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+# A SEMANTIC LOSS FUNCTION FOR DEEP LEARNING WITH SYMBOLIC KNOWLEDGE
+
+# ABSTRACT
+
+This paper develops a novel methodology for using symbolic knowledge in deep learning. From first principles, we derive a semantic loss function that bridges between neural output vectors and logical constraints. This loss function captures how close the neural network is to satisfying the constraints on its output. An experimental evaluation shows that our semantic loss function effectively guides the learner to achieve (near-)state-of-the-art results on semi-supervised multi-class classification. Moreover, it significantly increases the ability of the neural network to predict structured objects, such as rankings and paths. These discrete concepts are tremendously difficult to learn, and benefit from a tight integration of deep learning and symbolic reasoning methods.
+
+# 1 INTRODUCTION
+
+The widespread success of representation learning raises the question of which AI tasks are amenable to deep learning, which require classical model-based symbolic reasoning, and whether we can benefit from an integration of both. In recent years, significant effort has gone towards various ways of using representation learning to solve tasks that were previously tackled by symbolic methods. Such efforts include neural computers, Turing machines, and differentiable programming (e.g., Weston et al. (2014); Reed & De Freitas (2015); Graves et al. (2016); Riedel et al. (2016)), relational embeddings, deep learning for graph data, and neural theorem proving (e.g., Bordes et al. (2013); Neelakantan et al. (2015); Duvenaud et al. (2015); Niepert et al. (2016)), and many more. Other work has sought to augment deep learning with (symbolic) knowledge (e.g., Hu et al. (2016); Stewart & Ermon (2017); Marquez-Neila et al. (2017); Minervini et al. (2017); Wang et al. (2017)). ´
+
+This paper considers learning tasks where we have symbolic knowledge connecting the different outputs of a neural network. This knowledge takes the form of a constraint (or sentence) in Boolean logic. It can be as simple as an exactly-one constraint for one-hot output encodings, or as complex as a structured output prediction constraint for intricate combinatorial objects such as rankings, subgraphs, and paths. Our goal is to augment neural networks with the ability to learn how to make predictions subject to these constraints, and use the symbolic knowledge to improve its performance.
+
+Most neuro-symbolic approaches aim to simulate or learn symbolic reasoning in an end-to-end deep neural network, or capture symbolic knowledge in a vector-space embedding. This choice is partly motivated by the need for smooth differentiable models; adding symbolic reasoning code (e.g., SAT solvers) to a deep learning pipeline destroys this property. Unfortunately, while making reasoning differentiable, the precise logical meaning of the knowledge is often lost. In this paper, we take a distinctly different approach, and tackle the problem of differentiable but sound logical reasoning from first principles. Starting from a set of intuitive axioms, we derive a differentiable semantic loss function that captures how well the outputs of a neural network match a given constraint. This function precisely captures the meaning of the constraint, and is independent of its syntax.
+
+Next, we show how this semantic loss gives significant practical improvements in semi-supervised classification. The semantic loss defined over the exactly-one constraint in this setting permits us to obtain a learning signal from vast amounts of unlabeled data. The key idea is that the semantic loss helps us improve how consistently we are able to classify the unlabeled data. This simple addition to the loss function of standard deep learning architectures yields (near-)state-of-the-art performance in semi-supervised classification on MNIST, FASHION and CIFAR-10 datasets.
+
+![](images/908b06431175243b10447e79ffaca6771156931dd43e1b26edd7a157a8c2ba04.jpg)  
+Figure 1: Outputs of a neural network feed into semantic loss functions for constraints representing a one-hot encoding, a total ranking of preferences, and paths in a grid graph.
+
+Our final set of experiments study the benefits of the semantic loss function for complex structured output learning tasks, such as preference learning and path prediction in a graph (Daume et al., 2009; ´ Chang et al., 2013; Choi et al., 2015; Graves et al., 2016). In these scenarios, the task is two-fold: learn both the structure of the output space, and the actual classification function within that space. By capturing the structure of the output space with logical constraints, and minimizing the semantic loss for this constraint during learning, we are able to learn networks that are much more likely to correctly predict structured objects.
+
+# 2 BACKGROUND AND NOTATION
+
+To formally define semantic loss, we make use of concepts in propositional logic. We write uppercase letters $( X , Y )$ for Boolean variables and lowercase letters $( x , y )$ for their instantiation $X = 0$ or $X = 1$ ). Sets of variables are written in bold uppercase $( \mathbf { X } , \mathbf { Y } )$ , and their joint instantiation in bold lowercase $\mathbf { \Gamma } ( \mathbf { x } , \mathbf { y } )$ . A literal is a variable $( x )$ or its negation $( \neg x )$ . A logical sentence ( $\alpha$ or $\beta$ ) is constructed in the usual way, from variables and logical connectives $( \wedge , \vee$ , etc.), and is also called a formula or constraint. A state or world $\mathbf { x }$ is an instantiation to all variables $\mathbf { X }$ . A state $\mathbf { x }$ satisfies a sentence $\alpha$ , denoted $\mathbf { x } \Vdash \alpha$ , if the sentence evaluates to be true in that world, as defined in the usual way. A sentence $\alpha$ entails another sentence $\beta$ , denoted $\alpha \models \beta$ if all worlds that satisfy $\alpha$ also satisfy $\beta$ . A sentence $\alpha$ is logically equivalent to sentence $\beta$ , denoted $\alpha \equiv \beta$ , if both $\alpha \models \beta$ and $\beta \models \alpha$ .
+
+The output row vector of a neural net is denoted $\mathsf { p }$ . Each value in $\mathsf { p }$ represents the probability of an output and falls in $[ 0 , 1 ]$ . We use both softmax and sigmoid units for our output activation functions. The notation for states $\mathbf { x }$ is used to refer the an assignment, the logical sentence enforcing the assignment, or the binary vector capturing that same assignment, as these are all equivalent notions.
+
+Figure 1 illustrates the three different concrete output constraints of varying difficulty that are studied in our experiments. First, we examine the exactly-one or one-hot constraint capturing the encoding used in multi-class classification. It states that for a set of indicators $\mathbf { X } = \{ X _ { 1 } , \ldots , X _ { n } \}$ , one and exactly one of those indicators must be true, with the rest being false. This is enforced through a logical constraint $\alpha$ by conjoining sentences of the form $\neg X _ { 1 } \lor \neg X _ { 2 }$ for all pairs of variables (at most one variable is true), and a single sentence $X _ { 1 } \lor \cdots \lor X _ { n }$ (at least one variable is true). Our experiments further examine the valid simple path constraint. It states for a given source-destination pair and edge indicators, that the edge indicators which are set to true must form a valid simple path from source to destination. Finally, we explore the ordering constraint, which requires that a set of $n ^ { 2 }$ indicator variables represent a total ordering over $n$ variables, effectively encoding a permutation matrix. For a full description of the path and ordering constraints, we refer to Section 5.
+
+# 3 SEMANTIC LOSS
+
+Our goal in this section is to find a semantic loss function that bridges the gap between the continuous world of neural networks, and the symbolic world of propositional logic. We do so by first postulating intuitive high-level properties that we seek in such a function, and that illustrate its desired behavior. A second set of postulates establish a correspondence between constraints and data. Finally, we uniquely define the semantic loss function used throughout this paper.
+
+The semantic loss $\mathrm { L } ^ { \mathrm { s } } ( \alpha , \mathsf { p } )$ is a function of a sentence $\alpha$ in propositional logic, defined over variables $\mathbf { X } = \{ X _ { 1 } , \ldots , X _ { n } \}$ , and a vector of probabilities $\mathsf { p }$ for the same variables X. Element ${ \mathsf p } _ { i }$ denotes the predicted probability of variable $X _ { i }$ , and corresponds to a single output of the neural net. For example, the semantic loss between the one-hot constraint from the previous section, and a neural net output vector $\mathsf { p }$ , is intended to capture how close the prediction p is to having exactly one output set to true (that is, 1), and all others set to false (that is, 0), regardless of which output is correct.
+
+# 3.1 HIGH-LEVEL PROPERTIES
+
+The first axiom says that there is no loss when the logical constraint $\alpha$ is always true (it is a logical tautology), independent of the predicted probabilities p.
+
+Axiom 1 (Truth). The semantic loss of a true sentence is zero: $\forall \mathsf { p } , \mathrm { L } ^ { \mathrm { s } } ( t r u e , \mathsf { p } ) = 0 .$ .
+
+Next, when enforcing two constraints on disjoint sets of variables, we want the ability to compute the semantic loss of the two constraints separately, and sum the results for their joint semantic loss.
+
+Axiom 2 (Additive Independence). Let $\alpha$ be a sentence over $\mathbf { X }$ with probabilities $\mathsf { p }$ . Let $\beta$ be a sentence over $\mathbf { Y }$ disjoint from $\mathbf { X }$ with probabilities $\mathsf { q }$ . The semantic loss between sentence $\alpha \wedge \beta$ and the joint probability vector [p q] decomposes additively: $\operatorname { L } ^ { \mathrm { s } } ( \alpha \wedge \beta , [ { \mathsf { p q } } ] ) = \operatorname { L } ^ { \mathrm { s } } ( \alpha , { \mathsf { p } } ) + \operatorname { L } ^ { \mathrm { s } } ( \beta , { \mathsf { q } } )$ .
+
+It directly follows from Axioms 1 and 2 that the probabilities of variables that are not used on the constraint do not affect the semantic loss. Proposition 6 in Appendix A formalizes this intuition.
+
+To maintain logical meaning, we postulate that semantic loss is monotone in the order of implication.
+
+Axiom 3 (Monotonicity). If $\alpha \models \beta$ , then the semantic loss $\mathrm { L } ^ { \mathrm { s } } ( \alpha , { \mathfrak { p } } ) \geq \mathrm { L } ^ { \mathrm { s } } ( \beta , { \mathfrak { p } } )$ for any vector $\mathsf { p }$ .
+
+Intuitively, as we add stricter requirements to the logical constraint, going from $\beta$ to $\alpha$ and making it harder to satisfy, the semantic loss cannot decrease. For example, when $\beta$ enforces the output of an neural network to encode a subtree of a graph, and we tighten that requirement in $\alpha$ to be a path, the semantic loss cannot decrease. Every path is also a tree and any solution to $\alpha$ is a solution to $\beta$ .
+
+A first consequence following the monotonicity axiom is that logically equivalent sentences must incur an identical semantic loss for the same probability vector p. Hence, the semantic loss is indeed a semantic property of the logical sentence, and does not depend on the syntax of the sentence.
+
+Proposition 1. If $\alpha \equiv \beta$ , then the semantic loss $\mathrm { L } ^ { \mathrm { s } } ( \alpha , { \mathfrak { p } } ) = \mathrm { L } ^ { \mathrm { s } } ( \beta , { \mathfrak { p } } )$ for any vector p.
+
+A second consequence is that semantic loss must be non-negative (see Proposition 5 in Appendix A).
+
+# 3.2 DATA-SENTENCE CORRESPONDENCE
+
+A state $\mathbf { x }$ is equivalently represented as a data vector, as well as a logical constraint that enforces a value for every variable in $\mathbf { X }$ . When both the constraint and the predicted vector represent the same state (for example, $X _ { 1 } \wedge { \neg { X _ { 2 } } } \wedge X _ { 3 }$ vs. [1 0 1]), there should be no semantic loss.
+
+Axiom 4 (Identity). For any state $\mathbf { x }$ , there is zero semantic loss between its representation as a sentence, and its representation as a deterministic vector: $\forall \mathbf { x } , \mathrm { L } ^ { \mathrm { s } } ( \mathbf { x } , \mathbf { x } ) = 0$ .
+
+The axioms above together imply that any vector satisfying the constraint must incur zero loss. For example, when our constraint $\alpha$ requires that the output vector encodes an arbitrary total ranking, and the vector $\mathbf { x }$ correctly represents a single specific total ranking, there is no semantic loss.
+
+Proposition 2 (Satisfaction). If $\mathbf { \dot { x } } \Vdash \alpha$ , then the semantic loss $\mathrm { L } ^ { \mathrm { s } } ( \alpha , \mathbf { x } ) = 0$ .
+
+As a special case, logical literals $\scriptstyle { \dot { x } }$ or $\neg x$ ) constrain a single variable to take on a single value, and thus play a role similar to the labels used in supervised learning. Such constraints require an even tighter correspondence: the semantic loss must act like a classical loss function (i.e., cross entropy).
+
+Axiom 5 (Label-Literal Correspondence). The semantic loss of a single literal is proportionate to the cross-entropy loss for the equivalent data label: $\mathrm { L } ^ { \mathrm { s } } ( x , p ) \propto - \log ( p )$ and $\mathrm { L } ^ { \mathrm { s } } ( \neg { x } , \bar { p } ) \overset { \cdot } { \propto } - \log ( 1 - p )$ .
+
+Appendix A states Axioms 7 and 8, on the symmetry between values and the symmetry between variables, as well as a Lemma 7 that ties together the multiplicative constants mentioned in Axiom 5. Finally, this allows us to prove the following form of the semantic loss for a state $\mathbf { x }$ .
+
+Lemma 3. For state x and vector p, we have $\begin{array} { r } { \mathrm { L } ^ { \mathrm { s } } ( \mathbf { x } , \mathsf { p } ) \propto - \sum _ { i : \mathbf { x } | = X _ { i } } \log \mathsf { p } _ { i } - \sum _ { i : \mathbf { x } | = \lnot X _ { i } } \log ( 1 - \mathsf { p } _ { i } ) . } \end{array}$
+
+![](images/5fcbfe708dc5ef9963d1119ec14ca3d17f5eace11a8e49df2a20fcaf583e6a6f.jpg)  
+Figure 2: Binary classification toy example: a linear classifier without and with semantic loss.
+
+# 3.3 A GENERAL DEFINITION
+
+Lemma 3 falls short as a full definition of semantic loss for arbitrary sentences. One can define additional axioms to pin down $\mathrm { L } ^ { \mathrm { s } }$ . For example, the following axiom is highly desirable.
+
+Axiom 6 (Differentiability). For any fixed $\alpha$ , the semantic loss $\mathrm { L } ^ { \mathrm { s } } ( \alpha , \mathsf { p } )$ is monotone in each probability in p, continuous and differentiable.
+
+Appendix A makes the notion of semantic loss precise by stating one additional axiom. It is based on the observation that the state loss of Lemma 3 is proportionate to a log-probability. In particular, it corresponds to the probability of obtaining state $\mathbf { x }$ after independently sampling each $X _ { i }$ with probability ${ \mathsf { p } } _ { i }$ . We have now derived the semantic loss function from first principles as follows.
+
+Definition 1 (Semantic Loss). Let $\mathsf { p }$ be a vector of probabilities, one for each variable in $\mathbf { X }$ , and let $\alpha$ be a sentence over $\mathbf { X }$ . The semantic loss between $\alpha$ and $\mathsf { p }$ is
+
+$$
+\mathrm { L } ^ { \mathrm { s } } ( \alpha , \mathfrak { p } ) \propto - \log \sum _ { \mathbf { x } \in \alpha } \ \prod _ { i : \mathbf { x } | = X _ { i } } \mathfrak { p } _ { i } \prod _ { i : \mathbf { x } | = \ l - X _ { i } } ( 1 - \mathfrak { p } _ { i } ) .
+$$
+
+Theorem 4 (Uniqueness). The semantic loss function in Definition 1 satisfies Axioms 1–9 and is the only function that does so, up to a multiplicative constant.
+
+Intuitively, the semantic loss is proportionate to a negative logarithm of the probability of generating a state that satisfies the constraint, when sampling values according to p. Hence, it is the selfinformation (or “surprise”) of obtaining an assignment that satisfies the constraint (Jones, 1979).
+
+# 4 SEMI-SUPERVISED CLASSIFICATION
+
+The most straightforward constraint that is ubiquitous in multi-class classification is mutual exclusion over one-hot-encoded outputs. That is, for a given example, exactly one class and therefore exactly one binary indicator must be true. The machine learning community has made great strides in this machine learning task, due to the invention of assorted deep learning representations and their associated regularization terms (Krizhevsky et al., 2012; He et al., 2016). Many of these models take large amounts of fully labeled data for granted, and big data is indispensable for discovering accurate representations (Hastie et al., 2009). To sustain this progress, and alleviate the need for more labeled data, there is a growing interest into utilizing unlabeled data to augment the predictive power of classifiers (Stewart & Ermon, 2017; Bilenko et al., 2004). This section shows why semantic loss naturally qualifies for this task.
+
+Illustrative Example To illustrate the benefit of semantic loss in the semi-supervised setting, we begin our discussion with a small toy example. Consider a binary classification task as depicted in Figure 2. Ignoring the unlabeled examples, a simple linear classifier learns to distinguish the two classes by separating the labeled examples in Figure 2a. However, the unlabeled examples are also informative, as they must carry some properties that give them a particular label. This is the crux of semantic loss: a model must confidently assign a consistent class even to unlabeled data. Encouraging the model to do so results in a more accurate decision boundary, as illustrated in Figure 2b. Next, we further explore this idea and apply it to real-world image classification tasks.
+
+Table 1: MNIST. Previously reported test accuracies followed by semantic loss results $\pm$ stddev)   
+
+<table><tr><td>Accuracy % with # of used labels</td><td>100</td><td>1000</td><td>ALL</td></tr><tr><td>AtlasRBF (Pitelis et al., 2014) Deep Generative (Kingma et al., 2014) Virtual Adversarial (Miyato et al., 2016) Ladder Net (Rasmus et al., 2015)</td><td>91.9 (± 0.95) 96.67(± 0.14) 97.67 98.94(±0.37)</td><td>96.32 (± 0.12) 97.60(± 0.02) 98.64 99.16 (±0.08)</td><td>98.69 99.04 99.36 99.43 (± 0.02)</td></tr><tr><td>Baseline: MLP, Gaussian Noise Baseline: Self-Training MLP with Semantic Loss (our)</td><td>78.46 (±1.94) 72.55 (±4.21) 98.38 (±0.51)</td><td>94.26 (±0.31) 87.43 (±3.07) 98.78 (±0.17)</td><td>99.34(±0.08) 99.34 (±0.08) 99.36 (±0.02)</td></tr></table>
+
+# 4.1 METHOD
+
+Our proposed method intends to be generally applicable and compatible with any feedforward neural network. The semantic loss is simply another regularization term that can directly be plugged into an existing loss function. More specifically, for some weight $w$ , the new overall loss becomes
+
+When the constraint over the output space is simple (for example, there is a small number of solutions ${ \textbf { x } } | = \alpha )$ ), the semantic loss can be directly computed from Definition 1. Concretely, for the exactly-one constraint used in $n$ -class classification, the semantic loss reduces to
+
+$$
+\mathrm { L } ^ { \mathrm { s } } ( \mathrm { e x a c t l y - o n e } , \mathsf { p } ) \propto - \log \sum _ { i = 0 } ^ { n - 1 } \mathsf { p } _ { i } \prod _ { j = 0 , j \neq i } ^ { n - 1 } ( 1 - \mathsf { p } _ { j } ) ,
+$$
+
+where he values ${ \mathsf { p } } _ { i }$ denote the probability of class $i$ as predicted by the neural net. The semantic loss for the exactly-one constraint is efficient and causes no noticeable overhead in our experiments.
+
+In general, for arbitrary constraints $\alpha$ , the semantic loss is not efficient to compute using Definition 1, and more advanced automated reasoning is required. Section 5 discusses this issue in more detail.
+
+# 4.2 EXPERIMENTAL EVALUATION
+
+In this section, we evaluate semantic loss in the semi-supervised setting by comparing it with several competitive models. As most semi-supervised learners build on a supervised learner, changing the underlying model significantly affects the semi-supervised learner’s performance. For comparison, we add semantic loss to the same base models used in ladder nets (Rasmus et al., 2015), which currently achieve state-of-the-art results on semi-supervised MNIST and CIFAR-10 (Krizhevsky & Hinton, 2009). Specifically, the MNIST base model is a fully-connected multilayer perceptron (MLP), with layers of size 784-1000-500-250-250-250-10. On CIFAR-10, it is a 10-layer convolutional neural network (CNN) with 3-by-3 padded filters. After every 3 layers, features are subject to a 2-by-2 max-pool layer with strides of 2. Furthermore, we use ReLu (Nair & Hinton, 2010), batch normalization (Ioffe & Szegedy, 2015), and Adam optimization (Kingma & Ba, 2015) with a learning rate of 0.002. We refer to Appendix B and C for a specification of the CNN model and additional details about hyper-parameter tuning.
+
+For all semi-supervised experiments, we use the standard 10,000 held-out test examples provided in the original datasets and randomly pick 10,000 from the standard 60,000 training examples (50,000 for CIFAR-10) as validation set. For values of $N$ that depend on the experiment, we retain $N$ randomly chosen labeled examples from the training set, and remove labels from the rest. We balance classes in the labeled samples to ensure no particular class is over-represented. Images are preprocessed for standardization and Gaussian noise (standard deviation 0.3) is added to every pixel.
+
+MNIST The permutation invariant MNIST classification task is commonly used as a test-bed for general semi-supervised learning algorithms. This setting does not use any prior information about the spatial arrangement of the input pixels. Therefore, it excludes many data augmentation techniques that involve geometric distortion of images, as well as convolutional neural networks.
+
+When evaluating on MNIST, we run experiments for 10 epochs, with a batch size of 10 labeled and 10 unlabeled examples. Experiments are repeated 10 times with different random seeds. Table 1 compares semantic loss to two baselines and state-of-the-art results from the literature. The first baseline is a purely supervised MLP, which makes no use of unlabeled data. The second is the classic self-training method for semi-supervised learning, which operates as follows. After every 1000 iterations, the unlabeled examples that are predicted by the MLP to have more than $9 5 \%$ probability of belonging to a single class, are assigned a psuedo-label and become labeled data.
+
+Table 2: FASHION. Test accuracy comparison between MLP with semantic loss and ladder nets.   
+
+<table><tr><td>Accuracy % with # of used labels</td><td>100</td><td>500</td><td>1000</td><td>ALL</td></tr><tr><td>Ladder Net (Rasmus et al., 2015)</td><td>81.46(±0.64)</td><td>85.18 (±0.27)</td><td>86.48 (± 0.15)</td><td>90.46</td></tr><tr><td>Baseline:MLP,GaussianNoise</td><td>69.45 (±2.03)</td><td>78.12 (±1.41)</td><td>80.94(±0.84)</td><td>89.87</td></tr><tr><td>MLP with Semantic Loss (our)</td><td>86.74 (±0.71)</td><td>89.49 (±0.24)</td><td>89.67 (±0.09)</td><td>89.81</td></tr></table>
+
+Table 3: CIFAR. Test accuracy comparison between CNN with semantic loss and ladder nets.   
+
+<table><tr><td rowspan=1 colspan=1>Accuracy % with # of used labels</td><td rowspan=1 colspan=1>4000</td><td rowspan=1 colspan=1>ALL</td></tr><tr><td rowspan=2 colspan=1>CNNBaselineinLadderNetLadder Net (Rasmus et al., 2015)Baseline: CNN,Whitening, CroppingCNN with Semantic Loss (our)</td><td rowspan=1 colspan=1>76.67 (± 0.61)79.60 (±0.47)</td><td rowspan=1 colspan=1>90.73</td></tr><tr><td rowspan=1 colspan=1>77.1381.79</td><td rowspan=1 colspan=1>90.9690.92</td></tr></table>
+
+When given 100 labeled examples ( $N = 1 0 0 \mathrm { \ : }$ ), MLP with semantic loss gains around $2 0 \%$ improvement over the purely supervised baseline. The improvement is even larger $( 2 5 \% )$ compared to self-training. Considering the only change is an additional loss term, this result is very encouraging. Compared to the state of the art, ladder nets slightly outperform semantic loss by $0 . 5 \%$ accuracy. This difference may be an artifact of the excessive tuning of architectures, hyper-parameters and learning rates that the MNIST dataset has been subject to. In the coming experiments, we extend our work to more challenging datasets, in order to provide a clearer comparison with ladder nets.
+
+First, we want to share a few more thoughts on how semantic loss works. A classical softmax layer interprets its output as representing a categorical distribution. Hence, by normalizing its outputs, softmax enforces the same mutual exclusion constraint enforced in our semantic loss function. However, there does not exist a natural way to extend softmax loss to unlabeled samples. In contrast, semantic loss does provide a learning signal on unlabeled samples, by forcing the underlying classifier to make an decision and construct a confident hypothesis for all data. However, for the fully supervised case $N = \mathrm { a l l }$ ), the semantic loss does not significantly affect accuracy. Because the MLP has enough capacity to almost perfectly fit the training data, where the constraint is always satisfied, the semantic loss is almost always zero. This is a direct consequence of Proposition 2.
+
+FASHION The FASHION (Xiao et al., 2017) dataset consists of Zalando’s article images, aiming to serve as a more challenging drop-in replacement for MNIST. Arguably, it has not been overused and requires more advanced techniques to achieve good performance. As in the previous experiment, we run our method for 10 epochs, whereas ladder nets need 100 epochs to converge. Again, experiments are repeated 10 times and Table 2 reports the classification accuracy and its standard deviation (except for $N =$ all where it is close to 0 and omitted for space).
+
+Experiments show that utilizing semantic loss results in a very large $1 7 \%$ improvement over the baseline when only 100 labels are provided. Moreover, our method compares favorably to ladder nets, except when the setting degrades to be fully supervised. Note that our method already nearly reaches its maximum accuracy with 500 labeled examples, which is only $1 \%$ of the training dataset.
+
+CIFAR-10 To show the general applicability of semantic loss, we evaluate it on CIFAR-10. This dataset consisting of 32-by-32 RGB images in 10 classes. A simple MLP would not have enough representation power to capture the huge variance across objects within the same class. To cope with this spike in difficulty, we switch our underlying model to a 10-layer CNN as described earlier. We use a batch size of 100 samples of which half are unlabeled. Experiments are run for 100 epochs. However, due to our limited computational resources, we report on a single trial. Note that we make slight modifications to the underlying model used in ladder nets to reproduce similar baseline performance. Please refer to Appendix B for the details of this experimental setup.
+
+As shown in Table 3, our method compares favorably to ladder nets. However, due to the slight difference in performance between the supervised base models, a direct comparison would be methodologically flawed. Instead, we compare the net improvements over baselines. In terms of this measure, our method scores a gain of $4 . { \bar { 6 } } 6 \%$ whereas ladder nets gain $2 . 9 3 \%$ .
+
+# 4.3 DISCUSSION
+
+Overall, the experiments so far have demonstrated the competitiveness and general applicability of our proposed method on semi-supervised learning tasks. It surpassed the previous state of the art (i.e. ladder nets ) on FASHION and CIFAR-10, while being close on MNIST. Considering the simplicity of our method, such results are encouraging. Indeed, a key advantage of semantic loss is that it only requires a simple additional loss term. Without changing the network architecture itself, we incur almost no computational overhead. Conversely, this property makes our method sensitive to the underlying model’s performance. Without the underlying predictive power of a strong supervised learning model, we do not expect to see the same benefits we observe here. Recently, we became aware that Miyato et al. (2016) extended their work to CIFAR-10 and achieved state-of-the-art results (Miyato et al., 2017), surpassing our performance by $5 \%$ . In future work, we plan to investigate whether applying semantic loss on their architecture would yield an even stronger performance.
+
+![](images/684ff625c16715167670292375e1562c7bf4528618063e9e1026bc8e042c3a58.jpg)  
+Figure 3: FASHION pictures grouped by how confidently the supervised base model classifies them correctly. With semantic loss, the final semi-supervised model predicts all correctly and confidently.
+
+Figure 3 illustrates the effect of semantic loss on FASHION pictures whose correct label was hidden from the learner. Pictures 3a and 3b are correctly classified by the supervised base model, and on the first set it is confident about this prediction $( { \mathsf p } _ { i } > 0 . 8 )$ . The semantic loss rarely diverts the model from these initially correct labels. However, it bootstraps these unlabeled examples to achieve higher confidence in the learned concepts. With this additional learning signal, the model changes its beliefs about Pictures 3c, which it was previously uncertain about. Finally, even on confidently misclassified Pictures 3d, the semantic loss is able to fully correct the mistakes of the base model.
+
+# 5 LEARNING WITH COMPLEX CONSTRAINTS
+
+While much of current machine learning research is focused on problems such as multi-class classification, there remain a multitude of difficult problems involving highly constrained output domains. As mentioned in the previous section, semantic loss has little effect on the fully-supervised exactly-one classification problem. This leads us to seek out more difficult problems to illustrate that semantic loss can also be highly informative in the supervised case, provided the output domain is a sufficiently complex space. Because semantic loss is defined by a Boolean formula, it can be used on any output domain that can be fully described in this manner. Here, we develop a framework for tractable semantic loss on highly complex constraints, and evaluate it on some difficult examples.
+
+# 5.1 A TRACTABLE SEMANTIC LOSS
+
+Our goal here is to develop a method for computing both the semantic loss and its gradient in a tractable manner. Examining Definition 1 of semantic loss, we see that the right-hand side is a wellknown automated reasoning task called weighted model counting (WMC) (Chavira & Darwiche, 2008; Sang et al., 2005). A key property of WMC is that its partial derivatives can be computed in terms of other, slightly modified WMCs. Furthermore, we know of circuit languages that compute WMCs, and that are amenable to backpropagation (Darwiche, 2003). We use the language and circuit compilation techniques described in Darwiche (2011) to build a Boolean circuit representing our semantic loss. We refer to the literature for details of this compilation approach. Due to certain properties of this circuit form, we can use it to compute both the values and the gradients of the semantic loss in time linear in the size of the circuit (Darwiche & Marquis, 2002). Once we have constructed this function, we can add it to our standard loss function as described in Section 4.1.
+
+Table 4: Grid shortest path test results: coherent, incoherent and constraint accuracy.   
+
+<table><tr><td rowspan=1 colspan=1>Test accuracy %</td><td rowspan=1 colspan=1>Coherent</td><td rowspan=1 colspan=1>Incoherent</td><td rowspan=1 colspan=1>Constraint</td></tr><tr><td rowspan=1 colspan=1>5-layer MLP</td><td rowspan=1 colspan=1>5.62</td><td rowspan=1 colspan=1>85.91</td><td rowspan=1 colspan=1>6.99</td></tr><tr><td rowspan=1 colspan=1>With semantic loss (our)</td><td rowspan=1 colspan=1>28.51</td><td rowspan=1 colspan=1>83.14</td><td rowspan=1 colspan=1>69.89</td></tr></table>
+
+Table 5: Preference prediction test results: coherent, incoherent and constraint accuracy.   
+
+<table><tr><td rowspan=1 colspan=1>Test accuracy %</td><td rowspan=1 colspan=1>Coherent</td><td rowspan=1 colspan=1>Incoherent</td><td rowspan=1 colspan=1>Constraint</td></tr><tr><td rowspan=1 colspan=1>3-layer MLP</td><td rowspan=1 colspan=1>1.01</td><td rowspan=1 colspan=1>75.78</td><td rowspan=1 colspan=1>2.72</td></tr><tr><td rowspan=1 colspan=1>With semantic loss (our)</td><td rowspan=1 colspan=1>13.59</td><td rowspan=1 colspan=1>72.43</td><td rowspan=1 colspan=1>55.28</td></tr></table>
+
+# 5.2 EXPERIMENTAL EVALUATION
+
+Our ambition when evaluating semantic loss’ performance on complex constraints is not to achieve state-of-the-art performance on any particular problem, but rather to highlight its effect. To this end, we evaluate our method on problems with a difficult output space, where the model could no longer be fit directly from data, and purposefully use simple MLPs for evaluation. The details of hyper-parameter tuning are again given in Appendix C.
+
+# 5.2.1 GRIDS
+
+We begin with a classic algorithmic problem, finding the shortest path in a graph. Specifically, we use a 4-by-4 grid $G = ( V , E )$ with uniform edge weights. We randomly remove edges for each example to increase difficulty. Formally, our input is a binary vector of length $\vert V \vert + \vert E \vert$ , with the first $| V |$ variables indicating sources and destinations, and the next $| E |$ which edges are removed. Similarly, each label is a binary vector of length $| E |$ indicating which edges are in the shortest path. Finally, we require through our constraint $\alpha$ that the output form a valid simple path between the desired source and destination. To compile this constraint, we use the method of Nishino et al. (2017) to encode pairwise simple paths, and logically merge them to enforce the correct source and destination. For more details on the constraint and data generation process, see Appendix D.
+
+To evaluate, we use our generated dataset of 1600 examples, with a 60/20/20 train/validation/test split. Table 4 compares test accuracy between a 5-layer MLP baseline, and the same model augmented with semantic loss. We report three different accuracies that illustrate the effect of semantic loss: “Coherent” indicates the percentage of examples for which the classifier gets the entire configuration right, while “Incoherent” measures the percentage of individually correct binary labels, which as a whole may not constitute a valid path at all. Finally, “Constraint” describes the percentage of predictions given by the model that satisfy the constraint associated with the problem. In the case of incoherent accuracy, semantic loss has little effect, and in fact slightly reduces the accuracy as it combats the standard sigmoid cross entropy. In regard to coherent accuracy, however, the semantic loss has a very large effect in guiding the network to jointly learn true paths, rather than optimizing each binary output individually. We further see this by observing the large increase in the percentage of predictions which really are paths between the desired nodes in the graph.
+
+# 5.2.2 PREFERENCE LEARNING
+
+The next problem we examine is that of predicting a complete order of preferences. That is, for a given set of user features, we would like to predict how the user would rank their preference over a fixed set of items. We encode a preference ordering over $n$ items as a flattened binary matrix $\{ X _ { i j } \}$ , where for each $i , j \in \{ 1 , \ldots , n \}$ , $X _ { i j }$ denotes that item $i$ is at position $j$ (Choi et al., 2015). Clearly, not all configurations of outputs correspond to a valid ordering.
+
+For data, we use preference rankings over 10 types of sushi for 5000 individuals, taken from PREFLIB (Mattei & Walsh, 2013). We take the ordering over 6 types of sushi as input features to predict the ordering over the remaining 4 types, with splits identical to those in Shen et al. (2017). We again split the data 60/20/20 into train/test/split, and employ a 3 layer MLP as our baseline. Table 5 compares the baseline to the same MLP augmented with semantic loss for valid total orderings. Again, we see that semantic loss has a marginal effect on incoherent accuracy, but massively improves the network’s ability to predict valid, correct orderings. Remarkably, without semantic loss, the network is only able to output a valid ordering on $0 . 0 1 \%$ of the test examples.
+
+# 6 RELATED WORK
+
+Incorporating symbolic background knowledge into machine learning is a long-standing challenge (Srinivasan et al., 1995). It has received considerable attention for structured prediction in natural language processing, in both supervised and semi-supervised settings. For example, constrained conditional models extend linear models with constraints that are enforced through integer linear programming (Chang et al., 2008; 2013). Constraints have also been studied in the context of probabilistic graphical models (Mateescu & Dechter, 2008; Ganchev et al., 2010). Kisa et al. (2014) utilize a circuit language called the probabilistic sentential decision diagram to induce distributions over arbitrary logical formulas. They learn generative models that satisfy preference and path constraints (Choi et al., 2015; 2016), which we both study in a discriminative setting.
+
+Various deep learning techniques have been proposed to enforce either arithmetic constraints (Pathak et al., 2015; Marquez-Neila et al., 2017) or logical constraints (Rockt ´ aschel et al., 2015; Hu et al., ¨ 2016; Demeester et al., 2016; Stewart & Ermon, 2017; Minervini et al., 2017; Diligenti et al., 2017; Donadello et al., 2017) on the output of a neural network. The common approach is to reduce logical constraints into differentiable arithmetic objectives by replacing logical operators with their fuzzy t-norms and logical implications with simple inequalities. A downside of this fuzzy relaxation is that the logical sentences lose their precise meaning. The learning objective becomes a function of the syntax rather than the semantics. Moreover, these relaxations are often only applied to Horn clauses. One alternative is to encode the logic into a factor graph and perform loopy belief propagation to compute a loss function (Naradowsky & Riedel, 2017), which is known to have issues in the presence of complex logical constraints (Smith & Gogate, 2014).
+
+Several specialized techniques have been proposed to exploit the rich structure of real world labels. Deng et al. (2014) propose hierarchy and exclusion graphs that allow a flexible joint modeling of hierarchical categories. It is a method invented to address examples whose labels are not provided at the most specific level. Finally, the objective of semantic loss to increase the confidence of predictions on unlabeled data is in common with information-theoretic approaches to semi-supervised learning (Grandvalet & Bengio, 2005; Erkan & Altun, 2010), and approaches that increase robustness to output perturbation (Miyato et al., 2016). A key difference between semantic loss and these information-theoretic losses is that semantic loss generalizes to arbitrary output constraints.
+
+# 7 CONCLUSIONS
+
+Both reasoning and semi-supervised learning are often identified as key challenges for deep learning going forward. In this paper, we developed a principled way of combining automated reasoning for propositional logic with existing deep learning architectures. Moreover, we showed that our semantic loss function provides significant benefits during semi-supervised classification, as well as deep structured prediction for highly complex output spaces.
+
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+
+A AXIOMATIZATION OF SEMANTIC LOSS: DETAILS
+
+This appendix provides further details on our axiomatization of the semantic loss.   
+Proposition 5 (Non-Negativity). Semantic loss is non-negative.
+
+Proof. Because $\alpha \models$ true for all $\alpha$ , the monotonicity axiom implies that $\forall \mathsf { p } , \mathrm { L } ^ { \mathrm { s } } ( \alpha , \mathsf { p } ) \geq \mathrm { L } ^ { \mathrm { s } } ( t r u e , \mathsf { p } )$ . By the truth axiom, $\mathrm { L } ^ { \mathrm { s } } ( t r u e , \mathsf { p } ) = 0$ , and therefore $\mathrm { L } ^ { \mathrm { s } } ( \alpha , { \mathsf { p } } ) \geq 0$ for all choices of $\alpha$ and $\mathsf { p }$ . □
+
+Proposition 6 (Locality). Let α be a sentence over $\mathbf { X }$ with probabilities p. For any $\mathbf { Y }$ disjoint from $\mathbf { X }$ with probabilities q, the semantic loss $\operatorname { L } ^ { \mathrm { s } } ( \alpha , [ \mathsf { p q } ] ) = \operatorname { L } ^ { \mathrm { s } } ( \alpha , \mathsf { p } )$ .
+
+Proof. Follows from the additive independence and truth axioms. Set $\beta \ = \ t r u e$ in the additive independence axiom, and observe that this sets $\mathrm { L } ^ { \mathrm { s } } ( \beta , { \mathsf { q } } ) = 0$ because of the truth axiom. □
+
+Proof of Proposition 2. The monotonicity axiom specializes to say that if ${ \textbf { x } } | = \alpha$ , we have that $\forall \mathsf { p } , \mathrm { L } ^ { \mathrm { s } } ( \mathbf { x } , \mathsf { p } ) \ \geq \ \mathrm { L } ^ { \mathrm { s } } ( \alpha , \mathsf { p } )$ . By choosing $\mathsf { p }$ to be $\mathbf { x }$ , this implies $\mathrm { L } ^ { \mathrm { s } } ( \mathbf { x } , \mathbf { x } ) \ \geq \ \mathrm { L } ^ { \mathrm { s } } ( \alpha , \mathbf { x } )$ . From the identity axiom, $\mathrm { L } ^ { \mathrm { s } } ( { \bf x } , { \bf x } ) = 0$ , and therefore $0 \geq \operatorname { L } ^ { \mathrm { s } } ( \alpha , \mathbf { x } )$ . Proposition 5 bounds the loss from below as $\mathrm { L } ^ { \mathrm { s } } ( \alpha , \mathbf { x } ) \geq 0$ . □
+
+Axiom 7 (Value Symmetry). For all $\mathsf { p }$ and $\alpha$ , we have that $\mathrm { L } ^ { \mathrm { s } } ( \alpha , { \mathfrak { p } } ) = \mathrm { L } ^ { \mathrm { s } } ( { \bar { \alpha } } , 1 - { \mathfrak { p } } )$ where $\bar { \alpha }$ replaces every variable in $\alpha$ by its negation.
+
+Axiom 8 (Variable Symmetry). Let $\alpha$ be a sentence over $\mathbf { X }$ with probabilities $\mathsf { p }$ . Let $\pi$ be a permutation of the variables $\mathbf { X }$ , let $\pi ( \alpha )$ be the sentence obtained by replacing variables $x$ by $\pi ( x )$ , and let $\pi ( { \mathfrak { p } } )$ be the corresponding permuted vector of probabilities. Then, $\operatorname { L } ^ { \mathrm { s } } ( \alpha , { \mathsf { p } } ) = \operatorname { L } ^ { \mathrm { s } } ( \pi ( \alpha ) , \pi ( { \mathsf { p } } ) )$ .
+
+The value and variable symmetry axioms together imply the equality of the multiplicative constants in the label-literal duality axiom for all literals.
+
+Lemma 7. There exists a single constant $K$ such that $\operatorname { L } ^ { \mathrm { s } } ( X , p ) = - K \log ( p )$ and $\operatorname { L } ^ { s } ( \lnot X , p ) =$ $- K \log ( 1 - p )$ for any literal $x$ .
+
+Proof. Value symmetry implies that $\operatorname { L } ^ { \mathrm { s } } ( X _ { i } , \mathfrak { p } ) = \operatorname { L } ^ { \mathrm { s } } ( \lnot X _ { i } , 1 - \mathfrak { p } )$ . Using label-literal correspondence, this implies $K _ { 1 } \log ( p _ { i } ) = K _ { 2 } \log ( 1 - ( 1 - p _ { i } ) )$ for the multiplicative constants $K _ { 1 }$ and $K _ { 2 }$ that are left unspecified by that axiom. This implies that the constants are identical. A similar argument based on variable symmetry proves equality between the multiplicative constants for different $i$ .
+
+Proof of Lemma 3. A state $\mathbf { x }$ is a conjunction of independent literals, and therefore subject to the additive independence axiom. Each literal’s loss in this sum is defined by Lemma 7. □
+
+The following and final axiom requires that the semantic loss is proportionate to the logarithm of a function that is additive for mutually exclusive sentences.
+
+Axiom 9 (Exponential Additivity). Let $\alpha$ and $\beta$ be mutually exclusive sentences (i.e., $\alpha \wedge \beta$ is unsatisfiable), and let $f ^ { s } ( K , \alpha , { \mathsf { p } } ) = K ^ { - \mathtt { L } ^ { \mathtt { s } } ( \alpha , { \mathsf { p } } ) }$ . Then, there exists a positive constant $K$ such that $f ^ { s } ( K , \alpha \vee \beta , \mathsf { p } ) = f ^ { s } ( K , \alpha , \mathsf { p } ) + f ^ { s } ( K , \beta , \mathsf { p } )$ .
+
+Proof of Theorem 4. The truth axiom states that $\forall \mathsf { p }$ $, f ^ { s } ( K , t r u e , \mathfrak { p } ) = 1$ for all positive constants $K$ . This is the first Kolmogorov axiom of probability. The second Kolmogorov axiom for $f ^ { s } ( K , . , { \mathsf { p } } )$ follows from the additive independence axiom of semantic loss. The third Kolmogorov axiom (for the finite discrete case) is given by the exponential additivity axiom of semantic loss. Hence, $f ^ { s } ( K , . , { \mathsf { p } } )$ is a probability distribution for some choice of $K$ , which implies the definition up to a multiplicative constant. □
+
+# B SPECIFICATION OF THE CONVOLUTIONAL NEURAL NETWORK MODEL
+
+Table 6 shows the slight architectural difference between the CNN used in ladder nets and ours. The major difference lies in the choice of ReLu. Note we add standard padded cropping to preprocess images and an additional fully connected layer at the end of the model, neither is used in ladder nets. We only make those slight modification so that the baseline performance reported by Rasmus et al. (2015) can be reproduced.
+
+# C HYPER-PARAMETER TUNING DETAILS
+
+Validation sets are used for tuning the weight associated with semantic loss, the only hyperparameter that causes noticeable difference in performance for our method. For our semi-supervised classification experiments, we perform a grid search over $\{ 0 . 0 0 1 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 5 , 0 . 1 \}$ to find the optimal value. Empirically, 0.005 always gives the best or nearly the best results and we report its results on all experiments.
+
+For the FASHION dataset specifically, because MNIST and FASHION share the same image size and structure, methods developed in MNIST should be able to directly perform on FASHION without heavy modifications. Because of this, we use the same hyper-parameters when evaluating our method. However, for the sake of fairness, we subject ladder nets to a small-scale parameter tuning in case its performance is more volatile.
+
+For the grids experiment, the only hyper parameter that needed to be tuned was again the weight given to semantic loss, which after trying $\{ 0 . 0 0 1 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 5 , 0 . 1 , 0 . 5 , 1 \}$ was selected to be
+
+Table 6: Specifications of CNNs in Ladder Net and our proposed method.   
+
+<table><tr><td>CNNinLadderNet</td><td>CNN in this paper</td></tr><tr><td colspan="2">Input 32×32 RGB image</td></tr><tr><td></td><td>Resizing to 36 × 36 with padding Cropping Back</td></tr><tr><td colspan="2">Whitening Contrast Normalization</td></tr><tr><td colspan="2">Gaussian Noise with std.of 0.3</td></tr><tr><td>3 ×3 conv. 96 BNLeakyReLU</td><td>3×3 conv. 96 BN ReLU</td></tr><tr><td>3×3 conv. 96 BNLeakyReLU</td><td>3 ×3 conv. 96 BN ReLU</td></tr><tr><td>3 ×3 conv. 96 BNLeakyReLU</td><td>3×3 conv. 96 BN ReLU</td></tr><tr><td colspan="2">2×2 max-pooling stride 2 BN</td></tr><tr><td>3×3 conv.192BNLeakyReLU</td><td>3×3 conv.192 BN ReLU</td></tr><tr><td>3×3 conv.192 BNLeakyReLU</td><td>3×3 conv. 192 BN ReLU</td></tr><tr><td>3×3 conv.192 BN LeakyReLU</td><td>3×3 conv.192 BN ReLU</td></tr><tr><td colspan="2">2×2 max-pooling stride 2 BN</td></tr><tr><td>3×3conv.192BNLeakyReLU</td><td>3×3conv.192 BNReLU</td></tr><tr><td>1×1 conv.192 BN LeakyReLU</td><td>3×3 conv. 192 BN ReLU</td></tr><tr><td>1×1conv.10 BNLeakyReLU</td><td>1×1 conv. 10 BN ReLu</td></tr><tr><td colspan="2">global meanpool BN</td></tr><tr><td></td><td>fully connected BN</td></tr><tr><td colspan="2">10-way softmax</td></tr></table>
+
+0.5 based on validation results. For the preference learning experiment, we initially chose the semantic loss weight from $\{ 0 . 0 0 1 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 5 , 0 . 1 , 0 . 5 , 1 \}$ to be 0.1 based on validation, and then further tuned the weight to 0.25.
+
+# D SPECIFICATION OF COMPLEX CONSTRAINT MODELS
+
+Grids To compile our grid constraint, we first use Nishino et al. (2017) to generate a constraint for each source destination pair. Then, we conjoin each of these with indicators specifying which source and destination pair must be used, and finally we disjoin all of these together to form our constraint.
+
+To generate the data, we begin by randomly removing one third of edges. We then filter out connected components with fewer than 5 nodes to reduce degenerate cases, and proceed with randomly selecting pairs of point to create data points.
+
+The predictive model we employ as our baseline is a 5 layer MLP with 50 hidden sigmoid units per layer. It is trained using Adam Optimizer, with full data batches (Kingma & Ba, 2015). Early stopping with respect to validation loss is used as a regularizer.
+
+Preference Learning We split each user’s ordering into their ordering over sushis 1,2,3,5,7,8, which we use as the features, and their ordering over 4,6,9,10 which are the labels we predict. The constraint is compiled directly from logic, as this can be done in a straightforward manner for an n-item ordering.
+
+The predictive model we use here is a 3 layer MLP with 25 hidden sigmoid units per layer. It is trained using Adam Optimizer with full data batches (Kingma & Ba, 2015). Early stopping with respect to validation loss is used as a regularizer.

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+        "text": "This paper develops a novel methodology for using symbolic knowledge in deep learning. From first principles, we derive a semantic loss function that bridges between neural output vectors and logical constraints. This loss function captures how close the neural network is to satisfying the constraints on its output. An experimental evaluation shows that our semantic loss function effectively guides the learner to achieve (near-)state-of-the-art results on semi-supervised multi-class classification. Moreover, it significantly increases the ability of the neural network to predict structured objects, such as rankings and paths. These discrete concepts are tremendously difficult to learn, and benefit from a tight integration of deep learning and symbolic reasoning methods. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The widespread success of representation learning raises the question of which AI tasks are amenable to deep learning, which require classical model-based symbolic reasoning, and whether we can benefit from an integration of both. In recent years, significant effort has gone towards various ways of using representation learning to solve tasks that were previously tackled by symbolic methods. Such efforts include neural computers, Turing machines, and differentiable programming (e.g., Weston et al. (2014); Reed & De Freitas (2015); Graves et al. (2016); Riedel et al. (2016)), relational embeddings, deep learning for graph data, and neural theorem proving (e.g., Bordes et al. (2013); Neelakantan et al. (2015); Duvenaud et al. (2015); Niepert et al. (2016)), and many more. Other work has sought to augment deep learning with (symbolic) knowledge (e.g., Hu et al. (2016); Stewart & Ermon (2017); Marquez-Neila et al. (2017); Minervini et al. (2017); Wang et al. (2017)). ´ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This paper considers learning tasks where we have symbolic knowledge connecting the different outputs of a neural network. This knowledge takes the form of a constraint (or sentence) in Boolean logic. It can be as simple as an exactly-one constraint for one-hot output encodings, or as complex as a structured output prediction constraint for intricate combinatorial objects such as rankings, subgraphs, and paths. Our goal is to augment neural networks with the ability to learn how to make predictions subject to these constraints, and use the symbolic knowledge to improve its performance. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Most neuro-symbolic approaches aim to simulate or learn symbolic reasoning in an end-to-end deep neural network, or capture symbolic knowledge in a vector-space embedding. This choice is partly motivated by the need for smooth differentiable models; adding symbolic reasoning code (e.g., SAT solvers) to a deep learning pipeline destroys this property. Unfortunately, while making reasoning differentiable, the precise logical meaning of the knowledge is often lost. In this paper, we take a distinctly different approach, and tackle the problem of differentiable but sound logical reasoning from first principles. Starting from a set of intuitive axioms, we derive a differentiable semantic loss function that captures how well the outputs of a neural network match a given constraint. This function precisely captures the meaning of the constraint, and is independent of its syntax. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, we show how this semantic loss gives significant practical improvements in semi-supervised classification. The semantic loss defined over the exactly-one constraint in this setting permits us to obtain a learning signal from vast amounts of unlabeled data. The key idea is that the semantic loss helps us improve how consistently we are able to classify the unlabeled data. This simple addition to the loss function of standard deep learning architectures yields (near-)state-of-the-art performance in semi-supervised classification on MNIST, FASHION and CIFAR-10 datasets. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/908b06431175243b10447e79ffaca6771156931dd43e1b26edd7a157a8c2ba04.jpg",
+        "image_caption": [
+            "Figure 1: Outputs of a neural network feed into semantic loss functions for constraints representing a one-hot encoding, a total ranking of preferences, and paths in a grid graph. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our final set of experiments study the benefits of the semantic loss function for complex structured output learning tasks, such as preference learning and path prediction in a graph (Daume et al., 2009; ´ Chang et al., 2013; Choi et al., 2015; Graves et al., 2016). In these scenarios, the task is two-fold: learn both the structure of the output space, and the actual classification function within that space. By capturing the structure of the output space with logical constraints, and minimizing the semantic loss for this constraint during learning, we are able to learn networks that are much more likely to correctly predict structured objects. ",
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+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND AND NOTATION",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To formally define semantic loss, we make use of concepts in propositional logic. We write uppercase letters $( X , Y )$ for Boolean variables and lowercase letters $( x , y )$ for their instantiation $X = 0$ or $X = 1$ ). Sets of variables are written in bold uppercase $( \\mathbf { X } , \\mathbf { Y } )$ , and their joint instantiation in bold lowercase $\\mathbf { \\Gamma } ( \\mathbf { x } , \\mathbf { y } )$ . A literal is a variable $( x )$ or its negation $( \\neg x )$ . A logical sentence ( $\\alpha$ or $\\beta$ ) is constructed in the usual way, from variables and logical connectives $( \\wedge , \\vee$ , etc.), and is also called a formula or constraint. A state or world $\\mathbf { x }$ is an instantiation to all variables $\\mathbf { X }$ . A state $\\mathbf { x }$ satisfies a sentence $\\alpha$ , denoted $\\mathbf { x } \\Vdash \\alpha$ , if the sentence evaluates to be true in that world, as defined in the usual way. A sentence $\\alpha$ entails another sentence $\\beta$ , denoted $\\alpha \\models \\beta$ if all worlds that satisfy $\\alpha$ also satisfy $\\beta$ . A sentence $\\alpha$ is logically equivalent to sentence $\\beta$ , denoted $\\alpha \\equiv \\beta$ , if both $\\alpha \\models \\beta$ and $\\beta \\models \\alpha$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The output row vector of a neural net is denoted $\\mathsf { p }$ . Each value in $\\mathsf { p }$ represents the probability of an output and falls in $[ 0 , 1 ]$ . We use both softmax and sigmoid units for our output activation functions. The notation for states $\\mathbf { x }$ is used to refer the an assignment, the logical sentence enforcing the assignment, or the binary vector capturing that same assignment, as these are all equivalent notions. ",
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+    {
+        "type": "text",
+        "text": "Figure 1 illustrates the three different concrete output constraints of varying difficulty that are studied in our experiments. First, we examine the exactly-one or one-hot constraint capturing the encoding used in multi-class classification. It states that for a set of indicators $\\mathbf { X } = \\{ X _ { 1 } , \\ldots , X _ { n } \\}$ , one and exactly one of those indicators must be true, with the rest being false. This is enforced through a logical constraint $\\alpha$ by conjoining sentences of the form $\\neg X _ { 1 } \\lor \\neg X _ { 2 }$ for all pairs of variables (at most one variable is true), and a single sentence $X _ { 1 } \\lor \\cdots \\lor X _ { n }$ (at least one variable is true). Our experiments further examine the valid simple path constraint. It states for a given source-destination pair and edge indicators, that the edge indicators which are set to true must form a valid simple path from source to destination. Finally, we explore the ordering constraint, which requires that a set of $n ^ { 2 }$ indicator variables represent a total ordering over $n$ variables, effectively encoding a permutation matrix. For a full description of the path and ordering constraints, we refer to Section 5. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 SEMANTIC LOSS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our goal in this section is to find a semantic loss function that bridges the gap between the continuous world of neural networks, and the symbolic world of propositional logic. We do so by first postulating intuitive high-level properties that we seek in such a function, and that illustrate its desired behavior. A second set of postulates establish a correspondence between constraints and data. Finally, we uniquely define the semantic loss function used throughout this paper. ",
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+    },
+    {
+        "type": "text",
+        "text": "The semantic loss $\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , \\mathsf { p } )$ is a function of a sentence $\\alpha$ in propositional logic, defined over variables $\\mathbf { X } = \\{ X _ { 1 } , \\ldots , X _ { n } \\}$ , and a vector of probabilities $\\mathsf { p }$ for the same variables X. Element ${ \\mathsf p } _ { i }$ denotes the predicted probability of variable $X _ { i }$ , and corresponds to a single output of the neural net. For example, the semantic loss between the one-hot constraint from the previous section, and a neural net output vector $\\mathsf { p }$ , is intended to capture how close the prediction p is to having exactly one output set to true (that is, 1), and all others set to false (that is, 0), regardless of which output is correct. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 HIGH-LEVEL PROPERTIES",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The first axiom says that there is no loss when the logical constraint $\\alpha$ is always true (it is a logical tautology), independent of the predicted probabilities p. ",
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+    {
+        "type": "text",
+        "text": "Axiom 1 (Truth). The semantic loss of a true sentence is zero: $\\forall \\mathsf { p } , \\mathrm { L } ^ { \\mathrm { s } } ( t r u e , \\mathsf { p } ) = 0 .$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Next, when enforcing two constraints on disjoint sets of variables, we want the ability to compute the semantic loss of the two constraints separately, and sum the results for their joint semantic loss. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Axiom 2 (Additive Independence). Let $\\alpha$ be a sentence over $\\mathbf { X }$ with probabilities $\\mathsf { p }$ . Let $\\beta$ be a sentence over $\\mathbf { Y }$ disjoint from $\\mathbf { X }$ with probabilities $\\mathsf { q }$ . The semantic loss between sentence $\\alpha \\wedge \\beta$ and the joint probability vector [p q] decomposes additively: $\\operatorname { L } ^ { \\mathrm { s } } ( \\alpha \\wedge \\beta , [ { \\mathsf { p q } } ] ) = \\operatorname { L } ^ { \\mathrm { s } } ( \\alpha , { \\mathsf { p } } ) + \\operatorname { L } ^ { \\mathrm { s } } ( \\beta , { \\mathsf { q } } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "It directly follows from Axioms 1 and 2 that the probabilities of variables that are not used on the constraint do not affect the semantic loss. Proposition 6 in Appendix A formalizes this intuition. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To maintain logical meaning, we postulate that semantic loss is monotone in the order of implication. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Axiom 3 (Monotonicity). If $\\alpha \\models \\beta$ , then the semantic loss $\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , { \\mathfrak { p } } ) \\geq \\mathrm { L } ^ { \\mathrm { s } } ( \\beta , { \\mathfrak { p } } )$ for any vector $\\mathsf { p }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, as we add stricter requirements to the logical constraint, going from $\\beta$ to $\\alpha$ and making it harder to satisfy, the semantic loss cannot decrease. For example, when $\\beta$ enforces the output of an neural network to encode a subtree of a graph, and we tighten that requirement in $\\alpha$ to be a path, the semantic loss cannot decrease. Every path is also a tree and any solution to $\\alpha$ is a solution to $\\beta$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A first consequence following the monotonicity axiom is that logically equivalent sentences must incur an identical semantic loss for the same probability vector p. Hence, the semantic loss is indeed a semantic property of the logical sentence, and does not depend on the syntax of the sentence. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. If $\\alpha \\equiv \\beta$ , then the semantic loss $\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , { \\mathfrak { p } } ) = \\mathrm { L } ^ { \\mathrm { s } } ( \\beta , { \\mathfrak { p } } )$ for any vector p. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A second consequence is that semantic loss must be non-negative (see Proposition 5 in Appendix A). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 DATA-SENTENCE CORRESPONDENCE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A state $\\mathbf { x }$ is equivalently represented as a data vector, as well as a logical constraint that enforces a value for every variable in $\\mathbf { X }$ . When both the constraint and the predicted vector represent the same state (for example, $X _ { 1 } \\wedge { \\neg { X _ { 2 } } } \\wedge X _ { 3 }$ vs. [1 0 1]), there should be no semantic loss. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Axiom 4 (Identity). For any state $\\mathbf { x }$ , there is zero semantic loss between its representation as a sentence, and its representation as a deterministic vector: $\\forall \\mathbf { x } , \\mathrm { L } ^ { \\mathrm { s } } ( \\mathbf { x } , \\mathbf { x } ) = 0$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The axioms above together imply that any vector satisfying the constraint must incur zero loss. For example, when our constraint $\\alpha$ requires that the output vector encodes an arbitrary total ranking, and the vector $\\mathbf { x }$ correctly represents a single specific total ranking, there is no semantic loss. ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2 (Satisfaction). If $\\mathbf { \\dot { x } } \\Vdash \\alpha$ , then the semantic loss $\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , \\mathbf { x } ) = 0$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As a special case, logical literals $\\scriptstyle { \\dot { x } }$ or $\\neg x$ ) constrain a single variable to take on a single value, and thus play a role similar to the labels used in supervised learning. Such constraints require an even tighter correspondence: the semantic loss must act like a classical loss function (i.e., cross entropy). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Axiom 5 (Label-Literal Correspondence). The semantic loss of a single literal is proportionate to the cross-entropy loss for the equivalent data label: $\\mathrm { L } ^ { \\mathrm { s } } ( x , p ) \\propto - \\log ( p )$ and $\\mathrm { L } ^ { \\mathrm { s } } ( \\neg { x } , \\bar { p } ) \\overset { \\cdot } { \\propto } - \\log ( 1 - p )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Appendix A states Axioms 7 and 8, on the symmetry between values and the symmetry between variables, as well as a Lemma 7 that ties together the multiplicative constants mentioned in Axiom 5. Finally, this allows us to prove the following form of the semantic loss for a state $\\mathbf { x }$ . ",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3. For state x and vector p, we have $\\begin{array} { r } { \\mathrm { L } ^ { \\mathrm { s } } ( \\mathbf { x } , \\mathsf { p } ) \\propto - \\sum _ { i : \\mathbf { x } | = X _ { i } } \\log \\mathsf { p } _ { i } - \\sum _ { i : \\mathbf { x } | = \\lnot X _ { i } } \\log ( 1 - \\mathsf { p } _ { i } ) . } \\end{array}$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/5fcbfe708dc5ef9963d1119ec14ca3d17f5eace11a8e49df2a20fcaf583e6a6f.jpg",
+        "image_caption": [
+            "Figure 2: Binary classification toy example: a linear classifier without and with semantic loss. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 A GENERAL DEFINITION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Lemma 3 falls short as a full definition of semantic loss for arbitrary sentences. One can define additional axioms to pin down $\\mathrm { L } ^ { \\mathrm { s } }$ . For example, the following axiom is highly desirable. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Axiom 6 (Differentiability). For any fixed $\\alpha$ , the semantic loss $\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , \\mathsf { p } )$ is monotone in each probability in p, continuous and differentiable. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Appendix A makes the notion of semantic loss precise by stating one additional axiom. It is based on the observation that the state loss of Lemma 3 is proportionate to a log-probability. In particular, it corresponds to the probability of obtaining state $\\mathbf { x }$ after independently sampling each $X _ { i }$ with probability ${ \\mathsf { p } } _ { i }$ . We have now derived the semantic loss function from first principles as follows. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Semantic Loss). Let $\\mathsf { p }$ be a vector of probabilities, one for each variable in $\\mathbf { X }$ , and let $\\alpha$ be a sentence over $\\mathbf { X }$ . The semantic loss between $\\alpha$ and $\\mathsf { p }$ is ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5cdaa907f91cf65ce0836950e29b48878daf757f6c8fdd7e1bf203317f92f496.jpg",
+        "text": "$$\n\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , \\mathfrak { p } ) \\propto - \\log \\sum _ { \\mathbf { x } \\in \\alpha } \\ \\prod _ { i : \\mathbf { x } | = X _ { i } } \\mathfrak { p } _ { i } \\prod _ { i : \\mathbf { x } | = \\ l - X _ { i } } ( 1 - \\mathfrak { p } _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4 (Uniqueness). The semantic loss function in Definition 1 satisfies Axioms 1–9 and is the only function that does so, up to a multiplicative constant. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Intuitively, the semantic loss is proportionate to a negative logarithm of the probability of generating a state that satisfies the constraint, when sampling values according to p. Hence, it is the selfinformation (or “surprise”) of obtaining an assignment that satisfies the constraint (Jones, 1979). ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 SEMI-SUPERVISED CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The most straightforward constraint that is ubiquitous in multi-class classification is mutual exclusion over one-hot-encoded outputs. That is, for a given example, exactly one class and therefore exactly one binary indicator must be true. The machine learning community has made great strides in this machine learning task, due to the invention of assorted deep learning representations and their associated regularization terms (Krizhevsky et al., 2012; He et al., 2016). Many of these models take large amounts of fully labeled data for granted, and big data is indispensable for discovering accurate representations (Hastie et al., 2009). To sustain this progress, and alleviate the need for more labeled data, there is a growing interest into utilizing unlabeled data to augment the predictive power of classifiers (Stewart & Ermon, 2017; Bilenko et al., 2004). This section shows why semantic loss naturally qualifies for this task. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Illustrative Example To illustrate the benefit of semantic loss in the semi-supervised setting, we begin our discussion with a small toy example. Consider a binary classification task as depicted in Figure 2. Ignoring the unlabeled examples, a simple linear classifier learns to distinguish the two classes by separating the labeled examples in Figure 2a. However, the unlabeled examples are also informative, as they must carry some properties that give them a particular label. This is the crux of semantic loss: a model must confidently assign a consistent class even to unlabeled data. Encouraging the model to do so results in a more accurate decision boundary, as illustrated in Figure 2b. Next, we further explore this idea and apply it to real-world image classification tasks. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/382d2a4abe15b2139dd29370c328c0091c8c8c4c4c01dfa3666c36f7e07b0f84.jpg",
+        "table_caption": [
+            "Table 1: MNIST. Previously reported test accuracies followed by semantic loss results $\\pm$ stddev) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Accuracy % with # of used labels</td><td>100</td><td>1000</td><td>ALL</td></tr><tr><td>AtlasRBF (Pitelis et al., 2014) Deep Generative (Kingma et al., 2014) Virtual Adversarial (Miyato et al., 2016) Ladder Net (Rasmus et al., 2015)</td><td>91.9 (± 0.95) 96.67(± 0.14) 97.67 98.94(±0.37)</td><td>96.32 (± 0.12) 97.60(± 0.02) 98.64 99.16 (±0.08)</td><td>98.69 99.04 99.36 99.43 (± 0.02)</td></tr><tr><td>Baseline: MLP, Gaussian Noise Baseline: Self-Training MLP with Semantic Loss (our)</td><td>78.46 (±1.94) 72.55 (±4.21) 98.38 (±0.51)</td><td>94.26 (±0.31) 87.43 (±3.07) 98.78 (±0.17)</td><td>99.34(±0.08) 99.34 (±0.08) 99.36 (±0.02)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.1 METHOD ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our proposed method intends to be generally applicable and compatible with any feedforward neural network. The semantic loss is simply another regularization term that can directly be plugged into an existing loss function. More specifically, for some weight $w$ , the new overall loss becomes ",
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+        "type": "text",
+        "text": "When the constraint over the output space is simple (for example, there is a small number of solutions ${ \\textbf { x } } | = \\alpha )$ ), the semantic loss can be directly computed from Definition 1. Concretely, for the exactly-one constraint used in $n$ -class classification, the semantic loss reduces to ",
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+        "type": "equation",
+        "img_path": "images/353f7d32bbc0b6553e939f45db8e9abcc0faed8d6e0a32bd8bee3c093b1cac7f.jpg",
+        "text": "$$\n\\mathrm { L } ^ { \\mathrm { s } } ( \\mathrm { e x a c t l y - o n e } , \\mathsf { p } ) \\propto - \\log \\sum _ { i = 0 } ^ { n - 1 } \\mathsf { p } _ { i } \\prod _ { j = 0 , j \\neq i } ^ { n - 1 } ( 1 - \\mathsf { p } _ { j } ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where he values ${ \\mathsf { p } } _ { i }$ denote the probability of class $i$ as predicted by the neural net. The semantic loss for the exactly-one constraint is efficient and causes no noticeable overhead in our experiments. ",
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+        "text": "In general, for arbitrary constraints $\\alpha$ , the semantic loss is not efficient to compute using Definition 1, and more advanced automated reasoning is required. Section 5 discusses this issue in more detail. ",
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+        "text": "4.2 EXPERIMENTAL EVALUATION ",
+        "text_level": 1,
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+        "text": "In this section, we evaluate semantic loss in the semi-supervised setting by comparing it with several competitive models. As most semi-supervised learners build on a supervised learner, changing the underlying model significantly affects the semi-supervised learner’s performance. For comparison, we add semantic loss to the same base models used in ladder nets (Rasmus et al., 2015), which currently achieve state-of-the-art results on semi-supervised MNIST and CIFAR-10 (Krizhevsky & Hinton, 2009). Specifically, the MNIST base model is a fully-connected multilayer perceptron (MLP), with layers of size 784-1000-500-250-250-250-10. On CIFAR-10, it is a 10-layer convolutional neural network (CNN) with 3-by-3 padded filters. After every 3 layers, features are subject to a 2-by-2 max-pool layer with strides of 2. Furthermore, we use ReLu (Nair & Hinton, 2010), batch normalization (Ioffe & Szegedy, 2015), and Adam optimization (Kingma & Ba, 2015) with a learning rate of 0.002. We refer to Appendix B and C for a specification of the CNN model and additional details about hyper-parameter tuning. ",
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+        "text": "For all semi-supervised experiments, we use the standard 10,000 held-out test examples provided in the original datasets and randomly pick 10,000 from the standard 60,000 training examples (50,000 for CIFAR-10) as validation set. For values of $N$ that depend on the experiment, we retain $N$ randomly chosen labeled examples from the training set, and remove labels from the rest. We balance classes in the labeled samples to ensure no particular class is over-represented. Images are preprocessed for standardization and Gaussian noise (standard deviation 0.3) is added to every pixel. ",
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+        "type": "text",
+        "text": "MNIST The permutation invariant MNIST classification task is commonly used as a test-bed for general semi-supervised learning algorithms. This setting does not use any prior information about the spatial arrangement of the input pixels. Therefore, it excludes many data augmentation techniques that involve geometric distortion of images, as well as convolutional neural networks. ",
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+        "type": "text",
+        "text": "When evaluating on MNIST, we run experiments for 10 epochs, with a batch size of 10 labeled and 10 unlabeled examples. Experiments are repeated 10 times with different random seeds. Table 1 compares semantic loss to two baselines and state-of-the-art results from the literature. The first baseline is a purely supervised MLP, which makes no use of unlabeled data. The second is the classic self-training method for semi-supervised learning, which operates as follows. After every 1000 iterations, the unlabeled examples that are predicted by the MLP to have more than $9 5 \\%$ probability of belonging to a single class, are assigned a psuedo-label and become labeled data. ",
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+        "table_caption": [
+            "Table 2: FASHION. Test accuracy comparison between MLP with semantic loss and ladder nets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Accuracy % with # of used labels</td><td>100</td><td>500</td><td>1000</td><td>ALL</td></tr><tr><td>Ladder Net (Rasmus et al., 2015)</td><td>81.46(±0.64)</td><td>85.18 (±0.27)</td><td>86.48 (± 0.15)</td><td>90.46</td></tr><tr><td>Baseline:MLP,GaussianNoise</td><td>69.45 (±2.03)</td><td>78.12 (±1.41)</td><td>80.94(±0.84)</td><td>89.87</td></tr><tr><td>MLP with Semantic Loss (our)</td><td>86.74 (±0.71)</td><td>89.49 (±0.24)</td><td>89.67 (±0.09)</td><td>89.81</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [
+            "Table 3: CIFAR. Test accuracy comparison between CNN with semantic loss and ladder nets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Accuracy % with # of used labels</td><td rowspan=1 colspan=1>4000</td><td rowspan=1 colspan=1>ALL</td></tr><tr><td rowspan=2 colspan=1>CNNBaselineinLadderNetLadder Net (Rasmus et al., 2015)Baseline: CNN,Whitening, CroppingCNN with Semantic Loss (our)</td><td rowspan=1 colspan=1>76.67 (± 0.61)79.60 (±0.47)</td><td rowspan=1 colspan=1>90.73</td></tr><tr><td rowspan=1 colspan=1>77.1381.79</td><td rowspan=1 colspan=1>90.9690.92</td></tr></table>",
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+        "type": "text",
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+        "type": "text",
+        "text": "When given 100 labeled examples ( $N = 1 0 0 \\mathrm { \\ : }$ ), MLP with semantic loss gains around $2 0 \\%$ improvement over the purely supervised baseline. The improvement is even larger $( 2 5 \\% )$ compared to self-training. Considering the only change is an additional loss term, this result is very encouraging. Compared to the state of the art, ladder nets slightly outperform semantic loss by $0 . 5 \\%$ accuracy. This difference may be an artifact of the excessive tuning of architectures, hyper-parameters and learning rates that the MNIST dataset has been subject to. In the coming experiments, we extend our work to more challenging datasets, in order to provide a clearer comparison with ladder nets. ",
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+        "type": "text",
+        "text": "First, we want to share a few more thoughts on how semantic loss works. A classical softmax layer interprets its output as representing a categorical distribution. Hence, by normalizing its outputs, softmax enforces the same mutual exclusion constraint enforced in our semantic loss function. However, there does not exist a natural way to extend softmax loss to unlabeled samples. In contrast, semantic loss does provide a learning signal on unlabeled samples, by forcing the underlying classifier to make an decision and construct a confident hypothesis for all data. However, for the fully supervised case $N = \\mathrm { a l l }$ ), the semantic loss does not significantly affect accuracy. Because the MLP has enough capacity to almost perfectly fit the training data, where the constraint is always satisfied, the semantic loss is almost always zero. This is a direct consequence of Proposition 2. ",
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+    {
+        "type": "text",
+        "text": "FASHION The FASHION (Xiao et al., 2017) dataset consists of Zalando’s article images, aiming to serve as a more challenging drop-in replacement for MNIST. Arguably, it has not been overused and requires more advanced techniques to achieve good performance. As in the previous experiment, we run our method for 10 epochs, whereas ladder nets need 100 epochs to converge. Again, experiments are repeated 10 times and Table 2 reports the classification accuracy and its standard deviation (except for $N =$ all where it is close to 0 and omitted for space). ",
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+        "type": "text",
+        "text": "Experiments show that utilizing semantic loss results in a very large $1 7 \\%$ improvement over the baseline when only 100 labels are provided. Moreover, our method compares favorably to ladder nets, except when the setting degrades to be fully supervised. Note that our method already nearly reaches its maximum accuracy with 500 labeled examples, which is only $1 \\%$ of the training dataset. ",
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+    {
+        "type": "text",
+        "text": "CIFAR-10 To show the general applicability of semantic loss, we evaluate it on CIFAR-10. This dataset consisting of 32-by-32 RGB images in 10 classes. A simple MLP would not have enough representation power to capture the huge variance across objects within the same class. To cope with this spike in difficulty, we switch our underlying model to a 10-layer CNN as described earlier. We use a batch size of 100 samples of which half are unlabeled. Experiments are run for 100 epochs. However, due to our limited computational resources, we report on a single trial. Note that we make slight modifications to the underlying model used in ladder nets to reproduce similar baseline performance. Please refer to Appendix B for the details of this experimental setup. ",
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+    {
+        "type": "text",
+        "text": "As shown in Table 3, our method compares favorably to ladder nets. However, due to the slight difference in performance between the supervised base models, a direct comparison would be methodologically flawed. Instead, we compare the net improvements over baselines. In terms of this measure, our method scores a gain of $4 . { \\bar { 6 } } 6 \\%$ whereas ladder nets gain $2 . 9 3 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "4.3 DISCUSSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Overall, the experiments so far have demonstrated the competitiveness and general applicability of our proposed method on semi-supervised learning tasks. It surpassed the previous state of the art (i.e. ladder nets ) on FASHION and CIFAR-10, while being close on MNIST. Considering the simplicity of our method, such results are encouraging. Indeed, a key advantage of semantic loss is that it only requires a simple additional loss term. Without changing the network architecture itself, we incur almost no computational overhead. Conversely, this property makes our method sensitive to the underlying model’s performance. Without the underlying predictive power of a strong supervised learning model, we do not expect to see the same benefits we observe here. Recently, we became aware that Miyato et al. (2016) extended their work to CIFAR-10 and achieved state-of-the-art results (Miyato et al., 2017), surpassing our performance by $5 \\%$ . In future work, we plan to investigate whether applying semantic loss on their architecture would yield an even stronger performance. ",
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+    {
+        "type": "image",
+        "img_path": "images/684ff625c16715167670292375e1562c7bf4528618063e9e1026bc8e042c3a58.jpg",
+        "image_caption": [
+            "Figure 3: FASHION pictures grouped by how confidently the supervised base model classifies them correctly. With semantic loss, the final semi-supervised model predicts all correctly and confidently. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figure 3 illustrates the effect of semantic loss on FASHION pictures whose correct label was hidden from the learner. Pictures 3a and 3b are correctly classified by the supervised base model, and on the first set it is confident about this prediction $( { \\mathsf p } _ { i } > 0 . 8 )$ . The semantic loss rarely diverts the model from these initially correct labels. However, it bootstraps these unlabeled examples to achieve higher confidence in the learned concepts. With this additional learning signal, the model changes its beliefs about Pictures 3c, which it was previously uncertain about. Finally, even on confidently misclassified Pictures 3d, the semantic loss is able to fully correct the mistakes of the base model. ",
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+    {
+        "type": "text",
+        "text": "5 LEARNING WITH COMPLEX CONSTRAINTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "While much of current machine learning research is focused on problems such as multi-class classification, there remain a multitude of difficult problems involving highly constrained output domains. As mentioned in the previous section, semantic loss has little effect on the fully-supervised exactly-one classification problem. This leads us to seek out more difficult problems to illustrate that semantic loss can also be highly informative in the supervised case, provided the output domain is a sufficiently complex space. Because semantic loss is defined by a Boolean formula, it can be used on any output domain that can be fully described in this manner. Here, we develop a framework for tractable semantic loss on highly complex constraints, and evaluate it on some difficult examples. ",
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+    {
+        "type": "text",
+        "text": "5.1 A TRACTABLE SEMANTIC LOSS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our goal here is to develop a method for computing both the semantic loss and its gradient in a tractable manner. Examining Definition 1 of semantic loss, we see that the right-hand side is a wellknown automated reasoning task called weighted model counting (WMC) (Chavira & Darwiche, 2008; Sang et al., 2005). A key property of WMC is that its partial derivatives can be computed in terms of other, slightly modified WMCs. Furthermore, we know of circuit languages that compute WMCs, and that are amenable to backpropagation (Darwiche, 2003). We use the language and circuit compilation techniques described in Darwiche (2011) to build a Boolean circuit representing our semantic loss. We refer to the literature for details of this compilation approach. Due to certain properties of this circuit form, we can use it to compute both the values and the gradients of the semantic loss in time linear in the size of the circuit (Darwiche & Marquis, 2002). Once we have constructed this function, we can add it to our standard loss function as described in Section 4.1. ",
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+        "type": "table",
+        "img_path": "images/ba42d7903b36e4520dcf1d5821e29357a1cf1180cc8d69b4a291aa60a57545f5.jpg",
+        "table_caption": [
+            "Table 4: Grid shortest path test results: coherent, incoherent and constraint accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Test accuracy %</td><td rowspan=1 colspan=1>Coherent</td><td rowspan=1 colspan=1>Incoherent</td><td rowspan=1 colspan=1>Constraint</td></tr><tr><td rowspan=1 colspan=1>5-layer MLP</td><td rowspan=1 colspan=1>5.62</td><td rowspan=1 colspan=1>85.91</td><td rowspan=1 colspan=1>6.99</td></tr><tr><td rowspan=1 colspan=1>With semantic loss (our)</td><td rowspan=1 colspan=1>28.51</td><td rowspan=1 colspan=1>83.14</td><td rowspan=1 colspan=1>69.89</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/fc01d806839dcd7d00573563b65a339ed59db17e31ff3c1aafedf315b18dc1ca.jpg",
+        "table_caption": [
+            "Table 5: Preference prediction test results: coherent, incoherent and constraint accuracy. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Test accuracy %</td><td rowspan=1 colspan=1>Coherent</td><td rowspan=1 colspan=1>Incoherent</td><td rowspan=1 colspan=1>Constraint</td></tr><tr><td rowspan=1 colspan=1>3-layer MLP</td><td rowspan=1 colspan=1>1.01</td><td rowspan=1 colspan=1>75.78</td><td rowspan=1 colspan=1>2.72</td></tr><tr><td rowspan=1 colspan=1>With semantic loss (our)</td><td rowspan=1 colspan=1>13.59</td><td rowspan=1 colspan=1>72.43</td><td rowspan=1 colspan=1>55.28</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.2 EXPERIMENTAL EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "Our ambition when evaluating semantic loss’ performance on complex constraints is not to achieve state-of-the-art performance on any particular problem, but rather to highlight its effect. To this end, we evaluate our method on problems with a difficult output space, where the model could no longer be fit directly from data, and purposefully use simple MLPs for evaluation. The details of hyper-parameter tuning are again given in Appendix C. ",
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+    {
+        "type": "text",
+        "text": "5.2.1 GRIDS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We begin with a classic algorithmic problem, finding the shortest path in a graph. Specifically, we use a 4-by-4 grid $G = ( V , E )$ with uniform edge weights. We randomly remove edges for each example to increase difficulty. Formally, our input is a binary vector of length $\\vert V \\vert + \\vert E \\vert$ , with the first $| V |$ variables indicating sources and destinations, and the next $| E |$ which edges are removed. Similarly, each label is a binary vector of length $| E |$ indicating which edges are in the shortest path. Finally, we require through our constraint $\\alpha$ that the output form a valid simple path between the desired source and destination. To compile this constraint, we use the method of Nishino et al. (2017) to encode pairwise simple paths, and logically merge them to enforce the correct source and destination. For more details on the constraint and data generation process, see Appendix D. ",
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+    {
+        "type": "text",
+        "text": "To evaluate, we use our generated dataset of 1600 examples, with a 60/20/20 train/validation/test split. Table 4 compares test accuracy between a 5-layer MLP baseline, and the same model augmented with semantic loss. We report three different accuracies that illustrate the effect of semantic loss: “Coherent” indicates the percentage of examples for which the classifier gets the entire configuration right, while “Incoherent” measures the percentage of individually correct binary labels, which as a whole may not constitute a valid path at all. Finally, “Constraint” describes the percentage of predictions given by the model that satisfy the constraint associated with the problem. In the case of incoherent accuracy, semantic loss has little effect, and in fact slightly reduces the accuracy as it combats the standard sigmoid cross entropy. In regard to coherent accuracy, however, the semantic loss has a very large effect in guiding the network to jointly learn true paths, rather than optimizing each binary output individually. We further see this by observing the large increase in the percentage of predictions which really are paths between the desired nodes in the graph. ",
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+    {
+        "type": "text",
+        "text": "5.2.2 PREFERENCE LEARNING ",
+        "text_level": 1,
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "The next problem we examine is that of predicting a complete order of preferences. That is, for a given set of user features, we would like to predict how the user would rank their preference over a fixed set of items. We encode a preference ordering over $n$ items as a flattened binary matrix $\\{ X _ { i j } \\}$ , where for each $i , j \\in \\{ 1 , \\ldots , n \\}$ , $X _ { i j }$ denotes that item $i$ is at position $j$ (Choi et al., 2015). Clearly, not all configurations of outputs correspond to a valid ordering. ",
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+    {
+        "type": "text",
+        "text": "For data, we use preference rankings over 10 types of sushi for 5000 individuals, taken from PREFLIB (Mattei & Walsh, 2013). We take the ordering over 6 types of sushi as input features to predict the ordering over the remaining 4 types, with splits identical to those in Shen et al. (2017). We again split the data 60/20/20 into train/test/split, and employ a 3 layer MLP as our baseline. Table 5 compares the baseline to the same MLP augmented with semantic loss for valid total orderings. Again, we see that semantic loss has a marginal effect on incoherent accuracy, but massively improves the network’s ability to predict valid, correct orderings. Remarkably, without semantic loss, the network is only able to output a valid ordering on $0 . 0 1 \\%$ of the test examples. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Incorporating symbolic background knowledge into machine learning is a long-standing challenge (Srinivasan et al., 1995). It has received considerable attention for structured prediction in natural language processing, in both supervised and semi-supervised settings. For example, constrained conditional models extend linear models with constraints that are enforced through integer linear programming (Chang et al., 2008; 2013). Constraints have also been studied in the context of probabilistic graphical models (Mateescu & Dechter, 2008; Ganchev et al., 2010). Kisa et al. (2014) utilize a circuit language called the probabilistic sentential decision diagram to induce distributions over arbitrary logical formulas. They learn generative models that satisfy preference and path constraints (Choi et al., 2015; 2016), which we both study in a discriminative setting. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Various deep learning techniques have been proposed to enforce either arithmetic constraints (Pathak et al., 2015; Marquez-Neila et al., 2017) or logical constraints (Rockt ´ aschel et al., 2015; Hu et al., ¨ 2016; Demeester et al., 2016; Stewart & Ermon, 2017; Minervini et al., 2017; Diligenti et al., 2017; Donadello et al., 2017) on the output of a neural network. The common approach is to reduce logical constraints into differentiable arithmetic objectives by replacing logical operators with their fuzzy t-norms and logical implications with simple inequalities. A downside of this fuzzy relaxation is that the logical sentences lose their precise meaning. The learning objective becomes a function of the syntax rather than the semantics. Moreover, these relaxations are often only applied to Horn clauses. One alternative is to encode the logic into a factor graph and perform loopy belief propagation to compute a loss function (Naradowsky & Riedel, 2017), which is known to have issues in the presence of complex logical constraints (Smith & Gogate, 2014). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Several specialized techniques have been proposed to exploit the rich structure of real world labels. Deng et al. (2014) propose hierarchy and exclusion graphs that allow a flexible joint modeling of hierarchical categories. It is a method invented to address examples whose labels are not provided at the most specific level. Finally, the objective of semantic loss to increase the confidence of predictions on unlabeled data is in common with information-theoretic approaches to semi-supervised learning (Grandvalet & Bengio, 2005; Erkan & Altun, 2010), and approaches that increase robustness to output perturbation (Miyato et al., 2016). A key difference between semantic loss and these information-theoretic losses is that semantic loss generalizes to arbitrary output constraints. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Both reasoning and semi-supervised learning are often identified as key challenges for deep learning going forward. In this paper, we developed a principled way of combining automated reasoning for propositional logic with existing deep learning architectures. Moreover, we showed that our semantic loss function provides significant benefits during semi-supervised classification, as well as deep structured prediction for highly complex output spaces. ",
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+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A AXIOMATIZATION OF SEMANTIC LOSS: DETAILS ",
+        "bbox": [
+            173,
+            664,
+            617,
+            681
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "This appendix provides further details on our axiomatization of the semantic loss.   \nProposition 5 (Non-Negativity). Semantic loss is non-negative. ",
+        "bbox": [
+            173,
+            695,
+            709,
+            729
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. Because $\\alpha \\models$ true for all $\\alpha$ , the monotonicity axiom implies that $\\forall \\mathsf { p } , \\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , \\mathsf { p } ) \\geq \\mathrm { L } ^ { \\mathrm { s } } ( t r u e , \\mathsf { p } )$ . By the truth axiom, $\\mathrm { L } ^ { \\mathrm { s } } ( t r u e , \\mathsf { p } ) = 0$ , and therefore $\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , { \\mathsf { p } } ) \\geq 0$ for all choices of $\\alpha$ and $\\mathsf { p }$ . □ ",
+        "bbox": [
+            173,
+            743,
+            823,
+            773
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proposition 6 (Locality). Let α be a sentence over $\\mathbf { X }$ with probabilities p. For any $\\mathbf { Y }$ disjoint from $\\mathbf { X }$ with probabilities q, the semantic loss $\\operatorname { L } ^ { \\mathrm { s } } ( \\alpha , [ \\mathsf { p q } ] ) = \\operatorname { L } ^ { \\mathrm { s } } ( \\alpha , \\mathsf { p } )$ . ",
+        "bbox": [
+            171,
+            780,
+            823,
+            809
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof. Follows from the additive independence and truth axioms. Set $\\beta \\ = \\ t r u e$ in the additive independence axiom, and observe that this sets $\\mathrm { L } ^ { \\mathrm { s } } ( \\beta , { \\mathsf { q } } ) = 0$ because of the truth axiom. □ ",
+        "bbox": [
+            173,
+            824,
+            825,
+            853
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Proof of Proposition 2. The monotonicity axiom specializes to say that if ${ \\textbf { x } } | = \\alpha$ , we have that $\\forall \\mathsf { p } , \\mathrm { L } ^ { \\mathrm { s } } ( \\mathbf { x } , \\mathsf { p } ) \\ \\geq \\ \\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , \\mathsf { p } )$ . By choosing $\\mathsf { p }$ to be $\\mathbf { x }$ , this implies $\\mathrm { L } ^ { \\mathrm { s } } ( \\mathbf { x } , \\mathbf { x } ) \\ \\geq \\ \\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , \\mathbf { x } )$ . From the identity axiom, $\\mathrm { L } ^ { \\mathrm { s } } ( { \\bf x } , { \\bf x } ) = 0$ , and therefore $0 \\geq \\operatorname { L } ^ { \\mathrm { s } } ( \\alpha , \\mathbf { x } )$ . Proposition 5 bounds the loss from below as $\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , \\mathbf { x } ) \\geq 0$ . □ ",
+        "bbox": [
+            174,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Axiom 7 (Value Symmetry). For all $\\mathsf { p }$ and $\\alpha$ , we have that $\\mathrm { L } ^ { \\mathrm { s } } ( \\alpha , { \\mathfrak { p } } ) = \\mathrm { L } ^ { \\mathrm { s } } ( { \\bar { \\alpha } } , 1 - { \\mathfrak { p } } )$ where $\\bar { \\alpha }$ replaces every variable in $\\alpha$ by its negation. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Axiom 8 (Variable Symmetry). Let $\\alpha$ be a sentence over $\\mathbf { X }$ with probabilities $\\mathsf { p }$ . Let $\\pi$ be a permutation of the variables $\\mathbf { X }$ , let $\\pi ( \\alpha )$ be the sentence obtained by replacing variables $x$ by $\\pi ( x )$ , and let $\\pi ( { \\mathfrak { p } } )$ be the corresponding permuted vector of probabilities. Then, $\\operatorname { L } ^ { \\mathrm { s } } ( \\alpha , { \\mathsf { p } } ) = \\operatorname { L } ^ { \\mathrm { s } } ( \\pi ( \\alpha ) , \\pi ( { \\mathsf { p } } ) )$ . ",
+        "bbox": [
+            174,
+            137,
+            825,
+            179
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The value and variable symmetry axioms together imply the equality of the multiplicative constants in the label-literal duality axiom for all literals. ",
+        "bbox": [
+            174,
+            185,
+            823,
+            214
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Lemma 7. There exists a single constant $K$ such that $\\operatorname { L } ^ { \\mathrm { s } } ( X , p ) = - K \\log ( p )$ and $\\operatorname { L } ^ { s } ( \\lnot X , p ) =$ $- K \\log ( 1 - p )$ for any literal $x$ . ",
+        "bbox": [
+            171,
+            218,
+            823,
+            247
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. Value symmetry implies that $\\operatorname { L } ^ { \\mathrm { s } } ( X _ { i } , \\mathfrak { p } ) = \\operatorname { L } ^ { \\mathrm { s } } ( \\lnot X _ { i } , 1 - \\mathfrak { p } )$ . Using label-literal correspondence, this implies $K _ { 1 } \\log ( p _ { i } ) = K _ { 2 } \\log ( 1 - ( 1 - p _ { i } ) )$ for the multiplicative constants $K _ { 1 }$ and $K _ { 2 }$ that are left unspecified by that axiom. This implies that the constants are identical. A similar argument based on variable symmetry proves equality between the multiplicative constants for different $i$ . ",
+        "bbox": [
+            174,
+            265,
+            825,
+            321
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof of Lemma 3. A state $\\mathbf { x }$ is a conjunction of independent literals, and therefore subject to the additive independence axiom. Each literal’s loss in this sum is defined by Lemma 7. □ ",
+        "bbox": [
+            173,
+            340,
+            823,
+            368
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The following and final axiom requires that the semantic loss is proportionate to the logarithm of a function that is additive for mutually exclusive sentences. ",
+        "bbox": [
+            174,
+            387,
+            823,
+            415
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Axiom 9 (Exponential Additivity). Let $\\alpha$ and $\\beta$ be mutually exclusive sentences (i.e., $\\alpha \\wedge \\beta$ is unsatisfiable), and let $f ^ { s } ( K , \\alpha , { \\mathsf { p } } ) = K ^ { - \\mathtt { L } ^ { \\mathtt { s } } ( \\alpha , { \\mathsf { p } } ) }$ . Then, there exists a positive constant $K$ such that $f ^ { s } ( K , \\alpha \\vee \\beta , \\mathsf { p } ) = f ^ { s } ( K , \\alpha , \\mathsf { p } ) + f ^ { s } ( K , \\beta , \\mathsf { p } )$ . ",
+        "bbox": [
+            174,
+            420,
+            825,
+            464
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof of Theorem 4. The truth axiom states that $\\forall \\mathsf { p }$ $, f ^ { s } ( K , t r u e , \\mathfrak { p } ) = 1$ for all positive constants $K$ . This is the first Kolmogorov axiom of probability. The second Kolmogorov axiom for $f ^ { s } ( K , . , { \\mathsf { p } } )$ follows from the additive independence axiom of semantic loss. The third Kolmogorov axiom (for the finite discrete case) is given by the exponential additivity axiom of semantic loss. Hence, $f ^ { s } ( K , . , { \\mathsf { p } } )$ is a probability distribution for some choice of $K$ , which implies the definition up to a multiplicative constant. □ ",
+        "bbox": [
+            174,
+            481,
+            825,
+            565
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B SPECIFICATION OF THE CONVOLUTIONAL NEURAL NETWORK MODEL ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            587,
+            792,
+            603
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 6 shows the slight architectural difference between the CNN used in ladder nets and ours. The major difference lies in the choice of ReLu. Note we add standard padded cropping to preprocess images and an additional fully connected layer at the end of the model, neither is used in ladder nets. We only make those slight modification so that the baseline performance reported by Rasmus et al. (2015) can be reproduced. ",
+        "bbox": [
+            174,
+            618,
+            825,
+            689
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C HYPER-PARAMETER TUNING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            709,
+            529,
+            726
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Validation sets are used for tuning the weight associated with semantic loss, the only hyperparameter that causes noticeable difference in performance for our method. For our semi-supervised classification experiments, we perform a grid search over $\\{ 0 . 0 0 1 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 5 , 0 . 1 \\}$ to find the optimal value. Empirically, 0.005 always gives the best or nearly the best results and we report its results on all experiments. ",
+        "bbox": [
+            174,
+            741,
+            825,
+            811
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the FASHION dataset specifically, because MNIST and FASHION share the same image size and structure, methods developed in MNIST should be able to directly perform on FASHION without heavy modifications. Because of this, we use the same hyper-parameters when evaluating our method. However, for the sake of fairness, we subject ladder nets to a small-scale parameter tuning in case its performance is more volatile. ",
+        "bbox": [
+            174,
+            818,
+            825,
+            888
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For the grids experiment, the only hyper parameter that needed to be tuned was again the weight given to semantic loss, which after trying $\\{ 0 . 0 0 1 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 5 , 0 . 1 , 0 . 5 , 1 \\}$ was selected to be ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/358350f5d93ca940af966331f2a65fafdad421d46e88d1ecac1a9dcd0bdc9a55.jpg",
+        "table_caption": [
+            "Table 6: Specifications of CNNs in Ladder Net and our proposed method. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CNNinLadderNet</td><td>CNN in this paper</td></tr><tr><td colspan=\"2\">Input 32×32 RGB image</td></tr><tr><td></td><td>Resizing to 36 × 36 with padding Cropping Back</td></tr><tr><td colspan=\"2\">Whitening Contrast Normalization</td></tr><tr><td colspan=\"2\">Gaussian Noise with std.of 0.3</td></tr><tr><td>3 ×3 conv. 96 BNLeakyReLU</td><td>3×3 conv. 96 BN ReLU</td></tr><tr><td>3×3 conv. 96 BNLeakyReLU</td><td>3 ×3 conv. 96 BN ReLU</td></tr><tr><td>3 ×3 conv. 96 BNLeakyReLU</td><td>3×3 conv. 96 BN ReLU</td></tr><tr><td colspan=\"2\">2×2 max-pooling stride 2 BN</td></tr><tr><td>3×3 conv.192BNLeakyReLU</td><td>3×3 conv.192 BN ReLU</td></tr><tr><td>3×3 conv.192 BNLeakyReLU</td><td>3×3 conv. 192 BN ReLU</td></tr><tr><td>3×3 conv.192 BN LeakyReLU</td><td>3×3 conv.192 BN ReLU</td></tr><tr><td colspan=\"2\">2×2 max-pooling stride 2 BN</td></tr><tr><td>3×3conv.192BNLeakyReLU</td><td>3×3conv.192 BNReLU</td></tr><tr><td>1×1 conv.192 BN LeakyReLU</td><td>3×3 conv. 192 BN ReLU</td></tr><tr><td>1×1conv.10 BNLeakyReLU</td><td>1×1 conv. 10 BN ReLu</td></tr><tr><td colspan=\"2\">global meanpool BN</td></tr><tr><td></td><td>fully connected BN</td></tr><tr><td colspan=\"2\">10-way softmax</td></tr></table>",
+        "bbox": [
+            282,
+            126,
+            714,
+            401
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "0.5 based on validation results. For the preference learning experiment, we initially chose the semantic loss weight from $\\{ 0 . 0 0 1 , 0 . 0 0 5 , 0 . 0 1 , 0 . 0 5 , 0 . 1 , 0 . 5 , 1 \\}$ to be 0.1 based on validation, and then further tuned the weight to 0.25. ",
+        "bbox": [
+            176,
+            428,
+            821,
+            469
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D SPECIFICATION OF COMPLEX CONSTRAINT MODELS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            489,
+            648,
+            506
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Grids To compile our grid constraint, we first use Nishino et al. (2017) to generate a constraint for each source destination pair. Then, we conjoin each of these with indicators specifying which source and destination pair must be used, and finally we disjoin all of these together to form our constraint. ",
+        "bbox": [
+            174,
+            521,
+            825,
+            577
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To generate the data, we begin by randomly removing one third of edges. We then filter out connected components with fewer than 5 nodes to reduce degenerate cases, and proceed with randomly selecting pairs of point to create data points. ",
+        "bbox": [
+            174,
+            584,
+            825,
+            627
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The predictive model we employ as our baseline is a 5 layer MLP with 50 hidden sigmoid units per layer. It is trained using Adam Optimizer, with full data batches (Kingma & Ba, 2015). Early stopping with respect to validation loss is used as a regularizer. ",
+        "bbox": [
+            174,
+            633,
+            823,
+            675
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Preference Learning We split each user’s ordering into their ordering over sushis 1,2,3,5,7,8, which we use as the features, and their ordering over 4,6,9,10 which are the labels we predict. The constraint is compiled directly from logic, as this can be done in a straightforward manner for an n-item ordering. ",
+        "bbox": [
+            174,
+            690,
+            825,
+            747
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The predictive model we use here is a 3 layer MLP with 25 hidden sigmoid units per layer. It is trained using Adam Optimizer with full data batches (Kingma & Ba, 2015). Early stopping with respect to validation loss is used as a regularizer. ",
+        "bbox": [
+            176,
+            753,
+            823,
+            796
+        ],
+        "page_idx": 13
+    }
+]

parse/train/K9uApq7iyyI/K9uApq7iyyI.md

@@ -0,0 +1,500 @@
+# ResMLP: Feedforward networks for image classification with data-efficient training
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+We present ResMLP, an architecture built entirely upon multi-layer perceptrons for image classification. It is a simple residual network that alternates (i) a linear layer in which image patches interact, independently and identically across channels, and (ii) a two-layer feed-forward network in which channels interact independently per patch. When trained with a modern training strategy using heavy data-augmentation and optionally distillation, it attains surprisingly good accuracy/complexity tradeoffs on ImageNet. We also train ResMLP models in a self-supervised setup, to further remove priors from employing a labelled dataset. Finally, by adapting our model to machine translation we achieve surprisingly good results.
+
+We will share our code based on the Timm library and pre-trained models.
+
+# 11 1 Introduction
+
+12 Recently, the transformer architecture [60], adapted from its original use in natural language pro  
+13 cessing with only minor changes, has achieved performance competitive with the state of the art on   
+14 ImageNet-1k [50] when pre-trained with a sufficiently large amount of data [16]. Retrospectively,   
+15 this achievement is yet another step towards less priors: convolutional neural networks had removed   
+16 a lot of hand-made choices compared to hand-designed pre-CNN approaches, moving the paradigm   
+17 of hard-wired features to hand-designed architectural choices. Vision transformers avoid making   
+18 assumptions inherent to convolutional architectures and noticeably the translation invariance.   
+19 What these recent transformer-based works suggest is that longer training schedules, more parameters,   
+20 more data [16] and/or more regularization [56], are sufficient to recover the important priors for tasks   
+21 as complex as ImageNet classification. See also our discussion of related work in Section 4. This   
+22 concurs with recent studies [2, 15] that better disentangle the benefits from the architectures from   
+23 those of the training scheme.   
+24 In this paper, we push this trend further, and propose Residual Multi-Layer Perceptrons (ResMLP):   
+25 a purely multi-layer perceptron (MLP) based architecture for image classification. We outline our   
+26 architecture in Figure 1 and detail it further in Section 2. It is intended to be simple: it takes image   
+27 patches as input, projects them with a linear layer, and sequentially updates them in turn with two   
+28 residual operations: (i) a simple linear layer that provides interaction between the patches, which   
+29 is applied to all channels independently; and (ii) an MLP with a single hidden layer, which is   
+30 independently applied to all patches. At the end of the network, the patches are average pooled, and   
+31 fed to a linear classifier.   
+32 This architecture is strongly inspired by the vision transformers (ViT) [16], yet it is much simpler   
+33 in several ways: we do not use any form of attention, only linear layers along with the GELU   
+34 non-linearity [25]. Since our architecture is much more stable to train than transformers, we do not   
+35 need batch-specific or cross-channel normalizations such as BatchNorm, GroupNorm or LayerNorm.   
+36 Our training procedure mostly follows the one initially introduced for DeiT [56] and CaiT [57].   
+37 Due to its linear nature, the patch interactions in our model can be easily visualised and interpreted.   
+38 While the interaction pattern learned in the first layer is very similar to a small convolutional filter,   
+39 we observe more subtle interactions across patches in deeper layers. These include some form of   
+40 axial filters, and long-range interactions early in the network.
+
+![](images/abb017542f27a5fed6ae93c2da8dd9eea31072f0baaa1cd9b7e27ffed8ccf434.jpg)  
+Figure 1: The ResMLP architecture: After linearly projecting the image patches, our network alternately processes them by (1) a communication layer between vectors implemented as a linear layer; (2) a two-layer residual perceptron. We denote by A the Affine element-wise transformation, and by T the transposition.
+
+41 In summary, in this paper, we show that
+
+• despite their simplicity, Residual Multi-Layer Perceptrons reach surprisingly good accuracy/complexity trade-offs with ImageNet-1k training only1, without requiring normalization based on batch or channel statistics;
+
+• these models benefit significantly from distillation methods [56]; they are also compatible with modern self-supervised learning methods based on data augmentation, such as DINO [6];
+
+• thank to its design where patch embeddings simply “communicate” through a linear layer, we can make observations on the spatial interaction that the network learns across layers;
+
+• we adapt ResMLP to machine translation, and again obtain surprisingly good results.
+
+# 50 2 Method
+
+51 Our model, depicted in Figure 1, is inspired by the ViT model, from which we adopt the columnar   
+52 structure with fixed-resolution feature maps. We proceed two drastic simplifications. We refer the   
+53 reader to Dosovitskiy et al. [16] for more details about the ViT architecture.   
+54 The overall ResMLP architecture. Our model, denoted by ResMLP, takes a grid of $N \times N$ non  
+55 overlapping patches as input, where the patch size is typically equal to $1 6 \times 1 6$ . The patches are then   
+56 independently passed through a linear layer to form a set of $\overline { { N ^ { 2 } d } }$ -dimensional embeddings.   
+57 The resulting set of $N ^ { 2 }$ embeddings are fed to a sequence of Residual Multi-Layer Perceptron layers   
+58 to produce a set of $N ^ { 2 } d .$ -dimensional output embeddings. These output embeddings are then averaged   
+59 as a $d$ -dimension vector to represent the image, which is fed to a linear classifier to predict the label   
+60 associated with the image. Training uses the cross-entropy loss.   
+61 The Residual Multi-Perceptron Layer. Our network is a sequence of layers that all have the same   
+62 structure: a linear sublayer followed by a feedforward sublayer. Similar to the Transformer layer,   
+63 each sublayer is paralleled with a skip-connection [23]. We do not apply Layer Normalization [1]   
+64 because training is stable without it when using the following Affine transformation:
+
+$$
+\begin{array} { r } { \mathbb { A } \mathbb { f } \mathbb { f } _ { \alpha , \beta } ( \mathbf { x } ) = \mathbb { D } \mathrm { i } \mathtt { a } \mathbb { g } ( \pmb { \alpha } ) \mathbf { x } + \beta , } \end{array}
+$$
+
+65 where $_ { \pmb { \alpha } }$ and $\beta$ are learnable weight vectors. This operation simply rescales and shifts the input   
+66 element-wise. Moreover, it has no cost at inference time, as it can absorbed in the adjacent linear   
+67 layer. Note, when writing $\operatorname { \mathbb { A } f f } ( \mathbf { X } )$ the operation is applied independently to each column of $\mathbf { X }$ .   
+68 While similar to BatchNorm [30] and Layer Normalization [1], the Aff operator does not depend on   
+69 any batch statistics. Therefore, it is closer to the recent LayerScale method [57], which improves the   
+70 optimization of deep transformers when initializing $_ { \pmb { \alpha } }$ to a small value. Note, LayerScale does not   
+71 have a bias term.   
+72 We apply this transformation twice for each residual block. As as a pre-normalization Aff replaces   
+73 the LayerNormalization, and avoids using channel-wise statistics. Here, we initialize $\alpha = 1$ , and   
+74 $\beta = { \bf 0 }$ . As a post-processing of the residual block, Aff implements LayerScale and therefore we   
+75 follow the same small value initialization for $_ { \pmb { \alpha } }$ as in [57] for the post-normalization.   
+76 Finally, we follow the same structure for the feedforward sublayer as in the Transformer; we only   
+77 replace the ReLU non-linearity by a GELU function [25].   
+78 Overall, our Multi-layer perceptron takes a set of $N ^ { 2 } \ d .$ -dimensional input features stacked in a   
+79 $d \times N ^ { 2 }$ matrix $\mathbf { X }$ , and outputs a set of $N ^ { 2 }$ $d$ -dimension output features, stacked in a matrix $\mathbf { Y }$ with   
+80 the following set of transformations:
+
+$$
+\begin{array} { r c l } { \mathbf { Z } } & { = } & { \mathbf { X } + \mathbb { A } \mathbb { f } \mathbb { f } \left( \left( \mathbf { A } \mathbb { A } \mathbb { f } \mathbb { f } \left( \mathbf { X } \right) ^ { \top } \right) ^ { \top } \right) , } \\ { \mathbf { Y } } & { = } & { \mathbf { Z } + \mathbb { A } \mathbb { f } \mathbb { f } \left( \mathbf { C } \mathbb { G } \mathbb { E } \mathbb { L } \mathbb { U } \left( \mathbf { B } \mathbb { A } \mathbb { f } \mathbb { f } \left( \mathbf { Z } \right) \right) \right) , } \end{array}
+$$
+
+81 where A, $\mathbf { B }$ and $\mathbf { C }$ are the main learnable weight matrices of the layer. The dimensions of the   
+82 parameter matrix $\mathbf { A }$ are $N ^ { 2 } \times N ^ { 2 }$ , i.e., this sublayer exchanges information across all the locations,   
+83 while the feedforward sublayer works per location. As a consequence, the intermediate activation   
+84 matrix $\mathbf { Z }$ has the same dimensions as the matrices $\mathbf { X }$ and $\mathbf { Y }$ . Finally, the weight matrices $\mathbf { B }$ and $\mathbf { C }$   
+85 have the same dimensions as in a Transformer layer, which are $4 d { \times } d$ and $d \times 4 d$ , respectively.   
+86 The main difference compared to a Transformer layer is that we replace the self-attention by the   
+87 linear interaction defined in Eq. (2). While self-attention computes a convex combination of other   
+88 features with coefficients that are data dependent, the linear interaction layer in Eq. (2) computes   
+89 a general linear combination using learned coefficients that are not data dependent. As compared   
+90 to a convolutional layers which have local support and share weights across space, our linear   
+91 patch interaction layer offers a global support and does not share weights, moreover it is applied   
+92 independently across channels.   
+93 Relationship to the Vision Transformer. Our model can be regarded as a drastic simplification of   
+94 the ViT model by Dosovitskiy et al. [16]. We depart from this model as follows:
+
+• We do not include any self-attention block. Instead we have a linear patch interaction layer without any non-linearity.
+
+• We do not have the extra “class” token that is typically used in these models to aggregate information via attention. Instead, we simply use average pooling. We do, however, also consider a specific aggregation layer as a variant, which we describe in the next paragraph.
+
+• We do not include any form of positional embedding: the linear communication module between patches implicitly takes into account the patch position.
+
+• Instead of pre-LayerNormalization, we use a simple learnable affine transform, thus avoiding any form of batch and channel-wise statistics.
+
+104 Class-MLP. As an alternative to average pooling, we also experimented with an adaptation of the   
+105 class-attention introduced in CaiT [57]. In CaiT, this consists of two layers that have the same   
+106 structure as the transformer, but in which only the class token is updated based on the frozen patch   
+107 embeddings. We translate this method to our architecture, except that, after aggregating the patches   
+108 with a linear layer, we replace the attention-based interaction between the class and patch embeddings   
+109 by simple linear layers, still keeping the patch embeddings frozen. This increases the performance, at   
+110 the expense of adding some parameters and computational cost. We refer to this pooling variant as   
+111 “class-MLP”, since the purpose of these few layers is to replace average pooling.
+
+# 112 3 Experiments
+
+113 In this section, we present experimental results for our ResMLP architecture for image classification.   
+114 We also study the impact of the different components in a series of ablations. We consider three   
+115 training paradigms in our experiments:   
+6 • Supervised learning: We train ResMLP from labeled images with a softmax classifier and cross  
+7 entropy loss. This paradigm is the main focus of our work.
+
+Table 1: Comparison between architectures on ImageNet classification. We compare different architectures based on convolutional networks, Transformers and feedforward networks with comparable FLOPs and number of parameters. We report Top-1 accuracy on the validation set of ImageNet-1k with different measure of complexity: throughput, FLOPs, number of parameters and peak memory usage. All the models use $2 2 4 \times 2 2 4$ images as input. By default the Transformers and feedforward networks uses $1 4 \times 1 4$ patches of size $1 6 \times 1 6$ , see Table 3 for the detailed specification of our main models. The throughput is measured on a single V100-32GB GPU with batch size fixed to 32. For reference, we include the state of the art with ImageNet training only.   
+
+<table><tr><td></td><td>Arch.</td><td>#params (x106)</td><td>throughput (im/s)</td><td>FLOPS (x109)</td><td>Peak Mem (MB)</td><td>Top-1 Acc.</td></tr><tr><td rowspan="2">State of the art</td><td>CaiT-M48↑448Y [57]</td><td>356</td><td>5.4</td><td>329.6</td><td>5477.8</td><td>86.5</td></tr><tr><td>NfNet-F6 SAM [5]</td><td>438</td><td>16.0</td><td>377.3</td><td>5519.3</td><td>86.5</td></tr><tr><td rowspan="6">Convolutional networks</td><td>EfficientNet-B3 [53]</td><td>12</td><td>661.8</td><td>1.8</td><td>1174.0</td><td>81.1</td></tr><tr><td>EfficientNet-B4 [53]</td><td>19</td><td>349.4</td><td>4.2</td><td>1898.9</td><td>82.6</td></tr><tr><td>EfficientNet-B5 [53]</td><td>30</td><td>169.1</td><td>9.9</td><td>2734.9</td><td>83.3</td></tr><tr><td>RegNetY-4GF[47]</td><td>21</td><td>861.0</td><td>4.0</td><td>568.4</td><td>80.0</td></tr><tr><td>RegNetY-8GF[47]</td><td>39</td><td>534.4</td><td>8.0</td><td>841.6</td><td>81.7</td></tr><tr><td>RegNetY-16GF[47]</td><td>84</td><td>334.7</td><td>16.0</td><td>1329.6</td><td>82.9</td></tr><tr><td rowspan="3">Transformer networks</td><td>DeiT-S [56]</td><td>22</td><td>940.4</td><td>4.6</td><td>217.2</td><td>79.8</td></tr><tr><td>DeiT-B [56]</td><td>86</td><td>292.3</td><td>17.5</td><td>573.7</td><td>81.8</td></tr><tr><td>CaiT-XS24 [57]</td><td>27</td><td>447.6</td><td>5.4</td><td>245.5</td><td>81.8</td></tr><tr><td rowspan="3">Feedforward networks</td><td>ResMLP-S12</td><td>15</td><td>1415.1</td><td>3.0</td><td>179.5</td><td>76.6</td></tr><tr><td>ResMLP-S24</td><td>30</td><td>715.4</td><td>6.0</td><td>235.3</td><td>79.4</td></tr><tr><td>ResMLP-B24</td><td>116</td><td>231.3</td><td>23.0</td><td>663.0</td><td>81.0</td></tr></table>
+
+• Self-supervised learning: We train the ResMLP architecture without labels. We consider the DINO method of Caron et al. [6] that trains a network by distilling knowledge from previous instances of the same network, leading to a form of self-distillation without labels.
+
+• Knowledge distillation: We employ the knowledge distillation procedure proposed by Touvron et al. [56] to guide the supervised training of ResMLP with a convnet.
+
+# 123 3.1 Experimental setting
+
+Datasets. We train our models on the ImageNet-1k dataset [50], that contains 1.2M images evenly spread over 1,000 object categories. In the absence of an available test set for this benchmark, we follow the standard practice in the community by reporting performance on the validation set. This is not ideal since the validation set was originally designed to select hyper-parameters. Comparing methods on this set may not be conclusive enough because an improvement in performance may not be caused by better modeling, but by a better selection of hyper-parameters. To mitigate this risk, we report additional results in transfer learning and on two alternative versions of ImageNet that have been built to have distinct validation and test sets, namely the ImageNet-real [3] and ImageNet-v2 [49] datasets. We also report a few data-points when training on ImageNet-21k. Our hyper-parameters are mostly adopted from Touvron et al. [56, 57].
+
+134 Hyper-parameter settings. In the case of supervised learning, we train our network with the Lamb   
+135 optimizer [63] with a learning rate of $5 \times 1 0 ^ { - 3 }$ and weight decay 0.2. We initialize the LayerScale   
+136 parameters as a function of the depth by following CaiT [57]. The rest of the hyper-parameters follow   
+137 the default setting used in DeiT [56]. For the knowledge distillation paradigm, we use the same   
+138 RegNety-16GF [48] as in DeiT with the same training schedule. The majority of our models take two   
+139 days to train on eight V100-32GB GPUs.
+
+# 3.2 Main Results
+
+In this section, we compare our architecture to models with more conventional network architectures of comparable size and throughput on ImageNet.
+
+43 Comparison with Transformers and convnets in a supervised setting. In Table 1, we compare   
+44 ResMLP with different convolutional and Transformer architectures. For completeness, we report   
+145 the best-published numbers obtained with a model trained on ImageNet alone. As expected, in   
+146 terms of the trade-off between accuracy, FLOPs, and throughput, ResMLP is not as efficient as   
+147 convolutional networks or Transformers. However, their accuracy is encouraging: we compare them   
+148 with architectures that have benefited from years of research and careful optimization towards these   
+149 trade-offs. Overall, our results suggest that the structural constraints imposed by the layer design do   
+150 not have a drastic influence on performance, especially when training models with enough data and   
+151 recent advances in training and regularization.   
+152 Self-supervised pre-training of ResMLP. We explore the possibility of training ResMLP using   
+153 DINO, a recent self-supervised learning approach [6]. We pre-train ResMLP-S12 models with this   
+154 approach during 300 epochs. We report our results in Table 2. As expected given the supervised   
+155 classification results, the accuracies obtained with ResMLP are less good than with ViT. Nevertheless,   
+156 the performance is surprisingly high for a pure MLP architecture and competitive with Convnet in   
+157 knn evaluation. We hope that these result will serve as a baseline for future work.   
+158 After self-supervised pre-training, we also fine-tune the network on ImageNet using ground truth   
+159 labels. This pre-training substantially improves the accuracy, when comparing with the same model   
+160 ResMLP-S24 solely trained with labels (top-1 acc. of $7 9 . 9 \%$ on ImageNet-val instead of $7 9 . 4 \%$ for   
+161 ResMLP-S24, with the same total number of epochs). Results on ImageNet-v2 suggest that it reduces   
+162 overfitting $6 8 . 6 \%$ on ImageNet-v2, vs $6 7 . 9 \%$ with supervised training only).   
+163 Improving models with knowledge distillation. We study our model when training following the   
+164 knowledge distillation approach of Touvron et al. [56]. In their work, the authors show the impact of   
+165 training a ViT model by distilling it from a RegNet. In this experiment, we explore if ResMLP also   
+166 benefits from this procedure and summarize our results in Table 3 (Blocks “Baseline models” and   
+167 “Training”). We observe that similar to DeiT models, ResMLP greatly benefits from distilling from a   
+168 convnet. This result concurs with the observations made by d’Ascoli et al. [13], who used convnets   
+169 to initialize feedforward networks. Even though our setting differs from theirs in scale, the problem   
+170 of overfitting for feedforward networks is still present on ImageNet. The additional regularization   
+171 obtained from the distillation is a possible explanation for this improvement.   
+172 Visualisation. Because they are linear, our patch interaction layers from Eq. (2) are easily inter  
+173 pretable. In Figure 2 we visualise the rows of the interaction matrices A as $N \times N$ images, for our   
+174 ResMLP-S24 model. The early layers show convolution-like patterns: the learned weights resemble   
+175 shifted versions of each other and have local support. Interestingly, in many layers, the support also   
+176 extends along both axes, most prominently seen in layer seven. The last seven layers of the network   
+177 are different: they consist of a spike for the patch itself and a diffuse response across other patches   
+178 with larger or smaller magnitude; see layer 20.
+
+Table 2: Self-supervised learning with DINO [6]. Classification accuracy on ImageNet-1k val. ResMLPs evaluated with linear and $k$ -NN evaluation on ImageNet are comparable to convnets but inferior to ViT.   
+
+<table><tr><td>Models</td><td>ResNet-50</td><td>ViT-S/16</td><td>ViT-S/8</td><td>ViT-B/16</td><td>ResMLP-S12</td><td>ResMLP-S24</td></tr><tr><td>Params. (×106)</td><td>25</td><td>22</td><td>22</td><td>87</td><td>15</td><td>30</td></tr><tr><td>FLOPS (×109)</td><td>4.1</td><td>4.6</td><td>22.4</td><td>17.5</td><td>3.0</td><td>6.0</td></tr><tr><td>Linear</td><td>75.3</td><td>77.0</td><td>79.7</td><td>78.2</td><td>67.5</td><td>72.8</td></tr><tr><td>k-NN</td><td>67.5</td><td>74.5</td><td>78.3</td><td>76.1</td><td>62.6</td><td>69.4</td></tr></table>
+
+# 179 3.3 Visualization & analysis of the linear interaction between patches
+
+180 Measuring sparsity of the weights. The visualizations described above suggest that the linear   
+181 communication layers are sparse. We analyze this quantitatively in more detail in Figure 3. We   
+182 measure the sparsity of the matrix A, and compare it to the sparsity of $\mathbf { B }$ and $\mathbf { C }$ from the per-patch   
+183 MLP. Since there are no exact zeros, we measure the rate of components whose absolute value is   
+184 lower than $5 \%$ of the maximum value. Note, discarding the small values is analogous to the case   
+185 where we normalize the matrix by its maximum and use a finite-precision representation of weights.   
+186 For instance, with a 4-bits representation of weight, one would typically round to zero all weights   
+187 whose absolute value is below $6 . 2 5 \%$ of the maximum value.   
+188 The measurements in Figure 3 show that all three matrices are sparse, with the layers implementing   
+189 the patch communication being significantly more so. This suggests that they may be compatible with   
+190 parameter pruning, or better, with modern quantization techniques that induce sparsity at training   
+191 time, such as Quant-Noise [20] and DiffQ [19]. The sparsity structure, in particular in earlier layers,   
+192 see Figure. 2, hints that we could implement the patch interaction linear layer with a convolution. We   
+193 provide some results for convolutional variants in our ablation study. Further research on network   
+194 compression is beyond the scope of this paper, yet we believe it worth investigating in the future.   
+195 Communication across patches if we remove the linear interaction layer (linear none), we   
+196 obtain substantially lower accuracy ( $- 2 0 \%$ top-1 acc.) for a “bag-of-patches” approach. We have tried   
+197 several alternatives for the linear patch interaction layer, which are presented in Table 3 (block “patch   
+198 communication”). Amongst them, using the same MLP structure as for patch processing (linear   
+199 MLP), which we analyze in more details in the supplementary material. The simpler choice of a   
+200 single linear square layer led to a better accuracy/performance trade-off – considering that the MLP   
+201 variant requires compute halfway between ResMLP-S12 and ResMLP-S24 – and requires fewer   
+202 parameters than a residual MLP block.   
+203 The visualization in Figure 2 indicates that many linear interaction layers look like convolutions. In   
+204 our ablation, we replaced the linear layer with different types of $3 \times 3$ convolutions. The depth-wise   
+205 convolution does not implement interaction across channels – as our linear patch communication   
+206 layer – and yields similar performance at a comparable number of parameters and FLOPs. While   
+207 full $3 \times 3$ convolutions yield best results, they come with roughly double the number of parameters
+
+![](images/1a5c45d9a0efc551807e3e5b32b664add8eacee830756aaeaa0f33c734b6d1d1.jpg)  
+Figure 2: Visualisation of the linear layers in ResMLP-S24. For each layer we visualise the rows of the matrix A as a set of $1 4 \times 1 4$ pixel images, for sake of space we only show the rows corresponding to the $6 \times 6$ central patches. We observe patterns in the linear layers that share similarities with convolutions. In appendix B we provide comparable visualizations for all layers of a ResMLP-S12 model.
+
+![](images/b5f35749759ac14eaf9aeb24d25e43c14d1a40bfe0b0669024fbc8482b3a3fe3.jpg)  
+Figure 3: Sparsity of linear interaction layers. For each layer (linear and MLP), we show the rate of components whose absolute value is lower than $5 \%$ of the maximum. Linear interaction layers are sparser than the matrices involved in the per-patch MLP.
+
+![](images/60d9f8bbf12c5cd094917c6d04868c6e5af73178211cc8057c5716ff70c22b03.jpg)  
+Figure 4: Top-1 accuracy on ImageNet-V2 vs. ImageNet-val. ResMLPs tend to overfit slightly more under identical training method. This is partially alleviated with by introducing more regularization (more data or distillation, see e.g., ResMLP-B24/8-distil).
+
+Table 3: Ablation. Our default configurations are presented in the three first rows. By default we train during 400 epochs. The “old-fashioned” is similar to what was employed for ResNet [23]: SGD, 90-epochs waterfall schedule, same augmentations up to variations due to library used.   
+
+<table><tr><td rowspan="2">Ablation</td><td rowspan="2">Model</td><td rowspan="2">Patch size</td><td rowspan="2">Params ×106</td><td rowspan="2">FLOPs ×109</td><td rowspan="2">Variant</td><td colspan="3">top-1 acc.on ImageNet</td></tr><tr><td>val</td><td>real[3]</td><td>v2 [49]</td></tr><tr><td rowspan="3">Baseline models</td><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>12 layers, working dimension 384</td><td>76.6</td><td>83.3</td><td>64.4</td></tr><tr><td>ResMLP-S24</td><td>16</td><td>30.0</td><td>6.0</td><td>24 layers, working dimension 384</td><td>79.4</td><td>85.3</td><td>67.9</td></tr><tr><td>ResMLP-B24</td><td>16</td><td>115.7</td><td>23.0</td><td>24 layers, working dimension 768</td><td>81.0</td><td>86.1</td><td>69.0</td></tr><tr><td>Normalization</td><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>Aff →Layernorm</td><td>77.7</td><td>84.1</td><td>65.7</td></tr><tr><td>Pooling</td><td>ResMLP-S12</td><td>16</td><td>17.7</td><td>3.0</td><td>average pooling →Class-MLP</td><td>77.5</td><td>84.0</td><td>66.1</td></tr><tr><td rowspan="5">Patch communication</td><td>ResMLP-S12</td><td>16</td><td>14.9</td><td>2.8</td><td>linear→ none</td><td>56.5</td><td>63.4</td><td>43.1</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>18.6</td><td>4.3</td><td>linear→MLP</td><td>77.3</td><td>84.0</td><td>65.7</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>30.8</td><td>6.0</td><td>linear → conv 3x3</td><td>77.3</td><td>84.4</td><td>65.7</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>14.9</td><td>2.8</td><td>linear →conv 3x3 depth-wise</td><td>76.3</td><td>83.4</td><td>64.6</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>16.7</td><td>3.2</td><td>linear→conv 3x3 depth-separable</td><td>77.0</td><td>84.0</td><td>65.5</td></tr><tr><td rowspan="3">Patch size</td><td>ResMLP-S12/14</td><td>14</td><td>15.6</td><td>4.0</td><td>patch size 16×16-→14×14</td><td>76.9</td><td>83.7</td><td>65.0</td></tr><tr><td>ResMLP-S12/8</td><td>8</td><td>22.1</td><td>14.0</td><td>patch size 16×16-8×8</td><td>79.1</td><td>85.2</td><td>67.2</td></tr><tr><td>ResMLP-B24/8</td><td>8</td><td>129.1</td><td>100.2</td><td> patch size 16×16-&gt;8 ×8</td><td>81.0</td><td>85.7</td><td>68.6</td></tr><tr><td rowspan="7">Training</td><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>old-fashioned (90 epochs)</td><td>69.2</td><td>76.0</td><td>56.1</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>pre-trained SSL (DINO)</td><td>76.5</td><td>83.6</td><td>64.5</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>distillation</td><td>77.8</td><td>84.6</td><td>66.0</td></tr><tr><td>ResMLP-S24</td><td>16</td><td>30.0</td><td>6.0</td><td> pre-trained SSL (DINO)</td><td>79.9</td><td>85.9</td><td>68.6</td></tr><tr><td>ResMLP-S24</td><td>16</td><td>30.0</td><td>6.0</td><td>distillation</td><td>80.8</td><td>86.6</td><td>69.8</td></tr><tr><td>ResMLP-B24/8</td><td>8</td><td>129.1</td><td>100.2</td><td>distillation</td><td>83.6</td><td>88.4</td><td>73.4</td></tr><tr><td>ResMLP-B24/8</td><td>8</td><td>129.1</td><td>100.2</td><td>pre-trained ImageNet-21k (60 epochs)</td><td>84.4</td><td>88.9</td><td>74.2</td></tr></table>
+
+and FLOPs. Interestingly, the depth-separable convolutions combine accuracy close to that of full $3 \times 3$ convolutions with a number of parameters and FLOPs comparable to our linear layer. This suggests that convolutions on low-resolution feature maps at all layers is an interesting alternative to the common pyramidal design of convnets, where early layers operate at higher resolution and smaller feature dimension.
+
+# 3.4 Ablation studies
+
+Table 3 reports the ablation study of our base network and a summary of our preliminary exploratory studies. We discuss the ablation below and give more detail about early experiments in Appendix A.
+
+Control of overfitting. Since MLPs are subject to overfitting, we show in Fig. 4 a control experiment to probe for problems with generalization. We explicitly analyze the differential of performance between the ImageNet-val and the distinct ImageNet-V2 test set. The relative offsets between curves reflect to which extent models are overfitted to ImageNet-val w.r.t. hyper-parameter selection. The degree of overfitting of our MLP-based model is overall neutral or slightly higher to that of other transformer-based architectures or convnets with same training procedure.
+
+Normalization & activation. Our network configuration does not contain any batch normalizations. Instead, we use the affine per-channel transform Aff. This is akin to Layer Normalization [1], typically used in transformers, except that we avoid to collect any sort of statistics, since we do no need it it for convergence. In preliminary experiments with pre-norm and post-norm [24], we observed that both choices converged. Pre-normalization in conjunction with Batch Normalization could provide an accuracy gain in some cases, see Appendix A.
+
+We choose to use a GELU [25] function. In Appendix A we also analyze the activation function: ReLU [22] also gives a good performance, but it was a bit more unstable in some settings. We did not manage to get good results with SiLU [25] and HardSwish [28].
+
+Pooling. Replacing average pooling with Class-MLP, see Section 2, brings a significant gain for a negligible computational cost. We do not include it by default to keep our models more simple.
+
+Patch size. Smaller patches significantly increase the performance, but also increase the number of flops (see Block "Patch size" in Table 3). Smaller patches benefit more to larger models, but only with an improved optimization scheme involving more regularization (distillation) or more data.
+
+6 Training. Consider the Block “Training’ in Table 3. ResMLP significantly benefits from modern training procedures such as those used in DeiT. For instance, the DeiT training procedure improves
+
+<table><tr><td>Architecture</td><td>FLOPs</td><td>Res.</td><td>CIFAR10</td><td>CIFAR100</td><td>Flowers102</td><td>Cars</td><td>iNat18</td><td>iNat19</td></tr><tr><td>EfficientNet-B7 [53]</td><td>37.0B</td><td>600</td><td>98.9</td><td>91.7</td><td>98.8</td><td>94.7</td><td>1</td><td>1</td></tr><tr><td>ViT-B/16 [16]</td><td>55.5B</td><td>384</td><td>98.1</td><td>87.1</td><td>89.5</td><td>1</td><td>1</td><td>1</td></tr><tr><td>ViT-L/16 [16]</td><td>190.7B</td><td>384</td><td>97.9</td><td>86.4</td><td>89.7</td><td></td><td></td><td></td></tr><tr><td>Deit-B/16 [56]</td><td>17.5B</td><td>224</td><td>99.1</td><td>90.8</td><td>98.4</td><td>92.1</td><td>73.2</td><td>77.7</td></tr><tr><td>ResNet50 [58]</td><td>4.1B</td><td>224</td><td>1</td><td>1</td><td>96.2</td><td>90.0</td><td>68.4</td><td>73.7</td></tr><tr><td>Grafit/ResNet50 [58]</td><td>4.1B</td><td>224</td><td>1</td><td>1</td><td>97.6</td><td>92.7</td><td>68.5</td><td>74.6</td></tr><tr><td>ResMLP-S12</td><td>3.0B</td><td>224</td><td>98.1</td><td>87.0</td><td>97.4</td><td>84.6</td><td>60.2</td><td>71.0</td></tr><tr><td>ResMLP-S24</td><td>6.0B</td><td>224</td><td>98.7</td><td>89.5</td><td>97.9</td><td>89.5</td><td>64.3</td><td>72.5</td></tr></table>
+
+Table 4: Evaluation on transfer learning. Classification accuracy (top-1) of models trained on ImageNet-1k for transfer to datasets covering different domains. The ResMLP architecture takes $2 2 4 \times 2 2 4$ images during training and transfer, while ViTs and EfficientNet-B7 work with higher resolutions, see “Res.” column.
+
+238 the performance of ResMLP-S12 by $7 . 4 \%$ compared to the training employed for ResNet $[ 2 3 ]$ .   
+239 This is in line with recent work pointing out the importance of the training strategy over the model   
+240 choice [2, 48]. Pre-training on more data and distillation also improve the performance of ResMLP,   
+241 especially for the bigger models, e.g., distillation improves the accuracy of ResMLP-B24/8 by $2 . 6 \%$ .
+
+Other analysis. In our early exploration, we evaluated several alternative design choices. As in transformers, we could use positional embeddings mixed with the input patches. In our experiments we did not see any benefit from using these features, see Appendix A. This observation suggests that our linear patch interaction layer provides sufficient spatial communication, and referencing absolute positions obviates the need for any form of positional encoding.
+
+# 3.5 Transfer learning
+
+We evaluate the quality of features obtained from a ResMLP architecture when transferring them to other domains. The goal is to assess if the features generated from a feedforward network are more prone to overfitting on the training data distribution. We adopt the typical setting where we pre-train a model on ImageNet-1k and fine-tune it on the training set associated with a specific domain. We report the performance with different architectures on various image benchmarks in Table 4, namely CIFAR-10 and CIFAR-100 [34], Flowers-102 [42], Stanford Cars [33] and iNaturalist [27]. We refer the reader to the corresponding references for a more detailed description of the datasets. We observe that the performance of our ResMLP is competitive with the existing architectures, showing that pretraining feedforward models with enough data and regularization via data augmentation greatly reduces their tendency to overfit on the original distribution. Interestingly, this regularization also prevents them from overfitting on the training set of smaller dataset during the fine-tuning stage.
+
+# 3.6 Machine translation
+
+We also evaluate the ResMLP transpose-mechanism to replace the self-attention in the encoder and decoder of a neural machine translation system. We train models on the WMT 2014 English-German and English-French tasks, following the setup from Ott et al. [45]. We consider models of dimension 512, with a hidden MLP size of 2048, and with 6 or 12 layers. Note that the current state of the art employs much larger models: our 6 layers model is more comparable to the base transformer model from Vaswani et al. [60], which serves as a baseline, along with pre-transformer architectures such as Recurrent and convolutional neural networks. We use Adagrad with learning rate 0.2, $3 2 \mathrm { k }$ steps of linear warmup, label smoothing 0.1, dropout rate 0.15 for en-de and 0.1 for en-fr. We initialize the LayerScale parameter to 0.2. We generate translations with the beam search algorithm, with a beam of size 4. As shown in Table 5, the results are at least on par with other architectures:
+
+Table 5: Machine translation on WMT 2014 translation tasks. We report tokenized BLEU on newstest2014.   
+
+<table><tr><td>Models</td><td>GNMT[61]</td><td></td><td>ConvS2S [21]Transf. (base)[60]</td><td>ResMLP-6</td><td>ResMLP-12</td></tr><tr><td>EN-DE</td><td>24.6</td><td>25.2</td><td>27.3</td><td>26.4</td><td>26.8</td></tr><tr><td>EN-FR</td><td>39.9</td><td>40.5</td><td>38.1</td><td>40.3</td><td>40.6</td></tr></table>
+
+2Interestingly, if trained with this “old-fashion” setting, ResMLP-S12 outperforms AlexNet [35] by a margin.
+
+We review the research on applying Fully Connected Network (FCN) for computer vision problems as well as other architectures that shares common modules with our model.
+
+Fully-connected network for images. Many studies have shown that FCNs are competitive with convnets for the tasks of digit recognition [11, 51], keyword spotting [7] and handwritting recognition [4]. Several works [37, 40, 59] have questioned if FCNs are also competitive on natural image datasets, such as CIFAR-10 [34]. More recently, d’Ascoli et al. [13] have shown that a FCN initialized with the weights of a pretrained convnet achieves performance that are superior than the original convnet. Neyshabur [41] further extend this line of work by achieving competitive performance by training an FCN from scratch but with a regularizer that constrains the models to be close to a convnet. These studies have been conducted on small scale datasets with the purpose of studying the impact of architectures on generalization in terms of sample complexity [18] and energy landscape [31]. In our work, we show that, in the larger scale setting of ImageNet, FCNs can attain surprising accuracy without any constraint or initialization inspired by convnets.
+
+Finally, the application of FCN networks in computer vision have also emerged in the study of the properties of networks with infinite width [43], or for inverse scattering problems [32]. More interestingly, the Tensorizing Network [44] is an approximation of very large FCN that shares similarity with our model, in that they intend to remove prior by approximating even more general tensor operations, i.e., not arbitrarily marginalized along some pre-defined sharing dimensions. However, their method is designed to compress the MLP layers of a standard convnets.
+
+Other architectures with similar components. Our FCN architecture shares several components with other architectures, such as convnets [35, 36] or transformers [60]. A fully connected layer is equivalent to a convolution layer with a $1 \times 1$ receptive field, and several work have explored convnet architectures with small receptive fields. For instance, the VGG model [52] uses $3 \times 3$ convolutions, and later, other architectures such as the ResNext [62] or the Xception [10] mix $1 \times 1$ and $3 \times 3$ convolutions. In contrast to convnets, in our model interaction between patches is obtained via a linear layer that is shared across channels, and that relies on absolute rather than relative positions.
+
+More recently, transformers have emerged as a promising architecture for computer vision [9, 17, 46, 56, 66]. In particular, our architecture takes inspiration from the structure used in the Vision Transformer (ViT) [17], and as consequence, shares many components. Our model takes a set of non-overlapping patches as input and passes them through a series of MLP layers that share the same structure as ViT, replacing the self-attention layer with a linear patch interaction layer. Both layers have a global field-of-view, unlike convolutional layers. Whereas in self-attention the weights to aggregate information from other patches are data dependent through queries and keys, in ResMLP the weights are not data dependent and only based on absolute positions of patches. In our implementation we follow the improvements of DeiT [56] to train vision transformers, use the skip-connections from ResNets [23] with pre-normalization of the layers [8, 24].
+
+307 Finally, our work questions the importance of self-attention in existing architectures. Similar ob  
+308 servations have been made in natural language processing. Notably, Synthesizer [54] shows that   
+309 dot-product self-attention can be replaced by a feedforward network, with competitive performance   
+310 on sentence representation benchmarks. As opposed to our work, Synthesizer does use data dependent   
+311 weights, but in contrast to transformers the weights are determined from the queries only.
+
+# 312 5 Conclusion
+
+In this paper we have shown that a simple residual architecture, whose residual blocks consist of a one-hidden layer feed-forward network and a linear patch interaction layer, achieves an unexpectedly high performance on ImageNet classification benchmarks, provided that we adopt a modern training strategy such as those recently introduced for transformer-based architectures. Thanks to their simple structure, with linear layers as the main mean of communication between patches, we can vizualize the filters learned by this simple MLP. While some of the layers are similar to convolutional filters, we also observe sparse long-range interactions as early as the second layer of the network. We hope that our model free of spatial priors will contribute to further understanding of what networks with less priors learn, and potentially guide the design choices of future networks without the pyramidal design prior adopted by most convolutional neural networks.
+
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+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [No] We only use MLPs for image classification. There is already other system much more powerful therefore the impact is negligible   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We provide code in appendix   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] Because it’s too expensive in computational resources. But for some experiments we run multiple times and we have $\sim 0 . 1$ as standard deviation   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] The majority of our experiments take 2 days on 8 GPUs V100-32GB
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [No] But license can be found with the references we mention   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] We only use datasets that are widely used in computer vision such as ImageNet or CIFAR   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] We only use datasets that are widely used in computer vision such as ImageNet or CIFAR
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+        "text": "ResMLP: Feedforward networks for image classification with data-efficient training ",
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+        "text": "We present ResMLP, an architecture built entirely upon multi-layer perceptrons for image classification. It is a simple residual network that alternates (i) a linear layer in which image patches interact, independently and identically across channels, and (ii) a two-layer feed-forward network in which channels interact independently per patch. When trained with a modern training strategy using heavy data-augmentation and optionally distillation, it attains surprisingly good accuracy/complexity tradeoffs on ImageNet. We also train ResMLP models in a self-supervised setup, to further remove priors from employing a labelled dataset. Finally, by adapting our model to machine translation we achieve surprisingly good results. ",
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+        "text": "11 1 Introduction ",
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+        "text": "12 Recently, the transformer architecture [60], adapted from its original use in natural language pro  \n13 cessing with only minor changes, has achieved performance competitive with the state of the art on   \n14 ImageNet-1k [50] when pre-trained with a sufficiently large amount of data [16]. Retrospectively,   \n15 this achievement is yet another step towards less priors: convolutional neural networks had removed   \n16 a lot of hand-made choices compared to hand-designed pre-CNN approaches, moving the paradigm   \n17 of hard-wired features to hand-designed architectural choices. Vision transformers avoid making   \n18 assumptions inherent to convolutional architectures and noticeably the translation invariance.   \n19 What these recent transformer-based works suggest is that longer training schedules, more parameters,   \n20 more data [16] and/or more regularization [56], are sufficient to recover the important priors for tasks   \n21 as complex as ImageNet classification. See also our discussion of related work in Section 4. This   \n22 concurs with recent studies [2, 15] that better disentangle the benefits from the architectures from   \n23 those of the training scheme.   \n24 In this paper, we push this trend further, and propose Residual Multi-Layer Perceptrons (ResMLP):   \n25 a purely multi-layer perceptron (MLP) based architecture for image classification. We outline our   \n26 architecture in Figure 1 and detail it further in Section 2. It is intended to be simple: it takes image   \n27 patches as input, projects them with a linear layer, and sequentially updates them in turn with two   \n28 residual operations: (i) a simple linear layer that provides interaction between the patches, which   \n29 is applied to all channels independently; and (ii) an MLP with a single hidden layer, which is   \n30 independently applied to all patches. At the end of the network, the patches are average pooled, and   \n31 fed to a linear classifier.   \n32 This architecture is strongly inspired by the vision transformers (ViT) [16], yet it is much simpler   \n33 in several ways: we do not use any form of attention, only linear layers along with the GELU   \n34 non-linearity [25]. Since our architecture is much more stable to train than transformers, we do not   \n35 need batch-specific or cross-channel normalizations such as BatchNorm, GroupNorm or LayerNorm.   \n36 Our training procedure mostly follows the one initially introduced for DeiT [56] and CaiT [57].   \n37 Due to its linear nature, the patch interactions in our model can be easily visualised and interpreted.   \n38 While the interaction pattern learned in the first layer is very similar to a small convolutional filter,   \n39 we observe more subtle interactions across patches in deeper layers. These include some form of   \n40 axial filters, and long-range interactions early in the network. ",
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+            "Figure 1: The ResMLP architecture: After linearly projecting the image patches, our network alternately processes them by (1) a communication layer between vectors implemented as a linear layer; (2) a two-layer residual perceptron. We denote by A the Affine element-wise transformation, and by T the transposition. "
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+        "text": "41 In summary, in this paper, we show that ",
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+        "text": "• despite their simplicity, Residual Multi-Layer Perceptrons reach surprisingly good accuracy/complexity trade-offs with ImageNet-1k training only1, without requiring normalization based on batch or channel statistics; ",
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+        "text": "• these models benefit significantly from distillation methods [56]; they are also compatible with modern self-supervised learning methods based on data augmentation, such as DINO [6]; ",
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+        "text": "• thank to its design where patch embeddings simply “communicate” through a linear layer, we can make observations on the spatial interaction that the network learns across layers; ",
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+        "text": "• we adapt ResMLP to machine translation, and again obtain surprisingly good results. ",
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+        "text": "50 2 Method ",
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+        "text": "51 Our model, depicted in Figure 1, is inspired by the ViT model, from which we adopt the columnar   \n52 structure with fixed-resolution feature maps. We proceed two drastic simplifications. We refer the   \n53 reader to Dosovitskiy et al. [16] for more details about the ViT architecture.   \n54 The overall ResMLP architecture. Our model, denoted by ResMLP, takes a grid of $N \\times N$ non  \n55 overlapping patches as input, where the patch size is typically equal to $1 6 \\times 1 6$ . The patches are then   \n56 independently passed through a linear layer to form a set of $\\overline { { N ^ { 2 } d } }$ -dimensional embeddings.   \n57 The resulting set of $N ^ { 2 }$ embeddings are fed to a sequence of Residual Multi-Layer Perceptron layers   \n58 to produce a set of $N ^ { 2 } d .$ -dimensional output embeddings. These output embeddings are then averaged   \n59 as a $d$ -dimension vector to represent the image, which is fed to a linear classifier to predict the label   \n60 associated with the image. Training uses the cross-entropy loss.   \n61 The Residual Multi-Perceptron Layer. Our network is a sequence of layers that all have the same   \n62 structure: a linear sublayer followed by a feedforward sublayer. Similar to the Transformer layer,   \n63 each sublayer is paralleled with a skip-connection [23]. We do not apply Layer Normalization [1]   \n64 because training is stable without it when using the following Affine transformation: ",
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+        "text": "$$\n\\begin{array} { r } { \\mathbb { A } \\mathbb { f } \\mathbb { f } _ { \\alpha , \\beta } ( \\mathbf { x } ) = \\mathbb { D } \\mathrm { i } \\mathtt { a } \\mathbb { g } ( \\pmb { \\alpha } ) \\mathbf { x } + \\beta , } \\end{array}\n$$",
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+        "text": "65 where $_ { \\pmb { \\alpha } }$ and $\\beta$ are learnable weight vectors. This operation simply rescales and shifts the input   \n66 element-wise. Moreover, it has no cost at inference time, as it can absorbed in the adjacent linear   \n67 layer. Note, when writing $\\operatorname { \\mathbb { A } f f } ( \\mathbf { X } )$ the operation is applied independently to each column of $\\mathbf { X }$ .   \n68 While similar to BatchNorm [30] and Layer Normalization [1], the Aff operator does not depend on   \n69 any batch statistics. Therefore, it is closer to the recent LayerScale method [57], which improves the   \n70 optimization of deep transformers when initializing $_ { \\pmb { \\alpha } }$ to a small value. Note, LayerScale does not   \n71 have a bias term.   \n72 We apply this transformation twice for each residual block. As as a pre-normalization Aff replaces   \n73 the LayerNormalization, and avoids using channel-wise statistics. Here, we initialize $\\alpha = 1$ , and   \n74 $\\beta = { \\bf 0 }$ . As a post-processing of the residual block, Aff implements LayerScale and therefore we   \n75 follow the same small value initialization for $_ { \\pmb { \\alpha } }$ as in [57] for the post-normalization.   \n76 Finally, we follow the same structure for the feedforward sublayer as in the Transformer; we only   \n77 replace the ReLU non-linearity by a GELU function [25].   \n78 Overall, our Multi-layer perceptron takes a set of $N ^ { 2 } \\ d .$ -dimensional input features stacked in a   \n79 $d \\times N ^ { 2 }$ matrix $\\mathbf { X }$ , and outputs a set of $N ^ { 2 }$ $d$ -dimension output features, stacked in a matrix $\\mathbf { Y }$ with   \n80 the following set of transformations: ",
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+        "text": "$$\n\\begin{array} { r c l } { \\mathbf { Z } } & { = } & { \\mathbf { X } + \\mathbb { A } \\mathbb { f } \\mathbb { f } \\left( \\left( \\mathbf { A } \\mathbb { A } \\mathbb { f } \\mathbb { f } \\left( \\mathbf { X } \\right) ^ { \\top } \\right) ^ { \\top } \\right) , } \\\\ { \\mathbf { Y } } & { = } & { \\mathbf { Z } + \\mathbb { A } \\mathbb { f } \\mathbb { f } \\left( \\mathbf { C } \\mathbb { G } \\mathbb { E } \\mathbb { L } \\mathbb { U } \\left( \\mathbf { B } \\mathbb { A } \\mathbb { f } \\mathbb { f } \\left( \\mathbf { Z } \\right) \\right) \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "text": "81 where A, $\\mathbf { B }$ and $\\mathbf { C }$ are the main learnable weight matrices of the layer. The dimensions of the   \n82 parameter matrix $\\mathbf { A }$ are $N ^ { 2 } \\times N ^ { 2 }$ , i.e., this sublayer exchanges information across all the locations,   \n83 while the feedforward sublayer works per location. As a consequence, the intermediate activation   \n84 matrix $\\mathbf { Z }$ has the same dimensions as the matrices $\\mathbf { X }$ and $\\mathbf { Y }$ . Finally, the weight matrices $\\mathbf { B }$ and $\\mathbf { C }$   \n85 have the same dimensions as in a Transformer layer, which are $4 d { \\times } d$ and $d \\times 4 d$ , respectively.   \n86 The main difference compared to a Transformer layer is that we replace the self-attention by the   \n87 linear interaction defined in Eq. (2). While self-attention computes a convex combination of other   \n88 features with coefficients that are data dependent, the linear interaction layer in Eq. (2) computes   \n89 a general linear combination using learned coefficients that are not data dependent. As compared   \n90 to a convolutional layers which have local support and share weights across space, our linear   \n91 patch interaction layer offers a global support and does not share weights, moreover it is applied   \n92 independently across channels.   \n93 Relationship to the Vision Transformer. Our model can be regarded as a drastic simplification of   \n94 the ViT model by Dosovitskiy et al. [16]. We depart from this model as follows: ",
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+        "text": "• We do not include any self-attention block. Instead we have a linear patch interaction layer without any non-linearity. ",
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+        "text": "• We do not have the extra “class” token that is typically used in these models to aggregate information via attention. Instead, we simply use average pooling. We do, however, also consider a specific aggregation layer as a variant, which we describe in the next paragraph. ",
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+        "text": "• We do not include any form of positional embedding: the linear communication module between patches implicitly takes into account the patch position. ",
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+        "text": "• Instead of pre-LayerNormalization, we use a simple learnable affine transform, thus avoiding any form of batch and channel-wise statistics. ",
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+        "text": "104 Class-MLP. As an alternative to average pooling, we also experimented with an adaptation of the   \n105 class-attention introduced in CaiT [57]. In CaiT, this consists of two layers that have the same   \n106 structure as the transformer, but in which only the class token is updated based on the frozen patch   \n107 embeddings. We translate this method to our architecture, except that, after aggregating the patches   \n108 with a linear layer, we replace the attention-based interaction between the class and patch embeddings   \n109 by simple linear layers, still keeping the patch embeddings frozen. This increases the performance, at   \n110 the expense of adding some parameters and computational cost. We refer to this pooling variant as   \n111 “class-MLP”, since the purpose of these few layers is to replace average pooling. ",
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+        "type": "text",
+        "text": "112 3 Experiments ",
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+        "text": "113 In this section, we present experimental results for our ResMLP architecture for image classification.   \n114 We also study the impact of the different components in a series of ablations. We consider three   \n115 training paradigms in our experiments:   \n6 • Supervised learning: We train ResMLP from labeled images with a softmax classifier and cross  \n7 entropy loss. This paradigm is the main focus of our work. ",
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+        "table_caption": [
+            "Table 1: Comparison between architectures on ImageNet classification. We compare different architectures based on convolutional networks, Transformers and feedforward networks with comparable FLOPs and number of parameters. We report Top-1 accuracy on the validation set of ImageNet-1k with different measure of complexity: throughput, FLOPs, number of parameters and peak memory usage. All the models use $2 2 4 \\times 2 2 4$ images as input. By default the Transformers and feedforward networks uses $1 4 \\times 1 4$ patches of size $1 6 \\times 1 6$ , see Table 3 for the detailed specification of our main models. The throughput is measured on a single V100-32GB GPU with batch size fixed to 32. For reference, we include the state of the art with ImageNet training only. "
+        ],
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+        "table_body": "<table><tr><td></td><td>Arch.</td><td>#params (x106)</td><td>throughput (im/s)</td><td>FLOPS (x109)</td><td>Peak Mem (MB)</td><td>Top-1 Acc.</td></tr><tr><td rowspan=\"2\">State of the art</td><td>CaiT-M48↑448Y [57]</td><td>356</td><td>5.4</td><td>329.6</td><td>5477.8</td><td>86.5</td></tr><tr><td>NfNet-F6 SAM [5]</td><td>438</td><td>16.0</td><td>377.3</td><td>5519.3</td><td>86.5</td></tr><tr><td rowspan=\"6\">Convolutional networks</td><td>EfficientNet-B3 [53]</td><td>12</td><td>661.8</td><td>1.8</td><td>1174.0</td><td>81.1</td></tr><tr><td>EfficientNet-B4 [53]</td><td>19</td><td>349.4</td><td>4.2</td><td>1898.9</td><td>82.6</td></tr><tr><td>EfficientNet-B5 [53]</td><td>30</td><td>169.1</td><td>9.9</td><td>2734.9</td><td>83.3</td></tr><tr><td>RegNetY-4GF[47]</td><td>21</td><td>861.0</td><td>4.0</td><td>568.4</td><td>80.0</td></tr><tr><td>RegNetY-8GF[47]</td><td>39</td><td>534.4</td><td>8.0</td><td>841.6</td><td>81.7</td></tr><tr><td>RegNetY-16GF[47]</td><td>84</td><td>334.7</td><td>16.0</td><td>1329.6</td><td>82.9</td></tr><tr><td rowspan=\"3\">Transformer networks</td><td>DeiT-S [56]</td><td>22</td><td>940.4</td><td>4.6</td><td>217.2</td><td>79.8</td></tr><tr><td>DeiT-B [56]</td><td>86</td><td>292.3</td><td>17.5</td><td>573.7</td><td>81.8</td></tr><tr><td>CaiT-XS24 [57]</td><td>27</td><td>447.6</td><td>5.4</td><td>245.5</td><td>81.8</td></tr><tr><td rowspan=\"3\">Feedforward networks</td><td>ResMLP-S12</td><td>15</td><td>1415.1</td><td>3.0</td><td>179.5</td><td>76.6</td></tr><tr><td>ResMLP-S24</td><td>30</td><td>715.4</td><td>6.0</td><td>235.3</td><td>79.4</td></tr><tr><td>ResMLP-B24</td><td>116</td><td>231.3</td><td>23.0</td><td>663.0</td><td>81.0</td></tr></table>",
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+        "text": "• Self-supervised learning: We train the ResMLP architecture without labels. We consider the DINO method of Caron et al. [6] that trains a network by distilling knowledge from previous instances of the same network, leading to a form of self-distillation without labels. ",
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+        "text": "• Knowledge distillation: We employ the knowledge distillation procedure proposed by Touvron et al. [56] to guide the supervised training of ResMLP with a convnet. ",
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+    {
+        "type": "text",
+        "text": "123 3.1 Experimental setting ",
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+        "text": "Datasets. We train our models on the ImageNet-1k dataset [50], that contains 1.2M images evenly spread over 1,000 object categories. In the absence of an available test set for this benchmark, we follow the standard practice in the community by reporting performance on the validation set. This is not ideal since the validation set was originally designed to select hyper-parameters. Comparing methods on this set may not be conclusive enough because an improvement in performance may not be caused by better modeling, but by a better selection of hyper-parameters. To mitigate this risk, we report additional results in transfer learning and on two alternative versions of ImageNet that have been built to have distinct validation and test sets, namely the ImageNet-real [3] and ImageNet-v2 [49] datasets. We also report a few data-points when training on ImageNet-21k. Our hyper-parameters are mostly adopted from Touvron et al. [56, 57]. ",
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+        "text": "134 Hyper-parameter settings. In the case of supervised learning, we train our network with the Lamb   \n135 optimizer [63] with a learning rate of $5 \\times 1 0 ^ { - 3 }$ and weight decay 0.2. We initialize the LayerScale   \n136 parameters as a function of the depth by following CaiT [57]. The rest of the hyper-parameters follow   \n137 the default setting used in DeiT [56]. For the knowledge distillation paradigm, we use the same   \n138 RegNety-16GF [48] as in DeiT with the same training schedule. The majority of our models take two   \n139 days to train on eight V100-32GB GPUs. ",
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+        "text": "3.2 Main Results ",
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+        "text": "In this section, we compare our architecture to models with more conventional network architectures of comparable size and throughput on ImageNet. ",
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+        "text": "43 Comparison with Transformers and convnets in a supervised setting. In Table 1, we compare   \n44 ResMLP with different convolutional and Transformer architectures. For completeness, we report   \n145 the best-published numbers obtained with a model trained on ImageNet alone. As expected, in   \n146 terms of the trade-off between accuracy, FLOPs, and throughput, ResMLP is not as efficient as   \n147 convolutional networks or Transformers. However, their accuracy is encouraging: we compare them   \n148 with architectures that have benefited from years of research and careful optimization towards these   \n149 trade-offs. Overall, our results suggest that the structural constraints imposed by the layer design do   \n150 not have a drastic influence on performance, especially when training models with enough data and   \n151 recent advances in training and regularization.   \n152 Self-supervised pre-training of ResMLP. We explore the possibility of training ResMLP using   \n153 DINO, a recent self-supervised learning approach [6]. We pre-train ResMLP-S12 models with this   \n154 approach during 300 epochs. We report our results in Table 2. As expected given the supervised   \n155 classification results, the accuracies obtained with ResMLP are less good than with ViT. Nevertheless,   \n156 the performance is surprisingly high for a pure MLP architecture and competitive with Convnet in   \n157 knn evaluation. We hope that these result will serve as a baseline for future work.   \n158 After self-supervised pre-training, we also fine-tune the network on ImageNet using ground truth   \n159 labels. This pre-training substantially improves the accuracy, when comparing with the same model   \n160 ResMLP-S24 solely trained with labels (top-1 acc. of $7 9 . 9 \\%$ on ImageNet-val instead of $7 9 . 4 \\%$ for   \n161 ResMLP-S24, with the same total number of epochs). Results on ImageNet-v2 suggest that it reduces   \n162 overfitting $6 8 . 6 \\%$ on ImageNet-v2, vs $6 7 . 9 \\%$ with supervised training only).   \n163 Improving models with knowledge distillation. We study our model when training following the   \n164 knowledge distillation approach of Touvron et al. [56]. In their work, the authors show the impact of   \n165 training a ViT model by distilling it from a RegNet. In this experiment, we explore if ResMLP also   \n166 benefits from this procedure and summarize our results in Table 3 (Blocks “Baseline models” and   \n167 “Training”). We observe that similar to DeiT models, ResMLP greatly benefits from distilling from a   \n168 convnet. This result concurs with the observations made by d’Ascoli et al. [13], who used convnets   \n169 to initialize feedforward networks. Even though our setting differs from theirs in scale, the problem   \n170 of overfitting for feedforward networks is still present on ImageNet. The additional regularization   \n171 obtained from the distillation is a possible explanation for this improvement.   \n172 Visualisation. Because they are linear, our patch interaction layers from Eq. (2) are easily inter  \n173 pretable. In Figure 2 we visualise the rows of the interaction matrices A as $N \\times N$ images, for our   \n174 ResMLP-S24 model. The early layers show convolution-like patterns: the learned weights resemble   \n175 shifted versions of each other and have local support. Interestingly, in many layers, the support also   \n176 extends along both axes, most prominently seen in layer seven. The last seven layers of the network   \n177 are different: they consist of a spike for the patch itself and a diffuse response across other patches   \n178 with larger or smaller magnitude; see layer 20. ",
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+            "Table 2: Self-supervised learning with DINO [6]. Classification accuracy on ImageNet-1k val. ResMLPs evaluated with linear and $k$ -NN evaluation on ImageNet are comparable to convnets but inferior to ViT. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>ResNet-50</td><td>ViT-S/16</td><td>ViT-S/8</td><td>ViT-B/16</td><td>ResMLP-S12</td><td>ResMLP-S24</td></tr><tr><td>Params. (×106)</td><td>25</td><td>22</td><td>22</td><td>87</td><td>15</td><td>30</td></tr><tr><td>FLOPS (×109)</td><td>4.1</td><td>4.6</td><td>22.4</td><td>17.5</td><td>3.0</td><td>6.0</td></tr><tr><td>Linear</td><td>75.3</td><td>77.0</td><td>79.7</td><td>78.2</td><td>67.5</td><td>72.8</td></tr><tr><td>k-NN</td><td>67.5</td><td>74.5</td><td>78.3</td><td>76.1</td><td>62.6</td><td>69.4</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "179 3.3 Visualization & analysis of the linear interaction between patches ",
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+        "text": "180 Measuring sparsity of the weights. The visualizations described above suggest that the linear   \n181 communication layers are sparse. We analyze this quantitatively in more detail in Figure 3. We   \n182 measure the sparsity of the matrix A, and compare it to the sparsity of $\\mathbf { B }$ and $\\mathbf { C }$ from the per-patch   \n183 MLP. Since there are no exact zeros, we measure the rate of components whose absolute value is   \n184 lower than $5 \\%$ of the maximum value. Note, discarding the small values is analogous to the case   \n185 where we normalize the matrix by its maximum and use a finite-precision representation of weights.   \n186 For instance, with a 4-bits representation of weight, one would typically round to zero all weights   \n187 whose absolute value is below $6 . 2 5 \\%$ of the maximum value.   \n188 The measurements in Figure 3 show that all three matrices are sparse, with the layers implementing   \n189 the patch communication being significantly more so. This suggests that they may be compatible with   \n190 parameter pruning, or better, with modern quantization techniques that induce sparsity at training   \n191 time, such as Quant-Noise [20] and DiffQ [19]. The sparsity structure, in particular in earlier layers,   \n192 see Figure. 2, hints that we could implement the patch interaction linear layer with a convolution. We   \n193 provide some results for convolutional variants in our ablation study. Further research on network   \n194 compression is beyond the scope of this paper, yet we believe it worth investigating in the future.   \n195 Communication across patches if we remove the linear interaction layer (linear none), we   \n196 obtain substantially lower accuracy ( $- 2 0 \\%$ top-1 acc.) for a “bag-of-patches” approach. We have tried   \n197 several alternatives for the linear patch interaction layer, which are presented in Table 3 (block “patch   \n198 communication”). Amongst them, using the same MLP structure as for patch processing (linear   \n199 MLP), which we analyze in more details in the supplementary material. The simpler choice of a   \n200 single linear square layer led to a better accuracy/performance trade-off – considering that the MLP   \n201 variant requires compute halfway between ResMLP-S12 and ResMLP-S24 – and requires fewer   \n202 parameters than a residual MLP block.   \n203 The visualization in Figure 2 indicates that many linear interaction layers look like convolutions. In   \n204 our ablation, we replaced the linear layer with different types of $3 \\times 3$ convolutions. The depth-wise   \n205 convolution does not implement interaction across channels – as our linear patch communication   \n206 layer – and yields similar performance at a comparable number of parameters and FLOPs. While   \n207 full $3 \\times 3$ convolutions yield best results, they come with roughly double the number of parameters ",
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+        "image_caption": [
+            "Figure 2: Visualisation of the linear layers in ResMLP-S24. For each layer we visualise the rows of the matrix A as a set of $1 4 \\times 1 4$ pixel images, for sake of space we only show the rows corresponding to the $6 \\times 6$ central patches. We observe patterns in the linear layers that share similarities with convolutions. In appendix B we provide comparable visualizations for all layers of a ResMLP-S12 model. "
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+        "image_caption": [
+            "Figure 3: Sparsity of linear interaction layers. For each layer (linear and MLP), we show the rate of components whose absolute value is lower than $5 \\%$ of the maximum. Linear interaction layers are sparser than the matrices involved in the per-patch MLP. "
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+        "image_caption": [
+            "Figure 4: Top-1 accuracy on ImageNet-V2 vs. ImageNet-val. ResMLPs tend to overfit slightly more under identical training method. This is partially alleviated with by introducing more regularization (more data or distillation, see e.g., ResMLP-B24/8-distil). "
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+    {
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+        "table_caption": [
+            "Table 3: Ablation. Our default configurations are presented in the three first rows. By default we train during 400 epochs. The “old-fashioned” is similar to what was employed for ResNet [23]: SGD, 90-epochs waterfall schedule, same augmentations up to variations due to library used. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Ablation</td><td rowspan=\"2\">Model</td><td rowspan=\"2\">Patch size</td><td rowspan=\"2\">Params ×106</td><td rowspan=\"2\">FLOPs ×109</td><td rowspan=\"2\">Variant</td><td colspan=\"3\">top-1 acc.on ImageNet</td></tr><tr><td>val</td><td>real[3]</td><td>v2 [49]</td></tr><tr><td rowspan=\"3\">Baseline models</td><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>12 layers, working dimension 384</td><td>76.6</td><td>83.3</td><td>64.4</td></tr><tr><td>ResMLP-S24</td><td>16</td><td>30.0</td><td>6.0</td><td>24 layers, working dimension 384</td><td>79.4</td><td>85.3</td><td>67.9</td></tr><tr><td>ResMLP-B24</td><td>16</td><td>115.7</td><td>23.0</td><td>24 layers, working dimension 768</td><td>81.0</td><td>86.1</td><td>69.0</td></tr><tr><td>Normalization</td><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>Aff →Layernorm</td><td>77.7</td><td>84.1</td><td>65.7</td></tr><tr><td>Pooling</td><td>ResMLP-S12</td><td>16</td><td>17.7</td><td>3.0</td><td>average pooling →Class-MLP</td><td>77.5</td><td>84.0</td><td>66.1</td></tr><tr><td rowspan=\"5\">Patch communication</td><td>ResMLP-S12</td><td>16</td><td>14.9</td><td>2.8</td><td>linear→ none</td><td>56.5</td><td>63.4</td><td>43.1</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>18.6</td><td>4.3</td><td>linear→MLP</td><td>77.3</td><td>84.0</td><td>65.7</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>30.8</td><td>6.0</td><td>linear → conv 3x3</td><td>77.3</td><td>84.4</td><td>65.7</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>14.9</td><td>2.8</td><td>linear →conv 3x3 depth-wise</td><td>76.3</td><td>83.4</td><td>64.6</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>16.7</td><td>3.2</td><td>linear→conv 3x3 depth-separable</td><td>77.0</td><td>84.0</td><td>65.5</td></tr><tr><td rowspan=\"3\">Patch size</td><td>ResMLP-S12/14</td><td>14</td><td>15.6</td><td>4.0</td><td>patch size 16×16-→14×14</td><td>76.9</td><td>83.7</td><td>65.0</td></tr><tr><td>ResMLP-S12/8</td><td>8</td><td>22.1</td><td>14.0</td><td>patch size 16×16-8×8</td><td>79.1</td><td>85.2</td><td>67.2</td></tr><tr><td>ResMLP-B24/8</td><td>8</td><td>129.1</td><td>100.2</td><td> patch size 16×16-&gt;8 ×8</td><td>81.0</td><td>85.7</td><td>68.6</td></tr><tr><td rowspan=\"7\">Training</td><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>old-fashioned (90 epochs)</td><td>69.2</td><td>76.0</td><td>56.1</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>pre-trained SSL (DINO)</td><td>76.5</td><td>83.6</td><td>64.5</td></tr><tr><td>ResMLP-S12</td><td>16</td><td>15.4</td><td>3.0</td><td>distillation</td><td>77.8</td><td>84.6</td><td>66.0</td></tr><tr><td>ResMLP-S24</td><td>16</td><td>30.0</td><td>6.0</td><td> pre-trained SSL (DINO)</td><td>79.9</td><td>85.9</td><td>68.6</td></tr><tr><td>ResMLP-S24</td><td>16</td><td>30.0</td><td>6.0</td><td>distillation</td><td>80.8</td><td>86.6</td><td>69.8</td></tr><tr><td>ResMLP-B24/8</td><td>8</td><td>129.1</td><td>100.2</td><td>distillation</td><td>83.6</td><td>88.4</td><td>73.4</td></tr><tr><td>ResMLP-B24/8</td><td>8</td><td>129.1</td><td>100.2</td><td>pre-trained ImageNet-21k (60 epochs)</td><td>84.4</td><td>88.9</td><td>74.2</td></tr></table>",
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+        "type": "text",
+        "text": "and FLOPs. Interestingly, the depth-separable convolutions combine accuracy close to that of full $3 \\times 3$ convolutions with a number of parameters and FLOPs comparable to our linear layer. This suggests that convolutions on low-resolution feature maps at all layers is an interesting alternative to the common pyramidal design of convnets, where early layers operate at higher resolution and smaller feature dimension. ",
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+    {
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+        "text": "3.4 Ablation studies ",
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+        "type": "text",
+        "text": "Table 3 reports the ablation study of our base network and a summary of our preliminary exploratory studies. We discuss the ablation below and give more detail about early experiments in Appendix A. ",
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+        "type": "text",
+        "text": "Control of overfitting. Since MLPs are subject to overfitting, we show in Fig. 4 a control experiment to probe for problems with generalization. We explicitly analyze the differential of performance between the ImageNet-val and the distinct ImageNet-V2 test set. The relative offsets between curves reflect to which extent models are overfitted to ImageNet-val w.r.t. hyper-parameter selection. The degree of overfitting of our MLP-based model is overall neutral or slightly higher to that of other transformer-based architectures or convnets with same training procedure. ",
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+    {
+        "type": "text",
+        "text": "Normalization & activation. Our network configuration does not contain any batch normalizations. Instead, we use the affine per-channel transform Aff. This is akin to Layer Normalization [1], typically used in transformers, except that we avoid to collect any sort of statistics, since we do no need it it for convergence. In preliminary experiments with pre-norm and post-norm [24], we observed that both choices converged. Pre-normalization in conjunction with Batch Normalization could provide an accuracy gain in some cases, see Appendix A. ",
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+    {
+        "type": "text",
+        "text": "We choose to use a GELU [25] function. In Appendix A we also analyze the activation function: ReLU [22] also gives a good performance, but it was a bit more unstable in some settings. We did not manage to get good results with SiLU [25] and HardSwish [28]. ",
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+    {
+        "type": "text",
+        "text": "Pooling. Replacing average pooling with Class-MLP, see Section 2, brings a significant gain for a negligible computational cost. We do not include it by default to keep our models more simple. ",
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+    {
+        "type": "text",
+        "text": "Patch size. Smaller patches significantly increase the performance, but also increase the number of flops (see Block \"Patch size\" in Table 3). Smaller patches benefit more to larger models, but only with an improved optimization scheme involving more regularization (distillation) or more data. ",
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+    {
+        "type": "text",
+        "text": "6 Training. Consider the Block “Training’ in Table 3. ResMLP significantly benefits from modern training procedures such as those used in DeiT. For instance, the DeiT training procedure improves ",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: Evaluation on transfer learning. Classification accuracy (top-1) of models trained on ImageNet-1k for transfer to datasets covering different domains. The ResMLP architecture takes $2 2 4 \\times 2 2 4$ images during training and transfer, while ViTs and EfficientNet-B7 work with higher resolutions, see “Res.” column. "
+        ],
+        "table_body": "<table><tr><td>Architecture</td><td>FLOPs</td><td>Res.</td><td>CIFAR10</td><td>CIFAR100</td><td>Flowers102</td><td>Cars</td><td>iNat18</td><td>iNat19</td></tr><tr><td>EfficientNet-B7 [53]</td><td>37.0B</td><td>600</td><td>98.9</td><td>91.7</td><td>98.8</td><td>94.7</td><td>1</td><td>1</td></tr><tr><td>ViT-B/16 [16]</td><td>55.5B</td><td>384</td><td>98.1</td><td>87.1</td><td>89.5</td><td>1</td><td>1</td><td>1</td></tr><tr><td>ViT-L/16 [16]</td><td>190.7B</td><td>384</td><td>97.9</td><td>86.4</td><td>89.7</td><td></td><td></td><td></td></tr><tr><td>Deit-B/16 [56]</td><td>17.5B</td><td>224</td><td>99.1</td><td>90.8</td><td>98.4</td><td>92.1</td><td>73.2</td><td>77.7</td></tr><tr><td>ResNet50 [58]</td><td>4.1B</td><td>224</td><td>1</td><td>1</td><td>96.2</td><td>90.0</td><td>68.4</td><td>73.7</td></tr><tr><td>Grafit/ResNet50 [58]</td><td>4.1B</td><td>224</td><td>1</td><td>1</td><td>97.6</td><td>92.7</td><td>68.5</td><td>74.6</td></tr><tr><td>ResMLP-S12</td><td>3.0B</td><td>224</td><td>98.1</td><td>87.0</td><td>97.4</td><td>84.6</td><td>60.2</td><td>71.0</td></tr><tr><td>ResMLP-S24</td><td>6.0B</td><td>224</td><td>98.7</td><td>89.5</td><td>97.9</td><td>89.5</td><td>64.3</td><td>72.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "238 the performance of ResMLP-S12 by $7 . 4 \\%$ compared to the training employed for ResNet $[ 2 3 ]$ .   \n239 This is in line with recent work pointing out the importance of the training strategy over the model   \n240 choice [2, 48]. Pre-training on more data and distillation also improve the performance of ResMLP,   \n241 especially for the bigger models, e.g., distillation improves the accuracy of ResMLP-B24/8 by $2 . 6 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "Other analysis. In our early exploration, we evaluated several alternative design choices. As in transformers, we could use positional embeddings mixed with the input patches. In our experiments we did not see any benefit from using these features, see Appendix A. This observation suggests that our linear patch interaction layer provides sufficient spatial communication, and referencing absolute positions obviates the need for any form of positional encoding. ",
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+    {
+        "type": "text",
+        "text": "3.5 Transfer learning ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate the quality of features obtained from a ResMLP architecture when transferring them to other domains. The goal is to assess if the features generated from a feedforward network are more prone to overfitting on the training data distribution. We adopt the typical setting where we pre-train a model on ImageNet-1k and fine-tune it on the training set associated with a specific domain. We report the performance with different architectures on various image benchmarks in Table 4, namely CIFAR-10 and CIFAR-100 [34], Flowers-102 [42], Stanford Cars [33] and iNaturalist [27]. We refer the reader to the corresponding references for a more detailed description of the datasets. We observe that the performance of our ResMLP is competitive with the existing architectures, showing that pretraining feedforward models with enough data and regularization via data augmentation greatly reduces their tendency to overfit on the original distribution. Interestingly, this regularization also prevents them from overfitting on the training set of smaller dataset during the fine-tuning stage. ",
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+    {
+        "type": "text",
+        "text": "3.6 Machine translation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also evaluate the ResMLP transpose-mechanism to replace the self-attention in the encoder and decoder of a neural machine translation system. We train models on the WMT 2014 English-German and English-French tasks, following the setup from Ott et al. [45]. We consider models of dimension 512, with a hidden MLP size of 2048, and with 6 or 12 layers. Note that the current state of the art employs much larger models: our 6 layers model is more comparable to the base transformer model from Vaswani et al. [60], which serves as a baseline, along with pre-transformer architectures such as Recurrent and convolutional neural networks. We use Adagrad with learning rate 0.2, $3 2 \\mathrm { k }$ steps of linear warmup, label smoothing 0.1, dropout rate 0.15 for en-de and 0.1 for en-fr. We initialize the LayerScale parameter to 0.2. We generate translations with the beam search algorithm, with a beam of size 4. As shown in Table 5, the results are at least on par with other architectures: ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/7c33884cad304759188150131d5a467acbad94c727a430b91de45bcf9cec553f.jpg",
+        "table_caption": [
+            "Table 5: Machine translation on WMT 2014 translation tasks. We report tokenized BLEU on newstest2014. "
+        ],
+        "table_footnote": [
+            "2Interestingly, if trained with this “old-fashion” setting, ResMLP-S12 outperforms AlexNet [35] by a margin. "
+        ],
+        "table_body": "<table><tr><td>Models</td><td>GNMT[61]</td><td></td><td>ConvS2S [21]Transf. (base)[60]</td><td>ResMLP-6</td><td>ResMLP-12</td></tr><tr><td>EN-DE</td><td>24.6</td><td>25.2</td><td>27.3</td><td>26.4</td><td>26.8</td></tr><tr><td>EN-FR</td><td>39.9</td><td>40.5</td><td>38.1</td><td>40.3</td><td>40.6</td></tr></table>",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "We review the research on applying Fully Connected Network (FCN) for computer vision problems as well as other architectures that shares common modules with our model. ",
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+        "text": "Fully-connected network for images. Many studies have shown that FCNs are competitive with convnets for the tasks of digit recognition [11, 51], keyword spotting [7] and handwritting recognition [4]. Several works [37, 40, 59] have questioned if FCNs are also competitive on natural image datasets, such as CIFAR-10 [34]. More recently, d’Ascoli et al. [13] have shown that a FCN initialized with the weights of a pretrained convnet achieves performance that are superior than the original convnet. Neyshabur [41] further extend this line of work by achieving competitive performance by training an FCN from scratch but with a regularizer that constrains the models to be close to a convnet. These studies have been conducted on small scale datasets with the purpose of studying the impact of architectures on generalization in terms of sample complexity [18] and energy landscape [31]. In our work, we show that, in the larger scale setting of ImageNet, FCNs can attain surprising accuracy without any constraint or initialization inspired by convnets. ",
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+        "text": "Finally, the application of FCN networks in computer vision have also emerged in the study of the properties of networks with infinite width [43], or for inverse scattering problems [32]. More interestingly, the Tensorizing Network [44] is an approximation of very large FCN that shares similarity with our model, in that they intend to remove prior by approximating even more general tensor operations, i.e., not arbitrarily marginalized along some pre-defined sharing dimensions. However, their method is designed to compress the MLP layers of a standard convnets. ",
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+        "text": "Other architectures with similar components. Our FCN architecture shares several components with other architectures, such as convnets [35, 36] or transformers [60]. A fully connected layer is equivalent to a convolution layer with a $1 \\times 1$ receptive field, and several work have explored convnet architectures with small receptive fields. For instance, the VGG model [52] uses $3 \\times 3$ convolutions, and later, other architectures such as the ResNext [62] or the Xception [10] mix $1 \\times 1$ and $3 \\times 3$ convolutions. In contrast to convnets, in our model interaction between patches is obtained via a linear layer that is shared across channels, and that relies on absolute rather than relative positions. ",
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+        "text": "More recently, transformers have emerged as a promising architecture for computer vision [9, 17, 46, 56, 66]. In particular, our architecture takes inspiration from the structure used in the Vision Transformer (ViT) [17], and as consequence, shares many components. Our model takes a set of non-overlapping patches as input and passes them through a series of MLP layers that share the same structure as ViT, replacing the self-attention layer with a linear patch interaction layer. Both layers have a global field-of-view, unlike convolutional layers. Whereas in self-attention the weights to aggregate information from other patches are data dependent through queries and keys, in ResMLP the weights are not data dependent and only based on absolute positions of patches. In our implementation we follow the improvements of DeiT [56] to train vision transformers, use the skip-connections from ResNets [23] with pre-normalization of the layers [8, 24]. ",
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+    {
+        "type": "text",
+        "text": "307 Finally, our work questions the importance of self-attention in existing architectures. Similar ob  \n308 servations have been made in natural language processing. Notably, Synthesizer [54] shows that   \n309 dot-product self-attention can be replaced by a feedforward network, with competitive performance   \n310 on sentence representation benchmarks. As opposed to our work, Synthesizer does use data dependent   \n311 weights, but in contrast to transformers the weights are determined from the queries only. ",
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+        "text": "312 5 Conclusion ",
+        "text_level": 1,
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+        "text": "In this paper we have shown that a simple residual architecture, whose residual blocks consist of a one-hidden layer feed-forward network and a linear patch interaction layer, achieves an unexpectedly high performance on ImageNet classification benchmarks, provided that we adopt a modern training strategy such as those recently introduced for transformer-based architectures. Thanks to their simple structure, with linear layers as the main mean of communication between patches, we can vizualize the filters learned by this simple MLP. While some of the layers are similar to convolutional filters, we also observe sparse long-range interactions as early as the second layer of the network. We hope that our model free of spatial priors will contribute to further understanding of what networks with less priors learn, and potentially guide the design choices of future networks without the pyramidal design prior adopted by most convolutional neural networks. ",
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parse/train/UVH3Ucewd-IXZ/UVH3Ucewd-IXZ.md

@@ -0,0 +1,218 @@
+# Deep learning for neuroimaging: a validation study
+
+Sergey M. Plis The Mind Research Network Albuquerque, NM 87106 s.m.plis@gmail.com
+
+Devon R. Hjelm   
+University of New Mexico   
+Albuquerque, NM 87131   
+dhjelm@mrn.org
+
+Ruslan Salakhutdinov University of Toronto Toronto, Ontario M5S 2E4 rsalakhu@cs.toronto.edu
+
+Vince D. Calhoun The Mind Research Network Albuquerque, NM 87106 vcalhoun@mrn.org
+
+# Abstract
+
+Deep learning methods have recently made notable advances in the tasks of classification and representation learning. These tasks are important for brain imaging and neuroscience discovery, making the methods attractive for porting to a neuroimager’s toolbox. Success of these methods is, in part, explained by the flexibility of deep learning models. However, this flexibility makes the process of porting to new areas a difficult parameter optimization problem. In this work we demonstrate our results (and feasible parameter ranges) in application of deep learning methods to structural and functional brain imaging data. We also describe a novel constraint-based approach to visualizing high dimensional data. We use it to ana- lyze the effect of parameter choices on data transformations. Our results show that deep learning methods are able to learn physiologically important representations and detect latent relations in neuroimaging data.
+
+# 1 Introduction
+
+One of the main goals of brain imaging and neuroscience—and, possibly, of most natural sciences— is to improve understanding of the investigated system based on data. In our case, this amounts to inference of descriptive features of brain structure and function from non-invasive measurements. Brain imaging field has come a long way from anatomical maps and atlases towards data driven feature learning methods, such as seed-based correlation [2], canonical correlation analysis [33], and independent component analysis (ICA) [1, 24]. These methods are highly successful in revealing known brain features with new details [3] (supporting their credibility), in recovering features that differentiate patients and controls [28] (assisting diagnosis and disease understanding), and starting a “resting state” revolution after revealing consistent patters in data from uncontrolled resting experiments [29, 35]. Classification is often used merely as a correctness checking tool, as the main emphasis is on learning about the brain. A perfect oracle that does not explain its conclusions would be useful, but mainly to facilitate the inference of the ways the oracle draws these conclusions.
+
+As an oracle, deep learning methods are breaking records taken over the areas of speech, signal, image, video and text mining and recognition by improving state of the art classification accuracy by, sometimes, more than $30 \%$ where the prior decade struggled to obtain a $1 \%$ improvements [19, 21]. What differentiates them from other classifiers, however, is the automatic feature learning from data which largely contributes to improvements in accuracy. Presently, this seems to be the closest solution to an oracle that reveals its methods — a desirable tool for brain imaging.
+
+Another distinguishing feature of deep learning is the depth of the models. Based on already acceptable feature learning results obtained by shallow models—currently dominating neuroimaging field—it is not immediately clear what benefits would depth have. Considering the state of multimodal learning, where models are either assumed to be the same for analyzed modalities [26] or cross-modal relations are sought at the (shallow) level of mixture coefficients [23], deeper models better fit the intuitive notion of cross-modality relations, as, for example, relations between genetics and phenotypes should be indirect, happening at a deeper conceptual level.
+
+In this work we present our recent advances in application of deep learning methods to functional and structural magnetic resonance imaging (fMRI and sMRI). Each consists of brain volumes but for sMRI these are static volumes—one per subject/session,—while for fMRI a single subject dataset is comprised of multiple volumes capturing the changes during an experimental session. Our goal is to validate feasibility of this application by $a$ ) investigating if a building block of deep generative models—a restricted Boltzmann machine (RBM) [17]—is competitive with ICA (a representative model of its class) (Section 2); $b$ ) examining the effect of the depth in deep learning analysis of structural MRI data (Section 3.3); and $c$ ) determining the value of the methods for discovery of latent structure of a large-scale (by neuroimaging standards) dataset (Section 3.4). The measure of feature learning performance in a shallow model (a) is comparable with existing methods and known brain physiology. However, this measure cannot be used when deeper models are investigated. As we further demonstrate, classification accuracy does not provide the complete picture either. To be able to visualize the effect of depth and gain an insight into the learning process, we introduce a flexible constraint satisfaction embedding method that allows us to control the complexity of the constraints (Section 3.2). Deliberately choosing local constraints we are able to reflect the transformations that the deep belief network (DBN) [15] learns and applies to the data and gain additional insight.
+
+# 2 A shallow belief network for feature learning
+
+Prior to investigating the benefits of depth of a DBN in learning representations from fMRI and sMRI data, we would like to find out if a shallow (single hidden layer) model–which is the RBM— from this family meets the field’s expectations. As mentioned in the introduction, a number of methods are used for feature learning from neuroimaging data: most of them belong to the single matrix factorization (SMF) class. We do a quick comparison to a small subset of SMF methods on simulated data; and continue with a more extensive comparison against ICA as an approach trusted in the neuroimaging field. Similarly to RBM, ICA relies on the bipartite graph structure, or even is an artificial neural network with sigmoid hidden units as is in the case of Infomax ICA [1] that we compare against. Note the difference with RBM: ICA applies its weight matrix to the (shorter) temporal dimension of the data imposing independence on the spatial dimension while RBM applies its weight matrix (hidden units “receptive fields”) to the high dimensional spatial dimension instead (Figure 2).
+
+# 2.1 A restricted Boltzmann machine
+
+A restricted Boltzmann machine (RBM) is a Markov random field that models data distribution parameterizing it with the Gibbs distribution over a bipartite graph between visible $\pmb { v }$ and hidden variables $^ { h }$ [10]: $\begin{array} { r } { p ( \pmb { v } ) = \sum _ { \pmb { h } } p ( \pmb { v } , \pmb { h } ) = \sum _ { \pmb { h } } 1 / Z \exp ( - E ( \pmb { v } , \pmb { h } ) ) } \end{array}$ , where $\begin{array} { r } { Z = \sum v \sum _ { h } e ^ { - E ( { \pmb v } , { \pmb h } ) } } \end{array}$ is the normalization term (the partition function) and $E ( v , h )$ is the energy of the system. Each visible variable in the case of fMRI data represents a voxel of an fMRI scan with a real-valued and approximately Gaussian distribution. In this case, the energy is defined as:
+
+$$
+E ( v , h ) = - \sum _ { i j } \frac { v _ { j } } { \sigma _ { j } } W _ { j i } h _ { i } - \sum _ { j } \frac { ( a _ { j } - v _ { j } ) ^ { 2 } } { \sigma _ { j } ^ { 2 } } - \sum _ { i } b _ { i } h _ { i } ,
+$$
+
+where $a _ { j }$ and $b _ { j }$ are biases and $\sigma _ { j }$ is the standard deviation of a parabolic containment function for each visible variable $v _ { j }$ centered on the bias $a _ { j }$ . In general, the parameters $\sigma _ { i }$ need to be learned along with the other parameters. However, in practice normalizing the distribution of each voxel to have zero mean and unit variance is faster and yet effective [27]. A number of choices affect the quality of interpretation of the representations learned from fMRI by an RBM. Encouraging sparse features via the $L _ { 1 }$ -regularization: $\lambda \| W \| _ { 1 }$ ( $\lambda = 0 . 1$ gave best results) and using hyperbolic tangent for hidden units non-linearity are essential settings that respectively facilitate spatial and temporal interpretation of the result. The weights were updated using the truncated Gibbs sampling method called contrastive divergence (CD) with a single sampling step (CD-1). Further information on RBM model can be found in [16, 17].
+
+# 2.2 Synthetic data
+
+![](images/03e5dc7c4b4fc13894498eac8e135ded37c71d1069aa60fab7dab98ab2c17ad8.jpg)
+
+a: Average spatial map (SM) and time course (TC) correlations to ground truth for RBM and SMF models (gray box).
+
+b: Ground truth (GT) SMs and estimates obtained by RBM and ICA (thresholded at 0.4 height). Colors are consistent across the methods. Grey indicates background or areas without SMs above threshold.
+
+c: Spatial, temporal, and cross correlation (FNC) accuracy for ICA (red) and RBM (blue), as a function of spatial overlap of the true sources from 1b. Lines indicate the average correlation to GT, and the color-fill indicates $\pm 2$ standard errors around the mean.
+
+Figure 1: Comparison of RBM estimation accuracy of features and their time courses with SMFs.
+
+In this section we summarize our comparisons of RBM with SMF models—including Infomax ICA [1], PCA [14], sparse PCA (sPCA) [37], and sparse NMF (sNMF) [18]—on synthetic data with known spatial maps generated to simulate fMRI.
+
+Figure 1a shows the correlation of spatial maps (SM) and time course (TC) estimates to the ground truth for RBM, ICA, PCA, sPCA, and sNMF. Correlations are averaged across all sources and datasets. RBM and ICA showed the best overall performance. While sNMF also estimated SMs well, it showed inferior performance on TC estimation, likely due to the non-negativity constraint. Based on these results and the broad adoption of ICA in the field, we focus on comparing Infomax ICA and RBM.
+
+Figure 1b shows the full set of ground truth sources along with RBM and ICA estimates for a single representative dataset. SMs are thresholded and represented as contours for visualization. Results over all synthetic datasets showed similar performance for RBM and ICA (Figure 1c), with a slight advantage for ICA with regard to SM estimation, and a slight advantage for RBM with regards to TC estimation. RBM and ICA also showed comparable performance estimating cross correlations also called functional network connectivity (FNC).
+
+# 2.3 An fMRI data application
+
+![](images/a669dc66ad09d2234094b6f0a2361707082acab0f3b05effb83465e7d44334c7.jpg)  
+Figure 2: The processes of feature learning and time course computation from fMRI data by an RBM. The visible units are voxels and a hidden unit receptive field covers an fMRI volume.
+
+Data used in this work comprised of task-related scans from 28 (five females) healthy participants, all of whom gave written, informed, IRB-approved consent at Hartford Hospital and were compensated for participation1. All participants were scanned during an auditory oddball task (AOD) involving the detection of an infrequent target sound within a series of standard and novel sounds2.
+
+Scans were acquired at the Olin Neuropsychiatry Research Center at the Institute of Living/Hartford Hospital on a Siemens Allegra 3T dedicated head scanner equipped with $4 0 \mathrm { m T } / \mathrm { m }$ gradients and a standard quadrature head coil [4, 9]. The AOD consisted of two 8-min runs, and 249 scans (volumes) at 2 second TR $0 . 5 \ \mathrm { H z }$ sampling rate) were used for the final dataset. Data were post-processed using the SPM5 software package [12], motion corrected using INRIalign [11], and subsampled to $5 3 \times 6 3 \times 4 6$ voxels. The complete fMRI dataset was masked below mean and the mean image across the dataset was removed, giving a complete dataset of size 70969 voxels by 6972 volumes. Each voxel was then normalized to have zero mean and unit variance.
+
+The RBM was constructed using 70969 Gaussian visible units and 64 hyperbolic tangent hidden units. The hyper parameters $\epsilon$ (0.08 from the searched $[ 1 \times 1 0 ^ { - 4 } , 1 \times 1 \dot { 0 } ^ { - 1 } ]$ range) for learning rate and $\lambda$ (0.1 from the searched range $[ 1 \times \mathrm { { 1 0 ^ { - 2 } , 1 \times 1 0 ^ { - 1 } } } ] )$ for $L _ { 1 }$ weight decay were selected as those that showed a reduction of reconstruction error over training and a significant reduction in span of the receptive fields respectively. Parameter value outside the ranges either resulted in unstable or slow learning () or uninterpretable features $( \lambda )$ . The RBM was then trained with a batch size of 5 for approximately 100 epochs to allow for full convergence of the parameters.
+
+![](images/70ccc1f25a64c86a603394c7ffb5b0be3ebecdbb876b304f5e099240f7dd66be.jpg)  
+Figure 3: Intrinsic brain networks estimated by ICA and RBM.
+
+After flipping the sign of negative receptive fields, we then identified and labeled spatially distinct features as corresponding to brain regions with the aid of AFNI [5] excluding features which had a high probability of corresponding to white matter, ventricles, or artifacts (eg. motion, edges).
+
+We normalized the fMRI volume time series to mean zero and used the trained RBM in feed-forward mode to compute time series for each fMRI feature. This was done to better compare to ICA, where the mean is removed in PCA preprocessing.
+
+The work-flow is outlined in Figure 2, while Figure 3 shows comparison of resulting features with those obtained by Infomax ICA. In general, RBM performs competitively with ICA, while providing–perhaps, not surprisingly due to the used $L _ { 1 }$ regularization—sharper and more localized features. While we recognize that this is a subjective measure we list more features in Figure S2 of Section 5 and note that RBM features lack negative parts for corresponding features. Note, that in the case of $L _ { 1 }$ regularized weights RBM algorithms starts to resemble some of the ICA approaches (such as the recent RICA by Le at al. [20]), which may explain the similar performance. However, the differences and possible advantages are the generative nature of the RBM and no enforcement of component orthogonality (not explicit at the least). Moreover, the block structure of the correlation matrix (see below the Supplementary material section) of feature time courses provide a grouping that is more physiologically supported than that provided by ICA. For example, see Figure S1 in the supplementary material section below. Perhaps, because ICA working hard to enforce spatial independence subtly affects the time courses and their cross-correlations in turn. We have observed comparable running times of the (non GPU) ICA (http://www.nitrc.org/projects/gift) and a GPU implementation of the RBM (https://github.com/nitishsrivastava/deepnet).
+
+# 3 Validating the depth effect
+
+Since the RBM results demonstrate a feature-learning performance competitive with the state of the art (or better), we proceed to investigating the effects of the model depth. To do that we turn from fMRI to sMRI data. As it is commonly assumed in the deep learning literature [22] the depth is often improving classification accuracy. We investigate if that is indeed true in the sMRI case. Structural data is convenient for the purpose as each subject/session is represented only by a single volume that has a label: control or patient in our case. Compare to 4D data where hundreds of volumes belong to the same subject with the same disease state.
+
+# 3.1 A deep belief network
+
+A DBN is a sigmoidal belief network (although other activation functions may be used) with an RBM as the top level prior. The joint probability distribution of its visible and hidden units is
+
+parametrized as follows:
+
+$$
+P ( v , h ^ { 1 } , h ^ { 2 } , \dots , h ^ { l } ) = P ( v | h ^ { 1 } ) P ( h ^ { 1 } | h ^ { 2 } ) \cdots P ( h ^ { l - 2 } , h ^ { l - 1 } ) P ( h ^ { l - 1 } , h ^ { l } ) ,
+$$
+
+where $l$ is the number of hidden layers, $P ( h ^ { l - 1 } , h ^ { l } )$ is an RBM, and $P ( h ^ { i } | h ^ { i + 1 } )$ factor into individual conditionals:
+
+$$
+P ( h ^ { i } | h ^ { i + 1 } ) = \prod _ { j = 1 } ^ { n _ { i } } P ( h _ { j } ^ { i } | h ^ { i + 1 } )
+$$
+
+The important property of DBN for our goals of feature learning to facilitate discovery is its ability to operate in generative mode with fixed values on chosen hidden units thus allowing one to investigate the features that the model have learned and/or weighs as important in discriminative decisions. We, however, not going to use this property in this section, focusing instead on validating the claim that a network’s depth provides benefits for neuroimaging data analysis. And we will do this using discriminative mode of DBN’s operation as it provides an objective measure of the depth effect.
+
+DBN training splits into two stages: pre-training and discriminative fine tuning. A DBN can be pre-trained by treating each of its layers as an RBM—trained in an unsupervised way on inputs from the previous layer—and later fine-tuned by treating it as a feed-forward neural network. The latter allows supervised training via the error back propagation algorithm. We use this schema in the following by augmenting each DBN with a soft-max layer at the fine-tuning stage.
+
+# 3.2 Nonlinear embedding as a constraint satisfaction problem
+
+A DBN and an RBM operate on data samples, which are brain volumes in the fMRI and sMRI case. A five-minute fMRI experiment with 2 seconds sampling rate yields 150 of these volumes per subject. For sMRI studies number of participating subjects varies but in this paper we operate with a 300 and a 3500 subject-volumes datasets. Transformations learned by deep learning methods do not look intuitive in the hidden node space and generative sampling of the trained model does not provide a sense if a model have learned anything useful in the case of MRI data: in contrast to natural images, fMRI and sMRI images do not look very intuitive. Instead, we use a nonlinear embedding method to control whether a model learned useful information and to assist in investigation of what have it, in fact, learned.
+
+One of the purposes of an embedding is to display a complex high dimensional dataset in a way that is $i$ ) intuitive, and $\Ddot { u }$ ) representative of the data sample. The first requirement usually leads to displaying data samples as points in a 2-dimensional map, while the second is more elusive and each approach addresses it differently. Embedding approaches include relatively simple random linear projections—provably preserving some neighbor relations [6]—and a more complex class of nonlinear embedding approaches [30, 32, 34, 36]. In an attempt to organize the properties of this diverse family we have aimed at representing nonlinear embedding methods under a single constraint satisfaction problem (CSP) framework (see below). We hypothesize that each method places the samples in a map to satisfy a specific set of constraints. Although this work is not yet complete, it proven useful in our current study. We briefly outline the ideas in this section to provide enough intuition of the method that we further use in Section 3.
+
+Since we can control the constraints in the CSP framework, to study the effect of deep learning we choose them to do the least amount of work—while still being useful—letting the DBN do (or not) the hard part. A more complicated method such as t-SNE [36] already does complex processing to preserve the structure of a dataset in a 2D map – it is hard to infer if the quality of the map is determined by a deep learning method or the embedding. While some of the existing method may have provided the “least amount of work” solutions as well we chose to go with the CSP framework. It explicitly states the constraints that are being satisfied and thus lets us reason about deep learning effects within the constraints, while with other methods—where the constraints are implicit—this would have been harder.
+
+A constraint satisfaction problem (CSP) is one requiring a solution that satisfies a set of constraints. One of the well known examples is the boolean satisfiability problem (SAT). There are multiple other important CSPs such as the packing, molecular conformations, and, recently, error correcting codes [7]. Freedom to setup per point constraints without controlling for their global interactions makes a CSP formulation an attractive representation of the nonlinear embedding problem. Pursuing this property we use the iterative “divide and concur” (DC) algorithm [13] as the solver for our representation. In DC algorithm we treat each point on the solution map as a variable and assign a set of constraints that this variable needs to satisfy (more on these later). Then each points gets a “replica” for each constraint it is involved into. Then DC algorithm alternates the divide and concur projections. The divide projection moves each “replica” points to the nearest locations in the 2D map that satisfy the constraint they participate in. The concur projection concurs locations of all “replicas” of a point by placing them at the average location on the map. The key idea is to avoid local traps by combining the divide and concur steps within the difference map [8]. A single location update is represented by:
+
+$$
+\begin{array} { r l } & { x _ { c } = P _ { c } ( ( 1 + 1 / \beta ) * P _ { d } ( x ) - 1 / \beta * x ) } \\ & { x _ { d } = P _ { d } ( ( 1 - 1 / \beta ) * P _ { c } ( x ) + 1 / \beta * x ) } \\ & { x = x + \beta * ( x _ { c } - x _ { d } ) , } \end{array}
+$$
+
+where $P _ { d } ( \cdot )$ and $P _ { c } ( \cdot )$ denote the divide and concur projections and $\beta$ is a user-defined parameter.
+
+While the concur projection will only differ by subsets of “replicas” across different methods representable in DC framework, the divide projection is unique and defines the algorithm behavior. In this paper, we choose a divide projection that keeps $k$ nearest neighbors of each point in the higher dimensional space also its neighbors in the 2D map. This is a simple local neighborhood constraint that allows us to assess effects of deep learning transformation leaving most of the mapping decisions to the deep learning.
+
+Note, that for a general dataset we may not be able to satisfy this constraint: each point has exactly the same neighbors in 2D as in the original space (and this is what we indeed observe). The DC algorithm, however, is only guaranteed to find the solution if it exists and oscillates otherwise. Oscillating behavior is detectable and may be used to stop the algorithm. We found informative watching the 2D map in dynamics, as the points that keep oscillating provide additional information into the structure of the data. Another practically important feature of the algorithm: it is deterministic. Given the same parameters ( $\beta$ and the parameters of $P _ { d } ( \cdot )$ ) it converges to the same solution regardless of the initial point. If each of the points participates in each constraint then complexity of the algorithm is quadratic. With our simple $k$ neighborhood constraints it is $O ( k n )$ , for $n$ samples/points.
+
+# 3.3 A schizophrenia structural MRI dataset
+
+We use a combined data from four separate schizophrenia studies conducted at Johns Hopkins University (JHU), the Maryland Psychiatric Research Center (MPRC), the Institute of Psychiatry, London, UK (IOP), and the Western Psychiatric Institute and Clinic at the University of Pittsburgh (WPIC) (the data used in Meda et al. [25]). The combined sample comprised 198 schizophrenia patients and 191 matched healthy controls and contained both first episode and chronic patients [25]. At all sites, whole brain MRIs were obtained on a $1 . 5 \mathrm { T }$ Signa GE scanner using identical parameters and software. Original structural MRI images were segmented in native space and the resulting gray and white matter images then spatially normalized to gray and white matter templates respectively to derive the optimized normalization parameters. These parameters were then applied to the whole brain structural images in native space prior to a new segmentation. The obtained 60465 voxel gray matter images were used in this study. Figure 4 shows example orthogonal slice views of the gray matter data samples of a patient and a healthy control.
+
+![](images/c4b5a343fc1e4cb85c20d34f23973aa6ac9d95e1c2b63880dbf4b1bc1bf866da.jpg)  
+Figure 4: A smoothed gray matter segmentation of a patient and a healthy control: each is a training sample.
+
+The main question of this Section is to evaluate the effect of the depth of a DBN on sMRI. To answer this question, we investigate if classification rates improve with the depth. For that we sequentially investigate DBNs of 3 depth. From RBM experiments we have learned that even with a larger number of hidden units (72, 128 and 512) RBM tends to only keep around 50 features driving the rest to zero. Classification rate and reconstruction error still slightly improves, however, when the number of hidden units increases. These observations affected our choice of 50 hidden units of the first two layers and 100 for the third. Each hidden unit is connected to all units in the previous layer which results in an all to all connectivity structure between the layers, which is a more common and conventional approach to constructing these models. Note, larger networks (up to double the umber of units) lead to similar results. We pre-train each layer via an unsupervised RBM and discriminatively fine-tune models of depth 1 (50 hidden units in the top layer), 2 (50-50 hidden units in the first and the top layer respectively), and 3 (50-50-100 hidden units in the first, second and the top layer respectively) by adding a softmax layer on top of each of these models and training via the back propagation.
+
+We estimate the accuracy of classification via 10-fold cross validation on fine-tuned models splitting the 389 subject dataset into 10 approximately class-balanced folds. We train the rbf-kernel SVM, logistic regression and a k-nearest neighbors (knn) classifier using activations of the top-most hidden layers in fine-tuned models to the training data of each fold as their input. The testing is performed likewise but on the test data. We also perform the same 10-fold cross validation on the raw data. Table 1 summarizes the precision and recall values in the F-scores and their standard deviations.
+
+All models demonstrate a similar trend when the accuracy only slightly increases from depth-1 to depth-2 DBN and then improves signifi
+
+<table><tr><td>depth</td><td>raw</td><td>1</td><td>2</td><td>3</td></tr><tr><td>SVMF-score</td><td>0.68 ±0.01</td><td>0.66 ±0.09</td><td>0.62 ±0.12</td><td>0.90±0.14</td></tr><tr><td>LRF-score</td><td>0.63 ± 0.09</td><td>0.65 ±0.11</td><td>0.61±0.12</td><td>0.91±0.14</td></tr><tr><td>KNNF-score</td><td>0.61 ± 0.11</td><td>0.55 ± 0.15</td><td>0.58 ± 0.16</td><td>0.90±0.16</td></tr></table>
+
+Table 1: Classification on fine-tuned models (test data)
+
+cantly. Table 1 supports the general claim of deep learning community about improvement of classification rate with the depth even for sMRI data. Improvement in classification even for the simple knn classifier indicates the character of the transformation that the DBN learns and applies to the data: it may be changing the data manifold to organize classes by neighborhoods. Ideally, to make general conclusion about this transformation we need to analyze several representative datasets. However, even working with the same data we can have a closer view of the depth effect using the method introduced in Section 3.2. Although it may seem that the DBN does not provide significant improvements in sMRI classification from depth-1 to depth-2 in this model, it keeps on learning potentially useful transformaions of the data. We can see that using our simple local neighborhoodbased embedding. Figure 5 displays 2D maps of the raw data, as well as the depth 1, 2, and 3 activations (of a network trained on 335 subjects): the deeper networks place patients and control groups further apart. Additionally, Figure 5 displays the 54 subjects that the DBN was not train on. These hold out subjects are also getting increased separation with depth. This DBN’s behavior is potentially useful for generalization, when larger and more diverse data become available.
+
+![](images/476d7f9ec0679dcbfd04d1eaf746d79f5ea4d2c6d16e5a4e64ca1d42cdbab586.jpg)  
+Figure 5: Effect of a DBN’s depth on neighborhood relations. Each map is shown at the same iteration of the algorithm with the same $k = 5 0$ . The color differentiates the classes (patients and controls) and the training (335 subjects) from validation (54 subjects) data. Although the data becomes separable at depth 1 and more so at depth 2, the DBN continues distilling details that pull the classes further apart.
+
+Our new mapping method has two essential properties to facilitate the conclusion and provide confidence in the result: its already mentioned local properties and the deterministic nature of the algorithm. The latter leads to independence of the resulting maps from the starting point. The map only depends on the models parameter $k$ —the size of the neighborhood—and the data.
+
+# 3.4 A large-scale Huntington disease data
+
+In this section we focus on sMRI data collected from healthy controls and Huntington disease (HD) patients as part of the PREDICT-HD project (www.predict-hd. net). Huntington disease is a genetic neurodegenerative disease that results in degeneration of neurons in certain areas of the brain. The project is focused on identifying the earliest detectable changes in thinking skills, emotions and brain structure as a person begins the transition from health to being diagnosed with Huntington disease. We would like to know if deep learning methods can assist in answering that question.
+
+For this study T1-weighted scans were collected at multiple sites (32 international sites), representing multiple field strengths (1.5T and 3.0T) and multiple manufactures (Siemens, Phillips, and GE). The 1.5T T1 weighted scans were an axial 3D volumetric spoiled-gradient echo series $( \approx 1 \times 1 \times 1 . 5 ~ \mathrm { m m }$ voxels), and the 3.0T T1 weighted scans were a 3D Volumetric MPRAGE series $( \approx 1 \times 1 \times 1 \ : \mathrm { m n }$ m voxels).
+
+![](images/73650e39fcfeef56626cab5dffbf6d664648d47ea362e5822a60fc11d81800a5.jpg)  
+Figure 6: A gray matter of MRI scans of an HD patient and a healthy control.
+
+The images were segmented in the native space and the normalized to a common template. After correlating
+
+$$
+\begin{array}{c} \begin{array} { r l } { \mathsf { d e p t h } } & { { } \parallel \mathsf { r a w } } \\ { S V M F ^ { - s c o r e } } \\ { L R F ^ { - s c o r e } } \end{array} \left\| \begin{array} { l l } { \mathrm { \Delta ~ n w ~ } } & { { } 0 . 6 5 \pm 0 . 0 1 } \\ { 0 . 7 5 } & { { } 0 . 6 5 \pm 0 . 0 1 } \end{array} \right.  & { 0 . 6 5 \pm 0 . 0 1 \quad 1 . 0 0 \pm 0 . 0 0 } \\ { 0 . 6 5 \pm 0 . 0 1 } & { { } 0 . 6 5 \pm 0 . 0 1 } \end{array}
+$$
+
+Table 2: Classification on fine-tuned models (HD data)
+
+the normalized gray matter segmentation with the template and eliminating poorly correlating scans we obtain a dataset of 3500 scans, where 2641 were from patients and 859 from healthy controls. We have used all of the scans in this imbalanced sample to pre-train and fine tune the same model architecture (50-50-100) as in Section 3.3 for all three depths3.
+
+Table 2 lists the average F-score values for both classes at the raw data and all depth levels. Note the drop from the raw data and then a recovery at depth 3. The limited capacity of levels 1 and 2 has reduced the network ability to differentiate the groups but representational capacity of depth 3 network compensates for the initial bottleneck. This, confirms our previous observation on the depth effect, however, does not yet help the main question of the PREDICT-HD study. Note, however, while Table 1 in the previous section evaluates generalization ability of the DBN, Table 2 here only demonstrates changes in DBN’s representational capacity with the depth as we use no testing data. To further investigate utility of the deep learning approach for scientific discovery we again augment it with the embedding method of Section 3.2. Figure 7 shows the map of 3500 scans of HD patients and healthy controls. Each point on the map is an sMRI volume, shown in Figures 6 and 7. Although we have used the complete data to train the DBN, discriminative fine-tuning had access only to binary label: control or patient. In addition to that, we have information about severity of the disease from low to high. We have color coded this information in Figure 7 from bright yellow (low) through orange (medium) to red (high). The network4 discriminates the patients by disease severity which results in a spectrum on the map. Note, that neither t-SNE (not shown), nor our new embedding see the spectrum or even the patient groups in the raw data. This is a important property of the method that may help support its future use in discovery of new information about the disease.
+
+![](images/eeeb39731b80a20709ab71582b9bc533ca1ffc2bcf1813556881823eb58f2849.jpg)  
+Figure 7: Patients and controls group separation map with additional unsupervised spectral decomposition of sMRI scans by disease severity. The map represents 3500 scans.
+
+# 4 Conclusions
+
+Our investigations show that deep learning has a high potential in neuroimaging applications. Even the shallow RBM is already competitive with the model routinely used in the field: it produces physiologically meaningful features which are (desirably) highly focal and have time course cross correlations that connect them into meaningful functional groups (Section 5). The depth of the DBN does indeed help classification and increases group separation. This is apparent on two sMRI datasets collected under varying conditions, at multiple sites each, from different disease groups, and pre-processed differently. This is a strong evidence of DBNs robustness. Furthermore, our study shows a high potential of DBNs for exploratory analysis. As Figure 7 demonstrates, DBN in conjunction with our new mapping method can reveal hidden relations in data. We did find it difficult initially to find workable parameter regions, but we hope that other researchers won’t have this difficulty starting from the baseline that we provide in this paper.
+
+# References
+
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+[25] Shashwath A Meda, Nicole R Giuliani, Vince D Calhoun, Kanchana Jagannathan, David J Schretlen, Anne Pulver, Nicola Cascella, Matcheri Keshavan, Wendy Kates, Robert Buchanan, et al. A large scale $\mathit { n } { } = 4 0 0 { }$ ) investigation of gray matter differences in schizophrenia using optimized voxel-based morphometry. Schizophrenia research, 101(1):95–105, 2008.   
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+
+# 5 Supplementary material
+
+The correlation matrices for both RBM and ICA results on the fMRI dataset of Section 2.3 are provided in Figure S1, where the ordering of components is performed separately for each method. Each network is named by their physiological function but we do not go in depth explaining these in the current paper. For RBM, modularity is more apparent, both visually and quantitatively. Modularity, as defined in [31], averages $0 . 4 0 \pm 0 . 0 6 0$ across subjects for RBM, and $0 . 3 5 \pm 0 . 0 5 6$ for ICA. These values are significantly greater for RBM $( t = 7 . 1 5 , p < 1 e - 6$ per the paired t-test). Also note that the scale of correlation values for RBM and ICA is different, which highlights that RBM overestimated strong FNC values.
+
+![](images/ae7cce6d19fbb9d82bd5b077ad3a04815fb6c12d2a5a349b52cb5037cccb90d4.jpg)  
+Figure S1: Correlation matrices determined from RBM (left) and ICA (right), averaged over subjects. Note that the color scales for RBM and ICA are different (RBM shows a larger range in correlations). The correlation matrix for ICA on the same scale as RBM is also provided as an inset (upper right). Feature groupings for RBM and ICA were determined separately using the FNC matrices and known anatomical and functional properties.
+
+![](images/f9a151bf4c870b7e24e5b9860efde560866c036587adf63bded04a361b7f39fd.jpg)  
+Figure S2: Sample pairs consisting of RBM (top) and ICA (bottom) SMs thresholded at 2 standard deviations. Pairing was done with the aid of spatial correlations, temporal properties, and visual inspection. Values indicate the spatial correlation between RBM and ICA SMs.

parse/train/UVH3Ucewd-IXZ/UVH3Ucewd-IXZ_content_list.json

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+        "type": "text",
+        "text": "Deep learning for neuroimaging: a validation study ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Sergey M. Plis The Mind Research Network Albuquerque, NM 87106 s.m.plis@gmail.com ",
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+    {
+        "type": "text",
+        "text": "Devon R. Hjelm   \nUniversity of New Mexico   \nAlbuquerque, NM 87131   \ndhjelm@mrn.org ",
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+    {
+        "type": "text",
+        "text": "Ruslan Salakhutdinov University of Toronto Toronto, Ontario M5S 2E4 rsalakhu@cs.toronto.edu ",
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+    {
+        "type": "text",
+        "text": "Vince D. Calhoun The Mind Research Network Albuquerque, NM 87106 vcalhoun@mrn.org ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Deep learning methods have recently made notable advances in the tasks of classification and representation learning. These tasks are important for brain imaging and neuroscience discovery, making the methods attractive for porting to a neuroimager’s toolbox. Success of these methods is, in part, explained by the flexibility of deep learning models. However, this flexibility makes the process of porting to new areas a difficult parameter optimization problem. In this work we demonstrate our results (and feasible parameter ranges) in application of deep learning methods to structural and functional brain imaging data. We also describe a novel constraint-based approach to visualizing high dimensional data. We use it to ana- lyze the effect of parameter choices on data transformations. Our results show that deep learning methods are able to learn physiologically important representations and detect latent relations in neuroimaging data. ",
+        "bbox": [
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "One of the main goals of brain imaging and neuroscience—and, possibly, of most natural sciences— is to improve understanding of the investigated system based on data. In our case, this amounts to inference of descriptive features of brain structure and function from non-invasive measurements. Brain imaging field has come a long way from anatomical maps and atlases towards data driven feature learning methods, such as seed-based correlation [2], canonical correlation analysis [33], and independent component analysis (ICA) [1, 24]. These methods are highly successful in revealing known brain features with new details [3] (supporting their credibility), in recovering features that differentiate patients and controls [28] (assisting diagnosis and disease understanding), and starting a “resting state” revolution after revealing consistent patters in data from uncontrolled resting experiments [29, 35]. Classification is often used merely as a correctness checking tool, as the main emphasis is on learning about the brain. A perfect oracle that does not explain its conclusions would be useful, but mainly to facilitate the inference of the ways the oracle draws these conclusions. ",
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+        "text": "As an oracle, deep learning methods are breaking records taken over the areas of speech, signal, image, video and text mining and recognition by improving state of the art classification accuracy by, sometimes, more than $30 \\%$ where the prior decade struggled to obtain a $1 \\%$ improvements [19, 21]. What differentiates them from other classifiers, however, is the automatic feature learning from data which largely contributes to improvements in accuracy. Presently, this seems to be the closest solution to an oracle that reveals its methods — a desirable tool for brain imaging. ",
+        "bbox": [
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+        "text": "Another distinguishing feature of deep learning is the depth of the models. Based on already acceptable feature learning results obtained by shallow models—currently dominating neuroimaging field—it is not immediately clear what benefits would depth have. Considering the state of multimodal learning, where models are either assumed to be the same for analyzed modalities [26] or cross-modal relations are sought at the (shallow) level of mixture coefficients [23], deeper models better fit the intuitive notion of cross-modality relations, as, for example, relations between genetics and phenotypes should be indirect, happening at a deeper conceptual level. ",
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+        "text": "In this work we present our recent advances in application of deep learning methods to functional and structural magnetic resonance imaging (fMRI and sMRI). Each consists of brain volumes but for sMRI these are static volumes—one per subject/session,—while for fMRI a single subject dataset is comprised of multiple volumes capturing the changes during an experimental session. Our goal is to validate feasibility of this application by $a$ ) investigating if a building block of deep generative models—a restricted Boltzmann machine (RBM) [17]—is competitive with ICA (a representative model of its class) (Section 2); $b$ ) examining the effect of the depth in deep learning analysis of structural MRI data (Section 3.3); and $c$ ) determining the value of the methods for discovery of latent structure of a large-scale (by neuroimaging standards) dataset (Section 3.4). The measure of feature learning performance in a shallow model (a) is comparable with existing methods and known brain physiology. However, this measure cannot be used when deeper models are investigated. As we further demonstrate, classification accuracy does not provide the complete picture either. To be able to visualize the effect of depth and gain an insight into the learning process, we introduce a flexible constraint satisfaction embedding method that allows us to control the complexity of the constraints (Section 3.2). Deliberately choosing local constraints we are able to reflect the transformations that the deep belief network (DBN) [15] learns and applies to the data and gain additional insight. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 A shallow belief network for feature learning ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Prior to investigating the benefits of depth of a DBN in learning representations from fMRI and sMRI data, we would like to find out if a shallow (single hidden layer) model–which is the RBM— from this family meets the field’s expectations. As mentioned in the introduction, a number of methods are used for feature learning from neuroimaging data: most of them belong to the single matrix factorization (SMF) class. We do a quick comparison to a small subset of SMF methods on simulated data; and continue with a more extensive comparison against ICA as an approach trusted in the neuroimaging field. Similarly to RBM, ICA relies on the bipartite graph structure, or even is an artificial neural network with sigmoid hidden units as is in the case of Infomax ICA [1] that we compare against. Note the difference with RBM: ICA applies its weight matrix to the (shorter) temporal dimension of the data imposing independence on the spatial dimension while RBM applies its weight matrix (hidden units “receptive fields”) to the high dimensional spatial dimension instead (Figure 2). ",
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+    {
+        "type": "text",
+        "text": "2.1 A restricted Boltzmann machine ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A restricted Boltzmann machine (RBM) is a Markov random field that models data distribution parameterizing it with the Gibbs distribution over a bipartite graph between visible $\\pmb { v }$ and hidden variables $^ { h }$ [10]: $\\begin{array} { r } { p ( \\pmb { v } ) = \\sum _ { \\pmb { h } } p ( \\pmb { v } , \\pmb { h } ) = \\sum _ { \\pmb { h } } 1 / Z \\exp ( - E ( \\pmb { v } , \\pmb { h } ) ) } \\end{array}$ , where $\\begin{array} { r } { Z = \\sum v \\sum _ { h } e ^ { - E ( { \\pmb v } , { \\pmb h } ) } } \\end{array}$ is the normalization term (the partition function) and $E ( v , h )$ is the energy of the system. Each visible variable in the case of fMRI data represents a voxel of an fMRI scan with a real-valued and approximately Gaussian distribution. In this case, the energy is defined as: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/87cc444cf97cd9ccbd6b314dd81aef36eb2d1c11c788843f32f8d9f175ccb686.jpg",
+        "text": "$$\nE ( v , h ) = - \\sum _ { i j } \\frac { v _ { j } } { \\sigma _ { j } } W _ { j i } h _ { i } - \\sum _ { j } \\frac { ( a _ { j } - v _ { j } ) ^ { 2 } } { \\sigma _ { j } ^ { 2 } } - \\sum _ { i } b _ { i } h _ { i } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $a _ { j }$ and $b _ { j }$ are biases and $\\sigma _ { j }$ is the standard deviation of a parabolic containment function for each visible variable $v _ { j }$ centered on the bias $a _ { j }$ . In general, the parameters $\\sigma _ { i }$ need to be learned along with the other parameters. However, in practice normalizing the distribution of each voxel to have zero mean and unit variance is faster and yet effective [27]. A number of choices affect the quality of interpretation of the representations learned from fMRI by an RBM. Encouraging sparse features via the $L _ { 1 }$ -regularization: $\\lambda \\| W \\| _ { 1 }$ ( $\\lambda = 0 . 1$ gave best results) and using hyperbolic tangent for hidden units non-linearity are essential settings that respectively facilitate spatial and temporal interpretation of the result. The weights were updated using the truncated Gibbs sampling method called contrastive divergence (CD) with a single sampling step (CD-1). Further information on RBM model can be found in [16, 17]. ",
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+    {
+        "type": "text",
+        "text": "2.2 Synthetic data ",
+        "text_level": 1,
+        "bbox": [
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+        "img_path": "images/03e5dc7c4b4fc13894498eac8e135ded37c71d1069aa60fab7dab98ab2c17ad8.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "a: Average spatial map (SM) and time course (TC) correlations to ground truth for RBM and SMF models (gray box). ",
+        "bbox": [
+            179,
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+    },
+    {
+        "type": "text",
+        "text": "b: Ground truth (GT) SMs and estimates obtained by RBM and ICA (thresholded at 0.4 height). Colors are consistent across the methods. Grey indicates background or areas without SMs above threshold. ",
+        "bbox": [
+            387,
+            294,
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "c: Spatial, temporal, and cross correlation (FNC) accuracy for ICA (red) and RBM (blue), as a function of spatial overlap of the true sources from 1b. Lines indicate the average correlation to GT, and the color-fill indicates $\\pm 2$ standard errors around the mean. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figure 1: Comparison of RBM estimation accuracy of features and their time courses with SMFs. ",
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section we summarize our comparisons of RBM with SMF models—including Infomax ICA [1], PCA [14], sparse PCA (sPCA) [37], and sparse NMF (sNMF) [18]—on synthetic data with known spatial maps generated to simulate fMRI. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1a shows the correlation of spatial maps (SM) and time course (TC) estimates to the ground truth for RBM, ICA, PCA, sPCA, and sNMF. Correlations are averaged across all sources and datasets. RBM and ICA showed the best overall performance. While sNMF also estimated SMs well, it showed inferior performance on TC estimation, likely due to the non-negativity constraint. Based on these results and the broad adoption of ICA in the field, we focus on comparing Infomax ICA and RBM. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1b shows the full set of ground truth sources along with RBM and ICA estimates for a single representative dataset. SMs are thresholded and represented as contours for visualization. Results over all synthetic datasets showed similar performance for RBM and ICA (Figure 1c), with a slight advantage for ICA with regard to SM estimation, and a slight advantage for RBM with regards to TC estimation. RBM and ICA also showed comparable performance estimating cross correlations also called functional network connectivity (FNC). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 An fMRI data application ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+            681
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a669dc66ad09d2234094b6f0a2361707082acab0f3b05effb83465e7d44334c7.jpg",
+        "image_caption": [
+            "Figure 2: The processes of feature learning and time course computation from fMRI data by an RBM. The visible units are voxels and a hidden unit receptive field covers an fMRI volume. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Data used in this work comprised of task-related scans from 28 (five females) healthy participants, all of whom gave written, informed, IRB-approved consent at Hartford Hospital and were compensated for participation1. All participants were scanned during an auditory oddball task (AOD) involving the detection of an infrequent target sound within a series of standard and novel sounds2. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Scans were acquired at the Olin Neuropsychiatry Research Center at the Institute of Living/Hartford Hospital on a Siemens Allegra 3T dedicated head scanner equipped with $4 0 \\mathrm { m T } / \\mathrm { m }$ gradients and a standard quadrature head coil [4, 9]. The AOD consisted of two 8-min runs, and 249 scans (volumes) at 2 second TR $0 . 5 \\ \\mathrm { H z }$ sampling rate) were used for the final dataset. Data were post-processed using the SPM5 software package [12], motion corrected using INRIalign [11], and subsampled to $5 3 \\times 6 3 \\times 4 6$ voxels. The complete fMRI dataset was masked below mean and the mean image across the dataset was removed, giving a complete dataset of size 70969 voxels by 6972 volumes. Each voxel was then normalized to have zero mean and unit variance. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The RBM was constructed using 70969 Gaussian visible units and 64 hyperbolic tangent hidden units. The hyper parameters $\\epsilon$ (0.08 from the searched $[ 1 \\times 1 0 ^ { - 4 } , 1 \\times 1 \\dot { 0 } ^ { - 1 } ]$ range) for learning rate and $\\lambda$ (0.1 from the searched range $[ 1 \\times \\mathrm { { 1 0 ^ { - 2 } , 1 \\times 1 0 ^ { - 1 } } } ] )$ for $L _ { 1 }$ weight decay were selected as those that showed a reduction of reconstruction error over training and a significant reduction in span of the receptive fields respectively. Parameter value outside the ranges either resulted in unstable or slow learning (\u000f) or uninterpretable features $( \\lambda )$ . The RBM was then trained with a batch size of 5 for approximately 100 epochs to allow for full convergence of the parameters. ",
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+        "image_caption": [
+            "Figure 3: Intrinsic brain networks estimated by ICA and RBM. "
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+        "type": "text",
+        "text": "After flipping the sign of negative receptive fields, we then identified and labeled spatially distinct features as corresponding to brain regions with the aid of AFNI [5] excluding features which had a high probability of corresponding to white matter, ventricles, or artifacts (eg. motion, edges). ",
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+        "type": "text",
+        "text": "We normalized the fMRI volume time series to mean zero and used the trained RBM in feed-forward mode to compute time series for each fMRI feature. This was done to better compare to ICA, where the mean is removed in PCA preprocessing. ",
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+    {
+        "type": "text",
+        "text": "The work-flow is outlined in Figure 2, while Figure 3 shows comparison of resulting features with those obtained by Infomax ICA. In general, RBM performs competitively with ICA, while providing–perhaps, not surprisingly due to the used $L _ { 1 }$ regularization—sharper and more localized features. While we recognize that this is a subjective measure we list more features in Figure S2 of Section 5 and note that RBM features lack negative parts for corresponding features. Note, that in the case of $L _ { 1 }$ regularized weights RBM algorithms starts to resemble some of the ICA approaches (such as the recent RICA by Le at al. [20]), which may explain the similar performance. However, the differences and possible advantages are the generative nature of the RBM and no enforcement of component orthogonality (not explicit at the least). Moreover, the block structure of the correlation matrix (see below the Supplementary material section) of feature time courses provide a grouping that is more physiologically supported than that provided by ICA. For example, see Figure S1 in the supplementary material section below. Perhaps, because ICA working hard to enforce spatial independence subtly affects the time courses and their cross-correlations in turn. We have observed comparable running times of the (non GPU) ICA (http://www.nitrc.org/projects/gift) and a GPU implementation of the RBM (https://github.com/nitishsrivastava/deepnet). ",
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+        "type": "text",
+        "text": "3 Validating the depth effect ",
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+        "text": "Since the RBM results demonstrate a feature-learning performance competitive with the state of the art (or better), we proceed to investigating the effects of the model depth. To do that we turn from fMRI to sMRI data. As it is commonly assumed in the deep learning literature [22] the depth is often improving classification accuracy. We investigate if that is indeed true in the sMRI case. Structural data is convenient for the purpose as each subject/session is represented only by a single volume that has a label: control or patient in our case. Compare to 4D data where hundreds of volumes belong to the same subject with the same disease state. ",
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+        "text": "3.1 A deep belief network ",
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+        "text": "A DBN is a sigmoidal belief network (although other activation functions may be used) with an RBM as the top level prior. The joint probability distribution of its visible and hidden units is ",
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+        "type": "text",
+        "text": "parametrized as follows: ",
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+        "img_path": "images/034a9bca17b958ad18fe6b11458de31a470237beea190c0545436adf9c327f3f.jpg",
+        "text": "$$\nP ( v , h ^ { 1 } , h ^ { 2 } , \\dots , h ^ { l } ) = P ( v | h ^ { 1 } ) P ( h ^ { 1 } | h ^ { 2 } ) \\cdots P ( h ^ { l - 2 } , h ^ { l - 1 } ) P ( h ^ { l - 1 } , h ^ { l } ) ,\n$$",
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+        "type": "text",
+        "text": "where $l$ is the number of hidden layers, $P ( h ^ { l - 1 } , h ^ { l } )$ is an RBM, and $P ( h ^ { i } | h ^ { i + 1 } )$ factor into individual conditionals: ",
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+        "text": "$$\nP ( h ^ { i } | h ^ { i + 1 } ) = \\prod _ { j = 1 } ^ { n _ { i } } P ( h _ { j } ^ { i } | h ^ { i + 1 } )\n$$",
+        "text_format": "latex",
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+        "text": "The important property of DBN for our goals of feature learning to facilitate discovery is its ability to operate in generative mode with fixed values on chosen hidden units thus allowing one to investigate the features that the model have learned and/or weighs as important in discriminative decisions. We, however, not going to use this property in this section, focusing instead on validating the claim that a network’s depth provides benefits for neuroimaging data analysis. And we will do this using discriminative mode of DBN’s operation as it provides an objective measure of the depth effect. ",
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+        "type": "text",
+        "text": "DBN training splits into two stages: pre-training and discriminative fine tuning. A DBN can be pre-trained by treating each of its layers as an RBM—trained in an unsupervised way on inputs from the previous layer—and later fine-tuned by treating it as a feed-forward neural network. The latter allows supervised training via the error back propagation algorithm. We use this schema in the following by augmenting each DBN with a soft-max layer at the fine-tuning stage. ",
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+        "text": "3.2 Nonlinear embedding as a constraint satisfaction problem ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A DBN and an RBM operate on data samples, which are brain volumes in the fMRI and sMRI case. A five-minute fMRI experiment with 2 seconds sampling rate yields 150 of these volumes per subject. For sMRI studies number of participating subjects varies but in this paper we operate with a 300 and a 3500 subject-volumes datasets. Transformations learned by deep learning methods do not look intuitive in the hidden node space and generative sampling of the trained model does not provide a sense if a model have learned anything useful in the case of MRI data: in contrast to natural images, fMRI and sMRI images do not look very intuitive. Instead, we use a nonlinear embedding method to control whether a model learned useful information and to assist in investigation of what have it, in fact, learned. ",
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+        "type": "text",
+        "text": "One of the purposes of an embedding is to display a complex high dimensional dataset in a way that is $i$ ) intuitive, and $\\Ddot { u }$ ) representative of the data sample. The first requirement usually leads to displaying data samples as points in a 2-dimensional map, while the second is more elusive and each approach addresses it differently. Embedding approaches include relatively simple random linear projections—provably preserving some neighbor relations [6]—and a more complex class of nonlinear embedding approaches [30, 32, 34, 36]. In an attempt to organize the properties of this diverse family we have aimed at representing nonlinear embedding methods under a single constraint satisfaction problem (CSP) framework (see below). We hypothesize that each method places the samples in a map to satisfy a specific set of constraints. Although this work is not yet complete, it proven useful in our current study. We briefly outline the ideas in this section to provide enough intuition of the method that we further use in Section 3. ",
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+        "type": "text",
+        "text": "Since we can control the constraints in the CSP framework, to study the effect of deep learning we choose them to do the least amount of work—while still being useful—letting the DBN do (or not) the hard part. A more complicated method such as t-SNE [36] already does complex processing to preserve the structure of a dataset in a 2D map – it is hard to infer if the quality of the map is determined by a deep learning method or the embedding. While some of the existing method may have provided the “least amount of work” solutions as well we chose to go with the CSP framework. It explicitly states the constraints that are being satisfied and thus lets us reason about deep learning effects within the constraints, while with other methods—where the constraints are implicit—this would have been harder. ",
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+        "text": "A constraint satisfaction problem (CSP) is one requiring a solution that satisfies a set of constraints. One of the well known examples is the boolean satisfiability problem (SAT). There are multiple other important CSPs such as the packing, molecular conformations, and, recently, error correcting codes [7]. Freedom to setup per point constraints without controlling for their global interactions makes a CSP formulation an attractive representation of the nonlinear embedding problem. Pursuing this property we use the iterative “divide and concur” (DC) algorithm [13] as the solver for our representation. In DC algorithm we treat each point on the solution map as a variable and assign a set of constraints that this variable needs to satisfy (more on these later). Then each points gets a “replica” for each constraint it is involved into. Then DC algorithm alternates the divide and concur projections. The divide projection moves each “replica” points to the nearest locations in the 2D map that satisfy the constraint they participate in. The concur projection concurs locations of all “replicas” of a point by placing them at the average location on the map. The key idea is to avoid local traps by combining the divide and concur steps within the difference map [8]. A single location update is represented by: ",
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+        "img_path": "images/fa7ce1be22ae91ee20dbb025907340bc7b601263a1731f9b7344daa68d84c58c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { x _ { c } = P _ { c } ( ( 1 + 1 / \\beta ) * P _ { d } ( x ) - 1 / \\beta * x ) } \\\\ & { x _ { d } = P _ { d } ( ( 1 - 1 / \\beta ) * P _ { c } ( x ) + 1 / \\beta * x ) } \\\\ & { x = x + \\beta * ( x _ { c } - x _ { d } ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $P _ { d } ( \\cdot )$ and $P _ { c } ( \\cdot )$ denote the divide and concur projections and $\\beta$ is a user-defined parameter. ",
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+        "text": "While the concur projection will only differ by subsets of “replicas” across different methods representable in DC framework, the divide projection is unique and defines the algorithm behavior. In this paper, we choose a divide projection that keeps $k$ nearest neighbors of each point in the higher dimensional space also its neighbors in the 2D map. This is a simple local neighborhood constraint that allows us to assess effects of deep learning transformation leaving most of the mapping decisions to the deep learning. ",
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+        "type": "text",
+        "text": "Note, that for a general dataset we may not be able to satisfy this constraint: each point has exactly the same neighbors in 2D as in the original space (and this is what we indeed observe). The DC algorithm, however, is only guaranteed to find the solution if it exists and oscillates otherwise. Oscillating behavior is detectable and may be used to stop the algorithm. We found informative watching the 2D map in dynamics, as the points that keep oscillating provide additional information into the structure of the data. Another practically important feature of the algorithm: it is deterministic. Given the same parameters ( $\\beta$ and the parameters of $P _ { d } ( \\cdot )$ ) it converges to the same solution regardless of the initial point. If each of the points participates in each constraint then complexity of the algorithm is quadratic. With our simple $k$ neighborhood constraints it is $O ( k n )$ , for $n$ samples/points. ",
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+        "text": "3.3 A schizophrenia structural MRI dataset ",
+        "text_level": 1,
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+        "text": "We use a combined data from four separate schizophrenia studies conducted at Johns Hopkins University (JHU), the Maryland Psychiatric Research Center (MPRC), the Institute of Psychiatry, London, UK (IOP), and the Western Psychiatric Institute and Clinic at the University of Pittsburgh (WPIC) (the data used in Meda et al. [25]). The combined sample comprised 198 schizophrenia patients and 191 matched healthy controls and contained both first episode and chronic patients [25]. At all sites, whole brain MRIs were obtained on a $1 . 5 \\mathrm { T }$ Signa GE scanner using identical parameters and software. Original structural MRI images were segmented in native space and the resulting gray and white matter images then spatially normalized to gray and white matter templates respectively to derive the optimized normalization parameters. These parameters were then applied to the whole brain structural images in native space prior to a new segmentation. The obtained 60465 voxel gray matter images were used in this study. Figure 4 shows example orthogonal slice views of the gray matter data samples of a patient and a healthy control. ",
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+        "image_caption": [
+            "Figure 4: A smoothed gray matter segmentation of a patient and a healthy control: each is a training sample. "
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+        "text": "The main question of this Section is to evaluate the effect of the depth of a DBN on sMRI. To answer this question, we investigate if classification rates improve with the depth. For that we sequentially investigate DBNs of 3 depth. From RBM experiments we have learned that even with a larger number of hidden units (72, 128 and 512) RBM tends to only keep around 50 features driving the rest to zero. Classification rate and reconstruction error still slightly improves, however, when the number of hidden units increases. These observations affected our choice of 50 hidden units of the first two layers and 100 for the third. Each hidden unit is connected to all units in the previous layer which results in an all to all connectivity structure between the layers, which is a more common and conventional approach to constructing these models. Note, larger networks (up to double the umber of units) lead to similar results. We pre-train each layer via an unsupervised RBM and discriminatively fine-tune models of depth 1 (50 hidden units in the top layer), 2 (50-50 hidden units in the first and the top layer respectively), and 3 (50-50-100 hidden units in the first, second and the top layer respectively) by adding a softmax layer on top of each of these models and training via the back propagation. ",
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+        "text": "We estimate the accuracy of classification via 10-fold cross validation on fine-tuned models splitting the 389 subject dataset into 10 approximately class-balanced folds. We train the rbf-kernel SVM, logistic regression and a k-nearest neighbors (knn) classifier using activations of the top-most hidden layers in fine-tuned models to the training data of each fold as their input. The testing is performed likewise but on the test data. We also perform the same 10-fold cross validation on the raw data. Table 1 summarizes the precision and recall values in the F-scores and their standard deviations. ",
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+        "type": "text",
+        "text": "All models demonstrate a similar trend when the accuracy only slightly increases from depth-1 to depth-2 DBN and then improves signifi",
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+        "table_footnote": [
+            "Table 1: Classification on fine-tuned models (test data) "
+        ],
+        "table_body": "<table><tr><td>depth</td><td>raw</td><td>1</td><td>2</td><td>3</td></tr><tr><td>SVMF-score</td><td>0.68 ±0.01</td><td>0.66 ±0.09</td><td>0.62 ±0.12</td><td>0.90±0.14</td></tr><tr><td>LRF-score</td><td>0.63 ± 0.09</td><td>0.65 ±0.11</td><td>0.61±0.12</td><td>0.91±0.14</td></tr><tr><td>KNNF-score</td><td>0.61 ± 0.11</td><td>0.55 ± 0.15</td><td>0.58 ± 0.16</td><td>0.90±0.16</td></tr></table>",
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+        "type": "text",
+        "text": "cantly. Table 1 supports the general claim of deep learning community about improvement of classification rate with the depth even for sMRI data. Improvement in classification even for the simple knn classifier indicates the character of the transformation that the DBN learns and applies to the data: it may be changing the data manifold to organize classes by neighborhoods. Ideally, to make general conclusion about this transformation we need to analyze several representative datasets. However, even working with the same data we can have a closer view of the depth effect using the method introduced in Section 3.2. Although it may seem that the DBN does not provide significant improvements in sMRI classification from depth-1 to depth-2 in this model, it keeps on learning potentially useful transformaions of the data. We can see that using our simple local neighborhoodbased embedding. Figure 5 displays 2D maps of the raw data, as well as the depth 1, 2, and 3 activations (of a network trained on 335 subjects): the deeper networks place patients and control groups further apart. Additionally, Figure 5 displays the 54 subjects that the DBN was not train on. These hold out subjects are also getting increased separation with depth. This DBN’s behavior is potentially useful for generalization, when larger and more diverse data become available. ",
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+        "image_caption": [
+            "Figure 5: Effect of a DBN’s depth on neighborhood relations. Each map is shown at the same iteration of the algorithm with the same $k = 5 0$ . The color differentiates the classes (patients and controls) and the training (335 subjects) from validation (54 subjects) data. Although the data becomes separable at depth 1 and more so at depth 2, the DBN continues distilling details that pull the classes further apart. "
+        ],
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+        "type": "text",
+        "text": "Our new mapping method has two essential properties to facilitate the conclusion and provide confidence in the result: its already mentioned local properties and the deterministic nature of the algorithm. The latter leads to independence of the resulting maps from the starting point. The map only depends on the models parameter $k$ —the size of the neighborhood—and the data. ",
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+    {
+        "type": "text",
+        "text": "3.4 A large-scale Huntington disease data ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we focus on sMRI data collected from healthy controls and Huntington disease (HD) patients as part of the PREDICT-HD project (www.predict-hd. net). Huntington disease is a genetic neurodegenerative disease that results in degeneration of neurons in certain areas of the brain. The project is focused on identifying the earliest detectable changes in thinking skills, emotions and brain structure as a person begins the transition from health to being diagnosed with Huntington disease. We would like to know if deep learning methods can assist in answering that question. ",
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+    {
+        "type": "text",
+        "text": "For this study T1-weighted scans were collected at multiple sites (32 international sites), representing multiple field strengths (1.5T and 3.0T) and multiple manufactures (Siemens, Phillips, and GE). The 1.5T T1 weighted scans were an axial 3D volumetric spoiled-gradient echo series $( \\approx 1 \\times 1 \\times 1 . 5 ~ \\mathrm { m m }$ voxels), and the 3.0T T1 weighted scans were a 3D Volumetric MPRAGE series $( \\approx 1 \\times 1 \\times 1 \\ : \\mathrm { m n }$ m voxels). ",
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+        "image_caption": [
+            "Figure 6: A gray matter of MRI scans of an HD patient and a healthy control. "
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+        "text": "The images were segmented in the native space and the normalized to a common template. After correlating ",
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+        "img_path": "images/d5ccecaa66e79a2cca10e59f1c48307e58dfd75fc8bcad9ea8e5f17377a989ec.jpg",
+        "text": "$$\n\\begin{array}{c} \\begin{array} { r l } { \\mathsf { d e p t h } } & { { } \\parallel \\mathsf { r a w } } \\\\ { S V M F ^ { - s c o r e } } \\\\ { L R F ^ { - s c o r e } } \\end{array} \\left\\| \\begin{array} { l l } { \\mathrm { \\Delta ~ n w ~ } } & { { } 0 . 6 5 \\pm 0 . 0 1 } \\\\ { 0 . 7 5 } & { { } 0 . 6 5 \\pm 0 . 0 1 } \\end{array} \\right.  & { 0 . 6 5 \\pm 0 . 0 1 \\quad 1 . 0 0 \\pm 0 . 0 0 } \\\\ { 0 . 6 5 \\pm 0 . 0 1 } & { { } 0 . 6 5 \\pm 0 . 0 1 } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "table",
+        "img_path": "",
+        "table_caption": [
+            "Table 2: Classification on fine-tuned models (HD data) "
+        ],
+        "table_footnote": [],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "the normalized gray matter segmentation with the template and eliminating poorly correlating scans we obtain a dataset of 3500 scans, where 2641 were from patients and 859 from healthy controls. We have used all of the scans in this imbalanced sample to pre-train and fine tune the same model architecture (50-50-100) as in Section 3.3 for all three depths3. ",
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+        "text": "Table 2 lists the average F-score values for both classes at the raw data and all depth levels. Note the drop from the raw data and then a recovery at depth 3. The limited capacity of levels 1 and 2 has reduced the network ability to differentiate the groups but representational capacity of depth 3 network compensates for the initial bottleneck. This, confirms our previous observation on the depth effect, however, does not yet help the main question of the PREDICT-HD study. Note, however, while Table 1 in the previous section evaluates generalization ability of the DBN, Table 2 here only demonstrates changes in DBN’s representational capacity with the depth as we use no testing data. To further investigate utility of the deep learning approach for scientific discovery we again augment it with the embedding method of Section 3.2. Figure 7 shows the map of 3500 scans of HD patients and healthy controls. Each point on the map is an sMRI volume, shown in Figures 6 and 7. Although we have used the complete data to train the DBN, discriminative fine-tuning had access only to binary label: control or patient. In addition to that, we have information about severity of the disease from low to high. We have color coded this information in Figure 7 from bright yellow (low) through orange (medium) to red (high). The network4 discriminates the patients by disease severity which results in a spectrum on the map. Note, that neither t-SNE (not shown), nor our new embedding see the spectrum or even the patient groups in the raw data. This is a important property of the method that may help support its future use in discovery of new information about the disease. ",
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+        "img_path": "images/eeeb39731b80a20709ab71582b9bc533ca1ffc2bcf1813556881823eb58f2849.jpg",
+        "image_caption": [
+            "Figure 7: Patients and controls group separation map with additional unsupervised spectral decomposition of sMRI scans by disease severity. The map represents 3500 scans. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "4 Conclusions ",
+        "text_level": 1,
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+        "text": "Our investigations show that deep learning has a high potential in neuroimaging applications. Even the shallow RBM is already competitive with the model routinely used in the field: it produces physiologically meaningful features which are (desirably) highly focal and have time course cross correlations that connect them into meaningful functional groups (Section 5). The depth of the DBN does indeed help classification and increases group separation. This is apparent on two sMRI datasets collected under varying conditions, at multiple sites each, from different disease groups, and pre-processed differently. This is a strong evidence of DBNs robustness. Furthermore, our study shows a high potential of DBNs for exploratory analysis. As Figure 7 demonstrates, DBN in conjunction with our new mapping method can reveal hidden relations in data. We did find it difficult initially to find workable parameter regions, but we hope that other researchers won’t have this difficulty starting from the baseline that we provide in this paper. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+    },
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+        "text": "[1] A. J. Bell and T. J. Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7(6):1129–1159, November 1995. [2] Bharat Biswal, F Zerrin Yetkin, Victor M Haughton, and James S Hyde. Functional connectivity in the motor cortex of resting human brain using echo-planar mri. Magnetic resonance in medicine, 34(4):537–541, 1995.   \n[3] M.J. Brookes, M. Woolrich, H. Luckhoo, D. Price, J.R. Hale, M.C. Stephenson, G.R. Barnes, S.M. Smith, and P.G. Morris. Investigating the electrophysiological basis of resting state networks using magnetoencephalography. Proceedings of the National Academy of Sciences, 108(40):16783–16788, 2011. [4] V. D. Calhoun, K. A. Kiehl, and G. D. Pearlson. Modulation of temporally coherent brain networks estimated using ICA at rest and during cognitive tasks. Human Brain Mapping, 29(7):828–838, 2008.   \n[5] R. W. Cox et al. AFNI: software for analysis and visualization of functional magnetic resonance neuroimages. Computers and Biomedical Research, 29(3):162–173, 1996. [6] T. de Vries, S. Chawla, and M. E. Houle. Finding local anomalies in very high dimensional space. In Proceedings of the 10th $\\{ I E E E \\}$ international conference on data mining, pages 128–137. IEEE, IEEE Computer Society, 2010. [7] Nate Derbinsky, Jose Bento, Veit Elser, and Jonathan S Yedidia. An improved three-weight ´ message-passing algorithm. arXiv preprint arXiv:1305.1961, 2013. [8] V. Elser, I. Rankenburg, and P. Thibault. Searching with iterated maps. Proceedings of the National Academy of Sciences, 104(2):418, 2007. [9] Nathan Swanson et. al. Lateral differences in the default mode network in healthy controls and patients with schizophrenia. Human Brain Mapping, 32:654–664, 2011.   \n[10] Asja Fischer and Christian Igel. An introduction to restricted boltzmann machines. In Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, pages 14–36. Springer, 2012.   \n[11] L. Freire, A. Roche, and J. F. Mangin. What is the best similarity measure for motion correction in fMRI. IEEE Transactions in Medical Imaging, 21:470–484, 2002.   \n[12] Karl J Friston, Andrew P Holmes, Keith J Worsley, J-P Poline, Chris D Frith, and Richard SJ Frackowiak. Statistical parametric maps in functional imaging: a general linear approach. Human brain mapping, 2(4):189–210, 1994.   \n[13] S. Gravel and V. Elser. Divide and concur: A general approach to constraint satisfaction. Physical Review E, 78(3):36706, 2008.   \n[14] Trevor. Hastie, Robert. Tibshirani, and JH (Jerome H.) Friedman. The elements of statistical learning. Springer, 2009.   \n[15] G. Hinton and R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313(5786):504–507, 2006.   \n[16] G. E. Hinton, S. Osindero, and Y. W. Teh. A fast learning algorithm for deep belief nets. Neural computation, 18(7):1527–1554, 2006.   \n[17] Geoffrey Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14:2002, 2000.   \n[18] P. O. Hoyer. Non-negative sparse coding. In Neural Networks for Signal Processing, 2002. Proceedings of the 2002 12th IEEE Workshop on, pages 557–565, 2002.   \n[19] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. Imagenet classification with deep convolutional neural networks. In Neural Information Processing Systems, 2012.   \n[20] Quoc V Le, Alexandre Karpenko, Jiquan Ngiam, and Andrew Y Ng. Ica with reconstruction cost for efficient overcomplete feature learning. In NIPS, pages 1017–1025, 2011.   \n[21] Quoc V. Le, Rajat Monga, Matthieu Devin, Kai Chen, Greg S. Corrado, Jeff Dean, and Andrew Y. Ng. Building high-level features using large scale unsupervised learning. In International Conference on Machine Learning. 103, 2012.   \n[22] N. Le Roux and Y. Bengio. Deep belief networks are compact universal approximators. Neural computation, 22(8):2192–2207, 2010.   \n[23] Jingyu Liu and Vince Calhoun. Parallel independent component analysis for multimodal analysis: Application to fmri and eeg data. In Biomedical Imaging: From Nano to Macro, 2007. ISBI 2007. 4th IEEE International Symposium on, pages 1028–1031. IEEE, 2007.   \n[24] M. J. McKeown, S. Makeig, G. G. Brown, T. P. Jung, S. S. Kindermann, A. J. Bell, and T. J. Sejnowski. Analysis of fMRI data by blind separation into independent spatial components. Human Brain Mapping, 6(3):160–188, 1998.   \n[25] Shashwath A Meda, Nicole R Giuliani, Vince D Calhoun, Kanchana Jagannathan, David J Schretlen, Anne Pulver, Nicola Cascella, Matcheri Keshavan, Wendy Kates, Robert Buchanan, et al. A large scale $\\mathit { n } { } = 4 0 0 { }$ ) investigation of gray matter differences in schizophrenia using optimized voxel-based morphometry. Schizophrenia research, 101(1):95–105, 2008.   \n[26] M. Moosmann, T. Eichele, H. Nordby, K. Hugdahl, and V. D. Calhoun. Joint independent component analysis for simultaneous EEG-fMRI: principle and simulation. International Journal of Psychophysiology, 67(3):212–221, 2008.   \n[27] Vinod Nair and Geoffrey E Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th International Conference on Machine Learning (ICML-10), pages 807–814, 2010.   \n[28] V. K. Potluru and V. D. Calhoun. Group learning using contrast NMF : Application to functional and structural MRI of schizophrenia. Circuits and Systems, 2008. ISCAS 2008. IEEE International Symposium on, pages 1336–1339, May 2008.   \n[29] Marcus E. Raichle, Ann Mary MacLeod, Abraham Z. Snyder, William J. Powers, Debra A. Gusnard, and Gordon L. Shulman. A default mode of brain function. Proceedings of the National Academy of Sciences, 98(2):676–682, 2001.   \n[30] Sam T Roweis and Lawrence K Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, (5500):2323–2326, 2000.   \n[31] Mikail Rubinov and Olaf Sporns. Weight-conserving characterization of complex functional brain networks. Neuroimage, 56(4):2068–2079, 2011.   \n[32] John W Sammon Jr. A nonlinear mapping for data structure analysis. Computers, IEEE Transactions on, 100(5):401–409, 1969.   \n[33] Jing Sui, Tulay Adali, Qingbao Yu, Jiayu Chen, and Vince D. Calhoun. A review of multivariate methods for multimodal fusion of brain imaging data. Journal of Neuroscience Methods, 204(1):68–81, 2012.   \n[34] Joshua B Tenenbaum, Vin De Silva, and John C Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290(5500):2319–2323, 2000.   \n[35] M.P. van den Heuvel and H.E. Hulshoff Pol. Exploring the brain network: A review on restingstate fMRI functional connectivity. European Neuropsychopharmacology, 2010.   \n[36] Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9(2579-2605):85, 2008.   \n[37] Hui Zou, Trevor Hastie, and Robert Tibshirani. Sparse principal component analysis. Journal of computational and graphical statistics, 15(2):265–286, 2006. ",
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+    {
+        "type": "text",
+        "text": "5 Supplementary material ",
+        "text_level": 1,
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+            174,
+            102,
+            410,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The correlation matrices for both RBM and ICA results on the fMRI dataset of Section 2.3 are provided in Figure S1, where the ordering of components is performed separately for each method. Each network is named by their physiological function but we do not go in depth explaining these in the current paper. For RBM, modularity is more apparent, both visually and quantitatively. Modularity, as defined in [31], averages $0 . 4 0 \\pm 0 . 0 6 0$ across subjects for RBM, and $0 . 3 5 \\pm 0 . 0 5 6$ for ICA. These values are significantly greater for RBM $( t = 7 . 1 5 , p < 1 e - 6$ per the paired t-test). Also note that the scale of correlation values for RBM and ICA is different, which highlights that RBM overestimated strong FNC values. ",
+        "bbox": [
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+    {
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+        "img_path": "images/ae7cce6d19fbb9d82bd5b077ad3a04815fb6c12d2a5a349b52cb5037cccb90d4.jpg",
+        "image_caption": [
+            "Figure S1: Correlation matrices determined from RBM (left) and ICA (right), averaged over subjects. Note that the color scales for RBM and ICA are different (RBM shows a larger range in correlations). The correlation matrix for ICA on the same scale as RBM is also provided as an inset (upper right). Feature groupings for RBM and ICA were determined separately using the FNC matrices and known anatomical and functional properties. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
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+        ],
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+        "img_path": "images/f9a151bf4c870b7e24e5b9860efde560866c036587adf63bded04a361b7f39fd.jpg",
+        "image_caption": [
+            "Figure S2: Sample pairs consisting of RBM (top) and ICA (bottom) SMs thresholded at 2 standard deviations. Pairing was done with the aid of spatial correlations, temporal properties, and visual inspection. Values indicate the spatial correlation between RBM and ICA SMs. "
+        ],
+        "image_footnote": [],
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+        "page_idx": 10
+    }
+]

parse/train/r1gzoaNtvr/r1gzoaNtvr.md

@@ -0,0 +1,302 @@
+# EMERGENCE OF COMPOSITIONAL LANGUAGE WITH DEEP GENERATIONAL TRANSMISSION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Recent work has studied the emergence of language among deep reinforcement learning agents that must collaborate to solve a task. Of particular interest are the factors that cause language to be compositional—i.e. express meaning by combining words which themselves have meaning. Evolutionary linguists have found that in addition to structural priors like those already studied in deep learning, the dynamics of transmitting language from generation to generation contribute significantly to the emergence of compositionality. In this paper, we introduce these cultural evolutionary dynamics into language emergence by periodically replacing agents in a population to create a knowledge gap, implicitly inducing cultural transmission of language. We show that this implicit cultural transmission encourages the resulting languages to exhibit better compositional generalization.
+
+# 1 INTRODUCTION
+
+Compositionality is an important structure of language that reflects a disentangled understanding of the world – enabling the expression of infinitely many concepts using finitely many elements. Agents that have compositional understandings of the world generalize in obviously correct ways even in the face of limited training examples (Lake & Baroni, 2018). For example, an agent with a compositional understanding of blue squares and purple triangles should also understand purple squares without directly observing any of them. Developing artificial agents that can ground, understand, and produce compositional (and therefore more interpretable) language could greatly improve generalization to new instances and ease human-AI interactions.
+
+In building theories of how compositionality emerges in human languages, work in evolutionary linguistics looks to the process of cultural transmission (Kirby, 2001; Kirby et al., 2008). Cultural transmission of language occurs when a group of agents pass their language on to a new group of agents, e.g. parents who teach their children to speak as they do. Because this education is incomplete and biased, it allows the language itself to change over time via a process known as cultural evolution. This paradigm (Kirby et al., 2014) explains the emergence of compositionality as a result of expressivity and compressibility – i.e. to be most effective, a language should be expressive enough to differentiate between all possible meanings (e.g., objects) and compressible enough to be learned easily. Work in the evolutionary linguistics community has shown that over multiple ‘generations’ these competing pressures result in the emergence of compositional languages both in simulation (Kirby, 2001) and with human subjects (Kirby et al., 2008). These studies aim to understand humans whereas we want to understand and design artificial neural networks.
+
+Approaching the problem from another direction, recent work in AI has studied language emergence in such multi-agent, goal-driven tasks. These works have demonstrated that agent languages will emerge to enable coordination-centric tasks to be solved without direct or even indirect language supervision (Foerster et al., 2016; Sukhbaatar et al., 2016; Lazaridou et al., 2017; Das et al., 2017). However, the resulting languages are usually not compositional (Kottur et al., 2017) and are difficult to interpret, even by other machines (Andreas et al., 2017). Some existing work has studied means to encourage compositional language formation (Mordatch & Abbeel, 2018; Kottur et al., 2017), but these settings study fixed populations of agents – i.e. examining language within a single generation.
+
+In this work we bridge these two areas – examining the effect of generational cultural transmission on the compositionality of emergent languages in a multi-agent, goal-driven setting.
+
+![](images/d64c3dc18b10c42f786b8344e8cfe36b77969264468818f9ca45bddd10bb08bf.jpg)  
+(b) Implicit cultural transmission   
+Figure 1: We introduce cultural transmission into language emergence between neural agents. The starting point of our study is a goal-oriented dialog task (similar to that of Kottur et al. (2017)), summarized in Fig. 1a. During learning we periodically replace some agents with new ones (gray agents). These new agents do not know any language, but instead of creating one they learn it from older agents. This creates generations of language that become more compositional over time.
+
+We study this in the context of a cooperative dialog-based reference game involving two agents communicating in discrete symbols (Kottur et al., 2017); an example dialog is shown in Fig. 1a. To examine cultural transmission, we extend this setting to a population of agents (Fig. 1b) and introduce a simple mechanism to induce the expressivity and compressibility pressures inherent in cultural transmission. Specifically, we periodically re-initialize some subset of the agents in the population. In order to perform well at the task, the population’s emergent language must be sufficiently expressive to reference all the objects (expressivity) and must be easily learnable by these ‘new’ agents (compressibility). The new agents have a randomized language whereas the surviving agents already know a grounded language. This “knowledge gap” creates an implicit ‘teaching’ setting that is analogous to the explicit transmission stage in models of iterative learning (Kirby, 2001).
+
+Through our experiments and analysis, we show that periodic agent replacement is an effective way to induce cultural transmission and yields more compositionally generalizable language in our setting. To summarize, our contributions are:
+
+– We propose a method for inducing implicit cultural transmission in neural language models. – We introduce new metrics to measure the similarity between agent languages and verify cultural transmission has occurred as a result of our periodic agent replacement protocol. We show our cultural transmission procedure induces compositionality in neural language models, going from $13 \%$ accuracy on a compositionally novel test set to $46 \%$ in the best configuration. Further, we show this is complementary with previous priors which encourage compositionality.
+
+# 2 TASK & TALK: A TESTBED FOR COMPOSITIONAL LANGUAGE EMERGENCE
+
+We consider the cooperative Task & Talk reference game introduced in Kottur et al. (2017). Shown in Fig. 1a, the game is played by two agents – one who observes an attributed object – e.g. (purple, solid, square) – and another who is given a task to retrieve a subset of these attributes over the course of the dialog – e.g. (color,shape). The dialog itself consists of two rounds of agents exchanging single-token utterances from fixed vocabularies. At the end of the dialog, the task-aware agent must report the requested attributes and both agents are rewarded for correct predictions. This causes a language grounded in the objects to emerge because there is no other way to solve the task.
+
+A compositional solution to this task can look like a question-answer style dialog where the taskaware agent queries the other for specific attributes (Fig. 1a) – e.g. uttering “X” requesting the color to which the other agent replies “1” indicating purple. Importantly, this pattern would persist regardless of the other attribute values of the object (e.g. for all $( \mathtt { p u r p l e } , \star , \star )$ objects). However, as there is no grounding supervision provided, agents must learn to associate specific meanings to specific words and it is unlikely for compositional languages to emerge purely by chance. Given a color task, an agent might use “1” for (purple, solid, square) and then “2” for (purple, solid, circle). It is impossible for other agents to know that “2” means purple without having seen (purple, solid, circle), so compositional language is essential for generalization to compositionally novel instances.
+
+Models. To formalize this setting, let Q-bot and A-bot be agent policies parameterized by neural networks $Q$ and $A$ respectively. At each round $t$ , Q-bot observes the task $x _ { Q }$ and it’s memory of the dialog so far $h _ { Q } ^ { t - 1 }$ and produces a single-token utterance $m _ { Q } ^ { t } \in \mathcal { V }$ from the vocabulary $\nu$ . Functionally, $m _ { Q } ^ { t } , h _ { Q } ^ { t } \stackrel { \cdot } { = } Q ( m _ { A } ^ { t - 1 } , x _ { Q } , h _ { Q } ^ { t - 1 } )$ where $m _ { A } ^ { t - 1 }$ is A-bot’s reply in the previous round. Likewise, A-bot responds by computing $\quad \quad m _ { A ; } ^ { t } , h _ { A _ { . } } ^ { t } = A ( m _ { Q } ^ { t } , x _ { A } , h _ { A } ^ { t - 1 } )$ where $x _ { A }$ is the object instance represented symbolically by concatenating 3 one-hot vectors, one per attribute. After two rounds, Q-bot must respond to the task, predicting the requested attribute pair $\hat { u } = U ( x _ { Q } , h _ { Q } ^ { T } )$ as a function of the task and Q-bot’s final memory state. Both agents are rewarded if both attributes are correct (no partial credit). We follow the neural network architectures of $Q , A$ , and $U$ from Kottur et al. (2017).
+
+Measuring Compositional Generalization. Kottur et al. (2017) generated a synthetic dataset consisting of three attribute types (color, shape, style) each with four values (e.g. red, blue, square, star, dotted, solid, ...) and six tasks, one task for each ordered pair of different attribute types. In total, this results in 64 unique instances and 384 task-instance pairs. To evaluate compositionality, Kottur et al. (2017) held out 12 random instances for testing. Given the closed-world set of instances, these 12 are guaranteed to be never-before-seen triplets of attribute values; however, each individual value has been seen in training in other triplets. As such, accuracy on this set is a measure of compositional generalization.
+
+Shortcomings of Kottur et al. (2017) Evaluation. In our investigations, we found some shortcomings in the evaluation protocol of Kottur et al. (2017). First, the authors do not report variance over multiple runs or different random test-sets which we found to be significant. Second, the strategy of randomly selecting the test set can still reward some only partially compositional strategies. For instance, suppose agents develop a language that uses single words to refer to attribute pairs like (red, $\star$ , triangle) and (red, filled, $\star$ ). Such agents might generalize to an unseen instance (red, filled, triangle) by composing the ‘paired’ words above instead of disentangling individual attributes.
+
+We make two modifications to address these issues. Our results are reported as means and variances estimated from multiple training runs with different random seeds evaluated with 4-way crossvalidation. We also introduce a harder dataset where instead of withholding random individual instances (e.g., (green,dotted,triangle),. . . ) as in Kottur et al. (2017), we withhold all instances for a set of attribute pairs (e.g., (green,dotted, $\star$ ), $( \mathtt { r e d } , \mathtt { s o l i d } , \star ) , \dots )$ . We will refer to datasets generated in this fashion as novel pair and the original dataset as novel instance. We report on both settings for comparison, but find our new setting to be significantly more challenging in practice – requiring a stricter notion of compositionality that is more closely aligned with human intuitions about these attributes.
+
+# 3 COMPOSITIONAL LANGUAGE EMERGENCE WITH CULTURAL TRANSMISSION
+
+In iterative learning models of cultural transmission from evolutionary linguistics, competing pressures towards expressivity and compressibility have been shown to induce compositionality over multiple ‘generations’ of language transfer (Kirby, 2001; Kirby et al., 2008). The goal-driven nature of our reference game already encourages expressivity – agents must be able to refer to the objects in order to succeed. To introduce compressibility pressure and parallel literature in evolutionary linguistics, we introduce a population of agents which regularly has members replaced by new agents that lack any understanding of the remaining population’s language. As this paradigm lacks explicit teaching steps where new agents are trained to ground existing words, we consider this approach as a means of implicit cultural transmission.
+
+Populations of Agents. We consider a population of Q-bots $\{ Q ^ { 1 } , \ldots , Q ^ { N _ { Q } } \}$ and a population of Abots $\{ A ^ { 1 } , \ldots , A ^ { \tilde { N _ { A } } } \}$ with each agent having a different set of parameters. At each iteration during learning, we sample a random Q-bot-A-bot pair to interact and receive updates $\mathrm { ~ - ~ } i . e .$ . the red line (2) in Alg. 1. As any Q-bot may be made to communicate with any A-bot, there is pressure for the population to adopt a unified language. Likewise, when an agent is reinitialized it will receive positive reward much more quickly when it happens to use language that its conversational partners understand. Furthermore, ‘compressible’ languages that are easier to learn will result in greater reward for the population in the face of periodic re-initialization of agents.
+
+# Algorithm 1: Training with Replacement and Multiple Agents
+
+![](images/d9fb159b8995993138efdc0355babbc22213c082cff6da9b1e4522e7205d2628.jpg)
+
+11 return all Q-bots and A-bots.
+
+Introducing multiple agents may in itself add compressibility pressure and improve generalizations even without replacement (Raviv et al., 2018). Agents in a population have to model minor linguistic differences between conversational partners given the same memory capacity. Further, each agent provides another potential language variation that can be mimicked and perpetuated–increasing language diversity early in training. We examine these effects through no-replacement baselines, but find that generational pressure where some agents know less than others can also be important for compositionality in our setting.
+
+Replacement. In order to create a notion of ‘generations’ we replace agents periodically. Let $\pi$ be some replacement strategy that returns a subset of the population. Every $E$ epochs, we call $\pi$ and reinitialize the parameters and optimizers for the corresponding agents (blue lines 9-10 in Alg. 1). We investigate three settings of $\pi$ (appendix A.2 for more details):
+
+Uniform Random. Sample an A-bot and Q-bot from uniform random distributions. – Epsilon Greedy. With probability $1 - \varepsilon$ replace the A-bot and Q-bot with the lowest validation accuracy. We use $\varepsilon = 0 . 2$ in our experiments. – Oldest. Replace the oldest A-bot and Q-bot, breaking ties with uniform random sampling.
+
+# 4 EXPERIMENTAL SETTING
+
+Experimental Setting. We evaluate on both our modified Task & Talk dataset and the original from Kottur et al. (2017), as described in Section 2. All results are reported as means and variances computed from a total of 16 trials (four random seeds each with 4-way cross-validation). We report accuracy based on Q-bot getting both elements of the task correct – corresponding to the more restrictive “Both” setting from Kottur et al. (2017).
+
+Kottur et al. (2017) examined a series of increasingly restrictive settings in order to study conditions under which compositionality emerges. The primary variables are whether A-bot has memory (ablated by setting ${ h _ { A } ^ { \bar { t } } } { = } 0$ ) and the vocabulary sizes $\gamma _ { Q }$ and $\nu _ { A }$ for Q-bot and A-bot respectively. For comparison we also evaluate in these settings: Minimal Vocab ( $V _ { Q } { = } 3$ , $V _ { A } { = } 4$ ). Memoryless $^ +$ Minimal Vocab $V _ { Q } { = } 3$ , $V _ { A } { = } 4$ , ${ h _ { A } ^ { t } } \mathrm { { = } } 0 $ ), Overcomplete $V _ { Q } = V _ { A } = 6 \dot { 4 }$ ). We also introduce Memoryless $^ +$ Overcomplete $V _ { Q } = V _ { A } = 6 4$ , $h _ { A } ^ { t } { = } 0$ ) to complete the cross product of settings and examine the role of memory restriction in overcomplete vocabularies.
+
+The Memoryless $^ +$ Minimal Vocabulary setting results in the best compositional generalization; however, this is an extreme setting – requiring not only that the minimum number of groundable symbols be known but also that A-bot not be able to remember it’s previous utterance. While we do report these settings and see quite large performance gains due to cultural transmission, we are mainly interested in the more realistic Overcomplete setting where a large pool of possible tokens is provided and both dialog agents remember the conversation.
+
+Model and Training Details. Our A-bots and Q-bots have the same architectur as in Kottur et al. (2017). All agents are trained with $E = 2 5 0 0 0$ , a batch size of 1000, 1 and the Adam (Kingma & Ba, 2015) optimizer (one per bot) with learning rate 0.01. In the Multi Agent setting we use
+
+![](images/ca67aa287b00528c7c8e3eefeb9c5020788b825d6be972a2216a637b8e32a2a7.jpg)  
+Figure 2: Test set accuracies (with standard deviations) are reported against our new harder dataset using models similar to those in Kottur et al. (2017). Our variations on cultural transmission (darker blue bars) outperform the baselines where language does not change over generations.
+
+$N _ { A } = N _ { Q } = 5$ . We stop training after 8 generations (199000 epochs Multi Agent; 39000 epochs Single Agent). This differs from Kottur et al. (2017), which stopped once train accuracy reached $100 \%$ . Further, we do not perform negative mining.
+
+Baselines. We consider a set of baseline setting to isolate the effect of our approach.
+
+– Single Agent Populations. We ablate the effect of multi-agent populations by training individual A-bot-Q-bot pairs (i.e. populations with $N _ { A } = N _ { Q } = 1 _ { , }$ ). We apply the uniform random (either A-bot or Q-bot at random) and oldest (alternating between A-bot and Q-bot) replacement strategies to these agents; however, the epsilon greedy strategy is not well-defined here. In this setting we decrease $E$ from 25000 to 5000 to keep the average number of gradient updates for each agent constant with respect to the multi-agent experiments.
+
+– No Replacement. We also consider the effect of replacing no agents at all, but still allowing the agents to train for the full 199,000 (39,000) epochs. Improvement over this baseline shows the gains from our replacement strategy under identical computational budgets.
+
+# 5 RESULTS AND ANALYSIS
+
+# 5.1 IMPACT OF CULTURAL TRANSMISSION ON COMPOSITIONAL GENERALIZATION
+
+Results with standard deviations against our harder dataset are reported in Fig. 2. We compared methods and models using dependent paired t-tests and reported the resulting p-values in Section A.3. Result on the original Task & Talk dataset are in Section A.1.
+
+Cultural transmission induces compositionality. Our main result is that cultural transmission approaches outperform baselines without cultural transmission. This can be seen by noting that for each model type in Fig. 2, the 3 darker blue bars (Multi Agent Replacement approaches) are largest. After running a dependent paired t-test against all pairs of baselines and cultural transmission approaches we find a meaningful difference in all cases $( p \leq 0 . 0 5 )$ . This is strong support for our claim that our version of cultural transmission encourages compositional language because it causes better generalization to novel compositions of attributes.
+
+Next we go on to discuss some additional trends we hope the community will find useful.
+
+Population dynamics without replacement usually lead to some compositionality. The Multi Agent No Replacement policies usually outperform than the Single Agent No Replacement policies, though the difference isn’t very significant in the except in the Overcomplete and Minimal Vocab settings. This agrees with recent work from evolutionary linguistics, where multiple agents can lead to compositionality without generational transmission Raviv et al. (2018).
+
+Variations in replacement strategy tend to not affect performance. The Multi Agent Uniform Random/Epsilon Greedy/Oldest replacement strategies are not largely or consistently different from one another across model variations. This suggests that while some agent replacement needs to occur, it is not critical whether agents with worse language are replaced or whether there is a pool of similarly typed agents to remember knowledge lost from older generations. The main factor is that new agents learn in the presence of others who already know a language.
+
+Cultural transmission is complementary with other factors that encourage compositionality. As in Kottur et al. (2017), we find the Memoryless $^ +$ Small Vocab model is clearly the best. This agrees with factors noted elsewhere Kottur et al. (2017); Mordatch & Abbeel (2018); Nowak et al. (2000) and shows how many different factors can affect the emergence of compositionality.
+
+Removing memory makes only minor differences. Removing memory makes no difference (negative or positive) in Single Agent settings, but it can have a relatively small effect in Multi Agent settings, helping Small Vocab models and hurting Overcomplete models. While our approach is complementary with minimizing vocab size to increase compositionality, its makes memory removal less useful. As the Memoryless $^ +$ Overcomplete setting has not been reported before, these results suggest that the relationship between inter-round memory and compositionality is not clear.
+
+Overall, these results show that adding cultural transmission to neural dialog agents improves the compositional generalization of the languages learned by those agents in a way complementary to other priors. It thereby shows how to transfer the cultural transmission principle from evolutionary linguistics to deep learning.
+
+# 5.2 IS GENERATIONAL TRANSMISSION OCCURRING?
+
+Because it is implicit, cultural transmission may not actually be occurring; improvements may be from other sources. How can we measure cultural transmission? We focus on A-bots and take a simple approach. We assume that if two A-bots ‘speak the same language’ then that language was culturally transmitted. There is a combinatorial explosion of possible languages that could refer to all the objects of interest, so if the words that refer to the same object for two agents are the same then they were very likely transmitted from the other agents, rather than similar languages emerging from scratch just by chance. This leads to a simple approach: consider pairs of bots and see if they say similar things in the same context. If they do, then their language was likely transmitted.
+
+More formally, consider the distribution of tokens A-bot $A ^ { i }$ might use to describe its object $x _ { A }$ when talking to Q-bot $Q ^ { k }$ : $p _ { k , i } ( m _ { A } ^ { t } | x _ { A } )$ or $p _ { k , i }$ for short. We want to know how similar $\bar { A } ^ { i }$ ’s language is to that of another A-bot $A ^ { j }$ . We’ll start by comparing those two distributions by computing the KL divergence between them and then taking an average over context (objects, Q-bots, and dialog rounds) to get our pairwise agent language similarity metric $D _ { i j }$ :
+
+$$
+D _ { i j } = \hat { E } _ { x _ { A } , k , t } \left[ D _ { K L } \left( p _ { k , i } ( m _ { A } ^ { t } | x _ { A } ) , p _ { k , j } ( m _ { A } ^ { t } | x _ { A } ) \right) \right]
+$$
+
+Taking another average, this time over all pairs of bots (and also random seeds and cross-val folds), gives our final measure of language similarity reported in Fig. 3.
+
+$$
+D = \hat { E } _ { i , j \mathrm { ~ s . t . ~ } i \neq j } \left[ D _ { i j } \right]
+$$
+
+$D$ is smaller the more similar language is between bots. Note that even though $D _ { i j }$ is not symmetric (because KL divergence is not), $D$ is symmetric because it averages over both directions of pairs.
+
+We compute $D$ by sampling an empirical distribution over all messages and observations, taking 10 sample dialogues in each possible test state $( x _ { A } , x _ { Q } )$ of the world using the final populations of agents as in Fig. 2. Note that this metric applies to a group of agents, so we measure it for only the Multi Agent settings, including two new baselines colored red in Fig. 3. The Single Agents Combined baseline trains 4 Single Agent No Replacement models independently then puts them together and computes $D$ for that group. These agents only speak similar languages by chance, so $D$ is high. The Random Initialization baseline evaluates language similarity using newly initialized models. These agents have about a uniform distribution over words at every utterance, so their languages are both very similar and useless. For each model these baselines act like practical (not strict) upper and lower bounds on $D$ , respectively.
+
+Fig. 3 shows this language dissimilarity metric for all our settings. As we expect, the paired Single Agents are highly dissimilar compared to agents from Multi Agent populations. Further, all the replacement strategies result in increased language similarity—although the degree of this effect seems dependent on vocabulary setting. This provides some evidence that cultural transmission is occurring in Multi Agent settings and is encouraged by the replacement strategy in our approach. While all Multi Agent settings resulted in language transmission, our replacement strategies results in more compositional languages due to repeated teaching of new generations of agents.
+
+# 5.3 VISUALIZING EMERGENT LANGUAGES
+
+In this section we visualize the language learned by agents at various stages of training to reinforce our previous conclusions and build intuition. Each of the three sub-figures in Fig. 4 summarizes all of the conversations between a particular pair of bots for the (shape, color) task. To see how these summaries work, consider Fig. 4a. That sub-figure is divided into a $4 \times 4$ grid with 4 elements in each cell, so there is one element for each of the 64 possible objects. For this task, objects in each row of the $4 \times 4$ grid have the same shape and objects in each column have the same color. To the right of each object are the two tokens A-bot used to respond to Q-bot in the two dialog rounds. Ideally they should indicate the color and shape of the object. Finally, the check-marks or Xs to the right of A-bot’s utterances indicate whether Q-bot guessed correctly.
+
+![](images/e68419c7891d177e2ae37f512fe3dc70e24b66bef8877d8103b4dc83fed0b9e0.jpg)  
+Model Figure 3: Do bots in a population learn similar languages? On the y-axis (eq. (2)) lower values indicate similar language. Populations evolved with our method speak similar languages, but independently evolved agents do not. Thus our implicit procedure induces cultural transmission.
+
+![](images/383e1a2eec5bbbd3805c1ab8f39b1a203f63370349f09eb5189cc2d11b8bb83f.jpg)  
+Figure 4: Each sub-figure summarizes an A-bot’s language, as described in Section 5.3. By comparing the baseline of Fig. 4a to a similar pair of bots from our approach Section 4b we can see that our approach encourages compositional language to emerge. Furthermore, the similarity between Fig. 4b and Fig. 4c suggests language is indeed transmitted in our approach.
+
+From left to right: Fig. 4a summarizes the single pair from a Single Agent No Replacement run (3000 iterations old); Fig. 4b summarizes dialogs between an old Q-bot (about 23000 iterations) and a recently re-initialized A-bot (about 3000 iterations) at the 8th and final generation of a Multi Oldest run; Fig. 4c summarizes dialogs between the same old Q-bot as in Fig. 4b and an old A-bot (13000 iterations) from the same Multi Oldest experiment. Even though the A-bots in Fig. 4a and Fig. 4b have trained for about2 the same number of iterations, the A-bot trained in the presence of other bots which already know a functional language has already learned a somewhat compositional language whereas the Single Agent A-bot has not. Furthermore, by comparing the old A-bot’s language Fig. 4c with the new one Fig. 4b we can see that they are extremely similar. They even lead to the same mistakes. This again suggests that language is transmitted between bots, in agreement with our previous experiments.
+
+# 6 RELATED WORK
+
+Language Evolution Causes Structure. Researchers have spent decades studying how unique properties of human language like compositionality could have emerged. There is general agreement that people acquire language using a combination of innate cognitive capacity and learning from other language speakers (cultural transmission), with the degree of each being widely disputed Perfors (2002); Pinker & Bloom (1990). Both innate cognitive capacity and specific modern human languages like English co-evolved Briscoe (2000) via biological Pinker & Bloom (1990) and cultural Tomasello (1999); Smith (2006) evolution, respectively.
+
+In particular, explanations of how the cultural evolution of languages could cause structure like compositionality are in abundance Nowak & Krakauer (1999); Nowak et al. (2000); Smith et al. (2003); Brighton (2002); Vogt (2005); Kirby et al. (2014); Spike et al. (2017). An important piece of the explanation of linguistic structure is the iterated learning model Kirby et al. (2014); Kirby (2001); Kirby et al. (2008) used to motivate our approach. Indeed it shows that cultural transmission causes structure in computational Kirby (2001; 2002); Christiansen & Kirby (2003); Smith et al. (2003) and human Kirby et al. (2008); Cornish et al. (2009); Scott-Phillips & Kirby (2010) experiments. Even though cultural transmission may aid the emergence of compositionality, recent results in evolutionary linguistics Raviv et al. (2018) and deep learning Kottur et al. (2017); Mordatch & Abbeel (2018) also emphasize other factors.
+
+While existing work in deep learning has focused on biases that encourage compositionality, it has not considered settings where language is permitted to evolve over generations of agents. We have shown such an approach is viable and even complementary with other approaches.
+
+Language Emergence in Deep Learning. Recent work in deep learning has increasingly focused on multi-agent environments where deep agents learn to accomplish goals (possibly cooperative or competitive) by interacting appropriately with the environment and each other. Some of this work has shown that deep agents will develop their own language where none exists initially if driven by a task which requires communication Foerster et al. (2016); Sukhbaatar et al. (2016); Lazaridou et al. (2017). Most relevant is work which focuses on conditions under which compositional language emerges as deep agents learn to cooperate Mordatch & Abbeel (2018); Kottur et al. (2017). Both Mordatch & Abbeel (2018) and Kottur et al. (2017) find that limiting the vocabulary size so that there aren’t too many more words than there are objects to refer to encourages compositionality, which follows earlier results in evolutionary linguistics Nowak et al. (2000). Follow up work has continued to investigate the emergence of compositional language among neural agents, mainly focusing on perceptual as opposed to symbolic input and how the structure of the input relates to the tendency for compositional language to emerge Choi et al. (2018); Havrylov & Titov (2017); Lazaridou et al. (2018). Other work has shown that Multi Agent interaction leads to better emergent translation Lee et al. (2018), but it does not measure compositionality.
+
+Cultural Evolution and Neural Nets. Somewhat recently, Bengio (2012) suggested that culturally transmitted ideas may help in escaping from local minima. Experiments in Gulc¸ehre & Bengio ¨ (2016) support this idea by showing that supervision of intermediate representations allows a more complex toy task to be learned. Unlike our work, these experiments use direct supervision provided by the designed environment rather than indirect and implicit supervision provided by other agents.
+
+Two concurrent works examine the role of periodic agent replacement on language emergence – albeit in different environments. In Li & Bowling (2019) replacement is used to encourage languages to be easy to teach, and this in turn causes compositionality. In Dagan et al. (2019) neural language is transmitted through a bottleneck caused by replacement. The resulting language has increased efficiency and effectiveness, with further results showing that co-evolving the agents themselves with the language amplifies the effect. Both of these works support our central observations.
+
+# 7 CONCLUSION
+
+In this work we investigated cultural transmission in deep neural dialog agents, applying it to language emergence. The evolutionary linguistics community has long used cultural transmission to explain how compositional languages could have emerged. The deep learning community, having recently become interested in language emergence, has not investigated that link until now. Instead of explicit models of cultural transmission familiar in evolutionary linguistics, we favor an implicit model where language is transmitted from generation to generation only because it helps agents achieve their goals. We show that this does indeed cause cultural transmission and compositionality.
+
+Future work. While our work used an implicit version of cultural transmission, we are interested in investigating the effect of explicit versions of cultural transmission on language structure. In another direction, cultural transmission may also provide an appropriate prior for neural representations of non-language information.
+
+# REFERENCES
+
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+
+Yoshua Bengio. Evolving culture vs local minima. CoRR, abs/1203.2990, 2012.
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+
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+
+Hannah Cornish, Monica Tamariz, and Simon Kirby. Complex adaptive systems and the origins of adaptive structure: What experiments can tell us. Language Learning, 59(s1):187–205, 2009. doi: 10.1111/j.1467-9922.2009.00540.x. URL https://onlinelibrary.wiley.com/doi/ abs/10.1111/j.1467-9922.2009.00540.x.
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+
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+Simon Kirby, Hannah Cornish, and Kenny Smith. Cumulative cultural evolution in the laboratory: An experimental approach to the origins of structure in human language. Proceedings of the National Academy of Sciences, 2008. ISSN 0027-8424. doi: 10.1073/pnas.0707835105. URL https://www.pnas.org/content/early/2008/07/29/0707835105.
+
+Simon Kirby, Tom Griffiths, and Kenny Smith. Iterated learning and the evolution of language. Current Opinion in Neurobiology, 28:108–114, 2014.
+
+Satwik Kottur, Jose M. F. Moura, Stefan Lee, and Dhruv Batra. Natural language does not emerge ´ ’naturally’ in multi-agent dialog. In EMNLP, 2017.
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+Brenden M. Lake and Marco Baroni. Generalization without systematicity: On the compositional skills of sequence-to-sequence recurrent networks. In ICML, 2018.
+
+Angeliki Lazaridou, Alexander Peysakhovich, and Marco Baroni. Multi-agent cooperation and the emergence of (natural) language. In International Conference on Learning Representations (ICLR), 2017.
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+
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+
+# A APPENDIX
+
+# A.1 RESULTS ON SINGLE HELD OUT ATTRIBUTE DATASET OF KOTTUR ET AL. (2017)
+
+In Section 4 we proposed a new harder compositional dataset different from the one in Kottur et al. (2017). For comparison, in this section we train and evaluate our models on the original dataset from Kottur et al. (2017) to show that our approach also improves still improves compositionality in this setting and to show that our new dataset is indeed harder.
+
+![](images/3c1f5868d98f9e916b1c8cf1e9e36eb91faa16b49dcc985ae245616a9c280394.jpg)  
+Figure 5: Test set accuracies (with standard deviations) are reported by training and evaluating the same models as in our main results Fig. 2 against the dataset from Kottur et al. (2017). These results do not perform cross-validation, following Kottur et al. (2017). They only vary across 4 different random seeds. This dataset is significantly easier than our new dataset, as indicated by the Our proposed approach still outperforms models without replacement or multiple agents.
+
+# A.2 REPLACEMENT STRATEGIES
+
+Our approach to cultural transmission periodically replaces agents by re-initializing them. The approach section outlines various replacement strategies (policy $\pi$ ), but does not detail their implementation. We do so here.
+
+These strategies depend on a number of possible inputs:
+
+• $e$ the current epoch   
+• $E$ the period of agent replacement   
+• $v _ { i } ^ { Q } / v _ { i } ^ { A }$ the validation accuracy of agent $i$ for Q-bots/A-bots. For Q-bots this is averaged over all potential A-bot partners, and vice-versa for A-bots.   
+• $a _ { i } ^ { Q } / a _ { i } ^ { A }$ the age in epochs of agent $i$ for Q-bots/A-bots
+
+Single Agent strategies are given in Alg. 2 and Alg. 3. Multi Agent strategies are given in Alg. 4, Alg. 5, and Alg. 6. Note that Single Agent strategies always replace one agent while Multi Agent strategies always replace one Q-bot and one A-bot. An additional Replace All baseline strategy is given in Alg. ?? and generalizes to both Single and Multi Agent cases.
+
+# Algorithm 2: Single Agent - Random Replacement
+
+<table><tr><td>1 d~U{0,1}</td></tr><tr><td></td></tr><tr><td>2 if d=O then return {A-bot }</td></tr><tr><td></td></tr><tr><td>4 else return {Q-bot}</td></tr></table>
+
+# A.3 DETAILED RESULTS
+
+In our experiments we compare models and we compare replacement strategies. We ran dependent paired t-tests across random seeds, cross-val folds, and replacement strategies to compare models. We ran dependent paired t-tests across random seeds, cross-val folds, and models to compare replacement strategies. The p-values for all of these t-tests are reported here.
+
+Replacement strategy comparisons are in Fig. 7 (Single Agent) and Fig. 8 (Multi Agent). Model comparisons are in Fig. 6.
+
+![](images/fc000857cd1616211d7de8541016d7cbd9c818d97401baf12896a1033c618806.jpg)  
+Figure 6: Replacement strategy comparison p-values.
+
+![](images/3765b7e01ed3a5a0cfbba49a0af8fadc84510f5b295801cd24b995b74c104a2a.jpg)  
+Figure 7: Single Agent model comparison p-values.
+
+![](images/7ef9b3e377968b1755836c0ed298545d627e2ebf38fcd55e8c7a107744b84db2.jpg)  
+Figure 8: Multi Agent model comparison p-values.
+
+# Algorithm 3: Single Agent - Alternate Replacement
+
+1 Input: e   
+2 if $\lfloor e / E \rfloor = 0$ then   
+3 return { A-bot }   
+4 else   
+5 return { Q-bot }
+
+# Algorithm 4: Multi Agent - Uniform Random Replacement
+
+1 iA ∼ U{1, NA}   
+2 iQ ∼ U{1, NQ}   
+3 return { A-bot $i _ { A }$ , Q-bot iQ }
+
+# Algorithm 5: Multi Agent - Epsilon Greedy Replacement
+
+1 Input: $v _ { i } ^ { Q } \forall i$ , $v _ { i } ^ { A } \forall i$ , ε ∈ [0, 1) (usually 0.2)   
+2 $d \sim \mathcal { U } [ 0 , \overset { \cdot } { 1 } )$   
+3 if $d < \varepsilon$ then   
+4 $i _ { A } \sim \mathcal { U } \{ 1 , N _ { A } \}$   
+5 $i _ { Q } \sim \mathcal { U } \{ 1 , N _ { Q } \}$   
+$\mathbf { 6 }$ else   
+7 $i _ { A } = \mathrm { a r g m i n } _ { i } v _ { i } ^ { A }$ (unique in our experiments)   
+8 $i _ { Q } = \mathrm { a r g m i n } _ { i } v _ { i } ^ { Q }$ (unique in our experiments)   
+9 return { A-bot iA, Q-bot iQ }
+
+# Algorithm 6: Multi Agent - Oldest Replacement
+
+1 Input: $a _ { i } ^ { Q } \forall i$ , $a _ { i } ^ { V } \forall i$   
+2 $i _ { A } = \mathcal { U } \{ \mathrm { a r g m a x } _ { i } a _ { i } ^ { A } \}$   
+3 $i _ { Q } = \mathcal { U } \{ \mathrm { a r g m a x } _ { i } a _ { i } ^ { Q } \}$   
+4 return { A-bot $i _ { A }$ , Q-bot iQ }

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+        "text": "Recent work has studied the emergence of language among deep reinforcement learning agents that must collaborate to solve a task. Of particular interest are the factors that cause language to be compositional—i.e. express meaning by combining words which themselves have meaning. Evolutionary linguists have found that in addition to structural priors like those already studied in deep learning, the dynamics of transmitting language from generation to generation contribute significantly to the emergence of compositionality. In this paper, we introduce these cultural evolutionary dynamics into language emergence by periodically replacing agents in a population to create a knowledge gap, implicitly inducing cultural transmission of language. We show that this implicit cultural transmission encourages the resulting languages to exhibit better compositional generalization. ",
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+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Compositionality is an important structure of language that reflects a disentangled understanding of the world – enabling the expression of infinitely many concepts using finitely many elements. Agents that have compositional understandings of the world generalize in obviously correct ways even in the face of limited training examples (Lake & Baroni, 2018). For example, an agent with a compositional understanding of blue squares and purple triangles should also understand purple squares without directly observing any of them. Developing artificial agents that can ground, understand, and produce compositional (and therefore more interpretable) language could greatly improve generalization to new instances and ease human-AI interactions. ",
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+        "text": "In building theories of how compositionality emerges in human languages, work in evolutionary linguistics looks to the process of cultural transmission (Kirby, 2001; Kirby et al., 2008). Cultural transmission of language occurs when a group of agents pass their language on to a new group of agents, e.g. parents who teach their children to speak as they do. Because this education is incomplete and biased, it allows the language itself to change over time via a process known as cultural evolution. This paradigm (Kirby et al., 2014) explains the emergence of compositionality as a result of expressivity and compressibility – i.e. to be most effective, a language should be expressive enough to differentiate between all possible meanings (e.g., objects) and compressible enough to be learned easily. Work in the evolutionary linguistics community has shown that over multiple ‘generations’ these competing pressures result in the emergence of compositional languages both in simulation (Kirby, 2001) and with human subjects (Kirby et al., 2008). These studies aim to understand humans whereas we want to understand and design artificial neural networks. ",
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+        "text": "Approaching the problem from another direction, recent work in AI has studied language emergence in such multi-agent, goal-driven tasks. These works have demonstrated that agent languages will emerge to enable coordination-centric tasks to be solved without direct or even indirect language supervision (Foerster et al., 2016; Sukhbaatar et al., 2016; Lazaridou et al., 2017; Das et al., 2017). However, the resulting languages are usually not compositional (Kottur et al., 2017) and are difficult to interpret, even by other machines (Andreas et al., 2017). Some existing work has studied means to encourage compositional language formation (Mordatch & Abbeel, 2018; Kottur et al., 2017), but these settings study fixed populations of agents – i.e. examining language within a single generation. ",
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+        "text": "In this work we bridge these two areas – examining the effect of generational cultural transmission on the compositionality of emergent languages in a multi-agent, goal-driven setting. ",
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+            "(b) Implicit cultural transmission ",
+            "Figure 1: We introduce cultural transmission into language emergence between neural agents. The starting point of our study is a goal-oriented dialog task (similar to that of Kottur et al. (2017)), summarized in Fig. 1a. During learning we periodically replace some agents with new ones (gray agents). These new agents do not know any language, but instead of creating one they learn it from older agents. This creates generations of language that become more compositional over time. "
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+        "text": "We study this in the context of a cooperative dialog-based reference game involving two agents communicating in discrete symbols (Kottur et al., 2017); an example dialog is shown in Fig. 1a. To examine cultural transmission, we extend this setting to a population of agents (Fig. 1b) and introduce a simple mechanism to induce the expressivity and compressibility pressures inherent in cultural transmission. Specifically, we periodically re-initialize some subset of the agents in the population. In order to perform well at the task, the population’s emergent language must be sufficiently expressive to reference all the objects (expressivity) and must be easily learnable by these ‘new’ agents (compressibility). The new agents have a randomized language whereas the surviving agents already know a grounded language. This “knowledge gap” creates an implicit ‘teaching’ setting that is analogous to the explicit transmission stage in models of iterative learning (Kirby, 2001). ",
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+        "text": "Through our experiments and analysis, we show that periodic agent replacement is an effective way to induce cultural transmission and yields more compositionally generalizable language in our setting. To summarize, our contributions are: ",
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+        "text": "– We propose a method for inducing implicit cultural transmission in neural language models. – We introduce new metrics to measure the similarity between agent languages and verify cultural transmission has occurred as a result of our periodic agent replacement protocol. We show our cultural transmission procedure induces compositionality in neural language models, going from $13 \\%$ accuracy on a compositionally novel test set to $46 \\%$ in the best configuration. Further, we show this is complementary with previous priors which encourage compositionality. ",
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+        "text": "2 TASK & TALK: A TESTBED FOR COMPOSITIONAL LANGUAGE EMERGENCE ",
+        "text_level": 1,
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+        "text": "We consider the cooperative Task & Talk reference game introduced in Kottur et al. (2017). Shown in Fig. 1a, the game is played by two agents – one who observes an attributed object – e.g. (purple, solid, square) – and another who is given a task to retrieve a subset of these attributes over the course of the dialog – e.g. (color,shape). The dialog itself consists of two rounds of agents exchanging single-token utterances from fixed vocabularies. At the end of the dialog, the task-aware agent must report the requested attributes and both agents are rewarded for correct predictions. This causes a language grounded in the objects to emerge because there is no other way to solve the task. ",
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+        "text": "A compositional solution to this task can look like a question-answer style dialog where the taskaware agent queries the other for specific attributes (Fig. 1a) – e.g. uttering “X” requesting the color to which the other agent replies “1” indicating purple. Importantly, this pattern would persist regardless of the other attribute values of the object (e.g. for all $( \\mathtt { p u r p l e } , \\star , \\star )$ objects). However, as there is no grounding supervision provided, agents must learn to associate specific meanings to specific words and it is unlikely for compositional languages to emerge purely by chance. Given a color task, an agent might use “1” for (purple, solid, square) and then “2” for (purple, solid, circle). It is impossible for other agents to know that “2” means purple without having seen (purple, solid, circle), so compositional language is essential for generalization to compositionally novel instances. ",
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+        "text": "Models. To formalize this setting, let Q-bot and A-bot be agent policies parameterized by neural networks $Q$ and $A$ respectively. At each round $t$ , Q-bot observes the task $x _ { Q }$ and it’s memory of the dialog so far $h _ { Q } ^ { t - 1 }$ and produces a single-token utterance $m _ { Q } ^ { t } \\in \\mathcal { V }$ from the vocabulary $\\nu$ . Functionally, $m _ { Q } ^ { t } , h _ { Q } ^ { t } \\stackrel { \\cdot } { = } Q ( m _ { A } ^ { t - 1 } , x _ { Q } , h _ { Q } ^ { t - 1 } )$ where $m _ { A } ^ { t - 1 }$ is A-bot’s reply in the previous round. Likewise, A-bot responds by computing $\\quad \\quad m _ { A ; } ^ { t } , h _ { A _ { . } } ^ { t } = A ( m _ { Q } ^ { t } , x _ { A } , h _ { A } ^ { t - 1 } )$ where $x _ { A }$ is the object instance represented symbolically by concatenating 3 one-hot vectors, one per attribute. After two rounds, Q-bot must respond to the task, predicting the requested attribute pair $\\hat { u } = U ( x _ { Q } , h _ { Q } ^ { T } )$ as a function of the task and Q-bot’s final memory state. Both agents are rewarded if both attributes are correct (no partial credit). We follow the neural network architectures of $Q , A$ , and $U$ from Kottur et al. (2017). ",
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+        "type": "text",
+        "text": "Measuring Compositional Generalization. Kottur et al. (2017) generated a synthetic dataset consisting of three attribute types (color, shape, style) each with four values (e.g. red, blue, square, star, dotted, solid, ...) and six tasks, one task for each ordered pair of different attribute types. In total, this results in 64 unique instances and 384 task-instance pairs. To evaluate compositionality, Kottur et al. (2017) held out 12 random instances for testing. Given the closed-world set of instances, these 12 are guaranteed to be never-before-seen triplets of attribute values; however, each individual value has been seen in training in other triplets. As such, accuracy on this set is a measure of compositional generalization. ",
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+        "type": "text",
+        "text": "Shortcomings of Kottur et al. (2017) Evaluation. In our investigations, we found some shortcomings in the evaluation protocol of Kottur et al. (2017). First, the authors do not report variance over multiple runs or different random test-sets which we found to be significant. Second, the strategy of randomly selecting the test set can still reward some only partially compositional strategies. For instance, suppose agents develop a language that uses single words to refer to attribute pairs like (red, $\\star$ , triangle) and (red, filled, $\\star$ ). Such agents might generalize to an unseen instance (red, filled, triangle) by composing the ‘paired’ words above instead of disentangling individual attributes. ",
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+        "type": "text",
+        "text": "We make two modifications to address these issues. Our results are reported as means and variances estimated from multiple training runs with different random seeds evaluated with 4-way crossvalidation. We also introduce a harder dataset where instead of withholding random individual instances (e.g., (green,dotted,triangle),. . . ) as in Kottur et al. (2017), we withhold all instances for a set of attribute pairs (e.g., (green,dotted, $\\star$ ), $( \\mathtt { r e d } , \\mathtt { s o l i d } , \\star ) , \\dots )$ . We will refer to datasets generated in this fashion as novel pair and the original dataset as novel instance. We report on both settings for comparison, but find our new setting to be significantly more challenging in practice – requiring a stricter notion of compositionality that is more closely aligned with human intuitions about these attributes. ",
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+        "type": "text",
+        "text": "3 COMPOSITIONAL LANGUAGE EMERGENCE WITH CULTURAL TRANSMISSION ",
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+        "text": "In iterative learning models of cultural transmission from evolutionary linguistics, competing pressures towards expressivity and compressibility have been shown to induce compositionality over multiple ‘generations’ of language transfer (Kirby, 2001; Kirby et al., 2008). The goal-driven nature of our reference game already encourages expressivity – agents must be able to refer to the objects in order to succeed. To introduce compressibility pressure and parallel literature in evolutionary linguistics, we introduce a population of agents which regularly has members replaced by new agents that lack any understanding of the remaining population’s language. As this paradigm lacks explicit teaching steps where new agents are trained to ground existing words, we consider this approach as a means of implicit cultural transmission. ",
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+        "text": "Populations of Agents. We consider a population of Q-bots $\\{ Q ^ { 1 } , \\ldots , Q ^ { N _ { Q } } \\}$ and a population of Abots $\\{ A ^ { 1 } , \\ldots , A ^ { \\tilde { N _ { A } } } \\}$ with each agent having a different set of parameters. At each iteration during learning, we sample a random Q-bot-A-bot pair to interact and receive updates $\\mathrm { ~ - ~ } i . e .$ . the red line (2) in Alg. 1. As any Q-bot may be made to communicate with any A-bot, there is pressure for the population to adopt a unified language. Likewise, when an agent is reinitialized it will receive positive reward much more quickly when it happens to use language that its conversational partners understand. Furthermore, ‘compressible’ languages that are easier to learn will result in greater reward for the population in the face of periodic re-initialization of agents. ",
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+        "type": "text",
+        "text": "Algorithm 1: Training with Replacement and Multiple Agents ",
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+        "img_path": "images/d9fb159b8995993138efdc0355babbc22213c082cff6da9b1e4522e7205d2628.jpg",
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+        "type": "text",
+        "text": "11 return all Q-bots and A-bots. ",
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+        "type": "text",
+        "text": "Introducing multiple agents may in itself add compressibility pressure and improve generalizations even without replacement (Raviv et al., 2018). Agents in a population have to model minor linguistic differences between conversational partners given the same memory capacity. Further, each agent provides another potential language variation that can be mimicked and perpetuated–increasing language diversity early in training. We examine these effects through no-replacement baselines, but find that generational pressure where some agents know less than others can also be important for compositionality in our setting. ",
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+        "type": "text",
+        "text": "Replacement. In order to create a notion of ‘generations’ we replace agents periodically. Let $\\pi$ be some replacement strategy that returns a subset of the population. Every $E$ epochs, we call $\\pi$ and reinitialize the parameters and optimizers for the corresponding agents (blue lines 9-10 in Alg. 1). We investigate three settings of $\\pi$ (appendix A.2 for more details): ",
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+        "type": "text",
+        "text": "Uniform Random. Sample an A-bot and Q-bot from uniform random distributions. – Epsilon Greedy. With probability $1 - \\varepsilon$ replace the A-bot and Q-bot with the lowest validation accuracy. We use $\\varepsilon = 0 . 2$ in our experiments. – Oldest. Replace the oldest A-bot and Q-bot, breaking ties with uniform random sampling. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL SETTING ",
+        "text_level": 1,
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+        "text": "Experimental Setting. We evaluate on both our modified Task & Talk dataset and the original from Kottur et al. (2017), as described in Section 2. All results are reported as means and variances computed from a total of 16 trials (four random seeds each with 4-way cross-validation). We report accuracy based on Q-bot getting both elements of the task correct – corresponding to the more restrictive “Both” setting from Kottur et al. (2017). ",
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+        "type": "text",
+        "text": "Kottur et al. (2017) examined a series of increasingly restrictive settings in order to study conditions under which compositionality emerges. The primary variables are whether A-bot has memory (ablated by setting ${ h _ { A } ^ { \\bar { t } } } { = } 0$ ) and the vocabulary sizes $\\gamma _ { Q }$ and $\\nu _ { A }$ for Q-bot and A-bot respectively. For comparison we also evaluate in these settings: Minimal Vocab ( $V _ { Q } { = } 3$ , $V _ { A } { = } 4$ ). Memoryless $^ +$ Minimal Vocab $V _ { Q } { = } 3$ , $V _ { A } { = } 4$ , ${ h _ { A } ^ { t } } \\mathrm { { = } } 0 $ ), Overcomplete $V _ { Q } = V _ { A } = 6 \\dot { 4 }$ ). We also introduce Memoryless $^ +$ Overcomplete $V _ { Q } = V _ { A } = 6 4$ , $h _ { A } ^ { t } { = } 0$ ) to complete the cross product of settings and examine the role of memory restriction in overcomplete vocabularies. ",
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+        "type": "text",
+        "text": "The Memoryless $^ +$ Minimal Vocabulary setting results in the best compositional generalization; however, this is an extreme setting – requiring not only that the minimum number of groundable symbols be known but also that A-bot not be able to remember it’s previous utterance. While we do report these settings and see quite large performance gains due to cultural transmission, we are mainly interested in the more realistic Overcomplete setting where a large pool of possible tokens is provided and both dialog agents remember the conversation. ",
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+        "type": "text",
+        "text": "Model and Training Details. Our A-bots and Q-bots have the same architectur as in Kottur et al. (2017). All agents are trained with $E = 2 5 0 0 0$ , a batch size of 1000, 1 and the Adam (Kingma & Ba, 2015) optimizer (one per bot) with learning rate 0.01. In the Multi Agent setting we use ",
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+    {
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+        "img_path": "images/ca67aa287b00528c7c8e3eefeb9c5020788b825d6be972a2216a637b8e32a2a7.jpg",
+        "image_caption": [
+            "Figure 2: Test set accuracies (with standard deviations) are reported against our new harder dataset using models similar to those in Kottur et al. (2017). Our variations on cultural transmission (darker blue bars) outperform the baselines where language does not change over generations. "
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+        "type": "text",
+        "text": "$N _ { A } = N _ { Q } = 5$ . We stop training after 8 generations (199000 epochs Multi Agent; 39000 epochs Single Agent). This differs from Kottur et al. (2017), which stopped once train accuracy reached $100 \\%$ . Further, we do not perform negative mining. ",
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+    {
+        "type": "text",
+        "text": "Baselines. We consider a set of baseline setting to isolate the effect of our approach. ",
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+        "type": "text",
+        "text": "– Single Agent Populations. We ablate the effect of multi-agent populations by training individual A-bot-Q-bot pairs (i.e. populations with $N _ { A } = N _ { Q } = 1 _ { , }$ ). We apply the uniform random (either A-bot or Q-bot at random) and oldest (alternating between A-bot and Q-bot) replacement strategies to these agents; however, the epsilon greedy strategy is not well-defined here. In this setting we decrease $E$ from 25000 to 5000 to keep the average number of gradient updates for each agent constant with respect to the multi-agent experiments. ",
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+        "type": "text",
+        "text": "– No Replacement. We also consider the effect of replacing no agents at all, but still allowing the agents to train for the full 199,000 (39,000) epochs. Improvement over this baseline shows the gains from our replacement strategy under identical computational budgets. ",
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+    {
+        "type": "text",
+        "text": "5 RESULTS AND ANALYSIS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "5.1 IMPACT OF CULTURAL TRANSMISSION ON COMPOSITIONAL GENERALIZATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Results with standard deviations against our harder dataset are reported in Fig. 2. We compared methods and models using dependent paired t-tests and reported the resulting p-values in Section A.3. Result on the original Task & Talk dataset are in Section A.1. ",
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+        "type": "text",
+        "text": "Cultural transmission induces compositionality. Our main result is that cultural transmission approaches outperform baselines without cultural transmission. This can be seen by noting that for each model type in Fig. 2, the 3 darker blue bars (Multi Agent Replacement approaches) are largest. After running a dependent paired t-test against all pairs of baselines and cultural transmission approaches we find a meaningful difference in all cases $( p \\leq 0 . 0 5 )$ . This is strong support for our claim that our version of cultural transmission encourages compositional language because it causes better generalization to novel compositions of attributes. ",
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+    {
+        "type": "text",
+        "text": "Next we go on to discuss some additional trends we hope the community will find useful. ",
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+        "type": "text",
+        "text": "Population dynamics without replacement usually lead to some compositionality. The Multi Agent No Replacement policies usually outperform than the Single Agent No Replacement policies, though the difference isn’t very significant in the except in the Overcomplete and Minimal Vocab settings. This agrees with recent work from evolutionary linguistics, where multiple agents can lead to compositionality without generational transmission Raviv et al. (2018). ",
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+        "type": "text",
+        "text": "Variations in replacement strategy tend to not affect performance. The Multi Agent Uniform Random/Epsilon Greedy/Oldest replacement strategies are not largely or consistently different from one another across model variations. This suggests that while some agent replacement needs to occur, it is not critical whether agents with worse language are replaced or whether there is a pool of similarly typed agents to remember knowledge lost from older generations. The main factor is that new agents learn in the presence of others who already know a language. ",
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+        "type": "text",
+        "text": "Cultural transmission is complementary with other factors that encourage compositionality. As in Kottur et al. (2017), we find the Memoryless $^ +$ Small Vocab model is clearly the best. This agrees with factors noted elsewhere Kottur et al. (2017); Mordatch & Abbeel (2018); Nowak et al. (2000) and shows how many different factors can affect the emergence of compositionality. ",
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+        "type": "text",
+        "text": "Removing memory makes only minor differences. Removing memory makes no difference (negative or positive) in Single Agent settings, but it can have a relatively small effect in Multi Agent settings, helping Small Vocab models and hurting Overcomplete models. While our approach is complementary with minimizing vocab size to increase compositionality, its makes memory removal less useful. As the Memoryless $^ +$ Overcomplete setting has not been reported before, these results suggest that the relationship between inter-round memory and compositionality is not clear. ",
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+        "text": "Overall, these results show that adding cultural transmission to neural dialog agents improves the compositional generalization of the languages learned by those agents in a way complementary to other priors. It thereby shows how to transfer the cultural transmission principle from evolutionary linguistics to deep learning. ",
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+        "type": "text",
+        "text": "5.2 IS GENERATIONAL TRANSMISSION OCCURRING? ",
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+        "text": "Because it is implicit, cultural transmission may not actually be occurring; improvements may be from other sources. How can we measure cultural transmission? We focus on A-bots and take a simple approach. We assume that if two A-bots ‘speak the same language’ then that language was culturally transmitted. There is a combinatorial explosion of possible languages that could refer to all the objects of interest, so if the words that refer to the same object for two agents are the same then they were very likely transmitted from the other agents, rather than similar languages emerging from scratch just by chance. This leads to a simple approach: consider pairs of bots and see if they say similar things in the same context. If they do, then their language was likely transmitted. ",
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+        "type": "text",
+        "text": "More formally, consider the distribution of tokens A-bot $A ^ { i }$ might use to describe its object $x _ { A }$ when talking to Q-bot $Q ^ { k }$ : $p _ { k , i } ( m _ { A } ^ { t } | x _ { A } )$ or $p _ { k , i }$ for short. We want to know how similar $\\bar { A } ^ { i }$ ’s language is to that of another A-bot $A ^ { j }$ . We’ll start by comparing those two distributions by computing the KL divergence between them and then taking an average over context (objects, Q-bots, and dialog rounds) to get our pairwise agent language similarity metric $D _ { i j }$ : ",
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+        "type": "equation",
+        "img_path": "images/953f3fe7c772cfca959ec44bc6081602d889917e668b9222ba8bfb790e0b2393.jpg",
+        "text": "$$\nD _ { i j } = \\hat { E } _ { x _ { A } , k , t } \\left[ D _ { K L } \\left( p _ { k , i } ( m _ { A } ^ { t } | x _ { A } ) , p _ { k , j } ( m _ { A } ^ { t } | x _ { A } ) \\right) \\right]\n$$",
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+        "type": "text",
+        "text": "Taking another average, this time over all pairs of bots (and also random seeds and cross-val folds), gives our final measure of language similarity reported in Fig. 3. ",
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+        "type": "equation",
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+        "text": "$$\nD = \\hat { E } _ { i , j \\mathrm { ~ s . t . ~ } i \\neq j } \\left[ D _ { i j } \\right]\n$$",
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+        "type": "text",
+        "text": "$D$ is smaller the more similar language is between bots. Note that even though $D _ { i j }$ is not symmetric (because KL divergence is not), $D$ is symmetric because it averages over both directions of pairs. ",
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+        "type": "text",
+        "text": "We compute $D$ by sampling an empirical distribution over all messages and observations, taking 10 sample dialogues in each possible test state $( x _ { A } , x _ { Q } )$ of the world using the final populations of agents as in Fig. 2. Note that this metric applies to a group of agents, so we measure it for only the Multi Agent settings, including two new baselines colored red in Fig. 3. The Single Agents Combined baseline trains 4 Single Agent No Replacement models independently then puts them together and computes $D$ for that group. These agents only speak similar languages by chance, so $D$ is high. The Random Initialization baseline evaluates language similarity using newly initialized models. These agents have about a uniform distribution over words at every utterance, so their languages are both very similar and useless. For each model these baselines act like practical (not strict) upper and lower bounds on $D$ , respectively. ",
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+        "type": "text",
+        "text": "Fig. 3 shows this language dissimilarity metric for all our settings. As we expect, the paired Single Agents are highly dissimilar compared to agents from Multi Agent populations. Further, all the replacement strategies result in increased language similarity—although the degree of this effect seems dependent on vocabulary setting. This provides some evidence that cultural transmission is occurring in Multi Agent settings and is encouraged by the replacement strategy in our approach. While all Multi Agent settings resulted in language transmission, our replacement strategies results in more compositional languages due to repeated teaching of new generations of agents. ",
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+    {
+        "type": "text",
+        "text": "5.3 VISUALIZING EMERGENT LANGUAGES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we visualize the language learned by agents at various stages of training to reinforce our previous conclusions and build intuition. Each of the three sub-figures in Fig. 4 summarizes all of the conversations between a particular pair of bots for the (shape, color) task. To see how these summaries work, consider Fig. 4a. That sub-figure is divided into a $4 \\times 4$ grid with 4 elements in each cell, so there is one element for each of the 64 possible objects. For this task, objects in each row of the $4 \\times 4$ grid have the same shape and objects in each column have the same color. To the right of each object are the two tokens A-bot used to respond to Q-bot in the two dialog rounds. Ideally they should indicate the color and shape of the object. Finally, the check-marks or Xs to the right of A-bot’s utterances indicate whether Q-bot guessed correctly. ",
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+    {
+        "type": "image",
+        "img_path": "images/e68419c7891d177e2ae37f512fe3dc70e24b66bef8877d8103b4dc83fed0b9e0.jpg",
+        "image_caption": [
+            "Model Figure 3: Do bots in a population learn similar languages? On the y-axis (eq. (2)) lower values indicate similar language. Populations evolved with our method speak similar languages, but independently evolved agents do not. Thus our implicit procedure induces cultural transmission. "
+        ],
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+        "text": "",
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+    {
+        "type": "image",
+        "img_path": "images/383e1a2eec5bbbd3805c1ab8f39b1a203f63370349f09eb5189cc2d11b8bb83f.jpg",
+        "image_caption": [
+            "Figure 4: Each sub-figure summarizes an A-bot’s language, as described in Section 5.3. By comparing the baseline of Fig. 4a to a similar pair of bots from our approach Section 4b we can see that our approach encourages compositional language to emerge. Furthermore, the similarity between Fig. 4b and Fig. 4c suggests language is indeed transmitted in our approach. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From left to right: Fig. 4a summarizes the single pair from a Single Agent No Replacement run (3000 iterations old); Fig. 4b summarizes dialogs between an old Q-bot (about 23000 iterations) and a recently re-initialized A-bot (about 3000 iterations) at the 8th and final generation of a Multi Oldest run; Fig. 4c summarizes dialogs between the same old Q-bot as in Fig. 4b and an old A-bot (13000 iterations) from the same Multi Oldest experiment. Even though the A-bots in Fig. 4a and Fig. 4b have trained for about2 the same number of iterations, the A-bot trained in the presence of other bots which already know a functional language has already learned a somewhat compositional language whereas the Single Agent A-bot has not. Furthermore, by comparing the old A-bot’s language Fig. 4c with the new one Fig. 4b we can see that they are extremely similar. They even lead to the same mistakes. This again suggests that language is transmitted between bots, in agreement with our previous experiments. ",
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+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Language Evolution Causes Structure. Researchers have spent decades studying how unique properties of human language like compositionality could have emerged. There is general agreement that people acquire language using a combination of innate cognitive capacity and learning from other language speakers (cultural transmission), with the degree of each being widely disputed Perfors (2002); Pinker & Bloom (1990). Both innate cognitive capacity and specific modern human languages like English co-evolved Briscoe (2000) via biological Pinker & Bloom (1990) and cultural Tomasello (1999); Smith (2006) evolution, respectively. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In particular, explanations of how the cultural evolution of languages could cause structure like compositionality are in abundance Nowak & Krakauer (1999); Nowak et al. (2000); Smith et al. (2003); Brighton (2002); Vogt (2005); Kirby et al. (2014); Spike et al. (2017). An important piece of the explanation of linguistic structure is the iterated learning model Kirby et al. (2014); Kirby (2001); Kirby et al. (2008) used to motivate our approach. Indeed it shows that cultural transmission causes structure in computational Kirby (2001; 2002); Christiansen & Kirby (2003); Smith et al. (2003) and human Kirby et al. (2008); Cornish et al. (2009); Scott-Phillips & Kirby (2010) experiments. Even though cultural transmission may aid the emergence of compositionality, recent results in evolutionary linguistics Raviv et al. (2018) and deep learning Kottur et al. (2017); Mordatch & Abbeel (2018) also emphasize other factors. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While existing work in deep learning has focused on biases that encourage compositionality, it has not considered settings where language is permitted to evolve over generations of agents. We have shown such an approach is viable and even complementary with other approaches. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Language Emergence in Deep Learning. Recent work in deep learning has increasingly focused on multi-agent environments where deep agents learn to accomplish goals (possibly cooperative or competitive) by interacting appropriately with the environment and each other. Some of this work has shown that deep agents will develop their own language where none exists initially if driven by a task which requires communication Foerster et al. (2016); Sukhbaatar et al. (2016); Lazaridou et al. (2017). Most relevant is work which focuses on conditions under which compositional language emerges as deep agents learn to cooperate Mordatch & Abbeel (2018); Kottur et al. (2017). Both Mordatch & Abbeel (2018) and Kottur et al. (2017) find that limiting the vocabulary size so that there aren’t too many more words than there are objects to refer to encourages compositionality, which follows earlier results in evolutionary linguistics Nowak et al. (2000). Follow up work has continued to investigate the emergence of compositional language among neural agents, mainly focusing on perceptual as opposed to symbolic input and how the structure of the input relates to the tendency for compositional language to emerge Choi et al. (2018); Havrylov & Titov (2017); Lazaridou et al. (2018). Other work has shown that Multi Agent interaction leads to better emergent translation Lee et al. (2018), but it does not measure compositionality. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Cultural Evolution and Neural Nets. Somewhat recently, Bengio (2012) suggested that culturally transmitted ideas may help in escaping from local minima. Experiments in Gulc¸ehre & Bengio ¨ (2016) support this idea by showing that supervision of intermediate representations allows a more complex toy task to be learned. Unlike our work, these experiments use direct supervision provided by the designed environment rather than indirect and implicit supervision provided by other agents. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Two concurrent works examine the role of periodic agent replacement on language emergence – albeit in different environments. In Li & Bowling (2019) replacement is used to encourage languages to be easy to teach, and this in turn causes compositionality. In Dagan et al. (2019) neural language is transmitted through a bottleneck caused by replacement. The resulting language has increased efficiency and effectiveness, with further results showing that co-evolving the agents themselves with the language amplifies the effect. Both of these works support our central observations. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work we investigated cultural transmission in deep neural dialog agents, applying it to language emergence. The evolutionary linguistics community has long used cultural transmission to explain how compositional languages could have emerged. The deep learning community, having recently become interested in language emergence, has not investigated that link until now. Instead of explicit models of cultural transmission familiar in evolutionary linguistics, we favor an implicit model where language is transmitted from generation to generation only because it helps agents achieve their goals. We show that this does indeed cause cultural transmission and compositionality. ",
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+    {
+        "type": "text",
+        "text": "Future work. While our work used an implicit version of cultural transmission, we are interested in investigating the effect of explicit versions of cultural transmission on language structure. In another direction, cultural transmission may also provide an appropriate prior for neural representations of non-language information. ",
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+    {
+        "type": "text",
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+        "text": "Paul Vogt. The emergence of compositional structures in perceptually grounded language games. Artificial intelligence, 167(1-2):206–242, 2005. ",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 RESULTS ON SINGLE HELD OUT ATTRIBUTE DATASET OF KOTTUR ET AL. (2017) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            127,
+            784,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In Section 4 we proposed a new harder compositional dataset different from the one in Kottur et al. (2017). For comparison, in this section we train and evaluate our models on the original dataset from Kottur et al. (2017) to show that our approach also improves still improves compositionality in this setting and to show that our new dataset is indeed harder. ",
+        "bbox": [
+            174,
+            151,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/3c1f5868d98f9e916b1c8cf1e9e36eb91faa16b49dcc985ae245616a9c280394.jpg",
+        "image_caption": [
+            "Figure 5: Test set accuracies (with standard deviations) are reported by training and evaluating the same models as in our main results Fig. 2 against the dataset from Kottur et al. (2017). These results do not perform cross-validation, following Kottur et al. (2017). They only vary across 4 different random seeds. This dataset is significantly easier than our new dataset, as indicated by the Our proposed approach still outperforms models without replacement or multiple agents. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            220,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 REPLACEMENT STRATEGIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            417,
+            415,
+            431
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Our approach to cultural transmission periodically replaces agents by re-initializing them. The approach section outlines various replacement strategies (policy $\\pi$ ), but does not detail their implementation. We do so here. ",
+        "bbox": [
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+            440,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "These strategies depend on a number of possible inputs: ",
+        "bbox": [
+            174,
+            488,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "• $e$ the current epoch   \n• $E$ the period of agent replacement   \n• $v _ { i } ^ { Q } / v _ { i } ^ { A }$ the validation accuracy of agent $i$ for Q-bots/A-bots. For Q-bots this is averaged over all potential A-bot partners, and vice-versa for A-bots.   \n• $a _ { i } ^ { Q } / a _ { i } ^ { A }$ the age in epochs of agent $i$ for Q-bots/A-bots ",
+        "bbox": [
+            215,
+            513,
+            825,
+            602
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Single Agent strategies are given in Alg. 2 and Alg. 3. Multi Agent strategies are given in Alg. 4, Alg. 5, and Alg. 6. Note that Single Agent strategies always replace one agent while Multi Agent strategies always replace one Q-bot and one A-bot. An additional Replace All baseline strategy is given in Alg. ?? and generalizes to both Single and Multi Agent cases. ",
+        "bbox": [
+            176,
+            612,
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+            669
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2: Single Agent - Random Replacement ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            685,
+            514,
+            702
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/ed91e29c2a138102a978b310760c77a94d968d29ef44b5a4f4271c450d890ece.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>1 d~U{0,1}</td></tr><tr><td></td></tr><tr><td>2 if d=O then return {A-bot }</td></tr><tr><td></td></tr><tr><td>4 else return {Q-bot}</td></tr></table>",
+        "bbox": [
+            160,
+            704,
+            823,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.3 DETAILED RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            809,
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+            824
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In our experiments we compare models and we compare replacement strategies. We ran dependent paired t-tests across random seeds, cross-val folds, and replacement strategies to compare models. We ran dependent paired t-tests across random seeds, cross-val folds, and models to compare replacement strategies. The p-values for all of these t-tests are reported here. ",
+        "bbox": [
+            174,
+            833,
+            825,
+            888
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Replacement strategy comparisons are in Fig. 7 (Single Agent) and Fig. 8 (Multi Agent). Model comparisons are in Fig. 6. ",
+        "bbox": [
+            173,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/fc000857cd1616211d7de8541016d7cbd9c818d97401baf12896a1033c618806.jpg",
+        "image_caption": [
+            "Figure 6: Replacement strategy comparison p-values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            251,
+            766,
+            746
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/3765b7e01ed3a5a0cfbba49a0af8fadc84510f5b295801cd24b995b74c104a2a.jpg",
+        "image_caption": [
+            "Figure 7: Single Agent model comparison p-values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            269,
+            142,
+            694,
+            849
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/7ef9b3e377968b1755836c0ed298545d627e2ebf38fcd55e8c7a107744b84db2.jpg",
+        "image_caption": [
+            "Figure 8: Multi Agent model comparison p-values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            180,
+            759,
+            819
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 3: Single Agent - Alternate Replacement ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            148,
+            519,
+            165
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1 Input: e   \n2 if $\\lfloor e / E \\rfloor = 0$ then   \n3 return { A-bot }   \n4 else   \n5 return { Q-bot } ",
+        "bbox": [
+            160,
+            169,
+            313,
+            250
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 4: Multi Agent - Uniform Random Replacement ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            353,
+            570,
+            369
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1 iA ∼ U{1, NA}   \n2 iQ ∼ U{1, NQ}   \n3 return { A-bot $i _ { A }$ , Q-bot iQ } ",
+        "bbox": [
+            160,
+            371,
+            375,
+            416
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 5: Multi Agent - Epsilon Greedy Replacement ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            521,
+            555,
+            536
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1 Input: $v _ { i } ^ { Q } \\forall i$ , $v _ { i } ^ { A } \\forall i$ , ε ∈ [0, 1) (usually 0.2)   \n2 $d \\sim \\mathcal { U } [ 0 , \\overset { \\cdot } { 1 } )$   \n3 if $d < \\varepsilon$ then   \n4 $i _ { A } \\sim \\mathcal { U } \\{ 1 , N _ { A } \\}$   \n5 $i _ { Q } \\sim \\mathcal { U } \\{ 1 , N _ { Q } \\}$   \n$\\mathbf { 6 }$ else   \n7 $i _ { A } = \\mathrm { a r g m i n } _ { i } v _ { i } ^ { A }$ (unique in our experiments)   \n8 $i _ { Q } = \\mathrm { a r g m i n } _ { i } v _ { i } ^ { Q }$ (unique in our experiments)   \n9 return { A-bot iA, Q-bot iQ } ",
+        "bbox": [
+            158,
+            540,
+            504,
+            685
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 6: Multi Agent - Oldest Replacement ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            787,
+            495,
+            803
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1 Input: $a _ { i } ^ { Q } \\forall i$ , $a _ { i } ^ { V } \\forall i$   \n2 $i _ { A } = \\mathcal { U } \\{ \\mathrm { a r g m a x } _ { i } a _ { i } ^ { A } \\}$   \n3 $i _ { Q } = \\mathcal { U } \\{ \\mathrm { a r g m a x } _ { i } a _ { i } ^ { Q } \\}$   \n4 return { A-bot $i _ { A }$ , Q-bot iQ }",
+        "bbox": [
+            160,
+            806,
+            375,
+            873
+        ],
+        "page_idx": 14
+    }
+]