Question: Represent the scientific article to find a short TLDR

Question: We leverage recent insights from second-order optimisation for neural networks to construct a Kronecker factored Laplace approximation to the posterior over the weights of a trained network. Our approximation requires no modification of the training procedure, enabling practitioners to estimate the uncertainty of their models currently used in production without having to retrain them. We extensively compare our method to using Dropout and a diagonal Laplace approximation for estimating the uncertainty of a network. We demonstrate that our Kronecker factored method leads to better uncertainty estimates on out-of-distribution data and is more robust to simple adversarial attacks. Our approach only requires calculating two square curvature factor matrices for each layer. Their size is equal to the respective square of the input and output size of the layer, making the method efficient both computationally and in terms of memory usage. We illustrate its scalability by applying it to a state-of-the-art convolutional network architecture. Neural networks are most commonly trained in a maximum a posteriori (MAP) setting, which only yields point estimates of the parameters, ignoring any uncertainty about them. This often leads to overconfident predictions, especially in regimes that are weakly covered by training data or far away from the data manifold. While the confidence of wrong predictions is usually irrelevant in a research context, it is essential that a Machine Learning algorithm knows when it does not know in the real world, as the consequences of mistakes can be fatal, be it when driving a car or diagnosing a disease. The Bayesian framework of statistics provides a principled way for avoiding overconfidence in the parameters by treating them as unknown quantities and integrating over all possible values. Specifically, for the prediction of new data under a model, it fits a posterior distribution over the parameters given the training data and weighs the contribution of each setting of the parameters to the prediction by the probability of the data under those parameters times their prior probability. However, the posterior of neural networks is usually intractable due to their size and nonlinearity. There has been previous interest in integrating neural networks into the Bayesian framework BID26 BID15 BID28 BID1, however these approaches were designed for small networks by current standards. Recent adaptations to architectures of modern scale rely on crude approximations of the posterior to become tractable. All of BID9 BID14 BID2 assume independence between the individual weights. While they achieve good on small datasets, this strong restriction of the posterior is susceptible to underestimating the uncertainty, in particular when optimising the variational bound. The approach in BID6 requires the use of certain stochastic regularisers which are not commonly present in most recent architectures. Furthermore, it is not clear if the approximate posterior defined by these regularisers is a good fit to the true posterior. Recent work on second-order optimisation of neural networks BID27 BID3 has demonstrated that the diagonal blocks of the curvature can be well approximated by a Kronecker product. We combine this insight with the idea of modelling the posterior over the weights as a Gaussian, using a Laplace approximation BID26 with Kronecker factored covariance matrices. This leads to a computationally efficient matrix normal posterior distribution BID11 over the weights of every layer. Since the Laplace approximation is applied after training, our approach can be used to obtain uncertainty estimates from existing networks. Our method is inspired by recent Kronecker factored approximations of the curvature of a neural network BID27 BID3 for optimisation and we give a high-level review of these in the following. While the two methods approximate the Gauss-Newton and Fisher matrix respectively, as they are guaranteed to be positive semi-definite (p.s.d.), we base all of our discussion on the Hessian in order to be as general as possible. We denote a feedforward network as taking an input a 0 = x and producing an output h L. The intermediate representations for layers λ = 1,..., L are denoted as h λ = W λ a λ−1 and a λ = f λ (h λ). We refer to a λ as the activations, and h λ as the (linear) pre-activations. The bias terms are absorbed into the W λ by appending a 1 to each a λ. The network parameters are optimised w.r.t. an error function E(y, h L) for targets y. Most commonly used error functions, such as squared error and categorical cross-entropy, can be interpreted as exponential family negative log likelihoods − log p(y|h L). Traditional second-order methods use either the Hessian matrix or a positive semi-definite approximation thereof to generate parameter updates of the form ∆ = C −1 g, where C is the chosen curvature matrix and g the gradient of the error function parameterised by the network. However, this curvature matrix is infeasbile to compute for modern neural networks as their number of parameters is often in the millions, rendering the size of C of the order of several terabytes. Recent work BID27 BID3 exploits that, for a single data point, the diagonal blocks of these curvature matrices are Kronecker factored: DISPLAYFORM0 where H λ is the Hessian w.r.t. the weights in layer λ. Q λ = a λ−1 a T λ−1 denotes the covariance of the incoming activations a λ−1 and H λ = ∂ 2 E ∂h λ ∂h λ the pre-activation Hessian, i.e. the Hessian of the error w.r.t. the linear pre-activations h λ in a layer. We provide the derivation for this as well as the recursion for calculating H in Appendix A.The Kronecker factorisation holds two key advantages: the matrices that need be computed and stored are much smaller -if we assume all layers to be of dimensionality D, the two factors are each of size D 2, whereas the full Hessian for the weights of only one layer would have D 4 elements. Furthermore, the inverse of a Kronecker product is equal to the Kronecker product of the inverses, so it is only necessary to invert those two moderately sized matrices. In order to maintain this structure over a minibatch of data, all Kronecker factored second-order methods make two core approximations: First, they only model the diagonal blocks corresponding to the weights of a layer, such that the curvature decomposes into L independent matrices. Second, they assume Q λ and H λ to be independent. This is in order to maintain the Kronecker factorisation in expectation, i.e. DISPLAYFORM1, since the expectation of a Kronecker product is not guaranteed to be Kronecker factored itself. The main difference between the Kronecker factored second-order optimisers lies in how they efficiently approximate E [H λ]. For exact calculation, it would be necessary to pass back an entire matrix per data point in a minibatch, which imposes infeasible memory and computational requirements. KFRA BID3 simply passes back the expectation at every layer, while KFAC BID27 utilises the Fisher identity to only propagate a vector rather than a matrix, approximating the Kronecker factors with a stochastic rank-one matrix for each data point. The diagonal blocks of the Hessian and Gauss-Newton matrix are equal for neural networks with piecewise linear activation functions BID3, thus both methods can be used to directly approximate the diagonal blocks of the Hessian of such networks, as the Gauss-Newton and Fisher are equivalent for networks that parameterise an exponential family log likelihood.3 A SCALABLE LAPLACE APPROXIMATION FOR NEURAL NETWORKS The standard Laplace approximation is obtained by taking the second-order Taylor expansion around a mode of a distribution. For a neural network, such a mode can be found using standard gradientbased methods. Specifically, if we approximate the log posterior over the weights of a network given some data D around a MAP estimate θ *, we obtain: DISPLAYFORM0 where DISPLAYFORM1 is the stacked vector of weights andH = E [H] the average Hessian of the negative log posterior 1. The first order term is missing because we expand the function around a maximum θ *, where the gradient is zero. If we exponentiate this equation, it is easy to notice that the right-hand side is of Gaussian functional form for θ, thus we obtain a normal distribution by integrating over it. The posterior over the weights is then approximated as Gaussian: DISPLAYFORM2 assumingH is p.s.d. We can then approximate the posterior mean when predicting on unseen data D * by averaging the predictions of T Monte Carlo samples θ (t) from the approximate posterior: DISPLAYFORM3 Unfortunately, it is not feasible to compute or invert the Hessian matrix w.r.t. all of the weights jointly. An approximation that is easy to compute in modern automatic differentiation frameworks is the diagonal of the Fisher matrix F, which is simply the expectation of the squared gradients: DISPLAYFORM0 where diag extracts the diagonal of a matrix or turns a vector into a diagonal matrix. Such diagonal approximations to the curvature of a neural network have been used successfully for pruning the weights BID22 and, more recently, for transfer learning BID19.This corresponds to modelling the weights with a Normal distribution with diagonal covariance: DISPLAYFORM1 Unfortunately, even if the Taylor approximation is accurate, this will place significant probability mass in low probability areas of the true posterior if some weights exhibit high covariance. So while it is desirable to model the covariance between the weights, some approximations are needed in order to remain computationally efficient. First, we assume the weights of the different layers to be independent. This corresponds to the block-diagonal approximation in KFAC and KFRA, which empirically preserves sufficient information about the curvature to obtain competitive optimisation performance. For our purposes this means that our posterior factorises over the layers. As discussed above, the Hessian of the log-likelihood for a single datapoint is Kronecker factored, and we denote the two factor matrices as H λ = Q λ ⊗ H λ.2 By further assuming independence between Q and H in all layers, we can approximate the expected Hessian of each layer as: DISPLAYFORM0 Hence, the Hessian of every layer is Kronecker factored over an entire dataset and the Laplace approximation can be approximated by a product of Gaussians. Each Gaussian has a Kronecker factored covariance, corresponding to a matrix normal distribution BID11, which considers the two Kronecker factors of the covariance to be the covariances of the rows and columns of a matrix. The two factors are much smaller than the full covariance and allow for significantly more efficient inversion and sampling (we review the matrix normal distribution in Appendix B).Our ing posterior for the weights in layer λ is then: DISPLAYFORM1 In contrast to optimisation methods, we do not need to approximate E [H λ] as it is only calculated once. However, when it is possible to augment the data (e.g. randomised cropping of images), it may be advantageous. We provide a more detailed discussion of this in Appendix C. Just as the log posterior, the Hessian decomposes into a term depending on the data log likelihood and one on the prior. For the commonly used L 2 -regularisation, corresponding to a Gaussian prior, the Hessian is equal to the precision of the prior times the identity matrix. We approximate this by adding a multiple of the identity to each of the Kronecker factors from the log likelihood: DISPLAYFORM0 where τ is the precision of the Gaussian prior on the weights and N the size of the dataset. However, we can also treat them as hyperparameters and optimise them w.r.t. the predictive performance on a validation set. We emphasise that this can be done without retraining the network, so it does not impose a large computational overhead and is trivial to parallelise. Setting N to a larger value than the size of the dataset can be interpreted as including duplicates of the data points as pseudo-observations. Adding a multiple of the uncertainty to the precision matrix decreases the uncertainty about each parameter. This has a regularising effect both on our approximation to the true Laplace, which may be overestimating the variance in certain directions due to ignoring the covariances between the layers, as well as the Laplace approximation itself, which may be placing probability mass in low probability areas of the true posterior. Most recent attempts to approximating the posterior of a neural network are based on formulating an approximate distribution to the posterior and optimising the variational lower bound w.r.t. its parameters. BID9 BID2 BID18 as well as the expectation propagation based approaches of BID14 and BID7 assume independence between the individual weights which, particularly when optimising the KL divergence, often lets the model underestimate the uncertainty about the weights. BID6 interpret Dropout to approximate the posterior with a mixture of delta functions, assuming independence between the columns. BID21 suggest using an ensemble of networks for estimating the uncertainty. Our work is a scalable approximation of BID26. Since the per-layer Hessian of a neural network is infeasible to compute, we suggest a factorisation of the covariance into a Kronecker product, leading to a more efficient matrix normal distribution. The posterior that we obtain is reminiscent of BID24 and BID30, who optimise the parameters of a matrix normal distribution as their weights, which requires a modification of the training procedure. Since the Laplace approximation is a method for predicting in a Bayesian manner and not for training, we focus on comparing to uncertainty estimates obtained from Dropout BID6. The trained networks will be identical, but the prediction methods will differ. We also compare to a diagonal Laplace approximation to highlight the benefit from modelling the covariances between the weights. All experiments are implemented using Theano BID32 and BID4. For the diagonal and full Laplace approximation we use the Fisher identity and draw one sample per data point. We set the hyperparameters of the Laplace approximations (see Section 3.4) using a grid search over the likelihood of 20 validation points that are sampled the same way as the training set. The regularised Laplace approximations all give an overall good fit to the HMC predictive posterior. Their uncertainty is slightly higher close to the training data and increases more slowly away from the data than that of the HMC posterior. The diagonal and full Laplace approximation require stronger regularisation than our Kronecker factored one, as they have higher uncertainty when not regularised. In particular the full Laplace approximation vastly overestimates the uncertainty without additional regularisation, leading to a bad predictive mean (see Appendix E for the corresponding figures), as the Hessian of the log likelihood is underdetermined. This is commonly the case in deep learning, as the number of parameters is typically much larger than the number of data points. Hence restricting the structure of the covariance is not only a computational necessity for most architectures, but also allows for more precise estimation of the approximate covariance. For a more realistic test, similar to BID25, we assess the uncertainty of the predictions when classifying data from a different distribution than the training data. For this we train a network with two layers of 1024 hidden units and ReLU transfer functions to classify MNIST digits. We use a learning rate of 10 −2 and momentum of 0.9 for 250 epochs. We apply Dropout with p=0.5 after each inner layer, as our chief interest is to compare against its uncertainty estimates. We further use L 2 -regularisation with a factor of 10 −2 and randomly binarise the images during training according to their pixel intensities and draw 1, 000 such samples per datapoint for estimating the curvature factors. We use this network to classify the images in the notMNIST dataset 4, which contains 28×28 grey-scale images of the letters'A' to'J' from various computer fonts, i.e. not digits. An ideal classifier would make uniform predictions over its classes. We compare the uncertainty obtained by predicting the digit class of the notMNIST images using 1. a deterministic forward pass through the Dropout trained network, 2. by sampling different Dropout masks and averaging the predictions, and by sampling different weight matrices from 3. the matrix normal distribution obtained from our Kronecker factored Laplace approximation as well as 4. the diagonal one. As an additional baseline similar to BID2 BID9, we compare to a network with identical architecture with a fully factorised Gaussian (FFG) approximate posterior on the weights and a standard normal prior. We train the model on the variational lower bound using the reparametrisation trick BID17. We use 100 samples for the stochastic forward passes and optimise the hyperparameters of the Laplace approximations w.r.t. the cross-entropy on the validation set of MNIST.We measure the uncertainty of the different methods as the entropy of the predictive distribution, which has a minimal value of 0 when a single class is predicted with certainty and a maximum of about 2.3 for uniform predictions. FIG2 shows the inverse empirical cumulative distribution of the entropy values obtained from the four methods. Consistent with the in BID6, averaging the probabilities of multiple passes through the network yields predictions with higher uncertainty than a deterministic pass that approximates the geometric average BID29. However, there still are some images that are predicted to be a digit with certainty. Our Kronecker factored Laplace approximation makes hardly any predictions with absolute certainty and assigns high uncertainty to most of the letters as desired. The diagonal Laplace approximation required stronger regularisation towards predicting deterministically, yet it performs similarly to Dropout. As shown in TAB0, however, the network makes predictions on the test set of MNIST with similar accuracy to the deterministic forward pass and MC Dropout when using our approximation. The variational factorised Gaussian posterior has low uncertainty as expected. To further test the robustness of our prediction method close to the data distribution, we perform an adversarial attack on a neural network. As first demonstrated in BID31, neural networks are prone to being fooled by gradient-based changes to their inputs. BID23 suggest, and provide empirical support, that Bayesian models may be more robust to such attacks, since they implicitly form an infinitely large ensemble by integrating over the model parameters. For our experiments, we use the fully connected net trained on MNIST from the previous section and compare the sensitivity of the different prediction methods for two kinds of adversarial attacks. First, we use the untargeted Fast Gradient Sign method x adv = x − η sgn(∇ x max y log p (M) (y|x)) suggested in BID8, which takes the gradient of the class predicted with maximal probability by method M w.r.t. the input x and reduces this probability with varying step size η. This step size is rescaled by the difference between the maximal and minimal value per dimension in the dataset. It is to be expected that this method generates examples away from the data manifold, as there is no clear subset of the data that corresponds to e.g. "not ones". FIG3 shows the average predictive uncertainty and the accuracy on the original class on the MNIST test set as the step size η increases. The Kronecker factored Laplace approximation achieves significantly higher uncertainty than any other prediction method as the images move away from the data. Both the diagonal and the Kronecker factored Laplace maintain higher accuracy than MC Dropout on their original predictions. Interestingly, the deterministic forward pass appears to be most robust in terms of accuracy, however it has much smaller uncertainty on the predictions it makes and will confidently predict a false class for most images, whereas the other methods are more uncertain. Furthermore, we perform a targeted attack that attempts to force the network to predict a specific class, in our case'0' following BID23. Hence, for each method, we exclude all data points in the test set that are already predicted as'0'. The updates are of similar form to the untargeted attack, however they increase the probability of the pre-specified class y rather than decreasing the current maximum as x DISPLAYFORM0 We use a step size of η=10 −2 for the targeted attack. The uncertainty and accuracy on the original and target class are shown in FIG4. Here, the Kronecker factored Laplace approximation has slightly smaller uncertainty at its peak in comparison to the other methods, however it appears to be much more robust. It only misclassifies over 50% of the images after about 20 steps, whereas for the other methods this is the case after roughly 10 steps and reaches 100% accuracy on the target class after almost 50 updates, whereas the other methods are fooled on all images after about 25 steps. In conjunction with the experiment on notMNIST, it appears that the Laplace approximation achieves higher uncertainty than Dropout away from the data, as in the untargeted attack. In the targeted attack it exhibits smaller uncertainty than Dropout, yet it is more robust to having its prediction changed. The diagonal Laplace approximation again performs similarly to Dropout. To highlight the scalability of our method, we apply it to a state-of-the-art convolutional network architecture. Recently, deep residual networks BID12 b) have been the most successful ones among those. As demonstrated in BID10, Kronecker factored curvature methods are applicable to convolutional layers by interpreting them as matrix-matrix multiplications. We compare our uncertainty estimates on wide residual networks BID33, a recent variation that achieved competitive performance on CIFAR100 BID20 while, in contrast to most other residual architectures, including Dropout at specific points. While this does not correspond to using Dropout in the Bayesian sense BID5, it allows us to at least compare our method to the uncertainty estimates obtained from Dropout. We note that it is straightforward to incorporate batch normalisation BID16 into the curvature backpropagation algorithms, so we apply a standard Laplace approximation to its parameters as well. We are not aware of any interpretation of Dropout as performing Bayesian inference on the parameters of batch normalisation. Further implementation details are in Appendix G.Again, the accuracy of the prediction methods is comparable (see TAB1 in Appendix F). For calculating the curvature factors, we draw 5, 000 samples per image using the same data augmentation as during training, effectively increasing the dataset size to 2.5×10 8. The diagonal approximation had to be regularised to the extent of becoming deterministic, so we omit it from the . In FIG5 we compare the distribution of the predictive uncertainty on the test set. 5 We distinguish between the uncertainty on correct and incorrect classifications, as the mistakes of a system used in practice may be less severe if the network can at least indicate that it is uncertain. Thus, high uncertainty on misclassifications and low uncertainty on correct ones would be desirable, such that a system could return control to a human expert when it can not make a confident decision. In general, the network tends to be more uncertain on its misclassifcations than its correct ones regardless of whether it was trained with or without Dropout and of the method used for prediction. Both Dropout and the Laplace approximation similarly increase the uncertainty in the predictions, however this is irrespective of the correctness of the classification. Yet, our experiments show that the Kronecker factored Laplace approximation can be scaled to modern convolutional networks and maintain good classification accuracy while having similar uncertainty about the predictions as Dropout. We had to use much stronger regularisation for the Laplace approximation on the wide residual network, possibly because the block-diagonal approximation becomes more inaccurate on deep networks, possibly because the number of parameters is much higher relative to the number of data. It would be interesting to see how the Laplace approximations behaves on a much larger dataset like ImageNet for similarly sized networks, where we have a better ratio of data to parameters and curvature directions. However, even on a relatively small dataset like CIFAR we did not have to regularise the Laplace approximation to the degree of the posterior becoming deterministic. We presented a scalable approximation to the Laplace approximation for the posterior of a neural network and provided experimental suggesting that the uncertainty estimates are on par with current alternatives like Dropout, if not better. It enables practitioners to obtain principled uncertainty estimates from their models, even if they were trained in a maximum likelihood/MAP setting. There are many possible extensions to this work. One would be to automatically determine the scale and regularisation hyperparameters of the Kronecker factored Laplace approximation using the model evidence similar to how BID26 interpolates between the data log likelihood and the width of the prior. The model evidence could further be used to perform Bayesian model averaging on ensembles of neural networks, potentially improving their generalisation ability and uncertainty estimates. A challenging application would be active learning, where only little data is available relative to the number of curvature directions that need to be estimated. Here, we provide the basic derivation of the factorisation of the diagonal blocks of the Hessian in Eq. 1 and the recursive formula for calculating H as presented in BID3.The Hessian of a neural network with parameters θ as defined in the main text has elements: DISPLAYFORM0 For a given layer λ, the gradient w.r.t. a weight W λ a,b is: DISPLAYFORM1 Keeping λ fixed and differentiating again, we find that the per-sample Hessian of that layer is: DISPLAYFORM2 where DISPLAYFORM3 is the pre-activation Hessian. We can reexpress this in matrix notation as a Kronecker product as in Eq. 1: DISPLAYFORM4 The pre-activation Hessian can be calculated recursively as: DISPLAYFORM5 where the diagonal matrices B and D are defined as: DISPLAYFORM6 DISPLAYFORM7 f and f denote the first and second derivative of the transfer function. The recursion is initialised with the Hessian of the error w.r.t. the linear network outputs. For further details and on how to calculate the diagonal blocks of the Gauss-Newton and Fisher matrix, we refer the reader to BID3 and BID27. The matrix normal distribution BID11 ) is a multivariate distribution over an entire matrix of shape n × p rather than just a vector. In contrast to the multivariate normal distribution, it is parameterised by two p.s.d. covariance matrices, U: n × n and V: p × p, which indicate the covariance of the rows and columns respectively. In addition it has a mean matrix M: n × p. A vectorised sample from a matrix normal distribution X ∼ MN (M, U, V) corresponds to a sample from a normal distribution vec(X) ∼ N (vec(M), U ⊗ V ). However, samples can be drawn more efficiently as X = M + AZB with Z ∼ MN (0, I, I), and AA T = U and B T B = V. The sample Z corresponds to a sample from a normal distribution of length np that has been reshaped to a n × p matrix. This is more efficient in the sense that we only need to calculate two matrix-matrix products of small matrices, rather than a matrix-vector product with one big one. While the square root of Q λ is calculated during the forward pass on all layers, H requires an additional backward pass. Strictly speaking, it is not essential to approximate E [H] for the Kronecker factored Laplace approximation, as in contrast to optimisation procedures the curvature only needs to be calculated once and is thus not time critical. For datasets of the scale of ImageNet and the networks used for such datasets, it would still be impractically slow to perform the calculation for every data point individually. Furthermore, as most datasets are augmented during training, e.g. random cropping or reflections of images, the curvature of the network can be estimated using the same augmentations, effectively increasing the size of the dataset by orders of magnitude. Thus, we make use of the minibatch approximation in our experiments -as we make use of data augmentationin order to demonstrate its practical applicability. We note that E [H] can be calculated exactly by running KFRA BID3 with a minibatchsize of one, and then averaging the . KFAC BID27, in contrast, stochastically approximates the Fisher matrix, so even when run for every datapoint separately, it cannot calculate the curvature factor exactly. In the following, we also show figures for the adversarial experiments in which we calculate the curvature per datapoint and without data augmentation: Figure 6: Untargeted adversarial attack for Kronecker factored Laplace approximation with the curvature calculated with and without data augmentation/approximating the activation Hessian. Fig. 6 and FIG6 show how the Laplace approximation with the curvature estimated from 1000 randomly sampled binary MNIST images and the activation Hessian calculated with a minibatch size of 100 performs in comparison to the curvature factor being calculated without any data augmentation with a batch size of 100 or exactly. We note that without data augmentation we had to use much stronger regularisation of the curvature factors, in particular we had to add a non-negligible multiple of the identity to the factors, whereas with data augmentation it was only needed to ensure that the matrices are invertible. The Kronecker factored Laplace approximation reaches particularly high uncertainty on the untargeted adversarial attack and is most robust on the targeted attack when using data augmentation, suggesting that it is particularly well suited for large datasets and ones where some form of data augmentation can be applied. The difference between approximating the activation Hessian over a minibatch and calculating it exactly appears to be negligible. If we denote the dimensionality of the input to layer λ as D λ−1 and its output as D λ, the curvature factors correspond to the two precision matrices with DISPLAYFORM0'parameters' to estimate, since they are symmetric. So across a network, the number of curvature directions that we are estimating grows linearly in the number of layers and quadratically in the dimension of the layers, i.e. the number of columns of the weight matrices. The size of the full Hessian, on the other hand, grows quadratically in the number of layers and with the fourth power in the dimensionality of the layers (assuming they are all the same size).Once the curvature factors are calculated, which only needs to be done once, we use their Cholesky decomposition to solve two triangular linear systems when sampling weights from the matrix normal distribution. We use the same weight samples for each minibatch, i.e. we do not sample a weight matrix per datapoint. This is for computational efficiency and does not change the expectation. One possibility to save computation time would be to sample a fixed set of weight matrices from the approximate posterior -in order to avoid solving the linear system on every forward pass -and treat the networks that they define as an ensemble. The individual ensemble members can be evaluated in parallel and their outputs averaged, which can be done with a small overhead over evaluating a single network given sufficient compute resources. A further speed up can be achieved by distilling the predictive distributions of the Laplace network into a smaller, deterministic feedforward network as successfully demonstrated in BID0 for posterior samples using HMC.E COMPLEMENTARY FIGURES FOR THE TOY DATASET FIG7 shows the different Laplace approximations (Kronecker factored, diagonal, full) from the main text without any hyperparameter tuning. The figure of the uncertainty obtained from samples using HMC is repeated. Note that the scale is larger than in the main text due to the high uncertainty of the Laplace approximations. The Laplace approximations are increasingly uncertain away from the data, as the true posterior estimated from HMC samples, however they all overestimate the uncertainty without regularisation. This is easy to fix by optimising the hyperparameters on a validation set as discussed in the main text, ing in posterior uncertainty much more similar to the true posterior. As previously discussed in BID3, the Hessian of a neural network is usually underdetermined as the number of data points is much smaller than the number of parameters -in our case we have 20 data points to estimate a 78×78 precision matrix. This leads to the full Laplace approximation vastly overestimating the uncertainty and a bad predictive mean. Both the Kronecker factored and the diagonal approximation exhibit smaller variance than the full Laplace approximation as they restrict the structure of the precision matrix. Consistently with the other experiments, we find the diagonal Laplace approximation to place more mass in low probability areas of the posterior than the Kronecker factored approximation, ing in higher variance on the regression problem. This leads to a need for greater regularisation of the diagonal approximation to obtain acceptable predictive performance, and underestimating the uncertainty. This section shows the accuracy values obtained from the different predictions methods on the feedforward networks for MNIST and the wide residual network for CIFAR100. The for MNIST are shown in TAB0 and the for CIFAR in TAB1.In all cases, neither MC Dropout nor the Laplace approximation significantly change the classification accuracy of the network in comparison to a deterministic forward pass. Our wide residual network has n=3 block repetitions and a width factor of k=8 on CIFAR100 with and without Dropout using hyperparameters taken from BID33: the network parameters are trained on a cross-entropy loss using Nesterov momentum with an initial learning rate of 0.1 and momentum of 0.9 for 200 epochs with a minibatch size of 128. We decay the learning rate every 50 epochs by a factor of 0.2, which is slightly different to the schedule used in BID33 ) (they decay after 60, 120 and 160 epochs). As the original authors, we use L 2 -regularisation with a factor of 5×10 −4.We make one small modification to the architecture: instead of downsampling with 1×1 convolutions with stride 2, we use 2×2 convolutions. This is due to Theano not supporting the transformation of images into the patches extracted by a convolution for 1×1 convolutions with stride greater than 1, which we require for our curvature backpropagation through convolutions. We apply a standard Laplace approximation to the batch normalisation parameters -a Kronecker factorisation is not needed, since the parameters are one-dimensional. When calculating the curvature factors, we use the moving averages for the per-layer means and standard deviations obtained after training, in order to maintain independence between the data points in a minibatch. We need to make a further approximation to the ones discussed in Section 2.2 when backpropagating the curvature for residual networks. The residual blocks compute a function of the form res(x) = x + f φ (x), where f φ typically is a sequence of convolutions, batch normalisation and elementwise nonlinearities. This means that we would need to pass back two curvature matrices, one for each summand. However, this would double the number of backpropagated matrices for each residual connection, hence the computation time/memory requirements would grow exponentially in the number of residual blocks. Therefore, we simply add the curvature matrices after each residual connection.

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We construct a Kronecker factored Laplace approximation for neural networks that leads to an efficient matrix normal distribution over the weights.

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Question: Spectral embedding is a popular technique for the representation of graph data. Several regularization techniques have been proposed to improve the quality of the embedding with respect to downstream tasks like clustering. In this paper, we explain on a simple block model the impact of the complete graph regularization, whereby a constant is added to all entries of the adjacency matrix. Specifically, we show that the regularization forces the spectral embedding to focus on the largest blocks, making the representation less sensitive to noise or outliers. We illustrate these on both on both synthetic and real data, showing how regularization improves standard clustering scores. Spectral embedding is a standard technique for the representation of graph data . Given the adjacency matrix A ∈ R n×n + of the graph, it is obtained by solving either the eigenvalue problem: or the generalized eigenvalue problem: where D = diag(A1 n) is the degree matrix, with 1 n the all-ones vector of dimension n, L = D − A is the Laplacian matrix of the graph, Λ ∈ R k×k is the diagonal matrix of the k smallest (generalized) eigenvalues of L and X ∈ R n×k is the corresponding matrix of (generalized) eigenvectors. In this paper, we only consider the generalized eigenvalue problem, whose solution is given by the spectral decomposition of the normalized Laplacian matrix L norm = I − D −1/2 AD −1/2 . The spectral embedding can be interpreted as equilibrium states of some physical systems (; ;), a desirable property in modern machine learning. However, it tends to produce poor on real datasets if applied directly on the graph . One reason is that real graphs are most often disconnected due to noise or outliers in the dataset. In order to improve the quality of the embedding, two main types of regularization have been proposed. The first artificially increases the degree of each node by a constant factor , while the second adds a constant to all entries of the original adjacency matrix (; ;). In the practically interesting case where the original adjacency matrix A is sparse, the regularized adjacency matrix is dense but has a so-called sparse + low rank structure, enabling the computation of the spectral embedding on very large graphs . While explains the effects of regularization through graph conductance and through eigenvector perturbation on the Stochastic Block Model, there is no simple interpretation of the benefits of graph regularization. In this paper, we show on a simple block model that the complete graph regularization forces the spectral embedding to separate the blocks in decreasing order of size, making the embedding less sensitive to noise or outliers in the data. Indeed, identified that, without regularization, the cuts corresponding to the first dimensions of the spectral embedding tend to separate small sets of nodes, so-called dangling sets, loosely connected to the rest of the graph. Our work shows more explicitly that regularization forces the spectral embedding to focus on the largest clusters. Moreover, our analysis involves some explicit characterization of the eigenvalues, allowing us to quantify the impact of the regularization parameter. The rest of this paper is organized as follows. Section 2 presents block models and an important preliminary about their aggregation. Section 3 presents the main of the paper, about the regularization of block models, while Section 4 extends this to bipartite graphs. Section 5 presents the experiments and Section 6 concludes the paper. Let A ∈ R n×n + be the adjacency matrix of an undirected, weight graph, that is a symmetric matrix such that A ij > 0 if and only if there is an edge between nodes i and j, with weight A ij. Assume that the n nodes of the graph can be partitioned into K blocks of respective sizes n 1,..., n K so that any two nodes of the same block have the same neighborhood, i.e., the corresponding rows (or columns) of A are the same. Without any loss of generality, we assume that the matrix A has rank K. We refer to such a graph as a block model. Let Z ∈ R n×K be the associated membership matrix, with Z ij = 1 if index i belongs to block j and 0 otherwise. We denote by W = Z T Z ∈ R K×K the diagonal matrix of block sizes.. This is the adjacency matrix of the aggregate graph, where each block of the initial graph is replaced by a single node; two nodes in this graph are connected by an edge of weight equal to the total weight of edges between the corresponding blocks in the original graph. We denote byD = diag(Ā1 K) the degree matrix and byL =D −Ā the Laplacian matrix of the aggregate graph. The following shows that the solution to the generalized eigenvalue problem follows from that of the aggregate graph: Proposition 1. Let x be a solution to the generalized eigenvalue problem: Then either Z T x = 0 and λ = 1 or x = Zy where y is a solution to the generalized eigenvalue problem:L y = λDy. Proof. Consider the following reformulation of the generalized eigenvalue problem: Since the rank of A is equal to K, there are n − K eigenvectors x associated with the eigenvalue λ = 1, each satisfying Z T x = 0. By orthogonality, the other eigenvectors satisfy x = Zy for some vector y ∈ R K. We get: Thus y is a solution to the generalized eigenvalue problem. Let A be the adjacency matrix of some undirected graph. We consider a regularized version of the graph where an edge of weight α is added between all pairs of nodes, for some constant α > 0. The corresponding adjacency matrix is given by: where J = 1 n 1 T n is the all-ones matrix of same dimension as A. We denote by D α = diag(A α 1 n) the corresponding degree matrix and by L α = D α − A α the Laplacian matrix. We first consider a simple block model where the graph consists of K disjoint cliques of respective sizes n 1 > n 2 > · · · > n K nodes, with n K ≥ 1. In this case, we have A = ZZ T, where Z is the membership matrix. The objective of this section is to demonstrate that, in this setting, the k-th dimension of the spectral embedding isolates the k − 1 largest cliques from the rest of the graph, for any k ∈ {2, . . ., K} Lemma 1. Let λ 1 ≤ λ 2 ≤... ≤ λ n be the eigenvalues associated with the generalized eigenvalue problem: Proof. Since the Laplacian matrix L α is positive semi-definite, all eigenvalues are non-negative . We know that the eigenvalue 0 has multiplicity 1 on observing that the regularized graph is connected. Now for any vector x, so that the matrix A α is positive semi-definite. In view of, this shows that λ ≤ 1 for any eigenvalue λ. The proof then follows from Proposition 1, on observing that the eigenvalue 1 has multiplicity n − K. Lemma 2. Let x be a solution to the generalized eigenvalue problem with λ ∈. There exists some s ∈ {+1, −1} such that for each node i in block j, Proof. In view of Proposition 1, we have x = Zy where y is a solution to the generalized eigenvalue problem of the aggregate graph, with adjacency matrix: where I K is the identity matrix of dimension K × K. We deduce the degree matrix: and the Laplacian matrix:L The generalized eigenvalue problem associated with the aggregate graph is: After multiplication by W −1, we get: Observing that and since W = diag(n 1, . . ., n K), The then follows from the fact that x = Zy. Lemma 3. The K smallest eigenvalues satisfy: where for all j = 1,..., K, µ j = αn αn + n j. Proof. We know from Lemma 1 that the K smallest eigenvalues are in. Let x be a solution to the generalized eigenvalue problem with λ ∈. We know that x = Zy where y is an eigenvector associated with the same eigenvalue λ for the aggregate graph. Since 1 K is an eigenvector for the eigenvalue 0, we have y We then deduce from and This condition cannot be satisfied if λ < µ 1 or λ > µ K as the terms of the sum would be either all positive or all negative. Now let y be another eigenvector for the aggregate graph, with y TD α y = 0, for the eigenvalue λ ∈. By the same argument, we get: with λ ∈ {µ 1, . . ., µ K}. This condition cannot be satisfied if λ and λ are in the same interval (µ j, µ j+1) for some j as the terms in the sum would be all positive. There are K − 1 eigenvalues in for K − 1 such intervals, that is one eigenvalue per interval. The main of the paper is the following, showing that the k − 1 largest cliques of the original graph can be recovered from the spectral embedding of the regularized graph in dimension k. Theorem 1. Let X be the spectral embedding of dimension k, as defined by, for some k in the set {2, . . ., K}. Then sign(X) gives the k − 1 largest blocks of the graph. Proof. Let x be the j-th column of the matrix X, for some j ∈ {2, . . ., k}. In view of Lemma 3, this is the eigenvector associated with eigenvalue λ j ∈ (µ j−1, µ j), so that In view of Lemma 2, all entries of x corresponding to blocks of size n 1, n 2..., n j−1 have the same sign, the other having the opposite sign. Theorem 1 can be extended in several ways. First, the assumption of distinct block sizes can easily be relaxed. If there are L distinct values of block sizes, say m 1,..., m L blocks of sizes n 1 >... > n L, there are L distinct values for the thresholds µ j and thus L distinct values for the eigenvalues λ j in, the multiplicity of the j-th smallest eigenvalue being equal to m j. The spectral embedding in dimension k still gives k − 1 cliques of the largest sizes. Second, the graph may have edges between blocks. Taking A = ZZ T + εJ for instance, for some parameter ε ≥ 0, the are exactly the same, with α replaced by +α. A key observation is that regularization really matters when ε → 0, in which case the initial graph becomes disconnected and, in the absence of regularization, the spectral embedding may isolate small connected components of the graph. In particular, the regularization makes the spectral embedding much less sensitive to noise, as will be demonstrated in the experiments. Finally, degree correction can be added by varying the node degrees within blocks. Taking A = θZZ T θ, for some arbitrary diagonal matrix θ with positive entries, similar can be obtained under the regularization A α = A + αθJθ. Interestingly, the spectral embedding in dimension k then recovers the k − 1 largest blocks in terms of normalized weight, the ratio of the total weight of the block to the number of nodes in the block. Let B = R n×m + be the biadjacency matrix of some bipartite graph with respectively n, m nodes in each part, i.e., B ij > 0 if and only if there is an edge between node i in the first part of the graph and node j in the second part of the graph, with weight B ij. This is an undirected graph of n + m nodes with adjacency matrix: The spectral embedding of the graph can be written in terms of the biadjacency matrix as follows: where X 1, X 2 are the embeddings of each part of the graph, with respective dimensions n × k and In particular, the spectral embedding of the graph follows from the generalized SVD of the biadjacency matrix B. The complete regularization adds edges between all pairs of nodes, breaking the bipartite structure of the graph. Another approach consists in applying the regularization to the biadjacency matrix, i.e., in considering the regularized bipartite graph with biadjacency matrix: B α = B + αJ, where J = 1 n 1 T m is here the all-ones matrix of same dimension as B. The spectral embedding of the regularized graph is that associated with the adjacency matrix: As in Section 3, we consider a block model so that the biadjacency matrix B is block-diagonal with all-ones block matrices on the diagonal. Each part of the graph consists of K groups of nodes of respective sizes n 1 >... > n K and m 1 >... > m K, with nodes of block j in the first part connected only to nodes of block j in the second part, for all j = 1,..., K. We consider the generalized eigenvalue problem associated with the above matrix A α. In view of, this is equivalent to the generalized SVD of the regularized biadjacency matrix B α. We have the following , whose proofs are deferred to the appendix: Lemma 4. Let λ 1 ≤ λ 2 ≤... ≤ λ n be the eigenvalues associated with the generalized eigenvalue problem. We have λ 1 = 0 < λ 2 ≤... ≤ λ K < λ K+1 =... = λ n−2K <... < λ n = 2. Lemma 5. Let x be a solution to the generalized eigenvalue problem with λ ∈. There exists s 1, s 2 ∈ {+1, −1} such that for each node i in block j of part p ∈ {1, 2}, Lemma 6. The K smallest eigenvalues satisfy: Published as a conference paper at ICLR 2020 Theorem 2. Let X be the spectral embedding of dimension k, as defined by, for some k in the set {2, . . ., K}. Then sign(X) gives the k − 1 largest blocks of each part of the graph. Like Theorem 1, the assumption of decreasing block sizes can easily be relaxed. Assume that block pairs are indexed in decreasing order of µ j. Then the spectral embedding of dimension k gives the k − 1 first block pairs for that order. It is interesting to notice that the order now depends on α: when α → 0 +, the block pairs j of highest value (n nj + m mj) −1 (equivalently, highest harmonic mean of proportions of nodes in each part of the graph) are isolated first; when α → +∞, the block pairs j of highest value nj mj nm (equivalently, the highest geometric mean of proportions of nodes in each part of the graph) are isolated first. The also extend to non-block diagonal biadjacency matrices B and degree-corrected models, as for Theorem 1. We now illustrate the impact of regularization on the quality of spectral embedding. We focus on a clustering task, using both synthetic and real datasets where the ground-truth clusters are known. In all experiments, we skip the first dimension of the spectral embedding as it is not informative (the corresponding eigenvector is the all-ones vector, up to some multiplicative constant). The code to reproduce these experiments is available online 1. We first illustrate the theoretical of the paper with a toy graph consisting of 3 cliques of respective sizes 5, 3, 2. We compute the spectral embeddings in dimension 1, using the second smallest eigenvalue. Denoting by Z the membership matrix, we get X ≈ Z(−0.08, 0.11, 0.05) T for α = 1, showing that the embedding isolates the largest cluster; this is not the case in the absence of regularization, where X ≈ Z(0.1, −0.1, 0.41) T. This section describes the datasets used in our experiments. All graphs are considered as undirected. Table 1 presents the main features of the graphs. Stochastic Block-Model (SBM) We generate 100 instances of the same stochastic block model . There are 100 blocks of size 20, with intra-block edge probability set to 0.5 for the first 50 blocks and 0.05 for the other blocks. The inter-block edge probability is set to 0.001 Other sets of parameters can be tested using the code available online. The ground-truth cluster of each node corresponds to its block. This dataset consists of around 18000 newsgroups posts on 20 topics. This defines a weighted bipartite graph between documents and words. The label of each document corresponds to the topic. . This is the graph of hyperlinks between a subset of Wikipedia pages. The label of each page is its category (e.g., countries, mammals, physics). We consider a large set of metrics from the clustering literature. All metrics are upper-bounded by 1 and the higher the score the better. Homogeneity (H), Completeness (C) and V-measure score (V) . Supervised metrics. A cluster is homogeneous if all its data points are members of a single class in the ground truth. A clustering is complete if all the members of a class in the ground truth belong to the same cluster in the prediction. Harmonic mean of homogeneity and completeness. Adjusted Rand Index (ARI) . Supervised metric. This is the corrected for chance version of the Rand Index which is itself an accuracy on pairs of samples. Adjusted Mutual Information (AMI) Supervised metric. Adjusted for chance version of the mutual information. Fowlkes-Mallows Index (FMI) . Supervised metric. Geometric mean between precision and recall on the edge classification task, as described for the ARI. Modularity (Q) . Unsupervised metric. Fraction of edges within clusters compared to that is some null model where edges are shuffled at random. Normalized Standard Deviation (NSD) Unsupervised metric. 1 minus normalized standard deviation in cluster size. All graphs are embedded in dimension 20, with different regularization parameters. To compare the impact of this parameter across different datasets, we use a relative regularization parameter (w/n 2)α, where w = 1 T n A1 n is the total weight of the graph. We use the K-Means algorithm with to cluster the nodes in the embedding space. The parameter K is set to the ground-truth number of clusters (other experiments with different values of K are reported in the Appendix). We use the Scikit-learn implementation of KMeans and the metrics, when available. The spectral embedding and the modularity are computed with the Scikit-network package, see the documentation for more details 2. We report the in Table 2 for relative regularization parameter α = 0, 0.1, 1, 10. We see that the regularization generally improves performance, the optimal value of α depending on both the dataset and the score function. As suggested by Lemma 3, the optimal value of the regularization parameter should depend on the distribution of cluster sizes, on which we do not have any prior knowledge. To test the impact of noise on the spectral embedding, we add isolated nodes with self loop to the graph and compare the clustering performance with and without regularization. The number of isolated nodes is given as a fraction of the initial number of nodes in the graph. Scores are computed only on the initial nodes. The are reported in Table 3 for the Wikipedia for Schools dataset. We observe that, in the absence of regularization, the scores drop even with only 1% noise. The computed clustering is a trivial partition with all initial nodes in the same cluster. This means that the 20 first dimensions of the spectral embedding focus on the isolated nodes. On the other hand, the scores remain approximately constant in the regularized case, which suggests that regularization makes the embedding robust to this type of noise. In this paper, we have provided a simple explanation for the well-known benefits of regularization on spectral embedding. Specifically, regularization forces the embedding to focus on the largest clusters, making the embedding more robust to noise. This was obtained through the explicit characterization of the embedding for a simple block model, and extended to bipartite graphs. An interesting perspective of our work is the extension to stochastic block models, using for instance the concentration proved in . Another problem of interest is the impact of regularization on other downstream tasks, like link prediction. Finally, we would like to further explore the impact of the regularization parameter, exploiting the theoretical presented in this paper. We provide of proof of Theorem 2 as well as a complete set of experimental . The proof of Theorem 2 follows the same workflow as that of Theorem 1. Let Z 1 ∈ R n×K and Z 2 ∈ R m×K be the left and right membership matrices for the block matrix B ∈ R n×m. The aggregated matrix isB = Z T 1 BZ 2 ∈ R K×K. The diagonal matrices of block sizes are We have the equivalent of Proposition 1: Proposition 2. Let x 1, x 2 be a solution to the generalized singular value problem: x 2 = 0 and σ = 0 or x 1 = Z 1 y 1 and x 2 = Z 2 y 2 where y 1, y 2 is a solution to the generalized singular value problem: B y 2 = σD 1 y 1, B T y 1 = σD 2 y 2. Proof. Since the rank of B is equal to K, there are n−K pairs of singular vectors (x 1, x 2) associated with the singular values 0, each satisfying Z T 1 x 1 = 0 and Z T 2 x 2 = 0. By orthogonality, the other pairs of singular vectors satisfy x 1 = Z 1 y 1 and x 2 = Z 2 y 2 for some vectors y 1, y 2 ∈ R K. By replacing these in the original generalized singular value problem, we get that (y 1, y 2) is a solution to the generalized singular value problem for the aggregate graph. In the following, we focus on the block model described in Section 4, where Proof of Lemma 4. The generalized eigenvalue problem associated with the regularized matrix A α is equivalent to the generalized SVD of the regularized biadjacency matrix B α: In view of Proposition 2, the singular value σ = 0 has multiplicity n − K, meaning that the eigenvalue λ = 1 has multiplicity n − K. Since the graph is connected, the eigenvalue 0 has multiplicity 1. The proof then follows from the observation that if (x 1, x 2) is a pair of singular vectors for the singular value σ, then the vectors x = (x 1, ±x 2) T are eigenvectors for the eigenvalues 1 − σ, 1 + σ. Proof of Lemma 5. By Proposition 2, we can focus on the generalized singular value problem for the aggregate graph: we have: Observing that J K W 1 y 1 ∝ 1 K and J K W 2 y 2 ∝ 1 K, we get: As two diagonal matrices commute, we obtain: for some constants η 1, η 2, and Letting s 1 = −sign(η 1 (n j + αn) + η 2 m j ) and s 2 = −sign(η 1 n j + η 2 (m j + αm)), we get: and the follows from the fact that x 1 = Z 1 y 1 and x 2 = Z 2 y 2. Proof of Lemma 6. The proof is the same as that of Lemma 3, where the threshold values follow from Lemma 5: Proof of Theorem 2. Let x be the j-th column of the matrix X, for some j ∈ {2, . . ., k}. In view of Lemma 6, this is the eigenvector associated with eigenvalue λ j ∈ (µ j−1, µ j). In view of Lemma 4, all entries of x corresponding to blocks of size n 1, n 2..., n j−1 have the same sign, the other having the opposite sign. In this section, we present more extensive experimental . Tables 4 and 5 present for the same experiment as in Table 2 but for different values of K, namely K = 2 (bisection of the graph) and K = K truth /2 (half of the ground-truth value). As for K = K true, regularization generally improves clustering performance. However, the optimal value of α remains both dataset dependent and metric dependent. Note that, for the NG and WS datasets, the clustering remains trivial in the case K = 2, one cluster containing all the nodes, until a certain amount of regularization. Table 6 presents the different scores for both types of regularization on the NG dataset. As we can see, preserving the bipartite structure of the graph leads to slightly better performance. Finally, Table 7 shows the impact of regularization in the presence of noise for the NG dataset. The are similar as for the WS dataset: regularization makes the spectral embedding much more robust to noise.

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Graph regularization forces spectral embedding to focus on the largest clusters, making the representation less sensitive to noise.

We establish that the scaling laws derived in (Bora et al., 2017) are optimal or near-optimal in the absence of further assumptions.


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Question: The exposure bias problem refers to the training-inference discrepancy caused by teacher forcing in maximum likelihood estimation (MLE) training for auto-regressive neural network language models (LM). It has been regarded as a central problem for natural language generation (NLG) model training. Although a lot of algorithms have been proposed to avoid teacher forcing and therefore to alleviate exposure bias, there is little work showing how serious the exposure bias problem is. In this work, we first identify the auto-recovery ability of MLE-trained LM, which casts doubt on the seriousness of exposure bias. We then develop a precise, quantifiable definition for exposure bias. However, according to our measurements in controlled experiments, there's only around 3% performance gain when the training-inference discrepancy is completely removed. Our suggest the exposure bias problem could be much less serious than it is currently assumed to be. Language model (LM) is a central module for natural language generation (NLG) tasks such as machine translation , dialogue response generation, image captioning , etc. For decades, maximum likelihood estimation (MLE) has been the the most widely used objective for LM training. However, there is a popular belief in the natural language processing (NLP) community that standard MLE training will cause "exposure bias" and lead to a performance degradation during the test-time language generation. The exposure bias problem refers to the following discrepancy between MLE training and test-time generation for language models: During training, the language model predicts the next word conditioned on history words sampled from the groundtruth data distribution. And during generation, the model generates words conditioned on history sequences generated by the model itself. However, due to the exposure to real data during training, the language model is biased to only perform well on the ground-truth history distribution. As a , during generation the errors will accumulate along the generated sequence, and the distribution generated by the model will be distorted. The forced exposure to ground-truth data during training is also referred to as "teacher forcing". Given its defintion, the exposure bias problem could rise in the general cases when the model needs to make a sequence of decisions or generations (e.g. music/pixel/speech generation ). In this work, we focus on the task of language generation, because the exposure bias problem is originally proposed in this field , and has since attracted huge research attention. In order to avoid teacher forcing, many training algorithms (; ; ; ; ; ; ; ; ; ; ;) have been proposed as alternatives to MLE training. Most of these works utilize techniques from generative adversarial network (GAN) or reinforcement learning (RL) . In this paper, we refer to these algorithms as non-MLE methods or text GANs. Despite the huge research efforts devoted to alleviate exposure bias, surprisingly, its existence or significance is much less studied. In particular, to the best of our knowledge, no existing work Table 1: Samples of a MLE-trained STOA transformer LM when fed with different types of length-10 history prefix. To save space, we omitted the first 7 words of the random history. attempts to directly show the seriousness of exposure bias in an empirical or theoretical way. This work is motivated by the belief that a good solution should be built upon a testable and quantifiable problem definition. In this rest of this paper, we first identify the "self-recovery" ability of popular LM models, which casts doubt on the original claim of exposure bias. We then develop a precise and quantifiable definition of exposure bias, and validate its seriousness in controlled experiments. To study the seriousness of exposure bias in standard MLE LM training, we first stress that the following methodology, although tempting, is wrong: If we can rigorously show that the non-MLE methods proposed to avoid teacher forcing do indeed bring solid generation performance gain, then we can conclude exposure bias is a meaningful problem for the original MLE training. The reason is that we typically do not know the exact underlying reason for the performance gain. For example, despite the huge success of the batch normalization technique in deep learning, whether "internal covariate shift" (which is the motivation of batch norm) exists in deep neural network training remains a question . Therefore, in this work we seek a direct way to validate the seriousness of exposure bias. We focus on the following informal claim that immediately follows from the original definition of exposure bias: During generation, if we set the history distribution to be the ground-truth data distribution instead of the model's own distribution (now that there is no discrepancy between training and testing), then the model's language generation quality should be much better (we will formalize this notion in Section 4 and 5). We start with the following qualitative analysis. We feed a MLE-trained transformer LM on wiki-103 data-set with four kinds of prefixes: model's own samples, data samples, shuffled (word-level) data samples or samples from a uniform random distribution. Then we let the model complete the sentence given these prefixes as history. We list some samples in Table 1 and more in Appendix A (this experiment is also repeated for a LSTM LM). Assuming the seriousness of exposure bias, we expect the quality of generated sentence-completion samples with real-data prefixes to be significantly better than the ones from prefixes of model samples. However, by manual inspection, we do not observe noticeable differences in sample quality. More surprisingly, the model is still able to generate relevant and fairly high-quality samples from shuffled prefixes. Even in the extreme case where random sequences are fed, the model is able to generate reasonable sentences. Due to the recent increasing interest of solving exposure bias in the field of neural machine translation (NMT) , we repeat the above experiment in a standard NMT setting in Appendix A, and get very similar observations. These experiments clearly show that the MLE-trained auto-regressive LMs have the self-recovery ability, i.e. the model is able to recover from artificially distorted history input, and generate reasonably high-quality samples. This phenomenon is clearly in contradiction with the popular claim of exposure bias, that the error induced by the mismatch between history and data distribution should accumulate during the generation process. Motivated by these experiments, in the following sections, we turn to more rigorous methods to quantify the significance of exposure bias. Note that our quantification approaches will be independent of the training procedure and only require inference from the trained model. The task of auto-regressive language modelling is to learn the probability distribution of the (l + 1) th word W l+1 in a sentence conditioned on the word history W 1:l:= (W 1, . . ., W l). Here, we use the uppercase W i ∈ V to denote a discrete random variable distributed across the vocabulary V. The lower-case w is used to denote some particular word in V. Given a training data-set D consisting of sentences of length L, the standard MLE training minimizes the negative log-likelihood below: Note that in this work we assume all sentences are of length L for simplicity. We denote the generation distribution of the trained LM as P M, and the ground-truth data distribution as P D. Readers can assume P M refers to the generation distribution of a LSTM LM or a transformer LM trained with MLE objective, which is the major subject of this study. We will mainly present on LSTM based models to facilitate comparison with text-GAN works (listed in Section 1), which are mostly implemented on LSTM models. We will also provide with the transformer model, with very similar observations or measurements. Our quantification mainly relies on the measurements of the distance from the model's generation distribution to the data distribution. Hence we define the following notations to simplify expressions. Let P denote the set of probability distributions on the vocabulary V. Let d denote a distance measure between distributions (e.g. total variation distance), d: P × P → R ≥0. In this section, we propose an intuitive and seemingly correct quantification approach using marginal distributions. The approach can be applied to real-world text data experiments, but it has some lethal weak point. The discussion will lead us to our final precise definition of exposure bias in Section 5. Assuming a given history length l, we consider the marginal distribution of W l+1 from the following three random process: • Draw word sequences of length L from the data distribution P D. Denote the marginal distribution of the random variable at position l + 1 (W l+1) as P l+1 D|D, where • Draw word sequences of length L from the model distribution P M. Denote the marginal distribution of the random variable at position l + 1 as Under review as a conference paper at ICLR 2020 Denote the marginal distribution of the random variable at position l + 1 as P l+1 M |D, where By the definition of exposure bias, P l+1 M |M suffers from the training-testing discrepancy, while P l+1 M |D should be closer to the true distribution P l+1 D|D. To measure this discrepancy, define the marginal generation deviation (MGD) at history length l of history distribution P H with metric d as where P H ∈ {P M, P D} denotes the history distribution. MGD measures the deviation of the marginal distribution of W l+1 from ground-truth data distribution. Finally, we define the rate of exposure bias (EB-M) at history length l of model P M as the ratio (discrepancy) between the MGD measurements when two different history distributions are fed: For MLE-trained models, EB-M 1 is expected to be larger than 1, and larger EB-M indicates a more serious exposure bias problem for the trained model. For the metric d, we consider two popular probability metrics: total variation distance (denoted as d T V), and Jensen-Shannon divergence (denoted as d JS). In this section, we focus on answering the following question: "Does the EB-M measurement correctly reflect the significance of exposure bias?" In short, our answer is not really. The problem is that the distortion of the marginal P l+1 M |M is not only affected by the presumably existing exposure bias problem alone, but also by the mismatch between the history distribution P M from P D for W 1:l, which grows with the length of the history. Therefore, even if the measured EB-M is significantly larger than one, we can not conclude that exposure bias causes serious deterioration. We provide an example to illustrate this argument: Example 1. Suppose L = 2, and V = {A, B}. P D and P M are crafted as follows: However, the only problem P M has is the mismatch between the history distributions (P M and P D) for W 1. The next set of experiments also suggest that EB-M does not precisely reflect exposure bias. On the EMNLP-news data-set (specified in Appendix B), we compare EB-M measurements for several non-MLE training methods with the baseline MLE model. We include for Scheduled Sampling (SS) , Cooperative Training (CoT) , and Adversarial Ranking (RankGAN) . We provide implementation details for non-MLE methods in Appendix C. Intuitively, these methods will cause the model to be biased to behave well with model samples as history, instead of data samples. Therefore, we expect EB-M measurement for non-MLE trained models to be smaller than MLE trained models. However, Figure 1 shows that the measurements for different training frameworks are almost the same. We believe the reason is that the EB-M measurements are only reflecting the trivial mismatch between the history distributions. Is it possible that the original definition of exposure bias exactly refers to this mismatch between the model and data history distributions? However, note that this mismatch is inevitable for any imperfect model, and non-MLE training algorithms can not solve it. We believe a better, more precise definition is needed to discriminate exposure bias from this trivial mismatch. Motivated by this view, we propose a second approach in the section below. Following the discussion in the last section, we wish our measurement to be independent of the quality of the history distribution. In light of that, we design a quantity to measure the model's conditional generation quality. Let P H ∈ {P M, P D} denote the history distribution as in the MGD definition. With history length l fixed, we define the conditional generation deviation (CGD) with history distribution P H for P M using metric d as: where we assume that P D (· | W 1:l)) is computable, and use it to measure the quality of the model's conditional distribution. For the choice of the distribution distance d, in addition to d T V and d JS, we introduce greedy decoding divergence (d GD) defined as: where 1 is the indicator function, and P, Q ∈ P. The distance d GD 2 reflects the model's accuracy during greedy decoding. Similar to MGD, exposure bias should imply a significant gap between CGD(P M |M, l, d) and CGD(P M |D, l, d). We again define rate of exposure bias at history length l with metric d to be: For our definition of EB-C, a natural question is why we only focus on the generation distribution of the very next word. The reason is we want to precisely measure how the error caused by the history part affect the generation part, by keeping them separate. If we measure the deviation of, for example, two sampled tokens, the definition will be confusing: Because the second sampled token will be affected not only by the accumulated error induced by the history (sampled from the model), but also by the first generated token as history. To get a better understanding of the intuition behind the definition of EB-C, we recommend readers to read Appendix A about our NMT experiment. Since CGD requires inference for ground-truth data distribution P D, we first consider experiments in a synthetic setting. In text-GAN literature , a randomly-initialized one-layer LSTM model with hidden dimension of 32 is usually used as P D in synthetic experiments (we denote this setting as M random 32). However, the model is small-scale and does not reflect any structure existing in real-world text. To improve upon this approach, we take the MLE baseline model trained on EMNLP-news data (described in Appendix B) as P D in this synthetic setting. We denote the data model (P D) as M news 512. We then train two LSTM LM (P M) with different capacities using samples from the data model, with the standard MLE objective. One is a one-layer LSTM with hidden width of 512 (denoted as LSTM-512), the other one is with hidden width of 32 (denoted as LSTM-32). We train P M for 100 epochs using the Adam optimizer with learning rate 0.001. In each epoch, 250k sentences (same to the size of the original EMNLP-news data) of length L = 50 are sampled from M news-512 as training data to avoid over-fitting. We show perplexity (PPL) of the trained models in Appendix F. Finally, EB-C is calculated using 100k 3 samples from P M and P D. In Figure 2, we show EB-C measurements with different metrics d m, and the two models give similar . It is shown that EB-C has a steady but slow increasing trend as history length increases. This is expected as a consequence of exposure bias, because P M deviates farther from P D as history length increases. However, the average value of EB-C is less than 1.03 (the largest average value is from d JS for the LSTM-512 experiment), meaning that the gap between CGD(P M |M, l, d) and CGD(P M |D, l, d) is not large. Also, note that in most NLG applications (such as machine translation or image captioning), the generated sequence typically has short length (less than 20). In that range of history length, the EB-C measurements that exposure bias only has minimal influence. In Appendix E, we repeat the experiment for a transformer LM , and get very similar EB-C measurements. These measurements imply a striking : (Informal) Even if all the bad effects from exposure bias for MLE LM training are removed, the relative performance gain is at most 3%. If the sequence length is not very long, the gain is less than 1%.. To dive deeper into the cause of the gap in CGD, we experiment with corrupted versions of P M as history distribution. We first specify a corrupt rate c ∈, and randomly substitute words in a history sample from P M to a "noise" word drawn uniformly from the vocabulary with probability c. Consequently, larger c will cause the history distribution to deviate farther from the groundtruth P D. In Figure 3, we show CGD measurement versus the corrupted history P corrupt M. Large gaps are observed between CGD(P M |M corrupt) and CGD(P M |D). Therefore, the small gap between CGD(P M |M) and CGD(P M |D) in Figure 2 from the small deviation between the history distribution P M and P D. In other word, P M has learned a "good enough" distribution that is able to keep it in the well-behaving region during sampling. With these observations, we conclude that, in the synthetic setting considered, exposure bias does exist, but is much less serious than it is presumed to be. Although there exists mismatch between the history distribution P M and P D, the mismatch is still in the model's "comfortable zone". In other words, the LSTM LM is more robust than exposure bias claims it to be. To concretize the this argument, we provide an example LM and show that MLE training is unlikely to generate models with a large EB-C value. Example 2. Again suppose L = 2, and V = {A, B}, the ground-truth data distribution is uniform on {AA, AB, BB, BA}. P M is crafted as follows:. Note that the model behaves bad when W 1 = A, which is of high probability during sampling. However, this crafted model is unlikely to be an outcome of MLE training. The fact that P M (·|W 1 = B) is better modeled indicates that in the training data more sentences begin with W 1 = B than W 1 = A. So MLE training should assign more probability to P M (W 1 = B), not the other way around 4. setting 5, as we find it hard to do a fast implementation of RankGAN for the LSTM-512 setting. We find that RankGAN and CoT gives lower EB-C measurements than MLE, which is expected, as these methods avoid teacher forcing. For CoT, at short 4 If we change to PM (W1 = A) = 0.1, then EB-C(PM, 1, dT V) will be 0.2, meaning that the model has better conditional generation performance during sampling 5 The MLE model is used as the pre-trained model for the RankGAN generator. The MLE model has an oracle NLL of 8.67, and RankGAN's oracle NLL is 8.55. Table 2: An illustration for the next word collection process. The choices are shuffled. The first history sample is from real data, and the second history sample is from the trained model. Table 3: EB-C measurements with human as P D. hisotry length, EB-C is even less than 1. We believe the reason is that CoT trys to make the model be biased to behave better when fed with model samples. However, SS gives worse EB-C measurements comparing to MLE, which we currently do not have a good explanation. We refer readers to for a discussion about the SS objective. To the best of our knowledge, this is the first direct empirical evidence that text GAN does indeed alleviate the exposure bias problem. It also indicates that EB-C correctly reflect the significance of exposure bias. We believe the reason for why EB-C is still not less than 1 is that, text GANs still rely on MLE pre-training a lot. In this section, we design experiments to efficiently estimate EB-C for a SOTA transformer LM with real human as P D, by utilizing the Amazon Mechanical Turk (AMT) platform. Given a MLE-trained LM as P M, by examining the definition of EB-C in Equation 9 and 7, it is clear the only obstacle is that we don't have access to P D (· | W 1:l) with a given history W 1:l. So, in this section, we focus on the greedy decoding divergence (d GD) metric (Equation 8), which only requires the turkers to give the most probable next word prediction, instead of the full distribution (which is clearly intractable). In our preliminary trials, we find it is still very hard for a person to guess the next word, even with real data history samples. The reason is that the vocabulary is very big, and the turkers may be not familiar with the context (e.g. wikipedia). To alleviate that problem, we design the following simplification: For a given history, we let the model output its top-5 next word prediction, then we only ask the turkers to choose among the 5 choices (the turker can also express that he/she thinks none of them is likely). Finally, we examine whether the turker's choice is indeed the model's top-1 prediction. We illustrate this process in Table 2. We use the code of Transformer-XL to train a SOTA transformer LM on the wiki-103 data-set. We favour the wiki-103 data-set because it is large-scale and has long (over 30 words) paragraphs, which is useful for the measurements of exposure bias. The model is a 16-layer transformer-xl model with hidden dimension of 410. Since the estimation of CGD(P M |D, l, d) requires large amounts of unseen real data samples, we use half of the wiki-103 training data (around 900k sentences and 50m words) to train the model P M, and save the other half as samples from P D. Other training configurations (learning rate, batch size, etc.) are not changed 6. The ing model P M has a test-set PPL of 27.81 (if trained on full training data, the PPL will be 24.02). We collect data to estimate EB-C at history length 10, 20, and 30. For each length and history model (P M or P D) pair, we collect 10k d GD samples (via next-word prediction) from turkers on the AMT platform. More details about the AMT setup are provided in Appendix D. The are shown in Table 3. The EB-C measurements are strikingly similar to the in our synthetic experiments in that, removing the training-testing discrepancy only gives around 2% of relative performance gain. This further strengthens our claim that exposure bias is only a minor problem for MLE-based LM training. Several recent works attempt to carefully evaluate whether the non-MLE training methods (e.g. adversarial training) can give superior NLG performance than standard MLE training for RNN LM. tunes a "temperature" parameter in the softmax output, and evaluate models over the whole quality-diversity spectrum. proposes to use "Reverse Language Model score" or "Frechet InferSent Distance" to evaluate the model's generation performance. proposes a method for approximating a distribution over tokens from a GAN, and then evaluate the model with standard LM metrics. These works arrive at a similar : The general performance of Text GANs is not convincingly better, or even worse, than standard MLE training. Hence to some extent, they imply that exposure bias may be not a serious problem in MLE training. However, as we argued in Section 2, one can not draw direct about exposure bias with these . For example, it is also possible that exposure bias is indeed serious for MLE training, but text GAN does not solve the problem well enough. In this work, we first identify the self-recovery ability of MLE-trained LM, which casts doubt on the seriousness of exposure bias, which has been regarded as a central problem for MLE training by the LM community. We then explore two intuitive approaches to quantify the significance of exposure bias for LM training. The first quantification EB-M relies on the marginal generation distribution and reveals some vagueness in the original definition of exposure bias. We argue that we should focus on the model's generation performance in terms of its conditional distribution and propose a second quantification EB-C, which we regard as the precise definition for exposure bias. We design a evaluation of EB-C at different history length with real human (turkers from AMT) as the data model, for a SOTA transformer LM. It is shown that removing the training-testing discrepancy only gives around 2% of performance gain. Our synthetic experiments also gives very similar measurements. By analyzing EB-C measurements with perturbed history samples, we hypothesise that although the mismatch between the data and model distribution for history prefix exists, it is still in the model's "comfortable zone". With these , we claim that on the contrary to the popular belief, exposure bias is only a minor problem in MLE-based LM training. To wrap up, we discuss the fundamental question "Is MLE training really biased?", from the perspective of objective functions. Note that the MLE objective can be re-written as: where D KL denotes the Kullback-Leibler divergence, and θ denotes the trainable parameters in P M. Therefore, MLE training is minizing the divergence from P M, which is exactly the model's sampling distribution, from P D. While it's true that the training is "exposed" to data samples, we can not simply deduce the objective is "biased". We want to end our discussion with two remarks. First, the proposed quantification approaches should not be used as the only metric for NLG. For example, a position-aware uni-gram LM, which generates words independent of previous context, has no exposure bias problem and can pass our test easily. Second, the intention of this work is not to discourage researchers from exploring non-MLE training algorithms for LM. It is completely possible that an training objective different from, can lead to better generation performance . However, though non-MLE algorithms avoid teacher forcing, these algorithms (using GAN or RL for example) are usually less stable and more difficult to tune. Given that the quantified measurement of exposure bias is insignificant, we think it should be questioned whether adopting these techniques to avoid exposure bias is a wise trade-off. In Table 6, we provide more samples of a MLE-trained transformer LM model (discussed in Section 2) when fed with different kinds of history. And in Table 7 we repeat the experiment for a LSTM-LM trained on the EMNLP-News data. In Table 4 we repeat the preliminary experiment in Section 2 for a standard NMT setting. We train a 6-layer transformer model with hidden dimension 1024 on the IWSLT'14 German to English data set. We feed the trained model with types of prefix during decoding which represents different level of training-decoding discrepancy. Note that the source input is kept intact. The is very similar (or more striking) to our language model experiment, the data prefix does not seem to help, and in the extreme case of random prefix, the model still generates fairly good translation. In Section 2 we summarize this observation as the auto-recovery ability. To interpret the UNREL3 , we should not directly compare the translation generated from unrelated prefix to the target translation. In fact, we cannot even compare part of it (e.g. the part after the length-3 prefix). Instead, we highlight the surprising fact that although the model is forced to begin (conditioned) with a wrong prefix, it still comes up with a reasonable translation. This is not an Table 4: A standard NMT transformer model fed with different types of length-3 history prefix. We did not do any cherry picking. The "@@" is because BPE tokenization is used. "DATA" means the first three output tokens are forced to be correct. "NORMAL" means no prefix is forced during decoding. "UNREL" means the first three tokens are forced to be from another random unrelated sentence (which is wrong but grammatical). "RAND" means the first three tokens are completely random words. easy task even for human translators, yet the model does fairly well. Again, this contradicts with the "exposure bias" hypothesis that a MLE-trained LM will produce a increasingly deviated sequence when initiated with a non-perfect prefix. Actually, during generation the model self-corrects the error in the prefix. It is also the major motivation of our proposed EB-C measurement (Section 5), which is based on the view of measuring distances between conditional distributions. One problem in the implementation of EB-M is to estimate the described marginal distributions of W l+1. We adopt a simple sample-and-count method: P l+1 D|D is estimated by the distribution (histogram) of W l+1 from a number (to be specified below) of sentences sampled from the data distribution. For P l+1 M |M and P l+1 M |D, we first draw a number of history samples W 1:l from the corresponding history model (model distribution and data distribution respectively). We then feed sampled history sequences into the trained model and estimate the marginal distribution of the (l + 1) th word by averaging the predicted distribution P M (·|W 1:l). We measure EB-M for MLE-trained LSTM LM on two popular data-sets: EMNLP-news (EMNLP 2017 WMT News Section), and wikitext-103 7. For EMNLP-news we set L = 20, and only use data samples whose length is longer than L. The ing training/validation/test set has 268k/10k/10k sentences. The vocabulary is of size 5k. We use the 10k samples in the test set for evaluation of EB-M. Note that the EMNLP-news data-set is widely used in text GAN literatures;. We train a one-layer LSTM LM of hidden dimension 512 as the MLE baseline model for EMNLP-news. For wikitext-103, we set L = 50, and regard a paragraph in the original data as a long sentence. Further, we use half of the data for LM training, and utilize the other half for EB-M evaluation. The ing training/validation/test/evaluation set has 300k/1.5k/1.5k/300k sentences. The vocabulary is of size 50k. We train a two-layer LSTM LM of hidden dimension 1024 as the MLE baseline model for wikitext-103. For MLE baseline model training, the Adam optimizer is used with learning rate 0.001, no Dropout is applied. The model is trained for 100 epochs. We first measure EB-M on the wikitext-103 data-set, which has large amount of evaluation data. The are shown in Figure 5. We provide EB-M measurements with metric d T V in Appendix E, as they are similar to those using metric d JS. It is shown that the measurements become stable when using 100k data/model samples. EB-M has an average value of 1.10, indicating a significant gap between the model's MGD when fed with history from P D or P M. Further, we observe a steady growth of EB-M along the length of history, which is expected as an outcome of exposure bias. However, as discussed in Section 4.2, these measurements can only show that the LM does have better (marginal) generation quality when fed with data prefixes, but does not provide informative information for the significance of exposure bias. We implement our MLE baseline and scheduled sampling (SS) in PyTorch. For SS, we use a linear decay schedule to move from complete teacher forcing to replace-sample rate of 0.1. We find that larger rate will give worse performance. For CoT, we use a PyTorch implementation in https://github.com/pclucas14/ GansFallingShort. We use a mediator model that has twice the size of the generator. We set M-step to be 4, and G-step to be 1. For RankGAN, we use a TensorFlow implementation in https://github.com/ desire2020/RankGAN. Note that in our non-MLE experiments, the generator model is set to be the same size with the baseline MLE model. We tune the non-MLE methods using the corpus-BLEU metric, which is widely used in text GAN literature. In this section we provide more details for the AMT evaluation discussed in Section 5.3. We show the HIT interface in Figure 6. Each HIT will include 10 pairs of context and its corresponding choices. Five of them are history samples from real data, and the other five is from the trained model. The history samples are mixed, so that the turker doesn't know whether the history sample is from real data or the model. The next-word choices are also shuffled. The history length of the context could be 10, 20, or 30. We collect around 10k HITs for each history length configuration. The same history sample is not repeated across the HITs. We limit each turker to do at most 200 HITs. For all history length configurations, there are around 300 unique turkers. As shown by Figure 7, most turkers conduct less than 20 HITs. In Figure 8, we show that we are able to get stable measurements of EB-C with 100k samples for the LSTM-512 synthetic experiment. In Figure 9 and Figure 10 we provide EB-M measurements with metric d T V discussed in Section 4.2, the are similar to those using metric d JS. In Figure 11, we provide EB-C measurements of a 3-layer transformer LM with 512 hidden dimension, in the synthetic setting. We show PPL for model trained on EMNLP-news data-set in Table 5. The MLE model for wiki-103 data-set discussed in Section 4.2 has PPL 84.58. Note that due to our special setting 8, our PPL is not directly comparable to state-of-art LM on these data-sets. 8 We only keep sentences of length longer than L, and for wiki-103, only half of training data is used. At the same time, she responded against the package of short-form compatible boats... Table 6: More samples of a STOA MLE-trained transformer LM (on the wiki-103 data-set) when fed with different kinds of history. To save space, we omitted the first 7 words of the random history. Model Samples as Hisotry → Model Samples it was only a pieces that had gone up to → the forest and forces the shoppers about their chronic young i mean we didn' t know what i haven →' t considered through, " she told bbc radio if he were the president -elect, he was → known that he would run a force in business at but these are not as tired of " the same → message that the harry actor does have been hours in first opinion the agent have taken four seconds, or → if they don' t only know anything, were " the economy of the uk is low enough of → people of defending where americans think that " brexit, the economy grew on 1. 6 % since the → us voted, and when it turned around 200 streets i was able to produce on my own, which → is good; now that the theatre i' ve " i' ve not buying boys i addressed many → nervous times before, as a teenager made me is we think about one -third of the struggles we → actually want to see those very well that even more the story of a album -which made public -→ was still fantastic, and for the second time in " the test comes up before tuesday and when we →' re feeling ahead again soon, " she posted a year on when he was last seen in his → home and he did not see him, his suffering brady has forced the 9 -known targets to get → all -of -12 gun migration and performing communication i asked if he himself did, i managed to → show all my charges at all, it used to Data Samples as Hisotry → Model Samples what this group does is to take down various different → players in the future and we play in paris we over 1, 600 a day have reached greece this → gone in 2013 and it planned to allow civilians on " we' re working through a legacy period, → and i am proud of the experience of the worker' the first time anyone says you need help, → you don' t have put accurate press into the out of those who came last year, 69 per → cent of women can really take the drive to avoid he has not played for tottenham' s first team → this season then and sits down 15 -0 with so you have this man who seems to represent this → bad story, which he plays minutes -because he cnn: you made that promise, but it wasn →' t necessarily at all the features he had in this is a part of the population that is unk → lucky to have no fault today, and it would they picked him off three times and kept him out → of the game and was in the field, the the treatment was going to cost $ 12, 000 → as a of the request of anyone who was but if black political power is so important, why → doesn' t we becomes the case that either stands local media reported the group were not looking to hurt → the animals, but would never be seen to say Table 7: Samples of a MLE-trained LSTM LM (on the EMNLP-news data-set) when fed with different kinds of history. To save space, we omitted the first 7 words of the random history.

represent the tldr for finding the corresponding scientific article

We show that exposure bias could be much less serious than it is currently assumed to be for MLE LM training.

We introduce and analyze the phenomenon of "hallucinations" in NMT, or spurious translations unrelated to source text, and propose methods to reduce its frequency.