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Question: We study SGD and Adam for estimating a rank one signal planted in matrix or tensor noise. The extreme simplicity of the problem setup allows us to isolate the effects of various factors: signal to noise ratio, density of critical points, stochasticity and initialization. We observe a surprising phenomenon: Adam seems to get stuck in local minima as soon as polynomially many critical points appear (matrix case), while SGD escapes those. However, when the number of critical points degenerates to exponentials (tensor case), then both algorithms get trapped. Theory tells us that at fixed SNR the problem becomes intractable for large $d$ and in our experiments SGD does not escape this. We exhibit the benefits of warm starting in those situations. We conclude that in this class of problems, warm starting cannot be replaced by stochasticity in gradients to find the basin of attraction. Reductionism consists of breaking down the study of complex systems and phenomena into their atomic components. While the use of stochastic gradient based algorithms has shown tremendous success at minimizing complicated loss functions arising in deep learning, our understanding of why, when and how this happens is still limited. Statements such as stochastic gradients escape from isolated critical points along the road to the best basin of attraction, or SGD generalizes better because it does not get stuck in steep local minima still need to be better understood. Can we prove or replicate these phenomena in the simplest instances of the problem? We study the behavior of stochastic gradient descent (SGD) BID11 and an adaptive variant (Adam) BID8 under a class of well studied non-convex problems. The single spiked models were originally designed for studying principal component analysis on matrices BID12 BID3 BID5 and have also been extended to higher order tensors BID10. Adaptive stochastic optimization methods have been gaining popularity in the deep learning community thanks to fast training on some benchmarks. However, it has been observed that despite reaching a low value of the loss function, the solutions found by Adam do not generalize as well as SGD solutions do. An assumption, widely spread and adopted in the community, has been that SGD's randomness helps escaping local critical points [WRS + 17]. While the problem has been thoroughly studied theoretically [MR14, HSS15, HSSS16, BAGJ18], our contribution is to propose experimenting with this simple model to challenge claims such as those on randomized gradient algorithms in this very simple setup. It is noteworthy that the landscape of non-global critical points of these toy datasets are studied BID0 BID2 BID1 and formally linked to the neural nets empirical loss functions BID2 BID9. For this problem, the statistical properties of the optimizers are well understood, and in the more challenging tensor situation, also the impact of (spectral) warm start has been discussed BID10. We will examine the solutions found by SGD and Adam and compare them with spectral and power methods. This allows to empirically elucidate the existence of multiple regimes: the strong signal regime where all first order methods seem to find good solutions when polynomially many critical points appear, in the matrix case, SGD converges while Adam gets trapped, unless if initialized in the basin of attraction in the presence of exponentially many critical points (the tensor case), all algorithms fail, unless if d is moderately small and the SNR large enough to allow for proper initialization.2 Single spiked models, and stochastic gradients Even though proving strong about non-convex loss functions is in general challenging, a class of nonconvex statistical problems is very well studied and relatively well understood. Principal component analysis (PCA) or finding the leading eigenvector of a covariance matrix is a problem of interest in statistics and machine learning. The proof of convergence of power method to the leading principal component and the geometry of critical points of the maximum likelihood problem maximize u, Au s.t. u 2 = 1, Rayleigh quotient for matrix PCA are well established using eigenvalue decomposition. In addition, more recently, a class of extremely simplified models have shed light on the phase transitions of the problem difficulty as a function of the signal-to-noise ratio in the model. The so-called single spiked models consist of considering a symmetric normalized noise matrix to which a rank one signal is added. DISPLAYFORM0 It is known BID3 that the spectrum of the noise matrix asymptotically forms a semi-circle situated between −2 and 2. When the signal to noise ratio is weak λ ∈ then the signal dilutes in noise, while the leading principal component pops out of the semi-circle as soon as the signal to noise ratio λ is above the critical value λ > λ c = 1, in which case the solution of the problem forms asymptotically a cosine value of 1 − λ −2 1/2 with the signal and the optimal value of the Rayleigh quotient is λ + λ −1 BID12 BID3 BID5. It is proven that the power method allows to obtain the solution after logarithmically many steps, as a function of the problem dimension d. We will minimize the unconstrained objective function (DISPLAYFORM1 We set the value of γ to the theoretical asymptotic value of the leading eigenvalue, γ = 2 for λ < 1 or λ + λ −1 for larger λ, and will add random normal noise to the gradient for stochasticity: ∇ σ (u) = −Au + γu + σz where z i ∼ N (0, 1/d). This function has a constant Hessian H (u) = −λu 0 u 0 T − Z + γI d which is positive semi-definite as soon as γ is equal or larger than the value of the leading eigenvalue of A. The tensor version of the problem (see BID10 for notations and more discussion on problem setting) DISPLAYFORM0 u 2 = 1 Rayleigh quotient tensor PCA under the tensor single spiked model defined for a symmetric (π is a permutation of 3 elements) How does data abundance explain the success of first order methods? On the positive side, we discuss that in this model and considering a large dataset of i.id. samples, weak signals in individual observations accumulate and allow to solve the problem if n √ d. The counter part to the strong requirement λ d 1/4 (conjectured in BID10 and proven in BID6 BID7) is that accumulation of observations compensate low signal to noise ratio in each individual sample. Formally, Remark 2.1. Assume n sample of data according to model "Single spiked tensor", with the same signal u 0 and different i.i.d. noises Z q are observed: DISPLAYFORM1 DISPLAYFORM2 There exists constants c 0, c 1 such that if n ≥ c 0 √ d, then, warm started power iteration produces a vector u, such that with high probability u 0, u > 1 − c 1 /λ. This is established using Theorems 5 in BID10 and 6.3 in BID7 and considering the average tensorĀ DISPLAYFORM3 SinceZ is symmetric andZ i,j,k ∼ N (0, 1/d), the tensorĀ is sampled from a similar distribution as Single spiked tensor with a SNR λ n = √ nλ. This means that the requirement λ d 1/4 BID7, in the average tensor case, relaxes to n √ d. In words, this means that if we are solving a problem with a tensor PCA complexity, and if the number of i.i.d. observations grows quadratically as a function of the problem dimension, we can compute the solution reliably using spectral warm start, even though the original problem looks intractable. Our numerical report performance of different algorithms at solving simulations of matrix and tensor PCA problems. Under various problem generation parameter choices, we report values of• cosine or u, u 0. This is measures the quality of planted (hidden) signal recovery from the noisy observation. Higher values are preferred, and it cannot exceed 1, since both u 0 and u are normalized. This quantity is to be qualitatively compared with the test error in standard learning problems where the true value of the parameter is not available.• Rayleigh is the value of the log-likelihood objective function that we are maximizing. Higher values are preferred. The theoretical maximum value of this objective is the operator norm of the observed tensor A. This is comparable with (minus) the training loss. The signal to noise ratio λ is to be compared with the number of observations in a supervised learning problem. The stochasticity of the gradients σ is to be compared with the number of sample points in each minibatch of data: large stochasticity mimics small minibatch situations. In Figure 1 we plot the values of the objective function or Rayleigh quotient and the cosine of the ground truth with the solution as a function of the iterates. We replicated these plots at values of the SNR parameter λ < λ c = 1, at the critical value λ = 1 and above it for λ = 2 where the problems is considered to be easy. The learning rate and stochasticity σ were set by generating instances of the problem with different noise matrices. These plots allow to compare the optimization power of different algorithms and also keep track of the quality of the solution found. We can see in these plots that Adam gets stuck around the wrong region very fast. Note that with the value (or larger) of γ = 2 for λ < 1 and γ = λ + λ −1 for λ ≥ 1 that we can set given the true value of λ, and knowing the concentration around the asymptotics BID12 BID3 BID5, the objective function is strongly convex so we expect first order methods to show the same convergence rates as the power method. One can also observe that gradient descent corresponds to performing power iteration on a shifted matrix A + αI d. Figure 2 shows the value of the objective and the correlation with the ground truth as a function of SNR λ. We can see that SGD is superior to Adam, uniformly along λ, while power method (also a first order method) rivals with SGD. These plots exhibit the instability around and below the critical value λ c = 1 while above λ c the behavior is more stable. In the tensor setting we experimented with spectral initialization. Spectral initialization consists of flattening the tensor to a d×d 2 matrix and initializing tensor algorithms with the left singular vector of the flattened tensor. We observe benefits of spectral initialization at locating the initial point in a basin of attraction that leads to better solutions when λ is large enough. For λ = 1.0 spectral initialization does not in better estimates in Figure 3, while for larger values of λ we can see the benefit of warm start. In FIG2 we also plot values of the gradient and the number of positive eigenvalues of the Hessian along the optimization iterates for d = 100, λ = 2, γ = 2. We observe that spectral initialization located the initial point of the iterations in the basin of attraction where the problem is convex (all eigenvalues of the Hessian are positive). Adam, while starting in this region, fails at finding a solution as good as SGD's. We experimented with the amount of noise added to each gradient evaluation and mapped median values of the estimate and optimization problems for values of λ and σ over 100 instances of the problem generated with different noise matrices and stochastic gradients. Stochasticity does not seem to remedy to the problem difficulty. Numerical experiments suggest that irrespective of the magnitude of the stochastic component added to the gradient, the first order methods, initialized at random, fail at finding the best basin of attraction. In the same setup, spectral initialized first order methods successfully find the solutions. We propose to study algorithms used for minimizing deep learning loss functions, at optimizing a non-convex objective on simple synthetic datasets. Studying simplified problems has the advantage that the problem's properties, and the behavior of the optimizer and the solution, can be studied rigorously. The use of such datasets can help to perform sanity checks on improvement ideas to the algorithms, or to mathematically prove or disprove intuitions. The properties of the toy data sets align with some properties of deep learning loss functions. From the optimization standpoint, the ing tensor problems may appear to be even harder than deep learning problems. We observe that finding good solutions is hard unless if proper initialization is performed, while the value of stochasticity in gradient estimates seems too narrow and does not appear to compensate for poor initialization heuristics. Each column represents the values of those quantities along iterations of the algorithm. The prefix sp. refers to spectral initialization and l. refers to a decreasing learning weight scheduled in 1/ √ t. We observe the value of warm starting as soon as λ is large enough. Even at high SNR λ = 6, randomly initialized SGD fails while spectrally initialized SGD succeeds. Adam drifts to a non optimal critical point in that regime, even with spectral warm start.

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SGD and Adam under single spiked model for tensor PCA

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Provided: "There's non-consensual and pornographic images in the ImageNet dataset" Match: "The main goal of this short paper is to inform the neural art community at large on the ethical ramifications of using models trained on the imagenet dataset, or using seed images from classes 445 -n02892767- [’bikini, two-piece’] and 459- n02837789- [’brassiere, bra, bandeau’] of the same. We discovered that many of the images belong to these classes were verifiably pornographic, shot in a non-consensual setting, voyeuristic and also entailed underage nudity. Akin to the \textit{ivory carving-illegal poaching} and \textit{diamond jewelry art-blood diamond} nexuses, we posit there is a similar moral conundrum at play here and would like to instigate a conversation amongst the neural artists in the community. The emergence of tools such as BigGAN and GAN-breeder has ushered in an exciting new flavor of generative digital art, generated using deep neural networks (See for a survey). A cursory search on twitter 1 reveals hundreds of interesting art-works created using BigGANs. There are many detailed blog-posts 2 on generating neural art by beginning with seed images and performing nifty experiments in the latent space of BigGANs. At the point of authoring this paper, (8 September 2019, 4 :54 PM PST),users on the GanBreeder app 3 had discovered 49652500 images. Further, Christie's, the British auction house behemoth, recently hailed the selling of the neural network generated Portrait of Edmond Belamy for an incredible $432, 500 as signalling the arrival of AI art on the world auction stage. Given the rapid growth of this field, we believe this to be the right time to have a conversation about a particularly dark ethical consequence of using such frameworks that entail models trained on the ImageNet dataset which has many images that are pornographic, non-consensual, voyeuristic and also entail underage nudity. We argue that this lack of consent in the seed images used to train the models trickles down to the final artform in a way similar to the blood-diamond syndrome in jewelry art. An example: Consider the neural art image in Fig 1 we generated using the GanBreeder app. On first appearance, it is not very evident as to what the constituent seed classes are that went into the creation of this neural artwork image. When we solicited volunteers online to critique the artwork (See the collection of responses (Table 2) in the supplementary material), none had an inkling regarding a rather sinister trickle down effect at play here. As it turns out, we craftily generated this image using hand-picked specific instances of children images emanating from what we will showcase are two problematic seed image classes: Bikini and Brassiere. More specifically, for this particular image, we set the Gene weights to be: [Bikini: 42.35, Brassiere: 31.66, Comic Book -84.84]. We'd like to strongly emphasize at this juncture that the problem does not emanate from a visual patriarchal mindset, whereby we associate female undergarment imagery to be somehow unethical, but the root cause lies in the fact that many of the images were curated into the dataset (at least with regards to the 2 above mentioned classes) were voyeuristic, pornographic, non-consensual and also entailed underage nudity. 2 Root cause: Absence of referencing consent during the curation of the imagenet dataset The emergence of the ImageNet dataset is widely considered to be a pivotal moment 4 in the deep learning revolution that transformed the domain computer vision. Two highly cited papers (with more than 10000 citations each), authored by Deng et al in 2009 and authored by Russakovsky et al in 2015, provide deep insights into the procedure used to curate the dataset. In the 2009 paper, subsections 3.1-Collecting Candidate Images and 3.2-Cleaning Candidate Images are dedicated towards the algorithms used to collect and clean the dataset and also to elucidate the specific ways in which the Amazon Mechanical Turk (AMT) platform was harnessed to scale the dataset. Similarly the entirety of Section-3-Dataset construction at large scale in is dedicated towards extending the procedures for the 2015 release. It is indeed disappointing that neither the 2009 nor the 2015 versions of the endeavors required the AMT workers to check if the images they were asked to categorize and draw bounding boxes over, were ethically viable for usage. More specifically, in imagery pertaining to anthropocentric content, such as undergarment clothing, there was no attempt made towards assessing if the images entailed explicit consent given by the people in the images. In fact, none of the following words in the set [ethics, permission, voyeurism, consent] are mentioned in either of the two papers. As such, we have a plethora of images specifically belonging to the two categories detailed in Table 1, that have serious ethical shortcomings. In Fig 2, we showcase the gallery of images from the two classes categorized into four sub-categories: Non-consensual/Voyeuristic, Personal, Verifiably pornographic and Underage / Children. In Fig 2, we also include images that were also incorrectly categorized (Specifically there was no brassieres being sported by the subjects in the images) (Sub-figure (a) ) and those that involved male subjects indulging in lecherous tomfoolery (Sub-figure (e) ). In this paper, we expose a certain unethical facet of neural art that emerges from usage of nonconsensual images that are present in certain specific classes of the imagenet dataset. These images born out of an unethical (and in some cases, illegal) act of voyeuristic non-consensual photography predominantly targeting women (as well as children), might implicitly poison the sanctity of the artworks that eventually emerge. This work is complementary to works such as [10; 11] that have explored the unspoken ethical dimensions of harnessing cheap crowd-sourced platforms such as AMT for scientific research in the first place, which we firmly believe is also an important issue to be considered. We'd like to begin by admitting the self-contradictory facet of raising this specific issue by using a flagship example image that was generated using the very same ethically dubious procedure that we are targeting to root out. Secondly, we'd like to inform the reader that we have indeed raised the issue with curators of the imagenet dataset, but to no avail and plan to update this dissemination if and when they respond. This one reminds me of a mix between graffiti and paper mache using newspaper with color images or magazines. My attention is immediately drawn to near the top of the image which, at first glance, appears to be a red halo of sorts, but upon further consideration, looks to be long black branching horns on a glowing red . My attention then went to the center top portion, where the "horns" were coming from, which appeared to be the head or skull of a moose or something similar. The body of the creature appears to be of human-like form in a crucifix position, of sorts. The image appears more and more chaotic the further down one looks. Antisymmetric: left side is very artistic, rich in flavor and shades; right is more monotonic but has more texture. Reminds me of the two different sides of the brain through the anti-symmetry C-Data Scientist, Facebook Inc Futurism It's visually confusing in the sense that I couldn't tell if I was looking at a 3D object with a colorful or a painting. It's not just abstract, but also mysteriously detailed in areas to the point that I doubt that a human created these E -Senior software engineer, Mt View The symmetry implies a sort of intentionally. I get a sense of Picasso mixed with Frieda Callo here. F-Data Scientist, SF Reminds me of a bee and very colorful flowers, but with some nightmarish masks hidden in some places. Very tropical Table 2: Responses received for the neural art image in Fig 1"

In this paper, a new method we call Centered Initial Attack (CIA) is provided. It insures by construction the maximum perturbation to be smaller than a threshold fixed beforehand, without the clipping process.


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Question: This paper presents a method to explain the knowledge encoded in a convolutional neural network (CNN) quantitatively and semantically. How to analyze the specific rationale of each prediction made by the CNN presents one of key issues of understanding neural networks, but it is also of significant practical values in certain applications. In this study, we propose to distill knowledge from the CNN into an explainable additive model, so that we can use the explainable model to provide a quantitative explanation for the CNN prediction. We analyze the typical bias-interpreting problem of the explainable model and develop prior losses to guide the learning of the explainable additive model. Experimental have demonstrated the effectiveness of our method. Convolutional neural networks (CNNs) BID17 BID15 BID10 have achieved superior performance in various tasks, such as object classification and detection. Besides the discrimination power of neural networks, the interpretability of neural networks has received an increasing attention in recent years. In this paper, we focus on a new problem, i.e. explaining the specific rationale of each network prediction semantically and quantitatively. " Semantic explanations" and "quantitative explanations" are two core issues of understanding neural networks. We hope to explain the logic of each network prediction using clear visual concepts, instead of using middle-layer features without clear meanings or simply extracting pixel-level correlations between network inputs and outputs. We believe that semantic explanations may satisfy specific demands in real applications. In contrast to traditional qualitative explanations for neural networks, quantitative explanations enable people to diagnose feature representations inside neural networks and help neural networks earn trust from people. We expect the neural network to provide the quantitative rationale of the prediction, i.e. clarifying which visual concepts activate the neural network and how much they contribute to the prediction score. Above two requirements present significant challenges to state-of-the-art algorithms. To the best of our knowledge, no previous studies simultaneously explained network predictions using clear visual concepts and quantitatively decomposed the prediction score into value components of these visual concepts. Task: Therefore, in this study, we propose to learn another neural network, namely an explainer network, to explain CNN predictions. Accordingly, we can call the target CNN a performer network. Besides the performer, we also require a set of models that are pre-trained to detect different visual concepts. These visual concepts will be used to explain the logic of the performer's prediction. We are also given input images of the performer, but we do not need any additional annotations on the images. Then, the explainer is learned to mimic the logic inside the performer, i.e. the explainer receives the same features as the performer and is expected to generate similar prediction scores. As shown in Fig. 1, the explainer uses pre-trained visual concepts to explain each prediction. The explainer is designed as an additive model, which decomposes the prediction score into the sum of Figure 1: Explainer. We distill knowledge of a performer into an explainer as a paraphrase of the performer's representations. The explainer decomposes the prediction score into value components of semantic concepts, thereby obtaining quantitative semantic explanations for the performer.multiple value components. Each value component is computed based on a specific visual concept. In this way, we can roughly consider these value components as quantitative contributions of the visual concepts to the final prediction score. More specifically, we learn the explainer via knowledge distillation. Note that we do not use any ground-truth annotations on input images to supervise the explainer. It is because the task of the explainer is not to achieve a high prediction accuracy, but to mimic the performer's logic in prediction, no matter whether the performer's prediction is correct or not. Thus, the explainer can be regarded as a semantic paraphrase of feature representations inside the performer, and we can use the explainer to understand the logic of the performer's prediction. Theoretically, the explainer usually cannot recover the exact prediction score of the performer, owing to the limit of the representation capacity of visual concepts. The difference of the prediction score between the performer and the explainer corresponds to the information that cannot be explained by the visual concepts. Challenges: Distilling knowledge from a pre-trained neural network into an additive model usually suffers from the problem of bias-interpreting. When we use a large number of visual concepts to explain the logic inside the performer, the explainer may biasedly select very few visual concepts, instead of all visual concepts, as the rationale of the prediction (Fig. 4 in the appendix visualizes the bias-interpreting problem). Just like the typical over-fitting problem, theoretically, the bias interpreting is an ill-defined problem. To overcome this problem, we propose two types of losses for prior weights of visual concepts to guide the learning process. The prior weights push the explainer to compute a similar Jacobian of the prediction score w.r.t. visual concepts as the performer in early epochs, in order to avoid bias-interpreting. Originality: Our "semantic-level" explanation for CNN predictions has essential differences from traditional studies of "pixel-level" interpreting neural networks, such as the visualization of features in neural networks BID36 BID21 BID27 BID6 BID7 BID24, the extraction of pixellevel correlations between network inputs and outputs BID14 BID22 BID20, and the learning of neural networks with interpretable middle-layer features BID38 BID23.In particular, the explainer explains the performer without affecting the original discrimination power of the performer. As discussed in BID1, the interpretability of features is not equivalent to, and usually even conflicts with the discrimination power of features. Compared to forcing the performer to learn interpretable features, our strategy of explaining the performer solves the dilemma between the interpretability and the discriminability. In addition, our quantitative explanation has special values beyond the qualitative analysis of CNN predictions BID39.Potential values of the explainer: Quantitatively and semantically explaining a performer is of considerable practical values when the performer needs to earn trust from people in critical applications. As mentioned in BID37, owing to the potential bias in datasets and feature representations, a high testing accuracy still cannot fully ensure correct feature representations in neural networks. Thus, semantically and quantitatively clarifying the logic of each network prediction is a direct way to diagnose feature representations of neural networks. Fig. 3 shows example explanations for the performer's predictions. Predictions whose explanations conflict people's common sense may reflect problematic feature representations inside the performer. Contributions of this study are summarized as follows. (i) In this study, we focus on a new task, i.e. semantically and quantitatively explaining CNN predictions. (ii) We propose a new method to explain neural networks, i.e. distilling knowledge from a pre-trained performer into an interpretable additive explainer. Our strategy of using the explainer to explain the performer avoids hurting the discrimination power of the performer. (iii) We develop novel losses to overcome the typical biasinterpreting problem. Preliminary experimental have demonstrated the effectiveness of the proposed method. (iv) Theoretically, the proposed method is a generic solution to the problem of interpreting neural networks. We have applied our method to different benchmark CNNs for different applications, which has proved the broad applicability of our method. In this paper, we limit our discussion within the scope of understanding feature representations of neural networks. Network visualization: The visualization of feature representations inside a neural network is the most direct way of opening the black-box of the neural network. Related techniques include gradient-based visualization BID36 BID21 BID27 BID35 and up-convolutional nets BID6 BID20 BID13 BID7 BID24 extracted rough pixel-level correlations between network inputs and outputs, i.e. estimating image regions that directly contribute the network output. Network-attack methods BID14 BID29 computed adversarial samples to diagnose a CNN. BID16 ) discovered knowledge blind spots of a CNN in a weakly-supervised manner. BID37 ) examined representations of conv-layers and automatically discover biased representations of a CNN due to the dataset bias. However, above methods usually analyzed a neural network at the pixel level and did not summarize the network knowledge into clear visual concepts. BID1 defined six types of semantics for CNN filters, i.e. objects, parts, scenes, textures, materials, and colors. Then, BID41 proposed a method to compute the image-resolution receptive field of neural activations in a feature map. Other studies retrieved middle-layer features from CNNs representing clear concepts. BID25 retrieved features to describe objects from feature maps, respectively. BID41 selected neural units to describe scenes. Note that strictly speaking, each CNN filter usually represents a mixture of multiple semantic concepts. Unlike previous studies, we are more interested in analyzing the quantitative contribution of each semantic concept to each prediction, which was not discussed in previous studies. A new trend in the scope of network interpretability is to learn interpretable feature representations in neural networks BID12 BID28 BID18 in an un-/weakly-supervised manner. Capsule nets BID23 and interpretable RCNN BID33 learned interpretable features in intermediate layers. InfoGAN BID4 and β-VAE BID11 ) learned well-disentangled codes for generative networks. Interpretable CNNs BID38 learned filters in intermediate layers to represent object parts without given part annotations. However, as mentioned in BID1 BID39, interpretable features usually do not have a high discrimination power. Therefore, we use the explainer to interpret the pre-trained performer without hurting the discriminability of the performer. Explaining neural networks via knowledge distillation: Distilling knowledge from a black-box model into an explainable model is an emerging direction in recent years. BID40 used a tree structure to summarize the inaccurate 1 rationale of each CNN prediction into generic decision-making models for a number of samples. In contrast, we pursue the explicitly quantitative explanation for each CNN prediction. BID5 learned an explainable additive model, and BID31 ) distilled knowledge of a network into an additive model. BID30 BID2 BID32 ) distilled representations of neural networks into tree structures. These methods did not explain the network knowledge using humaninterpretable semantic concepts. More crucially, compared to previous additive models BID31, our research successfully overcomes the bias-interpreting problem, which is the core challenge when there are lots of visual concepts for explanation. In this section, we distill knowledge from a pre-trained performer f to an explainable additive model. We are given a performer f and n neural networks {f i |i = 1, 2, . . ., n} that are pre-trained to detect n different visual concepts. We learn the n neural networks along with the performer, and the n neural networks are expected to share low-layer features with the performer. Our method also requires a set of training samples for the performer f. The goal of the explainer is to use inference values of the n visual concepts to explain prediction scores of the performer. Note that we do not need any annotations on training samples w.r.t. the task, because additional supervision will push the explainer towards a good performance of the task, instead of objectively reflecting the knowledge in the performer. Given an input image I, letŷ = f (I) denote the output of the performer. Without loss of generality, we assume thatŷ is a scalar. If the performer has multiple outputs (e.g. a neural network for multicategory classification), we can learn an explainer to interpret each scalar output of the performer. In particular, when the performer takes a softmax layer as the last layer, we use the feature score before the softmax layer asŷ, so thatŷ's neighboring scores will not affect the value ofŷ. We design the following additive explainer model, which uses a mixture of visual concepts to approximate the function of the performer. The explainer decomposes the prediction scoreŷ into value components of pre-defined visual concepts. DISPLAYFORM0 Quantitative contribution from the first visual concept DISPLAYFORM1 where y i and α i (I) denote the scalar value and the weight for the i-th visual concept, respectively. b is a bias term. y i is given as the strength or confidence of the detection of the i-th visual concept. We can regard the value of α i (I) · y i as the quantitative contribution of the i-th visual concept to the final prediction. In most cases, the explainer cannot recover all information of the performer. The prediction difference between the explainer and the performer reflects the limit of the representation capacity of visual concepts. According to the above equation, the core task of the explainer is to estimate a set of weights α = [α 1, α 2, . . ., α n], which minimizes the difference of the prediction score between the performer and the explainer. Different input images may obtain different weights α, which correspond to different decision-making modes of the performer. For example, a performer may mainly use head patterns to classify a standing bird, while it may increase the weight for the wing concept to classify a flying bird. Therefore, we design another neural network g with parameters θ g (i.e. the explainer), which uses the input image I to estimate the n weights. We learn the explainer with the following knowledge-distillation loss. DISPLAYFORM2 However, without any prior knowledge about the distribution of the weight α i, the learning of g usually suffers from the problem of bias-interpreting. The neural network g may biasedly select very few visual concepts to approximate the performer as a shortcut solution, instead of sophisticatedly learning relationships between the performer output and all visual concepts. Thus, to overcome the bias-interpreting problem, we use a loss L for priors of α to guide the learning process in early epochs. DISPLAYFORM3 Loss, where w denotes prior weights, which represent a rough relationship between the performer's prediction value and n visual concepts. Just like α, different input images also have different prior weights w. The loss L(α, w) penalizes the dissimilarity between α and w. DISPLAYFORM4 Note that the prior weights w are approximated with strong assumptions (we will introduce two different ways of computing w later). We use inaccurate w to avoid significant bias-interpreting, rather than pursue a high accuracy. Thus, we set a decreasing weight for L, i.e. λ(t) = DISPLAYFORM5, where β is a scalar constant, and t denotes the epoch number. In this way, we mainly apply the prior loss L in early epochs. Then, in late epochs, the influence of L gradually decreases, and our method gradually shifts its attention to the distillation loss for a high distillation accuracy. We design two types of losses for prior weights, as follows. DISPLAYFORM6 Some applications require a positive relationship between the prediction of the performer and each visual concept, i.e. each weight α i must be a positive scalar. In this case, we use the cross-entropy between α and w as the prior loss. In other cases, the MSE loss between α and w is used as the loss. · 1 and · 2 denote the L-1 norm and L-2 norm, respectively. In particular, in order to ensure α i ≥ 0 in certain applications, we add a non-linear activation layer as the last layer of g, i.e. α = log[1 + exp(x)], where x is the output of the last conv-layer. In this subsection, we will introduce two techniques to efficiently compute rough prior weights w, which are oriented to the following two cases in application. Case 1, filters in intermediate conv-layers of the performer are interpretable: As shown in FIG0, learning a neural network with interpretable filters is an emerging research direction in recent years. For example, BID38 proposed a method to learn CNNs for object classification, where each filter in a high conv-layer is exclusively triggered by the appearance of a specific object part (see FIG4 in the appendix for the visualization of filters). Thus, we can interpret the classification score of an object as a linear combination of elementary scores for the detection of object parts. Because such interpretable filters are automatically learned without part annotations, the quantitative explanation for the CNN (i.e. the performer) can be divided into the following two tasks: (i) annotating the name of the object part that is represented by each filter, and (ii) learning an explainer to disentangle the exact additive contribution of each filter (or each object part) to the performer output. In this way, each f i, i = 1, 2,..., n, is given as an interpretable filter of the performer. According to BID37, we can roughly represent the network prediction aŝ DISPLAYFORM0 where x ∈ R H×W ×n denotes a feature map of the interpretable conv-layer, and x hwi is referred to as the activation unit in the location (h, w) of the i-th channel. y i measures the confidence of detecting the object part corresponding to the i-th filter. Here, we can roughly use the Jacobian of the network output w.r.t. the filter to approximate the weight w i of the filter. Z is for normalization. Considering that the normalization operation in Equation eliminates Z, we can directly use h,w ∂ŷ ∂x hwi as prior weights w in Equation without a need to compute the exact value of Z.Case 2, neural networks for visual concepts share features in intermediate layers with the performer: As shown in FIG0, given a neural network for the detection of multiple visual concepts, using certain visual concepts to explain a new visual concept is a generic way to interpret network predictions with broad applicability. Let us take the detection of a certain visual concept as the targetŷ and use other visual concepts as {y i} to explainŷ. All visual concepts share features in intermediate layers. Then, we estimate a rough numerical relationship betweenŷ and the score of each visual concept y i. Let x be a middle-layer feature shared by both the target and the i-th visual concept. When we modify the feature x, we can represent the value change of y i using a Taylor series, ∆y i = We designed two experiments to use our explainers to interpret different benchmark CNNs oriented to two different applications, in order to demonstrate the broad applicability of our method. In the first experiment, we used the detection of object parts to explain the detection of the entire object. In the second experiment, we used various face attributes to explain the prediction of another face attribute. We evaluated explanations obtained by our method qualitatively and quantitatively. In this experiment, we used the method proposed in BID38 to learn a CNN, where each filter in the top conv-layer represents a specific object part. We followed exact experimental settings in BID38, which used the Pascal-Part dataset BID3 to learn six CNNs for the six animal 2 categories in the dataset. Each CNN was learned to classify the target animal from random images. We considered each CNN as a performer and regarded its interpretable filters in the top conv-layer as visual concepts to interpret the classification score. Following experimental settings in BID38, we applied our method to four types of CNNs, including the AlexNet BID15, the VGG-M, VGG-S, and VGG-16 networks BID26, i.e. we learned CNNs for six categories based on each network structure. Note that as discussed in BID38, skip connections in residual networks BID10 increased the difficulty of learning part features, so they did not learn interpretable filters in residual networks. The AlexNet/VGG-M/VGG-S/VGG-16 performer had 256/512/512/512 filters in its top conv-layer, so we set n = 256, 512, 512, 512 for these networks. We used the masked output of the top conv-layer as x and plugged x to Equation to compute {y i} 1. We used the 152-layer ResNet BID10 3 as g to estimate weights of visual concepts 4. We set β = 10 for the learning of all explainers. Note that all interpretable filters in the performer represented object parts of the target category on positive images, instead of describing random (negative) images. Table 2: Classification accuracy and relative deviations of the explainer and the performer. We used relative deviations and the decrease of the classification accuracy to measure the information that could not be explained by pre-defined visual concepts. Please see the appendix for more . Intuitively, we needed to ensure a positive relationship betweenŷ and y i. Thus, we filtered out negative prior weights w i ← max{w i, 0} and applied the cross-entropy loss in Equation FORMULA6 to learn the explainer. Evaluation metric: The evaluation has two aspects. Firstly, we evaluated the correctness of the estimated explanation for the performer prediction. In fact, there is no ground truth about exact reasons for each prediction. We showed example explanations of for a qualitative evaluation of explanations. We also used grad-CAM visualization BID24 of feature maps to prove the correctness of our explanations (see the appendix). In addition, we normalized the absolute contribution from each visual concept as a distribution of contributions c i = |α i y i |/ j |α j y j |. We used the entropy of contribution distribution H(c) as an indirect evaluation metric for biasinterpreting. A biased explainer usually used very few visual concepts, instead of using most visual concepts, to approximate the performer, which led to a low entropy H(c).Secondly, we also measured the performer information that could not be represented by the visual concepts, which was unavoidable. We proposed two metrics for evaluation. The first metric is the prediction accuracy. We compared the prediction accuracy of the performer with the prediction accuracy of using the explainer's output i α i y i + b. Another metric is the relative deviation, which measures a normalized output difference between the performer and the explainer. The relative deviation of the image I is normalized as |ŷ I − i α I,i y I,i − b|/(max I ∈IŷI − min I ∈IŷI), whereŷ I denotes the performer's output for the image I.Considering the limited representation power of visual concepts, the relative deviation on an image reflected inference patterns, which were not modeled by the explainer. The average relative deviation over all images was reported to evaluate the overall representation power of visual concepts. Note that our objective was not to pursue an extremely low relative deviation, because the limit of the representation power is an objective existence. In this experiment, we learned a CNN based on the VGG-16 structure to estimate face attributes. We used the Large-scale CelebFaces Attributes (CelebA) dataset BID19 to train a CNN to estimate 40 face attributes. We selected a certain attribute as the target and used its prediction score asŷ. Other 39 attributes were taken as visual concepts to explain the score ofŷ (n = 39). The target attribute was selected from those representing global features of the face, i.e. attractive, heavy makeup, male, and young. It is because global features can usually be described by local visual concepts, but the inverse is not. We learned an explainer for each target attribute. We used the same 152-layer ResNet structure as in Experiment 1 (expect for n = 39) as g to estimate weights. We followed the Case-2 implementation in Section 3.1 to compute prior weights w, in which we used the 4096-dimensional output of the first fully-connected layer as the shared feature x. We set β = 0.2 and used the L-2 norm loss in Equation FORMULA6 to learn all explainers. We used the same evaluation metric as in Experiment 1. The quantitative explanation for the prediction of the attractive attribute. Figure 3: Quantitative explanations for the object classification (top) and the face-attribution prediction (bottom) made by performers. For performers oriented to object classification, we annotated the part that was represented by each interpretable filter in the performer, and we assigned contributions of filters α i y i to object parts (see the appendix). Thus, this figure illustrates contributions of different object parts. All object parts made positive contributions to the classification score. Note that in the bottom, bars indicate elementary contributions α i y i from features of different face attributes, rather than prediction values y i of these attributes. For example, the network predicts a negative goatee attribute y goatee < 0, and this information makes a positive contribution to the target attractive attribute, α i y i > 0. Please see the appendix for more . We compared our method with the traditional baseline of only using the distillation loss to learn the explainer. TAB2 evaluates bias-interpreting of explainers that were learned using our method and the baseline. In addition, Table 2 uses the classification accuracy and relative deviations of the explainer to measure the representation capacity of visual concepts. Our method suffered much less from the bias-interpreting problem than the baseline. Fig. 3 shows examples of quantitative explanations for the prediction made by the performer. We also used the grad-CAM visualization BID24 of feature maps of the performer to demonstrate the correctness of our explanations in Fig. 9 in the appendix. In particular, Fig. 4 in the appendix illustrates the distribution of contributions of visual concepts {c i} when we learned the explainer using different methods. Compared to our method, the distillation baseline usually used very few visual concepts for explanation and ignored most strongly activated interpretable filters, which could be considered as bias-interpreting. In this paper, we focus on a new task, i.e. explaining the logic of each CNN prediction semantically and quantitatively, which presents considerable challenges in the scope of understanding neural networks. We propose to distill knowledge from a pre-trained performer into an interpretable additive explainer. We can consider that the performer and the explainer encode similar knowledge. The additive explainer decomposes the prediction score of the performer into value components from semantic visual concepts, in order to compute quantitative contributions of different concepts. The strategy of using an explainer for explanation avoids decreasing the discrimination power of the performer. In preliminary experiments, we have applied our method to different benchmark CNN performers to prove the broad applicability. Note that our objective is not to use pre-trained visual concepts to achieve super accuracy in classification/prediction. Instead, the explainer uses these visual concepts to mimic the logic of the performer and produces similar prediction scores as the performer. In particular, over-interpreting is the biggest challenge of using an additive explainer to interpret another neural network. In this study, we design two losses to overcome the bias-interpreting problems. Besides, in experiments, we also measure the amount of the performer knowledge that could not be represented by visual concepts in the explainer. Table 4: Classification accuracy of the explainer and the performer. We use the the classification accuracy to measure the information loss when using an explainer to interpret the performer. Note that the additional loss for bias-interpreting successfully overcame the bias-interpreting problem, but did not decrease the classification accuracy of the explainer. Another interesting finding of this research is that sometimes, the explainer even outperformed the performer in classification. A similar phenomenon has been reported in BID9. A possible explanation for this phenomenon is given as follows. When the student network in knowledge distillation had sufficient representation power, the student network might learn better representations than the teacher network, because the distillation process removed abnormal middle-layer features corresponding to irregular samples and maintained common features, so as to boost the robustness of the student network. Table 5: Relative deviations of the explainer. The additional loss for bias-interpreting successfully overcame the bias-interpreting problem and just increased a bit (ignorable) relative deviation of the explainer. BID40 ) used a tree structure to summarize the inaccurate rationale of each CNN prediction into generic decision-making models for a number of samples. This method assumed the significance of a feature to be proportional to the Jacobian w.r.t. the feature, which is quite problematic. This assumption is acceptable for BID40, because the objective of BID40 ) is to learn a generic explanation for a group of samples, and the inaccuracy in the explanation for each specific sample does not significantly affect the accuracy of the generic explanation. In comparisons, our method focuses on the quantitative explanation for each specific sample, so we design an additive model to obtain more convincing explanations. Baseline Our method Figure 4: We compared the contribution distribution of different visual concepts (filters) that was estimated by our method and the distribution that was estimated by the baseline. The baseline usually used very few visual concepts to make predictions, which was a typical case of bias-interpreting. In comparisons, our method provided a much more reasonable contribution distribution of visual concepts. Legs & feet Tail Figure 9: Quantitative explanations for object classification. We assigned contributions of filters to their corresponding object parts, so that we obtained contributions of different object parts. According to top figures, we found that different images had similar explanations, i.e. the CNN used similar object parts to classify objects. Therefore, we showed the grad-CAM visualization of feature maps BID24 on the bottom, which proved this finding. We visualized interpretable filters in the top conv-layer of a CNN, which were learned based on BID38. We projected activation regions on the feature map of the filter onto the image plane for visualization. Each filter represented a specific object part through different images. BID38 ) learned a CNN, where each filter in the top conv-layer represented a specific object part. Thus, we annotated the name of the object part that corresponded to each filter based on visualization (see FIG4 for examples). We simply annotate each filter of the top conv-layer in a performer once, so the total annotation cost was O(N), where N is the filter number. Then, we assigned the contribution of a filter to its corresponding part, i.e. Contri part = i:i-th filter represents the part α i y i. We changed the order of the ReLU layer and the mask layer after the top conv-layer, i.e. placing the mask layer between the ReLU layer and the top conv-layer. According to BID38, this operation did not affect the performance of the pre-trained performer. We used the output of the mask layer as x and plugged x to Equation to compute {y i}.Because the distillation process did not use any ground-truth class labels, the explainer's output i α i y i + b was not sophisticatedly learned for classification. Thus, we used a threshold i α i y i + b > τ (τ ≈ 0), instead of 0, as the decision boundary for classification. τ was selected as the one that maximized the accuracy. Such experimental settings made a fairer comparison between the performer and the explainer.

This paper presents a method to explain the knowledge encoded in a convolutional neural network (CNN) quantitatively and semantically.

Represent the TLDR for finding the corresponding scientific article.

We provide insightful understanding of sequence-labeling NER and propose to use two types of cross structures, both of which bring theoretical and empirical improvements.


Question: represent the article for finding a single-sentence tldr!

Question: We present methodology for using dynamic evaluation to improve neural sequence models. Models are adapted to recent history via a gradient descent based mechanism, causing them to assign higher probabilities to re-occurring sequential patterns. Dynamic evaluation outperforms existing adaptation approaches in our comparisons. Dynamic evaluation improves the state-of-the-art word-level perplexities on the Penn Treebank and WikiText-2 datasets to 51.1 and 44.3 respectively, and the state-of-the-art character-level cross-entropies on the text8 and Hutter Prize datasets to 1.19 bits/char and 1.08 bits/char respectively. Sequence generation and prediction tasks span many modes of data, ranging from audio and language modelling, to more general timeseries prediction tasks. Applications of such models include speech recognition, machine translation, dialogue generation, speech synthesis, forecasting, and music generation, among others. Neural networks can be applied to these tasks by predicting sequence elements one-by-one, conditioning on the history of sequence elements, forming an autoregressive model. Convolutional neural networks (CNNs) and recurrent neural networks (RNNs), including long-short term memory (LSTM) networks BID7 in particular, have achieved many successes at these tasks. However, in their basic form, these models have a limited ability to adapt to recently observed parts of a sequence. Many sequences contain repetition; a pattern that occurs once is more likely to occur again. For instance, a word that occurs once in a document is much more likely to occur again. A sequence of handwriting will generally stay in the same handwriting style. A sequence of speech will generally stay in the same voice. Although RNNs have a hidden state that can summarize the recent past, they are often unable to exploit new patterns that occur repeatedly in a test sequence. This paper concerns dynamic evaluation, which we investigate as a candidate solution to this problem. Our approach adapts models to recent sequences using gradient descent based mechanisms. We show several ways to improve on past dynamic evaluation approaches in Section 5, and use our improved methodology to achieve state-of-the-art in Section 7. In Section 6 we design a method to dramatically to reduce the number of adaptation parameters in dynamic evaluation, making it practical in a wider range of situations. In Section 7.4 we analyse dynamic evaluation's performance over varying time-scales and distribution shifts, and demonstrate that dynamically evaluated models can generate conditional samples that repeat many patterns from the conditioning data. Generative models can assign probabilities to sequences by modelling each term in the factorization given by the product rule. The probability of a sequence x 1:T = {x 1, . . ., x T} factorizes as P (x 1:T) = P (x 1)P (x 2 |x 1)P (x 3 |x 2, x 1) · · · P (x T |x 1 . . . x T −1).Methods that apply this factorization either use a fixed context when predicting P (x t |x 1:t−1), for instance as in N-grams or CNNs, or use a recurrent hidden state to summarize the context, as in an RNN. However, for longer sequences, the history x 1:t−1 often contains re-occurring patterns that are difficult to capture using models with fixed parameters (static models).In many domains, in a dataset of sequences {x contains useful information about the generating distribution for that specific sequence P (x i 1:T). Therefore adapting the model parameters learned during training θ g is justified. We aim to infer a set of model parameters θ l from x i 1:t−1 that will better approximate P (x i t |x i 1:t−1) within sequence i. Many sequence modelling tasks are characterised by sequences generated from slightly different distributions as in the scenario described above. The generating distribution may also change continuously across a single sequence; for instance, a text excerpt may change topic. Furthermore, many machine learning benchmarks do not distinguish between sequence boundaries, and concatenate all sequences into one continuous sequence. Thus, many sequence modelling tasks could be seen as having a local distribution P l (x) as well as a global distribution P g (x):= P (l)P l (x) dl. During training time, the goal is to find the best fixed model possible for P g (x). However, during evaluation time, a model that can infer the current P l (x) from the recent history has an advantage. Dynamic evaluation methods continuously adapt the model parameters θ g, learned at training time, to parts of a sequence during evaluation. The goal is to learn adapted parameters θ l that provide a better model of the local sequence distribution, P l (x). When dynamic evaluation is applied in the present work, a long test sequence x 1:T is divided up into shorter sequences of length n. We define s 1:M to be a sequence of shorter sequence segments s i DISPLAYFORM0 The initial adapted parameters θ 0 l are set to θ g, and used to compute the probability of the first segment, P (s 1 |θ 0 l). This probability gives a cross entropy loss L(s 1), with gradient ∇L(s 1), which is computed using truncated back-propagation through time BID31. The gradient ∇L(s 1) is used to update the model, ing in adapted parameters θ 1 l, before evaluating P (s 2 |θ 1 l). The same procedure is then repeated for s 2, and for each s i in the sequence as shown in FIG1. Gradients for each loss L(s i) are only backpropagated to the beginning of s i, so computation is linear in the sequence length. Each update applies one maximum likelihood training step to approximate the current local distribution P l (x). The computational cost of dynamic evaluation is one forward pass and one gradient computation through the data, with some slight overhead to apply the update rule for every sequence segment. As in all autoregressive models, dynamic evaluation only conditions on sequence elements that it has already predicted, and so evaluates a valid log-probability for each sequence. Dynamic evaluation can also be used while generating sequences. In this case, the model generates each sequence segment s i using fixed weights, and performs a gradient descent based update step on L(s i). Applying dynamic evaluation for sequence generation could in generated sequences with more consistent regularities, meaning that patterns that occur in the generated sequence are more likely to occur again. Adaptive language modelling was first considered for n-grams, adapting to recent history via caching BID9 BID15, and other methods BID2. More recently, the neural cache approach and the closely related pointer sentinel-LSTM (b) have been used to for adaptive neural language modelling. Neural caching has recently been used to improve the state-of-the-art at word-level language modelling BID18. The neural cache model learns a type of non-parametric output layer on the fly at test time, which allows the network to adapt to recent observations. Each past hidden state h i is paired with the next input x i+1, and is stored as a tuple (h i, x i+1). When a new hidden state h t is observed, the output probabilities are adjusted to give a higher weight to output words that coincided with past hidden states with a large inner product (h where e (xi+1) is a one hot encoding of x i+1, and ω is a scaling parameter. The cache probabilities are interpolated with the base network probabilities to adapt the base network at test time. The neural cache closely relates to dynamic evaluation, as both methods can be added on top of a base model for adaptation at test time. The main difference is the mechanism used to fit to recent history: the neural cache approach uses a non-parametric, nearest neighbours-like method, whereas dynamic evaluation uses a gradient descent based method to change model parameters dynamically. Both methods rely on an autoregressive factorisation, as they depend on observing sequence elements after they are predicted in order to perform adaptation. Dynamic evaluation and neural caching methods are therefore both applicable to sequence prediction and generation tasks, but not directly to more general supervised learning tasks. One drawback of the neural cache method is that it cannot adjust the recurrent hidden state dynamics. As a , the neural cache's ability to capture information that occurs jointly between successive sequence elements is limited. This capability is critical for adapting to sequences where each element has very little independent meaning, e.g. character level language modelling. Another related approach is fast weights, BID1 BID27. Fast weights feature recurrent architectures with dynamically changing weight matrices as a function of recent sequence history. Thus, dynamic evaluation as applied at test time, could be considered a form of fast-weights. In traditional fast weights, the network learns to control changes to the weights during training time, allowing it to be applied to more general sequence problems including sequence labeling. In dynamic evaluation, the procedure to change the weights is automated at test time via gradient descent optimization, making it only directly applicable to autoregressive sequence modelling. As dynamic evaluation leverages gradient descent, it has the potential to generalize better to previously unseen pattern repetitions at test time. Dynamic evaluation of neural language models was proposed by BID20. Their approach simply used stochastic gradient descent (SGD) updates at every time step, computing the gradient with fully truncated backpropagation through time, which is equivalent to setting n = 1 in equation FORMULA1. Dynamic evaluation has since been applied to character and word-level language models (; ; BID24). Previous work using dynamic evaluation considered it as an aside, and did not explore it in depth. We propose several changes to BID20's dynamic evaluation method with SGD and fully truncated backpropagation, which we refer to as traditional dynamic evaluation. The first modification reduces the update frequency, so that gradients are backpropagated over more timesteps. This change provides more accurate gradient information, and also improves the computational efficiency of dynamic evaluation, since the update rule is applied much less often. We use sequence segments of length 5 for word-level tasks and 20 for character-level tasks. Next, we add a global decay prior to bias the model towards the parameters θ g learned during training. Our motivation for dynamic evaluation assumes that the local generating distribution P l (x) is constantly changing, so it is potentially desirable to weight recent sequence history higher in adaptation. Adding a global decay prior accomplishes this by causing previous adaptation updates to decay exponentially over time. The use of a decay prior for dynamic evaluation relates to the update rule used for fast weights in BID1, which decayed fast weights towards zero exponentially over time. For SGD with a global prior, learning rate η and decay rate λ; we form the update rule DISPLAYFORM0 We then consider using an RMSprop BID28 derived update rule for the learning rule in place of SGD. RMSprop uses a moving average of recent squared gradients to scale learning rates for each weight. In dynamic evaluation, near the start of a test sequence, RMSprop has had very few gradients to average, and therefore may not be able to leverage its updates as effectively. For this reason, we collect mean squared gradients, MS g, on the training data rather than on recent test data (which is what RMSprop would do). MS g is given by DISPLAYFORM1 where N b is the number of training batches and ∇L k is the gradient on the kth training batch. The mini-batch size for this computation becomes a hyper-parameter, as larger mini-batches will in smaller mean squared gradients. The update rule, which we call RMS with a global prior in our experiments, is then DISPLAYFORM2 where is a stabilization parameter. For the decay step of our update rule, we also consider scaling the decay rate for each parameter proportionally to MS g. Parameters with a high RMS gradient affect the dynamics of the network more, so it makes sense to decay them faster. RMS norm is MS g divided by its mean, ing in a normalized version of MS g with a mean of 1: DISPLAYFORM3 We clip the values of RMS norm to be no greater than 1 /λ to be sure that the decay rate does not exceed 1 for any parameter. Combining the learning component and the regularization component in the final update equation, which we refer to as RMS with an RMS global prior DISPLAYFORM4 6 SPARSE DYNAMIC EVALUATION Mini-batching over sequences is desirable for some test-time sequence modelling applications because it allows faster processing of multiple sequences in parallel. Dynamic evaluation has a high memory cost for mini-batching because it is necessary to store a different set of parameters for each sequence in the mini-batch. Therefore, we consider a sparse dynamic evaluation variant that updates a smaller number of parameters. We introduce a new adaptation matrix M which is initialized to zeros. M multiplies hidden state vector h t of an RNN at every time-step to get a new hidden state h t, via h t = h t + Mh t. h t then replaces h t and is propagated throughout the network via both recurrent and feed-forward connections. In a stacked RNN, this formulation could be applied to every layer or just one layer. Applying dynamic evaluation to M avoids the need to apply dynamic evaluation to the original parameters of the network, reduces the number of adaptation parameters, and makes mini-batching less memory intensive. We reduce the number of adaptation parameters further by only using M to transform an arbitrary subset of H hidden units. This in M being an H ×H matrix with d = H 2 adaptation parameters. If H is chosen to be much less than the number of hidden units, this reduces the number of adaptation parameters dramatically. In Section 7.3 we experiment with sparse dynamic evaluation for character-level language models. We applied dynamic evaluation to word-level and character-level language modelling. In all tasks, we evaluate dynamic evaluation on top of a base model. After training the base model, we tune hyper-parameters for dynamic evaluation on the validation set, and evaluate both the static and dynamic versions of the model on the test set. We also consider follow up experiments that analyse the sequence lengths for which dynamic evaluation is useful. We train base models on the Penn Treebank (PTB, BID16, WikiText-2 (b) datasets, and compare the performance of static and dynamic evaluation. These experiments compare dynamic evaluation against past approaches such as the neural cache and measure dynamic evaluation's general performance across different models and datasets. PTB is derived from articles of the Wall Street Journal. It contains 929k training tokens and a vocab size limited to 10k words. It is one of the most commonly used benchmarks in language modelling. We consider two baseline models on PTB, a standard LSTM implementation with recurrent dropout BID33, and the recent state-of-the-art averaged SGD (ASGD) weight-dropped LSTM (AWD-LSTM, BID18 .Our standard LSTM was taken from the Chainer BID29 tutorial on language modelling 1, and used two LSTM layers with 650 units each, trained with SGD and regularized with recurrent dropout. On our standard LSTM, we experiment with traditional dynamic evaluation as applied by BID20, as well as each modification we make building up to our final update rule as described in Section 5. As our final update rule (RMS + RMS global prior) worked best, we use this for all other experiments and use "dynamic eval" by default to refer to this update rule in tables. We applied dynamic evaluation on an AWD-LSTM BID18. The AWD-LSTM is a vanilla LSTM that combines the use of drop-connect BID30 on recurrent weights for regularization, and a variant of ASGD BID25 for optimisation. Our model, which used 3 layers and tied input and output embeddings BID26 BID8, was intended to be a direct replication of AWD-LSTM, using code from their implementation 2. Results are given in Table 1.Dynamic evaluation gives significant overall improvements to both models on this dataset. Dynamic evaluation also achieves better final than the neural cache on both a standard LSTM and the AWD-LSTM reimplementation, and improves the state-of-the-art on PTB. RNN+LDA+kN-5+cache 92.0 CharCNN BID11 19M 78.9 LSTM BID33 66M 82.2 78.4 Variational LSTM BID4 66M 73.4 Pointer sentinel-LSTM (b) 21M 72.4 70.9 Variational LSTM + augmented loss BID8 51M 71.1 68.5 Variational RHN BID34 23M 67.9 65.4 NAS cell 54M 62.4 Variational LSTM + gradual learning 105M 61.7 LSTM + BB tuning BID17 24M 60.9 58.3 LSTM 86.9 82.3 LSTM + neural cache 74. BID8 28M 91.5 87.0 Pointer sentinel-LSTM (b) 84.8 80.8 LSTM + BB tuning BID17 24M 69.1 65.9LSTM 104.2 99.3 LSTM + neural cache 72 WikiText-2 is roughly twice the size of PTB, with 2 million training tokens and a vocab size of 33k. It features articles in a non-shuffled order, with dependencies across articles that adaptive methods should be able to exploit. For this dataset, we use the same baseline LSTM implementation and AWD-LSTM re-implementation as on PTB. Results are given in TAB3.Dynamic evaluation improves the state-of-the-art perplexity on WikiText-2, and provides a significantly greater improvement than neural caching to both base models. This suggests that dynamic evaluation is effective at exploiting regularities that co-occur across non-shuffled documents. model valid test LSTM 121.8 LSTM + neural cache 99.9AWD-LSTM 80.0 87.5 AWD-LSTM + neural cache 67.5 75.1 AWD-LSTM + dynamic eval 63.3 70.3 We benchmark the performance of dynamic evaluation against static evaluation and the neural cache on the larger text8 dataset. Like WikiText-2, text8 is derived from Wikipedia text. Text8 was introduced for word level language modelling by BID22, which preprocessed the data by mapping rare words to an'<unk>' token, ing in a vocab of 44k and 17M training tokens. We use the same test set as in BID22, but also hold out the final 100k training tokens as a validation set to allow for fair hyper-parameter tuning (the original task did not have a validation set). We trained an AWD-LSTM with 52M parameters using the implementation from BID18. We then compare the performance of static evaluation, dynamic evaluation, and neural caching at test time. To ensure a fair comparison between dynamic evaluation and the neural cache, we used robust hyper-parameter tuning on the validation set for both methods. For dynamic evaluation, we used the hyper-parameter settings found on PTB, and only tuned the learning rate (to 2 significant figures). The neural cache uses 3 hyper-parameters: the cache length, a mixing parameter and a flatness parameter. Starting from a cache size of 3000, we used a series of grid searches to find optimal values for the mixing parameter and flatness parameter (to 2 significant figures). We then varied the cache size in the range of 2000-4000, and found that the affect of this was negligible, so we kept the cache size at 3000. Results are given in table 3, with the from that used the same test set given for context. Dynamic evaluation soundly outperforms static evaluation and the neural cache method, demonstrating that the benefits of dynamic evaluation do not wash away when using a stronger model with more training data. We consider dynamic evaluation on the character-level text8, and Hutter Prize datasets. The Hutter Prize dataset is comprised of Wikipedia text, and includes XML and characters from non-Latin languages. It is 100 million UTF-8 bytes long and contains 205 unique bytes. Similarly to other reported , we use a 90-5-5 split for training, validation, and testing. The text8 dataset is derived the Hutter Prize dataset, but has all XML removed, and is lower cased to only have 26 characters of English text plus spaces. The character-level text8 task corresponds to the unprocessed version of the text8 data used for our medium-scale word level language modelling experiments. As with Hutter Prize, we use the standard 90-5-5 split for training, validation, and testing for text8. We used a multiplicative LSTM (mLSTM) BID13 3 as our base model for both datasets. The mLSTMs for both tasks used 2800 hidden units, an embedding layer of 400 units, weight normalization , variational dropout BID4, and ADAM BID12 for training. We also consider sparse dynamic evaluation, as described in Section 6, on the Hutter Prize dataset. For sparse dynamic evaluation, we adapted a subset of 500 hidden units, ing in a 500×500 adaptation matrix and 250k adaptation parameters. Our mLSTM only contained one recurrent layer, so only one adaptation matrix was used for sparse dynamic evaluation. All of our dynamic evaluation in this section use the final update rule given in Section 5. Results for Hutter Prize are given in TAB7, and for text8 are given in BID32 17M 1.44 HyperLSTM BID6 27M 1.34 Hierarchical multiscale LSTM BID3 1.32 Bytenet decoder BID10 1.31 LSTM + BB tuning BID17 46M 1.30 Recurrent highway networks BID34 46M 1.27 Fast-slow LSTM BID23 BID32 4M 1.44 LSTM BID3 1.43 Batch normalised LSTM BID3 1.36 Hierarchical multiscale LSTM BID3 1.29 Recurrent highway networks BID34 Dynamic evaluation achieves large improvements to our base models and state-of-the-art on both datasets. Sparse dynamic evaluation also achieves significant improvements on Hutter Prize using only 0.5% of the adaptation parameters of regular dynamic evaluation. We measure time-scales at which dynamic evaluation gains an advantage over static evaluation. Starting from the model trained on Hutter Prize, we plot the performance of static and dynamic evaluation against the number of characters processed on sequences from the Hutter Prize test set, and sequences in Spanish from the European Parliament dataset BID12.The Hutter Prize data experiments show the timescales at which dynamic evaluation gained the advantage observed in TAB7. We divided the Hutter Prize test set into 500 sequences of length 10000, and applied static and dynamic evaluation to these sequences using the same model and methodology used to obtain in TAB7. Losses were averaged across these 500 sequences to obtain average losses at each time step. Plots of the average cross-entropy errors against the number of Hutter characters sequenced are given in FIG2.The Spanish experiments measure how dynamic evaluation handles large distribution shifts between training and test time, as Hutter Prize contains very little Spanish. We used the first 5 million characters of the Spanish European Parliament data in place of the Hutter Prize test set. The Spanish experiments used the same base model and dynamic evaluation settings as Hutter Prize. Plots of the average cross-entropy errors against the number of Spanish characters sequenced are given in FIG2.On both datasets, dynamic evaluation gave a very noticeable advantage after a few hundred characters. For Spanish this advantage continued to grow as more of the sequence was processed, whereas for Hutter, this advantage was maximized after viewing around 2-3k characters. The advantage of dynamic evaluation was also much greater on Spanish sequences than Hutter sequences. We also drew 300 character conditional samples from the static and dynamic versions of our model after viewing 10k characters of Spanish. For the dynamic model, we continued to apply dynamic evaluation during sampling as well, by the process described in Section 3. The conditional samples are given in the appendix. The static samples quickly switched to English that resembled Hutter Prize data. The dynamic model generated data with some Spanish words and a number of made up words with characteristics of Spanish words for the entirety of the sample. This is an example of the kinds of features that dynamic evaluation was able to learn to model on the fly. This work explores and develops methodology for applying dynamic evaluation to sequence modelling tasks. Experiments show that the proposed dynamic evaluation methodology gives large test time improvements across character and word level language modelling. Our improvements to language modelling have applications to speech recognition and machine translation over longer contexts, including broadcast speech recognition and paragraph level machine translation. Overall, dynamic evaluation is shown to be an effective method for exploiting pattern re-occurrence in sequences. A.1 DYNAMIC SAMPLES CONDITIONED ON SPANISH 300 character samples generated from the dynamic version of the model trained on Hutter Prize, conditioned on 10k of Spanish characters. The final sentence fragment of the 10k conditioning characters is given to the reader, with the generated text given in bold:Tiene importancia este compromiso en la medida en que la Comisión es un organismo que tiene el montembre tas procedíns la conscriptione se ha Tesalo del Pómienda que et hanemos que Pe la Siemina. De la Pedrera Orden es Señora Presidente civil, Orden de siemin presente relevante frónmida que esculdad pludiore e formidad President de la Presidenta Antidorne Adamirmidad i ciemano de el 200'. Fo 300 character samples generated from the static version of the model trained on Hutter Prize, conditioned on 10k of Spanish characters. The final sentence fragment of the 10k conditioning characters is given to the reader, with the generated text given in bold:Tiene importancia este compromiso en la medida en que la Comisión es un organismo que tiene el monde, <br>There is a secret act in the world except Cape Town, seen in now flat comalo and ball market and has seen the closure of the eagle as imprints in a dallas within the country." Is a topic for an increasingly small contract saying Allan Roth acquired the government in [ ].===

Paper presents dynamic evaluation methodology for adaptive sequence modelling

Represent this text to find a much much longer article about it with say hundreds-thousands of words.

LSTMs learn long-range dependencies compositionally by building them from shorter constituents over the course of training.