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labeled. for a recent example of multi - instance learning with deep models, see kotzias 2015 et al. ( ). some machine learning algorithms do not just experience a fixed dataset. for example, reinforcement learning algorithms interact with an environment, so there is a feedback loop between the learning system and its e... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 121 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##set contains 150 examples with four features for each example. this means we can represent the dataset with a design matrix x ∈r150 4 ×, where xi, 1 is the sepal length of plant i, xi, 2 is the sepal width of plant i, etc. we will describe most of the learning algorithms in this book in terms of how they operate on d... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 121 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics types of such heterogeneous data. in cases like these, rather than describing the dataset as a matrix with m rows, we will describe it as a set containing m elements : { x ( 1 ), x ( 2 ),..., x ( ) m }. this notation does not imply that any two example vectors x ( ) i and x ( ) j have... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 122 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
i of course, sometimes the label may be more than just a single number. for example, if we want to train a speech recognition system to transcribe entire sentences, then the label for each example sentence is a sequence of words. just as there is no formal definition of supervised and unsupervised learning, there is no ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 122 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
a system that can take a vector x ∈rn as input and predict the value of a scalar y ∈r as its output. in the case of linear regression, the output is a linear function of the input. let [UNK] be the value that our model predicts should take on. we define the output to be y [UNK] = wx ( 5. 3 ) where w ∈rn is a vector of. ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 122 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics then increasing the value of that feature increases the value of our prediction [UNK]. if a feature receives a negative weight, then increasing the value of that feature decreases the value of our prediction. if a feature ’ s weight is large in magnitude, then it has a large [UNK] on ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 123 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
targets as y ( ) test. one way of measuring the performance of the model is to compute the mean squared error of the model on the test set. if [UNK] ( ) test gives the predictions of the model on the test set, then the mean squared error is given by msetest = 1 m i ( [UNK] ( ) test −y ( ) test ) 2 i. ( 5. 4 ) intuitive... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 123 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
intuitive way of doing this ( which we will justify later, in section ) is just to 5. 5. 1 minimize the mean squared error on the training set, msetrain. to minimize msetrain, we can simply solve for where its gradient is : 0 ∇wmsetrain = 0 ( 5. 6 ) ⇒∇w 1 m | | [UNK] ( ) train −y ( ) train | | 2 2 = 0 ( 5. 7 ) ⇒1 m∇w |... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 123 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics − − 1 0. 0 5 0 0 0 5 1 0.... x1 −3 −2 −1 0 1 2 3 y linear regression example 0 5 1 0 1 5... w1 0 20. 0 25. 0 30. 0 35. 0 40. 0 45. 0 50. 0 55. mse ( train ) optimization of w figure 5. 1 : a linear regression problem, with a training set consisting of ten data points, each containing ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 124 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( ) train x ( ) train w y − ( ) train = 0 ( 5. 9 ) ⇒∇w wx ( ) train x ( ) train w w −2 x ( ) train y ( ) train + y ( ) train y ( ) train = 0 ( 5. 10 ) ⇒2x ( ) train x ( ) train w x −2 ( ) train y ( ) train = 0 ( 5. 11 ) ⇒w = x ( ) train x ( ) train −1 x ( ) train y ( ) train ( 5. 12 ) the system of equations whose solu... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 124 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
. 13 ) so the mapping from parameters to predictions is still a linear function but the mapping from features to predictions is now an [UNK] function. this extension to [UNK] functions means that the plot of the model ’ s predictions still looks like a line, but it need not pass through the origin. instead of adding th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 124 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics b, one can continue to use the model with only weights but augment x with an extra entry that is always set to. the weight corresponding to the extra entry 1 1 plays the role of the bias parameter. we will frequently use the term “ linear ” when referring to [UNK] functions throughout... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 125 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
, overfitting and underfitting the central challenge in machine learning is that we must perform well on new, previously unseen inputs — not just those on which our model was trained. the ability to perform well on previously unobserved inputs is called generalization. typically, when training a machine learning model, w... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 125 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
set. in our linear regression example, we trained the model by minimizing the training error, 1 m ( ) train | | x ( ) train w y − ( ) train | | 2 2, ( 5. 14 ) but we actually care about the test error, 1 m ( ) test | | x ( ) test w y − ( ) test | | 2 2. how can we [UNK] performance on the test set when we get to observ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 125 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics training set? the field of statistical learning theory provides some answers. if the training and the test set are collected arbitrarily, there is indeed little we can do. if we are allowed to make some assumptions about how the training and test set are collected, then we can make som... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 126 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the i. i. d. assumptions allow us to mathematically study the relationship between training error and test error. one immediate connection we can observe between the training and test error is that the expected training error of a randomly selected model is equal to the expected test error of that model. suppose we hav... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 126 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
equal to the expected value of training error. the factors determining how well a machine learning algorithm will perform are its ability to : 1. make the training error small. 2. make the gap between training and test error small. these two factors correspond to the two central challenges in machine learning : underfit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 126 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics functions. models with low capacity may struggle to fit the training set. models with high capacity can overfit by memorizing properties of the training set that do not serve them well on the test set. one way to control the capacity of a learning algorithm is by choosing its hypothesis... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 127 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##x2. ( 5. 16 ) though this model implements a quadratic function of its, the output is input still a linear function of the parameters, so we can still use the normal equations to train the model in closed form. we can continue to add more powers of x as additional features, for example to obtain a polynomial of degre... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 127 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
in the true underlying problem, so it underfits. the degree - 9 predictor is capable of representing the correct function, but it is also capable of representing infinitely many other functions that pass exactly through the training points, because we 112 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 127 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics have more parameters than training examples. we have little chance of choosing a solution that generalizes well when so many wildly [UNK] solutions exist. in this example, the quadratic model is perfectly matched to the true structure of the task so it generalizes well to new data. fi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 128 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##determined normal equations. the solution passes through all of the training points exactly, but we have not been lucky enough for it to extract the correct structure. it now has a deep valley in between two training points that does not appear in the true underlying function. it also increases sharply on the left si... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 128 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics the imperfection of the optimization algorithm, mean that the learning algorithm ’ s [UNK] capacity may be less than the representational capacity of the model family. our modern ideas about improving the generalization of machine learning models are refinements of thought dating back ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 129 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
- chervonenkis dimension, or vc dimension. the vc dimension measures the capacity of a binary classifier. the vc dimension is defined as being the largest possible value of m for which there exists a training set of m [UNK] x points that the classifier can label arbitrarily. quantifying the capacity of the model allows st... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 129 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
can be quite [UNK] to determine the capacity of deep learning algorithms. the problem of determining the capacity of a deep learning model is especially [UNK] because the [UNK] capacity is limited by the capabilities of the optimization algorithm, and we have little theoretical understanding of the very general non - c... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 129 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics 0 optimal capacity capacity error underfitting zone overfitting zone generalization gap training error generalization error figure 5. 3 : typical relationship between capacity and error. training and test error behave [UNK]. at the left end of the graph, training error and generalizatio... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 130 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
that searches over all possible probability distributions ) that cannot be implemented in practice. however, we can also design practical non - parametric models by making their complexity a function of the training set size. one example of such an algorithm is nearest neighbor regression. unlike linear regression, whi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 130 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics parametric learning algorithm inside another algorithm that increases the number of parameters as needed. for example, we could imagine an outer loop of learning that changes the degree of the polynomial learned by linear regression on top of a polynomial expansion of the input. the i... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 131 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
best possible error is achieved. any fixed parametric model with less than optimal capacity will asymptote to an error value that exceeds the bayes error. see figure for an illustration. note that it is possible for the model to have optimal 5. 4 capacity and yet still have a large gap between training and generalization... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 131 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
logical reasoning. machine learning promises to find rules that are probably most correct about members of the set they concern. unfortunately, even this does not resolve the entire problem. the no free lunch theorem for machine learning ( wolpert 1996, ) states that, averaged over all possible data generating distribut... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 131 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics figure 5. 4 : the [UNK] of the training dataset size on the train and test error, as well as on the optimal model capacity. we constructed a synthetic regression problem based on adding a moderate amount of noise to a degree - 5 polynomial, generated a single test set, and then genera... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 132 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
so its test error asymptotes to a high value. the test error at optimal capacity asymptotes to the bayes error. the training error can fall below the bayes error, due to the ability of the training algorithm to memorize specific instances of the training set. as the training size increases to infinity, the training error... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 132 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics performance ( over all possible tasks ) as merely predicting that every point belongs to the same class. fortunately, these results hold only when we average over possible data all generating distributions. if we make assumptions about the kinds of probability distributions we encount... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 133 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
it performs better. so far, the only method of modifying a learning algorithm that we have discussed concretely is to increase or decrease the model ’ s representational capacity by adding or removing functions from the hypothesis space of solutions the learning algorithm is able to choose. we gave the specific example ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 133 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
predict sin ( x ) from x. we can thus control the performance of our algorithms by choosing what kind of functions we allow them to draw solutions from, as well as by controlling the amount of these functions. we can also give a learning algorithm a preference for one solution in its hypothesis space to another. this m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 133 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics data significantly better than the preferred solution. for example, we can modify the training criterion for linear regression to include weight decay. to perform linear regression with weight decay, we minimize a sum comprising both the mean squared error on the training and a criteri... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 134 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##fit via weight decay, we can train a high - degree polynomial regression model with [UNK] values of. see figure for the results. λ 5. 5 figure 5. 5 : we fit a high - degree polynomial regression model to our example training set from figure. the true function is quadratic, but here we use only models with degree 9. 5. 2 ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 134 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics more generally, we can regularize a model that learns a function f ( x ; θ ) by adding a penalty called a regularizer to the cost function. in the case of weight decay, the regularizer is ω ( w ) = w w. in chapter, we will see that many other 7 regularizers are possible. expressing pr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 135 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
its generalization error but not its training error. regularization is one of the central concerns of the field of machine learning, rivaled in its importance only by optimization. the no free lunch theorem has made it clear that there is no best machine learning algorithm, and, in particular, no best form of regulariza... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 135 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
learns the best hyperparameters for another learning algorithm ). in the polynomial regression example we saw in figure, there is a single 5. 2 hyperparameter : the degree of the polynomial, which acts as a capacity hyper - parameter. the λ value used to control the strength of weight decay is another example of a hyper... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 135 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics setting must be a hyperparameter because it is not appropriate to learn that hyperparameter on the training set. this applies to all hyperparameters that control model capacity. if learned on the training set, such hyperparameters would always choose the maximum possible model capacit... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 136 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
its hyperparameters. for this reason, no example from the test set can be used in the validation set. therefore, we always construct the validation set from the training data. specifically, we split the training data into two disjoint subsets. one of these subsets is used to learn the parameters. the other subset is our... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 136 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
typically by a smaller amount than the training error. after all hyperparameter optimization is complete, the generalization error may be estimated using the test set. in practice, when the same test set has been used repeatedly to evaluate performance of [UNK] algorithms over many years, and especially if we consider ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 136 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics 5. 3. 1 cross - validation dividing the dataset into a fixed training set and a fixed test set can be problematic if it results in the test set being small. a small test set implies statistical uncertainty around the estimated average test error, making it [UNK] to claim that algorithm ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 137 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
then be estimated by taking the average test error across k trials. on trial i, the i - th subset of the data is used as the test set and the rest of the data is used as the training set. one problem is that there exist no unbiased estimators of the variance of such average error estimators ( bengio and grandvalet 2004... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 137 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
model, such as the weights in our linear regression example in section, but it can also be a whole function. 5. 1. 4 in order to distinguish estimates of parameters from their true value, our convention will be to denote a point estimate of a parameter by θ [UNK]. let { x ( 1 ),..., x ( ) m } be a set of m independent ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 137 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics algorithm 5. 1 the k - fold cross - validation algorithm. it can be used to estimate generalization error of a learning algorithm a when the given dataset d is too small for a simple train / test or train / valid split to yield accurate estimation of generalization error, because the ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 138 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
). while these confidence intervals are not well - justified after the 5. 47 use of cross - validation, it is still common practice to use them to declare that algorithm a is better than algorithm b only if the confidence interval of the error of algorithm a lies below and does not intersect the confidence interval of algo... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 138 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics ( i. i. d. ) data points. a or is any function of the data : point estimator statistic [UNK] = ( g x ( 1 ),..., x ( ) m ). ( 5. 19 ) the definition does not require that g return a value that is close to the true θ or even that the range of g is the same as the set of allowable values ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 139 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
is drawn from a random process, any function of the data is random. therefore [UNK] is a random variable. point estimation can also refer to the estimation of the relationship between input and target variables. we refer to these types of point estimates as function estimators. function estimation as we mentioned above... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 139 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
regression example ( discussed in section ) are both examples of 5. 2 scenarios that may be interpreted either as estimating a parameter w or estimating a function [UNK] y mapping from to x. we now review the most commonly studied properties of point estimators and discuss what they tell us about these estimators. 5. 4... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 139 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics where the expectation is over the data ( seen as samples from a random variable ) and θ is the true underlying value of θ used to define the data generating distri - bution. an estimator [UNK] is said to be unbiased if bias ( [UNK] ) = 0, which implies that e ( [UNK] ) = θ. an estimato... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 140 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
estimator for the θ parameter of this distribution is the mean of the training samples : [UNK] = 1 m m i = 1 x ( ) i. ( 5. 22 ) to determine whether this estimator is biased, we can substitute equation 5. 22 into equation : 5. 20 bias ( [UNK] ) = [ e [UNK] ] −θ ( 5. 23 ) = e 1 m m i = 1 x ( ) i −θ ( 5. 24 ) = 1 m m i =... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 140 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
: gaussian distribution estimator of the mean now, consider a set of samples { x ( 1 ),..., x ( ) m } that are independently and identically distributed according to a gaussian distribution p ( x ( ) i ) = n ( x ( ) i ; µ, σ2 ), where i ∈ { 1,..., m }. 125 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 140 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics recall that the gaussian probability density function is given by p x ( ( ) i ; µ, σ 2 ) = 1 √ 2πσ2 exp −1 2 ( x ( ) i −µ ) 2 σ2. ( 5. 29 ) a common estimator of the gaussian mean parameter is known as the sample mean : [UNK] = 1 m m i = 1 x ( ) i ( 5. 30 ) to determine the bias of th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 141 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##biased estimator of gaussian mean parameter. example : estimators of the variance of a gaussian distribution as an example, we compare two [UNK] estimators of the variance parameter σ2 of a gaussian distribution. we are interested in knowing if either estimator is biased. the first estimator of σ2 we consider is known... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 141 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics we begin by evaluating the term e [ [UNK] 2 m ] : e [ [UNK] m ] = e 1 m m i = 1 x ( ) i [UNK] 2 ( 5. 38 ) = m −1 m σ2 ( 5. 39 ) returning to equation, we conclude that the bias of 5. 37 [UNK] 2 m is −σ2 / m. therefore, the sample variance is a biased estimator. the unbiased sample var... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 142 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
σ 2 ( 5. 43 ) = σ2. ( 5. 44 ) we have two estimators : one is biased and the other is not. while unbiased estimators are clearly desirable, they are not always the “ best ” estimators. as we will see we often use biased estimators that possess other important properties. 5. 4. 3 variance and standard error another prop... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 142 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics the variance or the standard error of an estimator provides a measure of how we would expect the estimate we compute from data to vary as we independently resample the dataset from the underlying data generating process. just as we might like an estimator to exhibit low bias we would ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 143 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
square root of the sample variance nor the square root of the unbiased estimator of the variance provide an unbiased estimate of the standard deviation. both approaches tend to underestimate the true standard deviation, but are still used in practice. the square root of the unbiased estimator of the variance is less of... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 143 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
µm + 1 96se ( [UNK]. µm ) ), ( 5. 47 ) under the normal distribution with mean [UNK] and variance se ( [UNK] ) 2. in machine learning experiments, it is common to say that algorithma is better than algorithm b if the upper bound of the 95 % confidence interval for the error of algorithm a is less than the lower bound of... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 143 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics example : bernoulli distribution we once again consider a set of samples { x ( 1 ),..., x ( ) m } drawn independently and identically from a bernoulli distribution ( recall p ( x ( ) i ; θ ) = θ x ( ) i ( 1 −θ ) ( 1−x ( ) i ) ). this time we are interested in computing the variance of... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 144 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
in the dataset. this is a common property of popular estimators that we will return to when we discuss consistency ( see section ). 5. 4. 5 5. 4. 4 trading [UNK] and variance to minimize mean squared error bias and variance measure two [UNK] sources of error in an estimator. bias measures the expected deviation from th... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 144 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##ly, cross - validation is highly successful on many real - world tasks. alter - natively, we can also compare the mean squared error ( mse ) of the estimates : mse = [ ( e [UNK] −θ ) 2 ] ( 5. 53 ) = bias ( [UNK] ) 2 + var ( [UNK] ) ( 5. 54 ) 129 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 144 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics the mse measures the overall expected deviation — in a squared error sense — between the estimator and the true value of the parameter θ. as is clear from equation, evaluating the mse incorporates both the bias and the variance. 5. 54 desirable estimators are those with small mse and ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 145 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##e. 5. 2 5. 3 the relationship between bias and variance is tightly linked to the machine learning concepts of capacity, underfitting and overfitting. in the case where gen - eralization error is measured by the mse ( where bias and variance are meaningful components of generalization error ), increasing capacity tends ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 145 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics value of the corresponding parameters. more formally, we would like that [UNK] = θ. ( 5. 55 ) the symbol plim indicates convergence in probability, meaning that for any > 0, p ( | [UNK] − | θ > ) →0 as m →∞. the condition described by equation is 5. 55 known as consistency. it is some... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 146 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
), with a dataset consisting of m samples : { x ( 1 ),..., x ( ) m }. we could use the first sample x ( 1 ) of the dataset as an unbiased estimator : [UNK] = x ( 1 ). in that case, e ( [UNK] m ) = θ so the estimator is unbiased no matter how many data points are seen. this, of course, implies that the estimate is asympt... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 146 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
are good estimators for [UNK] models. the most common such principle is the maximum likelihood principle. consider a set of m examples x = { x ( 1 ),..., x ( ) m } drawn independently from the true but unknown data generating distribution pdata ( ) x. let pmodel ( x ; θ ) be a parametric family of probability distribut... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 146 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics this product over many probabilities can be inconvenient for a variety of reasons. for example, it is prone to numerical underflow. to obtain a more convenient but equivalent optimization problem, we observe that taking the logarithm of the likelihood does not change its arg max but do... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 147 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##fined by the training set and the model distribution, with the degree of dissimilarity between the two measured by the kl divergence. the kl divergence is given by dkl ( [UNK] ) = [UNK] [ log [UNK] ( ) log x − pmodel ( ) ] x. ( 5. 60 ) the term on the left is a function only of the data generating process, not the mod... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 147 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
- likelihood is a cross - entropy between the empirical distribution defined by the training set and the probability distribution defined by model. for example, mean squared error is the cross - entropy between the empirical distribution and a gaussian model. we can thus see maximum likelihood as an attempt to make the m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 147 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics are [UNK]. in software, we often phrase both as minimizing a cost function. maximum likelihood thus becomes minimization of the negative log - likelihood ( nll ), or equivalently, minimization of the cross entropy. the perspective of maximum likelihood as minimum kl divergence becomes... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 148 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
( 5. 62 ) if the examples are assumed to be i. i. d., then this can be decomposed into θml = arg max θ m i = 1 log ( p y ( ) i | x ( ) i ; ) θ. ( 5. 63 ) example : linear regression as maximum likelihood linear regression, introduced earlier in section, may be justified as a maximum likelihood 5. 1. 4 procedure. previou... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 148 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the same input value x but [UNK] values of y. the goal of the learning algorithm is now to fit the distribution p ( y | x ) to all of those [UNK] y values that are all compatible with x. to derive the same linear regression algorithm we obtained before, we define p ( y | x ) = n ( y ; [UNK] ( x ; w ), σ2 ). the function ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 148 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics examples are assumed to be i. i. d., the conditional log - likelihood ( equation ) is 5. 63 given by m i = 1 log ( p y ( ) i | x ( ) i ; ) θ ( 5. 64 ) = log −m σ −m 2 log ( 2 ) π − m i = 1 [UNK] ( ) i −y ( ) i 2 2σ2, ( 5. 65 ) where [UNK] ( ) i is the output of the linear regression o... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 149 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
same location of the optimum. this justifies the use of the mse as a maximum likelihood estimation procedure. as we will see, the maximum likelihood estimator has several desirable properties. 5. 5. 2 properties of maximum likelihood the main appeal of the maximum likelihood estimator is that it can be shown to be the b... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 149 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
other - wise, maximum likelihood can recover the correct pdata, but will not be able to determine which value of was used by the data generating processing. θ there are other inductive principles besides the maximum likelihood estima - tor, many of which share the property of being consistent estimators. however, 134 | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 149 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics consistent estimators can [UNK] in their statistic [UNK], meaning that one consistent estimator may obtain lower generalization error for a fixed number of samples m, or equivalently, may require fewer examples to obtain a fixed level of generalization error. statistical [UNK] is typica... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 150 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
reasons ( consistency and [UNK] ), maximum likelihood is often considered the preferred estimator to use for machine learning. when the number of examples is small enough to yield overfitting behavior, regularization strategies such as weight decay may be used to obtain a biased version of maximum likelihood that has le... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 150 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
of certainty of states of knowledge. the dataset is directly observed and so is not random. on the other hand, the true parameter θ is unknown or uncertain and thus is represented as a random variable. before observing the data, we represent our knowledge of θ using the prior probability distribution, p ( θ ) ( sometim... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 150 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics value of θ before observing any data. for example, one might assume that a priori θ lies in some finite range or volume, with a uniform distribution. many priors instead reflect a preference for “ simpler ” solutions ( such as smaller magnitude [UNK], or a function that is closer to bei... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 151 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
scenarios where bayesian estimation is typically used, the prior begins as a relatively uniform or gaussian distribution with high entropy, and the observation of the data usually causes the posterior to lose entropy and concentrate around a few highly likely values of the parameters. relative to maximum likelihood est... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 151 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
by the posterior density itself. after having observed { x ( 1 ),..., x ( ) m }, if we are still quite uncertain about the value of θ, then this uncertainty is incorporated directly into any predictions we might make. in section, we discussed how the frequentist approach addresses the uncer - 5. 4 tainty in a given poi... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 151 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics prior distribution. the prior has an influence by shifting probability mass density towards regions of the parameter space that are preferred. in practice, a priori the prior often expresses a preference for models that are simpler or more smooth. critics of the bayesian approach ident... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 152 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
over the entire training set as : y [UNK] ( ) train = x ( ) train w. ( 5. 70 ) expressed as a gaussian conditional distribution on y ( ) train, we have p ( y ( ) train | x ( ) train, w y ) = ( n ( ) train ; x ( ) train w i, ) ( 5. 71 ) [UNK] −1 2 ( y ( ) train −x ( ) train w ) ( y ( ) train −x ( ) train w ), ( 5. 72 ) ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 152 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
or unnatural to express our prior beliefs in terms of the parameters of the model, in practice we typically assume a fairly broad distribution expressing a high degree of uncertainty about θ. for real - valued parameters it is common to use a gaussian as a prior distribution : p ( ) = ( ; w n w µ0, λ0 ) exp [UNK] −1 2 ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 152 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics where µ0 and λ0 are the prior distribution mean vector and covariance matrix respectively. 1 with the prior thus specified, we can now proceed in determining the posterior distribution over the model parameters. p, p, p ( w x | y ) [UNK] ( y x | w ) ( ) w ( 5. 74 ) [UNK] −1 2 ( ) y xw ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 153 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
y ) exp [UNK] −1 2 ( w µ − m ) λ−1 m ( w µ − m ) + 1 2µ mλ−1 m µm ( 5. 77 ) [UNK] −1 2 ( w µ − m ) λ−1 m ( w µ − m ). ( 5. 78 ) all terms that do not include the parameter vector w have been omitted ; they are implied by the fact that the distribution must be normalized to integrate to. 1 equation shows how to normaliz... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 153 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
##esian learning process with an infinitely wide prior on w. the more important [UNK] is that the bayesian estimate provides a covariance matrix, showing how likely all the [UNK] values of are, rather than providing only the estimate w µm. 5. 6. 1 maximum ( map ) estimation a posteriori while the most principled approac... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 153 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics single point estimate. one common reason for desiring a point estimate is that most operations involving the bayesian posterior for most interesting models are intractable, and a point estimate [UNK] a tractable approximation. rather than simply returning to the maximum likelihood est... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 154 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
distribution. log ( ) p θ as an example, consider a linear regression model with a gaussian prior on the weights w. if this prior is given by n ( w ; 0, 1 λi2 ), then the log - prior term in equation is proportional to the familiar 5. 79 λww weight decay penalty, plus a term that does not depend on w and does not [UNK]... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 154 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
view applies when the regularization consists of adding an extra term to the objective function that corresponds to log p ( θ ). not all regularization penalties correspond to map bayesian inference. for example, some regularizer terms may not be the logarithm of a probability distribution. other regularization terms d... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 154 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics 5. 7 supervised learning algorithms recall from section that supervised learning algorithms are, roughly speaking, 5. 1. 3 learning algorithms that learn to associate some input with some output, given a training set of examples of inputs x and outputs y. in many cases the outputs y m... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 155 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
scenario by defining a [UNK] family of probability distributions. if we have two classes, class 0 and class 1, then we need only specify the probability of one of these classes. the probability of class 1 determines the probability of class 0, because these two values must add up to 1. the normal distribution over real ... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 155 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
the case of linear regression, we were able to find the optimal weights by solving the normal equations. logistic regression is somewhat more [UNK]. there is no closed - form solution for its optimal weights. instead, we must search for them by maximizing the log - likelihood. we can do this by minimizing the negative l... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 155 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics this same strategy can be applied to essentially any supervised learning problem, by writing down a parametric family of conditional probability distributions over the right kind of input and output variables. 5. 7. 2 support vector machines one of the most influential approaches to su... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 156 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
can be shown that the linear function used by the support vector machine can be re - written as wx + = + b b m i = 1 αixx ( ) i ( 5. 82 ) where x ( ) i is a training example and α is a vector of [UNK]. rewriting the learning algorithm this way allows us to replace x by the output of a given feature function φ ( x ) and... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 156 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
evaluations, we can make predictions using the function f b ( ) = x + i α ik, ( x x ( ) i ). ( 5. 83 ) this function is nonlinear with respect to x, but the relationship between φ ( x ) and f ( x ) is linear. also, the relationship between α and f ( x ) is linear. the kernel - based function is exactly equivalent to pr... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 156 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
chapter 5. machine learning basics guaranteed to converge [UNK]. this is possible because we consider φ fixed and optimize only α, i. e., the optimization algorithm can view the decision function as being linear in a [UNK] space. second, the kernel function k often admits an implementation that is significantly more comp... | /home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf | 157 | Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org) | 0 |
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