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##ily close to zero if the model is able to correctly classify every example in the training set, but it is impossible to actually reach the value of zero. likewise, a model of real values p ( y | x ) = n ( y ; f ( θ ), β−1 ) can have negative log - likelihood that asymptotes to negative infinity — if f ( θ ) is able to... | /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 |
chapter 8. optimization for training deep models θ j ( ) θ figure 8. 4 : optimization based on local downhill moves can fail if the local surface does not point toward the global solution. here we provide an example of how this can occur, even if there are no saddle points and no local minima. this example cost functio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 307 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
making small, local moves. the previous sections have primarily focused on how the correct direction of these local moves can be [UNK] to compute. we may be able to compute some properties of the objective function, such as its gradient, only approximately, with bias or variance in our estimate of the correct direction... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 307 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models high computational cost. sometimes local information provides us no guide, when the function has a wide flat region, or if we manage to land exactly on a critical point ( usually this latter scenario only happens to methods that solve explicitly for critical points, such ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 308 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
that local descent can follow, and if we are able to initialize learning within that well - behaved region. this last view suggests research into choosing good initial points for traditional optimization algorithms to use. 8. 2. 8 theoretical limits of optimization several theoretical results show that there are limits... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 308 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
using a larger network for which many more parameter settings correspond to an acceptable solution. moreover, in the context of neural network training, we usually do not care about finding the exact minimum of a function, but seek only to reduce its value [UNK] to obtain good generalization error. theoretical analysis ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 308 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models 8. 3 basic algorithms we have previously introduced the gradient descent ( section ) algorithm that 4. 3 follows the gradient of an entire training set downhill. this may be accelerated considerably by using stochastic gradient descent to follow the gradient of randomly ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 309 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
: initial parameter θ while do stopping criterion not met sample a minibatch of m examples from the training set { x ( 1 ),..., x ( ) m } with corresponding targets y ( ) i. compute gradient estimate : [UNK] ← + 1 m∇θ i l f ( ( x ( ) i ; ) θ, y ( ) i ) apply update : θ θ ← [UNK] end while a crucial parameter for the sg... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 309 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models ∞ k = 1 2 k <. ∞ ( 8. 13 ) in practice, it is common to decay the learning rate linearly until iteration : τ k = ( 1 ) −α 0 + ατ ( 8. 14 ) with α = k τ. after iteration, it is common to leave constant. τ the learning rate may be chosen by trial and error, but it is usual... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 310 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##cillations, with the cost function often increasing significantly. gentle oscillations are fine, especially if training with a stochastic cost function such as the cost function arising from the use of dropout. if the learning rate is too low, learning proceeds slowly, and if the initial learning rate is too low, learn... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 310 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of training examples becomes very large. for a large enough dataset, sgd may converge to within some fixed tolerance of its final test set error before it has processed the entire training set. to study the convergence rate of an optimization algorithm it is common to measure the excess error j ( θ ) −minθ j ( θ ), which... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 310 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models and bousquet 2008 ( ) argue that it therefore may not be worthwhile to pursue an optimization algorithm that converges faster than o ( 1 k ) for machine learning tasks — faster convergence presumably corresponds to overfitting. moreover, the asymptotic analysis obscures m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 311 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
2 momentum while stochastic gradient descent remains a very popular optimization strategy, learning with it can sometimes be slow. the method of momentum ( polyak 1964, ) is designed to accelerate learning, especially in the face of high curvature, small but consistent gradients, or noisy gradients. the momentum algori... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 311 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
momentum of the particle. a hyperparameter α ∈ [ 0, 1 ) determines how quickly the contributions of previous gradients exponentially decay. the update rule is given by : v v ←α −∇ θ 1 m m i = 1 l ( ( f x ( ) i ; ) θ, y ( ) i ), ( 8. 15 ) θ θ v ← +. ( 8. 16 ) the velocity v accumulates the gradient elements ∇θ 1 m m i =... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 311 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models − − − 30 20 10 0 10 20 −30 −20 −10 0 10 20 figure 8. 5 : momentum aims primarily to solve two problems : poor conditioning of the hessian matrix and variance in the stochastic gradient. here, we illustrate how momentum overcomes the first of these two problems. the contou... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 312 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models previously, the size of the step was simply the norm of the gradient multiplied by the learning rate. now, the size of the step depends on how large and how aligned a sequence of gradients are. the step size is largest when many successive gradients point in exactly the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 313 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
over time than to shrink over time. algorithm 8. 2 stochastic gradient descent ( sgd ) with momentum require : learning rate, momentum parameter. α require : initial parameter, initial velocity. θ v while do stopping criterion not met sample a minibatch of m examples from the training set { x ( 1 ),..., x ( ) m } with ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 313 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( ) = t ∂2 ∂t2 θ ( ) t. ( 8. 18 ) rather than viewing this as a second - order [UNK] equation of the position, we can introduce the variable v ( t ) representing the velocity of the particle at time t and rewrite the newtonian dynamics as a first - order [UNK] equation : v ( ) = t ∂ ∂tθ ( ) t, ( 8. 19 ) 298 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 313 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models f ( ) = t ∂ ∂tv ( ) t. ( 8. 20 ) the momentum algorithm then consists of solving the [UNK] equations via numerical simulation. a simple numerical method for solving [UNK] equations is euler ’ s method, which simply consists of simulating the dynamics defined by the equati... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 314 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
continues sliding in that direction until it begins to go uphill again. one other force is necessary. if the only force is the gradient of the cost function, then the particle might never come to rest. imagine a hockey puck sliding down one side of a valley and straight up the other side, oscillating back and forth for... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 314 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
traveling through the air experiences turbulent drag, with force proportional to the square of the velocity, while a particle moving along the ground experiences dry friction, with a force of constant magnitude. we can reject each of these options. turbulent drag, proportional to the square of the velocity, becomes ver... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 314 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models that the gradient can continue to cause motion until a minimum is reached, but strong enough to prevent motion if the gradient does not justify moving. 8. 3. 3 nesterov momentum sutskever 2013 et al. ( ) introduced a variant of the momentum algorithm that was inspired by... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 315 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the standard method of momentum. the complete nesterov momentum algorithm is presented in algorithm. 8. 3 in the convex batch gradient case, nesterov momentum brings the rate of convergence of the excess error from o ( 1 / k ) ( after k steps ) to o ( 1 / k2 ) as shown by nesterov 1983 ( ). unfortunately, in the stocha... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 315 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models 8. 4 parameter initialization strategies some optimization algorithms are not iterative by nature and simply solve for a solution point. other optimization algorithms are iterative by nature but, when applied to the right class of optimization problems, converge to accep... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 316 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
error, and the initial point can [UNK] the generalization as well. modern initialization strategies are simple and heuristic. designing improved initialization strategies is a [UNK] task because neural network optimization is not yet well understood. most initialization strategies are based on achieving some nice prope... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 316 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to a deterministic cost and model will constantly update both of these units in the same way. even if the model or training algorithm is capable of using stochasticity to compute [UNK] updates for [UNK] units ( for example, if one trains with dropout ), it is usually best to initialize each unit to compute a [UNK] func... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 316 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models motivates random initialization of the parameters. we could explicitly search for a large set of basis functions that are all mutually [UNK] from each other, but this often incurs a noticeable computational cost. for example, if we have at most as many outputs as inputs,... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 317 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##ian or uniform distribution. the choice of gaussian or uniform distribution does not seem to matter very much, but has not been exhaustively studied. the scale of the initial distribution, however, does have a large [UNK] on both the outcome of the optimization procedure and on the ability of the network to generaliz... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 317 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
descent step ). large weights may also result in extreme values that cause the activation function to saturate, causing complete loss of gradient through saturated units. these competing factors determine the ideal initial scale of the weights. the perspectives of regularization and optimization can give very [UNK] ins... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 317 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models due to triggering some early stopping criterion based on overfitting ) expresses a prior that the final parameters should be close to the initial parameters. recall from section that gradient descent with early stopping is equivalent to weight 7. 8 decay for some models. i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 318 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
should interact with each other, and how they should interact. some heuristics are available for choosing the initial scale of the weights. one heuristic is to initialize the weights of a fully connected layer with m inputs and n outputs by sampling each weight from u ( − 1 √m, 1 √m ), while glorot and bengio ( ) sugge... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 318 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a carefully chosen scaling or gain factor g that accounts for the nonlinearity applied at each layer. they derive specific values of the scaling factor for [UNK] types of nonlinear activation functions. this initialization scheme is also motivated by a model of a deep network as a sequence of matrix multiplies without n... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 318 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models 1, 000 layers, without needing to use orthogonal initializations. a key insight of this approach is that in feedforward networks, activations and gradients can grow or shrink on each step of forward or back - propagation, following a random walk behavior. this is because... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 319 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of optimization but inadvertently increase generalization error. in practice, we usually need to treat the scale of the weights as a hyperparameter whose optimal value lies somewhere roughly near but not exactly equal to the theoretical predictions. one drawback to scaling rules that set all of the initial weights to h... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 319 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
scheme can cause problems for units such as maxout units that have several filters that must be carefully coordinated with each other. when computational resources allow it, it is usually a good idea to treat the initial scale of the weights for each layer as a hyperparameter, and to choose these scales using a hyperpar... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 319 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models increasing its weights, it is possible to eventually obtain a network with reasonable initial activations throughout. if learning is still too slow at this point, it can be useful to look at the range or standard deviation of the gradients as well as the activations. thi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 320 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
there are a few situations where we may set some biases to non - zero values : • if a bias is for an output unit, then it is often beneficial to initialize the bias to obtain the right marginal statistics of the output. to do this, we assume that the initial weights are small enough that the output of the unit is determ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 320 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, and it can be very helpful to initialize the biases of such layers to match the marginal distribution over. x • sometimes we may want to choose the bias to avoid causing too much saturation at initialization. for example, we may set the bias of a relu hidden unit to 0. 1 rather than 0 to avoid saturating the relu at ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 320 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models can view h as a gate that determines whether uh u ≈ or uh ≈0. in these situations, we want to set the bias for h so that h ≈1 most of the time at initialization. otherwise u does not have a chance to learn. for example, jozefowicz 2015 et al. ( ) advocate setting the bia... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 321 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
correct marginal mean of the output, and set the variance parameters to the marginal variance of the output in the training set. besides these simple constant or random methods of initializing model parame - ters, it is possible to initialize model parameters using machine learning. a common strategy discussed in part ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 321 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##parameters that is the most [UNK] to set because it has a significant impact on model performance. as we have discussed in sections and, the 4. 3 8. 2 cost is often highly sensitive to some directions in parameter space and insensitive to others. the momentum algorithm can mitigate these issues somewhat, but does so a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 321 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models rate for each parameter, and automatically adapt these learning rates throughout the course of learning. the algorithm (, ) is an early heuristic approach delta - bar - delta jacobs 1988 to adapting individual learning rates for model parameters during training. the appr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 322 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to the square root of the sum of all of their historical squared values (, ). the duchi et al. 2011 parameters with the largest partial derivative of the loss have a correspondingly rapid decrease in their learning rate, while parameters with small partial derivatives have a relatively small decrease in their learning ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 322 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to a convex function. when applied to a non - convex function to train a neural network, the learning trajectory may pass through many [UNK] structures and eventually arrive at a region that is a locally convex bowl. adagrad shrinks the learning rate according to the entire history of the squared gradient and may 307 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 322 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models algorithm 8. 4 the adagrad algorithm require : global learning rate require : initial parameter θ require : small constant, perhaps δ 10−7, for numerical stability initialize gradient accumulation variable r = 0 while do stopping criterion not met sample a minibatch of m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 323 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
as if it were an instance of the adagrad algorithm initialized within that bowl. rmsprop is shown in its standard form in algorithm and combined with 8. 5 nesterov momentum in algorithm. compared to adagrad, the use of the 8. 6 moving average introduces a new hyperparameter, ρ, that controls the length scale of the mov... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 323 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
estimate of the first order moment ( with exponential weighting ) of the gradient. the most straightforward way to add momentum to rmsprop is to apply momentum to the rescaled gradients. the use of momentum in combination with rescaling does not have a clear theoretical motivation. second, adam includes 308 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 323 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models algorithm 8. 5 the rmsprop algorithm require : global learning rate, decay rate. ρ require : initial parameter θ require : small constant δ, usually 10−6, used to stabilize division by small numbers. initialize accumulation variables r = 0 while do stopping criterion not... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 324 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
moments to account for their initialization at the origin ( see algorithm ). rmsprop also incorporates an estimate of the 8. 7 ( uncentered ) second - order moment, however it lacks the correction factor. thus, unlike in adam, the rmsprop second - order moment estimate may have high bias early in training. adam is gene... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 324 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
adaptive learning rates ( represented by rmsprop and adadelta ) performed fairly robustly, no single best algorithm has emerged. currently, the most popular optimization algorithms actively in use include sgd, sgd with momentum, rmsprop, rmsprop with momentum, adadelta and adam. the choice of which algorithm to use, at... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 324 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models algorithm 8. 6 rmsprop algorithm with nesterov momentum require : global learning rate, decay rate, momentum [UNK]. ρ α require : initial parameter, initial velocity. θ v initialize accumulation variable r = 0 while do stopping criterion not met sample a minibatch of m e... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 325 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
we discuss the application of second - order methods to the training of deep networks. see ( ) for an earlier treatment of this subject. lecun et al. 1998a for simplicity of exposition, the only objective function we examine is the empirical risk : j ( ) = θ ex, [UNK] ( ) x, y [ ( ( ; ) ) ] = l f x θ, y 1 m m i = 1 l f... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 325 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models algorithm 8. 7 the adam algorithm require : step size ( suggested default : ) 0 001. require : exponential decay rates for moment estimates, ρ1 and ρ2 in [ 0, 1 ). ( suggested defaults : and respectively ) 0 9. 0 999. require : small constant δ used for numerical stabili... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 326 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
←ρ2r + ( 1 −ρ2 ) g g correct bias in first moment : [UNK] ← s 1−ρt 1 correct bias in second moment : [UNK] ← r 1−ρt 2 compute update : ∆ = θ − [UNK] √ [UNK] + δ ( operations applied element - wise ) apply update : θ θ θ ← + ∆ end while newton ’ s method is an optimization scheme based on using a second - order tay - lor... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 326 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
θ 0 ) ( 8. 27 ) thus for a locally quadratic function ( with positive definite h ), by rescaling the gradient by h −1, newton ’ s method jumps directly to the minimum. if the objective function is convex but not quadratic ( there are higher - order terms ), this update can be iterated, yielding the training algorithm as... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 326 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models algorithm 8. 8 newton ’ s method with objective j ( θ ) = 1 m m i = 1 l f ( ( x ( ) i ; ) θ, y ( ) i ). require : initial parameter θ0 require : training set of examples m while do stopping criterion not met compute gradient : g ←1 m∇θ i l f ( ( x ( ) i ; ) θ, y ( ) i ) ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 327 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
by updat - ing the quadratic approximation ). second, update the parameters according to equation. 8. 27 in section, we discussed how newton ’ s method is appropriate only when 8. 2. 3 the hessian is positive definite. in deep learning, the surface of the objective function is typically non - convex with many features, ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 327 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
’ s method, such as the levenberg – marquardt algorithm ( levenberg 1944 marquardt 1963, ;, ), and works fairly well as long as the negative eigenvalues of the hessian are still relatively close to zero. in cases where there are more extreme directions of curvature, the value of α would have to be [UNK] large to [UNK] ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 327 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models such as saddle points, the application of newton ’ s method for training large neural networks is limited by the significant computational burden it imposes. the number of elements in the hessian is squared in the number of parameters, so with k parameters ( and for even ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 328 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
calculation of the inverse hessian by iteratively descending conjugate directions. the inspiration for this approach follows from a careful study of the weakness of the method of steepest descent ( see section for details ), where line searches are applied iteratively in 4. 3 the direction associated with the gradient.... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 328 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##−1. this relationship between dt−1 and d t is illustrated in figure for 8. 6 multiple iterations of steepest descent. as demonstrated in the figure, the choice of orthogonal directions of descent do not preserve the minimum along the previous search directions. this gives rise to the zig - zag pattern of progress, wher... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 328 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models figure 8. 6 : the method of steepest descent applied to a quadratic cost surface. the method of steepest descent involves jumping to the point of lowest cost along the line defined by the gradient at the initial point on each step. this resolves some of the problems seen ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 329 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, where h is the hessian matrix. the straightforward way to impose conjugacy would involve calculation of the eigenvectors of h to choose βt, which would not satisfy our goal of developing a method that is more computationally viable than newton ’ s method for large problems. can we calculate the conjugate directions w... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 329 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models 2. polak - ribiere : βt = ( ∇θj ( θt ) −∇θj ( θt−1 ) ) ∇θj ( θt ) ∇θj ( θt−1 ) ∇θj ( θt−1 ) ( 8. 31 ) for a quadratic surface, the conjugate directions ensure that the gradient along the previous direction does not increase in magnitude. we therefore stay at the minimum ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 330 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
i l f ( ( x ( ) i ; ) θ, y ( ) i ) compute βt = ( g t−gt−1 ) gt g t−1gt−1 ( polak - ribiere ) ( nonlinear conjugate gradient : optionally reset βt to zero, for example if t is a multiple of some constant, such as ) k k = 5 compute search direction : ρt = −gt + βtρt−1 perform line search to find : ∗ = argmin1 m m i = 1 l... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 330 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
primary interest in this chapter is to explore optimization methods for training neural networks and other related deep learning models where the corresponding objective function is far from quadratic. perhaps surprisingly, the method of conjugate gradients is still applicable in this setting, though with some modificat... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 330 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models are no longer assured to remain at the minimum of the objective for previous directions. as a result, the nonlinear conjugate gradients algorithm includes occasional resets where the method of conjugate gradients is restarted with line search along the unaltered gradient... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 331 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##n – fletcher – goldfarb – shanno ( bfgs ) algorithm attempts to bring some of the advantages of newton ’ s method without the computational burden. in that respect, bfgs is similar to the conjugate gradient method. however, bfgs takes a more direct approach to the approximation of newton ’ s update. recall that newto... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 331 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
luenberger 1984 ( ). once the inverse hessian approximation mt is updated, the direction of descent ρt is determined by ρt = mtgt. a line search is performed in this direction to determine the size of the step, ∗, taken in this direction. the final update to the parameters is given by : θt + 1 = θt + ∗ρt. ( 8. 33 ) like... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 331 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models unlike conjugate gradients, the success of the approach is not heavily dependent on the line search finding a point very close to the true minimum along the line. thus, relative to conjugate gradients, bfgs has the advantage that it can spend less time refining each line s... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 332 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
used with exact line searches, the directions defined by l - bfgs are mutually conjugate. however, unlike the method of conjugate gradients, this procedure remains well behaved when the minimum of the line search is reached only approximately. the l - bfgs strategy with no storage described here can be generalized to in... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 332 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of training very deep models. very deep models involve the composition of several functions or layers. the gradient tells how to update each parameter, under the assumption that the other layers do not change. in practice, we update all of the layers simultaneously. when we make the update, unexpected results can happe... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 332 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models example, suppose we have a deep neural network that has only one unit per layer and does not use an activation function at each hidden layer : [UNK] = xw1w2w3... wl. here, wi provides the weight used by layer i. the output of layer i is hi = hi−1wi. the output [UNK] is a... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 333 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
include second - order and third - order [UNK], on up to [UNK] of order l. the new value of [UNK] is given by x w ( 1 −g1 ) ( w2 −g 2 ) (... wl −gl ). ( 8. 34 ) an example of one second - order term arising from this update is 2g1 g2 l i = 3 wi. this term might be negligible if l i = 3wi is small, or might be exponenti... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 333 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
approximations that prevent them from truly accounting for all significant second - order interactions. building an n - th order optimization algorithm for n > 2 thus seems hopeless. what can we do instead? batch normalization provides an elegant way of reparametrizing almost any deep network. the reparametrization sign... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 333 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models and dividing by σj. the rest of the network then operates on h in exactly the same way that the original network operated on. h at training time, µ = 1 m i hi, : ( 8. 36 ) and σ = δ + 1 m i ( ) h µ − 2 i, ( 8. 37 ) where δ is a small positive value such as 10−8 imposed t... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 334 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##ized activation statistics or involved intervening to renormalize unit statistics after each gradient descent step. the former approach usually resulted in imperfect normalization and the latter usually resulted in significant wasted time as the learning algorithm repeatedly proposed changing the mean and variance and... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 334 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
gaussian, because the transformation from x to hl is linear. however, h l−1 will no longer have zero mean and unit variance. after applying batch normalization, we obtain the normalized [UNK] that restores the zero mean and unit variance properties. for almost any update to the lower layers, [UNK] will remain a unit ga... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 334 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models of one of the lower weights can flip the relationship between [UNK] and y. these situations are very rare. without normalization, nearly every update would have an extreme [UNK] on the statistics of hl−1. batch normalization has thus made this model significantly easier to... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 335 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
to change. because the final layer of the network is able to learn a linear transformation, we may actually wish to remove all linear relationships between units within a layer. indeed, this is the approach taken by ( ), who provided desjardins et al. 2015 the inspiration for batch normalization. unfortunately, eliminat... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 335 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
be set back to any arbitrary value β? the answer is that the new parametrization can represent the same family of functions of the input as the old parametrization, but the new parametrization has [UNK] learning dynamics. in the old parametrization, the mean of h was determined by a complicated interaction between the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 335 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models the latter. more specifically, xw + b should be replaced by a normalized version of xw. the bias term should be omitted because it becomes redundant with the β parameter applied by the batch normalization reparametrization. the input to a layer is usually the output of a ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 336 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
it with respect to another variable x j and so on, repeatedly cycling through all variables, we are guaranteed to arrive at a ( local ) minimum. this practice is known as coordinate descent, because we optimize one coordinate at a time. more generally, block coordinate descent refers to minimizing with respect to a sub... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 336 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
weight matrix w that can linearly decode a matrix of activation values h to reconstruct the training set x. most applications of sparse coding also involve weight decay or a constraint on the norms of the columns of w, in order to prevent the pathological solution with extremely small and large. h w the function j is n... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 336 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models us an optimization strategy that allows us to use [UNK] convex optimization algorithms, by alternating between optimizing w with h fixed, then optimizing h w with fixed. coordinate descent is not a very good strategy when the value of one variable strongly influences the op... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 337 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. 8. 7. 3 polyak averaging polyak averaging ( polyak and juditsky 1992, ) consists of averaging together several points in the trajectory through parameter space visited by an optimization algorithm. if t iterations of gradient descent visit points θ ( 1 ),..., θ ( ) t, then the output of the polyak averaging algorithm... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 337 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
points in parameter space from the distant past that may be separated from the current point by large barriers in the cost function does not seem like a useful behavior. as a result, when applying polyak averaging to non - convex problems, it is typical to use an exponentially decaying running average : [UNK] ( ) t = [... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 337 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models 8. 7. 4 supervised pretraining sometimes, directly training a model to solve a specific task can be too ambitious if the model is complex and hard to optimize or if the task is very [UNK]. it is sometimes more [UNK] to train a simpler model to solve the task, then make th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 338 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
greedy algorithms may also be followed by a fine - tuning stage in which a joint optimization algorithm searches for an optimal solution to the full problem. initializing the joint optimization algorithm with a greedy solution can greatly speed it up and improve the quality of the solution it finds. pretraining, and espe... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 338 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
pretraining one layer at a time, simonyan and zisserman ( ) pretrain a deep convolutional network ( eleven weight layers ) and then use 2015 the first four and last three layers from this network to initialize even deeper networks ( with up to nineteen layers of weights ). the middle layers of the new, very deep network... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 338 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models y h ( 1 ) h ( 1 ) x ( a ) u ( 1 ) u ( 1 ) w ( 1 ) w ( 1 ) y h ( 1 ) h ( 1 ) x ( b ) u ( 1 ) u ( 1 ) w ( 1 ) w ( 1 ) y h ( 1 ) h ( 1 ) x ( c ) u ( 1 ) u ( 1 ) w ( 1 ) w ( 1 ) h ( 2 ) h ( 2 ) y u ( 2 ) u ( 2 ) w ( 2 ) w ( 2 ) y h ( 1 ) h ( 1 ) x ( d ) u ( 1 ) u ( 1 ) w ( 1... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 339 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a [UNK] shallow architecture. another drawing of the ( b ) same architecture. we keep only the input - to - hidden layer of the original network and ( c ) discard the hidden - to - output layer. we send the output of the first hidden layer as input to another supervised single hidden layer mlp that is trained with the s... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 339 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models intermediate levels of a deep hierarchy. in general, pretraining may help both in terms of optimization and in terms of generalization. an approach related to supervised pretraining extends the idea to the context of transfer learning : yosinski 2014 et al. ( ) pretrain ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 340 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a network that has low enough depth and great enough width ( number of units per layer ) to be easy to train. this network then becomes a teacher for a second network, designated the student. the student network is much deeper and thinner ( eleven to nineteen layers ) and would be [UNK] to train with sgd under normal c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 340 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
their task, as well as to predict the intermediate layer of the teacher network. although a thin and deep network appears to be more [UNK] to train than a wide and shallow network, the thin and deep network may generalize better and certainly has lower computational cost if it is thin enough to have far fewer parameter... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 340 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models 8. 7. 5 designing models to aid optimization to improve optimization, the best strategy is not always to improve the optimization algorithm. instead, many improvements in the optimization of deep models have come from designing the models to be easier to optimize. in pri... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 341 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
have significant slope in large portions of their domain. in par - ticular, model innovations like the lstm, rectified linear units and maxout units have all moved toward using more linear functions than previous models like deep networks based on sigmoidal units. these models have nice properties that make optimization ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 341 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
vanishing gradient problem ( srivastava 2015 et al., ). a related idea to skip connections is adding extra copies of the output that are attached to the intermediate hidden layers of the network, as in googlenet (, ) szegedy et al. 2014a and deeply - supervised nets (, ). these “ auxiliary heads ” are trained lee et al... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 341 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 8. optimization for training deep models should do, via a shorter path. these hints provide an error signal to lower layers. 8. 7. 6 continuation methods and curriculum learning as argued in section, many of the challenges in optimization arise from the 8. 2. 7 global structure of the cost function and cannot b... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 342 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
[UNK], with j ( 0 ) being fairly easy to minimize, and j ( ) n, the most [UNK], being j ( θ ), the true cost function motivating the entire process. when we say that j ( ) i is easier than j ( + 1 ) i, we mean that it is well behaved over more of θ space. a random initialization is more likely to land in the region whe... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 342 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
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