] of dropout, it is also not [UNK]. to demonstrate this, warde - farley 2014 et al. ( ) designed control experiments using a method called dropout boosting that they designed to use exactly the same mask noise as traditional dropout but lac... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 281 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
approaches to training exponentially large ensembles of models that share weights. dropconnect is a special case of dropout where each product between a single scalar weight and a single hidden unit state is considered a unit that can be dropped ( wan 2013 et al., ). stochastic pooling is a form of randomized pooling (... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 281 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning dropout as bagging an ensemble of models formed by including or excluding units. however, there is no need for this model averaging strategy to be based on inclusion and exclusion. in principle, any kind of random modification is admissible. in practice, we must choose modifica... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 282 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
implements approximate inference in the ensemble, without needing any weight scaling. so far we have described dropout purely as a means of performing [UNK], approximate bagging. however, there is another view of dropout that goes further than this. dropout trains not just a bagged ensemble of models, but an ensemble o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 282 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
good in many contexts. warde - farley 2014 et al. ( ) compared dropout training to training of large ensembles and concluded that dropout [UNK] additional improvements to generalization error beyond those obtained by ensembles of independent models. it is important to understand that a large portion of the power of dro... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 282 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning the image is removed. destroying extracted features rather than original values allows the destruction process to make use of all of the knowledge about the input distribution that the model has acquired so far. another important aspect of dropout is that the noise is multipl... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 283 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. 13 adversarial training in many cases, neural networks have begun to reach human performance when evaluated on an i. i. d. test set. it is natural therefore to wonder whether these models have obtained a true human - level understanding of these tasks. in order to probe the level of understanding a network has of the... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 283 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
for example, in computer security, that are beyond the scope of this chapter. however, they are interesting in the context of regularization because one can reduce the error rate on the original i. i. d. test set via adversarial training — training on adversarially perturbed examples from the training set (, ; szegedy ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 283 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning +. 007 × = x sign ( ∇xj ( θ x,, y ) ) x + sign ( ∇xj ( θ x,, y ) ) y = “ panda ” “ nematode ” “ gibbon ” w / 57. 7 % confidence w / 8. 2 % confidence w / 99. 3 % confidence figure 7. 8 : a demonstration of adversarial example generation applied to googlenet (, ) on imagenet. by ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 284 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. if we change each input by, then a linear function with weights w can change by as much as | | | | w 1, which can be a very large amount if w is high - dimensional. adversarial training discourages this highly sensitive locally linear behavior by encouraging the network to be locally constant in the neighborhood of t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 284 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
is not associated with a label in the dataset, the model itself assigns some label [UNK]. the model ’ s label [UNK] may not be the true label, but if the model is high quality, then [UNK] has a high probability of providing the true label. we can seek an adversarial example xthat causes the classifier to output a label ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 284 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning robust to small changes anywhere along the manifold where the unlabeled data lies. the assumption motivating this approach is that [UNK] classes usually lie on disconnected manifolds, and a small perturbation should not be able to jump from one class manifold to another class... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 285 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the classifier should be invariant to the local factors of variation that correspond to movement on the manifold, it would make sense to use as nearest - neighbor distance between points x1 and x2 the distance between the manifolds m1 and m2 to which they respectively belong. although that may be computationally [UNK] (... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 285 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
locally invariant to known factors of variation. these factors of variation correspond to movement along the manifold near which examples of the same class concentrate. local invariance is achieved by requiring ∇xf ( x ) to be orthogonal to the known manifold tangent vectors v ( ) i at x, or equivalently that the direc... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 285 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning this regularizer can of course be scaled by an appropriate hyperparameter, and, for most neural networks, we would need to sum over many outputs rather than the lone output f ( x ) described here for simplicity. as with the tangent distance algorithm, the tangent vectors are ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 286 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##tesimal amount of these transformations. tangent propagation does not require explicitly visiting a new input point. instead, it analytically regularizes the model to resist perturbation in the directions corresponding to the specified transformation. while this analytical approach is intellectually elegant, it has tw... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 286 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to double backprop ( drucker and lecun, 1992 ) and adversarial training (, ;, ). szegedy et al. 2014b goodfellow et al. 2014b double backprop regularizes the jacobian to be small, while adversarial training finds inputs near the original inputs and trains the model to produce the same output on these as on the original ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 286 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning x1 x2 normal tangent figure 7. 9 : illustration of the main idea of the tangent prop algorithm (, simard et al. 1992 rifai 2011c ) and manifold tangent classifier ( et al., ), which both regularize the classifier output function f ( x ). each curve represents the manifold for a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 287 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
f ( x ) to not change very much asx moves along the manifold. tangent propagation requires the user to manually specify functions that compute the tangent directions ( such as specifying that small translations of images remain in the same class manifold ) while the manifold tangent classifier estimates the manifold tan... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 287 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##r to learn the manifold structure by unsupervised learning, and ( 2 ) use these tangents to regularize a neural net classifier as in tangent prop ( equation ). 7. 67 this chapter has described most of the general strategies used to regularize neural networks. regularization is a central theme of machine learning and a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 287 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 7. regularization for deep learning will be revisited periodically by most of the remaining chapters. another central theme of machine learning is optimization, described next. 273 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 288 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8 optimization for training deep models deep learning algorithms involve optimization in many contexts. for example, performing inference in models such as pca involves solving an optimization problem. we often use analytical optimization to write proofs or design algorithms. of all of the many optimization pro... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 289 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
evaluated on the entire training set as well as additional regularization terms. we begin with a description of how optimization used as a training algorithm for a machine learning task [UNK] from pure optimization. next, we present several of the concrete challenges that make optimization of neural networks [UNK]. we ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 289 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models the second derivatives of the cost function. finally, we conclude with a review of several optimization strategies that are formed by combining simple optimization algorithms into higher - level procedures. 8. 1 how learning [UNK] from pure optimization optimization algo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 290 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
x, y [UNK] f, y, ( ( ; ) x θ ) ( 8. 1 ) where l is the per - example loss function, f ( x ; θ ) is the predicted output when the input is x, [UNK] is the empirical distribution. in the supervised learning case, y is the target output. throughout this chapter, we develop the unregularized supervised case, where the argu... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 290 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( ( ; ) x θ ) ( 8. 2 ) 8. 1. 1 empirical risk minimization the goal of a machine learning algorithm is to reduce the expected generalization error given by equation. this quantity is known as the 8. 2 risk. we emphasize here that the expectation is taken over the true underlying distribution pdata. if we knew the true ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 290 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models solvable by an optimization algorithm. however, when we do not know pdata ( x, y ) but only have a training set of samples, we have a machine learning problem. the simplest way to convert a machine learning problem back into an op - timization problem is to minimize the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 291 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##imizing the risk directly, we optimize the empirical risk, and hope that the risk decreases significantly as well. a variety of theoretical results establish conditions under which the true risk can be expected to decrease by various amounts. however, empirical risk minimization is prone to overfitting. models with hig... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 291 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
function we actually care about ( say classification error ) is not one that can be optimized [UNK]. for example, exactly minimizing expected 0 - 1 loss is typically intractable ( exponential in the input dimension ), even for a linear classifier ( marcotte and savard 1992, ). in such situations, one typically optimizes ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 291 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models in some cases, a surrogate loss function actually results in being able to learn more. for example, the test set 0 - 1 loss often continues to decrease for a long time after the training set 0 - 1 loss has reached zero, when training using the log - likelihood surrogate.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 292 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##fied. typically the early stopping criterion is based 7. 8 on the true underlying loss function, such as 0 - 1 loss measured on a validation set, and is designed to cause the algorithm to halt whenever overfitting begins to occur. training often halts while the surrogate loss function still has large derivatives, which... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 292 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( x ( ) i, y ( ) i ; ) θ. ( 8. 4 ) maximizing this sum is equivalent to maximizing the expectation over the empirical distribution defined by the training set : j ( ) = θ ex, [UNK] log pmodel ( ; ) x, y θ. ( 8. 5 ) most of the properties of the objective function j used by most of our opti - mization algorithms are also... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 292 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models most commonly used property is the gradient : ∇θj ( ) = θ ex, [UNK] log pmodel ( ; ) x, y θ. ( 8. 6 ) computing this expectation exactly is very expensive because it requires evaluating the model on every example in the entire dataset. in practice, we can compute these e... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 293 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. most optimization algorithms converge much faster ( in terms of total computation, not in terms of number of updates ) if they are allowed to rapidly compute approximate estimates of the gradient rather than slowly computing the exact gradient. another consideration motivating statistical estimation of the gradient f... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 293 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##batch stochastic gradient descent. typically the term “ batch gradient descent ” implies the use of the full training set, while the use of the term “ batch ” to describe a group of examples does not. for example, it is very common to use the term “ batch size ” to describe the size of a minibatch. optimization algor... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 293 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models than one but less than all of the training examples. these were traditionally called minibatch or minibatch stochastic methods and it is now common to simply call them stochastic methods. the canonical example of a stochastic method is stochastic gradient descent, presen... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 294 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
specific sizes of arrays. especially when using gpus, it is common for power of 2 batch sizes to [UNK] better runtime. typical power of 2 batch sizes range from 32 to 256, with 16 sometimes being attempted for large models. • small batches can [UNK] a regularizing [UNK] (, ), wilson and martinez 2003 perhaps due to the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 294 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
information in ways that amplify sampling errors more. methods that compute updates based only on the gradient g are usually relatively robust and can handle smaller batch sizes like 100. second - order methods, which use also the hessian matrix h and compute updates such as h−1g, typically require much larger batch si... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 294 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models h or its inverse amplifies pre - existing errors, in this case, estimation errors in g. very small changes in the estimate of g can thus cause large changes in the update h−1g, even if h were estimated perfectly. of course, h will be estimated only approximately, so the u... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 295 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
that first we have five blood samples taken at [UNK] times from the first patient, then we have three blood samples taken from the second patient, then the blood samples from the third patient, and so on. if we were to draw examples in order from this list, then each of our minibatches would be extremely biased, because i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 295 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
trained thereafter will use, and each individual model will be forced to reuse this ordering every time it passes through the training data. however, this deviation from true random selection does not seem to have a significant detrimental [UNK]. failing to ever [UNK] the examples in any way can seriously reduce the [UN... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 295 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models descent [UNK] the dataset once and then pass through it multiple times. on the first pass, each minibatch is used to compute an unbiased estimate of the true generalization error. on the second pass, the estimate becomes biased because it is formed by re - sampling values... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 296 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
are discrete. in this case, the generalization error ( equation ) can be written as a sum 8. 2 j∗ ( ) = θ x y pdata ( ) ( ( ; ) ) x, y l f x θ, y, ( 8. 7 ) with the exact gradient g = ∇θj ∗ ( ) = θ x y pdata ( ) x, y ∇θl f, y. ( ( ; ) x θ ) ( 8. 8 ) we have already seen the same fact demonstrated for the log - likeliho... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 296 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##r - responding targets y ( ) i from the data generating distribution pdata, and computing the gradient of the loss with respect to the parameters for that minibatch : [UNK] = 1 m∇θ i l f ( ( x ( ) i ; ) θ, y ( ) i ). ( 8. 9 ) updating in the direction of θ [UNK] performs sgd on the generalization error. of course, th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 296 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models of course, the additional epochs usually provide enough benefit due to decreased training error to [UNK] the harm they cause by increasing the gap between training error and test error. with some datasets growing rapidly in size, faster than computing power, it is becomin... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 297 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
- convex case. even convex optimization is not without its complications. in this section, we summarize several of the most prominent challenges involved in optimization for training deep models. 8. 2. 1 ill - conditioning some challenges arise even when optimizing convex functions. of these, the most prominent is ill ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 297 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models −50 0 50 100 150 200 250 training time ( epochs ) −2 0 2 4 6 8 10 12 14 16 gradient norm 0 50 100 150 200 250 training time ( epochs ) 0 1. 0 2. 0 3. 0 4. 0 5. 0 6. 0 7. 0 8. 0 9. 1 0. classification error rate figure 8. 1 : gradient descent often does not arrive at a cri... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 298 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
) gradient, the training process is reasonably successful. the validation set classification error decreases to a low level. the ghg term. in many cases, the gradient norm does not shrink significantly throughout learning, but the ghg term grows by more than an order of magnitude. the result is that learning becomes very... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 298 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models guaranteed to be a global minimum. some convex functions have a flat region at the bottom rather than a single global minimum point, but any point within such a flat region is an acceptable solution. when optimizing a convex function, we know that we have reached a good so... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 299 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##fiable because we can obtain equivalent models by exchanging latent variables with each other. for example, we could take a neural network and modify layer 1 by swapping the incoming weight vector for unit i with the incoming weight vector for unit j, then doing the same for the outgoing weight vectors. if we have m l... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 299 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
or maxout network lies on an ( m n × ) - dimensional hyperbola of equivalent local minima. these model identifiability issues mean that there can be an extremely large or even uncountably infinite amount of local minima in a neural network cost function. however, all of these local minima arising from non - identifiabilit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 299 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models for networks of practical interest and whether optimization algorithms encounter them. for many years, most practitioners believed that local minima were a common problem plaguing neural network optimization. today, that does not appear to be the case. the problem remain... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 300 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
if the norm of the gradient does not shrink to insignificant size, the problem is neither local minima nor any other kind of critical point. this kind of negative test can rule out local minima. in high dimensional spaces, it can be very [UNK] to positively establish that local minima are the problem. many structures ot... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 300 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
think of a saddle point as being a local minimum along one cross - section of the cost function and a local maximum along another cross - section. see figure for an illustration. 4. 5 many classes of random functions exhibit the following behavior : in low - dimensional spaces, local minima are common. in higher dimensi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 300 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models be heads. see ( ) for a review of the relevant theoretical work. dauphin et al. 2014 an amazing property of many random functions is that the eigenvalues of the hessian become more likely to be positive as we reach regions of lower cost. in our coin tossing analogy, this... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 301 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
with higher cost than the global minimum. they observed without proof that these results extend to deeper networks without nonlinearities. the output of such networks is a linear function of their input, but they are useful to study as a model of nonlinear neural networks because their loss function is a non - convex f... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 301 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
that use only gradient information, the situation is unclear. the gradient can often become very small near a saddle point. on the other hand, gradient descent empirically seems to be able to escape saddle points in many cases. ( ) provided visualizations of goodfellow et al. 2015 several learning trajectories of state... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 301 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models projection 2 of θ projection 1 of θ j ( ) θ figure 8. 2 : a visualization of the cost function of a neural network. image adapted with permission from goodfellow 2015 et al. ( ). these visualizations appear similar for feedforward neural networks, convolutional networks,... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 302 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models gradient descent is designed to move “ downhill ” and is not explicitly designed to seek a critical point. newton ’ s method, however, is designed to solve for a point where the gradient is zero. without appropriate modification, it can jump to a saddle point. the prolife... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 303 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
maxima of many classes of random functions become exponentially rare in high dimensional space, just like minima do. there may also be wide, flat regions of constant value. in these locations, the gradient and also the hessian are all zero. such degenerate locations pose major problems for all numerical optimization alg... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 303 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models figure 8. 3 : the objective function for highly nonlinear deep neural networks or for recurrent neural networks often contains sharp nonlinearities in parameter space resulting from the multiplication of several parameters. these nonlinearities give rise to very high der... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 304 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
size to be small enough that it is less likely to go outside the region where the gradient indicates the direction of approximately steepest descent. [UNK] structures are most common in the cost functions for recurrent neural networks, because such models involve a multiplication of many factors, with one factor for ea... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 304 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models by repeatedly applying the same operation at each time step of a long temporal sequence. repeated application of the same parameters gives rise to especially pronounced [UNK]. for example, suppose that a computational graph contains a path that consists of repeatedly mul... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 305 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. vanishing gradients make it [UNK] to know which direction the parameters should move to improve the cost function, while exploding gradients can make learning unstable. the [UNK] structures described earlier that motivate gradient clipping are an example of the exploding gradient phenomenon. the repeated multiplicati... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 305 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. 6 inexact gradients most optimization algorithms are designed with the assumption that we have access to the exact gradient or hessian matrix. in practice, we usually only have a noisy or even biased estimate of these quantities. nearly every deep learning algorithm relies on sampling - based estimates at least insof... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 305 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models with the more advanced models in part. for example, contrastive divergence iii gives a technique for approximating the gradient of the intractable log - likelihood of a boltzmann machine. various neural network optimization algorithms are designed to account for imperfec... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 306 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
goodfellow 2015 et al. ( ) argue that much of the runtime of training is due to the length of the trajectory needed to arrive at the solution. figure shows that 8. 2 the learning trajectory spends most of its time tracing out a wide arc around a mountain - shaped structure. much of research into the [UNK] of optimizati... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 306 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
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