Add files using upload-large-folder tool

parse/train/BJ9fZNqle/BJ9fZNqle.md

@@ -0,0 +1,407 @@
+# MULTI-MODAL VARIATIONAL ENCODER-DECODERS
+
+Iulian V. Serban†∗, Alexander G. Ororbia $\mathbf { I I } ^ { \times }$ ∗, Joelle Pineau‡, Aaron Courville†
+
+† Department of Computer Science and Operations Research, Universite de Montreal   
+×College of Information Sciences & Technology, Penn State University   
+‡ School of Computer Science, McGill University   
+iulian[DOT]vlad[DOT]serban[AT]umontreal[DOT]ca   
+ago109[AT]psu[DOT]edu   
+jpineau[AT]cs[DOT]mcgill[DOT]ca   
+aaron[DOT]courville[AT]umontreal[DOT]ca
+
+# ABSTRACT
+
+Recent advances in neural variational inference have facilitated efficient training of powerful directed graphical models with continuous latent variables, such as variational autoencoders. However, these models usually assume simple, unimodal priors — such as the multivariate Gaussian distribution — yet many realworld data distributions are highly complex and multi-modal. Examples of complex and multi-modal distributions range from topics in newswire text to conversational dialogue responses. When such latent variable models are applied to these domains, the restriction of the simple, uni-modal prior hinders the overall expressivity of the learned model as it cannot possibly capture more complex aspects of the data distribution. To overcome this critical restriction, we propose a flexible, simple prior distribution which can be learned efficiently and potentially capture an exponential number of modes of a target distribution. We develop the multi-modal variational encoder-decoder framework and investigate the effectiveness of the proposed prior in several natural language processing modeling tasks, including document modeling and dialogue modeling.
+
+# 1 INTRODUCTION
+
+With the development of the variational autoencoding framework (Kingma & Welling, 2013; Rezende et al., 2014), a tremendous amount of progress has been made in learning large-scale, directed latent variable models. This approach has lead to improved performance in applications ranging from computer vision (Gregor et al., 2015; Larsen et al., 2015) to natural language processing (Mnih & Gregor, 2014; Miao et al., 2015; Bowman et al., 2015; Serban et al., 2016b). Furthermore, these models naturally incorporate a Bayesian modeling perspective, by enabling the integration of problem-dependent knowledge in the form of a prior on the generating distribution.
+
+However, the majority of models proposed assume an extremely simple prior in the form of a multivariate Gaussian distribution in order to maintain mathematical and computational tractability. Although this assumption on the prior has lead to favorable results on several tasks, it is clearly a restrictive and often unrealistic assumption. First, it imposes a strong uni-modal structure on the latent variable space; latent samples from the generating model (prior distribution) all cluster around a single mean. Second, it encourages local smoothness on the latent variables; the similarity between two latent variables decreases exponentially as their distance increase. Thus, for complex, multi-modal distributions — such as the distribution over topics in a text corpus, or natural language responses in a dialogue system — the uni-modal Gaussian prior inhibits the model’s ability to extract and represent important structure in the data. To learn more powerful and expressive models — in particular, models with multi-modal latent variable structures for natural language processing applications — we seek a suitable and flexible prior than can be automatically adapted to model multiple modes of a target distribution.
+
+In this paper, we propose the multi-modal variational encoder-decoder framework, introducing an efficient, flexible prior distribution that is suitable for distributions such as those found in natural language text. We demonstrate the effectiveness of our multi-modal variational architectures in two representative tasks: document modeling and dialogue modeling. We find that our prior is able to capture elements of a target distribution that simpler priors — such as the uni-modal Gaussian — cannot model, thus allowing neural latent variable models to extract richer structure from data. In particular, we achieve state-of-the-art results on several document modeling tasks.
+
+# 2 RELATED WORK
+
+The idea of using an artificial neural network to approximate an inference model dates back to the 90s (Hinton & Zemel, 1994; Hinton et al., 1995; Dayan & Hinton, 1996). However, initial attempts at such an approach were hindered by the lack of low-bias, low-variance estimators of parameter gradients. Traditionally, researchers resorted to Markov chain Monte Carlo methods (MCMC) (Neal, 1992) which do not scale well and mix slowly, or to variational approaches which require a tractable, factored distribution to approximate the true posterior distribution, usually under-fitting it (Jordan et al., 1999). Others have since proposed using feed-forward inference models to efficiently initialize the mean-field inference algorithm for incrementally training Boltzmann architectures (Salakhutdinov & Larochelle, 2010; Ororbia II et al., 2015b). However, these approaches are limited by the mean-field inference’s inability to model structured posteriors. Recently, Mnih & Gregor (2014) proposed the neural variational inference and learning (NVIL) approach to match the true posterior directly without resorting to approximate inference. NVIL allows for the joint training of an inference network and directed generative model, maximizing a variational lower-bound on the data log-likelihood and facilitating exact sampling of the variational posterior. Simultaneously with this work, the variational autoencoder framework was proposed by Kingma & Welling (2013) and Rezende et al. (2014). This framework is the motivation of this paper, and will be discussed in detail in the next section.
+
+With respect to document modeling, it has recently been demonstrated that neural architectures can outperform well-established, standard topic models such as Latent Dirichlet Allocation (LDA) (Blei et al., 2003). For example, it has been demonstrated that models based on the Boltzmann machine, which learn semantic binary vectors (binary latent variables), perform very well (Hofmann, 1999). Work involving discrete latent variables include the constrained Poisson model (Salakhutdinov & Hinton, 2009), the Replicated Softmax model (Hinton & Salakhutdinov, 2009) and the Over-Replicated Softmax model (Srivastava et al., 2013), as well as similar, auto-regressive neural architectures and deep directed graphical models (Larochelle & Lauly, 2012; Uria et al., 2014; Lauly et al., 2016; Bornschein & Bengio, 2014). In particular, Mnih & Gregor (2014) showed that using NVIL yields better generative models of documents than these previous approaches. The success of these discrete latent variable models — which are able to partition probability mass into separate regions — serve as the main motivation for investigating models with continuous multi-modal latent variables for document modeling. More recently, Miao et al. (2015) have proposed continuous latent variable representations for document modeling, which has achieved state-of-the-art results. This model will be described later.
+
+With respect to dialogue modeling, latent variable models were investigated by Bangalore et al. (2008), Crook et al. (2009) as well as others. More recently, Zhai & Williams (2014) have proposed three models combining hidden Markov models and topic models. The success of these discrete latent variable models also motivates our investigation into dialogue models with multi-modal latent variables. Most related to our work is the Variational Hierarchical Recurrent Encoder-Decoder (VHRED) model by Serban et al. (2016b), which is a neural architecture with latent multivariate Gaussian variables. This model will be described later.
+
+There has been some work exploring alternative distributions for the latent variables in the variational autoencoder framework, including multi-modal distributions. Rezende & Mohamed (2015) propose an approach called normalizing flows which computes a more complex, potentially multimodal distribution, by projecting standard Gaussian variables through a sequence of non-linear transformations. This approach is similar to the inverse auto-regressive flow proposed by Kingma et al. (2016). Unfortunately, both normalizing flows and auto-regressive flow are only applicable to the approximate posterior distribution; typically these approaches require fixing the prior distribution to a uni-modal multivariate Gaussian. Furthermore, to the best of our knowledge, neither of these approaches have been investigated in the context of larger scale text processing tasks, such as the document modeling and dialogue modeling tasks we evaluate on. A complementary approach is to combine variational inference with MCMC sampling (Salimans et al., 2015; Burda et al., 2015), however this is computationally expensive and therefore difficult to scale up to many real-world tasks. Enriching the latent variable distributions has also been investigated by Maaløe et al. (2016).
+
+# 2.1 APPROACHES FOR LEARNING MULTI-MODAL LATENT VARIABLES
+
+Mixture of Gaussians Perhaps the most direct and naive approach to learning multi-modal latent variables is to parametrize the latent variable prior and approximate posterior distributions as a mixture of Gaussians. However, the KL divergence between two mixtures of Gaussian distributions cannot be computed in closed form (Durrieu et al., 2012). To train such a model, one would have to either resort to MCMC sampling, which may slow down and hurt the training process due to the high variance it incurs, or resort to approximations of the KL divergence, which may also hurt the training process.1
+
+Deep Directed Models An alternative to a mixture of Gaussians parametrization is to construct a deep directed graphical model composed of multiple layers of uni-modal latent variables (e.g. multivariate Gaussians) (Rezende et al., 2014). Such models have the potential to capture highly complex, multi-modal latent variable representations through the marginal distribution of the toplayer latent variables. However, this approaches has two major drawbacks. First, the variance of the gradient estimator grows with the number of layers. This makes it difficult to learn highly multimodal latent representations. Second, it is not clear how many modes such models can represent or how their inductive biases will affect their performance on tasks containing multi-modal latent structure. The piecewise constant latent variables we propose do not suffer from either of these two drawbacks; the piecewise constant variables incur low variance in the gradient estimator, and can, in principle, represent a number of modes exponential in the number of latent variables.
+
+Discrete Latent Variables A third approach for learning multi-modal latent representations is to instead use discrete latent variables as discussed above. For example, the learning procedure proposed by Mnih & Gregor (2014) for discrete latent variables can easily be combined with the variational autoencoder framework to learn models with both discrete and continuous latent variables. However, the major drawback of discrete latent variables is the high variance in the gradient estimator. Without further approximations, it might be difficult to scale up models with discrete latent variables for real-world tasks.
+
+# 3 THE MULTI-MODAL VARIATIONAL ENCODER-DECODER FRAMEWORK
+
+We start by describing the general neural variational learning framework. Then we present our proposed prior model aimed at enhancing the model’s ability to learn multiple modes of data distributions. We focus on modeling discrete output variables in the context of natural language processing applications. However, the framework can easily be adapted to handle continuous output variables, such as images, video and audio.
+
+# 3.1 NEURAL VARIATIONAL LEARNING
+
+Let $w _ { 1 } , \ldots , w _ { N }$ be a sequence of $N$ words conditioned on a continuous latent variable $z$ . In the general framework, the distribution over the variables follows the directed graphical model:
+
+$$
+P _ { \theta } ( w _ { 1 } , \dots , w _ { N } , z ) = \int \prod _ { n = 1 } ^ { N } P _ { \theta } ( w _ { n } | w _ { < n } , z ) P _ { \theta } ( z ) d z ,
+$$
+
+where $\theta$ are the model parameters. The model first generates the higher-level, continuous latent variable $z$ , and then, conditioned on this, generates the word sequence. The document modeling
+
+task further simplifies the model by assuming the words are independent of each other:
+
+$$
+P _ { \theta } ( w _ { 1 } , \dots , w _ { N } , z ) = \int \prod _ { n = 1 } ^ { N } P _ { \theta } ( w _ { n } | z ) P _ { \theta } ( z ) d z .
+$$
+
+Following the variational autoencoder (VAE) framework (Kingma & Welling, 2013), the parameters can be learned using the variational lower-bound:
+
+$$
+\begin{array} { r } { \begin{array} { r } { \log P _ { \theta } ( w _ { 1 } , \dots , w _ { N } , z ) \geq \operatorname { E } _ { z \sim Q _ { \psi } ( z | w _ { 1 } , \dots , w _ { N } ) } [ \log P _ { \theta } ( w _ { n } | w _ { < n } , z ) ] - \mathrm { K L } \left[ Q _ { \psi } ( z | w _ { 1 } , \dots , w _ { N } ) \right| | P _ { \theta } ( z ) ] , } \end{array} } \end{array}
+$$
+
+where $Q _ { \psi } ( z | w _ { 1 } , \dots , w _ { N } )$ is the approximation to the posterior for $z$ , called the encoder, or sometimes the recognition model or inference model, with parameters $\psi$ . The distribution $P _ { \theta } ( z )$ is the prior model for $z$ . The variational autoencoder model further makes use of the re-parametrization trick, which allows one to move the derivative of the lower-bound to inside the expectation. To accomplish this, we need to parametrize $z$ as a transformation from a fixed (parameter-less) random distribution:
+
+$$
+z = f _ { \theta } ( \epsilon ) ,
+$$
+
+where $\epsilon$ is drawn from a random distribution, e.g. a standard Gaussian distribution (with zero mean and unit standard deviation) or a uniform distribution in the interval $[ 0 , 1 ]$ , and $f$ is some transformation of this variable, also parametrized by $\theta$ .
+
+The majority of work on VAEs that uses the re-parametrization trick propose to parametrize $z -$ both the prior and approximate posterior (encoder) — as a multivariate Gaussian variable. However, the multivariate Gaussian is a uni-modal distribution and can therefore only represent one mode in latent space. This means the mapping from latent variable to outputs — i.e. the conditional distribution $P _ { \theta } ( w _ { n } | z )$ — has to be highly non-linear in order to capture additional modes. However, in general, it is difficult to learn such non-linear mappings with existing stochastic optimization methods, such as mini-batch stochastic gradient descent and its variants. Learning such a non-linear mapping is particularly difficult using the variational bound in eq. (3), because it incurs additional variance from sampling the latent variable $z$ . Consequently, such a model is very likely to converge on a solution which does not model multi-modality which then leads to a poor approximation of the output distribution.
+
+# 3.2 THE PIECEWISE-CONSTANT PRIOR FOR LATENT VARIABLES
+
+In this work, we overcome the uni-modal restriction by parametrizing $z$ using a piecewise constant probability density function (PDF). This parametrization will allow $z$ to represent complex aspects of the data distribution in latent variable space, such as multiple modes and highly non-smooth regions of probability mass. From a manifold learning perspective, this extension translates into expanding the set of manifolds representable by the model parameters to include more non-linear manifolds – in particular, manifolds where there exists separate clusters of probability mass.
+
+Let $n \in \mathbb { N }$ be the number of piecewise constant components. We assume $z$ is drawn from the PDF:
+
+$$
+P ( z ) = \frac { 1 } { K } \sum _ { i = 1 } ^ { n } 1  { \Big ( } \frac { i - 1 } { n } \underline { { \leq z } } \leq \frac { i } { n } { \Big ) } ^ { a _ { i } } ,
+$$
+
+where $1 _ { ( x ) }$ is the indicator function (which is one whenever $x$ is true and otherwise zero), $a _ { i } > 0$ for $i = 1 , \ldots , n$ are the distribution parameters (which will be learned during training), and $K$ is the normalization constant:
+
+$$
+K = \sum _ { i = 1 } ^ { n } K _ { i } , \quad { \mathrm { w h e r e ~ } } K _ { 0 } : = 0 , K _ { i } : = { \frac { a _ { i } } { n } } { \mathrm { ~ f o r ~ } } i = 1 , \dots , n .
+$$
+
+To train the model using the re-parametrization trick, we need to generate $z = f ( \epsilon )$ where $\epsilon \sim$ Uniform $( 0 , 1 )$ . To do so, we employ inverse transform sampling (Devroye, 1986), which requires finding the inverse of the cumulative distribution function (CDF). We first derive the CDF of eq. (5):
+
+$$
+\phi ( z ) = { \frac { 1 } { K } } \sum _ { i = 1 } ^ { n } 1  { } _ { \left( { \frac { i } { n } } \leq z \right) } ^ { K _ { i } + 1 } { \Big ( } { \frac { i - 1 } { n } } \leq z \leq { \frac { i } { n } } { \Big ) } \left( z - { \frac { i - 1 } { n } } \right) a _ { i } .
+$$
+
+Next, we derive its inverse:
+
+$$
+\phi ^ { - 1 } ( \epsilon ) = \sum _ { i = 1 } ^ { n } 1 _ { \left( { \frac { 1 } { K } } \sum _ { j = 0 } ^ { i - 1 } K _ { j } \leq \epsilon \leq { \frac { 1 } { K } } \sum _ { j = 0 } ^ { i } K _ { j } \right) } \left( { \frac { i - 1 } { n } } + { \frac { K } { a _ { i } } } \left( \epsilon - { \frac { 1 } { K } } \sum _ { j = 0 } ^ { i - 1 } K _ { j } \right) \right)
+$$
+
+Armed with the inverse CDF, we can now draw a sample $z$
+
+$$
+z = \phi ^ { - 1 } ( \epsilon ) , \quad \mathrm { w h e r e } \epsilon \sim \mathrm { U n i f o r m } ( 0 , 1 ) .
+$$
+
+In addition to sampling, we need to compute the Kullback-Leibler (KL) divergence between the prior and approximate posterior distributions of the piecewise constant variables. We assume both the prior and the posterior are piecewise constant distributions. We use prior to denote prior parameters and post to denote posterior parameters (encoder model parameters). The KL divergence between the prior and posterior can be computed using a sum of integrals, where each integral inside the sum corresponds to one constant segment:
+
+$$
+\begin{array} { l } { { \displaystyle \mathbb { K L } \left[ \displaystyle Q _ { \psi } ( z | w _ { 1 } , \dots , w _ { N } ) \right] \left\| P _ { \theta } ( z ) \right\} = \int _ { 0 } ^ { 1 } Q _ { \psi } ( z | w _ { 1 } , \dots , w _ { N } ) \log \left( \frac { Q _ { \psi } ( z | w _ { 1 } , \dots , w _ { N } ) } { P _ { \theta } ( z ) } \right) d z } } \\ { { \displaystyle \qquad = \sum _ { i = 1 } ^ { n } \int _ { 0 } ^ { 1 / n } \frac { a _ { i } ^ { \mathrm { p o t } } } { K ^ { \mathrm { p o t } } } \log \left( \frac { a _ { i } ^ { \mathrm { p o t } } / K ^ { \mathrm { p o t } } } { a _ { i } ^ { \mathrm { p o t } } / K ^ { \mathrm { p o t } } } \right) d z } } \\ { ~ } \\ { { \displaystyle \qquad = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \frac { a _ { i } ^ { \mathrm { p o t } } } { K ^ { \mathrm { p o t } } } \log \left( \frac { a _ { i } ^ { \mathrm { p o t } } / K ^ { \mathrm { p o t } } } { a _ { i } ^ { \mathrm { p o t } } / K ^ { \mathrm { p o t } } } \right) } } \\ { { \displaystyle \qquad = \frac { 1 } { n } \frac { 1 } { K ^ { \mathrm { p o t } } } \frac { \sum _ { i = 1 } ^ { n } a _ { i } ^ { \mathrm { p o t } } } { i - 1 } \left( \log ( a _ { i } ^ { \mathrm { p o t } } ) - \log ( a _ { i } ^ { \mathrm { p o t } } ) \right) } } \\ { ~ } \\ { { \displaystyle \qquad + \log ( K ^ { \mathrm { p o t } } ) - \log ( K ^ { \mathrm { p o t } } ) } } \end{array}
+$$
+
+In order to train the model, we take partial derivatives of the variational bound in eq. (3) w.r.t. each parameter in $\theta$ and $\psi$ . These expressions involve derivatives of the indicator functions, which have derivatives zero everywhere except for the changing points where the derivative is undefined. However, the probability of sampling $\epsilon$ such that an indicator function is exactly at its changing point is effectively zero. Therefore, we fix their derivatives to zero.2 A similar approach is used for training neural networks with rectified linear units. Figure 1 illustrates how the piecewise constant latent variables can work with Gaussian latent variables in order to model multi-modality.
+
+# 4 LATENT VARIABLE PARAMETRIZATIONS
+
+The latent variable parametrizations are crucial to modeling the data effectively. In this section, we will develop the parametrizations for both the Gaussian variable and our proposed piecewise latent variable.
+
+For all parametrizations, let $c$ be the conditioning information for the prior. In document modeling there is no conditioning information available to the prior, so $c = \emptyset$ . In dialogue modeling $c$ is the vector representation of the dialogue context, namely all previous utterances until the current time step. Let $x$ be the current output sequence (observation), which the model must generate (e.g. $w _ { 1 } , \ldots , w _ { N }$ for document modeling).
+
+# 4.1 GAUSSIAN PARAMETRIZATION
+
+Let $\mu ^ { \mathrm { p r i o r } }$ and $\sigma ^ { 2 , \mathrm { p r i o r } }$ be the prior mean and variance, and let $\mu ^ { \mathrm { p o s t } }$ and $\sigma ^ { 2 , \mathrm { p o s t } }$ be the posterior mean and variance. For Gaussian latent variables, the prior distribution mean and variances are encoded using linear transformations of a hidden state. In particular, the prior distribution covariance is
+
+![](images/67554ddd11954cd275638c1a7e186936a68fb54053b5485afdf2b2aaa8e6e5d3.jpg)  
+Figure 1: The horizontal axis corresponds to $z _ { 1 }$ , which is a univariate Gaussian variable. The vertical axis corresponds to $z _ { 2 }$ , which is a piecewise constant variable. The PDF for each variable is shown along each axis, and their joint distribution is illustrated in grey color.
+
+encoded as a diagonal covariance matrix using a softplus function:
+
+$$
+\begin{array} { r l } & { \mu ^ { \mathrm { p r i o r } } = H _ { \mu } ^ { \mathrm { p r i o r } } \mathrm { E n c } ( c ) + b _ { \mu } ^ { \mathrm { p r i o r } } , } \\ & { \sigma ^ { \mathrm { 2 , p r i o r } } = \mathrm { d i a g } ( \log ( 1 + \exp ( H _ { \sigma } ^ { \mathrm { p r i o r } } \mathrm { E n c } ( c ) + b _ { \sigma } ^ { \mathrm { p r i o r } } ) ) ) , } \end{array}
+$$
+
+where $\operatorname { E n c } ( c )$ is an embedding/encoding of the context $c$ (e.g. given by a bag-of-words encoder or an LSTM encoder applied to $c$ ), which is shared across all latent variable dimensions. The parameters $H _ { \mu } ^ { \mathrm { p r i o r } } , b _ { \mu } ^ { \mathrm { p r i o r } } , H _ { \sigma } ^ { \mathrm { p r i o r } } , b _ { \sigma } ^ { \mathrm { p r i o r } }$ r, H priorσ , are to be learned.
+
+For the posterior distribution, our preliminary experiments have shown that it is much better to parametrize the posterior distribution by interpolating between the prior distribution mean and variance and a new estimate of the mean and variance. This interpolation is controlled by a gating mechanism, which makes it easy for the model to learn how to turn on/off latent dimensions:
+
+$$
+\begin{array} { r l } & { \mu ^ { \mathrm { p o s t } } = ( 1 - \alpha _ { \mu } ) \mu ^ { \mathrm { p r i o r } } + \alpha _ { \mu } \left( H _ { \mu } ^ { \mathrm { p o s t } } \mathrm { E n c } ( c , x ) + b _ { \mu } ^ { \mathrm { p o s t } } \right) , } \\ & { \sigma ^ { 2 , \mathrm { p o s t } } = ( 1 - \alpha _ { \sigma } ) \sigma ^ { 2 , \mathrm { p r i o r } } + \alpha _ { \sigma } \mathrm { d i a g } ( \log ( 1 + \exp ( H _ { \sigma } ^ { \mathrm { p o s t } } \mathrm { E n c } ( c , x ) + b _ { \sigma } ^ { \mathrm { p o s t } } ) ) ) , } \end{array}
+$$
+
+where $\operatorname { E n c } ( c , x )$ is an encoding/embedding of both $c$ and $x$ , and where the parameters are $H _ { \mu } ^ { \mathrm { p o s t } } , b _ { \mu } ^ { \mathrm { p o s t } } , H _ { \sigma } ^ { \mathrm { p o s t } } , b _ { \sigma } ^ { \mathrm { p o s t } } , \alpha _ { \mu } , \alpha _ { \sigma }$ . The interpolation mechanism is controlled by $\alpha _ { \mu }$ and $\alpha _ { \sigma }$ , which are initialized to zero (i.e. initialized such that the posterior is equal to the prior).3
+
+# 4.2 PIECEWISE CONSTANT PARAMETRIZATION
+
+Similar to the Gaussian variances, we propose to parametrize the piecewise constant prior parameters using an exponential function applied to a linear transformation of the context embedding/encoding:
+
+$$
+\begin{array} { r } { a _ { i } ^ { \mathrm { p r i o r } } = \exp ( { H _ { a , i } ^ { \mathrm { p r i o r } } } \mathrm { E n c } ( c ) + b _ { a , i } ^ { \mathrm { p r i o r } } ) , \quad i = 1 , \ldots , n , } \end{array}
+$$
+
+where $H _ { a } ^ { \mathrm { p r i o r } }$ and $b _ { a } ^ { \mathrm { p r i o r } }$ are the parameters to be learned.
+
+We may also constrain the piecewise constant posterior parameters to be an interpolation between the prior parameters and a new estimated parameter:
+
+$$
+a _ { i } ^ { \mathrm { p o s t } } = ( 1 - \alpha _ { a , i } ) a _ { i } ^ { \mathrm { p r i o r } } + \alpha _ { a , i } \exp ( H _ { a , i } ^ { \mathrm { p o s t } } \mathrm { E n c } ( c , x ) + b _ { a , i } ^ { \mathrm { p o s t } } ) , \quad i = 1 , \ldots , n ,
+$$
+
+where $H _ { a } ^ { \mathrm { p o s t } } , b _ { a } ^ { \mathrm { p o s t } }$ , $\alpha _ { a }$ are the parameters. However, we found that this interpolation hurt performance and therefore fixed $\alpha _ { a } = { \bf 1 }$ .
+
+To take advantage of the properties of both priors, the Gaussian and piecewise constant variables may be combined, as was suggested in Section 3.2. In this work, we primarily experimented with their concatenation to create a hybrid model.
+
+# 5 VARIATIONAL TEXT MODELING
+
+We now present two probabilistic models, the NVDM and the VHRED, which are extended to incorporate the latent variable parametrization and used for the document modeling and the dialogue modeling experiments described below.
+
+# 5.1 NEURAL VARIATIONAL DOCUMENT MODEL (NVDM)
+
+The NVDM framework (Mnih & Gregor, 2014; Miao et al., 2015) collapses the recurrent neural encoder into a simpler bag-of-words model (since no symbol order is taken into account), which may be defined as a multi-layer perceptron (MLP) for $\dot { E } n c ( c = \varnothing , x ) = E n c ( x )$ . Let $V$ be the vocabulary. Let $W$ represent a document matrix, where row $w _ { i }$ is the 1-of- $| V |$ binary encoding of the $\overrightarrow { \imath } ^ { \prime }$ th word in the document. $E n c ( W )$ is trained to compress a document vector into a continuous distributed representation upon which the posterior model is built.
+
+The NVDM parametrization requires only learning the parameters $b _ { a } ^ { \mathrm { p r i o r } } , W _ { a } ^ { \mathrm { p o s t } } , b _ { a } ^ { \mathrm { p o s t } }$ for the piecewise variables, and learning the parameters $b _ { \mu } ^ { \mathrm { p r i o r } } , b _ { \sigma } ^ { \mathrm { p r i o r } } , W _ { \mu } ^ { \mathrm { p o s t } } , b _ { \mu } ^ { \mathrm { p o s t } } , W _ { \sigma } ^ { \mathrm { p o s t } } , b _ { \sigma } ^ { \mathrm { p o s t } }$ for the Gaussian variables. We initialize the bias parameters to zero, in order for the NVDM to start with a centered Gaussian prior. This prior will be adapted by the parametric encoder as learning progresses, while also learning to turn on/off latent dimensions controlled through the gating mechanism. It is important to note that our particular instantiation of the NVDM is different from that of Mnih & Gregor (2014) and Miao et al. (2015); we jointly learn the prior mean and variance whereas in previous work it has been assumed to be a standard Gaussian. Furthermore, our models learn to interpolate between the generated prior and posterior models to calculate a new posterior.
+
+Based on preliminary experiments, we choose the encoder to be a 2-hidden layer perceptron, defined by parameters $\{ E ^ { 0 } , \mathbf { \dot { \boldsymbol { b } } ^ { 0 } } , \dot { E } ^ { 1 } , \boldsymbol { b } ^ { 1 } \}$ . The decoder is defined by parameters $\{ R , c \}$ . For example, in the case of the hybrid VAE we use eq. (15)–(20) to generate the distribution parameters. In this case, to draw a sample from the Gaussian prior, we draw a standard Gaussian variable and then multiply it by the standard deviation and add the mean of the Gaussian prior. To draw a sample from the piecewise prior, we use eq. (8). As such, the complete architecture is:
+
+$$
+\begin{array} { r l } & { ~ \pi ( W ) = f ^ { 0 } ( E ^ { 0 } W + b ^ { 0 } ) , } \\ & { ~ E n c ( W ) = f ^ { 1 } ( E ^ { 1 } \pi ( W ) + b ^ { 1 } ) , } \\ & { z _ { G a u s s i a n } = \mu ^ { \mathrm { p o s t } } + \sqrt { \sigma ^ { 2 } , \mathrm { p o s t } } \otimes \epsilon _ { 0 } , } \\ & { z _ { P i e c e w i s e } = \phi ^ { - 1 , p o s t } ( \epsilon _ { 1 } ) , } \\ & { ~ z = \langle z _ { G a u s s i a n } , z _ { P i e c e w i s e } \rangle , } \\ & { D e c ( w , z ) = g ( - w ^ { \mathsf { T } } R z ) , } \end{array}
+$$
+
+where $\otimes$ is the Hadamard product, $\langle \circ , \circ \rangle$ is an operator that combines the Gaussian and the Piecewise variables and $D e c ( w , z )$ is the decoder model. 4 As a result of using the re-parametrization trick and choice of prior, we calculate the latent variable $z$ through the two samples, $\epsilon _ { 0 }$ and $\epsilon _ { 1 }$ . $f ( \circ )$ is a non-linear activation function. We choose it to be the softsign function, or $f ( v ) = v / ( 1 + | v | )$ . The decoder model $D e c ( z )$ outputs a probability distribution over words conditioned on $z$ . In this case, we define $g ( \circ )$ as the softmax function (omitting the bias term $c$ for clarity) computed as:
+
+$$
+D e c ( w , z ) = P _ { \theta } ( w | z ) = \frac { \exp { ( - w ^ { \mathrm { T } } R z ) } } { \sum _ { w ^ { \prime } } \exp { ( - w ^ { \mathrm { T } } R z ) } } ,
+$$
+
+The decoder’s output is used to calculate the first term in the variational lower-bound: $\log { P _ { \theta } ( W | z ) }$ . The prior and posterior distributions are used to compute the KL term in the variational lower-bound. The lower-bound defined becomes:
+
+$$
+\mathcal { L } = \mathrm { E } _ { Q _ { \psi } ( z | W ) } \left[ \sum _ { i = 1 } ^ { N } \log P _ { \theta } ( w _ { i } | z ) \right] - \mathrm { K L } \left[ Q _ { \psi } ( z | W ) | | P _ { \theta } ( z ) \right] ,
+$$
+
+where the $\mathrm { K L }$ term is the sum of the Gaussian and piecewise KL-divergence measures:
+
+$$
+\mathrm { K L } \left[ Q ( z | W ) | | P ( z ) \right] = \mathrm { K L } _ { G a u s s i a n } \left[ Q ( z | W ) | | P ( z ) \right] + \mathrm { K L } _ { P i e c e w i s e } \left[ Q ( z | W ) | | P ( z ) \right] .
+$$
+
+The KL-terms may be interpreted as regularizers of the parameter updates for the encoder model (Kingma & Welling, 2013). These terms encourage the posterior distributions to be similar to their corresponding prior distributions, by limiting the amount of information the encoder model transmits regarding the output. For example, it encourages the uni-modal Gaussian posterior to move its mean close to the mean of the Gaussian prior, which makes it difficult for the Gaussian posterior to represent different modes conditioned on the observation. Similarly, this encourages the piecewise constant posterior to be similar to the piecewise constant prior. However, since the piecewise constant posterior is multi-modal, it may be able to shift some of its probability mass towards the prior distribution while keeping other probability mass on one or several modes dependent upon the output observation (e.g. if the prior distribution is a uniform distribution and the true posterior concentrates all its probability mass in several small regions, then the approximate posterior could interpolate between the prior and the true posterior).
+
+# 5.2 VARIATIONAL HIERARCHICAL RECURRENT ENCODER-DECODER (VHRED)
+
+The VHRED model is an extension of the hierarchical recurrent encoder-decoder model (HRED) for dialogue (Serban et al., 2016b;a). The model decomposes dialogues using a two-level hierarchy: sequences of utterances (e.g. sentences), and sub-sequences of tokens (words). Let ${ \bf w } _ { n }$ be the $n$ ’th utterance in a dialogue with $N$ utterances. Let $w _ { n , m }$ be the $m$ ’th word in the $n$ ’th utterance from vocabulary $V$ , and let $M _ { n }$ be the number of words in the $n$ ’th utterance. In addition to this, VHRED has a latent multivariate continuous variable $\mathbf { z } _ { n }$ for each utterance $n = 1 , \ldots , N$ . The probability distribution of the generative model factorizes as:
+
+$$
+\begin{array} { l } { { \displaystyle P _ { \theta } ( \mathbf { w } _ { 1 } , \ldots , \mathbf { w } _ { N } ) = \prod _ { n = 1 } ^ { N } P _ { \theta } ( \mathbf { w } _ { n } | \mathbf { w } _ { < n } , \mathbf { z } _ { n } ) P _ { \theta } ( \mathbf { z } _ { n } | \mathbf { w } _ { < n } ) } , } \\ { ~ } \\ { { \displaystyle \qquad = \prod _ { n = 1 } ^ { N } \prod _ { m = 1 } ^ { M _ { n } } P _ { \theta } ( w _ { n , m } | w _ { n , < m } , \mathbf { w } _ { < n } , \mathbf { z } _ { n } ) P _ { \theta } ( \mathbf { z } _ { n } | \mathbf { w } _ { < n } ) } , } \end{array}
+$$
+
+where $\theta$ are the model parameters. VHRED uses three RNN modules: an encoder RNN, a context RNN and a decoder RNN. First, each utterance is encoded into a vector by the encoder RNN:
+
+$$
+h _ { n , 0 } ^ { \mathrm { e n c } } = \mathbf { 0 } , \ h _ { n , m } ^ { \mathrm { e n c } } = f _ { \theta } ^ { \mathrm { e n c } } ( h _ { n , m - 1 } ^ { \mathrm { e n c } } , w _ { n , m } ) \forall m = 1 , \ldots , M _ { n } ,
+$$
+
+where $f _ { \theta } ^ { \mathrm { e n c } }$ is either a GRU or a bidirectional GRU function. The last hidden state of the encoder RNN is given as input to the context RNN. Then, the context RNN updates its internal hidden state to reflect all the information up until that utterance:
+
+$$
+h _ { 0 } ^ { \mathrm { c o n } } = { \bf 0 } , h _ { n } ^ { \mathrm { c o n } } = f _ { \theta } ^ { \mathrm { c o n } } ( h _ { n - 1 } ^ { \mathrm { c o n } } , h _ { n , M _ { n } } ^ { \mathrm { e n c } } ) ,
+$$
+
+where $f _ { \theta } ^ { \mathrm { c o n } }$ is a GRU function taking as input two vectors. This state is used to compute the prior distribution over the latent variable $\mathbf { z } _ { n }$ :
+
+$$
+P _ { \theta } ( \mathbf { z } _ { n } \mid \mathbf { w } _ { < n } ) = f _ { \theta } ^ { \mathrm { p r i o r } } ( h _ { n - 1 } ^ { c o n } ) ,
+$$
+
+where $f ^ { \mathrm { p r i o r } }$ is a PDF parametrized by both $\theta$ and $h _ { n } ^ { c o n }$ . Next, a sample is drawn from this distribution: ${ \bf z } _ { n } \sim P _ { \boldsymbol \theta } ( { \bf z } _ { n } | { \bf w } _ { < n } )$ . The sample and context state are given as input to the decoder RNN:
+
+$$
+\begin{array} { r l } & { h _ { n , 0 } ^ { \mathrm { d e c } } = \mathbf { 0 } , \ h _ { n , m } ^ { \mathrm { d e c } } = f _ { \theta } ^ { \mathrm { d e c } } ( h _ { n , m - 1 } ^ { \mathrm { d e c } } , h _ { n - 1 } ^ { \mathrm { c o n } } , \mathbf { z } _ { n } , w _ { n , m } ) } \\ & { \qquad \forall m = 1 , \ldots , M _ { n } , } \end{array}
+$$
+
+where $f _ { \theta } ^ { \mathrm { d e c } }$ is the LSTM gating function taking as input four vectors. The output distribution is computed by passing $h _ { n , m } ^ { \mathrm { d e c } }$ through an MLP $f _ { \theta } ^ { \mathrm { m l p } }$ , an affine transformation and a softmax function:
+
+$$
+P _ { \theta } ( w _ { n , m + 1 } | w _ { n , \le m } , { \mathbf w } _ { < n } , { \mathbf z } _ { n } ) = \frac { e ^ { ( O w _ { n , m + 1 } ) ^ { \mathrm { T } } f _ { \theta } ^ { \mathrm { m p } } ( h _ { n , m } ^ { \mathrm { d e c } } ) } } { \sum _ { w ^ { \prime } } e ^ { ( O w ^ { \prime } ) ^ { \mathrm { T } } f _ { \theta } ^ { \mathrm { m p } } ( h _ { n , m } ^ { \mathrm { d e c } } ) } } ,
+$$
+
+where $O \in \mathbb { R } ^ { | V | \times d }$ is the word embedding matrix for the output distribution with embedding dimensionality $d \in \mathbb { N }$ . The model is trained by maximizing the variational lower-bound, which factorizes into independent terms for each sub-sequence (utterance):
+
+$$
+\begin{array} { r } { \log P _ { \theta } ( \mathbf w _ { 1 } , \dots , \mathbf w _ { N } ) \geq \displaystyle \sum _ { n = 1 } ^ { N } - { \mathrm { K L } } \left[ Q _ { \psi } ( \mathbf z _ { n } \mid \mathbf w _ { 1 } , \dots , \mathbf w _ { n } ) \right| \left| P _ { \theta } ( \mathbf z _ { n } \mid \mathbf w _ { < n } ) \right] } \\ { + \mathbb { E } _ { Q _ { \psi } ( \mathbf z _ { n } \mid \mathbf w _ { 1 } , \dots , \mathbf w _ { n } ) } \left[ \log P _ { \theta } ( \mathbf w _ { n } \mid \mathbf z _ { n } , \mathbf w _ { < n } ) \right] , } \end{array}
+$$
+
+where distribution $Q _ { \psi }$ is the approximate posterior distribution with parameters $\psi$ , which is computed similar to the prior distribution but further conditioned on the future encoder RNN hidden state:
+
+$$
+Q _ { \psi } ( \mathbf { z } _ { n } \mid \mathbf { w } _ { \leq n } ) = f _ { \psi } ^ { \mathrm { p o s t } } ( h _ { n - 1 } ^ { c o n } , h _ { n , M _ { n } } ^ { e n c } ) ,
+$$
+
+where $f ^ { \mathrm { p o s t } }$ is a PDF. More details are given by Serban et al. (2016b).
+
+The original VHRED model as described by Serban et al. (2016b) used only Gaussian latent variables. We will refer to this model as Gaussian-VHRED (G-VHRED). The VHRED model with both Gaussian and piecewise constant latent variables will be referred to as Hybrid-VHRED (HVHRED). In this case, we combine the Gaussian and piecewise latent variables by concatenating them into one vector.5
+
+# 6 EXPERIMENTS
+
+In order to validate the ability of our piecewise latent variables to capture complex aspects of data distributions, we conduct experiments with both the NVDM and VHRED models.
+
+All models are trained using back-propagation to obtain parameter gradients with respect to the variational lower-bound on the log-likelihood or the exact log-likelihood. We used a standard firstorder gradient-descent optimizer, Adam (Kingma & Ba, 2015), for both models, where only hyperparameter choices varied depending on the task. The specifics of the design of the encoder and decoder differed between the two tasks (as described in Sections 5.1 and 5.2). For all models that used piecewise latent variables, we chose to fix $\alpha _ { a _ { i } } = 1$ , meaning the piecewise prior and posterior models are kept separate (instead of having the posterior be an interpolation between another distribution and the prior), since we found this to perform better6
+
+# 6.1 DOCUMENT MODELING
+
+For our experiments in document modeling, we make use of the 20 News-Groups dataset. We follow the pre-processing and set-up of Hinton & Salakhutdinov (2009). In addition, we make use of the Reuters corpus (RCV1-V2), using a version that contained a selected 5,000 term vocabulary. 7 Note that the features are a $\log ( 1 + T F )$ transform of the original frequency vectors. To test our document models on text from another language (in this case, Brazilian Portuguese), we make use of the CADE12 dataset (stop-word removed and stemmed) Cardoso-Cachopo (2007), where we further filtered terms that occurred less than 130 times to obtain a vocabulary of 3,736 terms (over 26,991 training and 13,486 test documents). For all datasets, we track the validation bound on a subset of 100 vectors randomly drawn from each training corpus.
+
+Table 1: Comparative test perplexities on various document datasets (50 latent variables). Note that document probabilities were calculated using 10 samples to estimate the variational lower bound.   
+
+<table><tr><td>20-NG</td><td>Sampled</td><td>SGD-Inf</td><td>RCV1</td><td>Sampled</td><td>SGD-Inf</td></tr><tr><td>LDA</td><td>1058</td><td></td><td>G-NVDM</td><td>905</td><td>837</td></tr><tr><td>RSM</td><td>953</td><td></td><td>H-NVDM-3</td><td>865</td><td>807</td></tr><tr><td>docNADE</td><td>896</td><td></td><td>H-NVDM-5</td><td>833</td><td>781</td></tr><tr><td>SBN</td><td>909</td><td></td><td></td><td></td><td></td></tr><tr><td>fDARN</td><td>917</td><td></td><td>CADE</td><td>Sampled</td><td>SGD-Inf</td></tr><tr><td>NVDM</td><td>836</td><td></td><td>G-NVDM</td><td>339</td><td>230</td></tr><tr><td>G-NVDM</td><td>651</td><td>588</td><td>H-NVDM-3</td><td>258</td><td>193</td></tr><tr><td>H-NVDM-3</td><td>607</td><td>546</td><td>H-NVDM-5</td><td>294</td><td>209</td></tr><tr><td>H-NVDM-5</td><td>566</td><td>496</td><td></td><td></td><td></td></tr></table>
+
+Table 2: Word query similarity test, where each (20 News-Group) document model’s decoder is given a query and must return the top 10 most relevant words. The first query was “government” while the second was “space”. It appears that the models with piecewise variables tend to associate more general/abstract terms to the query, which may or may not always be what is desired.   
+
+<table><tr><td>G-NVDM</td><td>H-NVDM-3</td><td>H-NVDM-5</td></tr><tr><td>governments</td><td>citizens</td><td>arms</td></tr><tr><td>citizens</td><td>rights</td><td>rights</td></tr><tr><td>country</td><td>governments</td><td>federal</td></tr><tr><td>threat</td><td>civil</td><td>country</td></tr><tr><td>private</td><td>freedom</td><td>policy</td></tr><tr><td>rights</td><td>legitimate</td><td>administration</td></tr><tr><td>individuals</td><td>constitution</td><td>protect</td></tr><tr><td>military</td><td>private</td><td>private</td></tr><tr><td>freedom</td><td>court</td><td>citizens</td></tr><tr><td>foreign</td><td>states</td><td>military</td></tr></table>
+
+<table><tr><td>G-NVDM</td><td>H-NVDM-3</td><td>H-NVDM-5</td></tr><tr><td>environment</td><td>project</td><td>science</td></tr><tr><td>project</td><td>gov</td><td>built</td></tr><tr><td>flight</td><td>major</td><td>high</td></tr><tr><td>lab</td><td>based</td><td>technology</td></tr><tr><td>mission</td><td>earth</td><td>world</td></tr><tr><td>launch</td><td>include</td><td>form</td></tr><tr><td>field</td><td>science</td><td>scale</td></tr><tr><td>working</td><td>nasa</td><td>sun</td></tr><tr><td>build</td><td>systems</td><td>special</td></tr><tr><td>gov</td><td>technical</td><td>area</td></tr></table>
+
+For the Gaussian NVDM $G$ -NVDM), we constrain the interpolated posterior variance to lie in the range of [0.01, 10.0]. For the hybrid NVDMs (H-NVDM) 8, we vary the number of components used in the PDF, investigating the effect that 3 and 5 pieces had on the final quality of the model. Parameter updates for all models were estimated using mini-batches of 100 samples drawn randomly without replacement from the training data over 150 epochs. A learning rate of 0.002 was used. Model selection and early stopping (the only additional form of regularization employed for this set of experiments) were conducted using the validation lower-bound, estimated using five stochastic samples per validation example. We rescale large gradients by their norm (Pascanu et al., 2012). Inference networks made use of 50 units in each hidden layer for 20 News-Groups and CADE and 100 for RCV1, while all performed best with 50 latent variables (chosen via preliminary experimentation with smaller models). On the 20 News-Groups, since we were able to use the same set-up (especially vocabulary) as Hinton & Salakhutdinov (2009), we also report the perplexities of a topic model ( $L D A$ , Hinton & Salakhutdinov (2009)), the Replicated Softmax (RSM, Hinton & Salakhutdinov (2009)), the document neural auto-regressive estimator (docNADE, Larochelle & Lauly (2012)), a sigmoid belief network (SBN, Mnih & Gregor (2014)), a deep auto-regressive neural network (fDARN, Mnih & Gregor (2014)), and a neural variational document model with a fixed standard Gaussian prior (NVDM, lowest reported perplexity, Miao et al. (2015)).
+
+In Table 1, we report the test document perplexity (under the Sampled column), calculated using the standard formula, $\begin{array} { r } { e x p ( - \frac { 1 } { D } \sum _ { n } \frac { 1 } { L _ { n } } \log \bar { P _ { \theta } } ( x _ { n } ) } \end{array}$ . Note that $\log { P _ { \theta } ( x _ { n } ) }$ , or the log-probability of a particular document, was approximated with an estimate of the variational lower-bound using 10 samples, as was done in Mnih $\&$ Gregor (2014). The second score (or column SGD-Inf ), refers to the model’s test-perplexity when the lower-bound is tightened using iterative inference to search for the optimal latent variable per document. In this paper, our iterative inference procedure consisted of simple stochastic gradient descent (no more than 100 steps), with a learning rate of 0.1 and the same gradient rescaling used in training, using early-stopping (for 20 News-Groups, the lookahead was 10, while on Reuters and CADE the lookahead was 5). The parameters of the model, as well as the well as the generated prior, are fixed, and the gradients of the variational lower bound with respect to generated posterior model parameters (i.e., the mean and variance of the Gaussian variables, and the piecewise components, $a _ { i }$ ) are used to update the posterior model for each document (using a freshly drawn sample each step).
+
+First and foremost, we note that the best baseline model (i.e., the NVDM) is more competitive when both the prior and posterior models are learnt together (i.e., the G-NVDM), as opposed to the fixed prior of Miao et al. (2015). However, we observe that integrating our proposed piecewise variables yields even better results in our document modeling experiments, substantially improving over the baselines. More importantly, in some cases, as in the 20 News-Groups and Reuters datasets, increasing the number of pieces from 3 to 5 can further reduce perplexity. Thus, we have achieved a new state-of-the-art perplexity on 20 News-Group task and — to the best of our knowledge – better perplexities on the CADE12 and RCV1 tasks compared to using a state-of-the-art model like the GNVDM. Furthermore, we observe iterative inference yields yet a further boost in performance since the bound estimated is tighter, however, this form of inference is expensive and requires additional meta-parameters (e.g., a step-size, an early-stopping criterion, etc.). We remark a simpler, and more accurate, approach to inference would be to use importance sampling.
+
+In Table 2, we examine the top ten highest ranked words given a query term, using the decoder parameter matrix (since the decoder is directly affected by the latent variables in our document models). It appears that the piecewise variables affect what is uncovered by the model with respect to the data, as each model returns different, but relevant results with respect to the query word. In our current examples, it appears that the H-NVDM with 5 pieces returns more general words. For example, in the case of “government”, the baseline seems to value the plural form of the word (which is largely based on morphology) while the hybrid model actually pulls out meaningful terms such as “federal”, “policy”, and “administration”. The case of “space” is interesting–the hybrid with 5 pieces seems to value two senses of the word–one related to “outer space” (e.g., “sun”, “world”, etc.) and another related to the dimensions of depth, height, and width within which things may exist and move (e.g., “area”, “form”, “scale”, etc.).
+
+# 6.2 DIALOGUE MODELING
+
+We experiment with VHRED for dialogue modeling. This is a difficult problem, extensively studied in the recent literature (Ritter et al., 2011; Lowe et al., 2015; Sordoni et al., 2015; Li et al., 2016; Serban et al., 2016a). Related systems for dialogue response generation have recently gained a significant amount of attention from industry, with high-profile projects such as Google’s SmartReply system (Kannan et al., 2016) and Microsoft’s chatbot Xiaolice (Markoff & Mozur, 2015). Even more recently, Amazon has announced the Alexa Prize Challenge for the research community with the goal of developing a natural and engaging chatbot system (Farber, 2016).
+
+We focus on non-goal-driven dialogue modeling and use the Twitter Dialogue Corpus (Ritter et al., 2011) based on public Twitter conversations. The dataset is split into training, validation, and test sets, containing respectively 749,060, 93,633 and 9,399 dialogues each. On average, each dialogue contains about 6 utterances (dialogue turns) and about 94 words. The dataset is the same as used by Serban et al. (2016b), but further pre-processed using byte-pair encoding (Sennrich et al., 2016) using a vocabulary consisting of 5000 sub-words.9 The dialogues are substantially longer than recent large-scale language modeling corpora, such as the 1 Billion Word Language Model Benchmark (Chelba et al., 2014), which usually focus on modeling single sentences.
+
+Parameter optimization was conducted with a learning rate of 0.0002 and mini-batches of size 40 or 80.10 We use a variant of truncated back-propagation and apply gradient clipping (Pascanu et al., 2012). Model selection and early stopping — the only additional form of regularization employed for this set of experiments — are conducted using the validation lower-bound, estimated using one stochastic sample per validation example.
+
+Table 3: Approximate posterior word encoding on Twitter. The numbers are computed by counting the number of times each word is among the 5 words with the largest sum of squared gradients of the Gaussian KL divergence (G-KL) and piecewise constant KL divergence (P-KL)   
+
+<table><tr><td rowspan="2">Word Time-related</td><td rowspan="2">G-VHRED G-KL</td><td colspan="2">H-VHRED</td><td rowspan="2">Word Event-related</td><td rowspan="2">G-VHRED</td><td colspan="2">H-VHRED</td></tr><tr><td>G-KL</td><td>P-KL</td><td>G-KL G-KL</td><td>P-KL</td></tr><tr><td>monday</td><td>3</td><td>5</td><td>10</td><td>school</td><td>9</td><td>16</td><td>50</td></tr><tr><td>tuesday</td><td>2</td><td>3</td><td>7</td><td>class</td><td>11</td><td>16</td><td>27</td></tr><tr><td>wednesday</td><td>4</td><td>11</td><td>13</td><td>game</td><td>20</td><td>26</td><td>41</td></tr><tr><td>thursday</td><td>2</td><td>3</td><td>9</td><td>movie</td><td>12</td><td>20</td><td>41</td></tr><tr><td>friday</td><td>9</td><td>18</td><td>26</td><td>club</td><td>13</td><td>22</td><td>28</td></tr><tr><td> saturday</td><td>6</td><td>6</td><td>13</td><td>party</td><td>8</td><td>10</td><td>32</td></tr><tr><td>sunday</td><td>2</td><td>2</td><td>9</td><td>wedding</td><td>7</td><td>13</td><td>23</td></tr><tr><td>weekend</td><td>8</td><td>16</td><td>32</td><td>birthday</td><td>12</td><td>20</td><td>23</td></tr><tr><td>today</td><td>18</td><td>28</td><td>56</td><td>easter</td><td>15</td><td>15</td><td>23</td></tr><tr><td>night</td><td>16</td><td>31</td><td>68</td><td>concert</td><td>7</td><td>16</td><td>20</td></tr><tr><td>tonight</td><td>32</td><td>36</td><td>47</td><td>dance</td><td>11</td><td>12</td><td>21</td></tr><tr><td>Word Sentiment</td><td>G-VHRED</td><td>H-VHRED</td><td></td><td>Word</td><td>G-VHRED</td><td>H-VHRED</td><td></td></tr><tr><td>-related</td><td>G-KL</td><td>G-KL</td><td>P-KL</td><td>Acronyms,Punctuation Marks&amp; Emoticons</td><td>G-KL</td><td>G-KL</td><td>P-KL</td></tr><tr><td>good</td><td>72</td><td>73</td><td>44</td><td>lol</td><td>394</td><td>358</td><td>312</td></tr><tr><td>love</td><td>102</td><td>101</td><td>38</td><td>omg</td><td>52</td><td>45</td><td>19</td></tr><tr><td>awesome</td><td>26</td><td>44</td><td>39</td><td>·</td><td>386</td><td>558</td><td>1009</td></tr><tr><td>cool</td><td>14</td><td>28</td><td>29</td><td>！</td><td>648</td><td>951</td><td>525</td></tr><tr><td>haha</td><td>132</td><td>101</td><td>75</td><td>？</td><td>507</td><td>851</td><td>221</td></tr><tr><td>hahaha</td><td>60</td><td>48</td><td>24</td><td>*</td><td>108</td><td>54</td><td>19</td></tr><tr><td>amazing</td><td>14</td><td>38</td><td>33</td><td>xd</td><td>28</td><td>42</td><td>26</td></tr><tr><td>thank</td><td>137</td><td>153</td><td>29</td><td>B</td><td>56</td><td>42</td><td>24</td></tr></table>
+
+Similar to Serban et al. (2016b), we use a bidirectional GRU RNN encoder, where the forward and backward RNNs each have 1000 hidden units. We experiment with context RNN encoders with 500 and 1000 hidden units, and find that that 1000 hidden units reach better performance w.r.t. the variational lower-bound on the validation set. The encoder and context RNNs use layer normalization (Ba et al., 2016). We experiment with decoder RNNs with 1000, 2000 and 4000 hidden units (LSTM cells), and find that 2000 hidden units reach better performance. For the G-VHRED model, we experiment with latent multivariate Gaussian variables with 100 and 300 dimensions, and find that 100 dimensions reach better performance. For the H-VHRED model, we experiment with latent multivariate Gaussian and piecewise constant variables each with 100 and 300 dimensions, and find that 100 dimensions reach better performance. We follow the training procedure of Serban et al. (2016b): we drop words in the decoder with a fixed drop rate of $2 5 \%$ and multiply the KL terms in the variational lower-bound by a scalar, which starts at zero and linearly increases to 1 over the first 60,000 training batches.
+
+We also experiment with an LSTM baseline model and a HRED baseline model (Serban et al., 2016a). For the LSTM model, we experiment with number of hidden units (LSTM cells) equal to 1000, 2000 and 4000 and find that 4000 hidden units perform best w.r.t. validation perplextiy. For the HRED model, we use the same encoder and context RNN architectures as the G-VHRED and H-VHRED models described earlier. We set the encoder RNN to have 1000 hidden units. We experiment with a context RNN with 500 and 1000 hidden units, and find that 1000 hidden units reach better performance. For the decoder RNN, we experiment with 1000 and 2000 hidden units (LSTM cells) and find that 2000 hidden units perform better.
+
+Approximate Posterior Analysis Our hypothesis is that the piecewise constant latent variables are able to capture multi-modal aspects of the dialogue. Therefore, we evaluate the models by analyzing what information they have learned to represent in the latent variables. For each test dialogue with $n$ utterances, we condition each model on the first $n - 1$ utterances and compute the latent posterior distributions using all $n$ utterances. We then compute the gradients of the KL terms of the multivariate Gaussian and piecewise constant latent variables w.r.t. each word in the dialogue. Since the words vectors are discrete, we compute the sum of the squared gradients w.r.t. each word embedding. The higher the sum of the squared gradients of a word is, the more influence it will have on the posterior approximation (encoder model). For every test dialogue, we count the top 5 words with highest squared gradients separately for the multivariate Gaussian and piecewise constant latent variables.11
+
+The results are shown in Table 3. The piecewise constant latent variables clearly capture different aspects of the dialogue compared to the Gaussian latent variables. The piecewise constant variable approximate posterior encodes words related to time (e.g. weekdays and times of day) and events (e.g. parties, concerts, Easter). On the other hand, the Gaussian variable approximate posterior encodes words related to sentiment (e.g. laughter and appreciation) and acronyms, punctuation marks and emoticons (i.e. smilies). We also conduct a similar analysis on the document models evaluated in Sub-section 6.1, the results of which may be found in the Appendix.
+
+Response Evaluation Non-goal-driven dialogue models are typically evaluated by asking humans to rate the quality of different responses. We follow the approach by Liu et al. (2016) by conducting an Amazon Mechanical Turk experiment to compare the G-VHRED and H-VHRED models. For each test dialogue, we use TF-IDF to extract 100 candidate responses (Lowe et al., 2015). We then rank the responses according to the G-VHRED model and H-VHRED model using the variational lowerbound.12 We ask three human evaluators to rate model responses for 45 dialogues on a Likert-type scale $1 - 5$ , with 1 representing an inappropriate response and 5 representing a highly appropriate response.13 For each dialogue, we show the human evaluators the top two responses ranked by the G-VHRED and H-VHRED models. We choose to evaluate the re-ranked responses for two reasons. First, it reduces variance in the output because it uses the approximate posterior model, compared to using beam search with samples from the high-entropy prior. Second, it decreases the number of generic responses, which are extremely common among generative models and which human evaluators tend to prefer despite not advancing the dialogue (Li et al., 2016).
+
+The results are as follows. The G-VHRED model achieves scores 1.88 and 2.13 for the first and second ranked responses on average, and the H-VHRED model achieves scores 1.93 and 2.04 on average. In other words, H-VHRED performs nominally better on the first ranked response while G-VHRED performs nominally better on the second ranked response. In conclusion, if there exists a difference between the two models, naive human evaluators cannot see it.
+
+Although naive human evaluators cannot distinguish between the model responses, based on our previous analysis we know that the two models encode different aspects of dialogue conversations. Therefore, we further investigate the probability of different responses to dialogue contexts related to time and events. Two examples are shown in Figure 2, where the dialogue contexts are “when do you want to meet this weekend?” and “where are you going tomorrow?”. H-VHRED assigns substantially more probability mass to relevant words compared to the G-VHRED as well as an LSTM baseline and HRED baseline. This confirms the ability of the piecewise constant latent variable to generate responses related to time and events.
+
+Finally, we also evaluate the diversity of the G-VHRED and H-VHRED model outputs w.r.t. the top ranked FF-IDF candidate responses. We measure the average word entropy (Serban et al., 2016b) as well as number of unique words for each response and unique words across all test responses, but did not find a significant difference between the two models. This indicates that the Gaussian latent variables alone are able to increase response diversity, while the piecewise constant latent variables instead help encode specific aspects of the dialogue such as time and events.
+
+![](images/622012d108e740cb7f598eb4920252e01e3b5bbb0affee750de533a33398b33b.jpg)  
+Figure 2: Probabilities for different responses related to time and events: left) probabilities for giving a one-word response with one of the weekdays (monday, tuesday, . . . , sunday) conditioned on the context utterance “when do you want to meet this weekend?”, right) probabilities forgiving a oneword response with one of several event-related nouns (school, class, . . . , wedding) on the context utterance “where are you going tomorrow?”. The probabilities have been normalized in log-space by the number of words in the response including end-of-utterance tokens. For G-VHRED and H-VHRED, the probabilities were estimated using the variational lower-bound over 10 samples.
+
+# 7 CONCLUSIONS
+
+In this paper, we have proposed the multi-modal variational encoder-decoder framework. In order to capture complex aspects of unknown data distributions, we developed the piecewise constant prior, which can be efficiently and flexibly adjusted to capture distributions with many modes, such as those over topics. In experiments on document modeling and dialogue modeling, we have shown the effectiveness of our framework in building models capable of learning richer structure from data. In particular, we have demonstrated new state-of-the-art results on several document modeling tasks.
+
+Future work should focus on exploring other natural language processing tasks, where multimodality plays an important role such as modeling technical help dialogues (Lowe et al., 2015) and online debates (Rosenthal & McKeown, 2015), and where additional information is available, such as in semi-supervised document categorization (Ororbia II et al., 2015a). Furthermore, the piecewise variables proposed in this work could prove useful in uncovering interesting and novel information in lesser-explored corpora.
+
+# REFERENCES
+
+J. L. Ba, J. R. Kiros, and G. E. Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016.   
+S. Bangalore, G. Di Fabbrizio, and A. Stent. Learning the structure of task-driven human–human dialogs. IEEE Transactions on Audio, Speech, and Language Processing, 16(7):1249–1259, 2008.   
+D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent dirichlet allocation. Journal of machine Learning research, 3(Jan):993–1022, 2003.   
+J. Bornschein and Y. Bengio. Reweighted wake-sleep. In ICLR 2015, 2014.   
+S. R. Bowman, L. Vilnis, O. Vinyals, A. M. Dai, R. Jozefowicz, and S. Bengio. Generating sentences from a continuous space. In Conference on Computational Natural Language Learning, 2015.   
+Y. Burda, R. Grosse, and R. Salakhutdinov. Importance weighted autoencoders. arXiv preprint arXiv:1509.00519, 2015.   
+A. Cardoso-Cachopo. Improving Methods for Single-label Text Categorization. PdD Thesis, Instituto Superior Tecnico, Universidade Tecnica de Lisboa, 2007.   
+C. Chelba, T. Mikolov, M. Schuster, Q. Ge, T. Brants, P. Koehn, and T. Robinson. One billion word benchmark for measuring progress in statistical language modeling. In INTERSPEECH, 2014.   
+N. Crook, R. Granell, and S. Pulman. Unsupervised classification of dialogue acts using a dirichlet process mixture model. In Proceedings of the SIGDIAL 2009 Conference: The 10th Annual Meeting of the Special Interest Group on Discourse and Dialogue, pp. 341–348. Association for Computational Linguistics, 2009.   
+P. Dayan and G. E. Hinton. Varieties of helmholtz machine. Neural Networks, 9(8):1385–1403, 1996.   
+L. Devroye. Sample-based non-uniform random variate generation. In Proceedings of the 18th conference on Winter simulation, pp. 260–265. ACM, 1986.   
+J. L. Durrieu, J. P. Thiran, and F. Kelly. Lower and upper bounds for approximation of the kullbackleibler divergence between gaussian mixture models. In 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4833–4836. Ieee, 2012.   
+M. Farber. Amazon’s ’Alexa Prize’ Will Give College Students Up To $\$ 2.5 M$ To Create A Socialbot. Fortune, 2016.   
+K. Gregor, I. Danihelka, A. Graves, and D. Wierstra. DRAW: A recurrent neural network for image generation. In International Conference on Learning Representations (ICLR), 2015.   
+G. E. Hinton and R. Salakhutdinov. Replicated softmax: an undirected topic model. In Y. Bengio, D. Schuurmans, J. D. Lafferty, C. K. I. Williams, and A. Culotta (eds.), Advances in Neural Information Processing Systems 22, pp. 1607–1614. Curran Associates, Inc., 2009.   
+G. E. Hinton and R. S. Zemel. Autoencoders, minimum description length and helmholtz free energy. In J. D. Cowan, G. Tesauro, and J. Alspector (eds.), Advances in Neural Information Processing Systems 6, pp. 3–10. Morgan-Kaufmann, 1994.   
+G. E. Hinton, P. Dayan, B. J. Frey, and R. M. Neal. The" wake-sleep" algorithm for unsupervised neural networks. Science, 268(5214):1158, 1995.   
+T. Hofmann. Probabilistic latent semantic indexing. In Proceedings of the 22nd annual international ACM SIGIR Conference on Research and Development in Information Retrieval, pp. 50–57. ACM, 1999.   
+M. I. Jordan, Z. Ghahramani, T. S. Jaakkola, and L. K. Saul. An introduction to variational methods for graphical models. Machine Learning, 37(2):183–233, 1999.   
+A. Kannan, K. Kurach, et al. Smart Reply: Automated Response Suggestion for Email. In KDD, 2016.   
+D. Kingma and J. Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations (ICLR, 2015.   
+D. P. Kingma and M. Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.   
+D. P. Kingma, T. Salimans, and M. Welling. Improving variational inference with inverse autoregressive flow. arXiv preprint arXiv:1606.04934, 2016.   
+H. Larochelle and S. Lauly. A neural autoregressive topic model. In Advances in Neural Information Processing Systems, pp. 2708–2716, 2012.   
+Anders Boesen Lindbo Larsen, Søren Kaae Sønderby, and Ole Winther. Autoencoding beyond pixels using a learned similarity metric. arXiv preprint arXiv:1512.09300, 2015.   
+S. Lauly, Y. Zheng, A. Allauzen, and H. Larochelle. Document neural autoregressive distribution estimation. arXiv preprint arXiv:1603.05962, 2016.   
+J. Li, M. Galley, C. Brockett, J. Gao, and B. Dolan. A diversity-promoting objective function for neural conversation models. In The North American Chapter of the Association for Computational Linguistics (NAACL), 2016.
+
+C.-W. Liu, R. Lowe, I. V. Serban, M. Noseworthy, L. Charlin, and J. Pineau. How NOT to evaluate your dialogue system: An empirical study of unsupervised evaluation metrics for dialogue response generation. arXiv:1603.08023, 2016.
+
+Ryan Lowe, Nissan Pow, Iulian Serban, and Joelle Pineau. The Ubuntu Dialogue Corpus: A Large Dataset for Research in Unstructured Multi-Turn Dialogue Systems. In Proceedings of the SIGDIAL 2015 Conference, 2015. In press.
+
+L. Maaløe, C. K. Sønderby, S. K. Sønderby, and O. Winther. Auxiliary deep generative models. arXiv preprint arXiv:1602.05473, 2016.
+
+J. Markoff and P. Mozur. For Sympathetic Ear, More Chinese Turn to Smartphone Program. New York Times, 2015.
+
+Yishu Miao, Lei Yu, and Phil Blunsom. Neural variational inference for text processing. arXiv preprint arXiv:1511.06038, 2015.
+
+A. Mnih and K. Gregor. Neural variational inference and learning in belief networks. arXiv preprint arXiv:1402.0030, 2014.
+
+R. M. Neal. Connectionist learning of belief networks. Artificial intelligence, 56(1):71–113, 1992.
+
+A. G. Ororbia II, C. L. Giles, and D. Reitter. Learning a deep hybrid model for semi-supervised text classification. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing (EMNLP), Lisbon, Portugal, 2015a., 2015a.
+
+A. G. Ororbia II, C. L. Giles, and D. Reitter. Online semi-supervised learning with deep hybrid boltzmann machines and denoising autoencoders. arXiv preprint arXiv:1511.06964, 2015b.
+
+R. Pascanu, T. Mikolov, and Y. Bengio. On the difficulty of training recurrent neural networks. ICML, 28, 2012.
+
+D. J. Rezende and S. Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015.
+
+D. J. Rezende, S. Mohamed, and D. Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning (ICML), 2014.
+
+A. Ritter, C. Cherry, and W. B. Dolan. Data-driven response generation in social media. In Proceedings of the Conference on Empirical Methods in Natural Language Processing, pp. 583–593. Association for Computational Linguistics (ACL), 2011.
+
+S. Rosenthal and K. McKeown. I couldn’t agree more: The role of conversational structure in agreement and disagreement detection in online discussions. In 16th Annual Meeting of the Special Interest Group on Discourse and Dialogue, pp. 168, 2015.
+
+R. Salakhutdinov and G. E. Hinton. Semantic hashing. International Journal of Approximate Reasoning, 50(7):969–978, 2009.
+
+R. Salakhutdinov and H. Larochelle. Efficient learning of deep boltzmann machines. In International Conference on Artificial Intelligence and Statistics (AISTATs), pp. 693–700, 2010.
+
+T. Salimans, D. P Kingma, M. Welling, et al. Markov chain monte carlo and variational inference: Bridging the gap. In International Conference on Machine Learning (ICML), pp. 1218–1226, 2015.
+
+R. Sennrich, B. Haddow, and A. Birch. Neural machine translation of rare words with subword units. In Association for Computational Linguistics (ACL), 2016.
+
+I. V. Serban, A. Sordoni, Y. Bengio, A. Courville, and J. Pineau. Building end-to-end dialogue systems using generative hierarchical neural network models. In Thirtieth AAAI Conference (AAAI), pp. 3776–3784, 2016a.
+
+I. V. Serban, A. Sordoni, R. Lowe, L. Charlin, J. Pineau, A. Courville, and Y. Bengio. A hierarchical latent variable encoder-decoder model for generating dialogues. arXiv preprint arXiv:1605.06069, 2016b.   
+A. Sordoni, M. Galley, M. Auli, C. Brockett, Y. Ji, M. Mitchell, J. Nie, J. Gao, and B. Dolan. A neural network approach to context-sensitive generation of conversational responses. In Conference of the North American Chapter of the Association for Computational Linguistics (NAACL-HLT 2015), 2015. In press.   
+N. Srivastava, R. R Salakhutdinov, and G. E. Hinton. Modeling documents with deep boltzmann machines. arXiv preprint arXiv:1309.6865, 2013.   
+B. Uria, I. Murray, and H. Larochelle. A deep and tractable density estimator. In International Conference on Machine Learning (ICML), pp. 467–475, 2014.   
+K. Zhai and J. D. Williams. Discovering latent structure in task-oriented dialogues. In Association for Computational Linguistics (ACL), pp. 36–46, 2014.
+
+# APPENDIX A: ANALYSIS OF DOCUMENT MODEL PIECEWISE VARIABLES
+
+We present an additional analysis of the learned 20 News-Groups document models in order to explore what each set of latent variables might be capturing. To calculate the gradient of the KL terms needed to formulate word scores, we follow the approach described in Sub-section 6.2, however, conditioning only on the (training) document bag-of-words to compute the latent posterior to then calculate the gradient of the KL-terms with respect to each word in the document.
+
+In Table 4, we observe results similar to those of Sub-section 6.2–the piecewise variables capture different aspects of the document data. It is worth noting, in this experiment, that the Gaussian variables alone were originally were sensitive to some of these words. However, in the hybrid model, nearly all of the temporal words that the Gaussian variables were once more sensitive to now more strongly affect the piecewise variables, which themselves also capture all of the words that were originally missed. This might indicate a shift in responsibility in which latent variables the document model decide are more suitable to capture certain aspects of the data. This effect appears to be even stronger in the case of certain nationality-based adjectives (e.g., “american”, “israeli”, etc.). While the G-NVDM could model multi-modality in the data to some degree, this work would be primarily done in the model’s decoder. In the H-NVDM, the piecewise variables provide an explicit mechanism for capturing modes in the unknown target distribution, so it makes sense that the model would learn to use the piecewise variables instead, thus freeing up the Gaussian variables to capture other aspects of the data, as we found was the case with names (e.g., “jesus”, “kent”, etc.).
+
+Table 4: Approximate posterior word encodings on 20 News-Groups. For P-KL, we also bold every case where the piecewise variables showed greater sensitivity to the word than the Gaussian variables within the same hybrid model.   
+
+<table><tr><td>Word Time-related</td><td>G-NVDM G-KL</td><td>H-NVDM-5 G-KL</td><td>P-KL</td><td>Word Names</td><td>G-NVDM G-KL</td><td>H-NVDM-5 G-KL</td><td>P-KL</td></tr><tr><td>months</td><td>23</td><td>33</td><td>40</td><td>henry</td><td>33</td><td>47</td><td>39</td></tr><tr><td>day</td><td>28</td><td>32</td><td>35</td><td>tim</td><td>32</td><td>27</td><td>11</td></tr><tr><td>time</td><td>55</td><td>22</td><td>40</td><td>mary</td><td>26</td><td>51</td><td>30</td></tr><tr><td>century</td><td>28</td><td>13</td><td>19</td><td></td><td>40</td><td>72</td><td>30</td></tr><tr><td>past</td><td>30</td><td>18</td><td>28</td><td>james jesus</td><td>28</td><td>87</td><td>39</td></tr><tr><td>days</td><td>37</td><td>14</td><td>19</td><td>george</td><td>26</td><td>56</td><td>29</td></tr><tr><td>ahead</td><td>33</td><td>20</td><td>33</td><td>keith</td><td>65</td><td>94</td><td></td></tr><tr><td>years</td><td>44</td><td>16</td><td>38</td><td>kent</td><td>51</td><td>56</td><td>61</td></tr><tr><td>today</td><td>46</td><td>27</td><td>71</td><td>chris</td><td>38</td><td></td><td>15</td></tr><tr><td>back</td><td>31</td><td>30</td><td>47</td><td>thomas</td><td>19</td><td>55</td><td>28</td></tr><tr><td>future</td><td>20</td><td>15</td><td>20</td><td>hitler</td><td></td><td>35</td><td>19</td></tr><tr><td>order</td><td>42</td><td>14</td><td>26</td><td></td><td>10</td><td>14</td><td>9</td></tr><tr><td>minute</td><td>15</td><td>34</td><td>40</td><td>paul</td><td>25</td><td>52</td><td>18</td></tr><tr><td>began</td><td>16</td><td>5</td><td>13</td><td>mike</td><td>38</td><td>76</td><td>40</td></tr><tr><td>night</td><td>49</td><td>12</td><td>18</td><td>bush</td><td>21</td><td>20</td><td>14</td></tr><tr><td>hour</td><td>18</td><td>17</td><td>16</td><td></td><td></td><td></td><td></td></tr><tr><td>early</td><td>42</td><td>42</td><td>69</td><td>Adjectives</td><td>G-KL</td><td>G-KL</td><td>P-KL</td></tr><tr><td>yesterday</td><td>25 60</td><td>26</td><td>36</td><td>american</td><td>50</td><td>12</td><td>40</td></tr><tr><td>year</td><td></td><td>17</td><td>21</td><td>german</td><td>25</td><td>21</td><td>22</td></tr><tr><td>week</td><td>28</td><td>54</td><td>58</td><td>european</td><td>20</td><td>17</td><td>27</td></tr><tr><td>hours</td><td>20</td><td>26</td><td>31</td><td>muslim</td><td>19</td><td>7</td><td>23</td></tr><tr><td>minutes</td><td>40</td><td>34</td><td>38</td><td>french</td><td>11</td><td>17</td><td>17</td></tr><tr><td>months</td><td>23</td><td>33</td><td>40</td><td>canadian</td><td>18</td><td>10</td><td>16</td></tr><tr><td>history</td><td>32</td><td>18</td><td>28</td><td>japanese</td><td>16</td><td>9</td><td>24</td></tr><tr><td>late</td><td>41</td><td>45</td><td>31</td><td>jewish</td><td>56</td><td>37</td><td>54</td></tr><tr><td>moment</td><td>23</td><td>17</td><td>16</td><td>english</td><td>19</td><td>16</td><td>26</td></tr><tr><td>season</td><td>45 29</td><td>29</td><td>37</td><td>islamic</td><td>14</td><td>18</td><td>28</td></tr><tr><td>summer</td><td>30</td><td>28</td><td>31 38</td><td>israeli</td><td>24</td><td>14</td><td>18</td></tr><tr><td>start</td><td>21</td><td>14</td><td>34</td><td>british</td><td>35</td><td>15</td><td>17</td></tr><tr><td>continue happened</td><td>22</td><td>32 27</td><td>35</td><td>russian</td><td>14</td><td>19</td><td>20</td></tr></table>

parse/train/EAdJEN8xKUl/EAdJEN8xKUl.md

@@ -0,0 +1,301 @@
+# Confident-Anchor-Induced Multi-Source-Free Domain Adaptation
+
+Jiahua Dong1, 2∗, Zhen Fang3∗, Anjin Liu3, Gan $\mathbf { S u n ^ { 1 \dagger } }$ , Tongliang Liu4
+
+1State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences. 2University of Chinese Academy of Sciences. 3DeSI Lab, AAII, University of Technology Sydney. 4TML Lab, University of Sydney. {dongjiahua1995, fzjlyt, sungan1412}@gmail.com, anjin.liu@uts.edu.au, tongliang.liu@sydney.edu.au
+
+# Abstract
+
+Unsupervised domain adaptation has attracted appealing academic attentions by transferring knowledge from labeled source domain to unlabeled target domain. However, most existing methods assume the source data are drawn from a single domain, which cannot be successfully applied to explore complementarily transferable knowledge from multiple source domains with large distribution discrepancies. Moreover, they require access to source data during training, which are inefficient and unpractical due to privacy preservation and memory storage. To address these challenges, we develop a novel Confident-Anchor-induced multisource-free Domain Adaptation (CAiDA) model, which is a pioneer exploration of knowledge adaptation from multiple source domains to the unlabeled target domain without any source data, but with only pre-trained source models. Specifically, a source-specific transferable perception module is proposed to automatically quantify the contributions of the complementary knowledge transferred from multi-source domains to the target domain. To generate pseudo labels for the target domain without access to the source data, we develop a confident-anchorinduced pseudo label generator by constructing a confident anchor group and assigning each unconfident target sample with a semantic-nearest confident anchor. Furthermore, a class-relationship-aware consistency loss is proposed to preserve consistent inter-class relationships by aligning soft confusion matrices across domains. Theoretical analysis answers why multi-source domains are better than a single source domain, and establishes a novel learning bound to show the effectiveness of exploiting multi-source domains. Experiments on several representative datasets illustrate the superiority of our proposed CAiDA model. The code is available at https://github.com/Learning-group123/CAiDA.
+
+# 1 Introduction
+
+Unsupervised Domain Adaptation (UDA) [22, 59, 63] captures transferable knowledge from labeled data in a source domain to classify unlabeled data in a target domain. UDA has achieved remarkable successes in many applications, e.g., object detection [23], medical diagnose [9, 12], sentiment analysis [34], etc. Generally, most of existing methods employ adversarial learning [17] to encourage the learned source and target features to be indistinguishable from each other [10, 15], or minimize the distribution discrepancy across domains by matching the statistical moments of distributions [41].
+
+However, the above-mentioned methods have a strong assumption that the source data are merely drawn from a single domain. Unfortunately, the source data are often collected under different deployed environments (i.e., multiple source domains with large distribution discrepancies) in realworld applications, which makes them difficult to explore complementarily transferable knowledge from the multi-source domains for target prediction. To achieve this, Multi-Source Domain Adaptation (MSDA) [32, 34, 61] is proposed to match the features across domains and then quantify the contributions of source domains [2, 41, 60]. Additionally, [29, 58] aim to weight the source contributions by normalizing the distance similarities between source and target domains.
+
+Unfortunately, recent MSDA methods [29, 34, 41, 44] require massive labeled source data when adapting source domains to the target domain. This could make them inefficient and unpractical in real-world applications with sensitive information (e.g., medical diagnosis [12] and recommendation system [24]), due to privacy preservation issues, storage and security concerns [50, 51]. To this end, a new challenging and practical problem named Multi-Source-Free Domain Adaptation (MSFDA) is researched, which explores transferable knowledge from multiple source domains to target domain with only pre-trained source models and without access to any source data. The trivial solutions for tackling MSFDA via using existing single-source-free domain adaptation methods [25, 30, 33, 57] are to adapt each source model individually and simply take an average prediction of source models. However, they cannot explore the contributions of the complementary information transferred from different source domains, due to the lack of source data. Therefore, tackling the MSFDA problem is a challenging but rarely-researched task.
+
+To address the MSFDA problem, we develop a novel Confident-Anchor-induced multi-source-free Domain Adaptation (CAiDA) model, which is a pioneer exploration to capture transferable information from multiple source models to promote target prediction without access to source data. Specifically, a source-specific transferable perception module is designed to calibrate the contributions of the transferability from multiple source domains. We develop a confident-anchor-induced pseudo label generator to mine pseudo labels for the unlabeled target data, by incorporating with the quantified source transferability contributions. We construct a confident anchor group to assign each target sample with a semantic-nearest confident anchor, and perform feature augmentation between them to generate confident target pseudo label. A class-relationship-aware consistency loss is proposed to ensure the semantic consistency of underlying inter-class relationships across domains via the alignment of soft confusion matrices. Furthermore, based on some mild assumptions, theoretical analysis guarantees that multiple source models could help generate more reliable pseudo labels. Our theoretical analysis also provides a novel learning bound for MSFDA, which reveals that multiple source models help achieve a tighter generalization error bound for the target domain. We verify the effectiveness of our proposed model via comparison experiments on benchmark datasets. The main contributions of our work are summarized as follows:
+
+• We develop a novel Confident-Anchor-induced multi-source-free Domain Adaptation (CAiDA) model to explore transferable knowledge from multiple source domains to assist target prediction with pre-trained source models and without access to source data. To our best knowledge, this paper is a pioneer exploration of multi-source-free domain adaptation in the field of transfer learning.
+
+• We propose a novel MSFDA theory, which shows that multiple source models could improve the possibility of obtaining more reliable pseudo labels under some mild assumptions. Our theoretical analysis also provides a novel generalization bound for MSFDA to show the effect of multiple source models. This novel bound implies a positive answer to the solvability of MSFDA problem.
+
+• A source-specific transferable perception module and a class-relationship-aware consistency loss are designed to quantify the contributions of the transferability of multiple source domains and ensure the semantic consistency of underlying inter-class relationships across domains, respectively.
+
+• Based on the confident pseudo labeling strategy in theoretical analysis, a confident-anchor-induced pseudo label generator is proposed to generate pseudo labels for the target domain by establishing a confident anchor group and assigning each target sample with a semantic-nearest confident anchor.
+
+# 2 Related Work
+
+Unsupervised Domain Adaptation aims to borrow transferable knowledge from source domain to promote the prediction of the unlabeled target domain. After Hoffman et al. [22] introduce adversarial learning [17] into domain adaptation, enormous adversarial-based methods [8, 11, 15, 59] are proposed to perform feature-level or pixel-level distribution alignment. Besides, some moment matching-based methods [14, 35, 41] focus on matching the distribution statistical moments at different orders to minimize the distribution discrepancy across domains. Furthermore, some researches design the adversarial dropout [28], batch normalization [52] and auxiliary reconstruction tasks [5,16] to narrow the domain discrepancy. Unfortunately, the methods mentioned above assume massive labeled source data are available. This is unpractical due to privacy and security concerns.
+
+Source-Free Domain Adaptation (SFDA) [33] is studied to tackle the above challenge. A common strategy in SFDA methods is to mine the confident pseudo labels for target domain. To this end, [33] uses a self-supervised pseudo labeling strategy, and [25] designs a confidence-based sample filtering method. [42] alleviates the negative transfer brought by noisy pseudo labels through confidence reweighting and regularization. In addition to the pseudo labeling strategy, the model adaptation strategy has also been studied. For example, [30, 57] employ an adversarial learning strategy to perform model adaptation with only pre-trained source models. However, they cannot be applied to tackle the MSFDA problem, due to the distribution discrepancies across different source domains.
+
+Multi-Source Domain Adaptation is extended from vanilla domain adaptation [12, 15, 41] by exploring transferable knowledge from multiple sources. To capture the relationship between different source domains and a given target domain, Guo et al. [19] design a mixture-of-experts model for unsupervised domain adaptation from multiple sources. [56] focuses on determining which source domain is the best for target prediction via dynamic curriculum learning. Some discrepancy-based methods aim to narrow the distribution discrepancy across domains by minimizing different measures such as the Rényi-divergence [21] and maximum mean discrepancy [19]. Moreover, some adversarial-based methods focus on optimizing the $\mathcal { H }$ -divergence [41, 60], generative adversarial loss [55, 61] and Wasserstein distance [32] to make features from multiple sources indistinguishable for a shared discriminator. [34, 44] perform knowledge adaptation at the pixel-level by replaying multiple source domains. Due to lack of source data, above strategies in MSDA may be invalid and unsuitable to address the challenging MSFDA problem. To this end, Ahmed et al. [1] utilize nearest distance measure to mine target pseudo labels, and weight the predictions from multiple source models for the MSFDA task. It may result in that the generation process has high probability to obtain noisy labels [4, 31, 54] when the strategy is not matched with the target data, while our model could generate confident pseudo labels from two different perspectives, i.e., geometry and probability.
+
+# 3 Problem Setting
+
+Let $\mathcal { X }$ and $y = [ K ] : = \{ 1 , . . . , K \}$ denote the feature space and label space. A domain is a joint distribution $P _ { X Y }$ on $\mathcal { X } \times \mathcal { V }$ . There are $n$ source domains $\{ P _ { X Y } ^ { i } \} _ { i = 1 } ^ { n }$ . For any source domain $\bar { P } _ { X Y } ^ { i }$ a corresponding neural-network-based predictor (model) $h ^ { i } : \mathcal { X } \to \mathbb { R } ^ { K }$ is given. Given a target domain $\mathsf { \bar { P } } _ { X Y } ^ { t }$ with unlabeled target data $T = \{ \mathbf { x } ^ { j } \} _ { j = 1 } ^ { m } \sim P _ { X } ^ { t }$ , i.i.d., the aim of multi-source-free domain adaptation (MSFDA) is to classify the unlabeled target data by utilizing $T$ and $\{ h ^ { i } \} _ { i = 1 } ^ { n }$ .
+
+Let $\ell$ denote a non-negative loss function defined over $\mathbb { R } ^ { K } \times \mathbb { R } ^ { K }$ . Given a hypothesis space $\mathcal { H } \subset \{ h \ : \ \mathcal { X } \ \to \ \overline { { \mathbb { R } } } ^ { K } \}$ , we denote $\mathcal { L } _ { s } ^ { i } ( h ) ~ = ~ \mathbb { E } _ { ( \mathbf { x } , y ) \sim P _ { X Y } ^ { i } } \ell ( h ( \mathbf { x } ) , \Phi ( y ) )$ and $\begin{array} { r l } { \mathcal { L } _ { t } ( \pmb { h } ) } & { { } = } \end{array}$ $\mathbb { E } _ { ( \mathbf { x } , y ) \sim P _ { X Y } ^ { t } } \ell ( h ( \mathbf { x } ) , \Phi ( y ) )$ as the risks with respect to the $i$ -th source domain and a given target domain, where $\Phi : \mathcal { V } \to \mathbb { R } ^ { K }$ maps any label $y$ to a corresponding one-hot vector.
+
+The source predictor $h ^ { i }$ is a vector-valued function, i.e., $\pmb { h } ^ { i } ( \mathbf { x } ) = [ h _ { 1 } ^ { i } ( \mathbf { x } ) , . . . , h _ { K } ^ { i } ( \mathbf { x } ) ] ^ { \top }$ , and consists of two basic components: feature extractor $\mathbf { f } ^ { i } : \mathcal { X }  \mathbb { R } ^ { d }$ and classifier $\mathbf { c } ^ { i } : \mathbb { R } ^ { d }  \mathbb { R } ^ { K }$ , where $d$ is the dimension of extracted features. Therefore, $h ^ { i }$ can be rewritten as $\mathbf { c } ^ { i } \circ \mathbf { f } ^ { i }$ . After using softmax as the activation function in the output layer, we have hik ≥ 0 and ∑Kk=1 hik = 1. To ensure that each predictor $h ^ { i }$ is a relatively accurate predictor for the domain $P _ { X Y } ^ { i }$ , we assume that the predictor $h ^ { i }$ is $\epsilon$ -accurate under $\ell _ { 1 }$ loss, i.e., $\mathbb { E } _ { ( \mathbf { x } , y ) \sim P _ { X Y } ^ { i } } \Vert \pmb { h } ^ { i } ( \mathbf { x } ) - \pmb { \Phi } ( y ) \Vert _ { \ell _ { 1 } } < \epsilon$ , for any $i \in [ n ]$ .
+
+# 4 Theoretical Analysis
+
+Without any relations between source domains and target domain, MSFDA cannot be effectively addressed from the theoretical view. To bridge source and target domains, Mansour et al. [40] and
+
+Miraj Ahmed et al. [1] assume that $\begin{array} { r } { P _ { X Y } = \sum _ { i = 1 } ^ { n } \lambda _ { i } P _ { X Y } ^ { i } } \end{array}$ and $P _ { Y | X } ^ { t } = P _ { Y | X } ^ { i } = P _ { Y | X } ^ { j }$ , for any $i , j \in [ n ]$ , where $\lambda _ { i } \geq 0$ and $\textstyle \sum _ { i = 1 } ^ { n } \lambda _ { i } = 1$ . This assumption is very strong and may be unrealistic in many real-world applications. Motivated by the meta learning [43] and domain generalization [3], in this paper, we propose some novel and mild assumptions to address MSFDA problem.
+
+Assumption 1 (Meta Assumption.) $P _ { X Y } ^ { t } , P _ { X Y } ^ { 1 } , . . . , P _ { X Y } ^ { n }$ are drawn (i.i.d.) from a meta distribution $\mathcal { P }$ , which is defined over a joint distribution space $\mathcal { P } _ { X Y }$ .
+
+Assumption 2 (Regular Domain.) Let the joint distribution space ce , w $d _ { \mathrm { T V } } ( \cdot , \cdot )$ $P _ { X Y } ^ { t }$ a regular domain, i.e., for any . $\mathcal { P } _ { X Y }$ is endowed with total $\sigma > 0$ $\mathcal { P } ( \mathcal { N } _ { P _ { X Y } ^ { t } } ^ { \sigma } ) > 0$ $\mathcal { N } _ { P _ { X Y } ^ { t } } ^ { \sigma } = \{ P : d _ { \mathrm { T V } } ( P , P _ { X Y } ^ { t } ) \stackrel {  } { < } \sigma \}$
+
+It is easy to check, if the meta distribution $\mathcal { P }$ is discrete or continuous with continuous density function, then the target domain $P _ { X Y } ^ { t }$ drawn by $\mathcal { P }$ is a regular domain with probability 1. In addition, to weaken the assumption $P _ { Y | X } ^ { t } = P _ { Y | X } ^ { i } = P _ { Y | X } ^ { j }$ , for any $i , j \in [ n ]$ , our key strategy is to consider the anchor point assumption that is studied in label-noise learning [36].
+
+We say a given point $\mathbf { x }$ is a $\tau$ -anchor point, if there exists a predictor $h ^ { i }$ with the largest score $h _ { k } ^ { i } ( { \bf x } )$ , such that $h _ { k } ^ { i } ( { \bf x } ) - h _ { c } ^ { i } ( { \bf x } ) \geq \tau$ , for any $c \in [ K ]$ and $c \neq k$ . $h ^ { i }$ is the $\tau$ -anchor predictor for $\mathbf { x }$ .
+
+Assumption 3 ( $\tau$ -Anchor Point Assumption.) Given a $\tau$ -anchor point x, suppose that all $\tau$ - anchor predictors for $\mathbf { x }$ are $h ^ { i _ { 1 } } , . . . , h ^ { i _ { l } }$ , then the true label of the $\tau$ -anchor point $\mathbf { x }$ is $k _ { : }$ , if $h _ { k } ^ { j } ( { \bf x } )$ is the largest score among scores $h _ { c } ^ { i _ { 1 } } , . . . , h _ { c } ^ { i _ { l } }$ , for any $c \in [ K ]$ .
+
+When the source predictors are accurate enough, Assumption 3 implies that the source and target conditional distributions are similar in the high confidence region. Hence, it is much weaker than the traditional assumption $P _ { Y | X } ^ { t } = P _ { Y | X } ^ { i } = P _ { Y | X } ^ { j }$ , for $i , j \in [ n ]$ . $\tau$ is regarded as a threshold to distinguish which data has highly confident prediction. In general, $\tau$ is close to 1.
+
+Given any $h ^ { i }$ , it is easy to check that there exists a matrix function $\mathbf { A } ^ { i } ( \mathbf { x } ) = [ a _ { k l } ^ { i } ( \mathbf { x } ) ]$ such that $\begin{array} { r } { h _ { k } ^ { i } ( \mathbf { x } ) = \sum _ { l = 1 } ^ { K } a _ { k l } ^ { i } ( \mathbf { x } ) P _ { Y | X } ^ { t } ( l | \mathbf { x } ) } \end{array}$ with $\textstyle \sum _ { k = 1 } ^ { K } a _ { k l } ^ { i } = 1$ . We say the diagonal elements $\{ a _ { l l } ^ { i } \} _ { l = 1 } ^ { K }$ as the transfer factor for $h ^ { i }$ and $P _ { Y | X } ^ { t }$ . Generally, $\mathbf { A } ^ { i } ( \mathbf { x } ) = [ a _ { k l } ^ { i } ( \mathbf { x } ) ]$ is not unique, thus the transfer factor may be not unique. The following theorem indicates that Assumption 3 holds if we give proper assumptions for transfer factor.
+
+Theorem 1 Suppose that the Bayesian label is true label [7]. If there exist transfer factors and $a$ constant $B < K$ such that $\begin{array} { r } { \operatorname* { m a x } _ { i \in [ n ] , l \in [ K ] } a _ { l l } ^ { i } \geq B } \end{array}$ , then Assumption 3 holds with $\tau > 1 - B / K$ .
+
+Theorem 1 provides a theoretical support for Assumption 3 and indicates that when the transfer factors are positive, Assumption 3 always holds with a proper $\tau$ . To further study the highly confident pseudo labeling strategy, the following theorem provides a lower bound to estimate the number of highly confident pseudo labels, i.e., the number of $\tau$ -anchor points.
+
+Theorem 2 Assume Assumptions $I$ and 2 hold and the conditional distribution $P _ { Y \mid X } ^ { t }$ can be presented as a labeling function, i.e., $P _ { Y | X } ^ { t } ( y | \mathbf { x } ) = 0 o r 1$ . Given $\eta > 0$ , if $m \geq n$ and $( 1 - \eta ) ( 1 - \tau ) >$ $\epsilon + 2 \sigma + 2 \sqrt { \log ( 2 m / \delta ) / 2 m }$ , then with probability at least $1 - \delta - ( 1 - \mathcal { P } ( \mathcal { N } _ { P _ { X _ { t } Y _ { t } } } ^ { \sigma } ) ) ^ { n } > 0 .$ , at least ηm target data are $\tau$ -anchor points, where $\epsilon$ is the upper bound of the accuracies of source predictors, and $\sigma$ is introduced in Assumption 2.
+
+Theorem 2 indicates that multi-source predictors improve the probability to obtain more $\tau$ -anchor points. To further understand the effect of multi-source domains, we build a novel learning bound for the MSFDA task. Let $A _ { \tau }$ be the set consisting of all $\tau$ -anchor points. Denote the empirical risk $\widehat { \mathcal { L } } _ { s } ^ { \tau } ( h )$ by $\begin{array} { r } { \frac { 1 } { \left| A _ { \tau } \cap T \right| } \sum _ { \mathbf x \in A _ { \tau } \cap T } \ell ( \pmb h ( \mathbf x ) , \pmb \Phi ( y ) ) } \end{array}$ , where $y$ is the label of the anchor point $\mathbf { x }$ .
+
+Theorem 3 Given Assumption 3 and some assumptions used in Theorem 2, and suppose that the loss $\ell$ has upper bound $M > 0$ and hypothesis space $\mathcal { H }$ has finite Natarajan dimension, for $\eta > 0$ if $m \geq n$ and $( 1 - \eta ) ( 1 - \tau ) > \epsilon + 2 \sigma + 2 \sqrt { \log ( 2 m / \delta ) / 2 m }$ , then for any $\boldsymbol { h } \in \mathcal { H }$ and $b \in ( 0 , 1 )$ , there exists a constant $C ( b , K )$ such that with the probability at least $1 - 2 \delta - 2 ( 1 - \mathcal { P } ( \mathcal { N } _ { P _ { X Y } ^ { t } } ^ { \sigma } ) ) ^ { n }$ :
+
+$$
+\bigl | \mathcal { L } _ { t } ( h ) - \widehat { \mathcal { L } } _ { s } ^ { \tau } ( h ) \bigr | \leq M C ( b , K ) \sqrt { \frac { \log ( 2 / \delta ) } { \eta ^ { 1 - b } m ^ { 1 - b } } } + M \frac { 2 \sigma + \epsilon } { 1 - \tau - 2 \sigma - \epsilon } ,
+$$
+
+where $\epsilon$ is the upper bound of accuracies of source predictors, and $\sigma$ is introduced in Assumption 2.
+
+![](images/0453deba0c69280540053b9de2994f6079c9762a0913511b6a690b06346682d6.jpg)  
+Figure 1: Overview of our model, mainly including a source-specific transferable perception strategy to quantify the contributions of the transferability of source domains, a confident-anchorinduced pseudo label generator to generate pseudo labels for target domain, and a class-relationshipaware consistency loss to ensure the semantic consistency of underlying inter-class relationships.
+
+Theorem 3 shows that multiple source domains improve the probability to ensure a tighter generalization bound, i.e., Eq. (1) holds. Note that the Natarajan dimension used in Theorem 3 is a bit outdated. However, the Natarajan dimension can be replaced and Theorem 3 can be updated without any technical barriers, if there exists better generalization theory for supervised learning.
+
+Summary of Theoretical Analysis: The reasons to develop MSFDA are to study the solvability of MSFDA and understand how multi-source predictors benefit the target domain’s classification. By Theorems 2 and 3, we realize that multi-source predictors improve the probability to obtain more highly confident pseudo labels, resulting in a tighter generalization bound. The generalization bound in Theorem 3 gives a positive answer to the solvability of MSFDA. Additionally, Theorems 1 and 2 also imply two interesting and important results: Theorem 1 provides the first theoretical support to the confident pseudo labeling strategy, and Theorem 2 provides the first lower bound of the number of highly confident pseudo labels. As we know, the theoretical results in Theorems 1 and 2 are novel.
+
+# 5 The Proposed CAiDA Model
+
+The graphical illustration of our proposed model is depicted in Figure 1. It mainly consists of three significant components: source-specific transferable perception, confident-anchor-induced pseudo label generator and class-relationship-aware consistency loss, which are elaborated as follows.
+
+# 5.1 Source-Specific Transferable Perception
+
+Generally, in multi-source domain adaptation (MSDA), different source domains have different contributions to improve the performance on target domain [21, 61]. To this end, many previous MSDA methods match the features across different domains, and then quantify the contributions of source domains by taking the average of the trained source predictors [41, 60], or weight the trained source predictors by normalizing the distance similarities [29,58] between source and target domains. However, due to the lack of source data, these methods cannot employ source data to match features and cannot be successfully applied to multi-source-free domain adaptation (MSFDA) tasks.
+
+Therefore, a source-specific transferable perception module is developed to automatically quantify the contributions of the transferability of source domains, as shown in Figure 1. Specifically, the one-hot encoding vector $\mathbf { u } ^ { i } \in \mathbb { R } ^ { n }$ of the $i$ -th source domain can be considered as the unique domain characterization, which is then employed as network input to quantify the contribution of the transferability. We then concatenate all source domains’ one-hot characterizations together to obtain $\mathbf { U } = [ \mathbf { u } ^ { 1 } , \cdot \cdot \overline { { \mathbf { \Omega } } } , \mathbf { u } ^ { n } ] ^ { \top } \in \mathbb { R } ^ { n \times n }$ . U is then forwarded into a Multi-Layer Perceptron (MLP) network $\pmb { \Omega }$ to automatically quantify the contribution of the transferability $\pmb { \mu } \in \mathbb { R } ^ { n }$ of $n$ source domains: $\pmb { \mu } = \Omega ( \mathbf { U } ) = \bar { \Omega ( [ \bar { \mathbf { u } } ^ { 1 } , \mathbf { u } ^ { 2 } , \cdots , \mathbf { u } ^ { n } ] ^ { \top } ) }$ , such that $\textstyle \sum _ { i = 1 } ^ { n } \mu _ { i } \ = \ 1$ , where $\mu _ { i }$ denotes the quantified contribution of the $i$ -th source domain to the target prediction.
+
+When the source data are unavailable, it is difficult to narrow distribution discrepancy across domains, due to lack of any supervised information. Inspired by [33], we freeze the network parameters of classifiers $\{ \mathbf { c } ^ { i } \} _ { i = 1 } ^ { n }$ and solely perform distribution adaptation across domains on feature extractors $\{ \mathbf { f } ^ { i } \} _ { i = 1 } ^ { n }$ via information maximization [26], since $\{ { \mathbf { c } } ^ { i } \} _ { i = 1 } ^ { n }$ contain class distribution information of source domains. However, Liang et al. [33] cannot be effectively applied to multi-source-free domain adaptation scenario, where different source domains have different transferable contributions on target prediction. Therefore, ${ \mathcal { L } } _ { \mathrm { e n t } }$ is proposed to minimize conditional entropy of target outputs by incorporating the source-specific transferable perception $\mu$ :
+
+$$
+\mathcal { L } _ { \mathrm { e n t } } = - \frac { 1 } { m } \sum _ { j = 1 } ^ { m } [ \sum _ { k = 1 } ^ { K } h _ { k } ( \mathbf { x } ^ { j } ) \log ( h _ { k } ( \mathbf { x } ^ { j } ) ) ] ,
+$$
+
+where $h _ { k } ( \mathbf { x } ^ { j } )$ is the $k$ -th coordinate value of $\pmb { h } ( \mathbf { x } ^ { j } )$ , here $\begin{array} { r } { \pmb { h } ( \mathbf { x } ^ { j } ) = \sum _ { i = 1 } ^ { n } \mu _ { i } \pmb { h } ^ { i } ( \mathbf { x } ^ { j } ) } \end{array}$ denotes the combination of source predictions. The larger value of $\mu _ { i }$ indicates the larger contribution of transferability of the $i$ -th source domain for target adaptation. Unfortunately, our proposed CAiDA model may suffer from the trivial solution of Eq. (2) by predicting all target data as a single class to minimize Eq. (2). To tackle this issue, ${ \mathcal { L } } _ { \mathrm { d i v } }$ is designed to consider class prediction diversity by maximizing the entropy of empirical label distribution [6] predicted by different source domains:
+
+$$
+\mathcal { L } _ { \mathrm { d i v } } = \sum _ { k = 1 } ^ { K } \hat { p } _ { k } \log \hat { p } _ { k } ,
+$$
+
+where $\begin{array} { r } { \hat { p } _ { k } = \frac { 1 } { m } \sum _ { j = 1 } ^ { m } h _ { k } ( \mathbf { x } ^ { j } ) } \end{array}$ denotes the mean prediction probability over target data.
+
+# 5.2 Confident-Anchor-Induced Pseudo Label Generator
+
+Although the minimization of both ${ \mathcal { L } } _ { \mathrm { e n t } }$ and ${ \mathcal { L } } _ { \mathrm { d i v } }$ promotes the class diversity and knowledge adaptation between multiple unseen source data and target data, it cannot circumvent erroneous label assignment due to the noisy target prediction brought by domain discrepancy. To alleviate this issue, based on Assumption 3, a confident-anchor-induced pseudo label generator is developed to mine confident pseudo labels for target data, as shown in Figure 1. Specifically, based on the $\tau$ -anchor assumption, we select target data $\mathbf { x } ^ { j }$ as confident anchor when the maximum category prediction probability $h _ { k ^ { * } } ( \mathbf { x } ^ { j } )$ is larger than the rest of category prediction probabilities $h _ { k } ( \mathbf { x } ^ { \bar { j } } ) \ \bar { ( } k \neq k ^ { * } )$ by a threshold $\tau _ { p }$ . For each confident anchor, we integrate the features extracted from multiple source extractors together to obtain a probability-based confident anchor group $\mathcal { C } _ { p }$ :
+
+$$
+\begin{array} { r } { \mathcal { C } _ { p } = \{ \mathbf { f } ( \mathbf { x } ^ { j } ) | h _ { k ^ { * } } ( \mathbf { x } ^ { j } ) - h _ { k } ( \mathbf { x } ^ { j } ) \geq \tau _ { p } , \forall k \neq k ^ { * } , j = 1 , 2 , \cdots m \} , } \end{array}
+$$
+
+where $\begin{array} { r } { \mathbf { f } ( \mathbf { x } ^ { j } ) = \sum _ { i = 1 } ^ { n } \mu _ { i } \mathbf { f } ^ { i } ( \mathbf { x } ^ { j } ) } \end{array}$ denotes the feature of the $j$ -th target data. $\tau _ { p }$ is defined as the median value of the probability difference between the largest and the second largest probabilities over all target data. Furthermore, inspired by the proposed $\tau$ -anchor assumption, to circumvent the noisy anchor in $\mathcal { C } _ { p }$ , we construct a distance-based confident anchor group $\mathcal { C } _ { d }$ . The feature centroid of the $k$ -th class induced by the $i$ -th source domain for the whole target data is defined as $\xi _ { k } ^ { i } =$ $\textstyle \sum _ { j = 1 } ^ { m } h _ { k } ^ { i } ( \mathbf { x } ^ { j } ) \mathbf { f } ^ { i } ( \mathbf { x } ^ { j } ) / \textstyle \sum _ { j = 1 } ^ { m } h _ { k } ^ { i } ( \mathbf { x } ^ { j } ) . ~ \{ \xi _ { k } ^ { i } \}$ $\{ \xi _ { k } ^ { i } \} _ { i = 1 } ^ { n }$ k are then weighted with source-specific transferability to obtain the feature centroid $\xi _ { k }$ of the $k$ -th class over all source domains via $\begin{array} { r } { \xi _ { k } = \sum _ { i = 1 } ^ { n } \mu _ { i } \xi _ { k } ^ { i } } \end{array}$ . The distance between the feature of $\mathbf { x } ^ { j }$ and the $k$ -th feature centroid $\xi _ { k }$ is denoted as $d ( \mathbf { f } ( \mathbf { x } ^ { j } ) , \pmb { \xi } _ { k } )$ , where $d ( \cdot , \cdot )$ is a distance measure function. When the minimum distance $d ( \mathbf { f } ( \mathbf { x } ^ { j } ) , \xi _ { k ^ { * } } )$ is shorter than the rest of distances $d ( \mathbf { f } ( \mathbf { x } ^ { j } ) , \xi _ { k } ) ( k \neq k ^ { * } )$ by a threshold $\tau _ { d }$ , we select the target data $\mathbf { x } ^ { j }$ into $\mathcal { C } _ { d }$ :
+
+$$
+\begin{array} { r } { \mathcal { C } _ { d } = \{ \mathbf { f } ( \mathbf { x } ^ { j } ) | d ( \mathbf { f } ( \mathbf { x } ^ { j } ) , \xi _ { k } ) - d ( \mathbf { f } ( \mathbf { x } ^ { j } ) , \xi _ { k * } ) \geq \tau _ { d } , \forall k \neq k ^ { * } , j = 1 , 2 , \cdot \cdot \cdot m \} , } \end{array}
+$$
+
+where we set $\tau _ { d }$ as the median value of the distance difference between the shortest and the second shortest distances over all target data. Therefore, the final confident anchor group $\mathcal { C }$ is obtained by performing an intersection operation between $\mathcal { C } _ { p }$ and $\mathcal { C } _ { d }$ , i.e., $\mathcal { C } = \mathcal { C } _ { p } \cap \mathcal { C } _ { d }$ .
+
+To generate confident pseudo labels for unconfident target data, we select a semantic-nearest confident anchor from $\mathcal { C }$ for each target data via continual similarity searching. To be specific, given an unconfident target data $\mathbf { x } ^ { j }$ , we search a serial of unconfident guiding data consecutively using the distance measure function $d ( \cdot , \cdot )$ in the feature space, until the confident anchor from $\mathcal { C }$ is detected. During each searching process, the previous searched guiding data are not considered in the following iteration. Denote $I$ as the set containing previously searched guiding data, $\mathbf { x } ^ { j c }$ as the searched confident anchor closest to $\mathbf { x } ^ { j }$ . The guiding sample $\mathbf { x } ^ { j ( t ) }$ in the $t$ -th search could be obtained by:
+
+$$
+\mathbf { x } ^ { j ( t ) } = \arg \operatorname* { m i n } _ { \mathbf { x } ^ { j } \in T } d ( \mathbf { f } ( \mathbf { x } ^ { j ( t - 1 ) } ) , \mathbf { f } ( \mathbf { x } ^ { j } ) ) , \mathrm { s u b j e c t } \mathrm { t o } \mathbf { x } ^ { j ( t ) } \neq \mathbf { x } ^ { j ( t - 1 ) } , \mathbf { x } ^ { j ( t ) } \notin I ,
+$$
+
+# Algorithm 1 The Searching Process of Semantic-Nearest Confident Anchor $\mathbf { x } ^ { j c }$ .
+
+1: Input: $\mathbf { x } ^ { j }$ , $\mathcal { C } , t = 1$ ;   
+2: Initialize: $\mathbf { x } ^ { j ( t - 1 ) } = \mathbf { x } ^ { j } , I = \emptyset$ ;   
+3: While $\mathbf { x } ^ { j ( t - 1 ) } \notin \mathcal { C }$ do   
+4: Obtain $\mathbf { x } ^ { j ( t ) }$ via Eq. (6);   
+5: Update $I$ via adding $\mathbf { x } ^ { j ( t ) }$ into $I$ ;   
+6: Update $\mathbf { x } ^ { j ( t - 1 ) }$ via $\mathbf { \bar { x } } ^ { j ( t - 1 ) }  \mathbf { x } ^ { j ( t ) }$ ;   
+7: End   
+8: Obtain $\mathbf { x } ^ { j c }$ via $\mathbf { x } ^ { j c }  \mathbf { x } ^ { j ( t - 1 ) }$   
+9: Return $\mathbf { x } ^ { j c }$ .
+
+# Algorithm 2 The Optimization of Our Model.
+
+1: Input: $\{ h ^ { i } \} _ { i = 1 } ^ { n } , T , I$ $E$ epoches, $B$ batches;   
+2: Initialize: $\kappa _ { 1 } , \kappa _ { 2 }$ , the parameters of $\pmb { \Omega }$ ;   
+3: For $e = 1 , \cdots , E$ do   
+4: Obtain pseudo labels via Eq. (7);   
+5: For $b = 1 , \cdots , B$ do   
+6: Select a mini-batch of target data;   
+7: Update $\{ \mathbf { f } ^ { i } \} _ { i = 1 } ^ { n }$ and $\pmb { \Omega }$ via Eq.(9);   
+8: End   
+9: End   
+10: Return: $\{ h ^ { i } \} _ { i = 1 } ^ { n }$ and $\pmb { \Omega }$ .
+
+where $\mathbf { x } ^ { j ( t - 1 ) }$ is the guiding data in the previous search. The searching process of $\mathbf { x } ^ { j c }$ for the $j$ -th target data is summarized in Algorithm 1. Moreover, we fuse the target data and their corresponding confident anchor in the feature space for feature augmentation, and the synthetic feature is denoted as $\mathbf { f } _ { \mathrm { s y n } } ( \mathbf { x } ^ { j } ) = ( 1 - \omega ) \mathbf { f } ( \mathbf { x } ^ { j } ) + \mathbf { \dot { \omega } } \mathbf { f } ( \mathbf { x } ^ { j c } )$ , where $\omega \in [ 0 , 1 ]$ is the random weight to determine the influence of confident anchor on ${ \bf f } _ { \mathrm { s y n } } ( { \bf x } ^ { j } )$ . Therefore, the confident-anchor-induced pseudo label $\hat { y } ^ { j }$ of the $j$ -th target data $\mathbf { x } ^ { j }$ and the classification loss $\mathcal { L } _ { \mathrm { c l s } }$ of whole target data are formulated as:
+
+$$
+\hat { y } ^ { j } = \arg \operatorname* { m i n } _ { k \in [ K ] } d ( \mathbf { f } _ { \mathrm { s y n } } ( \mathbf { x } ^ { j } ) , \boldsymbol { \xi } _ { k } ) ; \quad \mathcal { L } _ { \mathrm { c l s } } = \frac { 1 } { m } \sum _ { j = 1 } ^ { m } \sum _ { k = 1 } ^ { K } [ - \mathbb { 1 } _ { \hat { y } ^ { j } = k } \log ( h _ { k } ( \mathbf { x } ^ { j } ) ) ] .
+$$
+
+# 5.3 Class-Relationship-Aware Consistency Loss
+
+The inherent relationships between different classes have semantic consistency across domains, regardless of distribution discrepancy. In light of this, aligning class relationships could promote more shared transferable knowledge from source domains towards target adaptation. To achieve this, as depicted in Figure 1, a class-relationship-aware consistency loss $\mathcal { L } _ { \mathrm { c r c } }$ is designed to encourage target data from the same class to be compactly clustered together while preserving the intrinsic inter-class relationships via soft confusion matrix alignment. To be specific, the soft label distribution $s _ { k } ^ { i }$ of the $k$ -th class predicted via the $i$ -the source predictor is formulated as $\begin{array} { r } { s _ { k } ^ { i } = \frac { 1 } { m } \sum _ { j = 1 } ^ { m } [ \mathbb { 1 } _ { \hat { y } ^ { j } = k } \mu _ { i } \bar { h } ^ { i } ( \mathbf { x } ^ { j } ) ] } \end{array}$ . The collection of soft label distributions $\{ s _ { k } ^ { i } \} _ { k = 1 } ^ { K }$ represents a kind of soft con usion matrix associated with a particular domain, encoding inter-class relationships learned by the $i$ (e.g., computers have more similar semantic relationship with desks than horses). Without access to the source data, $\mathcal { L } _ { \mathrm { c r c } }$ aims to align soft confusion matrices from different source predictors:
+
+$$
+\mathcal { L } _ { \mathrm { c r c } } = \frac { 1 } { 2 n ^ { 2 } } \sum _ { i = 1 } ^ { n } \sum _ { i ^ { \prime } = 1 } ^ { n } \sum _ { k = 1 } ^ { K } \big ( \mathrm { K L } ( s _ { k } ^ { i } | | s _ { k } ^ { i ^ { \prime } } ) + \mathrm { K L } ( s _ { k } ^ { i ^ { \prime } } | | s _ { k } ^ { i } ) \big ) ,
+$$
+
+where $\begin{array} { r } { \mathrm { K L } ( p | | q ) = \sum _ { r } p _ { r } \log \frac { p _ { r } } { q _ { r } } } \end{array}$ denotes the Kullback-Leibler (KL) divergence. The complexity of $\mathcal { L } _ { \mathrm { c r c } }$ is not problematic in practice, due to the limited number of source domains.
+
+In summary, the overall optimization objective to optimize $\{ \mathbf { f } ^ { i } \} _ { i = 1 } ^ { n }$ and $\pmb { \Omega }$ is formulated as:
+
+$$
+\begin{array} { r } { \mathcal { L } = \mathcal { L } _ { \mathrm { e n t } } + \mathcal { L } _ { \mathrm { d i v } } + \kappa _ { 1 } \mathcal { L } _ { \mathrm { c l s } } + \kappa _ { 2 } \mathcal { L } _ { \mathrm { c r c } } , } \end{array}
+$$
+
+where $\kappa _ { 1 } , \kappa _ { 2 }$ are the balanced weights. The optimization of our model is presented in Algorithm 2.
+
+# 6 Experiments
+
+# 6.1 Datasets and Baseline Methods
+
+Datasets: Office-31 [22] consists of three representative domains with 31 shared object categories in the office environment, i.e., Amazon (A), Webcam (W) and DSLR (D). Office-Caltech [18] is an extension dataset of Office-31 [22] by adding an additional subset called Caltech-256 (C) on it and extracting 10 common object classes among them. Office-Home [46] is composed of four different domains including Product $( \mathrm { P r } )$ , Clipart (Cl), Art (Ar), and Realworld (Re). Each of these subsets consists of 65 shared object categories. Digits-Five [41] contains five digit recognition subsets including MNIST-M (MM), USPS (UP), MNIST (MT), SVHN (SV) and Synthetic Digits (SY).
+
+Table 1: Comparisons between our model and other competing methods on Office-31 [22] dataset (the left block) and Office-Caltech [18] dataset (the right block).   
+
+<table><tr><td>Methods</td><td>Source Data</td><td>A,D→WA,W→D</td><td></td><td>D,W→A</td><td>Avg.</td><td>A,D,C→WAC,W→DC,D,W→AA,D,W-→C</td><td></td><td></td><td></td><td>Avg.</td></tr><tr><td>Source only [20]</td><td>√</td><td>97.1</td><td>92.0</td><td>51.6</td><td>80.2</td><td>93.5</td><td>94.2</td><td>90.6</td><td>87.5</td><td>91.5</td></tr><tr><td>MDAN [60]</td><td>√</td><td>99.2</td><td>95.4</td><td>55.2</td><td>83.3</td><td>99.4</td><td>98.7</td><td>93.5</td><td>91.6</td><td>95.8</td></tr><tr><td>DCTN [55]</td><td>√</td><td>99.6</td><td>96.9</td><td>54.9</td><td>83.8</td><td>99.3</td><td>99.4</td><td>94.1</td><td>91.3</td><td>96.0</td></tr><tr><td>MSDA [41]</td><td>√</td><td>99.4</td><td>96.2</td><td>55.4</td><td>83.7</td><td>99.5</td><td>99.2</td><td>94.5</td><td>92.2</td><td>96.4</td></tr><tr><td>MDDA [62]</td><td>√</td><td>99.2</td><td>97.1</td><td>56.2</td><td>84.2</td><td>99.3</td><td>99.6</td><td>95.3</td><td>92.3</td><td>96.6</td></tr><tr><td>LtC-MSDA [47]</td><td>√</td><td>99.6</td><td>97.2</td><td>56.9</td><td>84.6</td><td>99.4</td><td>99.7</td><td>93.7</td><td>95.1</td><td>97.0</td></tr><tr><td>Source model only</td><td>X</td><td>95.4</td><td>97.5</td><td>60.2</td><td>84.4</td><td>98.0</td><td>99.5</td><td>96.3</td><td>92.1</td><td>96.5</td></tr><tr><td>BAIT [57]</td><td>X</td><td>98.5</td><td>98.8</td><td>71.1</td><td>89.5</td><td>98.0</td><td>97.5</td><td>97.5</td><td>95.7</td><td>97.2</td></tr><tr><td>PrDA[25]</td><td>X</td><td>93.8</td><td>96.7</td><td>73.2</td><td>87.9</td><td>97.6</td><td>97.1</td><td>97.3</td><td>94.6</td><td>96.7</td></tr><tr><td>SHOT[33]</td><td>X</td><td>94.9</td><td>97.8</td><td>75.0</td><td>89.3</td><td>99.6</td><td>96.8</td><td>95.7</td><td>95.8</td><td>97.0</td></tr><tr><td>MA [30]</td><td>X</td><td>96.1</td><td>97.3</td><td>75.2</td><td>89.5</td><td>99.8</td><td>97.2</td><td>95.7</td><td>95.6</td><td>97.1</td></tr><tr><td>DECISION[1]</td><td>X</td><td>98.4</td><td>99.6</td><td>75.4</td><td>91.1</td><td>99.6</td><td>100.0</td><td>95.9</td><td>95.9</td><td>98.0</td></tr><tr><td>Ours-w/oEnt</td><td>X</td><td>97.5</td><td>99.1</td><td>74.2</td><td>90.3</td><td>99.1</td><td>99.0</td><td>94.5</td><td>96.3</td><td>97.2</td></tr><tr><td>Ours-w/oDiv</td><td>X</td><td>97.2</td><td>98.6</td><td>73.7</td><td>89.8</td><td>98.6</td><td>99.3</td><td>94.1</td><td>95.7</td><td>96.9</td></tr><tr><td>Ours-w/oCls</td><td>X</td><td>96.7</td><td>98.4</td><td>73.0</td><td>89.4</td><td>97.3</td><td>98.4</td><td>93.6</td><td>95.2</td><td>96.1</td></tr><tr><td>Ours-w/oCrc</td><td>X</td><td>98.6</td><td>99.5</td><td>75.4</td><td>91.2</td><td>99.6</td><td>100.0</td><td>95.3</td><td>96.5</td><td>97.9</td></tr><tr><td>Ours</td><td>X</td><td>98.9</td><td>99.8</td><td>75.8</td><td>91.6</td><td>99.8</td><td>100.0</td><td>96.8</td><td>97.1</td><td>98.4</td></tr></table>
+
+![](images/44462ea49f71a5630bf1edd44af7bd7f9f3381e3d4936c9e5bb6400c5b82e80f.jpg)  
+Figure 2: t-SNE [45] visualizations on Office-31 [22] dataset when performing D, $\mathrm { W } { \to } \mathrm { A }$ (a)(b) and A, $\mathrm { W } {  } \mathrm { D }$ (c)(d) domain adaptation tasks.
+
+Baseline Methods: To validate the effectiveness of our model, we conduct comparison experiments with a wide array of baseline methods. Specifically, MDAN [60], DCTN [55], $\mathrm { \dot { M } ^ { 3 } S D A }$ [41], MDDA [62] and LtC-MSDA [47] are traditional representative multi-source domain adaptation methods with access to source data. BAIT [57], PrDA [25], SHOT [33] and MA [30] focuses on unsupervised single-source domain adaptation without access to source data. We compare against the multi-source extensions of [25, 30, 33, 57] by taking an average of target soft predictions from all adapted source models. DECISION [1] combines the source adaptation models with suitable weights automatically for multi-source-free domain adaptation. Furthermore, Source only denotes the performance of evaluation on target data when combining the rest of source data for training, and Source model only represents the average performance over the predictions of all source models.
+
+# 6.2 Experiments on Office-31 and Office-Caltech Datasets
+
+Performance Comparisons: The comparisons between our CAiDA model and other state-of-theart methods on Office-31 [22] and Office-Caltech [18] datasets are presented in Table 1. We have the following conclusions from the results in Table 1: 1) When compared with the multi-source domain adaptation methods [41, 47, 55, 60, 62] that employ source data for training, our proposed model without access to source data significantly outperforms them by a large margin of $1 . 4 \% \sim 8 . 3 \%$ in terms of mean accuracy. It verifies the superiority of our model to tackle multi-source-free domain adaptation. 2) Our model performs better than the multi-source extensions of single-source-free domain adaptation methods [25, 30, 33, 57], which validates the effectiveness of our pseudo label generation process. 3) The performance of our model is better than [1] for all evaluation tasks, since the confident-anchor-induced pseudo label generator and class-relationship-aware consistency loss promote the adaptation performance. Figure 2 shows that our model significantly narrows distribution discrepancy across domains on Office-31 [22] when compared with Source model only.
+
+Ablation Studies: This subsection introduces the effectiveness of each component in our model via ablation studies on Office-31 [22] and Office-Caltech [18] datasets, as shown in Table 1. Oursw/oEnt, Ours-w/oDiv, Ours-w/oCls and Ours-w/oCrc are the abbreviations of training our proposed model without ${ \mathcal { L } } _ { \mathrm { e n t } }$ , ${ \mathcal { L } } _ { \mathrm { d i v } }$ , ${ \mathcal L } _ { \mathrm { c l s } }$ and $\mathcal { L } _ { \mathrm { c r c } }$ , respectively. When any one of component of our model is abandoned, the performance degrades $0 . 4 \% { \sim } 2 . 3 \%$ in terms of average accuracy, which illustrates the rationality and effectiveness of all components to cooperate together. All modules play an indispensable role in improving performance, even though our model has no access to source data.
+
+![](images/e686049d5e90e164361abb1fc5d84dbeef401aa399fdc389df5bf95fa496ae5a.jpg)  
+Figure 3: Qualitative analysis about parameters $\{ \kappa _ { 1 } , \kappa _ { 2 } \}$ (a)(b), weights of domain A (c) and con vergence on Office-Caltech (d), where R in (c) denotes the rest of domains except for the target.
+
+Table 2: Comparisons between our model and other competing methods on Office-Home [46].   
+
+<table><tr><td>Methods</td><td>Source Data</td><td>Ar, Cl,Pr→Re</td><td>Ar,Cl,Re→Pr</td><td>Ar,Pr,Re→Cl</td><td>Cl, Pr,Re→Ar</td><td>Avg.</td></tr><tr><td>Source only [20]</td><td>√</td><td>67.8</td><td>71.3</td><td>51.8</td><td>53.4</td><td>61.1</td></tr><tr><td>MDAN[60]</td><td></td><td>77.3</td><td>77.6</td><td>62.2</td><td>65.4</td><td>70.6</td></tr><tr><td>DCTN [55]</td><td>√</td><td>78.7</td><td>78.3</td><td>63.8</td><td>66.4</td><td>71.8</td></tr><tr><td>MSDA [41]</td><td>√</td><td>79.4</td><td>79.1</td><td>63.5</td><td>67.2</td><td>72.3</td></tr><tr><td>MDDA [62]</td><td></td><td>79.6</td><td>79.5</td><td>62.3</td><td>66.7</td><td>71.0</td></tr><tr><td>LtC-MSDA [47]</td><td>√</td><td>80.1</td><td>79.2</td><td>64.1</td><td>67.4</td><td>72.7</td></tr><tr><td>Source model only</td><td>X</td><td>76.3</td><td>78.8</td><td>50.1</td><td>50.9</td><td>64.0</td></tr><tr><td>BAIT[57]</td><td>X</td><td>77.2</td><td>79.4</td><td>59.6</td><td>71.1</td><td>71.8</td></tr><tr><td>PrDA [25]</td><td>X</td><td>76.8</td><td>79.1</td><td>57.5</td><td>69.3</td><td>70.7</td></tr><tr><td>SHOT[33]</td><td>X</td><td>82.9</td><td>82.8</td><td>59.3</td><td>72.2</td><td>74.3</td></tr><tr><td>MA [30]</td><td>X</td><td>81.7</td><td>82.3</td><td>57.4</td><td>72.5</td><td>73.5</td></tr><tr><td>DECISION[1]</td><td>X</td><td>83.6</td><td>84.4</td><td>59.4</td><td>74.5</td><td>75.5</td></tr><tr><td>Ours-w/oEnt</td><td>X</td><td>82.6</td><td>83.0</td><td>58.7</td><td>74.2</td><td>74.6</td></tr><tr><td>Ours-w/oDiv</td><td>X</td><td>82.1</td><td>82.9</td><td>58.5</td><td>73.8</td><td>74.3</td></tr><tr><td>Ours-w/oCls</td><td>X</td><td>81.4</td><td>82.7</td><td>57.9</td><td>73.1</td><td>73.8</td></tr><tr><td>Ours-w/oCrc</td><td>X</td><td>83.5</td><td>84.4</td><td>59.7</td><td>74.9</td><td>75.6</td></tr><tr><td>Ours</td><td>X</td><td>84.2</td><td>84.7</td><td>60.5</td><td>75.2</td><td>76.2</td></tr></table>
+
+Parameter Investigation: This subsection investigates the effects of hyper-parameters $\kappa _ { 1 }$ in a range of $\{ 0 . 1 , 0 . 3 , 0 . 5 , 0 . \bar { 7 } , 0 . 9 \}$ and $\kappa _ { 2 }$ in a range of $\{ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , 1 \dot { 0 } ^ { - 1 } , 1 \dot  \} \}$ on Office-31 [22] and Office-Caltech [18] datasets, as shown in Figure 3 (a)(b). It validates that our proposed model achieves stable performance over a wide range of hyper-parameters selection. Moreover, the best performance of our proposed model on target domain is obtained when $\kappa _ { 1 } = 0 . 7$ and $\kappa _ { 2 } = 1 0 ^ { - 2 }$ .
+
+Contribution Weights and Convergence Analysis: Figure 3 (c)(d) present the contribution weights of domain A and convergence curves of our proposed model on Office-31 [22] and Office-Caltech [18] datasets. The source-specific transferable perception module could quantify the transferability contributions of complementary information from multiple source domains to target prediction. Furthermore, the accuracy on Office-Caltech [18] dataset converges to a stable value after a few epoches, which demonstrates the convergence effectiveness of our proposed CAiDA model.
+
+# 6.3 Experiments on Office-Home Dataset
+
+Performance Comparisons: As presented in Table 2, we conduct comparison experiments on Office-Home [46] dataset to illustrate the effectiveness of our model. We have the following observations from Table 2: 1) Without access to source data, our model significantly outperforms the representative multi-source domain adaptation methods [41, 47, 55, 60, 62] by $3 . 5 \% { \sim } 5 . 6 \%$ in terms of average accuracy. 2) The confident pseudo label generator encourages our proposed model to perform better than [1], which verifies the superiority of our model for multi-source-free domain adaptation. 3) The performance of our model improves a large margin of $1 . 9 \% \sim 5 . 5 \%$ in terms of mean accuracy, compared with [25, 30, 33, 57]. It validates the efficiency of source-specific transferable perception strategy and class-relationship-aware consistency to narrow distribution discrepancy.
+
+Ablation Studies: As introduced in Table 2, we conduct ablation studies of our model on OfficeHome [46] dataset to illustrate the rationality of all designed modules. When compared with Ours, the performances of Ours-w/oEnt, Ours-w/oDiv, Ours-w/oCls and Ours-w/oCrc degrades $1 . 6 \%$ , $1 . 9 \%$ , $2 . 4 \%$ and $0 . 6 \%$ in terms of average accuracy, respectively. It validates that all designed modules could cooperate well to address MSFDA task. Moreover, a confident-anchor-induced pseudo label generator could reduce the distribution discrepancy via mining confident pseudo labels.
+
+Table 3: Comparisons between our model and other competing methods on Digits-Five [41] dataset, where R denotes the rest of four domains except for the single target domain.   
+
+<table><tr><td>Methods</td><td>Source Data</td><td>R→MM</td><td>R→MT</td><td>R→UP</td><td>R→SV</td><td>R→SY</td><td>Avg.</td></tr><tr><td>Source only [27]</td><td>√</td><td>63.4</td><td>90.5</td><td>88.7</td><td>63.5</td><td>82.4</td><td>77.7</td></tr><tr><td>MDAN[60]</td><td></td><td>69.5</td><td>98.0</td><td>92.4</td><td>69.2</td><td>87.4</td><td>83.3</td></tr><tr><td>DCTN [55]</td><td></td><td>70.5</td><td>96.2</td><td>92.8</td><td>77.6</td><td>86.8</td><td>84.8</td></tr><tr><td>MSDA [41]</td><td></td><td>72.8</td><td>98.4</td><td>96.1</td><td>81.3</td><td>89.6</td><td>87.7</td></tr><tr><td>MDDA [62]</td><td></td><td>78.6</td><td>98.8</td><td>93.9</td><td>79.3</td><td>89.7</td><td>88.1</td></tr><tr><td>LtC-MSDA [47]</td><td>√</td><td>85.6</td><td>99.0</td><td>98.3</td><td>83.2</td><td>93.0</td><td>91.8</td></tr><tr><td>Source model only</td><td>√</td><td>25.2</td><td>90.0</td><td>93.3</td><td>42.8</td><td>77.8</td><td>65.8</td></tr><tr><td>BAIT[57]</td><td>X</td><td>87.6</td><td>96.2</td><td>96.7</td><td>60.6</td><td>90.5</td><td>86.3</td></tr><tr><td>PrDA [25]</td><td>X</td><td>86.2</td><td>95.4</td><td>95.8</td><td>57.4</td><td>84.8</td><td>83.9</td></tr><tr><td>SHOT[33]</td><td>X</td><td>90.4</td><td>98.9</td><td>97.7</td><td>58.3</td><td>83.9</td><td>85.8</td></tr><tr><td>MA[30]</td><td>X</td><td>90.8</td><td>98.4</td><td>98.0</td><td>59.1</td><td>84.5</td><td>86.2</td></tr><tr><td>DECISION [1]</td><td>X</td><td>93.0</td><td>99.2</td><td>97.8</td><td>82.6</td><td>97.5</td><td>94.0</td></tr><tr><td>Ours-w/oEnt</td><td>X</td><td>92.1</td><td>97.3</td><td>96.0</td><td>80.7</td><td>96.3</td><td>92.5</td></tr><tr><td>Ours-w/oDiv</td><td>X</td><td>91.7</td><td>97.0</td><td>96.8</td><td>82.2</td><td>96.5</td><td>92.8</td></tr><tr><td>Ours-w/oCls</td><td>X</td><td>91.3</td><td>96.6</td><td>96.4</td><td>80.5</td><td>95.8</td><td>92.1</td></tr><tr><td>Ours-w/oCrc</td><td>X</td><td>92.8</td><td>98.2</td><td>98.1</td><td>82.8</td><td>97.7</td><td>93.9</td></tr><tr><td>Ours</td><td>X</td><td>93.7</td><td>99.1</td><td>98.6</td><td>83.3</td><td>98.1</td><td>94.6</td></tr></table>
+
+# 6.4 Experiments on Digits-Five Dataset
+
+This subsection presents the ablation studies and comparison experiments between our model and other competing methods on Digits-Five [41] dataset, as introduced in Table 3. Some conclusions are drawn from the presented results in Table 3: 1) Our proposed model performs the best in terms of average accuracy when compared with multi-source domain adaptation methods [1,41,47,55,60,62] and single-source-free adaptation methods [25,30,33,57]. The significant performance improvement demonstrates the effectiveness of our model, even without access to source data. 2) Our model outperforms [25, 30, 33, 57] by $8 . 3 \% \sim 1 0 . 7 \%$ mean accuracy, since it automatically quantifies the contributions of different source domains to target adaptation. When we compare our model with [1, 41, 47, 55, 60, 62], the confident-anchor-induced pseudo label generator and class-relationshipaware consistency loss facilitate the performance improvement by mining confident pseudo labels and aligning soft confusion matrices across domains. 3) The performance degradation in ablation studies (i.e., Ours-w/oEnt, Ours-w/oDiv, Ours-w/oCls and Ours-w/oCrc) validates that all proposed components are designed effectively and reasonably to explore transferable knowledge.
+
+# 7 Conclusion and Future Work
+
+This paper proposes a novel confident-anchor-induced multi-source-free domain adaptation (CAiDA) model to capture transferable knowledge from multiple source domains without access to source data. To be specific, a source-specific transferable perception module is developed to automatically weight the contributions of the transferability of source domains. Meanwhile, we design a confident-anchor-induced pseudo label generator to mine confident pseudo labels for the target domain by establishing a confident anchor group, and develop a class-relationship-aware consistency loss to capture consistent inter-class relationships across domains. Theoretical analysis provides some new perspectives to the highly confident pseudo labeling strategy, and gives theoretical support for MSFDA task under some proper mild assumptions. Extensive experiments illustrate the superiority of our proposed model. In the future, we will extend MSFDA to the multi-label [48, 49] or open-set [13, 37, 53] scenarios and use MSFDA techniques to study pandemic [38, 39].
+
+# Acknowledgments
+
+This work was partially supported by National Nature Science Foundation of China under Grant 62003336; National Postdoctoral Innovative Talents Support Program under Grant BX20200353; Nature Foundation of Liaoning Province of China under Grant 2020-KF-11-01; and Australian Research Council Projects under Grant DP-180103424, DE-190101473, and IC-190100031.
+
+# References
+
+[1] Sk Miraj Ahmed, Dripta S Raychaudhuri, Sujoy Paul, Samet Oymak, and Amit K. Roy-Chowdhury. Unsupervised multi-source domain adaptation without access to source data. In CVPR, pages 10103– 10112, 2021.   
+[2] Zechen Bai, Zhigang Wang, Jian Wang, Di Hu, and Errui Ding. Unsupervised multi-source domain adaptation for person re-identification. In CVPR, pages 12914–12923, 2021.   
+[3] Yogesh Balaji, Swami Sankaranarayanan, and Rama Chellappa. Metareg: Towards domain generalization using meta-regularization. In NeurIPS, volume 31, pages 1006–1016, 2018.   
+[4] Antonin Berthon, Bo Han, Gang Niu, Tongliang Liu, and Masashi Sugiyama. Confidence scores make instance-dependent label-noise learning possible. In ICML, volume 139, pages 825–836, 2021.   
+[5] Konstantinos Bousmalis, George Trigeorgis, Nathan Silberman, Dilip Krishnan, and Dumitru Erhan. Domain separation networks. In NeurIPS, pages 343–351, 2016.   
+[6] John Bridle, Anthony Heading, and David MacKay. Unsupervised classifiers, mutual information and ’phantom targets. In NeurIPS, volume 4, pages 1096–1101, 1992.   
+[7] Jiacheng Cheng, Tongliang Liu, Kotagiri Ramamohanarao, and Dacheng Tao. Learning with bounded instance and label-dependent label noise. In ICML, volume 119, pages 1789–1799, 2020.   
+[8] Haoang Chi, Feng Liu, Wenjing Yang, Long Lan, Tongliang Liu, Bo Han, William K. Cheung, and James T. Kwok. TOHAN: A one-step approach towards few-shot hypothesis adaptation. 2021.   
+[9] Jiahua Dong, Yang Cong, Gan Sun, and Dongdong Hou. Semantic-transferable weakly-supervised endoscopic lesions segmentation. In ICCV, pages 10711–10720, 2019.   
+[10] Jiahua Dong, Yang Cong, Gan Sun, Yuyang Liu, and Xiaowei Xu. Cscl: Critical semantic-consistent learning for unsupervised domain adaptation. In ECCV, volume 12353, pages 745–762, 2020.   
+[11] Jiahua Dong, Yang Cong, Gan Sun, Yunsheng Yang, Xiaowei Xu, and Zhengming Ding. Weaklysupervised cross-domain adaptation for endoscopic lesions segmentation. IEEE Transactions on Circuits and Systems for Video Technology, 31:2020–2033, 2020.   
+[12] Jiahua Dong, Yang Cong, Gan Sun, Bineng Zhong, and Xiaowei Xu. What can be transferred: Unsupervised domain adaptation for endoscopic lesions segmentation. In CVPR, pages 4022–4031, 2020.   
+[13] Zhen Fang, Jie Lu, Anjin Liu, Feng Liu, and Guangquan Zhang. Learning bounds for open-set learning. In ICML, volume 139, pages 3122–3132, 2021.   
+[14] Zhen Fang, Jie Lu, Feng Liu, Junyu Xuan, and Guangquan Zhang. Open set domain adaptation: Theoretical bound and algorithm. IEEE Transactions on Neural Networks and Learning Systems, abs/1907.08375, 2019.   
+[15] Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial training of neural networks. J. Mach. Learn. Res., 17:20962030, 2016.   
+[16] Muhammad Ghifary, W. Bastiaan Kleijn, Mengjie Zhang, David Balduzzi, and Wen Li. Deep reconstruction-classification networks for unsupervised domain adaptation. In ECCV, volume 9908, pages 597–613, 2016.   
+[17] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NeurIPS, volume 27, 2014.   
+[18] Kristen Grauman. Geodesic flow kernel for unsupervised domain adaptation. In CVPR, page 20662073, 2012.   
+[19] Jiang Guo, Darsh Shah, and Regina Barzilay. Multi-source domain adaptation with mixture of experts. In EMNLP, pages 4694–4703, 2018.   
+[20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, pages 770–778, 2016.   
+[21] Judy Hoffman, Mehryar Mohri, and Ningshan Zhang. Algorithms and theory for multiple-source adaptation. In NeurIPS, volume 31, pages 8256–8266, 2018.   
+[22] Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei Efros, and Trevor Darrell. CyCADA: Cycle-consistent adversarial domain adaptation. In ICML, volume 80, pages 1989–1998, 2018.   
+[23] Han-Kai Hsu, Wei-Chih Hung, Hung-Yu Tseng, Chun-Han Yao, Yi-Hsuan Tsai, Maneesh Singh, and Ming-Hsuan Yang. Progressive domain adaptation for object detection. In CVPR Workshops, pages 738–746, 2019.   
+[24] Shatha Jaradat. Deep cross-domain fashion recommendation. In RECSYS, page 407410, 2017.   
+[25] Youngeun Kim, Donghyeon Cho, Priyadarshini Panda, and Sungeun Hong. Progressive domain adaptation from a source pre-trained model. arXiv preprint arXiv:2007.01524.   
+[26] Andreas Krause, Pietro Perona, and Ryan Gomes. Discriminative clustering by regularized information maximization. In NeurIPS, volume 23, pages 775–783, 2010.   
+[27] Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86:2278–2324, 1998.   
+[28] Seungmin Lee, Dongwan Kim, Namil Kim, and Seong-Gyun Jeong. Drop to adapt: Learning discriminative features for unsupervised domain adaptation. In ICCV, pages 91–100, 2019.   
+[29] Keqiuyin Li, Jie Lu, Hua Zuo, and Guangquan Zhang. Multi-source contribution learning for domain adaptation. IEEE Transactions on Neural Networks and Learning Systems, pages 1–15, 2021.   
+[30] Rui Li, Qianfen Jiao, Wenming Cao, Hau-San Wong, and Si Wu. Model adaptation: Unsupervised domain adaptation without source data. In CVPR, pages 9638–9647, 2020.   
+[31] Xuefeng Li, Tongliang Liu, Bo Han, Gang Niu, and Masashi Sugiyama. Provably end-to-end label-noise learning without anchor points. In ICML, volume 139, pages 6403–6413, 2021.   
+[32] Yitong Li, michael Murias, geraldine Dawson, and David E Carlson. Extracting relationships by multidomain matching. In NeurIPS, volume 31, pages 6799–6810, 2018.   
+[33] Jian Liang, Dapeng Hu, and Jiashi Feng. Do we really need to access the source data? source hypothesis transfer for unsupervised domain adaptation. In ICML, pages 6028–6039, 2020.   
+[34] Chuang Lin, Sicheng Zhao, Lei Meng, and Tat-Seng Chua. Multi-source domain adaptation for visual sentiment classification. AAAI, 34:2661–2668, 2020.   
+[35] Feng Liu, Wenkai Xu, Jie Lu, Guangquan Zhang, Arthur Gretton, and Danica J. Sutherland. Learning deep kernels for non-parametric two-sample tests. In ICML, volume 119, pages 6316–6326, 2020.   
+[36] Tongliang Liu and Dacheng Tao. Classification with noisy labels by importance reweighting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 38:447–461, 2016.   
+[37] Yadan Luo, Zijian Wang, Zi Huang, and Mahsa Baktashmotlagh. Progressive graph learning for open-set domain adaptation. In ICML, volume 119, pages 6468–6478, 2020.   
+[38] Qianqian Ma, Yang-Yu Liu, and Alex Olshevsky. Optimal lockdown for pandemic control. arXiv preprint arXiv:2010.12923, 2020.   
+[39] Qianqian Ma, Yang-Yu Liu, and Alex Olshevsky. Optimal vaccine allocation for pandemic stabilization. arXiv preprint arXiv: 2109.04612, 2021.   
+[40] Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh. Multiple source adaptation and the rényi divergence. In UAI, pages 367–374, 2009.   
+[41] Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang, Kate Saenko, and Bo Wang. Moment matching for multi-source domain adaptation. In ICCV, pages 1406–1415, 2019.   
+[42] Zhen Qiu, Yifan Zhang, Hongbin Lin, Shuaicheng Niu, Yanxia Liu, Qing Du, and Mingkui Tan. Sourcefree domain adaptation via avatar prototype generation and adaptation. In IJCAI, pages 2921–2927, 2021.   
+[43] Aravind Rajeswaran, Chelsea Finn, Sham M Kakade, and Sergey Levine. Meta-learning with implicit gradients. In NeurIPS, volume 32, pages 113–124, 2019.   
+[44] Paolo Russo, Tatiana Tommasi, and Barbara Caputo. Towards multi-source adaptive semantic segmentation. In Image Analysis and Processing, pages 292–301, 2019.   
+[45] Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9:2579–2605, 2008.   
+[46] Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. In CVPR, pages 5385–5394, 2017.   
+[47] Hang Wang, Minghao Xu, Bingbing Ni, and Wenjun Zhang. Learning to combine: Knowledge aggregation for multi-source domain adaptation. In ECCV, pages 727–744, 2020.   
+[48] Lichen Wang, Zhengming Ding, and Yun Fu. Generic multi-label annotation via adaptive graph and marginalized augmentation. ACM Transactions on Knowledge Discovery from Data, 16:1–20, 2021.   
+[49] Lichen Wang, Yunyu Liu, Can Qin, Gan Sun, and Yun Fu. Dual relation semi-supervised multi-label learning. In AAAI, volume 34, pages 6227–6234, 2020.   
+[50] Lixu Wang, Shichao Xu, Xiao Wang, and Qi Zhu. Eavesdrop the composition proportion of training labels in federated learning. arXiv preprint arXiv:1910.06044, 2019.   
+[51] Lixu Wang, Shichao Xu, Xiao Wang, and Qi Zhu. Towards class imbalance in federated learning. arXiv preprint arXiv:2008.06217, 2020.   
+[52] Ximei Wang, Ying Jin, Mingsheng Long, Jianmin Wang, and Michael I Jordan. Transferable normalization: Towards improving transferability of deep neural networks. In NeurIPS, volume 32, pages 1951– 1961, 2019.   
+[53] Zijian Wang, Yadan Luo, Ruihong Qiu, Zi Huang, and Mahsa Baktashmotlagh. Learning to diversify for single domain generalization. In ICCV, 2021.   
+[54] Songhua Wu, Xiaobo Xia, Tongliang Liu, Bo Han, Mingming Gong, Nannan Wang, Haifeng Liu, and Gang Niu. Class2simi: A noise reduction perspective on learning with noisy labels. In ICML, volume 139, pages 11285–11295, 2021.   
+[55] Ruijia Xu, Ziliang Chen, Wangmeng Zuo, Junjie Yan, and Liang Lin. Deep cocktail network: Multisource unsupervised domain adaptation with category shift. In CVPR, pages 3964–3973, 2018.   
+[56] Luyu Yang, Yogesh Balaji, Ser-Nam Lim, and Abhinav Shrivastava. Curriculum manager for source selection in multi-source domain adaptation. In ECCV, pages 608–624, 2020.   
+[57] Shiqi Yang, Yaxing Wang, Joost van de Weijer, Luis Herranz, and Shangling Jui. Unsupervised domain adaptation without source data by casting a bait. arXiv preprint arXiv:2010.12427, 2020.   
+[58] Jun Zhang, Weien Zhou, Xianqi Chen, Wen Yao, and Lu Cao. Multisource selective transfer framework in multiobjective optimization problems. IEEE Transactions on Evolutionary Computation, 24:424–438, 2020.   
+[59] Yiyang Zhang, Feng Liu, Zhen Fang, Bo Yuan, Guangquan Zhang, and Jie Lu. Clarinet: A one-step approach towards budget-friendly unsupervised domain adaptation. In IJCAI, pages 2526–2532, 2020.   
+[60] Han Zhao, Shanghang Zhang, Guanhang Wu, José M. F. Moura, Joao P Costeira, and Geoffrey J Gordon. Adversarial multiple source domain adaptation. In NeurIPS, volume 31, pages 8568–8579, 2018.   
+[61] Sicheng Zhao, Bo Li, Xiangyu Yue, Yang Gu, Pengfei Xu, Runbo Hu, Hua Chai, and Kurt Keutzer. Multi-source domain adaptation for semantic segmentation. In NeurIPS, volume 32, pages 7285–7298, 2019.   
+[62] Sicheng Zhao, Guangzhi Wang, Shanghang Zhang, Yang Gu, Yaxian Li, Zhichao Song, Pengfei Xu, Runbo Hu, Hua Chai, and Kurt Keutzer. Multi-source distilling domain adaptation. AAAI, 34:12975– 12983, 2020.   
+[63] Li Zhong, Zhen Fang, Feng Liu, Jie Lu, Bo Yuan, and Guangquan Zhang. How does the combined risk affect the performance of unsupervised domain adaptation approaches? In AAAI, pages 11079–11087, 2021.

parse/train/EAdJEN8xKUl/EAdJEN8xKUl_content_list.json

@@ -0,0 +1,1222 @@
+[
+    {
+        "type": "text",
+        "text": "Confident-Anchor-Induced Multi-Source-Free Domain Adaptation ",
+        "text_level": 1,
+        "bbox": [
+            222,
+            122,
+            779,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Jiahua Dong1, 2∗, Zhen Fang3∗, Anjin Liu3, Gan $\\mathbf { S u n ^ { 1 \\dagger } }$ , Tongliang Liu4 ",
+        "bbox": [
+            246,
+            224,
+            754,
+            241
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences. 2University of Chinese Academy of Sciences. 3DeSI Lab, AAII, University of Technology Sydney. 4TML Lab, University of Sydney. {dongjiahua1995, fzjlyt, sungan1412}@gmail.com, anjin.liu@uts.edu.au, tongliang.liu@sydney.edu.au ",
+        "bbox": [
+            215,
+            242,
+            782,
+            325
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            361,
+            535,
+            377
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised domain adaptation has attracted appealing academic attentions by transferring knowledge from labeled source domain to unlabeled target domain. However, most existing methods assume the source data are drawn from a single domain, which cannot be successfully applied to explore complementarily transferable knowledge from multiple source domains with large distribution discrepancies. Moreover, they require access to source data during training, which are inefficient and unpractical due to privacy preservation and memory storage. To address these challenges, we develop a novel Confident-Anchor-induced multisource-free Domain Adaptation (CAiDA) model, which is a pioneer exploration of knowledge adaptation from multiple source domains to the unlabeled target domain without any source data, but with only pre-trained source models. Specifically, a source-specific transferable perception module is proposed to automatically quantify the contributions of the complementary knowledge transferred from multi-source domains to the target domain. To generate pseudo labels for the target domain without access to the source data, we develop a confident-anchorinduced pseudo label generator by constructing a confident anchor group and assigning each unconfident target sample with a semantic-nearest confident anchor. Furthermore, a class-relationship-aware consistency loss is proposed to preserve consistent inter-class relationships by aligning soft confusion matrices across domains. Theoretical analysis answers why multi-source domains are better than a single source domain, and establishes a novel learning bound to show the effectiveness of exploiting multi-source domains. Experiments on several representative datasets illustrate the superiority of our proposed CAiDA model. The code is available at https://github.com/Learning-group123/CAiDA. ",
+        "bbox": [
+            232,
+            392,
+            766,
+            724
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            750,
+            312,
+            767
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised Domain Adaptation (UDA) [22, 59, 63] captures transferable knowledge from labeled data in a source domain to classify unlabeled data in a target domain. UDA has achieved remarkable successes in many applications, e.g., object detection [23], medical diagnose [9, 12], sentiment analysis [34], etc. Generally, most of existing methods employ adversarial learning [17] to encourage the learned source and target features to be indistinguishable from each other [10, 15], or minimize the distribution discrepancy across domains by matching the statistical moments of distributions [41]. ",
+        "bbox": [
+            174,
+            780,
+            825,
+            864
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, the above-mentioned methods have a strong assumption that the source data are merely drawn from a single domain. Unfortunately, the source data are often collected under different deployed environments (i.e., multiple source domains with large distribution discrepancies) in realworld applications, which makes them difficult to explore complementarily transferable knowledge from the multi-source domains for target prediction. To achieve this, Multi-Source Domain Adaptation (MSDA) [32, 34, 61] is proposed to match the features across domains and then quantify the contributions of source domains [2, 41, 60]. Additionally, [29, 58] aim to weight the source contributions by normalizing the distance similarities between source and target domains. ",
+        "bbox": [
+            174,
+            92,
+            825,
+            202
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, recent MSDA methods [29, 34, 41, 44] require massive labeled source data when adapting source domains to the target domain. This could make them inefficient and unpractical in real-world applications with sensitive information (e.g., medical diagnosis [12] and recommendation system [24]), due to privacy preservation issues, storage and security concerns [50, 51]. To this end, a new challenging and practical problem named Multi-Source-Free Domain Adaptation (MSFDA) is researched, which explores transferable knowledge from multiple source domains to target domain with only pre-trained source models and without access to any source data. The trivial solutions for tackling MSFDA via using existing single-source-free domain adaptation methods [25, 30, 33, 57] are to adapt each source model individually and simply take an average prediction of source models. However, they cannot explore the contributions of the complementary information transferred from different source domains, due to the lack of source data. Therefore, tackling the MSFDA problem is a challenging but rarely-researched task. ",
+        "bbox": [
+            174,
+            208,
+            825,
+            375
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To address the MSFDA problem, we develop a novel Confident-Anchor-induced multi-source-free Domain Adaptation (CAiDA) model, which is a pioneer exploration to capture transferable information from multiple source models to promote target prediction without access to source data. Specifically, a source-specific transferable perception module is designed to calibrate the contributions of the transferability from multiple source domains. We develop a confident-anchor-induced pseudo label generator to mine pseudo labels for the unlabeled target data, by incorporating with the quantified source transferability contributions. We construct a confident anchor group to assign each target sample with a semantic-nearest confident anchor, and perform feature augmentation between them to generate confident target pseudo label. A class-relationship-aware consistency loss is proposed to ensure the semantic consistency of underlying inter-class relationships across domains via the alignment of soft confusion matrices. Furthermore, based on some mild assumptions, theoretical analysis guarantees that multiple source models could help generate more reliable pseudo labels. Our theoretical analysis also provides a novel learning bound for MSFDA, which reveals that multiple source models help achieve a tighter generalization error bound for the target domain. We verify the effectiveness of our proposed model via comparison experiments on benchmark datasets. The main contributions of our work are summarized as follows: ",
+        "bbox": [
+            174,
+            381,
+            826,
+            601
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We develop a novel Confident-Anchor-induced multi-source-free Domain Adaptation (CAiDA) model to explore transferable knowledge from multiple source domains to assist target prediction with pre-trained source models and without access to source data. To our best knowledge, this paper is a pioneer exploration of multi-source-free domain adaptation in the field of transfer learning. ",
+        "bbox": [
+            174,
+            608,
+            825,
+            662
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose a novel MSFDA theory, which shows that multiple source models could improve the possibility of obtaining more reliable pseudo labels under some mild assumptions. Our theoretical analysis also provides a novel generalization bound for MSFDA to show the effect of multiple source models. This novel bound implies a positive answer to the solvability of MSFDA problem. ",
+        "bbox": [
+            174,
+            670,
+            825,
+            726
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• A source-specific transferable perception module and a class-relationship-aware consistency loss are designed to quantify the contributions of the transferability of multiple source domains and ensure the semantic consistency of underlying inter-class relationships across domains, respectively. ",
+        "bbox": [
+            174,
+            732,
+            825,
+            773
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• Based on the confident pseudo labeling strategy in theoretical analysis, a confident-anchor-induced pseudo label generator is proposed to generate pseudo labels for the target domain by establishing a confident anchor group and assigning each target sample with a semantic-nearest confident anchor. ",
+        "bbox": [
+            174,
+            780,
+            825,
+            821
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            848,
+            321,
+            864
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unsupervised Domain Adaptation aims to borrow transferable knowledge from source domain to promote the prediction of the unlabeled target domain. After Hoffman et al. [22] introduce adversarial learning [17] into domain adaptation, enormous adversarial-based methods [8, 11, 15, 59] are proposed to perform feature-level or pixel-level distribution alignment. Besides, some moment matching-based methods [14, 35, 41] focus on matching the distribution statistical moments at different orders to minimize the distribution discrepancy across domains. Furthermore, some researches design the adversarial dropout [28], batch normalization [52] and auxiliary reconstruction tasks [5,16] to narrow the domain discrepancy. Unfortunately, the methods mentioned above assume massive labeled source data are available. This is unpractical due to privacy and security concerns. ",
+        "bbox": [
+            174,
+            883,
+            823,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            189
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Source-Free Domain Adaptation (SFDA) [33] is studied to tackle the above challenge. A common strategy in SFDA methods is to mine the confident pseudo labels for target domain. To this end, [33] uses a self-supervised pseudo labeling strategy, and [25] designs a confidence-based sample filtering method. [42] alleviates the negative transfer brought by noisy pseudo labels through confidence reweighting and regularization. In addition to the pseudo labeling strategy, the model adaptation strategy has also been studied. For example, [30, 57] employ an adversarial learning strategy to perform model adaptation with only pre-trained source models. However, they cannot be applied to tackle the MSFDA problem, due to the distribution discrepancies across different source domains. ",
+        "bbox": [
+            174,
+            194,
+            825,
+            305
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multi-Source Domain Adaptation is extended from vanilla domain adaptation [12, 15, 41] by exploring transferable knowledge from multiple sources. To capture the relationship between different source domains and a given target domain, Guo et al. [19] design a mixture-of-experts model for unsupervised domain adaptation from multiple sources. [56] focuses on determining which source domain is the best for target prediction via dynamic curriculum learning. Some discrepancy-based methods aim to narrow the distribution discrepancy across domains by minimizing different measures such as the Rényi-divergence [21] and maximum mean discrepancy [19]. Moreover, some adversarial-based methods focus on optimizing the $\\mathcal { H }$ -divergence [41, 60], generative adversarial loss [55, 61] and Wasserstein distance [32] to make features from multiple sources indistinguishable for a shared discriminator. [34, 44] perform knowledge adaptation at the pixel-level by replaying multiple source domains. Due to lack of source data, above strategies in MSDA may be invalid and unsuitable to address the challenging MSFDA problem. To this end, Ahmed et al. [1] utilize nearest distance measure to mine target pseudo labels, and weight the predictions from multiple source models for the MSFDA task. It may result in that the generation process has high probability to obtain noisy labels [4, 31, 54] when the strategy is not matched with the target data, while our model could generate confident pseudo labels from two different perspectives, i.e., geometry and probability. ",
+        "bbox": [
+            174,
+            311,
+            826,
+            532
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Problem Setting ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            554,
+            341,
+            571
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathcal { X }$ and $y = [ K ] : = \\{ 1 , . . . , K \\}$ denote the feature space and label space. A domain is a joint distribution $P _ { X Y }$ on $\\mathcal { X } \\times \\mathcal { V }$ . There are $n$ source domains $\\{ P _ { X Y } ^ { i } \\} _ { i = 1 } ^ { n }$ . For any source domain $\\bar { P } _ { X Y } ^ { i }$ a corresponding neural-network-based predictor (model) $h ^ { i } : \\mathcal { X } \\to \\mathbb { R } ^ { K }$ is given. Given a target domain $\\mathsf { \\bar { P } } _ { X Y } ^ { t }$ with unlabeled target data $T = \\{ \\mathbf { x } ^ { j } \\} _ { j = 1 } ^ { m } \\sim P _ { X } ^ { t }$ , i.i.d., the aim of multi-source-free domain adaptation (MSFDA) is to classify the unlabeled target data by utilizing $T$ and $\\{ h ^ { i } \\} _ { i = 1 } ^ { n }$ . ",
+        "bbox": [
+            173,
+            587,
+            825,
+            662
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $\\ell$ denote a non-negative loss function defined over $\\mathbb { R } ^ { K } \\times \\mathbb { R } ^ { K }$ . Given a hypothesis space $\\mathcal { H } \\subset \\{ h \\ : \\ \\mathcal { X } \\ \\to \\ \\overline { { \\mathbb { R } } } ^ { K } \\}$ , we denote $\\mathcal { L } _ { s } ^ { i } ( h ) ~ = ~ \\mathbb { E } _ { ( \\mathbf { x } , y ) \\sim P _ { X Y } ^ { i } } \\ell ( h ( \\mathbf { x } ) , \\Phi ( y ) )$ and $\\begin{array} { r l } { \\mathcal { L } _ { t } ( \\pmb { h } ) } & { { } = } \\end{array}$ $\\mathbb { E } _ { ( \\mathbf { x } , y ) \\sim P _ { X Y } ^ { t } } \\ell ( h ( \\mathbf { x } ) , \\Phi ( y ) )$ as the risks with respect to the $i$ -th source domain and a given target domain, where $\\Phi : \\mathcal { V } \\to \\mathbb { R } ^ { K }$ maps any label $y$ to a corresponding one-hot vector. ",
+        "bbox": [
+            173,
+            667,
+            825,
+            732
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The source predictor $h ^ { i }$ is a vector-valued function, i.e., $\\pmb { h } ^ { i } ( \\mathbf { x } ) = [ h _ { 1 } ^ { i } ( \\mathbf { x } ) , . . . , h _ { K } ^ { i } ( \\mathbf { x } ) ] ^ { \\top }$ , and consists of two basic components: feature extractor $\\mathbf { f } ^ { i } : \\mathcal { X }  \\mathbb { R } ^ { d }$ and classifier $\\mathbf { c } ^ { i } : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { K }$ , where $d$ is the dimension of extracted features. Therefore, $h ^ { i }$ can be rewritten as $\\mathbf { c } ^ { i } \\circ \\mathbf { f } ^ { i }$ . After using softmax as the activation function in the output layer, we have hik ≥ 0 and ∑Kk=1 hik = 1. To ensure that each predictor $h ^ { i }$ is a relatively accurate predictor for the domain $P _ { X Y } ^ { i }$ , we assume that the predictor $h ^ { i }$ is $\\epsilon$ -accurate under $\\ell _ { 1 }$ loss, i.e., $\\mathbb { E } _ { ( \\mathbf { x } , y ) \\sim P _ { X Y } ^ { i } } \\Vert \\pmb { h } ^ { i } ( \\mathbf { x } ) - \\pmb { \\Phi } ( y ) \\Vert _ { \\ell _ { 1 } } < \\epsilon$ , for any $i \\in [ n ]$ . ",
+        "bbox": [
+            173,
+            736,
+            825,
+            832
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 Theoretical Analysis ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            849,
+            377,
+            867
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Without any relations between source domains and target domain, MSFDA cannot be effectively addressed from the theoretical view. To bridge source and target domains, Mansour et al. [40] and ",
+        "bbox": [
+            173,
+            882,
+            823,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Miraj Ahmed et al. [1] assume that $\\begin{array} { r } { P _ { X Y } = \\sum _ { i = 1 } ^ { n } \\lambda _ { i } P _ { X Y } ^ { i } } \\end{array}$ and $P _ { Y | X } ^ { t } = P _ { Y | X } ^ { i } = P _ { Y | X } ^ { j }$ , for any $i , j \\in [ n ]$ , where $\\lambda _ { i } \\geq 0$ and $\\textstyle \\sum _ { i = 1 } ^ { n } \\lambda _ { i } = 1$ . This assumption is very strong and may be unrealistic in many real-world applications. Motivated by the meta learning [43] and domain generalization [3], in this paper, we propose some novel and mild assumptions to address MSFDA problem. ",
+        "bbox": [
+            173,
+            89,
+            826,
+            151
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 1 (Meta Assumption.) $P _ { X Y } ^ { t } , P _ { X Y } ^ { 1 } , . . . , P _ { X Y } ^ { n }$ are drawn (i.i.d.) from a meta distribution $\\mathcal { P }$ , which is defined over a joint distribution space $\\mathcal { P } _ { X Y }$ . ",
+        "bbox": [
+            171,
+            155,
+            825,
+            184
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 2 (Regular Domain.) Let the joint distribution space ce , w $d _ { \\mathrm { T V } } ( \\cdot , \\cdot )$ $P _ { X Y } ^ { t }$ a regular domain, i.e., for any . $\\mathcal { P } _ { X Y }$ is endowed with total $\\sigma > 0$ $\\mathcal { P } ( \\mathcal { N } _ { P _ { X Y } ^ { t } } ^ { \\sigma } ) > 0$ $\\mathcal { N } _ { P _ { X Y } ^ { t } } ^ { \\sigma } = \\{ P : d _ { \\mathrm { T V } } ( P , P _ { X Y } ^ { t } ) \\stackrel {  } { < } \\sigma \\}$ ",
+        "bbox": [
+            173,
+            188,
+            826,
+            234
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It is easy to check, if the meta distribution $\\mathcal { P }$ is discrete or continuous with continuous density function, then the target domain $P _ { X Y } ^ { t }$ drawn by $\\mathcal { P }$ is a regular domain with probability 1. In addition, to weaken the assumption $P _ { Y | X } ^ { t } = P _ { Y | X } ^ { i } = P _ { Y | X } ^ { j }$ , for any $i , j \\in [ n ]$ , our key strategy is to consider the anchor point assumption that is studied in label-noise learning [36]. ",
+        "bbox": [
+            173,
+            237,
+            826,
+            300
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We say a given point $\\mathbf { x }$ is a $\\tau$ -anchor point, if there exists a predictor $h ^ { i }$ with the largest score $h _ { k } ^ { i } ( { \\bf x } )$ , such that $h _ { k } ^ { i } ( { \\bf x } ) - h _ { c } ^ { i } ( { \\bf x } ) \\geq \\tau$ , for any $c \\in [ K ]$ and $c \\neq k$ . $h ^ { i }$ is the $\\tau$ -anchor predictor for $\\mathbf { x }$ . ",
+        "bbox": [
+            173,
+            304,
+            825,
+            337
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Assumption 3 ( $\\tau$ -Anchor Point Assumption.) Given a $\\tau$ -anchor point x, suppose that all $\\tau$ - anchor predictors for $\\mathbf { x }$ are $h ^ { i _ { 1 } } , . . . , h ^ { i _ { l } }$ , then the true label of the $\\tau$ -anchor point $\\mathbf { x }$ is $k _ { : }$ , if $h _ { k } ^ { j } ( { \\bf x } )$ is the largest score among scores $h _ { c } ^ { i _ { 1 } } , . . . , h _ { c } ^ { i _ { l } }$ , for any $c \\in [ K ]$ . ",
+        "bbox": [
+            173,
+            338,
+            826,
+            386
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "When the source predictors are accurate enough, Assumption 3 implies that the source and target conditional distributions are similar in the high confidence region. Hence, it is much weaker than the traditional assumption $P _ { Y | X } ^ { t } = P _ { Y | X } ^ { i } = P _ { Y | X } ^ { j }$ , for $i , j \\in [ n ]$ . $\\tau$ is regarded as a threshold to distinguish which data has highly confident prediction. In general, $\\tau$ is close to 1. ",
+        "bbox": [
+            173,
+            388,
+            825,
+            450
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given any $h ^ { i }$ , it is easy to check that there exists a matrix function $\\mathbf { A } ^ { i } ( \\mathbf { x } ) = [ a _ { k l } ^ { i } ( \\mathbf { x } ) ]$ such that $\\begin{array} { r } { h _ { k } ^ { i } ( \\mathbf { x } ) = \\sum _ { l = 1 } ^ { K } a _ { k l } ^ { i } ( \\mathbf { x } ) P _ { Y | X } ^ { t } ( l | \\mathbf { x } ) } \\end{array}$ with $\\textstyle \\sum _ { k = 1 } ^ { K } a _ { k l } ^ { i } = 1$ . We say the diagonal elements $\\{ a _ { l l } ^ { i } \\} _ { l = 1 } ^ { K }$ as the transfer factor for $h ^ { i }$ and $P _ { Y | X } ^ { t }$ . Generally, $\\mathbf { A } ^ { i } ( \\mathbf { x } ) = [ a _ { k l } ^ { i } ( \\mathbf { x } ) ]$ is not unique, thus the transfer factor may be not unique. The following theorem indicates that Assumption 3 holds if we give proper assumptions for transfer factor. ",
+        "bbox": [
+            173,
+            454,
+            825,
+            535
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 Suppose that the Bayesian label is true label [7]. If there exist transfer factors and $a$ constant $B < K$ such that $\\begin{array} { r } { \\operatorname* { m a x } _ { i \\in [ n ] , l \\in [ K ] } a _ { l l } ^ { i } \\geq B } \\end{array}$ , then Assumption 3 holds with $\\tau > 1 - B / K$ . ",
+        "bbox": [
+            171,
+            537,
+            825,
+            569
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1 provides a theoretical support for Assumption 3 and indicates that when the transfer factors are positive, Assumption 3 always holds with a proper $\\tau$ . To further study the highly confident pseudo labeling strategy, the following theorem provides a lower bound to estimate the number of highly confident pseudo labels, i.e., the number of $\\tau$ -anchor points. ",
+        "bbox": [
+            173,
+            571,
+            826,
+            627
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 Assume Assumptions $I$ and 2 hold and the conditional distribution $P _ { Y \\mid X } ^ { t }$ can be presented as a labeling function, i.e., $P _ { Y | X } ^ { t } ( y | \\mathbf { x } ) = 0 o r 1$ . Given $\\eta > 0$ , if $m \\geq n$ and $( 1 - \\eta ) ( 1 - \\tau ) >$ $\\epsilon + 2 \\sigma + 2 \\sqrt { \\log ( 2 m / \\delta ) / 2 m }$ , then with probability at least $1 - \\delta - ( 1 - \\mathcal { P } ( \\mathcal { N } _ { P _ { X _ { t } Y _ { t } } } ^ { \\sigma } ) ) ^ { n } > 0 .$ , at least ηm target data are $\\tau$ -anchor points, where $\\epsilon$ is the upper bound of the accuracies of source predictors, and $\\sigma$ is introduced in Assumption 2. ",
+        "bbox": [
+            173,
+            631,
+            825,
+            713
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2 indicates that multi-source predictors improve the probability to obtain more $\\tau$ -anchor points. To further understand the effect of multi-source domains, we build a novel learning bound for the MSFDA task. Let $A _ { \\tau }$ be the set consisting of all $\\tau$ -anchor points. Denote the empirical risk $\\widehat { \\mathcal { L } } _ { s } ^ { \\tau } ( h )$ by $\\begin{array} { r } { \\frac { 1 } { \\left| A _ { \\tau } \\cap T \\right| } \\sum _ { \\mathbf x \\in A _ { \\tau } \\cap T } \\ell ( \\pmb h ( \\mathbf x ) , \\pmb \\Phi ( y ) ) } \\end{array}$ , where $y$ is the label of the anchor point $\\mathbf { x }$ . ",
+        "bbox": [
+            173,
+            717,
+            825,
+            779
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 Given Assumption 3 and some assumptions used in Theorem 2, and suppose that the loss $\\ell$ has upper bound $M > 0$ and hypothesis space $\\mathcal { H }$ has finite Natarajan dimension, for $\\eta > 0$ if $m \\geq n$ and $( 1 - \\eta ) ( 1 - \\tau ) > \\epsilon + 2 \\sigma + 2 \\sqrt { \\log ( 2 m / \\delta ) / 2 m }$ , then for any $\\boldsymbol { h } \\in \\mathcal { H }$ and $b \\in ( 0 , 1 )$ , there exists a constant $C ( b , K )$ such that with the probability at least $1 - 2 \\delta - 2 ( 1 - \\mathcal { P } ( \\mathcal { N } _ { P _ { X Y } ^ { t } } ^ { \\sigma } ) ) ^ { n }$ : ",
+        "bbox": [
+            174,
+            782,
+            825,
+            843
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a773dc51cba677764c5f14d9f23ec716e97e1461b4803a5ea7c5f8bfbe41055.jpg",
+        "text": "$$\n\\bigl | \\mathcal { L } _ { t } ( h ) - \\widehat { \\mathcal { L } } _ { s } ^ { \\tau } ( h ) \\bigr | \\leq M C ( b , K ) \\sqrt { \\frac { \\log ( 2 / \\delta ) } { \\eta ^ { 1 - b } m ^ { 1 - b } } } + M \\frac { 2 \\sigma + \\epsilon } { 1 - \\tau - 2 \\sigma - \\epsilon } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            849,
+            722,
+            892
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\epsilon$ is the upper bound of accuracies of source predictors, and $\\sigma$ is introduced in Assumption 2. ",
+        "bbox": [
+            174,
+            897,
+            823,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/0453deba0c69280540053b9de2994f6079c9762a0913511b6a690b06346682d6.jpg",
+        "image_caption": [
+            "Figure 1: Overview of our model, mainly including a source-specific transferable perception strategy to quantify the contributions of the transferability of source domains, a confident-anchorinduced pseudo label generator to generate pseudo labels for target domain, and a class-relationshipaware consistency loss to ensure the semantic consistency of underlying inter-class relationships. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            89,
+            825,
+            232
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 shows that multiple source domains improve the probability to ensure a tighter generalization bound, i.e., Eq. (1) holds. Note that the Natarajan dimension used in Theorem 3 is a bit outdated. However, the Natarajan dimension can be replaced and Theorem 3 can be updated without any technical barriers, if there exists better generalization theory for supervised learning. ",
+        "bbox": [
+            174,
+            301,
+            825,
+            357
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Summary of Theoretical Analysis: The reasons to develop MSFDA are to study the solvability of MSFDA and understand how multi-source predictors benefit the target domain’s classification. By Theorems 2 and 3, we realize that multi-source predictors improve the probability to obtain more highly confident pseudo labels, resulting in a tighter generalization bound. The generalization bound in Theorem 3 gives a positive answer to the solvability of MSFDA. Additionally, Theorems 1 and 2 also imply two interesting and important results: Theorem 1 provides the first theoretical support to the confident pseudo labeling strategy, and Theorem 2 provides the first lower bound of the number of highly confident pseudo labels. As we know, the theoretical results in Theorems 1 and 2 are novel. ",
+        "bbox": [
+            174,
+            363,
+            825,
+            474
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5 The Proposed CAiDA Model ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            492,
+            444,
+            511
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The graphical illustration of our proposed model is depicted in Figure 1. It mainly consists of three significant components: source-specific transferable perception, confident-anchor-induced pseudo label generator and class-relationship-aware consistency loss, which are elaborated as follows. ",
+        "bbox": [
+            174,
+            523,
+            825,
+            565
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.1 Source-Specific Transferable Perception ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            582,
+            491,
+            597
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Generally, in multi-source domain adaptation (MSDA), different source domains have different contributions to improve the performance on target domain [21, 61]. To this end, many previous MSDA methods match the features across different domains, and then quantify the contributions of source domains by taking the average of the trained source predictors [41, 60], or weight the trained source predictors by normalizing the distance similarities [29,58] between source and target domains. However, due to the lack of source data, these methods cannot employ source data to match features and cannot be successfully applied to multi-source-free domain adaptation (MSFDA) tasks. ",
+        "bbox": [
+            174,
+            607,
+            825,
+            704
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Therefore, a source-specific transferable perception module is developed to automatically quantify the contributions of the transferability of source domains, as shown in Figure 1. Specifically, the one-hot encoding vector $\\mathbf { u } ^ { i } \\in \\mathbb { R } ^ { n }$ of the $i$ -th source domain can be considered as the unique domain characterization, which is then employed as network input to quantify the contribution of the transferability. We then concatenate all source domains’ one-hot characterizations together to obtain $\\mathbf { U } = [ \\mathbf { u } ^ { 1 } , \\cdot \\cdot \\overline { { \\mathbf { \\Omega } } } , \\mathbf { u } ^ { n } ] ^ { \\top } \\in \\mathbb { R } ^ { n \\times n }$ . U is then forwarded into a Multi-Layer Perceptron (MLP) network $\\pmb { \\Omega }$ to automatically quantify the contribution of the transferability $\\pmb { \\mu } \\in \\mathbb { R } ^ { n }$ of $n$ source domains: $\\pmb { \\mu } = \\Omega ( \\mathbf { U } ) = \\bar { \\Omega ( [ \\bar { \\mathbf { u } } ^ { 1 } , \\mathbf { u } ^ { 2 } , \\cdots , \\mathbf { u } ^ { n } ] ^ { \\top } ) }$ , such that $\\textstyle \\sum _ { i = 1 } ^ { n } \\mu _ { i } \\ = \\ 1$ , where $\\mu _ { i }$ denotes the quantified contribution of the $i$ -th source domain to the target prediction. ",
+        "bbox": [
+            173,
+            710,
+            825,
+            835
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "When the source data are unavailable, it is difficult to narrow distribution discrepancy across domains, due to lack of any supervised information. Inspired by [33], we freeze the network parameters of classifiers $\\{ \\mathbf { c } ^ { i } \\} _ { i = 1 } ^ { n }$ and solely perform distribution adaptation across domains on feature extractors $\\{ \\mathbf { f } ^ { i } \\} _ { i = 1 } ^ { n }$ via information maximization [26], since $\\{ { \\mathbf { c } } ^ { i } \\} _ { i = 1 } ^ { n }$ contain class distribution information of source domains. However, Liang et al. [33] cannot be effectively applied to multi-source-free domain adaptation scenario, where different source domains have different transferable contributions on target prediction. Therefore, ${ \\mathcal { L } } _ { \\mathrm { e n t } }$ is proposed to minimize conditional entropy of target outputs by incorporating the source-specific transferable perception $\\mu$ : ",
+        "bbox": [
+            174,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            90,
+            825,
+            133
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b29a29f1cb4ca349b9a1e37a33a2d923842b743abb6a6e09bd5e0ea667d3399d.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { e n t } } = - \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } [ \\sum _ { k = 1 } ^ { K } h _ { k } ( \\mathbf { x } ^ { j } ) \\log ( h _ { k } ( \\mathbf { x } ^ { j } ) ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            140,
+            640,
+            185
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $h _ { k } ( \\mathbf { x } ^ { j } )$ is the $k$ -th coordinate value of $\\pmb { h } ( \\mathbf { x } ^ { j } )$ , here $\\begin{array} { r } { \\pmb { h } ( \\mathbf { x } ^ { j } ) = \\sum _ { i = 1 } ^ { n } \\mu _ { i } \\pmb { h } ^ { i } ( \\mathbf { x } ^ { j } ) } \\end{array}$ denotes the combination of source predictions. The larger value of $\\mu _ { i }$ indicates the larger contribution of transferability of the $i$ -th source domain for target adaptation. Unfortunately, our proposed CAiDA model may suffer from the trivial solution of Eq. (2) by predicting all target data as a single class to minimize Eq. (2). To tackle this issue, ${ \\mathcal { L } } _ { \\mathrm { d i v } }$ is designed to consider class prediction diversity by maximizing the entropy of empirical label distribution [6] predicted by different source domains: ",
+        "bbox": [
+            173,
+            191,
+            826,
+            276
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/06cde6936a09d90ee766587ea101cf13684b81aa12158021a37652e949f9a49e.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { d i v } } = \\sum _ { k = 1 } ^ { K } \\hat { p } _ { k } \\log \\hat { p } _ { k } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            282,
+            583,
+            311
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\hat { p } _ { k } = \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } h _ { k } ( \\mathbf { x } ^ { j } ) } \\end{array}$ denotes the mean prediction probability over target data. ",
+        "bbox": [
+            171,
+            318,
+            736,
+            335
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.2 Confident-Anchor-Induced Pseudo Label Generator ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            351,
+            576,
+            366
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Although the minimization of both ${ \\mathcal { L } } _ { \\mathrm { e n t } }$ and ${ \\mathcal { L } } _ { \\mathrm { d i v } }$ promotes the class diversity and knowledge adaptation between multiple unseen source data and target data, it cannot circumvent erroneous label assignment due to the noisy target prediction brought by domain discrepancy. To alleviate this issue, based on Assumption 3, a confident-anchor-induced pseudo label generator is developed to mine confident pseudo labels for target data, as shown in Figure 1. Specifically, based on the $\\tau$ -anchor assumption, we select target data $\\mathbf { x } ^ { j }$ as confident anchor when the maximum category prediction probability $h _ { k ^ { * } } ( \\mathbf { x } ^ { j } )$ is larger than the rest of category prediction probabilities $h _ { k } ( \\mathbf { x } ^ { \\bar { j } } ) \\ \\bar { ( } k \\neq k ^ { * } )$ by a threshold $\\tau _ { p }$ . For each confident anchor, we integrate the features extracted from multiple source extractors together to obtain a probability-based confident anchor group $\\mathcal { C } _ { p }$ : ",
+        "bbox": [
+            173,
+            376,
+            826,
+            502
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9fc9dad600b793b975555970744b6594bcffe121197c541d1d2c02d9b7840486.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { C } _ { p } = \\{ \\mathbf { f } ( \\mathbf { x } ^ { j } ) | h _ { k ^ { * } } ( \\mathbf { x } ^ { j } ) - h _ { k } ( \\mathbf { x } ^ { j } ) \\geq \\tau _ { p } , \\forall k \\neq k ^ { * } , j = 1 , 2 , \\cdots m \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            508,
+            712,
+            529
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\mathbf { f } ( \\mathbf { x } ^ { j } ) = \\sum _ { i = 1 } ^ { n } \\mu _ { i } \\mathbf { f } ^ { i } ( \\mathbf { x } ^ { j } ) } \\end{array}$ denotes the feature of the $j$ -th target data. $\\tau _ { p }$ is defined as the median value of the probability difference between the largest and the second largest probabilities over all target data. Furthermore, inspired by the proposed $\\tau$ -anchor assumption, to circumvent the noisy anchor in $\\mathcal { C } _ { p }$ , we construct a distance-based confident anchor group $\\mathcal { C } _ { d }$ . The feature centroid of the $k$ -th class induced by the $i$ -th source domain for the whole target data is defined as $\\xi _ { k } ^ { i } =$ $\\textstyle \\sum _ { j = 1 } ^ { m } h _ { k } ^ { i } ( \\mathbf { x } ^ { j } ) \\mathbf { f } ^ { i } ( \\mathbf { x } ^ { j } ) / \\textstyle \\sum _ { j = 1 } ^ { m } h _ { k } ^ { i } ( \\mathbf { x } ^ { j } ) . ~ \\{ \\xi _ { k } ^ { i } \\}$ $\\{ \\xi _ { k } ^ { i } \\} _ { i = 1 } ^ { n }$ k are then weighted with source-specific transferability to obtain the feature centroid $\\xi _ { k }$ of the $k$ -th class over all source domains via $\\begin{array} { r } { \\xi _ { k } = \\sum _ { i = 1 } ^ { n } \\mu _ { i } \\xi _ { k } ^ { i } } \\end{array}$ . The distance between the feature of $\\mathbf { x } ^ { j }$ and the $k$ -th feature centroid $\\xi _ { k }$ is denoted as $d ( \\mathbf { f } ( \\mathbf { x } ^ { j } ) , \\pmb { \\xi } _ { k } )$ , where $d ( \\cdot , \\cdot )$ is a distance measure function. When the minimum distance $d ( \\mathbf { f } ( \\mathbf { x } ^ { j } ) , \\xi _ { k ^ { * } } )$ is shorter than the rest of distances $d ( \\mathbf { f } ( \\mathbf { x } ^ { j } ) , \\xi _ { k } ) ( k \\neq k ^ { * } )$ by a threshold $\\tau _ { d }$ , we select the target data $\\mathbf { x } ^ { j }$ into $\\mathcal { C } _ { d }$ : ",
+        "bbox": [
+            173,
+            535,
+            825,
+            683
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb37f58c8f41be0e46e5e95f44ca198ff236e825276a19b971395d82d495be1f.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { C } _ { d } = \\{ \\mathbf { f } ( \\mathbf { x } ^ { j } ) | d ( \\mathbf { f } ( \\mathbf { x } ^ { j } ) , \\xi _ { k } ) - d ( \\mathbf { f } ( \\mathbf { x } ^ { j } ) , \\xi _ { k * } ) \\geq \\tau _ { d } , \\forall k \\neq k ^ { * } , j = 1 , 2 , \\cdot \\cdot \\cdot m \\} , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            689,
+            745,
+            708
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where we set $\\tau _ { d }$ as the median value of the distance difference between the shortest and the second shortest distances over all target data. Therefore, the final confident anchor group $\\mathcal { C }$ is obtained by performing an intersection operation between $\\mathcal { C } _ { p }$ and $\\mathcal { C } _ { d }$ , i.e., $\\mathcal { C } = \\mathcal { C } _ { p } \\cap \\mathcal { C } _ { d }$ . ",
+        "bbox": [
+            173,
+            715,
+            826,
+            760
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To generate confident pseudo labels for unconfident target data, we select a semantic-nearest confident anchor from $\\mathcal { C }$ for each target data via continual similarity searching. To be specific, given an unconfident target data $\\mathbf { x } ^ { j }$ , we search a serial of unconfident guiding data consecutively using the distance measure function $d ( \\cdot , \\cdot )$ in the feature space, until the confident anchor from $\\mathcal { C }$ is detected. During each searching process, the previous searched guiding data are not considered in the following iteration. Denote $I$ as the set containing previously searched guiding data, $\\mathbf { x } ^ { j c }$ as the searched confident anchor closest to $\\mathbf { x } ^ { j }$ . The guiding sample $\\mathbf { x } ^ { j ( t ) }$ in the $t$ -th search could be obtained by: ",
+        "bbox": [
+            173,
+            763,
+            826,
+            864
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/23d431b6a39f94023b86bfa640f74df813e2f5cc45277a9bdc8ecc3408556e72.jpg",
+        "text": "$$\n\\mathbf { x } ^ { j ( t ) } = \\arg \\operatorname* { m i n } _ { \\mathbf { x } ^ { j } \\in T } d ( \\mathbf { f } ( \\mathbf { x } ^ { j ( t - 1 ) } ) , \\mathbf { f } ( \\mathbf { x } ^ { j } ) ) , \\mathrm { s u b j e c t } \\mathrm { t o } \\mathbf { x } ^ { j ( t ) } \\neq \\mathbf { x } ^ { j ( t - 1 ) } , \\mathbf { x } ^ { j ( t ) } \\notin I ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            869,
+            754,
+            897
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 The Searching Process of Semantic-Nearest Confident Anchor $\\mathbf { x } ^ { j c }$ . ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            90,
+            493,
+            118
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1: Input: $\\mathbf { x } ^ { j }$ , $\\mathcal { C } , t = 1$ ;   \n2: Initialize: $\\mathbf { x } ^ { j ( t - 1 ) } = \\mathbf { x } ^ { j } , I = \\emptyset$ ;   \n3: While $\\mathbf { x } ^ { j ( t - 1 ) } \\notin \\mathcal { C }$ do   \n4: Obtain $\\mathbf { x } ^ { j ( t ) }$ via Eq. (6);   \n5: Update $I$ via adding $\\mathbf { x } ^ { j ( t ) }$ into $I$ ;   \n6: Update $\\mathbf { x } ^ { j ( t - 1 ) }$ via $\\mathbf { \\bar { x } } ^ { j ( t - 1 ) }  \\mathbf { x } ^ { j ( t ) }$ ;   \n7: End   \n8: Obtain $\\mathbf { x } ^ { j c }$ via $\\mathbf { x } ^ { j c }  \\mathbf { x } ^ { j ( t - 1 ) }$   \n9: Return $\\mathbf { x } ^ { j c }$ . ",
+        "bbox": [
+            179,
+            121,
+            450,
+            248
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 2 The Optimization of Our Model. ",
+        "text_level": 1,
+        "bbox": [
+            509,
+            90,
+            807,
+            106
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "1: Input: $\\{ h ^ { i } \\} _ { i = 1 } ^ { n } , T , I$ $E$ epoches, $B$ batches;   \n2: Initialize: $\\kappa _ { 1 } , \\kappa _ { 2 }$ , the parameters of $\\pmb { \\Omega }$ ;   \n3: For $e = 1 , \\cdots , E$ do   \n4: Obtain pseudo labels via Eq. (7);   \n5: For $b = 1 , \\cdots , B$ do   \n6: Select a mini-batch of target data;   \n7: Update $\\{ \\mathbf { f } ^ { i } \\} _ { i = 1 } ^ { n }$ and $\\pmb { \\Omega }$ via Eq.(9);   \n8: End   \n9: End   \n10: Return: $\\{ h ^ { i } \\} _ { i = 1 } ^ { n }$ and $\\pmb { \\Omega }$ . ",
+        "bbox": [
+            511,
+            109,
+            816,
+            251
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { x } ^ { j ( t - 1 ) }$ is the guiding data in the previous search. The searching process of $\\mathbf { x } ^ { j c }$ for the $j$ -th target data is summarized in Algorithm 1. Moreover, we fuse the target data and their corresponding confident anchor in the feature space for feature augmentation, and the synthetic feature is denoted as $\\mathbf { f } _ { \\mathrm { s y n } } ( \\mathbf { x } ^ { j } ) = ( 1 - \\omega ) \\mathbf { f } ( \\mathbf { x } ^ { j } ) + \\mathbf { \\dot { \\omega } } \\mathbf { f } ( \\mathbf { x } ^ { j c } )$ , where $\\omega \\in [ 0 , 1 ]$ is the random weight to determine the influence of confident anchor on ${ \\bf f } _ { \\mathrm { s y n } } ( { \\bf x } ^ { j } )$ . Therefore, the confident-anchor-induced pseudo label $\\hat { y } ^ { j }$ of the $j$ -th target data $\\mathbf { x } ^ { j }$ and the classification loss $\\mathcal { L } _ { \\mathrm { c l s } }$ of whole target data are formulated as: ",
+        "bbox": [
+            174,
+            277,
+            825,
+            366
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/900c29a950a3fe70d737f3e6f8fe42f016ba9a1c7dc0dc3ed9557fe0d0218063.jpg",
+        "text": "$$\n\\hat { y } ^ { j } = \\arg \\operatorname* { m i n } _ { k \\in [ K ] } d ( \\mathbf { f } _ { \\mathrm { s y n } } ( \\mathbf { x } ^ { j } ) , \\boldsymbol { \\xi } _ { k } ) ; \\quad \\mathcal { L } _ { \\mathrm { c l s } } = \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } \\sum _ { k = 1 } ^ { K } [ - \\mathbb { 1 } _ { \\hat { y } ^ { j } = k } \\log ( h _ { k } ( \\mathbf { x } ^ { j } ) ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            251,
+            371,
+            745,
+            416
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 Class-Relationship-Aware Consistency Loss ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            429,
+            517,
+            445
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The inherent relationships between different classes have semantic consistency across domains, regardless of distribution discrepancy. In light of this, aligning class relationships could promote more shared transferable knowledge from source domains towards target adaptation. To achieve this, as depicted in Figure 1, a class-relationship-aware consistency loss $\\mathcal { L } _ { \\mathrm { c r c } }$ is designed to encourage target data from the same class to be compactly clustered together while preserving the intrinsic inter-class relationships via soft confusion matrix alignment. To be specific, the soft label distribution $s _ { k } ^ { i }$ of the $k$ -th class predicted via the $i$ -the source predictor is formulated as $\\begin{array} { r } { s _ { k } ^ { i } = \\frac { 1 } { m } \\sum _ { j = 1 } ^ { m } [ \\mathbb { 1 } _ { \\hat { y } ^ { j } = k } \\mu _ { i } \\bar { h } ^ { i } ( \\mathbf { x } ^ { j } ) ] } \\end{array}$ . The collection of soft label distributions $\\{ s _ { k } ^ { i } \\} _ { k = 1 } ^ { K }$ represents a kind of soft con usion matrix associated with a particular domain, encoding inter-class relationships learned by the $i$ (e.g., computers have more similar semantic relationship with desks than horses). Without access to the source data, $\\mathcal { L } _ { \\mathrm { c r c } }$ aims to align soft confusion matrices from different source predictors: ",
+        "bbox": [
+            173,
+            454,
+            826,
+            613
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/69703f2b839c6dac88ac121d80ab7457ca1d97192a2576f23ef9e7ea9258c31e.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { c r c } } = \\frac { 1 } { 2 n ^ { 2 } } \\sum _ { i = 1 } ^ { n } \\sum _ { i ^ { \\prime } = 1 } ^ { n } \\sum _ { k = 1 } ^ { K } \\big ( \\mathrm { K L } ( s _ { k } ^ { i } | | s _ { k } ^ { i ^ { \\prime } } ) + \\mathrm { K L } ( s _ { k } ^ { i ^ { \\prime } } | | s _ { k } ^ { i } ) \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            618,
+            678,
+            662
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\mathrm { K L } ( p | | q ) = \\sum _ { r } p _ { r } \\log \\frac { p _ { r } } { q _ { r } } } \\end{array}$ denotes the Kullback-Leibler (KL) divergence. The complexity of $\\mathcal { L } _ { \\mathrm { c r c } }$ is not problematic in practice, due to the limited number of source domains. ",
+        "bbox": [
+            173,
+            667,
+            826,
+            699
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In summary, the overall optimization objective to optimize $\\{ \\mathbf { f } ^ { i } \\} _ { i = 1 } ^ { n }$ and $\\pmb { \\Omega }$ is formulated as: ",
+        "bbox": [
+            169,
+            703,
+            772,
+            720
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a9630e1f2f889813212a85b4fa7adad26e39f9615705c6406cc00eceaf4b2c36.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } = \\mathcal { L } _ { \\mathrm { e n t } } + \\mathcal { L } _ { \\mathrm { d i v } } + \\kappa _ { 1 } \\mathcal { L } _ { \\mathrm { c l s } } + \\kappa _ { 2 } \\mathcal { L } _ { \\mathrm { c r c } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            727,
+            622,
+            743
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $\\kappa _ { 1 } , \\kappa _ { 2 }$ are the balanced weights. The optimization of our model is presented in Algorithm 2. ",
+        "bbox": [
+            173,
+            751,
+            820,
+            767
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            785,
+            312,
+            803
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.1 Datasets and Baseline Methods ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            815,
+            429,
+            830
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Datasets: Office-31 [22] consists of three representative domains with 31 shared object categories in the office environment, i.e., Amazon (A), Webcam (W) and DSLR (D). Office-Caltech [18] is an extension dataset of Office-31 [22] by adding an additional subset called Caltech-256 (C) on it and extracting 10 common object classes among them. Office-Home [46] is composed of four different domains including Product $( \\mathrm { P r } )$ , Clipart (Cl), Art (Ar), and Realworld (Re). Each of these subsets consists of 65 shared object categories. Digits-Five [41] contains five digit recognition subsets including MNIST-M (MM), USPS (UP), MNIST (MT), SVHN (SV) and Synthetic Digits (SY). ",
+        "bbox": [
+            173,
+            842,
+            825,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1da60061ff2b91146d2abe52d9a256a677e6e00e53db9061c77938b524ce4172.jpg",
+        "table_caption": [
+            "Table 1: Comparisons between our model and other competing methods on Office-31 [22] dataset (the left block) and Office-Caltech [18] dataset (the right block). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Source Data</td><td>A,D→WA,W→D</td><td></td><td>D,W→A</td><td>Avg.</td><td>A,D,C→WAC,W→DC,D,W→AA,D,W-→C</td><td></td><td></td><td></td><td>Avg.</td></tr><tr><td>Source only [20]</td><td>√</td><td>97.1</td><td>92.0</td><td>51.6</td><td>80.2</td><td>93.5</td><td>94.2</td><td>90.6</td><td>87.5</td><td>91.5</td></tr><tr><td>MDAN [60]</td><td>√</td><td>99.2</td><td>95.4</td><td>55.2</td><td>83.3</td><td>99.4</td><td>98.7</td><td>93.5</td><td>91.6</td><td>95.8</td></tr><tr><td>DCTN [55]</td><td>√</td><td>99.6</td><td>96.9</td><td>54.9</td><td>83.8</td><td>99.3</td><td>99.4</td><td>94.1</td><td>91.3</td><td>96.0</td></tr><tr><td>MSDA [41]</td><td>√</td><td>99.4</td><td>96.2</td><td>55.4</td><td>83.7</td><td>99.5</td><td>99.2</td><td>94.5</td><td>92.2</td><td>96.4</td></tr><tr><td>MDDA [62]</td><td>√</td><td>99.2</td><td>97.1</td><td>56.2</td><td>84.2</td><td>99.3</td><td>99.6</td><td>95.3</td><td>92.3</td><td>96.6</td></tr><tr><td>LtC-MSDA [47]</td><td>√</td><td>99.6</td><td>97.2</td><td>56.9</td><td>84.6</td><td>99.4</td><td>99.7</td><td>93.7</td><td>95.1</td><td>97.0</td></tr><tr><td>Source model only</td><td>X</td><td>95.4</td><td>97.5</td><td>60.2</td><td>84.4</td><td>98.0</td><td>99.5</td><td>96.3</td><td>92.1</td><td>96.5</td></tr><tr><td>BAIT [57]</td><td>X</td><td>98.5</td><td>98.8</td><td>71.1</td><td>89.5</td><td>98.0</td><td>97.5</td><td>97.5</td><td>95.7</td><td>97.2</td></tr><tr><td>PrDA[25]</td><td>X</td><td>93.8</td><td>96.7</td><td>73.2</td><td>87.9</td><td>97.6</td><td>97.1</td><td>97.3</td><td>94.6</td><td>96.7</td></tr><tr><td>SHOT[33]</td><td>X</td><td>94.9</td><td>97.8</td><td>75.0</td><td>89.3</td><td>99.6</td><td>96.8</td><td>95.7</td><td>95.8</td><td>97.0</td></tr><tr><td>MA [30]</td><td>X</td><td>96.1</td><td>97.3</td><td>75.2</td><td>89.5</td><td>99.8</td><td>97.2</td><td>95.7</td><td>95.6</td><td>97.1</td></tr><tr><td>DECISION[1]</td><td>X</td><td>98.4</td><td>99.6</td><td>75.4</td><td>91.1</td><td>99.6</td><td>100.0</td><td>95.9</td><td>95.9</td><td>98.0</td></tr><tr><td>Ours-w/oEnt</td><td>X</td><td>97.5</td><td>99.1</td><td>74.2</td><td>90.3</td><td>99.1</td><td>99.0</td><td>94.5</td><td>96.3</td><td>97.2</td></tr><tr><td>Ours-w/oDiv</td><td>X</td><td>97.2</td><td>98.6</td><td>73.7</td><td>89.8</td><td>98.6</td><td>99.3</td><td>94.1</td><td>95.7</td><td>96.9</td></tr><tr><td>Ours-w/oCls</td><td>X</td><td>96.7</td><td>98.4</td><td>73.0</td><td>89.4</td><td>97.3</td><td>98.4</td><td>93.6</td><td>95.2</td><td>96.1</td></tr><tr><td>Ours-w/oCrc</td><td>X</td><td>98.6</td><td>99.5</td><td>75.4</td><td>91.2</td><td>99.6</td><td>100.0</td><td>95.3</td><td>96.5</td><td>97.9</td></tr><tr><td>Ours</td><td>X</td><td>98.9</td><td>99.8</td><td>75.8</td><td>91.6</td><td>99.8</td><td>100.0</td><td>96.8</td><td>97.1</td><td>98.4</td></tr></table>",
+        "bbox": [
+            176,
+            123,
+            825,
+            305
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/44462ea49f71a5630bf1edd44af7bd7f9f3381e3d4936c9e5bb6400c5b82e80f.jpg",
+        "image_caption": [
+            "Figure 2: t-SNE [45] visualizations on Office-31 [22] dataset when performing D, $\\mathrm { W } { \\to } \\mathrm { A }$ (a)(b) and A, $\\mathrm { W } {  } \\mathrm { D }$ (c)(d) domain adaptation tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            309,
+            825,
+            443
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            491,
+            823,
+            520
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Baseline Methods: To validate the effectiveness of our model, we conduct comparison experiments with a wide array of baseline methods. Specifically, MDAN [60], DCTN [55], $\\mathrm { \\dot { M } ^ { 3 } S D A }$ [41], MDDA [62] and LtC-MSDA [47] are traditional representative multi-source domain adaptation methods with access to source data. BAIT [57], PrDA [25], SHOT [33] and MA [30] focuses on unsupervised single-source domain adaptation without access to source data. We compare against the multi-source extensions of [25, 30, 33, 57] by taking an average of target soft predictions from all adapted source models. DECISION [1] combines the source adaptation models with suitable weights automatically for multi-source-free domain adaptation. Furthermore, Source only denotes the performance of evaluation on target data when combining the rest of source data for training, and Source model only represents the average performance over the predictions of all source models. ",
+        "bbox": [
+            174,
+            525,
+            825,
+            664
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.2 Experiments on Office-31 and Office-Caltech Datasets ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            683,
+            588,
+            699
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Performance Comparisons: The comparisons between our CAiDA model and other state-of-theart methods on Office-31 [22] and Office-Caltech [18] datasets are presented in Table 1. We have the following conclusions from the results in Table 1: 1) When compared with the multi-source domain adaptation methods [41, 47, 55, 60, 62] that employ source data for training, our proposed model without access to source data significantly outperforms them by a large margin of $1 . 4 \\% \\sim 8 . 3 \\%$ in terms of mean accuracy. It verifies the superiority of our model to tackle multi-source-free domain adaptation. 2) Our model performs better than the multi-source extensions of single-source-free domain adaptation methods [25, 30, 33, 57], which validates the effectiveness of our pseudo label generation process. 3) The performance of our model is better than [1] for all evaluation tasks, since the confident-anchor-induced pseudo label generator and class-relationship-aware consistency loss promote the adaptation performance. Figure 2 shows that our model significantly narrows distribution discrepancy across domains on Office-31 [22] when compared with Source model only. ",
+        "bbox": [
+            174,
+            710,
+            825,
+            877
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Ablation Studies: This subsection introduces the effectiveness of each component in our model via ablation studies on Office-31 [22] and Office-Caltech [18] datasets, as shown in Table 1. Oursw/oEnt, Ours-w/oDiv, Ours-w/oCls and Ours-w/oCrc are the abbreviations of training our proposed model without ${ \\mathcal { L } } _ { \\mathrm { e n t } }$ , ${ \\mathcal { L } } _ { \\mathrm { d i v } }$ , ${ \\mathcal L } _ { \\mathrm { c l s } }$ and $\\mathcal { L } _ { \\mathrm { c r c } }$ , respectively. When any one of component of our model is abandoned, the performance degrades $0 . 4 \\% { \\sim } 2 . 3 \\%$ in terms of average accuracy, which illustrates the rationality and effectiveness of all components to cooperate together. All modules play an indispensable role in improving performance, even though our model has no access to source data. ",
+        "bbox": [
+            173,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/e686049d5e90e164361abb1fc5d84dbeef401aa399fdc389df5bf95fa496ae5a.jpg",
+        "image_caption": [
+            "Figure 3: Qualitative analysis about parameters $\\{ \\kappa _ { 1 } , \\kappa _ { 2 } \\}$ (a)(b), weights of domain A (c) and con vergence on Office-Caltech (d), where R in (c) denotes the rest of domains except for the target. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            87,
+            825,
+            219
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/02e873804dc64b11eed3b38ade45c86d8a09ace5c6776b0dd2b3c30b6be8684b.jpg",
+        "table_caption": [
+            "Table 2: Comparisons between our model and other competing methods on Office-Home [46]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Source Data</td><td>Ar, Cl,Pr→Re</td><td>Ar,Cl,Re→Pr</td><td>Ar,Pr,Re→Cl</td><td>Cl, Pr,Re→Ar</td><td>Avg.</td></tr><tr><td>Source only [20]</td><td>√</td><td>67.8</td><td>71.3</td><td>51.8</td><td>53.4</td><td>61.1</td></tr><tr><td>MDAN[60]</td><td></td><td>77.3</td><td>77.6</td><td>62.2</td><td>65.4</td><td>70.6</td></tr><tr><td>DCTN [55]</td><td>√</td><td>78.7</td><td>78.3</td><td>63.8</td><td>66.4</td><td>71.8</td></tr><tr><td>MSDA [41]</td><td>√</td><td>79.4</td><td>79.1</td><td>63.5</td><td>67.2</td><td>72.3</td></tr><tr><td>MDDA [62]</td><td></td><td>79.6</td><td>79.5</td><td>62.3</td><td>66.7</td><td>71.0</td></tr><tr><td>LtC-MSDA [47]</td><td>√</td><td>80.1</td><td>79.2</td><td>64.1</td><td>67.4</td><td>72.7</td></tr><tr><td>Source model only</td><td>X</td><td>76.3</td><td>78.8</td><td>50.1</td><td>50.9</td><td>64.0</td></tr><tr><td>BAIT[57]</td><td>X</td><td>77.2</td><td>79.4</td><td>59.6</td><td>71.1</td><td>71.8</td></tr><tr><td>PrDA [25]</td><td>X</td><td>76.8</td><td>79.1</td><td>57.5</td><td>69.3</td><td>70.7</td></tr><tr><td>SHOT[33]</td><td>X</td><td>82.9</td><td>82.8</td><td>59.3</td><td>72.2</td><td>74.3</td></tr><tr><td>MA [30]</td><td>X</td><td>81.7</td><td>82.3</td><td>57.4</td><td>72.5</td><td>73.5</td></tr><tr><td>DECISION[1]</td><td>X</td><td>83.6</td><td>84.4</td><td>59.4</td><td>74.5</td><td>75.5</td></tr><tr><td>Ours-w/oEnt</td><td>X</td><td>82.6</td><td>83.0</td><td>58.7</td><td>74.2</td><td>74.6</td></tr><tr><td>Ours-w/oDiv</td><td>X</td><td>82.1</td><td>82.9</td><td>58.5</td><td>73.8</td><td>74.3</td></tr><tr><td>Ours-w/oCls</td><td>X</td><td>81.4</td><td>82.7</td><td>57.9</td><td>73.1</td><td>73.8</td></tr><tr><td>Ours-w/oCrc</td><td>X</td><td>83.5</td><td>84.4</td><td>59.7</td><td>74.9</td><td>75.6</td></tr><tr><td>Ours</td><td>X</td><td>84.2</td><td>84.7</td><td>60.5</td><td>75.2</td><td>76.2</td></tr></table>",
+        "bbox": [
+            176,
+            275,
+            823,
+            457
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            473,
+            825,
+            542
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Parameter Investigation: This subsection investigates the effects of hyper-parameters $\\kappa _ { 1 }$ in a range of $\\{ 0 . 1 , 0 . 3 , 0 . 5 , 0 . \\bar { 7 } , 0 . 9 \\}$ and $\\kappa _ { 2 }$ in a range of $\\{ 1 0 ^ { - 4 } , 1 0 ^ { - 3 } , 1 0 ^ { - 2 } , 1 \\dot { 0 } ^ { - 1 } , 1 \\dot  \\} \\}$ on Office-31 [22] and Office-Caltech [18] datasets, as shown in Figure 3 (a)(b). It validates that our proposed model achieves stable performance over a wide range of hyper-parameters selection. Moreover, the best performance of our proposed model on target domain is obtained when $\\kappa _ { 1 } = 0 . 7$ and $\\kappa _ { 2 } = 1 0 ^ { - 2 }$ . ",
+        "bbox": [
+            173,
+            549,
+            825,
+            618
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Contribution Weights and Convergence Analysis: Figure 3 (c)(d) present the contribution weights of domain A and convergence curves of our proposed model on Office-31 [22] and Office-Caltech [18] datasets. The source-specific transferable perception module could quantify the transferability contributions of complementary information from multiple source domains to target prediction. Furthermore, the accuracy on Office-Caltech [18] dataset converges to a stable value after a few epoches, which demonstrates the convergence effectiveness of our proposed CAiDA model. ",
+        "bbox": [
+            174,
+            625,
+            825,
+            708
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.3 Experiments on Office-Home Dataset ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            726,
+            472,
+            741
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Performance Comparisons: As presented in Table 2, we conduct comparison experiments on Office-Home [46] dataset to illustrate the effectiveness of our model. We have the following observations from Table 2: 1) Without access to source data, our model significantly outperforms the representative multi-source domain adaptation methods [41, 47, 55, 60, 62] by $3 . 5 \\% { \\sim } 5 . 6 \\%$ in terms of average accuracy. 2) The confident pseudo label generator encourages our proposed model to perform better than [1], which verifies the superiority of our model for multi-source-free domain adaptation. 3) The performance of our model improves a large margin of $1 . 9 \\% \\sim 5 . 5 \\%$ in terms of mean accuracy, compared with [25, 30, 33, 57]. It validates the efficiency of source-specific transferable perception strategy and class-relationship-aware consistency to narrow distribution discrepancy. ",
+        "bbox": [
+            174,
+            752,
+            825,
+            877
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ablation Studies: As introduced in Table 2, we conduct ablation studies of our model on OfficeHome [46] dataset to illustrate the rationality of all designed modules. When compared with Ours, the performances of Ours-w/oEnt, Ours-w/oDiv, Ours-w/oCls and Ours-w/oCrc degrades $1 . 6 \\%$ , $1 . 9 \\%$ , $2 . 4 \\%$ and $0 . 6 \\%$ in terms of average accuracy, respectively. It validates that all designed modules could cooperate well to address MSFDA task. Moreover, a confident-anchor-induced pseudo label generator could reduce the distribution discrepancy via mining confident pseudo labels. ",
+        "bbox": [
+            173,
+            882,
+            823,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/96264562184f6e0452232155affb59318df3070bab1c02d9dc9183a2b26687c5.jpg",
+        "table_caption": [
+            "Table 3: Comparisons between our model and other competing methods on Digits-Five [41] dataset, where R denotes the rest of four domains except for the single target domain. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Source Data</td><td>R→MM</td><td>R→MT</td><td>R→UP</td><td>R→SV</td><td>R→SY</td><td>Avg.</td></tr><tr><td>Source only [27]</td><td>√</td><td>63.4</td><td>90.5</td><td>88.7</td><td>63.5</td><td>82.4</td><td>77.7</td></tr><tr><td>MDAN[60]</td><td></td><td>69.5</td><td>98.0</td><td>92.4</td><td>69.2</td><td>87.4</td><td>83.3</td></tr><tr><td>DCTN [55]</td><td></td><td>70.5</td><td>96.2</td><td>92.8</td><td>77.6</td><td>86.8</td><td>84.8</td></tr><tr><td>MSDA [41]</td><td></td><td>72.8</td><td>98.4</td><td>96.1</td><td>81.3</td><td>89.6</td><td>87.7</td></tr><tr><td>MDDA [62]</td><td></td><td>78.6</td><td>98.8</td><td>93.9</td><td>79.3</td><td>89.7</td><td>88.1</td></tr><tr><td>LtC-MSDA [47]</td><td>√</td><td>85.6</td><td>99.0</td><td>98.3</td><td>83.2</td><td>93.0</td><td>91.8</td></tr><tr><td>Source model only</td><td>√</td><td>25.2</td><td>90.0</td><td>93.3</td><td>42.8</td><td>77.8</td><td>65.8</td></tr><tr><td>BAIT[57]</td><td>X</td><td>87.6</td><td>96.2</td><td>96.7</td><td>60.6</td><td>90.5</td><td>86.3</td></tr><tr><td>PrDA [25]</td><td>X</td><td>86.2</td><td>95.4</td><td>95.8</td><td>57.4</td><td>84.8</td><td>83.9</td></tr><tr><td>SHOT[33]</td><td>X</td><td>90.4</td><td>98.9</td><td>97.7</td><td>58.3</td><td>83.9</td><td>85.8</td></tr><tr><td>MA[30]</td><td>X</td><td>90.8</td><td>98.4</td><td>98.0</td><td>59.1</td><td>84.5</td><td>86.2</td></tr><tr><td>DECISION [1]</td><td>X</td><td>93.0</td><td>99.2</td><td>97.8</td><td>82.6</td><td>97.5</td><td>94.0</td></tr><tr><td>Ours-w/oEnt</td><td>X</td><td>92.1</td><td>97.3</td><td>96.0</td><td>80.7</td><td>96.3</td><td>92.5</td></tr><tr><td>Ours-w/oDiv</td><td>X</td><td>91.7</td><td>97.0</td><td>96.8</td><td>82.2</td><td>96.5</td><td>92.8</td></tr><tr><td>Ours-w/oCls</td><td>X</td><td>91.3</td><td>96.6</td><td>96.4</td><td>80.5</td><td>95.8</td><td>92.1</td></tr><tr><td>Ours-w/oCrc</td><td>X</td><td>92.8</td><td>98.2</td><td>98.1</td><td>82.8</td><td>97.7</td><td>93.9</td></tr><tr><td>Ours</td><td>X</td><td>93.7</td><td>99.1</td><td>98.6</td><td>83.3</td><td>98.1</td><td>94.6</td></tr></table>",
+        "bbox": [
+            176,
+            123,
+            825,
+            325
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            338,
+            826,
+            395
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6.4 Experiments on Digits-Five Dataset ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            410,
+            459,
+            426
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This subsection presents the ablation studies and comparison experiments between our model and other competing methods on Digits-Five [41] dataset, as introduced in Table 3. Some conclusions are drawn from the presented results in Table 3: 1) Our proposed model performs the best in terms of average accuracy when compared with multi-source domain adaptation methods [1,41,47,55,60,62] and single-source-free adaptation methods [25,30,33,57]. The significant performance improvement demonstrates the effectiveness of our model, even without access to source data. 2) Our model outperforms [25, 30, 33, 57] by $8 . 3 \\% \\sim 1 0 . 7 \\%$ mean accuracy, since it automatically quantifies the contributions of different source domains to target adaptation. When we compare our model with [1, 41, 47, 55, 60, 62], the confident-anchor-induced pseudo label generator and class-relationshipaware consistency loss facilitate the performance improvement by mining confident pseudo labels and aligning soft confusion matrices across domains. 3) The performance degradation in ablation studies (i.e., Ours-w/oEnt, Ours-w/oDiv, Ours-w/oCls and Ours-w/oCrc) validates that all proposed components are designed effectively and reasonably to explore transferable knowledge. ",
+        "bbox": [
+            173,
+            436,
+            825,
+            616
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion and Future Work ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            635,
+            449,
+            651
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper proposes a novel confident-anchor-induced multi-source-free domain adaptation (CAiDA) model to capture transferable knowledge from multiple source domains without access to source data. To be specific, a source-specific transferable perception module is developed to automatically weight the contributions of the transferability of source domains. Meanwhile, we design a confident-anchor-induced pseudo label generator to mine confident pseudo labels for the target domain by establishing a confident anchor group, and develop a class-relationship-aware consistency loss to capture consistent inter-class relationships across domains. Theoretical analysis provides some new perspectives to the highly confident pseudo labeling strategy, and gives theoretical support for MSFDA task under some proper mild assumptions. Extensive experiments illustrate the superiority of our proposed model. In the future, we will extend MSFDA to the multi-label [48, 49] or open-set [13, 37, 53] scenarios and use MSFDA techniques to study pandemic [38, 39]. ",
+        "bbox": [
+            174,
+            660,
+            825,
+            811
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            830,
+            328,
+            848
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This work was partially supported by National Nature Science Foundation of China under Grant 62003336; National Postdoctoral Innovative Talents Support Program under Grant BX20200353; Nature Foundation of Liaoning Province of China under Grant 2020-KF-11-01; and Australian Research Council Projects under Grant DP-180103424, DE-190101473, and IC-190100031. ",
+        "bbox": [
+            176,
+            856,
+            825,
+            911
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            267,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "[1] Sk Miraj Ahmed, Dripta S Raychaudhuri, Sujoy Paul, Samet Oymak, and Amit K. Roy-Chowdhury. Unsupervised multi-source domain adaptation without access to source data. In CVPR, pages 10103– 10112, 2021.   \n[2] Zechen Bai, Zhigang Wang, Jian Wang, Di Hu, and Errui Ding. Unsupervised multi-source domain adaptation for person re-identification. In CVPR, pages 12914–12923, 2021.   \n[3] Yogesh Balaji, Swami Sankaranarayanan, and Rama Chellappa. Metareg: Towards domain generalization using meta-regularization. In NeurIPS, volume 31, pages 1006–1016, 2018.   \n[4] Antonin Berthon, Bo Han, Gang Niu, Tongliang Liu, and Masashi Sugiyama. Confidence scores make instance-dependent label-noise learning possible. In ICML, volume 139, pages 825–836, 2021.   \n[5] Konstantinos Bousmalis, George Trigeorgis, Nathan Silberman, Dilip Krishnan, and Dumitru Erhan. Domain separation networks. In NeurIPS, pages 343–351, 2016.   \n[6] John Bridle, Anthony Heading, and David MacKay. Unsupervised classifiers, mutual information and ’phantom targets. In NeurIPS, volume 4, pages 1096–1101, 1992.   \n[7] Jiacheng Cheng, Tongliang Liu, Kotagiri Ramamohanarao, and Dacheng Tao. Learning with bounded instance and label-dependent label noise. In ICML, volume 119, pages 1789–1799, 2020.   \n[8] Haoang Chi, Feng Liu, Wenjing Yang, Long Lan, Tongliang Liu, Bo Han, William K. Cheung, and James T. Kwok. TOHAN: A one-step approach towards few-shot hypothesis adaptation. 2021.   \n[9] Jiahua Dong, Yang Cong, Gan Sun, and Dongdong Hou. Semantic-transferable weakly-supervised endoscopic lesions segmentation. In ICCV, pages 10711–10720, 2019.   \n[10] Jiahua Dong, Yang Cong, Gan Sun, Yuyang Liu, and Xiaowei Xu. Cscl: Critical semantic-consistent learning for unsupervised domain adaptation. In ECCV, volume 12353, pages 745–762, 2020.   \n[11] Jiahua Dong, Yang Cong, Gan Sun, Yunsheng Yang, Xiaowei Xu, and Zhengming Ding. Weaklysupervised cross-domain adaptation for endoscopic lesions segmentation. IEEE Transactions on Circuits and Systems for Video Technology, 31:2020–2033, 2020.   \n[12] Jiahua Dong, Yang Cong, Gan Sun, Bineng Zhong, and Xiaowei Xu. What can be transferred: Unsupervised domain adaptation for endoscopic lesions segmentation. In CVPR, pages 4022–4031, 2020.   \n[13] Zhen Fang, Jie Lu, Anjin Liu, Feng Liu, and Guangquan Zhang. Learning bounds for open-set learning. In ICML, volume 139, pages 3122–3132, 2021.   \n[14] Zhen Fang, Jie Lu, Feng Liu, Junyu Xuan, and Guangquan Zhang. Open set domain adaptation: Theoretical bound and algorithm. IEEE Transactions on Neural Networks and Learning Systems, abs/1907.08375, 2019.   \n[15] Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. Domain-adversarial training of neural networks. J. Mach. Learn. Res., 17:20962030, 2016.   \n[16] Muhammad Ghifary, W. Bastiaan Kleijn, Mengjie Zhang, David Balduzzi, and Wen Li. Deep reconstruction-classification networks for unsupervised domain adaptation. In ECCV, volume 9908, pages 597–613, 2016.   \n[17] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In NeurIPS, volume 27, 2014.   \n[18] Kristen Grauman. Geodesic flow kernel for unsupervised domain adaptation. In CVPR, page 20662073, 2012.   \n[19] Jiang Guo, Darsh Shah, and Regina Barzilay. Multi-source domain adaptation with mixture of experts. In EMNLP, pages 4694–4703, 2018.   \n[20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, pages 770–778, 2016.   \n[21] Judy Hoffman, Mehryar Mohri, and Ningshan Zhang. Algorithms and theory for multiple-source adaptation. In NeurIPS, volume 31, pages 8256–8266, 2018.   \n[22] Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei Efros, and Trevor Darrell. CyCADA: Cycle-consistent adversarial domain adaptation. In ICML, volume 80, pages 1989–1998, 2018.   \n[23] Han-Kai Hsu, Wei-Chih Hung, Hung-Yu Tseng, Chun-Han Yao, Yi-Hsuan Tsai, Maneesh Singh, and Ming-Hsuan Yang. Progressive domain adaptation for object detection. In CVPR Workshops, pages 738–746, 2019.   \n[24] Shatha Jaradat. Deep cross-domain fashion recommendation. In RECSYS, page 407410, 2017.   \n[25] Youngeun Kim, Donghyeon Cho, Priyadarshini Panda, and Sungeun Hong. Progressive domain adaptation from a source pre-trained model. arXiv preprint arXiv:2007.01524.   \n[26] Andreas Krause, Pietro Perona, and Ryan Gomes. Discriminative clustering by regularized information maximization. In NeurIPS, volume 23, pages 775–783, 2010.   \n[27] Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86:2278–2324, 1998.   \n[28] Seungmin Lee, Dongwan Kim, Namil Kim, and Seong-Gyun Jeong. Drop to adapt: Learning discriminative features for unsupervised domain adaptation. In ICCV, pages 91–100, 2019.   \n[29] Keqiuyin Li, Jie Lu, Hua Zuo, and Guangquan Zhang. Multi-source contribution learning for domain adaptation. IEEE Transactions on Neural Networks and Learning Systems, pages 1–15, 2021.   \n[30] Rui Li, Qianfen Jiao, Wenming Cao, Hau-San Wong, and Si Wu. Model adaptation: Unsupervised domain adaptation without source data. In CVPR, pages 9638–9647, 2020.   \n[31] Xuefeng Li, Tongliang Liu, Bo Han, Gang Niu, and Masashi Sugiyama. Provably end-to-end label-noise learning without anchor points. In ICML, volume 139, pages 6403–6413, 2021.   \n[32] Yitong Li, michael Murias, geraldine Dawson, and David E Carlson. Extracting relationships by multidomain matching. In NeurIPS, volume 31, pages 6799–6810, 2018.   \n[33] Jian Liang, Dapeng Hu, and Jiashi Feng. Do we really need to access the source data? source hypothesis transfer for unsupervised domain adaptation. In ICML, pages 6028–6039, 2020.   \n[34] Chuang Lin, Sicheng Zhao, Lei Meng, and Tat-Seng Chua. Multi-source domain adaptation for visual sentiment classification. AAAI, 34:2661–2668, 2020.   \n[35] Feng Liu, Wenkai Xu, Jie Lu, Guangquan Zhang, Arthur Gretton, and Danica J. Sutherland. Learning deep kernels for non-parametric two-sample tests. In ICML, volume 119, pages 6316–6326, 2020.   \n[36] Tongliang Liu and Dacheng Tao. Classification with noisy labels by importance reweighting. IEEE Transactions on Pattern Analysis and Machine Intelligence, 38:447–461, 2016.   \n[37] Yadan Luo, Zijian Wang, Zi Huang, and Mahsa Baktashmotlagh. Progressive graph learning for open-set domain adaptation. In ICML, volume 119, pages 6468–6478, 2020.   \n[38] Qianqian Ma, Yang-Yu Liu, and Alex Olshevsky. Optimal lockdown for pandemic control. arXiv preprint arXiv:2010.12923, 2020.   \n[39] Qianqian Ma, Yang-Yu Liu, and Alex Olshevsky. Optimal vaccine allocation for pandemic stabilization. arXiv preprint arXiv: 2109.04612, 2021.   \n[40] Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh. Multiple source adaptation and the rényi divergence. In UAI, pages 367–374, 2009.   \n[41] Xingchao Peng, Qinxun Bai, Xide Xia, Zijun Huang, Kate Saenko, and Bo Wang. Moment matching for multi-source domain adaptation. In ICCV, pages 1406–1415, 2019.   \n[42] Zhen Qiu, Yifan Zhang, Hongbin Lin, Shuaicheng Niu, Yanxia Liu, Qing Du, and Mingkui Tan. Sourcefree domain adaptation via avatar prototype generation and adaptation. In IJCAI, pages 2921–2927, 2021.   \n[43] Aravind Rajeswaran, Chelsea Finn, Sham M Kakade, and Sergey Levine. Meta-learning with implicit gradients. In NeurIPS, volume 32, pages 113–124, 2019.   \n[44] Paolo Russo, Tatiana Tommasi, and Barbara Caputo. Towards multi-source adaptive semantic segmentation. In Image Analysis and Processing, pages 292–301, 2019.   \n[45] Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9:2579–2605, 2008.   \n[46] Hemanth Venkateswara, Jose Eusebio, Shayok Chakraborty, and Sethuraman Panchanathan. Deep hashing network for unsupervised domain adaptation. In CVPR, pages 5385–5394, 2017.   \n[47] Hang Wang, Minghao Xu, Bingbing Ni, and Wenjun Zhang. Learning to combine: Knowledge aggregation for multi-source domain adaptation. In ECCV, pages 727–744, 2020.   \n[48] Lichen Wang, Zhengming Ding, and Yun Fu. Generic multi-label annotation via adaptive graph and marginalized augmentation. ACM Transactions on Knowledge Discovery from Data, 16:1–20, 2021.   \n[49] Lichen Wang, Yunyu Liu, Can Qin, Gan Sun, and Yun Fu. Dual relation semi-supervised multi-label learning. In AAAI, volume 34, pages 6227–6234, 2020.   \n[50] Lixu Wang, Shichao Xu, Xiao Wang, and Qi Zhu. Eavesdrop the composition proportion of training labels in federated learning. arXiv preprint arXiv:1910.06044, 2019.   \n[51] Lixu Wang, Shichao Xu, Xiao Wang, and Qi Zhu. Towards class imbalance in federated learning. arXiv preprint arXiv:2008.06217, 2020.   \n[52] Ximei Wang, Ying Jin, Mingsheng Long, Jianmin Wang, and Michael I Jordan. Transferable normalization: Towards improving transferability of deep neural networks. In NeurIPS, volume 32, pages 1951– 1961, 2019.   \n[53] Zijian Wang, Yadan Luo, Ruihong Qiu, Zi Huang, and Mahsa Baktashmotlagh. Learning to diversify for single domain generalization. In ICCV, 2021.   \n[54] Songhua Wu, Xiaobo Xia, Tongliang Liu, Bo Han, Mingming Gong, Nannan Wang, Haifeng Liu, and Gang Niu. Class2simi: A noise reduction perspective on learning with noisy labels. In ICML, volume 139, pages 11285–11295, 2021.   \n[55] Ruijia Xu, Ziliang Chen, Wangmeng Zuo, Junjie Yan, and Liang Lin. Deep cocktail network: Multisource unsupervised domain adaptation with category shift. In CVPR, pages 3964–3973, 2018.   \n[56] Luyu Yang, Yogesh Balaji, Ser-Nam Lim, and Abhinav Shrivastava. Curriculum manager for source selection in multi-source domain adaptation. In ECCV, pages 608–624, 2020.   \n[57] Shiqi Yang, Yaxing Wang, Joost van de Weijer, Luis Herranz, and Shangling Jui. Unsupervised domain adaptation without source data by casting a bait. arXiv preprint arXiv:2010.12427, 2020.   \n[58] Jun Zhang, Weien Zhou, Xianqi Chen, Wen Yao, and Lu Cao. Multisource selective transfer framework in multiobjective optimization problems. IEEE Transactions on Evolutionary Computation, 24:424–438, 2020.   \n[59] Yiyang Zhang, Feng Liu, Zhen Fang, Bo Yuan, Guangquan Zhang, and Jie Lu. Clarinet: A one-step approach towards budget-friendly unsupervised domain adaptation. In IJCAI, pages 2526–2532, 2020.   \n[60] Han Zhao, Shanghang Zhang, Guanhang Wu, José M. F. Moura, Joao P Costeira, and Geoffrey J Gordon. Adversarial multiple source domain adaptation. In NeurIPS, volume 31, pages 8568–8579, 2018.   \n[61] Sicheng Zhao, Bo Li, Xiangyu Yue, Yang Gu, Pengfei Xu, Runbo Hu, Hua Chai, and Kurt Keutzer. Multi-source domain adaptation for semantic segmentation. In NeurIPS, volume 32, pages 7285–7298, 2019.   \n[62] Sicheng Zhao, Guangzhi Wang, Shanghang Zhang, Yang Gu, Yaxian Li, Zhichao Song, Pengfei Xu, Runbo Hu, Hua Chai, and Kurt Keutzer. Multi-source distilling domain adaptation. AAAI, 34:12975– 12983, 2020.   \n[63] Li Zhong, Zhen Fang, Feng Liu, Jie Lu, Bo Yuan, and Guangquan Zhang. How does the combined risk affect the performance of unsupervised domain adaptation approaches? In AAAI, pages 11079–11087, 2021. ",
+        "bbox": [
+            171,
+            121,
+            828,
+            911
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            92,
+            830,
+            912
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            92,
+            828,
+            526
+        ],
+        "page_idx": 12
+    }
+]

parse/train/H1Yp-j1Cb/H1Yp-j1Cb.md

@@ -0,0 +1,634 @@
+# AN ONLINE LEARNING APPROACH TO GENERATIVEADVERSARIAL NETWORKS
+
+Paulina Grnarova, Kfir Y. Levy, Aurelien Lucchi, Thomas Hofmann, Andreas Krause   
+ETH Zürich   
+{paulina.grnarova,yehuda.levy,aurelien.lucchi,thomas.hofmann}@inf.ethz.ch   
+krausea@ethz.ch
+
+# ABSTRACT
+
+We consider the problem of training generative models with a Generative Adversarial Network (GAN). Although GANs can accurately model complex distributions, they are known to be difficult to train due to instabilities caused by a difficult minimax optimization problem. In this paper, we view the problem of training GANs as finding a mixed strategy in a zero-sum game. Building on ideas from online learning we propose a novel training method named CHEKHOV GAN 1. On the theory side, we show that our method provably converges to an equilibrium for semi-shallow GAN architectures, i.e. architectures where the discriminator is a one-layer network and the generator is arbitrary. On the practical side, we develop an efficient heuristic guided by our theoretical results, which we apply to commonly used deep GAN architectures. On several real-world tasks our approach exhibits improved stability and performance compared to standard GAN training.
+
+# 1 INTRODUCTION
+
+A recent trend in generative models is to use a deep neural network as a generator. Two notable approaches are variational auto-encoders (VAE) (Kingma & Welling, 2013; Rezende et al., 2014) as well as Generative Adversarial Networks (GAN) (Goodfellow et al., 2014). Unlike VAEs, the GAN approach offers a way to circumvent log-likelihood-based estimation and it also typically produces visually sharper samples (Goodfellow et al., 2014). The goal of the generator network is to generate samples that are indistinguishable from real samples, where indistinguishability is measured by an additional discriminative model. This creates an adversarial game setting where one pits a generator against a discriminator.
+
+Let us denote the data distribution by $p _ { \mathrm { d a t a } } ( \mathbf { x } )$ and the model distribution by $p _ { \mathbf { u } } ( \mathbf { x } )$ . A probabilistic discriminator is denoted by $h _ { \mathbf { v } } : \mathbf { x } \to [ 0 ; 1 ]$ and a generator by $G _ { \mathbf { u } } : \mathbf { z } \to \mathbf { x }$ . The GAN objective is:
+
+$$
+\operatorname* { m i n } _ { \mathbf { u } } \operatorname* { m a x } _ { \mathbf { v } } M ( \mathbf { u } , \mathbf { v } ) = \frac { 1 } { 2 } \mathbb { E } _ { \mathbf { x } \sim p _ { \mathrm { d a t a } } } \log h _ { \mathbf { v } } ( \mathbf { x } ) + \frac { 1 } { 2 } \mathbb { E } _ { \mathbf { z } \sim p _ { \mathbf { z } } } \log ( 1 - h _ { \mathbf { v } } ( G _ { \mathbf { u } } ( \mathbf { z } ) ) ) ~ .
+$$
+
+Each of the two players (generator/discriminator) tries to optimize their own objective, which is exactly balanced by the loss of the other player, thus yielding a two-player zero-sum minimax game. Standard GAN approaches aim at finding a pure Nash Equilibrium by using traditional gradient-based techniques to minimize each player’s cost in an alternating fashion. However, an update made by one player can repeatedly undo the progress made by the other one, without ever converging.
+
+In general, alternating gradient descent fails to converge even for very simple games as shown by Salimans et al. (2016). In the setting of GANs, one of the central open issues is this nonconvergence problem, which in practice leads to oscillations between different kinds of generated samples (Metz et al., 2016).
+
+While standard GAN methods seek to find pure minimax strategies, we propose to consider mixed strategies, which allows us to leverage online learning algorithms for mixed strategies in large games.
+
+![](images/44f305483c32abe020914c9284827a02ad4c4024c38600ba6909b7b4fb2bbb53.jpg)  
+Figure 1: Three types of GAN architectures. Left: shallow. Middle: semi-shallow. Right: deep.
+
+Building on the approach of Freund & Schapire (1999), we propose a novel training algorithm for GANs that we call CHEKHOV GAN .
+
+The standard GAN training method is not guaranteed to converge for general GAN architectures. Nevertheless, it does converge for shallow ones 2, i.e. for GAN architectures which consist of a single layer network as a discriminator, and a generator with one hidden layer (see Fig. 1(a)). Unfortunately, shallow GANs are very different from the deep GANs (Fig. 1(c)) which are used in practice, and ideally one would hope to understand models which are more similar to deep architectures.
+
+In this paper we make a step forward by considering semi-shallow GANs, an intermediate architecture where the generator is any arbitrary network and the discriminator consists of a single layer (Fig. 1(b)). Our contributions are:
+
+(1) We show that finding a Mixed Nash Equilibrium (MNE or simply equilibrium) gives rise to useful generators and discriminators.   
+(2) We provide an algorithm that provably converges to an equilibrium for semi-shallow architectures. (3) Guided by our theoretical results we devise a new GAN training algorithm that is applicable to standard deep GAN architectures.
+
+We first discuss the benefits of pursuing a mixed equilibrium. Based on results from game theory we show that by reaching an equilibrium we obtain useful (mixed) generator and discriminator. In the context of GANs, "usefulness" means that the mixed generator provided by the equilibrium solution will “fool" any adversary at least as good as any single generator (similar results hold for the discriminator).
+
+On the theory side, we show that GANs with semi-shallow architectures induce semi-concave games, i.e., games which are concave with respect to the max player, but need not have a special structure with respect to the min player. Then we show that in such games players may efficiently invoke regret minimization procedures in order to find an equilibrium; this in turn gives rise to a way of finding an equilibrium in semi-shallow GANs. To the best of our knowledge, this result is novel in the context of GANs and might also find uses in other scenarios where such structure may arise. We would like to emphasize that this is a significant step from a theoretical point of view as the standard approach to training GANs is only known to be theoretically sound for convex-concave games, which correspond to shallow-networks (though it is still used heuristically to train deep GAN architectures).
+
+On the practical side, we develop an efficient heuristic guided by our theoretical results, which we apply to commonly used deep GAN architectures. We provide experimental results demonstrating that our approach exhibits better empirical stability compared to the vanilla GAN and generates more diverse samples, while retaining the same level of visual quality.
+
+In Section 2, we briefly review necessary notions from online learning and zero-sum games. We then present our approach and its theoretical guarantees in Section 3, and our practical algorithm is presented in Section 4. Lastly, we present empirical results on standard benchmarks in Section 5.
+
+# 2 BACKGROUND & RELATED WORK
+
+# 2.1 GANS
+
+GAN Objectives: The classical way to learn a generative model consists of minimizing a divergence function between a parametrized model distribution $p _ { \mathbf { u } } ( \mathbf { x } )$ and the true data distribution $p _ { \mathrm { d a t a } } ( \mathbf { x } )$ The original GAN approach by Goodfellow et al. (2014) is for example known to be related to optimizing the Jensen-Shannon divergence. This was later generalized by Nowozin et al. (2016) who described a broader family of GAN objectives stemming from $f$ -divergences. A different popular type of GAN objectives is the family of Integral Probability Metrics (Müller, 1997), such as the kernel MMD (Gretton et al., 2012; Li et al., 2015) or the Wasserstein metric (Arjovsky & Bottou, 2017). All of these divergence measures yield a minimax objective.
+
+Training methods for GANs: In order to solve the minimax objective in Eq. 1, Goodfellow et al. (2014) suggested an approach that alternatively minimizes over u and v using mini-batch stochastic gradient descent. This approach can be shown to converge only when the updates are made in function space. In practice, this condition is not met - since this procedure works in the parameter space - and many issues arise during training (Arjovsky & Bottou, 2017; Radford et al., 2015), thus requiring careful initialization and proper regularization as well as other tricks (Metz et al., 2016; Pfau & Vinyals, 2016; Radford et al., 2015; Salimans et al., 2016). Even so, several problems are still commonly observed including a phenomena where the generator oscillates, without ever converging to a fixed point, or mode collapse when the generator maps many latent codes $z$ to the same point, thus failing to produce diverse samples.
+
+The closest work related to our approach is that of Arora et al. (2017) who showed the existence of an approximate mixed equilibrium with certain generalization properties; yet without providing a constructive way to find such equilibria. Instead, they advocate the use of mixed strategies, and suggest to do so by using the exponentiated gradient algorithm of Kivinen & Warmuth (1997). The work of Tolstikhin et al. (2017) also uses a similar mixture approach based on boosting. Other works have studied the problem of equilibrium and stabilization of GANs, often relying on the use of an auto-encoder as discriminator (Berthelot et al., 2017) or jointly with the GAN models (Che et al., 2016). In this work, we focus on providing convergence guarantees to a mixed equilibrium (definition in Section 3.2) using a technique from online optimization that relies on the players’ past actions.
+
+# 2.2 ONLINE LEARNING
+
+Online learning is a sequential decision making framework in which a player aims at minimizing a cumulative loss function revealed to her sequentially. The source of the loss functions may be arbitrary or even adversarial, and the player seeks to provide worst case guarantees on her performance. Formally, this framework can be described as a repeated game of $T$ rounds between a player $\mathcal { P } _ { 1 }$ and an adversary $\mathcal { P } _ { 2 }$ . At each round $t \in [ T ]$ :
+
+1. The player $( \mathcal { P } _ { 1 } )$ chooses a point $\mathbf { u } _ { t } \in \mathcal { K }$ according to some algorithm $\mathcal { A }$   
+2. The adversary $( \mathcal { P } _ { 2 } )$ chooses a loss function $f _ { t } \in \mathcal { F }$   
+3. The player $( \mathcal { P } _ { 1 } )$ suffers a loss $f _ { t } ( { \mathbf { u } } _ { t } )$ , and the loss function $f _ { t } ( \cdot )$ is revealed to her.
+
+The adversary is usually limited to choosing losses from a structured class of objectives $\mathcal { F }$ , most commonly linear/convex losses. Also, the decision set $\kappa$ is often assumed to be convex. The performance of the player’s strategy is measured by the regret, defined as,
+
+$$
+\mathrm { R e g r e t } _ { T } ^ { A } ( f _ { 1 } , \dots , f _ { T } ) = \sum _ { t = 1 } ^ { T } f _ { t } ( \mathbf { u } _ { t } ) - \operatorname* { m i n } _ { \mathbf { u } ^ { * } \in \mathcal { K } } \sum _ { t = 1 } ^ { T } f _ { t } ( \mathbf { u } ^ { * } ) \ .
+$$
+
+Thus, the regret measures the cumulative loss of the player compared to the loss of the best fixed decision in hindsight. A player aims at minimizing her regret, and we are interested in no-regret strategies for which players ensure regret which is sublinear in $T$ for any loss sequence 3.
+
+While there are several no-regret strategies, many of them may be seen as instantiations of the Follow-the-Regularized-Leader (FTRL) algorithm where
+
+$$
+\mathbf { u } _ { t } = \arg \operatorname* { m i n } _ { \mathbf { u } \in \mathcal { K } } \sum _ { \tau = 1 } ^ { t - 1 } f _ { \tau } ( \mathbf { u } ) + \eta _ { t } ^ { - 1 } R ( \mathbf { u } ) \qquad ( \mathbf { F T R L } )
+$$
+
+FTRL takes the accumulated loss observed up to time $t$ and then chooses the point in $\kappa$ that minimizes the accumulated loss plus a regularization term $\eta _ { t } ^ { - 1 } R ( \mathbf { u } )$ . The regularization term prevents the player from abruptly changing her decisions between consecutive rounds4. This property is often crucial to obtaining no-regret guarantees. Note that FTRL is not always guaranteed to yield no-regret, and is mainly known to provide such guarantees in the setting where losses are linear/convex (Hazan et al., 2016; Shalev-Shwartz et al., 2012).
+
+# 2.3 ZERO-SUM GAMES
+
+Consider two players, $\mathcal { P } _ { 1 } , \mathcal { P } _ { 2 }$ , which may choose pure decisions among the sets $\kappa _ { 1 }$ and $\displaystyle { { \cal { K } } _ { 2 } }$ , respectively. A zero-sum game is defined by a function $M : \mathcal { K } _ { 1 } \times \mathcal { K } _ { 2 } \mapsto \mathbb { R }$ which sets the utilities of the players. Concretely, upon choosing a pure strategy $( \mathbf { u } , \mathbf { v } ) \in \mathcal { K } _ { 1 } \times \mathcal { K } _ { 2 }$ the utility of $\mathcal { P } _ { 1 }$ is $- M ( \mathbf { u } , \mathbf { v } )$ , while the utility of $\mathcal { P } _ { 2 }$ is $M ( \mathbf { u } , \mathbf { v } )$ . The goal of either $\mathcal { P } _ { 1 } / \mathcal { P } _ { 2 }$ is to maximize their worst case utilities; thus,
+
+$$
+\operatorname* { m i n } _ { \mathbf { u } \in K _ { 1 } } \operatorname* { m a x } _ { \mathbf { v } \in K _ { 2 } } M ( \mathbf { u } , \mathbf { v } ) \quad ( \mathbf { G o a l o f } \mathcal { P } _ { 1 } ) , \qquad \& \quad \operatorname* { m a x } _ { \mathbf { v } \in K _ { 2 } } \operatorname* { m i n } _ { \mathbf { u } \in K _ { 1 } } M ( \mathbf { u } , \mathbf { v } ) \quad ( \mathbf { G o a l o f } \mathcal { P } _ { 2 } )
+$$
+
+This definition of a game makes sense if there exists a point $( \mathbf { u } ^ { * } , \mathbf { v } ^ { * } )$ , such that neither $\mathcal { P } _ { 1 }$ nor $\mathcal { P } _ { 2 }$ may increase their utility by unilateral deviation. Such a point $( \mathbf { u } ^ { * } , \mathbf { v } ^ { * } )$ is called a Pure Nash Equilibrium, which is formally defined as a point which satisfies the following conditions:
+
+$$
+M ( \mathbf { u } ^ { * } , \mathbf { v } ^ { * } ) \leq \operatorname* { m i n } _ { \mathbf { u } \in K _ { 1 } } M ( \mathbf { u } , \mathbf { v } ^ { * } ) , \ \& \ M ( \mathbf { u } ^ { * } , \mathbf { v } ^ { * } ) \geq \operatorname* { m a x } _ { \mathbf { v } \in K _ { 2 } } M ( \mathbf { u } ^ { * } , \mathbf { v } ) .
+$$
+
+While a pure Nash equilibrium does not always exist, the pioneering work of Nash et al. (1950) established that there always exists a Mixed Nash Equilibrium (MNE or simply equilibrium), i.e., there always exist two distributions $\mathcal { D } _ { 1 } , \mathcal { D } _ { 2 }$ such that,
+
+$$
+\boldsymbol { \mathfrak { L } } _ { ( \mathbf { u } , \mathbf { v } ) \sim \mathcal { D } _ { 1 } \times \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] \leq \operatorname* { m i n } _ { \mathbf { u } \in \mathcal { K } _ { 1 } } \mathbb { E } _ { \mathbf { v } \sim \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] , \ \& \ \mathbb { E } _ { ( \mathbf { u } , \mathbf { v } ) \sim \mathcal { D } _ { 1 } \times \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] \geq \operatorname* { m a x } _ { \mathbf { v } \in \mathcal { K } _ { 2 } } \mathbb { E } _ { \mathbf { u } \sim \mathcal { D } _ { 1 } } [ M ( \mathbf { u } , \mathbf { v } ) ] .
+$$
+
+Finding an exact MNE might be computationally hard, and we are usually satisfied with finding an approximate MNE. This is defined below,
+
+Definition 1. Let $\varepsilon > 0$ . Two distributions $\mathcal { D } _ { 1 } , \mathcal { D } _ { 2 }$ are called $\varepsilon$ -MNE if the following holds,
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { ( \mathbf { u } , \mathbf { v } ) \sim \mathcal { D } _ { 1 } \times \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] \leq \underset { \mathbf { u } \in \mathcal { K } _ { 1 } } { \operatorname* { m i n } } \mathbb { E } _ { \mathbf { v } \sim \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] + \varepsilon , } \\ & { \mathbb { E } _ { ( \mathbf { u } , \mathbf { v } ) \sim \mathcal { D } _ { 1 } \times \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] \geq \underset { \mathbf { v } \in \mathcal { K } _ { 2 } } { \operatorname* { m a x } } \mathbb { E } _ { \mathbf { u } \sim \mathcal { D } _ { 1 } } [ M ( \mathbf { u } , \mathbf { v } ) ] - \varepsilon . } \end{array}
+$$
+
+Terminology: In the sequel when we discuss zero-sum games, we shall sometimes use the GAN terminology, relating the min player $\mathcal { P } _ { 1 }$ as the generator, and the max player $\mathcal { P } _ { 2 }$ , as the discriminator.
+
+No-Regret & Zero-sum Games: In zero-sum games, no-regret algorithms may be used to find an approximate MNE. Unfortunately, computationally tractable no-regret algorithms do not always exist. An exception is the setting when $M$ is convex-concave. In this case, the players may invoke the powerful no-regret methods from online convex optimization to (approximately) solve the game. This seminal idea was introduced in Freund & Schapire (1999), where it was demonstrated how to√ invoke no-regret algorithms during $T$ rounds to obtain an approximation guarantee of $\varepsilon = O ( 1 / { \sqrt { T } } )$ in zero-sum matrix games. This was later improved by Daskalakis et al. (2015), and Rakhlin $\&$ Sridharan (2013), demonstrating a guarantee of $\varepsilon = O ( \log T / T )$ . The result that we are about to present builds on the scheme of Freund & Schapire (1999).
+
+# 3 FINDING EQUILIBRIUM IN GANS
+
+Why Mixed Equilibrium? In this work, our ultimate goal is to efficiently find an approximate MNE for the game. However, in GANs, we are usually interested in designing good generators, and one might ask whether finding an equilibrium serves this cause better than solving the minimax problem, i.e., finding $\mathbf { u } \in \mathrm { a r g m i n } _ { \mathbf { u } \in \mathcal { K } _ { 1 } } \operatorname* { m a x } _ { \mathbf { v } \in \mathcal { K } _ { 2 } } M ( \mathbf { u } , \mathbf { v } )$ . Interestingly, the minimax value of the equilibrium generator is always smaller than the minimax value of any pure strategy. Actually, the equilibrium strategy of the generator might be much better. This benefit of finding an equilibrium can be demonstrated on the following simple zero-sum game. Consider the following paper-rock-scissors game, i.e. a zero-sum game with the minimax objective
+
+$$
+\operatorname* { m i n } _ { i \in \{ 1 , 2 , 3 \} } \operatorname* { m a x } _ { j \in \{ 1 , 2 , 3 \} } M ( i , j ) { \mathrm { ~ ; ~ w h e r e ~ } } M = { \left[ \begin{array} { l l l } { 0 } & { - 1 } & { 1 } \\ { 1 } & { 0 } & { - 1 } \\ { - 1 } & { 1 } & { 0 } \end{array} \right] } .
+$$
+
+Solving for the minimax objective yields a pure strategy with a minimax value of 1; conversely, the equilibrium strategy of the min player is a uniform distribution over actions; and its minimax value is 0. Thus, finding an equilibrium by allowing mixed strategies implies a smaller minimax value, and as we show in the Section 3.3 this is true in general. In the context of GANs, this result means that the mixed generator provided by the equilibrium solution will “fool" any adversary at least as good as any single generator. Similarly, the mixed discriminator provided by the equilibrium solution will discern any generator at least as good as any single discriminator.
+
+The rest of this section presents a method that efficiently finds an equilibrium for semi-shallow GANs (see Fig. 1(b)). Such architectures do not induce a convex-concave game, and therefore the result of Freund & Schapire (1999) does not directly apply. Nevertheless, we show that semi-shallow GANs imply a game structure which gives rise to an efficient procedure for finding an equilibrium. In Sec. 3.1 we show that semi-shallow GANs define games with a property that we denote as semiconcave. Later, Sec. 3.2 provides an algorithm with provable guarantees for such games. Finally, in Section 3.3 we show that the minimax objective of the generator’s equilibrium strategy is optimal with respect to the minimax objective.
+
+# 3.1 SEMI-SHALLOW GANS
+
+Semi-shallow GANs do not lead to a convex-concave game. Nonetheless, here we show that for an appropriate choice of the activation function they induce a game that is concave with respect to the discriminator. Later, in Sec. 3.2, we show that this property allows to efficiently find an equilibrium.
+
+Proposition 1. Consider the GAN objective in Eq. (1) and assume that the discriminator is a singlelayer network with a sigmoid activation function, meaning $h _ { \mathbf { v } } ( \mathbf { x } ) = 1 / ( 1 + \exp ( - \mathbf { v } ^ { \top } \mathbf { x } ) )$ , where $\mathbf { v } \in \mathbb { R } ^ { n }$ . Then the GAN objective is concave in v.
+
+Note that the above is not restricted to the sigmoid activation function, and it also holds for other choices of the activation function 5.
+
+# 3.2 SEMI-CONCAVE ZERO-SUM GAMES
+
+Here we discuss the setting of zero-sum games (see Eq. (4)) which are semi-concave. Formally a game, $M$ , is semi-concave if for any fixed $\mathbf { u } _ { 0 } \in \mathcal { K } _ { 1 }$ the function $g ( \mathbf { v } ) : = M ( \mathbf { u } _ { 0 } , \mathbf { v } )$ is concave in $\mathbf { v }$ . Algorithm 1 presents our method for semi-concave games. This algorithm is an instantiation of the scheme derived by Freund & Schapire (1999), with specific choices of the online algorithms $\ A _ { 1 } , A _ { 2 }$ , used by the players. Note that both $A _ { 1 } , A _ { 2 }$ are two different instances of the FTRL approach presented in Eq. (3).
+
+Let us discuss Algorithm 1 and then present its guarantees. First note that each player calculates a sequence of $T$ points based on an online algorithm $\boldsymbol { A } _ { 1 } / \boldsymbol { A } _ { 2 }$ . Interestingly, the sequence of (loss/reward) functions given to the online algorithm is based on the game objective $M$ , and also on the decisions made by the other player. For example, the loss sequence that $\mathcal { P } _ { 1 }$ receives is $\{ f _ { t } ( \mathbf { u } ) : = M ( \mathbf { u } , \mathbf { v } _ { t } ) \} _ { t \in [ T ] }$ . After $T$ rounds we end up with two mixed strategies $\mathcal { D } _ { 1 } , \mathcal { D } _ { 2 }$ , each being a uniform distribution over the respective online decisions $\{ \mathbf { u } _ { t } \} _ { t \in [ T ] }$ , $\{ \mathbf { v } _ { t } \} _ { t \in [ T ] }$ . Note that the first decision points $\mathbf { u } _ { 1 } , \mathbf { v } _ { 1 }$ are set by $A _ { 1 } , A _ { 2 }$ before encountering any (loss/reward) function, and the dummy functions $f _ { 0 } ( { \mathbf { u } } ) = 0 , g _ { 0 } ( { \mathbf { v } } ) = 0$ are only introduced in order to simplify the exposition. Since $\mathcal { P } _ { 1 }$ ’s goal is to minimize, it is natural to think of the $f _ { t }$ ’s as loss functions, and measure the guarantees of $\boldsymbol { \mathcal { A } } _ { 1 }$ according to the regret as defined in Equation (2). Analogously, since $\mathcal { P } _ { 2 }$ ’s goal is to maximize, it is natural to think of the $g _ { t }$ ’s as reward functions, and measure the guarantees of $\boldsymbol { A } _ { 2 }$ according to the following appropriate definition of regret, $\begin{array} { r } { \mathrm { R e g r e t } _ { T } ^ { A _ { 2 } } = \operatorname* { m a x } _ { { \mathbf { v } } ^ { * } \in K _ { 2 } } \sum _ { t = 1 } ^ { T } g _ { t } ( { \mathbf { v } } ^ { * } ) - \sum _ { t = 1 } ^ { T } g _ { t } ( \bar { \mathbf { v } _ { t } } ) . } \end{array}$ .
+
+<table><tr><td>Algorithm1 CHEKHOv GAN</td><td></td></tr><tr><td>Input: #steps T,Game objective M(·,·) fort=1...T do</td><td></td></tr><tr><td>Calculate: t-1</td><td>t-1</td></tr><tr><td>∑fr(u)&amp; (Alg.A2)Vt ← argmax) (Alg. Ai）ut ←argmin u∈K1 T=0</td><td>VT Vgr(v)Tv &gt; /v||²2 2m0 VEK2 T=0</td></tr><tr><td>Update:</td><td>ft(-)=M(·,Vt) &amp; gt(·)=M(ut,·)</td></tr><tr><td>end for Output mixed strategies: D1 ~ Uni{u1,...,ur}, D2 ~ Uni{V1,..., VT}.</td><td></td></tr></table>
+
+The following theorem presents our guarantees for semi-concave games:
+
+Theorem 1. Let $\displaystyle { \mathcal { K } } _ { 2 }$ be a convex set. Also, let $M$ be a semi-concave zero-sum game, and assume $M$ is $L$ -Lipschitz continuous. Then upon invoking Alg. 1 for √ $T$ steps it outputs mixed strategies $( \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } )$ that are $\varepsilon$ -MNE, where $\varepsilon = O ( 1 / { \sqrt { T } } )$ .
+
+The most important point to note is that the accuracy of the approximation $\varepsilon$ improves as the number of iterations $T$ grows. This lets us obtain an arbitrarily good approximation for a large enough $T$ . As mentioned before, both $\ A _ { 1 } , A _ { 2 }$ are two different instances of the FTRL approach presented in Eq. (3). Concretely, Alg. $\boldsymbol { A } _ { 1 }$ is in fact follow-the-leader (FTL), i.e., FTRL without regularization. Alg. $\boldsymbol { A } _ { 2 }$ also uses the FTRL scheme. Yet, instead of the original reward functions, $g _ { t } ( \cdot )$ , it utilizes linear approximations $\tilde { g } _ { t } ( \mathbf { v } ) = \nabla g _ { t } ( \mathbf { v } _ { t } ) ^ { \top } \mathbf { v }$ . Also note the use of the (minus) square $\ell _ { 2 }$ norm as regularization6. The $\eta _ { 0 }$ parameter depends on the Lipschitz constant of $M$ as well as on the diameter of $\displaystyle { \mathcal { K } } _ { 2 }$ defined as, $d _ { 2 } : = \operatorname* { m a x } _ { \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } \in \mathcal { K } _ { 2 } } \| \mathbf { v } _ { 1 } - \mathbf { v } _ { 2 } \|$ . Concretely, $\eta _ { 0 } = d _ { 2 } / \sqrt { 2 } L$ .
+
+Next we provide a short proof sketch for Thm. 1. The full proof appears in Appendix A.
+
+Proof sketch. The proof makes use of a theorem due to Freund & Schapire (1999) which shows that if both $\mathcal { A } _ { 1 }$ and $\boldsymbol { A } _ { 2 }$ ensure no-regret then it implies convergence to an approximate MNE. Since the game is concave with respect to $\mathcal { P } _ { 2 }$ , it is well known that the FTRL version $\boldsymbol { A } _ { 2 }$ appearing in Thm. 1 is a no-regret strategy (see e.g. Hazan et al. (2016)). The challenge is therefore to show that $\mathcal { A } _ { 1 }$ is also a no-regret strategy. This is non-trivial, especially for semi-concave games that do not necessarily have any special structure with respect to the generator 7. However, the loss sequence received by the generator is not arbitrary, but rather it follows a special sequence based on the choices of the discriminator, $\{ f _ { t } ( \cdot ) = M ( \cdot , \mathbf { \bar { v } } _ { t } ) \} _ { t }$ . In the case of semi-concave games, the sequence of discriminator decisions, $\{ \mathbf { v } _ { t } \} _ { t }$ has a special property which “stabilizes" the loss sequence $\{ f _ { t } \} _ { t }$ , which in turn enables us to establish no-regret for $\boldsymbol { A } _ { 1 }$ . □
+
+Remark: Note that Alg. $\mathcal { A } _ { 1 }$ in Thm. 1 assumes the availability of an oracle that can efficiently find a global minimum for the FTL objective, $\textstyle \sum _ { \tau = 0 } ^ { t - 1 } f _ { \tau } ( \mathbf { u } )$ . This involves a minimization over a sum of generative networks. Therefore, our result may be seen as a reduction from the problem of finding an equilibrium to an offline optimization problem. This reduction is not trivial, especially in light of the negative results of Hazan & Koren (2016), which imply that in the general case finding an equilibrium is hard, even with such an efficient offline optimization oracle at hand. Thus, our result enables to take advantage of progress made in supervised deep learning in order to efficiently find an equilibrium for GANs.
+
+# 3.3 MINIMAX VALUE OF EQUILIBRIUM STRATEGY
+
+In GANs we are mainly interested in ensuring the performance of the generator (resp. discriminator) with respect to the minimax (resp. maximin) objective. Let $( \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } )$ be the pair of mixed strategies that Algorithm 1 outputs. Note that the minimax value of $\mathcal { D } _ { 1 }$ might be considerably smaller than the pure minimax value, as is shown in the example regarding the paper-rock-scissors game (see Sec. 3). The next lemma shows that the mixed strategy $\mathcal { D } _ { 1 }$ is always (approximately) better with respect to the pure minimax value (see proof in appendix B.2)
+
+Lemma 1. The mixed strategy $\mathcal { D } _ { 1 }$ that Algorithm $^ { l }$ outputs is $\varepsilon$ -optimal with respect to the minimax value, i.e.,
+
+$$
+\operatorname* { m a x } _ { \mathbf { v } \in K _ { 2 } } \mathbb { E } _ { \mathbf { u } \sim \mathcal { D } _ { 1 } } [ M ( \mathbf { u } , \mathbf { v } ) ] \leq \operatorname* { m i n } _ { \mathbf { u } \in K _ { 1 } } \operatorname* { m a x } _ { \mathbf { v } \in K _ { 2 } } M ( \mathbf { u } , \mathbf { v } ) + \varepsilon
+$$
+
+where $\varepsilon$ here is equal to the one defined in Thm. 2.
+
+Analogous result hold for $\mathcal { D } _ { 2 }$ with respect to the pure maximin objective.
+
+# 4 PRACTICAL CHEKHOV GAN ALGORITHM FOR DEEP ARCHITECTURES
+
+# Algorithm 2 Practical CHEKHOV GAN
+
+Input: #steps $T$ , Game objective $M ( \cdot , \cdot )$ , number of past states $K$ , spacing $m$ Initialize: Set loss/reward $f _ { 0 } ( \cdot ) = 0 , g _ { 0 } ( \cdot ) = 0$ , initialize queues $\mathcal { Q } _ { 1 }$ .insert $( f _ { 0 } )$ , $\mathcal { Q } _ { 2 }$ .insert(g0) for $t = 1 \dots T$ do
+
+Update generator and discriminator based on a mini-batch of noise samples and data samples:
+
+$$
+\mathbf { u } _ { t + 1 }  \mathbf { u } _ { t } - \eta _ { t } \cdot \nabla _ { \mathbf { u } _ { t } } ( { \frac { 1 } { | { \mathcal { Q } } _ { 1 } | } } \sum _ { f \in { \mathcal { Q } } _ { 1 } } f ( \mathbf { u } ) + { \frac { C } { { \sqrt { t } } } } \| \mathbf { u } \| ^ { 2 } ) \ \& \ \mathbf { v } _ { t + 1 }  \mathbf { v } _ { t } - \eta _ { t } \cdot \nabla _ { \mathbf { v } _ { t } } ( { \frac { 1 } { | { \mathcal { Q } } _ { 2 } | } } \sum _ { g \in { \mathcal { Q } } _ { 2 } } g ( \mathbf { v } ) - { \frac { C } { { \sqrt { t } } } } t \| \mathbf { u } \| ^ { 2 } ) .
+$$
+
+Calculate: $f _ { t } ( \cdot ) = M ( \cdot , \mathbf { v } _ { t } )$ & $g _ { t } ( \cdot ) = M ( \mathbf { u } _ { t } , \cdot )$
+
+$$
+\mathcal { D } _ { 1 } \sim \operatorname { U n i } \{ \mathbf { u } _ { 1 } , \dots , \mathbf { u } _ { K } \in \mathcal { Q } _ { 1 } \} , \mathcal { D } _ { 2 } \sim \operatorname { U n i } \{ \mathbf { v } _ { 1 } , \dots , \mathbf { v } _ { K } \in \mathcal { Q } _ { 2 } \} .
+$$
+
+In Section 3 we described a method (Alg. 1) which provably reaches an equilibrium for semi-shallow GANs. This method considers the whole history of generators and discriminators in making a decision at each round, which contrasts with the standard GAN training method that only considers the last generator and discriminator. Another difference is that our method outputs a mixed model (i.e., generator and discriminator) rather than a single model.
+
+Building on the theoretical approach introduced in Section 3, we now present a practical method (Alg. 2) which can be efficiently applied to train common deep GAN architectures. Algorithm 2 combines the ideas of (a) considering the history of generators and discriminators at each update, and (b) outputting a mixed strategy, while only requiring an access to gradient information which can be efficiently obtained by running back-propagation. Next we discuss Alg. 2 in more details and highlight the differences compared to the theoretical approach:
+
+(i) We use the FTRL objective (Eq. (3)) for both players. Note that Alg. $\mathcal { A } _ { 1 }$ appearing in Thm 1 uses FTRL with linear approximations, which is only appropriate for semi-concave games.
+
+(ii) As calculating the global minimizer of the FTRL objective is impractical, we instead update the weights based on the gradients of the FTRL objective. This can be done by using traditional optimization techniques such as SGD or Adam. Thus the update at each round depends on the gradients of the past generators and discriminators. This differs from the standard GAN training which only employs the gradient of the last generator and discriminator.
+
+(iii) The full FTRL algorithm requires saving the entire history of past generators/discriminators, which is computationally intractable. We find it sufficient to maintain a summary of the history using a small number of representative models. In order to capture a diverse subset of the history, we keep a queue $\mathcal { Q }$ containing $K : = | \mathcal { Q } |$ states (models). The spacing between consecutive models is determined by the following heuristic: every $m$ update steps we remove the oldest model in the queue and add the current one. The number of steps between switches, $m$ , can be set as a constant, but our experiments revealed it is more effective to keep $m$ small at the beginning and increase its value as the number of rounds increases. We hypothesize that as the training progresses and the individual models become more discriminative, we should switch the models at a lower rate, keeping them more spaced out. The pseudo-code and a detailed description of the algorithm appears in the Appendix.
+
+Intuition. In practice GANs commonly exhibit a non-convergent behavior. As a consequence, the generator oscillates between generating different modes from the target distribution. This is hypothesized to be due to the differences of the minimax and maximin solutions of the game (Goodfellow, 2016). If the order of the min and max operations switch, the minimization with respect to the generator’s parameters is performed in the inner loop. This causes the generator to map every latent code to one or very few points for which the discriminator believes are likely. As simultaneous gradient descent updates do not clearly prioritize any specific ordering of minimax or maximin, in practice we often obtain results that resemble the latter.
+
+In contrast, CHEKHOV GAN takes advantage of the history of the player’s actions which yields better gradient information. Intuitively, the generator is updated such that it fools the past discriminators. In order to do so, the generator has to spread its mass more fairly according to the true data distribution. The discriminator can no longer simply learn to put low probability on the few modes of generated samples, which causes oscillations.
+
+The mode collapse problems of GANs is also closely related to the phenomenon of catastrophic forgetting (Seff et al., 2017). When GANs are trained sequentially on samples coming from different modes, the discriminator tends to forget the previous modes it has learned about. This leads to having generated samples that focus only on the last or most prominent modes. By introducing a history of samples from previous generators, the discriminator is less likely to forget the part of the space that it has already learned.
+
+Fig. 2 illustrates the mode collapse problem. The data consists of a mixture of 7 Gaussians with different sampling probabilities whose centers are aligned in a circle. As two modes have higher probabilities and are seen more frequently, they attract the gradients towards them and cause mode collapse and forgetting. Chekhov GAN manages to recover the true data distribution in this case as well, unlike vanilla GANs.
+
+![](images/53f87b8331ee91a528a46197f7e6f2f06cb454627e7d564f6c0590459b7e1398.jpg)  
+Figure 2: Mode Collapse on a Gaussian Mixture. We show heat maps of the generator distribution over time, as well as the target data distribution in the last column. Standard GAN updates (top row) cause mode collapse, whereas CHEKHOV GAN using $K = 1 0$ past steps (bottom row) spreads its mass over all the modes of the target distribution.
+
+# 5 EXPERIMENTAL RESULTS
+
+We now demonstrate that CHEKHOV GAN yields improved stability and sample diversity. To do so, we use a comparable number of datasets and baselines as standard GAN approaches, e.g. Metz et al. (2016); Arjovsky & Bottou (2017). We test our method on models where the traditional GAN training has difficulties converging and engages in a behavior of mode collapse. We also perform experiments on harder tasks using the DCGAN architecture (Radford et al., 2015) which is commonly used in the literature. Note that the DCGAN architecture, when trained using standard techniques, still suffers from instabilities and mode collapse (Nagarajan & Kolter, 2017; Roth et al., 2017). We here demonstrate that CHEKHOV GAN reduces mode dropping while retaining high visual sample quality. For all of the experiments, we generate from the newest generator only. Experimental details and comparisons to additional baselines, as well as a set of recommended hyperparameters are available in Appendix D and Appendix C, respectively.
+
+# 5.1 NON-CONVERGENCE AND MODE DROPPING
+
+# 5.1.1 AUGMENTED MNIST
+
+We first evaluate the ability of our approach to avoid mode collapse on real image data coming from an augmented version of the MNIST dataset. Similarly to (Metz et al., 2016; Che et al., 2016), we combine three randomly selected MNIST digits to form 3-channel images, resulting in a dataset with 1000 different classes, one for each of the possible combinations of the ten MNIST digits.
+
+We train a simplified DCGAN architecture (see details in Appendix D) with both GAN and CHEKHOV GAN with a different number of saved past states. The evaluation of each model is done as follows. We generate a fixed amount of samples (25,600) from each model and classify them using a pretrained MNIST classifier with an accuracy of $9 9 . 9 9 \%$ . The models that exhibit less mode collapse are expected to generate samples from most of the 1000 modes.
+
+We report two different evaluation metrics in Table 1: i) the number of classes for which a model generated at least one sample, and ii) the reverse KL divergence. The reverse KL divergence between the model and the target data distribution is computed by considering that the data distribution is a uniform distribution over all classes.
+
+<table><tr><td rowspan=1 colspan=1>Models</td><td rowspan=1 colspan=1>0 states (GAN)</td><td rowspan=1 colspan=1>5 states</td><td rowspan=1 colspan=1>10 states</td></tr><tr><td rowspan=1 colspan=1>Generated Classes</td><td rowspan=1 colspan=1>629 ± 121.08</td><td rowspan=1 colspan=1>743 ± 64.31</td><td rowspan=1 colspan=1>795± 37</td></tr><tr><td rowspan=1 colspan=1>Reverse KL</td><td rowspan=1 colspan=1>1.96 ± 0.64</td><td rowspan=1 colspan=1>1.40 ± 0.21</td><td rowspan=1 colspan=1>1.24 ± 0.17</td></tr></table>
+
+Table 1: Stacked MNIST: Number of generated classes out of 1000 possible combinations, and the reverse KL divergence score. The results are averaged over 10 runs.
+
+# 5.2 IMAGE MODELING
+
+We turn to the evaluation of our model for the task of generating rich image data for which the modes of the data distribution are unknown. In the following, we perform experiments that indirectly measure mode coverage through metrics based on the sample diversity and quality.
+
+# 5.2.1 INFERENCE VIA OPTIMIZATION ON CIFAR10
+
+We train a DCGAN architecture on CIFAR10 (Krizhevsky & Hinton, 2009) and evaluate the performance of each model using the inference via optimization technique introduced in (Metz et al., 2016) and explained in Appendix D.3.3.
+
+The average MSE over 10 rounds using different seeds is reported in Table 2. Using CHEKHOV GAN with as few as 5 past states results in a significant gain which can be further improved by increasing the number of past states to 10 and 25. In addition, the training procedure becomes more stable as indicated by the decrease in the standard deviation. The percentage of mini-batches that achieve the lowest reconstruction loss with the different models is given in Table 2. This can also be visualized by comparing the closest images $x _ { c l o s e s t }$ from each model to real target images $x _ { t a r g e t }$ as shown in Figure 3. The images are randomly selected images from the batch which has the largest absolute difference in MSE between GAN and CHEKHOV GAN with 25 states. The samples obtained by the original GAN are often blurry while samples from CHEKHOV GAN are both sharper and exhibit more variety, suggesting a better coverage of the true data distribution.
+
+<table><tr><td rowspan=1 colspan=1>Target</td><td rowspan=1 colspan=1>Past States</td><td rowspan=1 colspan=1>0 (GAN)</td><td rowspan=1 colspan=1>5 states</td><td rowspan=1 colspan=1>10 states</td><td rowspan=1 colspan=1>25 states</td></tr><tr><td rowspan=2 colspan=1>TrainSet</td><td rowspan=1 colspan=1>MSE</td><td rowspan=1 colspan=1>61.13 ± 3.99</td><td rowspan=1 colspan=1>58.84 ± 3.67</td><td rowspan=1 colspan=1>56.99± 3.49</td><td rowspan=1 colspan=1>48.42 ± 2.99</td></tr><tr><td rowspan=1 colspan=1>Best Rank (%)</td><td rowspan=1 colspan=1>0%</td><td rowspan=1 colspan=1>0%</td><td rowspan=1 colspan=1>18.66 %</td><td rowspan=1 colspan=1>81.33 %</td></tr><tr><td rowspan=2 colspan=1>TestSet</td><td rowspan=1 colspan=1>MSE</td><td rowspan=1 colspan=1>59.5 ± 3.65</td><td rowspan=1 colspan=1>56.66 ± 3.60</td><td rowspan=1 colspan=1>53.75 ± 3.47</td><td rowspan=1 colspan=1>46.82 ± 2.96</td></tr><tr><td rowspan=1 colspan=1>Best Rank (%)</td><td rowspan=1 colspan=1>0%</td><td rowspan=1 colspan=1>0%</td><td rowspan=1 colspan=1>17.57 %</td><td rowspan=1 colspan=1>82.43 %</td></tr></table>
+
+Table 2: CIFAR10: MSE between target images from the train and test set and the best rank which consists of the percentage of minibatches containing target images that can be reconstructed with the lowest loss across various models. We use 20 different minibatches, each containing 64 target images. Increasing the number of past states for CHEKHOV GAN allows the model to better match the real images.
+
+![](images/02099c6a6f26f5ee425260fe52bef6039457b7da39cf0cd9c09d0a8068031e96.jpg)  
+Table 3: CIFAR10: Target images from the test set are shown on the left. The images from each model that best resemble the target image are shown for different number of past states: 0 (GAN), 10 and 25 (CHEKHOV GAN ). The reconstruction MSE loss is indicated above each image.
+
+Note that the numbers quoted in our paper can directly be compared to the ones reported in unrolled GAN (Metz et al., 2016) since we have used the same architecture and choice of hyper-parameters. We include a comparison in the appendix.
+
+# 5.2.2 ESTIMATION OF MISSING MODES ON CELEBA
+
+We estimate the number of missing modes on the CelebA dataset (Liu et al., 2015) by using an auxiliary discriminator as performed in (Che et al., 2016). The experiment consists of two phases. In the first phase we train GAN and CHEKHOV GAN models and generate a fixed number of images. In the second phase we independently train a noisy discriminator using the DCGAN architecture where the training data is the previously generated data from each of the models, respectively. The noisy discriminator is then used as a mode estimator. The test images from CelebA that are classified as fake by the mode estimator are considered as images belonging to a missing mode. Table 4 shows the number of missed modes for the two models. Generated samples from each model are given in the Appendix.
+
+<table><tr><td rowspan=1 colspan=1>0</td><td rowspan=1 colspan=1>0 states (GAN)</td><td rowspan=1 colspan=1>5 states (CHEKHOV GAN)</td></tr><tr><td rowspan=1 colspan=1>0.25</td><td rowspan=1 colspan=1>3004 ± 4154</td><td rowspan=1 colspan=1>1407 ± 1848</td></tr><tr><td rowspan=1 colspan=1>0.5</td><td rowspan=1 colspan=1>2568.25 ± 4148</td><td rowspan=1 colspan=1>1007 ± 1805</td></tr></table>
+
+Table 4: CelebA: Number of images from the test set that the auxiliary discriminator classifies as not real. Gaussian noise with variance $\sigma ^ { 2 }$ is added to the input of the auxiliary discriminator, with the standard deviation shown in the first row. The test set consists of 50,000 images.
+
+Interestingly, even with small number of past states $\scriptstyle ( \mathrm { K } = 5 )$ , CHEKHOV GAN manages to stabilize the training and generate more diverse samples on all the datasets. In terms of computational complexity, our algorithm scales linearly with $K$ . However, all the elements in the sum are independent and can be computed efficiently in a parallel manner.
+
+Table 5: Inception score (higher is better) and FID (lower is better) on Cifar10: The first section shows GAN variants and the second shows WGAN variants. The results are averaged over 8 runs.   
+
+<table><tr><td>Model</td><td>Inception Score</td><td>Fréchet Inception Distance</td></tr><tr><td>GAN</td><td>5.89 ± 0.15</td><td>42.99 ± 6.85</td></tr><tr><td>Chekhov GAN</td><td>6.02 ± 0.15</td><td>40.97 ± 1.03</td></tr><tr><td>Unrolled GAN</td><td>5.51 ± 0.13</td><td>53.83 ± 6.09</td></tr><tr><td>MIX+GAN</td><td>6.03 ± 0.16</td><td>41.79 ± 5.10</td></tr><tr><td>WGAN</td><td>4.85± 0.12</td><td>66.59 ± 1.28</td></tr><tr><td>Chekhov WGAN</td><td>5.13 ± 0.13</td><td>58.82 ± 1.34</td></tr><tr><td>MIX+WGAN</td><td>4.91 ± 0.12</td><td>64.89 ± 4.53</td></tr><tr><td>WGAN GP</td><td>5.31 ± 0.13</td><td>52.52 ± 5.28</td></tr><tr><td>Real Data</td><td>11.24 ± 0.12</td><td>5.19± 0.02</td></tr></table>
+
+# 5.3 INCEPTION SCORE AND FRÉCHET INCEPTION DISTANCE
+
+We compare our algorithm against several state-of-the-art GAN methods using the Inception Score (Salimans et al., 2016) as metric, as well as its improved version, the Fréchet Inception Distance (Heusel et al., 2017). All the models used in this experiment rely on the same architectures for both the generator and the discriminator. We generated 50,000 samples for the calculation of the two metrics. Table 5 shows the results on Cifar10 when all models are trained in an unsupervised fashion.
+
+We apply our algorithm on top of GAN and WGAN (Arjovsky et al., 2017), as both define a minimax game. Both Chekhov GAN and Chekhov WGAN are trained using 5 past states. For fair comparison, $\mathbf { M I X + G A N }$ and MIX $+$ WGAN (Arora et al., 2017) use a mixture of 5 generators and discriminators and the number of unrolling steps for Unrolled GAN (Metz et al., 2016) is set to 5. The number of parameters for MIX $\cdot +$ (GAN/WGAN) is 5 times the number of parameters for all the other baselines.
+
+We find that by applying our algorithm on top of WGAN, we get an improvement of 5.13 vs. 4.85 (inception score) and 58.82 vs. 66.59 (FID). Throughout the training, Chekhov WGAN consistently outperforms WGAN by achieving higher scores and lower distances, as shown in Figure 3. Across the WGAN-based variants, Chekhov WGAN consistently outperforms $\mathrm { M I X + W G A N }$ as well, while WGAN GP (Gulrajani et al., 2017) achieves the best scores within the group. However, as shown in Figure 4, Chekhov WGAN improves upon WGAN and reaches almost the same level of performance as WGAN GP.
+
+In terms of the GAN-based variants, Chekhov GAN consistently outperforms Unrolled GAN by a large margin across all epochs. MIX $^ +$ GAN, GAN and Chekhov GAN achieve comparable scores in terms of both metrics, but Chekhov GAN does so while reducing the variance significantly (see Figure 4).
+
+![](images/3b747abbc4db9a346ebe8889f12e549d00bee98b33090db7f8fddb4db82a606e.jpg)  
+Figure 3: Chekhov GAN consistently outperforms WGAN across both metrics: (left) Inception score, (right) Fréchet Inception Distance (FID). The shaded area denotes the standard deviation.
+
+![](images/52a0ae7d44b4601a1eb0e72ff61e934e7eef9563375d0ccaddd08cb2e1e9f8bf.jpg)  
+Figure 4: Comparison of FID: (left) WGAN-based variants, (right) GAN-based variants.The shaded area denotes the standard deviation.
+
+# 6 CONCLUSION
+
+We have presented a principled approach to training GANs, which is guaranteed to reach convergence to a mixed equilibrium for semi-shallow architectures. Empirically, our approach presents several advantages when applied to commonly used GAN architectures, such as improved stability or reduction in mode dropping. Our results open an avenue for the use of online-learning and gametheoretic techniques in the context of training GANs. One question that remains open is whether the theoretical guarantees can be extended to more complex architectures.
+
+# REFERENCES
+
+Martin Arjovsky and Léon Bottou. Towards principled methods for training generative adversarial networks. In NIPS 2016 Workshop on Adversarial Training. In review for ICLR, volume 2016, 2017.   
+Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein gan. arXiv preprint arXiv:1701.07875, 2017.   
+Sanjeev Arora, Rong Ge, Yingyu Liang, Tengyu Ma, and Yi Zhang. Generalization and equilibrium in generative adversarial nets (gans). arXiv preprint arXiv:1703.00573, 2017.   
+David Berthelot, Tom Schumm, and Luke Metz. Began: Boundary equilibrium generative adversarial networks. arXiv preprint arXiv:1703.10717, 2017.   
+Tong Che, Yanran Li, Athul Paul Jacob, Yoshua Bengio, and Wenjie Li. Mode regularized generative adversarial networks. arXiv preprint arXiv:1612.02136, 2016.   
+Constantinos Daskalakis, Alan Deckelbaum, and Anthony Kim. Near-optimal no-regret algorithms for zero-sum games. Games and Economic Behavior, 92:327–348, 2015.   
+Yoav Freund and Robert E Schapire. Adaptive game playing using multiplicative weights. Games and Economic Behavior, 29(1-2):79–103, 1999.   
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Aistats, volume 9, pp. 249–256, 2010.   
+Ian Goodfellow. Nips 2016 tutorial: Generative adversarial networks. arXiv preprint arXiv:1701.00160, 2016.   
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative Adversarial Nets. pp. 2672–2680, 2014.   
+Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. Journal of Machine Learning Research, 13(Mar):723–773, 2012.
+
+Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of wasserstein gans. arXiv preprint arXiv:1704.00028, 2017.
+
+Elad Hazan and Tomer Koren. The computational power of optimization in online learning. In Proc. STOC, pp. 128–141. ACM, 2016.
+
+Elad Hazan et al. Introduction to online convex optimization. Foundations and Trends $\textsuperscript { \textregistered }$ in Optimization, 2(3-4):157–325, 2016.
+
+Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, Günter Klambauer, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a nash equilibrium. arXiv preprint arXiv:1706.08500, 2017.
+
+Adam Kalai and Santosh Vempala. Efficient algorithms for online decision problems. Journal of Computer and System Sciences, 71(3):291–307, 2005.
+
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
+
+Diederik P Kingma and Max Welling. Auto-Encoding Variational Bayes. arXiv.org, December 2013.
+
+Jyrki Kivinen and Manfred K Warmuth. Exponentiated gradient versus gradient descent for linear predictors. Information and Computation, 132(1):1–63, 1997.
+
+Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. 2009.
+
+Yujia Li, Kevin Swersky, and Richard S Zemel. Generative moment matching networks. In ICML, pp. 1718–1727, 2015.
+
+Ziwei Liu, Ping Luo, Xiaogang Wang, and Xiaoou Tang. Deep learning face attributes in the wild. In Proceedings of the IEEE International Conference on Computer Vision, pp. 3730–3738, 2015.
+
+Peter McCullagh and James A Nelder. Generalized linear models, no. 37 in monograph on statistics and applied probability, 1989.
+
+Luke Metz, Ben Poole, David Pfau, and Jascha Sohl-Dickstein. Unrolled generative adversarial networks. arXiv preprint arXiv:1611.02163, 2016.
+
+Alfred Müller. Integral probability metrics and their generating classes of functions. Advances in Applied Probability, 29(02):429–443, 1997.
+
+Vaishnavh Nagarajan and J Zico Kolter. Gradient descent gan optimization is locally stable. arXiv preprint arXiv:1706.04156, 2017.
+
+John F Nash et al. Equilibrium points in n-person games. Proceedings of the national academy of sciences, 36(1):48–49, 1950.
+
+Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. In Advances in Neural Information Processing Systems, pp. 271–279, 2016.
+
+David Pfau and Oriol Vinyals. Connecting generative adversarial networks and actor-critic methods. arXiv preprint arXiv:1610.01945, 2016.
+
+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.
+
+Sasha Rakhlin and Karthik Sridharan. Optimization, learning, and games with predictable sequences. In Advances in Neural Information Processing Systems, pp. 3066–3074, 2013.
+
+D J Rezende, S Mohamed, and D Wierstra. Stochastic backpropagation and approximate inference in deep generative models. arXiv.org, 2014.
+
+Kevin Roth, Aurelien Lucchi, Sebastian Nowozin, and Thomas Hofmann. Stabilizing training of generative adversarial networks through regularization. arXiv preprint arXiv:1705.09367, 2017.
+
+Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, pp. 2226–2234, 2016.
+
+Ari Seff, Alex Beatson, Daniel Suo, and Han Liu. Continual learning in generative adversarial nets. arXiv preprint arXiv:1705.08395, 2017.
+
+Shai Shalev-Shwartz et al. Online learning and online convex optimization. Foundations and Trends $\textsuperscript { \textregistered }$ in Machine Learning, 4(2):107–194, 2012.
+
+Ilya Tolstikhin, Sylvain Gelly, Olivier Bousquet, Carl-Johann Simon-Gabriel, and Bernhard Schölkopf. Adagan: Boosting generative models. arXiv preprint arXiv:1701.02386, 2017.
+
+# A ANALYSIS
+
+Here we provide the proof of Thm. 1.
+
+Proof. We make a use of a theorem due to Freund & Schapire (1999) which shows that if both $\ A _ { 1 } , A _ { 2 }$ ensure no regret implies approximate MNE. For completeness we provide its proof in Sec. A.2.
+
+Theorem 2. The mixed strategies $( \mathcal { D } _ { 1 } , \mathcal { D } _ { 2 } )$ that Algorithm $^ { l }$ outputs are $\varepsilon$ -MNE, where
+
+$$
+\varepsilon : = \left( B _ { T } ^ { A _ { 1 } } + B _ { T } ^ { A _ { 2 } } \right) / T .
+$$
+
+here $B _ { T } ^ { A _ { 1 } } , B _ { T } ^ { A _ { 2 } }$ are bounds on the regret of $A _ { 1 } , A _ { 2 }$ .
+
+According to Thm. 2, it is sufficient to show that both $\mathcal { A } _ { 1 }$ , and $\mathcal { A } _ { 2 }$ ensure a regret bound of $O ( \sqrt { T } )$ .
+
+Guarantees for $\boldsymbol { A } _ { 2 }$ : this FTRL version is well known in online learning, and its regret guarantees can be found in the literature, (e.g, Theorem 5.1 in Hazan et al. (2016)). The following lemma provides its guarantees,
+
+Lemma 2. Let $d _ { 2 }$ be the diameter of $\displaystyle { \mathcal { K } } _ { 2 }$ . Invoking $\mathcal { A } _ { 2 }$ with $\eta _ { 0 } = d _ { 2 } / \sqrt { 2 } L$ , ensures the following regret bound over the sequence of concave functions $\{ g _ { t } \} _ { t \in [ T ] }$ ,
+
+$$
+R e g r e t _ { T } ^ { A _ { 2 } } ( g _ { 1 } , . . . , g _ { T } ) \leq L d _ { 2 } \sqrt { 2 T } .
+$$
+
+Moreover, the following applies for the sequence $\{ \mathbf { v } _ { t } \} _ { t \in [ T ] }$ generated by $\boldsymbol { A } _ { 2 }$ ,
+
+$$
+\left\| \mathbf { v } _ { t + 1 } - \mathbf { v } _ { t } \right\| \leq d _ { 2 } / \sqrt { 2 T } .
+$$
+
+Note that the proof heavily relies on the concavity of the $g _ { t } ( \cdot )$ ’s, which is due to the concavity of the game with respect to the discriminator. For completeness we provide a proof of the second part of the lemma in Sec. B.3.
+
+Guarantees for $\mathcal { A } _ { 1 }$ : By Lemma 2, the sequence generated by the discriminator not only ensures low regret but is also stable in the sense that consecutive decision points are close by. This is the key property which will enable us to establish a regret bound for algorithm $\mathcal { A } _ { 1 }$ . Next we state the guarantees of $\mathcal { A } _ { 1 }$ ,
+
+Lemma 3. Let $C : = \mathrm { m a x } _ { \mathbf { u } \in \mathcal { K } _ { 1 } , \mathbf { v } \in \mathcal { K } _ { 2 } } | M ( \mathbf { u } , \mathbf { v } ) |$ . Consider the loss sequence appearing in $A l g ~ I$ , $\{ f _ { t } ( \cdot ) : = M ( \cdot , \mathbf { v } _ { t } ) \} _ { t \in [ T ] }$ . Then algorithm $\mathcal { A } _ { 1 }$ ensures the following regret bound over this sequence,
+
+$$
+R e g r e t _ { T } ^ { A _ { 1 } } ( f _ { 1 } , \dots , f _ { T } ) \le \frac 1 { \sqrt { 2 } } L d _ { 2 } \sqrt { T } + 2 C .
+$$
+
+Combining the regret bounds of Lemmas 2, 3 and Theorem 2 concludes the proof of Thm. 1.
+
+# A.1 PROOF OF LEMMA 3
+
+Proof. We use the following regret bound regarding the FTL (follow-the-leader) decision rule, derived in Kalai & Vempala (2005) (see also Shalev-Shwartz et al. (2012)),
+
+Lemma 4. For any sequence of loss functions $\{ f _ { t } \} _ { t \in [ T ] }$ , the regret of FTL is bounded as follows,
+
+$$
+R e g r e t _ { T } ^ { F T L } ( f _ { 1 } , \dots , f _ { T } ) \leq \sum _ { t = 1 } ^ { T } \left( f _ { t } ( \mathbf { u } _ { t } ) - f _ { t } ( \mathbf { u } _ { t + 1 } ) \right) .
+$$
+
+Since $\mathcal { A } _ { 1 }$ is FTL, the above bound applies. Thus, using the above bound together with the stability of the $\{ \mathbf { v } _ { t } \} _ { t \in [ T ] }$ sequence we obtain,
+
+$$
+\begin{array} { r l } { \mathbb { R } \mathrm { e g r e r } _ { i } ^ { \mathbf { A } _ { 1 } ^ { \mathsf { * } } } } & { \overset { , } { \sum _ { t = 1 } ^ { \infty } } \big ( f _ { t } ( \mathbf { u } _ { t + 1 } ) - f _ { t } ( \mathbf { u } _ { t + 1 } ) \big ) } \\ & { \leq \displaystyle \sum _ { t = 1 } ^ { \infty - 1 } \big ( f _ { t } ( \mathbf { u } _ { t } ) - f _ { t } ( \mathbf { u } _ { t + 1 } ) + f _ { t + 1 } ( \mathbf { u } _ { t + 1 } ) - f _ { t + 1 } ( \mathbf { u } _ { t + 1 } ) \big ) + \big ( f _ { T } ( \mathbf { u } _ { T } ) - f _ { T } ( \mathbf { u } _ { T + 1 } ) \big ) } \\ & { = \displaystyle \sum _ { t = 1 } ^ { \infty - 1 } \big ( f _ { t + 1 } ( \mathbf { u } _ { t + 1 } ) - f _ { t } ( \mathbf { u } _ { t + 1 } ) \big ) + \displaystyle \sum _ { t = 1 } ^ { T - 1 } \big ( f _ { t } ( \mathbf { u } _ { t } ) - f _ { t + 1 } ( \mathbf { u } _ { t - 1 } ) \big ) + \big ( f _ { T } ( \mathbf { u } _ { T } ) - f _ { T } ( \mathbf { u } _ { T + 1 } ) } \\ & { \quad - \displaystyle \sum _ { t = 1 } ^ { \infty - 1 } \big ( W ( \mathbf { u } _ { t + 1 } , \mathbf { v } _ { t + 1 } ) - W ( \mathbf { u } _ { t + 1 } , \mathbf { v } _ { t + 1 } ) \big ) + \big ( f _ { t } ( \mathbf { u } _ { 1 } ) - f _ { T } ( \mathbf { u } _ { T + 1 } ) \big ) } \\ & { \leq \displaystyle \sum _ { t = 1 } ^ { \infty - 1 } | | \mathbf { v } _ { t + 1 } | , \quad \mathbf { u } _ { t + 1 } | , \quad \mathbf { v } _ { t + 1 } | , \quad \mathsf { a } _ { T } ( \mathbf { u } _ { T + 1 } ) - f _ { T } ( \mathbf { u } _ { T + 1 } ) \big ) } \\ & { \leq \displaystyle \sum _ { t = 1 } ^ { \infty - 1 } | \mathbf { v } _ { t } | , \quad \mathbf { v } _ { t } | - \mathbf { v } _ { t } | + \big ( f _ { t } ( \mathbf { u } _ { 1 } ) - f _ { T } ( \mathbf { u } _ { T + 1 } ) \big ) } \\ &  \leq \ \end{array}
+$$
+
+where the fourth line uses $f _ { t } ( \cdot ) : = M ( \cdot , \mathbf { v } _ { t } )$ , the fifth line uses the Lipschitz continuity of $M$ . And the sixth line used the stability of the $\mathbf { v } _ { t }$ ’s due to Lemma 2. Finally, we use $| f _ { t } ( u ) | = | M ( u , v _ { t } ) | \leq$ $C$ .
+
+# A.2 PROOF OF THEOREM 2
+
+Proof. Writing explicitly $f _ { t } ( \mathbf { u } ) : = M ( \mathbf { u } , \mathbf { v } _ { t } )$ and $g _ { t } ( \mathbf { v } ) : = M ( \mathbf { u } _ { t } , \mathbf { v } )$ , and plugging these into the regret guarantees of $\ A _ { 1 } , A _ { 2 }$ , we have,
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { t = 1 } ^ { T } M ( \mathbf { u } _ { t } , \mathbf { v } _ { t } ) - \mathop { \operatorname* { m i n } } _ { \mathbf { u } \in \mathcal { K } _ { 1 } } \sum _ { t = 1 } ^ { T } M ( \mathbf { u } , \mathbf { v } _ { t } ) \leq B _ { T } ^ { A _ { 1 } } , } \\ & { \displaystyle \sum _ { t = 1 } ^ { T } - M ( \mathbf { u } _ { t } , \mathbf { v } _ { t } ) - \mathop { \operatorname* { m i n } } _ { \mathbf { v } \in \mathcal { K } _ { 2 } } \sum _ { t = 1 } ^ { T } - M ( \mathbf { u } _ { t } , \mathbf { v } ) \leq B _ { T } ^ { A _ { 2 } } . } \end{array}
+$$
+
+By definition, $\begin{array} { r } { \operatorname* { m i n } _ { \mathbf { u } \in { \mathcal { K } } _ { 1 } } \sum _ { t = 1 } ^ { T } M ( \mathbf { u } , \mathbf { v } _ { t } ) \leq \mathbb { E } _ { \mathbf { u } \sim \mathcal { D } _ { 1 } } [ \sum _ { t = 1 } ^ { T } M ( \mathbf { u } , \mathbf { v } _ { t } ) ] } \end{array}$ . Using this together with Equation (5), we get,
+
+$$
+\sum _ { t = 1 } ^ { T } M ( \mathbf { u } _ { t } , \mathbf { v } _ { t } ) - \mathbb { E } _ { \mathbf { u } \sim \mathcal { D } _ { 1 } } [ \sum _ { t = 1 } ^ { T } M ( \mathbf { u } , \mathbf { v } _ { t } ) ] \leq B _ { T } ^ { A _ { 1 } } ,
+$$
+
+Summing Equations (6),(7), and dividing by $T$ , we get,
+
+$$
+\operatorname* { m a x } _ { \mathbf { v } \in \mathcal { K } _ { 2 } } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } M ( \mathbf { u } _ { t } , \mathbf { v } ) - \mathbb { E } _ { \mathbf { u } \sim \mathcal { D } _ { 1 } } \big [ \frac { 1 } { T } \sum _ { t = 1 } ^ { T } M ( \mathbf { u } , \mathbf { v } _ { t } ) \big ] \leq \frac { B _ { T } ^ { A _ { 1 } } } { T } + \frac { B _ { T } ^ { A _ { 2 } } } { T } ~ .
+$$
+
+Recalling that $\mathcal { D } _ { 1 } \sim \operatorname { U n i } \{ \mathbf { u } _ { 1 } , \dots , \mathbf { u } _ { t } \} , \mathcal { D } _ { 2 } \sim \operatorname { U n i } \{ \mathbf { v } _ { 1 } , \dots , \mathbf { v } _ { t } \}$ , and denoting $\begin{array} { r } { \varepsilon : = \frac { B _ { T } ^ { A _ { 1 } } } { T } + \frac { B _ { T } ^ { A _ { 2 } } } { T } } \end{array}$ , we conclude that,
+
+$$
+\mathbb { E } _ { ( \mathbf { u } , \mathbf { v } ) \sim \mathcal { D } _ { 1 } \times \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] \geq \operatorname* { m a x } _ { \mathbf { v } \in { \cal K } _ { 2 } } \mathbb { E } _ { \mathbf { u } \sim \mathcal { D } _ { 1 } } [ M ( \mathbf { u } , \mathbf { v } ) ] - \varepsilon .
+$$
+
+We can similarly show that,
+
+$$
+\mathbb { E } _ { ( \mathbf { u } , \mathbf { v } ) \sim \mathcal { D } _ { 1 } \times \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] \leq \operatorname* { m i n } _ { \mathbf { u } \in { \mathcal { K } _ { 1 } } } \mathbb { E } _ { \mathbf { v } \sim \mathcal { D } _ { 2 } } [ M ( \mathbf { u } , \mathbf { v } ) ] + \varepsilon .
+$$
+
+which concludes the proof.
+
+# B REMAINING PROOFS
+
+# B.1 PROOF OF PROPOSITION 1
+
+Proof. Look at the first term in the GAN objective, $\mathbb { E } _ { p _ { \mathrm { d a t a } } } \log h _ { \mathbf { v } } ( \mathbf { x } )$ . For a fixed $x$ we have,
+
+$$
+\log h _ { \mathbf { v } } ( x ) = - \log \left( 1 + \exp ( - \mathbf { v } ^ { \top } x ) \right) ,
+$$
+
+and it can be shown that the above expression is always concave in $\textbf { v } ^ { 8 }$ . Since an expectation over concave functions is also concave, this implies the concavity of the first term in $\mathcal { H }$ .
+
+Similarly, look at the second term in the GAN objective, $\mathbb { E } _ { { \mathbf { z } } \in p _ { \mathbf { z } } } \log ( 1 - h _ { \mathbf { v } } ( G _ { \mathbf { u } } ( \mathbf { z } ) ) )$ . For a fixed $G _ { \bf u } ( z )$ we have,
+
+$$
+\log ( 1 - h _ { \mathbf { v } } ( G _ { \mathbf { u } } ( \mathbf { z } ) ) ) = - \log \left( 1 + \exp ( + v ^ { \top } G _ { \mathbf { u } } ( \mathbf { z } ) ) \right)
+$$
+
+and it can be shown that the above expression is always concave in $\mathbf { v }$ . Since an expectation over concave functions is also concave, this implies the concavity of the second term in $\mathcal { H }$ .
+
+Thus $\mathcal { H }$ is a sum of two concave terms and is therefore concave in $\mathbf { v }$ .
+
+# B.2 PROOF OF LEMMA 1
+
+Proof. Writing explicitly $f _ { t } ( \mathbf { u } ) : = M ( \mathbf { u } , \mathbf { v } _ { t } )$ and $g _ { t } ( \mathbf { v } ) : = M ( \mathbf { u } _ { t } , \mathbf { v } )$ , and plugging these into the regret guarantees of $\ A _ { 1 } , A _ { 2 }$ , we have,
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { t = 1 } ^ { T } M ( \mathbf { u } _ { t } , \mathbf { v } _ { t } ) - \mathop { \operatorname* { m i n } } _ { \mathbf { u } \in \mathcal { K } _ { 1 } } \sum _ { t = 1 } ^ { T } M ( \mathbf { u } , \mathbf { v } _ { t } ) \leq B _ { T } ^ { A _ { 1 } } , } \\ & { \displaystyle \sum _ { t = 1 } ^ { T } - M ( \mathbf { u } _ { t } , \mathbf { v } _ { t } ) - \mathop { \operatorname* { m i n } } _ { \mathbf { v } \in \mathcal { K } _ { 2 } } \sum _ { t = 1 } ^ { T } - M ( \mathbf { u } _ { t } , \mathbf { v } ) \leq B _ { T } ^ { A _ { 2 } } . } \end{array}
+$$
+
+Summing the above equations and dividing by $T$ , we get,
+
+$$
+\operatorname* { m a x } _ { \mathbf { v } \in { \mathcal { K } } _ { 2 } } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } M ( \mathbf { u } _ { t } , \mathbf { v } ) - \operatorname* { m i n } _ { \mathbf { u } \in { \mathcal { K } } _ { 1 } } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } M ( \mathbf { u } , \mathbf { v } _ { t } ) \leq \frac { B _ { T } ^ { A _ { 1 } } } { T } + \frac { B _ { T } ^ { A _ { 2 } } } { T } : = \varepsilon .
+$$
+
+Next we show that the second term above is always smaller than the minimax value,
+
+$$
+\begin{array} { c } { \displaystyle \underset { \mathbf { u } \in K _ { 1 } } { \operatorname* { m i n } } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } M ( \mathbf { u } , \mathbf { v } _ { t } ) \leq \displaystyle \operatorname* { m i n } _ { \mathbf { u } \in K _ { 1 } } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \underset { \mathbf { v } \in K _ { 2 } } { \operatorname* { m a x } } M ( \mathbf { u } , \mathbf { v } ) } \\ { \displaystyle = \underset { \mathbf { u } \in K _ { 1 } } { \operatorname* { m i n } } \underset { \mathbf { v } \in K _ { 2 } } { \operatorname* { m a x } } M ( \mathbf { u } , \mathbf { v } ) } \end{array}
+$$
+
+Plugging the above into Equation (9), and recalling ${ \mathcal { D } } _ { 1 } \sim { \mathrm { U n i } } \{ \mathbf { u } _ { 1 } , \dots , \mathbf { u } _ { t } \}$ , we get,
+
+$$
+\operatorname* { m a x } _ { \mathbf { v } \in { \cal K } _ { 2 } } \mathbb { E } _ { \mathbf { u } \sim \mathcal { D } _ { 1 } } M ( \mathbf { u } , \mathbf { v } ) \leq \operatorname* { m i n } _ { \mathbf { u } \in { \cal K } _ { 1 } } \operatorname* { m a x } _ { \mathbf { v } \in { \cal K } _ { 2 } } M ( \mathbf { u } , \mathbf { v } ) + \varepsilon .
+$$
+
+which concludes the proof.
+
+B.3 PROOF OF THE SECOND PART OF LEMMA 2 (STABILITY OF FTRL SEQUENCE IN CONCAVE CASE)
+
+Proof. Here we establish the stability of the FTRL decision rule, $\boldsymbol { A } _ { 2 }$ , depicted in Theorem 1.
+
+Note that the following applies to this FTRL objective,
+
+$$
+\sum _ { \tau = 0 } ^ { t - 1 } \nabla g _ { \tau } ( \mathbf { v } _ { \tau } ) ^ { \top } \mathbf { v } - \frac { \sqrt { T } } { 2 \eta _ { 0 } } \| \mathbf { v } \| ^ { 2 } = - \frac { \sqrt { T } } { 2 \eta _ { 0 } } \left\| \mathbf { v } - \frac { \eta _ { 0 } } { \sqrt { T } } \sum _ { \tau = 0 } ^ { t - 1 } \nabla g _ { \tau } ( \mathbf { v } _ { \tau } ) \right\| ^ { 2 } + C
+$$
+
+Where $C$ is a constant independent of $\mathbf { v }$ .
+
+Let us denote by $\Pi _ { \mathbb { K } _ { 2 } }$ the projection operator onto ${ \mathcal { K } } _ { 2 } \subset \mathbb { R } ^ { n }$ , meaning,
+
+$$
+\Pi _ { K _ { 2 } } ( \mathbf { v } _ { 0 } ) = \operatorname* { m i n } _ { \mathbf { v } \in K _ { 2 } } \| \mathbf { v } _ { 0 } - \mathbf { v } \| , \qquad \forall \mathbf { v } _ { 0 } \in \mathbb { R } ^ { n }
+$$
+
+By Equation (10) the FTRL rule, $\boldsymbol { A } _ { 2 }$ , can be written as follows,
+
+$$
+\begin{array} { r l r } {  { \mathbf { v } _ { t } = \underset { \mathbf { v } \in \mathcal { K } _ { 2 } } { \mathrm { a r g m i n } } \| \mathbf { v } - \frac { \eta _ { 0 } } { \sqrt { T } } \sum _ { \tau = 0 } ^ { t - 1 } \nabla g _ { \tau } ( \mathbf { v } _ { \tau } ) \| ^ { 2 } } } \\ & { } & { = \Pi _ { \mathcal { K } _ { 2 } } ( - \frac { \eta _ { 0 } } { \sqrt { T } } \sum _ { \tau = 0 } ^ { t - 1 } \nabla g _ { \tau } ( \mathbf { v } _ { \tau } ) ) . } \end{array}
+$$
+
+The projection operator is a contraction (see e.g, Hazan et al. (2016)), using this together with the above implies,
+
+$$
+\begin{array} { l } { \displaystyle \| { \mathbf { v } } _ { t + 1 } - { \mathbf { v } } _ { t } \| \leq \left\| \Pi _ { { \boldsymbol { \kappa } } _ { 2 } } \left( - \frac { \eta _ { 0 } } { \sqrt { T } } \sum _ { \tau = 0 } ^ { t } \nabla g _ { \tau } ( { \mathbf { v } } _ { \tau } ) \right) - \Pi _ { { \boldsymbol { \kappa } } _ { 2 } } \left( - \frac { \eta _ { 0 } } { \sqrt { T } } \sum _ { \tau = 0 } ^ { t - 1 } \nabla g _ { \tau } ( { \mathbf { v } } _ { \tau } ) \right) \right\| } \\ { \leq \left\| - \frac { \eta _ { 0 } } { \sqrt { T } } \nabla g _ { t } ( { \mathbf { v } } _ { t } ) \right\| } \end{array}
+$$
+
+where we used $\left\| \nabla g _ { t } ( \mathbf { v } _ { t } ) \right\| \leq L$ which is due to the Lipschitz continuity of $M$ . We also used $\eta _ { 0 } = d _ { 2 } / \sqrt { 2 } L$ . □
+
+# C PRACTICAL CHEKHOV GAN ALGORITHM
+
+The pseudo-code of algorithms $\mathcal { A } _ { 1 }$ and $\boldsymbol { A } _ { 2 }$ is given in Algorithm 3. The algorithm is symmetric for both players and consists as follows. At every step $t$ if we are currently in the switching mode (i.e. $t$ mod $m = = 0$ ) and the queue is full, we remove a model from the end of the queue, which is the oldest one. Otherwise, we do not remove any model from the queue, but instead just override the head (first) element with the current update.
+
+Algorithm 3 Update queue for Algorithm $\boldsymbol { A } _ { 1 }$ and $\boldsymbol { A } _ { 2 }$   
+
+<table><tr><td>Input: Current step t, m &gt;0 if(t mod m== 0 and|Ql)== K) then Q.remove_last() Q.insert(ft) m=m+inc else Q.replace_first(ft) end if</td></tr></table>
+
+We set the initial spacing, $m$ , to $\textstyle { \frac { N } { K } }$ , where $N$ is the number of update steps per epoch, and $K$ is the number of past states we keep. The number of updates per epoch is just the number of the data points divided by the size of the minibatches we use. The default value of inc is 10. Depending on the dataset and number of total update steps, for higher values of $K$ , this is the only parameter that needs to be tuned. We find that our model is not sensitive to the regularization hyperparameters. For symmetric architectures of the generator and the discriminator (such as DCGAN), for practitioners, we recommend using the same regularization for both players. For our experiments we set the default regularization to 0.1.
+
+# D EXPERIMENTS
+
+# D.1 TOY DATASET: MIXTURE OF GAUSSIANS
+
+We perform several experiments on a toy dataset where we varied the architecture size and the sampling probabilities. The toy dataset consists of a mixture of 7 Gaussians with a standard deviation of 0.01 and means equally spaced around a unit circle.
+
+![](images/b3f339bff042358dc09a840baaf693a8c04933e6433cbacb70e359417fa1cacf.jpg)  
+Figure 5: Both GAN and CHEKHOV GAN converge to the true data distribution when the dimensionality of the noise vector is 2
+
+The architecture for the generator consists in two fully connected layers (of size 128) and a linear projection to the dimensionality of the data (i.e. 2). The activation functions for the fully connected layers are tanh. The discriminator is symmetric and hence, composed of two fully connected layers (of size 128) followed by a linear layer of size 1. The activation functions for the fully connected layers are tanh, whereas the final layer uses sigmoid as an activation function.
+
+Following Metz et al. (2016), we intialize the weights for both networks to be orthogonal with scaling of 0.8. AdamKingma & Ba (2014) was used as an optimizer for both the discriminator and the generator, with a learning rate of $1 e - 4$ and $\beta _ { 1 } = 0 . 5$ . The discriminator and generator respectively minimize and maximize the objective
+
+$$
+\begin{array} { r } { { \mathbb E } _ { x \sim p _ { \mathrm { d a t a } } ( \mathbf x ) } - \log ( D ( x ) ) - { \mathbb E } _ { z \sim \mathcal { N } ( 0 , I _ { 2 5 6 } ) } \log ( 1 - D ( G ( z ) ) ) . } \end{array}
+$$
+
+The setup is the same for both models. For CHEKHOV GAN we use $K = 5$ past states with L2 regularization on the network weights using an initial regularization parameter of 0.01.
+
+Effect of the latent dimension. We find that for the case where $z \sim \mathcal { N } ( 0 , I _ { 2 5 6 } )$ , GANs with the traditional updates can fail to cover all modes by either rotating around the modes (as shown in Metz et al. (2016)) or converge to only a subset of the modes. However, if we sample the latent code from a lower dimensional space, e.g. $z \sim N ( 0 , I _ { 2 } )$ , such that it matches the data dimensionality, the generator needs to learn a simpler mapping. We then observe that both GAN and CHEKHOV GAN are able to recover the true data distribution in this case (see Figure 5). Additionally, tuning the momentum of the optimiser can also stabilize the training of GANs, resulting in a generator that learns all the modes from the data distribution.
+
+Experiments with concave discriminator. As out theoretical guarantees hold for a general (i.e. deep) generator and a concave discriminator, we perform experiments using a concave discriminator as well. We use the same architecture as described previously. In order to make the discriminator concave, we keep the parameters of the first two layers of the discriminator fixed and update only the parameters of the last layer. We generally find this much harder to stabilize due to the great imbalance between the number of trainable parameters of the discriminator and generator. Increasing the size of the last layer of the discriminator significantly improves the performance. Figure 6 shows that CHEKHOV GAN can learn the true data distribution. The final layer is increased by 10 times and the number of past states, $K$ , is 10.
+
+Mode collapse. We run an additional experiment directly targeted at testing for mode collapse. We sample points $x$ from the data distribution $p _ { \mathrm { d a t a } }$ with different probabilities for each mode. We perform an experiment with 5 Gaussian mixtures, again of standard deviation 0.01 arranged in a circle. The probabilities to sample points from each of the modes are [0.35, 0.35, 0.1, 0.1, 0.1]. In this case two modes have higher probability and could potentially attract the gradients towards them and cause mode collapse. In order to make things harder, the size of the hidden layers is 16 instead of 128. CHEKHOV GAN with $K = 1 0$ manages to recover the true data distribution in this case as well, unlike vanilla GANs. Using the history helps CHEKHOV GAN to spread its mass and cover even the modes with low sampling probability. Unrolled GAN Metz et al. (2016) instead uses the way the discriminator would react to the current changes when updating the generator. A comparison to Unrolled GANs and vanilla GANs is given in Figure 7.
+
+![](images/1afddeec7934fbbd907920bbd4a8c7f37429f7dde96d9ba50863dc919f425618.jpg)  
+Figure 6: CHEKHOV GAN with concave discriminator is able to learn the toy data distribution with $K = 1 0$
+
+![](images/faadf09f1255c6d6b4ef28c01afb0b9d35b56ae99ae3ef8d79c52fe7d7d42d8a.jpg)  
+Figure 7: Mode Collapse on a Gaussian Mixture. Comparison between a) vanilla GANs; b) CHEKHOV GAN with $K = 1 0$ , increase ${ \boldsymbol { \mathbf { \rho } } } { \boldsymbol { \mathbf { \rho } } } { \boldsymbol { \mathbf { \rho } } } { \boldsymbol { \mathbf { \rho } } } { \boldsymbol { \mathbf { \rho } } } { \boldsymbol { \mathbf { \rho } } } { \boldsymbol { \mathbf { \rho } } } { \boldsymbol { \mathbf { \rho } } } { \boldsymbol { \mathbf { \rho } } }$ and regularization of 0.0005; c) Unrolled GAN with 10 unrolling steps and d) Unrolled GAN with 5 unrolling steps.
+
+Mixed Nash Equilibrium. For all the experiments we generate from the most current generator only. We empirically find that generating such that we first sample uniformly at random one of the generators from the past, and then generating from it (which corresponds to the MNE) does not lead to any significant improvement (Figure 8). We hypothesize that this is due to a single generator being powerful enough to learn the distribution when playing against the experts from the past.
+
+![](images/4e49bb0f9b53fa703d0e5370a84a36c71b9542c9cb0f756768461d22a36df399.jpg)  
+Figure 8: Comparison of CHEKHOV GAN for $K = 1 0$ when generating from the most current and all of the generators at the final step of training.
+
+# D.2 AUGMENTED MNIST
+
+We here detail the experiment on the Stacked MNIST dataset. The dataset is created by stacking three randomly selected MNIST images in the color channels, resulting in a 3-channel image that belongs to one out of 1000 possible classes. The architectures of the generator and discriminator are given in Table 6 and Table 7, respectively.
+
+Table 6: Stacked MNIST: Generator Architecture   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Numberof outputs</td></tr><tr><td rowspan=1 colspan=1>Input z ~ N(0,I256)</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Fully Connected</td><td rowspan=1 colspan=1>512 (reshape to[-1,4,4,64])</td></tr><tr><td rowspan=1 colspan=1>Deconvolution</td><td rowspan=1 colspan=1>32</td></tr><tr><td rowspan=1 colspan=1>Deconvolution</td><td rowspan=1 colspan=1>16</td></tr><tr><td rowspan=1 colspan=1>Deconvolution</td><td rowspan=1 colspan=1>8</td></tr><tr><td rowspan=1 colspan=1>Deconvolution</td><td rowspan=1 colspan=1>3</td></tr></table>
+
+Table 7: Stacked MNIST: Discriminator Architecture   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Number of outputs</td></tr><tr><td rowspan=1 colspan=1>Convolution</td><td rowspan=1 colspan=1>4</td></tr><tr><td rowspan=1 colspan=1>Convolution</td><td rowspan=1 colspan=1>8</td></tr><tr><td rowspan=1 colspan=1>Convolution</td><td rowspan=1 colspan=1>16</td></tr><tr><td rowspan=1 colspan=1>FlattenandFully Connected</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+We use a simplified version of the DCGAN architecture as suggested by Metz et al. (2016). It contains "deconvolutional layers" which are implemented as transposed convolutions. All convolutions and deconvolutions use kernel size of $3 \times 3$ with a stride of 2. The weights are initialized using the Xavier initialization Glorot & Bengio (2010). The activation units for the discriminator are leaky ReLUs with a leak of 0.3, whereas the generator uses standard ReLUs. We train all models for 20 epochs with a batch size of 32, using the RMSProp optimizer with batch normalization. The optimal learning rate for GAN is 0.001, and for CHEKHOV GAN is 0.01. For all CHEKHOV GAN models we use regularization of 0.1 for the discriminator and 0.0001 for the generator. The regularization is L2 regularization only on the fully connected layers. For $K = 5$ , the increase parameter inc is set to 50. For ${ \mathrm { K } } { = } 1 0$ , inc is 120.
+
+# D.3 CIFAR10 / CELEBA
+
+We use the full DCGAN architecture Radford et al. (2015) for the experiments on CIFAR10 and CelebA, detailed in Table 8 and Table 9.
+
+Table 8: CIFAR10/CelebA Generator Architecture  
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Number of outputs</td></tr><tr><td rowspan=1 colspan=1>Input z ~ N(0, I256)</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>Fully Connected</td><td rowspan=1 colspan=1>32,768 (reshape to [-1, 4, 4, 512])</td></tr><tr><td rowspan=1 colspan=1>Deconvolution</td><td rowspan=1 colspan=1>256</td></tr><tr><td rowspan=1 colspan=1>Deconvolution</td><td rowspan=1 colspan=1>128</td></tr><tr><td rowspan=1 colspan=1>Deconvolution</td><td rowspan=1 colspan=1>74</td></tr><tr><td rowspan=1 colspan=1>Deconvolution</td><td rowspan=1 colspan=1>3</td></tr></table>
+
+Table 9: CIFAR10/CelebA Discriminator Architecture   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Number of outputs</td></tr><tr><td rowspan=1 colspan=1>Convolution</td><td rowspan=1 colspan=1>64</td></tr><tr><td rowspan=1 colspan=1>Convolution</td><td rowspan=1 colspan=1>128</td></tr><tr><td rowspan=1 colspan=1>Convolution</td><td rowspan=1 colspan=1>256</td></tr><tr><td rowspan=1 colspan=1>Convolution</td><td rowspan=1 colspan=1>512</td></tr><tr><td rowspan=1 colspan=1>Flatten and Fully Connected</td><td rowspan=1 colspan=1>1</td></tr></table>
+
+As for MNIST, we apply batch normalization. The activation functions for the generator are ReLUs, whereas the discriminator uses leaky ReLUs with a leak of 0.3. The learning rate for all the models is 0.0002 for both the generator and the discriminator and the updates are performed using the Adam optimizer. The regularization for CHEKHOV GAN is 0.1 and the increase parameter inc is 10.
+
+![](images/dc6bb8ec097b957d0cae76e45c52fdfb506da0414bcc4da7042ff95a550f6c39.jpg)  
+Figure 9: Random batch of generated images from GAN after training for 30 epochs (on the left) and CHEKHOV GAN $( \mathrm { K } \mathrm { = } 2 5$ ) after training for 30 epochs (on the right)
+
+# D.3.1 RESULTS ON CIFAR10
+
+We train for 30 epochs, which we find to be the optimal number of training steps for vanilla GAN in terms of MSE on images from the validation set. Table 10 includes comparison to other baselines. The first set of baselines (given with purple color) consist of GAN where the updates in the inner loop (for the discriminator), the outer loop (for the generator), or both are performed 25 times. The baselines shown with green color are regularized versions of GANs, where we apply the same regularization as in our CHEKHOV GAN in order to show that the gain is not due to the regularization only. Figure 9 presents two randomly sampled batches from the generator trained with GAN and CHEKHOV GAN .
+
+![](images/24e73e9b7dc45a8b82f3c65ff596a972629acc14088156fa2461a9cf56e44ca0.jpg)  
+Table 10: CIFAR10: MSE for other baselines on target images that come from the training set. GAN 25 updates indicates that either the generator, the discriminator or both have been updated 25 times at each update step. Regularized GAN is vanilla GAN where the fully connected layers have regularization of 0.05.
+
+Table 11 gives comparison between CHEKHOV GAN and Unrolled GAN for different numbers of past states or unrolling steps.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>5 states/steps</td><td rowspan=1 colspan=1>10 states/steps</td><td rowspan=1 colspan=1>25 states/steps</td></tr><tr><td rowspan=1 colspan=1>CHEKHOV GAN</td><td rowspan=1 colspan=1>58.84 ± 3.67</td><td rowspan=1 colspan=1>56.99 ± 3.49</td><td rowspan=1 colspan=1>48.42 ± 2.99</td></tr><tr><td rowspan=1 colspan=1>Unrolled GAN</td><td rowspan=1 colspan=1>61.44 ± 7.06</td><td rowspan=1 colspan=1>55.60 ± 5.52</td><td rowspan=1 colspan=1>-</td></tr></table>
+
+Table 11: CIFAR10: Comparison of average MSE on the training set between Unrolled GAN and CHEKHOV GAN for 5 and 10 unrolling steps and past states, respectively. The numbers for Unrolled GAN are taken from Metz et al. (2016).
+
+# D.3.2 CELEBA
+
+All models are trained for 10 epochs. Randomly generated batches of images are shown in Figure 10.
+
+![](images/9adf87311a83d832464c7168345702f915a0c1537e1ff1e536ceba8f005f2334.jpg)  
+Figure 10: Random batch of generated images from GAN (left) and CHEKHOV GAN $K = 5$ ) (right) after training for 10 epochs.
+
+# D.3.3 DETAILS ABOUT INFERENCE VIA OPTIMIZATION ON CIFAR10
+
+This approach consists in finding a noise vector $z _ { c l o s e s t }$ that when used as input to the generator would produce an image that is the closest to a target image in terms of mean squared error (MSE):
+
+$$
+\begin{array} { r } { z _ { c l o s e s t } = \mathop { \operatorname { a r g m i n } } _ { z } \mathbf { M S E } ( G ( z ) , x _ { t a r g e t } ) \qquad \qquad x _ { c l o s e s t } = G ( z _ { c l o s e s t } ) . } \end{array}
+$$
+
+We report the MSE in image space between $x _ { c l o s e s t }$ and $x _ { t a r g e t }$ . This measures the ability of the generator to generate samples that look like real images. A model engaging in mode collapse would fail to generate (approximate) images from the real data. Conversely, if a model covers the true data distribution it should be able to generate any specific image from it.
+
+# D.4 INCEPTION SCORE AND FID ON CIFAR10
+
+Additional plots that showcase inception scores for Chekhov WGAN and the WGAN-based baselines (WGAN, WGAN GP, MIX $^ { + }$ WGAN) and Chekhov GAN and the GAN-based baselines (GAN, Unrolled GAN, $\mathbf { M I X + G A N }$ ) are given in Figure 11.
+
+For Chekhov WGAN we set the regularization to 0.3 and $i n c = 1$ , whereas for Chekhov GAN we set the regularization to 0.005 and inc to 50. All the baselines were run using their recommended set of hyperparameters.
+
+![](images/6f5157f646c3a7b6e8511cd70cf88c2c8aa384e017f9da7070735b0630c9f2f3.jpg)  
+Figure 11: Comparison of Inception score: (left) WGAN-based variants, (right) GAN-based variants.The shaded area denotes the standard deviation.

parse/train/S1viikbCW/S1viikbCW.md

@@ -0,0 +1,347 @@
+# TCAV: RELATIVE CONCEPT IMPORTANCE TESTING WITH LINEAR CONCEPT ACTIVATION VECTORS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Neural networks commonly offer high utility but remain difficult to interpret. Developing methods to explain their decisions is challenging due to their large size, complex structure, and inscrutable internal representations. This work argues that the language of explanations should be expanded from that of input features (e.g., assigning importance weightings to pixels) to include that of higher-level, humanfriendly concepts. For example, an understandable explanation of why an image classifier outputs the label “zebra” would ideally relate to concepts such as “stripes” rather than a set of particular pixel values. This paper introduces the “concept activation vector” (CAV) which allows quantitative analysis of a concept’s relative importance to classification, with a user-provided set of input data examples defining the concept. CAVs may be easily used by non-experts, who need only provide examples, and with CAVs the high-dimensional structure of neural networks turns into an aid to interpretation, rather than an obstacle. Using the domain of image classification as a testing ground, we describe how CAVs may be used to test hypotheses about classifiers and also generate insights into the deficiencies and correlations in training data. CAVs also provide us a directed approach to choose the combinations of neurons to visualize with the DeepDream technique, which traditionally has chosen neurons or linear combinations of neurons at random to visualize.
+
+# 1 INTRODUCTION
+
+Neural networks (NNs) are capable of impressively good performance, yet understanding and interpreting their behavior remains a significant challenge. Solving this challenge is an important problem for several reasons. For example, explaining a system’s behavior may be necessary to establish acceptability and see adoption for critical applications, such as those in the medical domain. For scientists and engineers, any greater understanding of how neural networks function is appreciated, since it may lead to better models and help with debugging (30; 19).
+
+Recent work suggests that linear combinations of neurons may encode meaningful, insightful information (2; 19; 27). However, we lack methods to 1) identify which linear combinations (if any) relate to a given concept, and 2) how these can aid in our quantitative understanding of concepts and classification decisions. For example, we may hypothesize that an image model that successfully classifies zebras may naturally encode concepts for ‘stripe’ and ‘animal’, somewhere in its internal representations, using a linear combination of neurons. How can we formalize this notion, and test such a hypothesis?
+
+Neural networks build internal representations that are far richer than the input features or output classes explicit in their training data. Unfortunately, many machine learning interpretation methods provide results only in terms of input features. For example, the learned coefficients in linear classifiers or logistic regression can be interpreted as each feature’s classification importance. Similar first-order importance measures for neural networks often use first derivatives as a proxy for input feature importance, as is done for pixel importance in saliency maps (8; 22).
+
+It is critical that model understanding and interpretation not be limited to only the concepts explicit in training data. This can be seen by considering classification fairness—an increasingly relevant, difficult problem where interpretability can be useful—and noting that no input features may identify discriminated-against groups. For example, the Inception model (26) has an output class for ’doctor’ but no input features identifying the concepts of ’man’ or ’woman’ in a way that would allow existing interpretability approaches to quantify gender bias in classification.
+
+This work introduces the method of concept activation vectors (CAV) for the following purposes. First, CAV can be used to identify linear combinations of neurons in a layer of a model that correspond to given semantic concepts, even for new, user-provided concepts not explicit in the model’s training data. Second, CAV provides quantitative measures of the relative importance of userprovided concepts, which allows for hypothesis testing of the relationship between given concepts and the model’s predictions.
+
+Testing with CAV (TCAV) is designed with the following desiderata in mind.
+
+1. accessibility: Requires little to no user expertise in machine learning.   
+2. customization: Adapt to any concept of interest (e.g., gender) on the fly without pre-listing a set of concepts before training.   
+3. plug-in readiness: Work without retraining or modifying the model.   
+4. quantification: Provide quantitative explanation that are tied to human-relatable concept, and not input features.
+
+One of key ideas for TCAV is that we can test the relative importance between small set of concepts, rather than ranking the importance of all possible features/concepts. For example, we can gain insights by testing whether the concept of gender was used more than the ‘wearing scrubs’ concept for the classification of doctor. We can also test whether or not a given concept was relevant to the classification of a certain class. Similar forms of sparsity (i.e., only considering a few concepts at a time) are used in many existing interpretable models (12; 7; 28; 31; 29; 4). Note that interpretability does not mean understanding the entire network’s behavior on every feature/concept of the input (6). Such a goal may not be achievable, particularly for ML models with super-human performance (23).
+
+TCAV satisfies these desiderata—accessibility, customization, plug-in readiness and quantification —it enables quantitative relative importance testing for non-ML experts, for user-provided concepts without retraining or modifying the network. Users express their concepts of interest using examples—a set of data points exemplifying the concept. For example, if gender is the concept of interest, users can collect pictures of women. The use of examples has been shown to be powerful medium for communication between machine learning (ML) models and non-expert users (16; 12; 13). Cognitive studies on experts also support this approach (e.g., experts think in terms of examples (15)).
+
+The structure of this paper is as follows: Section 2 relates this work to existing interpretability methods. Section 3 explains the details of the TCAV method. In Section 4, we show 1) how this framework can be used to identify semantically meaningful directions in a layer and 2) the relative importance testing results that measure the relevance of concepts of interest to the classification output by the network.
+
+# 2 RELATED WORK
+
+In this section, we provide a brief overview of existing related interpretability methods and their relation to our desiderata. We also discuss the need and the challenges of desiderata 3): plug-in readiness.
+
+# 2.1 SALIENCY MAP METHODS
+
+One of the most popular approaches in interpreting NN is saliency methods (24; 22; 25; 8; 5). These techniques seek to identify regions of the input most relevant to the classification output by the network. Qualitatively, these methods often successfully label regions of the input which seem semantically relevant to the classification.
+
+Unfortunately, these methods do not satisfy our desiderata of 2) customization and 4) quantification. The lack customization is clear, as the user has no control over what concepts of interest these maps pick up on. Regarding quantification, there is no way to meaningfully quantify and interpret the brightness of various regions in these maps. As a hypothetical example, consider two saliency maps of two different cat pictures, where the brightness of the ears of the two cats differ. It is unclear both how to quantify ‘brightness’ and second what kind of actionable insights this level of brightness gives.
+
+Recent work has also demonstrated that the saliency map these methods produced may be very sensitive to completely uninteresting properties of the data distribution. In particular, (14) showed that simply applying a mean shift to the dataset may cause some saliency methods to result in significant changes in the given explanation. (9) also showed that saliency methods can be easily tricked.
+
+# 2.2 DEEPDREAM, NEURON LEVEL INVESTIGATION METHODS
+
+There are techniques, such as DeepDream, which can be used to visualize patterns that maximally activates each neuron of a neural network. The technique starts from an image of random noise and iteratively modifies the image in order to maximally activate a neuron or a linear combination of neurons of interest (17; 18). This technique has offered some insights into the information encoded in a neuron’s activation. This technique also has opened up opportunities for AI-aided art (17).
+
+However, the DeepDream method does not satisfy our desiderata 1) accessibility, 2) customization, and 4) quantification. It does not satisfy 1) because in order to apply it a user must first understand what a neuron is, and second be familiar enough with the the internal architecture in order to choose which neurons to visualize. It does not satisfy 2) because no current method exists to find which neurons correspond to semantically meaningful concepts such as gender, and it is unclear whether or not such a neuron even exists. It does not satisfy 4) because we do not understand these pictures and there is currently no method to quantify how these pictures relate to the output of the network. This method again does not provide actionable insight.
+
+As we show later, DeepDream may be combined with TCAV in order to identify and visualize interesting directions in a layer. Prior work on DeepDream has typically chosen neurons or linear combinations of neurons at random to visualize. Note that a user of TCAV does need to pick a layer in the network for which to apply TCAV to. However, if only the final prediction is concerned, the last layer can be used by default.
+
+# 2.3 WHY WE NEED DESIDERATA 3 - PLUG-IN READINESS
+
+To achieve interpretability, we have two options: (1) restrict ourselves to inherently interpretable models or (2) post-process our models in way that yields insights.
+
+Users may choose option (1) as there are a number of methods to build inherently interpretable models (12; 7; 28; 31; 29; 4). If the users are willing and are able to adapt these models, then this is the gold standard. Although building inherently interpretable machine learning models may be possible for some domains, doing so may result in decreased performance. Furthermore, changing the model may be costly for users who already have a model implemented.
+
+A method that can be applied without retraining or modifying the network could be instantly used to interpret existing models. Increasingly, attention is turning to the importance of providing explanations for ML decisions (for one example, consider (10)). As a result, there is a growing need for interpretation techniques that can be applied “out of the box,” that is, without rebuilding or retraining existing systems.
+
+# 2.4 A NOTE ON DESIDERATA 3 - PLUG-IN READINESS AND LOCAL EXPLANATIONS
+
+One of many challenges of building a post-processing interpretion method is to ensure that the explanation is truthful to the model’s behavior. By “truthful” we mean that explanations are roughly consistent with model’s internal state. For example, if the explanations are created completely independently of the model (11), it has high probability of having such inconsistencies.
+
+The plug-in readiness desiderata poses interesting challenge for explanations to remain consistent with the model’s behavior. Recently, there has been work showing that saliency methods contains such inconsistencies. For instance, (14) show that saliency methods are vulnerable to constant shift in input that does not affect classification. It has also shown that the methods can be easily tricked (9).
+
+One way to improve consistency between explanations and the model’s reasoning is to use the generated explanation as an input, and check the network’s output for validation. This is typically used in perturbation-based interpretability methods (16; 20). These methods perturb the input, and use the network’s response to generate explanations. They maintain the consistency either locally or globally1 by construction. TCAV is a type of perturbation method.
+
+Even a truthful explanation may be misleading if it is only locally truthful (20). For example, since the importance of features only needs to be truthful in the vicinity of the data point of interest, there is no guarantee that the method will not generate two completely conflicting explanations. These inconsistencies may result in decreased user trust at a minimum. On the other hand, making a globally truthful explanation may be difficult as the networks decision boundaries may be complex. TCAV produces globally explanations, and uses model’s output to generate explanations to maintain consistency between explanations and the model’s reasoning.
+
+# 3 TCAV METHOD
+
+We introduce a method that allows for global linear interpretability of a highly flexible class of models, namely deep feedforward neural networks trained on classification tasks. As a form of explanation, TCAV uses concepts that are provided by users, instead of using predetermined set of input features or input classes. These concepts are tied to real-world data that represents concepts of interest. Users express their concepts of interest using examples — a set of data points exemplifying the concept. These concepts enable testing the relative importance of concepts used in classification.
+
+Informally, the key idea is the following: although feedforward networks learn highly nonlinear functions there is evidence that they work by gradually disentangling concepts of interest, layer by layer (2; 3). It has also been shown that representations are not necessarily contained in individual neurons but more generally in linear combinations of neurons (19; 27). Thus the space of neuron activations in layers of a neural network may have a meaningful global linear structure. Furthermore, if such a structure exists, we can uncover it by training a linear classifier mapping the representation in a single layer to a human selected set of concepts.
+
+We now formalize this intuition. First, we formally define a concept activation vector. Let us imagine that an analyst is interested in a given concept $C$ (e.g., striped textures) and has gathered two sets of data points, $P _ { C }$ and $N$ , that represent positive and negative examples of this concept (say, photos of striped objects, versus a set of random photos). There is a lot of flexibility in how to choose $N$ , we often choose a random set of images of the same size as $P _ { C }$ . Here we represent the input to the network as vector in $\mathbb { R } ^ { n }$ , and $P _ { C } , N \subset \mathbb { R } ^ { n }$ . Consider a layer $l$ of the feedforward network consisting of $m$ neurons. Then running inference on an input example and looking at the activations at layer $l$ yields a function $f _ { l } : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ . For a set of inputs $X \subseteq \mathbb { R } ^ { n }$ we denote by $f _ { l } ( X )$ to be the set of layer activations $\{ f _ { l } ( x ) : x \in X \}$ . Note that for convolutional networks we view a layer in it’s flattened form, thus a layer of width $w$ , height $h$ , and $c$ channels becomes a flat vector of $m = w \times h \times c$ activations.
+
+The two sets $P _ { C }$ and $N$ then naturally give rise to two sets of activation vectors in layer $l$ , namely $f _ { l } ( P _ { C } )$ and $f _ { l } ( N )$ . We can then train a linear classifier on the binary classification task of distinguishing between these two sets. The weights of this classifier are an element $\boldsymbol { v } _ { C } ^ { l } \in \mathbb { R } ^ { m }$ . We call this the concept activation vector for the concept $C$ .
+
+A variation on this idea is a relative concept activation vector. Here the analyst selects two sets of inputs that represent two different concepts, $C$ and $D$ . Training a classifier on $f _ { l } ( P _ { C } )$ and $f _ { l } ( P _ { D } )$ yields a vector $v _ { C , D } ^ { l } \in \mathbb { R } ^ { m }$ . The vector $\bar { v } _ { C , D } ^ { l }$ intuitively defines a $1 - d$ subspace in layer $l$ where the projection of an embedding $f _ { l } ( x )$ along this subspace measures whether $x$ is more relevant to concept $C$ or $D$ .
+
+A key benefit of this technique is the flexibility allowed in choosing the set $P _ { C }$ . Rather than being tied to the class labels in the training data, an analyst can–after training–select sets that correspond to any concept of interest.
+
+We now describe two ways that this technique be used to interpret a feedforward neural network trained on an image classification task. First, we show relative importance testing between $M$ number of concepts, $P _ { C i }$ , where $i \in M$ . In this case, each $v _ { C } ^ { l }$ is learned by treating all $P _ { C i } , i \neq j$ as $N$ . Second, we can also test one concept, $P _ { C }$ , against a random concept, where $N$ are set of random pictures. In the next section we discuss the results of experiments with these methods.
+
+# 3.1 APPLICATION: TESTING THE IMPORTANCE OF CONCEPTS IN CLASSIFICATION
+
+The real value of CAVs comes from the fact that they may be used to test the relative importance of concepts. With CAVs, we can formulate generating explanation as a task of performing two-tailed statistical significance test (in particular, z-test). Given samples of class images (e.g., zebra pictures) and two concept vectors A and B, we perform two-tailed $\mathbf { Z }$ -testing to invalidate the null hypothesis that there is no difference in importance of concepts A and B for the class. We perform this testing for each pair of concepts. For example, an analyst might ask about a photo of a zebra, “was the presence of stripes in the image relevant to the model’s classification of the image as a zebra?” In some cases, with tedious effort in a photo retouching program, it might be possible answer questions like this directly. However, CAVs provide a faster and more general technique.
+
+Consider an input $x$ and concept $C$ of interest. At inference time, this will give rise to activations $f _ { l } ( x )$ at layer $l$ . We can then modify this vector of activations in various ways using the concept vector $v _ { C } ^ { l }$ . For example, we might create a new vector $w ^ { - } = f _ { l } ( x ) - v _ { C } ^ { l }$ , which might be interpreted as a counterfactual of “ $x$ with less of concept $C ^ { \mathfrak { n } }$ . Performing inference starting at layer $l + 1$ based on $w ^ { - }$ will give a new classification result $y _ { w }$ . We can also create a new vector $w ^ { + } = f _ { l } ( x ) + v _ { C } ^ { l }$ , which might be interpreted as a counterfactual of $^ { 6 6 } x$ with more of concept $C ^ { \mathfrak { n } }$ .
+
+Thus to test the relevance of stripes to the classification of a zebra, we would either add or subtract a ‘stripe’ concept activation vector from the zebra image embedding, run forward propagation the resulting embedding and examine the output of the network. Large changes in the probability of zebra would indicate that stripes played a key role in the classification. Simply adding the vector is a bit ad-hoc, especially when the norm of the vector may depend on how exactly the CAV was trained. However, we found this naive method to empirically yield results that were consistently semantically meaningful. We also note that this addition is loosely related to directional derivative – saliency maps take the derivative of the logits with respect to each pixel, while our work takes derivatives with respect to a concept direction. Future work should explore more principled methods, to measure relevance of concepts to classification.
+
+Quantitatively, the metric of the influence of the concept $C$ to class $k$ can be measured by
+
+$$
+I _ { w } ^ { u p } = \frac { 1 } { N } \sum _ { i } ^ { N } \mathbb { 1 } ( p _ { k } ( y ) < p _ { k } ( y _ { w } ) ) \qquad I _ { w } ^ { d o w n } = \frac { 1 } { N } \sum _ { i } ^ { N } \mathbb { 1 } ( p _ { k } ( y ) > p _ { k } ( y _ { w } ) )
+$$
+
+where $p _ { k } ( y )$ represents the probability of class $k$ of prediction $y$ and $i = \{ 1 , 2 , \dots , N \}$ where $N$ is the number of images of the class for inspection (e.g., zebra). Intuitively speaking, $I _ { w } ^ { u p / d o w n }$ measures the ratio of data points that become ‘more/less like class $k '$ after the modification with concept vector $v _ { C } ^ { l }$ .
+
+We can perform statistical significance testing in order to quantify the concept importance; we can test a hypothesis that one concept is more/less important than another concept. In other words, the null hypothesis that no color is significant. To test this hypothesis we can do $\mathbf { z }$ -testing on the measured importance values, and ask the question what is the probability that random concept vectors would observe the measured difference.
+
+# 3.2 APPLICATION: WHERE ARE CONCEPTS LEARNED?
+
+As a simple demonstration of the value of CAVs, we describe a technique for localizing where a concept is disentangled in a network. A common view of feedforward networks is that different concepts are represented at different layers. For example, many image classification networks are said to detect textures at low levels, with increasingly complex concepts recognized at later layers. Typically evidence for this view comes from various methods of reverse-engineering (3; 30; 17) the behavior of individual neurons in the network.
+
+Using concept activation vectors, however, we can approach this question from the opposite direction. We can pick out a particular concept $C$ in advance, and consider the concept vectors $v _ { C } ^ { 1 } , v _ { C } ^ { 2 } , \ldots , v _ { C } ^ { \hat { k } }$ for each layer in the network. The accuracies of the linear classifiers, $a _ { C } ^ { 1 } , a _ { C } ^ { 2 } , \dots , a _ { C } ^ { k }$ then provide a measure of how well disentangled the concept is in each layer. This may be viewed as a generalization of the ideas in (2; 3; 19). In the next section we describe the results of an experiment based on this idea.
+
+# 4 RESULTS
+
+In this section, we first show evidence that the learned CAV indeed detect the concepts of interest. Then we show the results for hypothesis testing using these CAVs, and the insights one can gain from it.
+
+All our experiments are based on the model from (26) using the publicly available model parameters from (1). The pictures used to learn the CAVs are collected from the latest ImageNet Fall 2011 release (21). Each concept vector is learned using between 30-500 pictures as input. We intentionally chose not to use all the available pictures in order to mirror realistic scenarios where users may not be able to collect a large amount of data. The class ‘arms’ only had 30 pictures (all manually collected), since ImageNet does not include this as a part of the dataset. The pictures used to learn texture concepts are taken from the (3) data set. In order to learn concept vectors for colors, we generated 500 pictures synthetically by generating the color channels randomly. We provide experiments both where CAV’s are trained to distinguish between a set of concepts (e.g. red, yellow, green, blue) and when one CAV is trained per concept (with $N$ chosen to be a random collection of images).
+
+# 4.1 EVIDENCE OF THE VALIDITY OF LINEARITY ASSUMPTION
+
+In this section we describe experiments that indicate our linearly learned CAVsalign with their intended concepts. First, the linear maps used to construct the CAVsare accurate in predicting the concepts. The point in the network where these concepts in the networks are learned (i.e., when accuracy becomes high) is consistent with general knowledge about the representations learned by neural networks. Low level features such as edges and textures are detected in the early layers and higher level concepts are only reliably detected in the upper layers. Next, we use activation maximization techniques (18) to visualize each of the CAVs and observe that the patterns are consistent with the provided concept. Finally, we show the top $k$ images that are most similar in terms of cosine similarity to each CAV for further qualitative confirmation.
+
+# 4.1.1 WHERE EACH CONCEPT IS LEARNED.
+
+Figure 1 shows the accuracy of the linear classifiers at each layer in the network for each type of CAV. Each classifier is trained to distinguish one concept from other concepts (e.g., textures1 set contains ‘stripes’, ‘zigzagged’ and ‘dotted’ texture concepts). Overall, we observe high accuracy as measured by a held out test set of size 1/3 that of the training size. This is evidence that the given concepts are linearly separable in many layers of the network.
+
+Note that the accuracy of more abstract CAV (e.g., objects) increases in higher layers of the network. The accuracy of a simpler CAV, such as color, is high throughout the entire network. This agrees with prior work on visualizing the learned representations at different layers of neural networks (30).
+
+This experiment does not yet show that these CAVs align with the concepts that makes sense semantically to humans. We demonstrate this with the next set of experiments.
+
+![](images/1441b37dc50d080387d63509b7c11bc69d89d54a33c77b39193f5bb58be65338.jpg)  
+Figure 1: Where each of the CAVs are learned. Accuracy at each layer for CAV. Simple concepts (e.g., colors) achieve high performance in all layers than more abstract concepts (e.g. persons, objects)
+
+![](images/49b8d0a897236ccfa31ec4a26fc54cfdd0b00fd124a51d76cf7e67f7a0396894.jpg)  
+Figure 2: Deepdreamed fibrous texture, knitted texture, corgis and Siberian huskey CAVs.
+
+In this section, we use the activation maximization technique (17) to visualize the learned representations in the direction of the CAV. We use the same techniques from (18). As typically done, we use a random image as a starting point (17) for the optimization to avoid choosing an arbitrary image as a starting point. Figure 2 shows highly recognizable features, such as knitted textures, and corgi faces.
+
+![](images/ee49dc3172e84df9b8f15518b7c2ee18146d0eb960a82555b5e0fe4c645a38f9.jpg)  
+Figure 3: Deepdreamed CAV texture concepts for each layer (each row) in inception. We can identify textures better in the mid-layer (mixed4d), then later layers.
+
+We can also observe how each concept is represented as the depth increases. Images in Fig. 3 show the results for sets of textures. Interestingly, there is a layer (around mixed 4d) where textures are clearly identifiable, after which they become less and less recognizable.
+
+The left images in Fig. 4 show the results for set of colors — green, blue, yellow and red (each column) and for the set of layers (lower to higher layers from top to bottom). The training set for the color CAVs are generated by simply replacing each RGB channel with randomly sampled values around 0.5, while leaving other channels empty. Note how the color CAVs also contain many random textures, this is perhaps not surprising - there is no reason to expect any direction in a layer to be associated purely with one concept while being orthogonal to all other concepts.
+
+The right image in Fig. 4 shows higher level concepts. The columns represent zebra, siberian huskies, and corgis. It is interesting to note how the zebra CAVs include textures suggestive of water, trees, and grass. This suggests that the model associates all of these concepts with the classification of zebra and is likely a learned bias that results from the background of zebra pictures in the training data.
+
+![](images/5607ed63116e3a31d0927291a67f547d95acf985633cb45cd990a661de836fc5.jpg)  
+Figure 4: Deepdreamed CAV color and animal concepts for each layer (each row) in inception. Zebra CAVs include textures suggestive of water, trees and grass.
+
+These visualizations provide some qualitative confirmation that the learned directions align with the given concepts. This also shows that DeepDream may be combined with TCAV in order to identify and visualize interesting directions in a layer. Prior work on DeepDream has typically chosen neurons or linear combinations of neurons at random to visualize. The next section provides further evidence that the CAVs are indeed aligned with the concept of interest using real data.
+
+# 4.1.3 QUALITATIVE CONFIRMATION: PICTURES THAT ARE SIMILAR TO CAV
+
+In order to qualitatively confirm that the learned CAV aligns with meaningful concepts, we compute cosine similarity between a set of pictures (all from one class) to the CAV. Fig. 5 shows that the top corgi images that aligns with striped CAV selects pictures with striped objects (e.g., a striped tube or vertical blinds in the background). Thus being similar to the striped CAV meant being highly similar to one of the other concepts in the CAV training set. We also show that CAVs for more abstract concepts can be constructed (e.g., CEO). Recall that many of these learned concepts are not classes that the NN learned to classify.
+
+We also observe that if there is not much relation between pictures and the concept, the cosine similarity is low. This experiment qualitatively supports our argument that the alignment between CAVs and the meaningful concepts. Note that the images least similar to striped appear with a checkered or knitted pattern, likely due to the fact that the striped CAV was trained relative to checkered and knitted images.
+
+![](images/eeca4cd44f32c8fec820df8d34b0f4df61cc3640eaabae50ca682a1e05fdbeb4.jpg)  
+Figure 5: Top and bottom 3 similar pictures of each concept
+
+When using TCAV, in order to ensure that the provided examples were sufficient to learn CAV, we recommend users to perform this qualitative test by checking cosine similarity with each concepts to the class.
+
+# 4.2 TESTING RELATIVE IMPORTANCE OF CONCEPTS
+
+In this section, we describe how concept activation vectors may be used for quantitative testing of the relative importance of concepts.
+
+Inof 6, we show . This mea $I _ { w ^ { + } } ^ { u p }$ for eight classes. We do not list at when CAVs are added, the p $I _ { w ^ { - } } ^ { u p }$ , as they always appear to be the inversebility of the class rarely stays the same. $I _ { w ^ { + } } ^ { u p }$ This confirms that adding or subtracting $v _ { C } ^ { l }$ clearly impacts the classification task. In particular the $v _ { C } ^ { l }$ direction is aligned with the direction that measures the probability of the target class.
+
+Note that adding random directions should be expected to change the probability of the target class in some way. We find, however, that for several CAV’s which are semantically relevant to the class (e.g., red to fire engine) the CAVdirection will consistently increase the probability of class for most images of this class. On the other hand, we show that random directions (e.g., a concept learned from random set of images) tend to be much less consistent. This difference allows us to perform a more rigorous hypothesis test in a following section.
+
+This testing identifies relationships between concepts and the target class that agree with human intuition (e.g., fire engines are red, cucumbers are bumpy, zebras are striped, CEOs wear suits). In the next section, we describe insights we gained from these tests.
+
+![](images/870201a410f3cfcd791e0af99ea5534afbc501e22d44e82f189663921649aef0.jpg)  
+Figure 6: Testing Relative Importance of Concepts
+
+# 4.3 GAINING INSIGHTS
+
+In this section, we show results that confirm common-sense intuitions about training images, as a kind of sanity check that TCAV provides valid results. We then describe results that surprised us, but lead to insights about the dataset that we did not initially have.
+
+In Fig 6, the yellow color was more influential to cab class than any other colors. For people who have lived in cities where this is the standard color for taxis, this is no shock — and indeed, most ImageNet pictures of cabs are yellow. Similarly, the ‘women’ concept is important to ‘bikini’ class, likely due to the fact that training images have humans wearing the bikini rather than pictures of the clothing. More subtly, we also discovered that the ‘model women’ concept is important to ‘bikini’ This lead us to realize that the most of ‘bikini’ class pictures feature professional models, typically very slim, posing in their bikinis — probably not well representative samples of bikinis in the real world. Note that the network was trained with bikini and cab classes, and the concept ‘yellow’, ‘women’ and ‘model women’ are not part of classes used to train this network2.
+
+The graph for ‘dumbbell’ class in Fig 6 shows that ‘arms’ concept was more important to predict dumbbell class than other concepts. This finding is consistent with previous qualitative findings from (17), where they discovered that the DeepDream picture of a dumbbell also showed an arm holding it. TCAV allows for quantitative confirmation of this previously qualitative finding. Moreover, unlike all other concepts in this figure, we only collected 30 pictures of each concept (ImageNet did not have arms as a label). Despite the small number of examples, the TCAV method is able to identify this relationship.
+
+The flexibility of the TCAV method makes it easy to explore a network for other surprising associations. For example, we saw that ‘baby’ concept was important to ‘school bus’ class. It turns out that some school bus pictures include young children riding or standing in front of the bus. Another discovery was to find that ‘red’ concept is important for cucumber class. We believe this due to the fact that a number of cucumber pictures include other red vegetables, such as tomato and carrot.
+
+We believe that TCAV can be useful in discovering these insights for many applications, including for example, to improve fairness in the model.
+
+# 4.4 RANDOM IMAGE COMPARISONS TO FILTER SPURIOUS RESULTS
+
+When constructing the CAV, the choice of negative samples may cause spurious suggestions that a concept is relevant to a class when it actually is not. In other words, the learned CAV may accidentally be aligned with something related to the class, and cause high $I _ { w ^ { + } } ^ { u p }$ . For instance, when we tested unrelated concepts and classes, such as zebra to a set of textures, honeycombed, lace-like and bumpy textures, we found that lace-like concept shows high $I _ { w ^ { + } } ^ { u p }$ .
+
+![](images/6447cdbc97429574a74d25523ef26a8ac68c7afca3b9608e6f7718e25f29d984.jpg)  
+Figure 7: Histogram of the mean of and random concepts. $I _ { w ^ { + } } ^ { u p }$ for the relationship of the zebra class with striped, lace-like
+
+One might argue that lace is vaguely related to a zebra’s stripes, but in fact even if we choose both $P _ { C }$ and $N$ to be independent random sets of images we still often observe $I _ { w ^ { + } } ^ { u p }$ to be relatively high. This is not surprising: there are directions in a layer $l$ which are highly aligned with the zebra concept, and randomly chosen directions will typically have small but non-zero projection along these directions.
+
+One way to filter out these spurious results is to do statistical testing against random concepts. Using different sets of random images for the negative class and striped images for $P _ { C }$ , we make 50-100 CAVs, all of which represent concept ‘striped’. For each ‘striped’ CAV we can measure $I _ { w ^ { + } } ^ { u p }$ . We can also generate a set of ‘random’ CAVs by choosing random images for both $P _ { C }$ and $\ddot { N }$ . Then we performed a $z$ -test to see if the mean $I _ { w ^ { + } } ^ { u p }$ s from striped CAVs are statistically different from the mean Iup+ of random CAVs. We can successfully filter out some spurious correlations, including lace-like concepts with zebras by using this method, see the histogram 4.4.
+
+# 5 CONCLUSION
+
+We have introduced the notion of a “concept activation vector,” or CAV, which is a flexible way to probe the internal representation of a concept in a classification network. Since CAVs may be defined via a set of example inputs, rather than custom coding, they are well suited to use by non-experts. We then described a technique (Testing with CAVs, or TCAV) for quantifying the relation between a CAV and a particular class. The TCAV technique allows us to provide quantitative answers to questions such as, “How important are the stripes to the classification of a zebra?”
+
+To provide evidence for the value of the TCAV technique, we described a series of experiments which supported common-sense intuition, for example, that stripes are indeed important to the identification of zebras. In addition, we used the DeepDream technique to create images whose internal representations approximate certain CAVs. The resulting pictures were strongly evocative of the original concepts. Finally, we described how the TCAV technique may be used to find associations between concepts, both obvious (“yellow” and “taxi”) and non-obvious (“red” and “cucumber”).
+
+In addition to analyzing a single network, TCAV can be also used to compare and contrast a pair of networks. For example, one can compare the relative importance of concepts to determine how the different choices of training process or architecture influences learning of each concept. Based on the results, users can perform model selection based on the concepts that are more or less important for the task.
+
+An interesting direction for future work may be to explore applications of using CAVs to adjust the results of a network during inference time. Adding a scalar multiple of a CAV to the activations of an intermediate layer can, as shown in our experiments, allow us to deemphasize or enhance conceptual aspects of an input. One potential application, for example, might be to reduce bias the network has learned from training data.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank Chris Olah, Alexander Mordvintsev and Ludwig Schubert for generously allowing us to use their code for DeepDream.
+
+# REFERENCES
+
+[1] "http://storage.googleapis.com/download.tensorflow.org/models/ inception5h.zip", 2017.   
+[2] Guillaume Alain and Yoshua Bengio. Understanding intermediate layers using linear classifier probes. arXiv preprint arXiv:1610.01644, 2016.   
+[3] David Bau, Bolei Zhou, Aditya Khosla, Aude Oliva, and Antonio Torralba. Network dissection: Quantifying interpretability of deep visual representations. In Computer Vision and Pattern Recognition, 2017.   
+[4] Rich Caruana, Yin Lou, Johannes Gehrke, Paul Koch, Marc Sturm, and Noemie Elhadad. Intelligible models for healthcare: Predicting pneumonia risk and hospital 30-day readmission. In KDD, 2015.   
+[5] Piotr Dabkowski and Yarin Gal. Real time image saliency for black box classifiers. arXiv preprint arXiv:1705.07857, 2017.   
+[6] Been Doshi-Velez, Finale; Kim. Towards a rigorous science of interpretable machine learning. In eprint arXiv:1702.08608, 2017.   
+[7] Finale Doshi-Velez, Byron C Wallace, and Ryan Adams. Graph-sparse lda: A topic model with structured sparsity. In Aaai, pages 2575–2581, 2015.   
+[8] Dumitru Erhan, Yoshua Bengio, Aaron Courville, and Pascal Vincent. Visualizing higher-layer features of a deep network. University of Montreal, 1341:3, 2009.   
+[9] Amirata Ghorbani, Abubakar Abid, and James Zou. Interpretation of neural networks is fragile. arXiv preprint arXiv:1710.10547, 2017.   
+[10] Bryce Goodman and Seth Flaxman. European union regulations on algorithmic decisionmaking and a” right to explanation”. arXiv preprint arXiv:1606.08813, 2016.   
+[11] Lisa Anne Hendricks, Zeynep Akata, Marcus Rohrbach, Jeff Donahue, Bernt Schiele, and Trevor Darrell. Generating visual explanations. In European Conference on Computer Vision, pages 3–19. Springer, 2016.   
+[12] Been. Kim, Cynthia Rudin, and Julie Shah. The Bayesian Case Model: A generative approach for case-based reasoning and prototype classification. In NIPS, 2014.   
+[13] Been Kim, Julie Shah, and Finale Doshi-Velez. Mind the gap: A generative approach to interpretable feature selection and extraction. In Advances in Neural Information Processing Systems, 2015.   
+[14] Pieter-Jan Kindermans, Sara Hooker, Julius Adebayo, Maximilian Alber, Kristof T Schutt,¨ Sven Dahne, Dumitru Erhan, and Been Kim. The (un) reliability of saliency methods. ¨ arXiv preprint arXiv:1711.00867, 2017.   
+[15] G.A. Klein. Do decision biases explain too much. HFES, 1989.   
+[16] Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. arXiv preprint arXiv:1703.04730, 2017.   
+[17] Alexander Mordvintsev, Christopher Olah, and Mike Tyka. Inceptionism: Going deeper into neural networks. Google Research Blog. Retrieved June, 20:14, 2015.
+
+[18] Chris Olah, Alexander Mordvintsev, and Ludwig Schubert. Feature visualization. Distill, 2017. https://distill.pub/2017/feature-visualization.
+
+[19] Maithra Raghu, Justin Gilmer, Jason Yosinski, and Jascha Sohl-Dickstein. Svcca: Singular vector canonical correlation analysis for deep understanding and improvement. arXiv preprint arXiv:1706.05806, 2017.   
+[20] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. “why should i trust you?”: Explaining the predictions of any classifier. arXiv preprint arXiv:1602.04938, 2016.   
+[21] Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015.   
+[22] Ramprasaath R Selvaraju, Abhishek Das, Ramakrishna Vedantam, Michael Cogswell, Devi Parikh, and Dhruv Batra. Grad-cam: Why did you say that? arXiv preprint arXiv:1611.07450, 2016.   
+[23] David Silver, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George van den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalchbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu, Thore Graepel, and Demis Hassabis. Mastering the game of go with deep neural networks and tree search. 2016.   
+[24] Daniel Smilkov, Nikhil Thorat, Been Kim, Fernanda Viegas, and Martin Wattenberg. Smooth- ´ grad: removing noise by adding noise. arXiv preprint arXiv:1706.03825, 2017.   
+[25] Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. arXiv preprint arXiv:1703.01365, 2017.   
+[26] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Computer Vision and Pattern Recognition (CVPR), 2015.   
+[27] Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.   
+[28] Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58:267–288, 1994.   
+[29] Berk Ustun, Stefano Traca, and Cynthia Rudin. Supersparse linear integer models for inter- \` pretable classification. arXiv preprint arXiv:1306.6677, 2013.   
+[30] Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pages 818–833. Springer, 2014.   
+[31] Hui Zou, Trevor Hastie, and Robert Tibshirani. Sparse principal component analysis. Journal of Computational and Graphical Statistics, 15:2006, 2004.
+
+top 3 images of corgis similar to knitted concept top 3 images of salmon similar to dotted concept top 3 images of zebra similar to striped concept bottom 3 images of zebra similar to striped concept top 3 images of salmon similar to striped concept top 3 images of corgis similar to porous concept bottom 3 images of salmon similar to porous concept top 3 images of CEO similar to labcoat concept
+
+![](images/01b3822fb3eea757f6fbb991156dc9e825965ef8eac09c9a03467497b05c3c08.jpg)  
+bottom 3 images of corgis similar to knitted concept
+
+![](images/f5ddb8ddf0f6c759ef7df7ffefc00591c09695ac4900f21db33229d3d1dd6fcb.jpg)  
+top 3 images of salmon similar to knitted concept
+
+![](images/c1b24a920c554f159642c276a82f7cd711be40997195c96e7ef2b3f09899949e.jpg)  
+bottom 3 images of salmon similar to knitted concept
+
+![](images/bb600dc8143c4759f82ec90d2defd3e58f16acd566de3df5a16d3712964e62f7.jpg)  
+top 3 images of corgis similar to dotted concept
+
+![](images/4422827ed9964fad3b546446cb9b3718dfd6836d3abde969a3283b6cd40955fb.jpg)  
+bottom 3 images of corgis similar to dotted concept
+
+![](images/f31229d80a5b9117fe451fb3313d1cde2c55eed737bb5ac8637080bd6239dd33.jpg)
+
+![](images/eb15b1d1aef2abe5ed47e345643091095d978f926bfaa5ec9944f5a65177c190.jpg)  
+bottom 3 images of salmon similar to dotted concept
+
+![](images/42a7f50c718bd58e8ca7e7321a0752f1b45ba0d35e68a7aa4b3f8ede06d7a8ac.jpg)
+
+![](images/f18ecbe9b05f2a322a30aded5ce92211f9cb540bce5a55871d42a1c01724f544.jpg)
+
+![](images/af7f653dbbc97cf2f1003ac953309319d1daa94b98f4f810816f256fc210ca55.jpg)
+
+![](images/9eb868e11fe2c7c693de4e430d8a0fa4d917206b2d6c1cb6214886ae43acdeac.jpg)  
+bottom 3 images of salmon similar to striped concept
+
+![](images/a9194247c8e9bfef4d529a103e0b2c7ec250d9250a4a246da754d57c7888f300.jpg)
+
+![](images/659e698cd8c34a679eae7b4cb7507e3acb93c2395006885442ed31c1082f135c.jpg)
+
+![](images/22c4eba3ee7b905bc7c45217211f1ff6b0df98129aa23466cfe74fa12b5da9d0.jpg)  
+bottom 3 images of corgis similar to porous concept
+
+![](images/b1bcea7459e07dbcb2fdae25ba0a3c61806985b8d9b3dcbfcb2ce126c0dcf6a4.jpg)  
+top 3 images of salmon similar to porous concept
+
+![](images/2468eee00877786d08f73a6af3013c317a436a1597b64db2cfc027277286c31b.jpg)
+
+![](images/96b3ed5be1dfa39fc2b8d09d1f2f4cd2aac2baa42026271f35d95c6b108a4b9e.jpg)
+
+# bottom 3 images of CEO similar to labcoat concept
+
+![](images/416ccbc89685be2f129eb829d141d841dfa7762d0493c809ee76158f28dd70c4.jpg)
+
+# top 3 images of doctor similar to labcoat concept
+
+![](images/632b3bdc7bb382e19cdb36a9e3c36935ecf5e4ccfdc0d382c80af8ad83284abf.jpg)
+
+# bottom 3 images of doctor similar to labcoat concept
+
+![](images/27ecbafc0375cb6b45e2248924df4b7cf103a59f796661d4bff81dbddd5c9ef1.jpg)
+
+top 3 images of boss similar to labcoat concept
+
+![](images/22e90a163737e447c938bc9ada629835c39b4027152e744a2f409a698fb682c4.jpg)
+
+# bottom 3 images of boss similar to labcoat concept
+
+![](images/ad6ba7f5ef0a2c4f3856c002247f076228e571e888b8959f44cbc453e550b76b.jpg)
+
+top 3 images of model women similar to labcoat concept
+
+![](images/b78c3c794c3d27365046be17b7257c00c65df1cfc5320f01f3b951427683b284.jpg)
+
+# bottom 3 images of model women similar to labcoat concept
+
+![](images/c4d158efb5dba5094eedba6d313430cbb69614514117962ccee926838234bd71.jpg)
+
+# top 3 images of CEO similar to suit concept
+
+![](images/9e6c40a4d640ea959afdeac711037aeefc4f45015d875353d98914b0296707bd.jpg)  
+bottom 3 images of CEO similar to suit concept
+
+![](images/84b29d7baddfc7416441ffa1c84629941cafa7aeb63ffbc43f1314d56e3295c3.jpg)
+
+top 3 images of head nurse similar to suit concept
+
+![](images/7941aece2d284b2d12ae06e37c20b7418489af82cbdddf6678901e3ab7f4bee6.jpg)
+
+# bottom 3 images of head nurse similar to suit concept
+
+![](images/822a296548f1c1a1b0389763ba65f4a8fee539dc7c888616791c3fffb2974b05.jpg)
+
+top 3 images of boss similar to suit concept
+
+![](images/5a97202b4da32a100ff39dded02c3dc8baa1ed42fc1df81528eef7a0e4522bc4.jpg)
+
+# bottom 3 images of boss similar to suit concept
+
+![](images/5f017e47462b8d08f74f5449d1373eec586ee2a4081e300a7df27e3d123a668d.jpg)
+
+![](images/ed56d96c321b63e0a6e943406764402efa529b05def270c5c06b1e4dde41c051.jpg)
+
+![](images/0e1e21e63967bff36b2fb4201ac3e842dfd9a9b3d53c324eda59db1232c79eba.jpg)

parse/train/S1viikbCW/S1viikbCW_content_list.json

@@ -0,0 +1,1938 @@
+[
+    {
+        "type": "text",
+        "text": "TCAV: RELATIVE CONCEPT IMPORTANCE TESTING WITH LINEAR CONCEPT ACTIVATION VECTORS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            170,
+            400,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            236,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks commonly offer high utility but remain difficult to interpret. Developing methods to explain their decisions is challenging due to their large size, complex structure, and inscrutable internal representations. This work argues that the language of explanations should be expanded from that of input features (e.g., assigning importance weightings to pixels) to include that of higher-level, humanfriendly concepts. For example, an understandable explanation of why an image classifier outputs the label “zebra” would ideally relate to concepts such as “stripes” rather than a set of particular pixel values. This paper introduces the “concept activation vector” (CAV) which allows quantitative analysis of a concept’s relative importance to classification, with a user-provided set of input data examples defining the concept. CAVs may be easily used by non-experts, who need only provide examples, and with CAVs the high-dimensional structure of neural networks turns into an aid to interpretation, rather than an obstacle. Using the domain of image classification as a testing ground, we describe how CAVs may be used to test hypotheses about classifiers and also generate insights into the deficiencies and correlations in training data. CAVs also provide us a directed approach to choose the combinations of neurons to visualize with the DeepDream technique, which traditionally has chosen neurons or linear combinations of neurons at random to visualize. ",
+        "bbox": [
+            232,
+            263,
+            764,
+            527
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            551,
+            336,
+            568
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks (NNs) are capable of impressively good performance, yet understanding and interpreting their behavior remains a significant challenge. Solving this challenge is an important problem for several reasons. For example, explaining a system’s behavior may be necessary to establish acceptability and see adoption for critical applications, such as those in the medical domain. For scientists and engineers, any greater understanding of how neural networks function is appreciated, since it may lead to better models and help with debugging (30; 19). ",
+        "bbox": [
+            174,
+            582,
+            823,
+            666
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent work suggests that linear combinations of neurons may encode meaningful, insightful information (2; 19; 27). However, we lack methods to 1) identify which linear combinations (if any) relate to a given concept, and 2) how these can aid in our quantitative understanding of concepts and classification decisions. For example, we may hypothesize that an image model that successfully classifies zebras may naturally encode concepts for ‘stripe’ and ‘animal’, somewhere in its internal representations, using a linear combination of neurons. How can we formalize this notion, and test such a hypothesis? ",
+        "bbox": [
+            174,
+            672,
+            825,
+            770
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Neural networks build internal representations that are far richer than the input features or output classes explicit in their training data. Unfortunately, many machine learning interpretation methods provide results only in terms of input features. For example, the learned coefficients in linear classifiers or logistic regression can be interpreted as each feature’s classification importance. Similar first-order importance measures for neural networks often use first derivatives as a proxy for input feature importance, as is done for pixel importance in saliency maps (8; 22). ",
+        "bbox": [
+            174,
+            777,
+            823,
+            861
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "It is critical that model understanding and interpretation not be limited to only the concepts explicit in training data. This can be seen by considering classification fairness—an increasingly relevant, difficult problem where interpretability can be useful—and noting that no input features may identify discriminated-against groups. For example, the Inception model (26) has an output class for ’doctor’ but no input features identifying the concepts of ’man’ or ’woman’ in a way that would allow existing interpretability approaches to quantify gender bias in classification. ",
+        "bbox": [
+            174,
+            868,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This work introduces the method of concept activation vectors (CAV) for the following purposes. First, CAV can be used to identify linear combinations of neurons in a layer of a model that correspond to given semantic concepts, even for new, user-provided concepts not explicit in the model’s training data. Second, CAV provides quantitative measures of the relative importance of userprovided concepts, which allows for hypothesis testing of the relationship between given concepts and the model’s predictions. ",
+        "bbox": [
+            174,
+            138,
+            825,
+            223
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Testing with CAV (TCAV) is designed with the following desiderata in mind. ",
+        "bbox": [
+            176,
+            229,
+            678,
+            244
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. accessibility: Requires little to no user expertise in machine learning.   \n2. customization: Adapt to any concept of interest (e.g., gender) on the fly without pre-listing a set of concepts before training.   \n3. plug-in readiness: Work without retraining or modifying the model.   \n4. quantification: Provide quantitative explanation that are tied to human-relatable concept, and not input features. ",
+        "bbox": [
+            212,
+            255,
+            825,
+            352
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One of key ideas for TCAV is that we can test the relative importance between small set of concepts, rather than ranking the importance of all possible features/concepts. For example, we can gain insights by testing whether the concept of gender was used more than the ‘wearing scrubs’ concept for the classification of doctor. We can also test whether or not a given concept was relevant to the classification of a certain class. Similar forms of sparsity (i.e., only considering a few concepts at a time) are used in many existing interpretable models (12; 7; 28; 31; 29; 4). Note that interpretability does not mean understanding the entire network’s behavior on every feature/concept of the input (6). Such a goal may not be achievable, particularly for ML models with super-human performance (23). ",
+        "bbox": [
+            174,
+            363,
+            825,
+            474
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "TCAV satisfies these desiderata—accessibility, customization, plug-in readiness and quantification —it enables quantitative relative importance testing for non-ML experts, for user-provided concepts without retraining or modifying the network. Users express their concepts of interest using examples—a set of data points exemplifying the concept. For example, if gender is the concept of interest, users can collect pictures of women. The use of examples has been shown to be powerful medium for communication between machine learning (ML) models and non-expert users (16; 12; 13). Cognitive studies on experts also support this approach (e.g., experts think in terms of examples (15)). ",
+        "bbox": [
+            174,
+            481,
+            825,
+            593
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The structure of this paper is as follows: Section 2 relates this work to existing interpretability methods. Section 3 explains the details of the TCAV method. In Section 4, we show 1) how this framework can be used to identify semantically meaningful directions in a layer and 2) the relative importance testing results that measure the relevance of concepts of interest to the classification output by the network. ",
+        "bbox": [
+            174,
+            599,
+            825,
+            669
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            689,
+            341,
+            705
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide a brief overview of existing related interpretability methods and their relation to our desiderata. We also discuss the need and the challenges of desiderata 3): plug-in readiness. ",
+        "bbox": [
+            174,
+            719,
+            825,
+            762
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 SALIENCY MAP METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            779,
+            395,
+            792
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "One of the most popular approaches in interpreting NN is saliency methods (24; 22; 25; 8; 5). These techniques seek to identify regions of the input most relevant to the classification output by the network. Qualitatively, these methods often successfully label regions of the input which seem semantically relevant to the classification. ",
+        "bbox": [
+            174,
+            805,
+            823,
+            861
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Unfortunately, these methods do not satisfy our desiderata of 2) customization and 4) quantification. The lack customization is clear, as the user has no control over what concepts of interest these maps pick up on. Regarding quantification, there is no way to meaningfully quantify and interpret the brightness of various regions in these maps. As a hypothetical example, consider two saliency maps of two different cat pictures, where the brightness of the ears of the two cats differ. It is unclear both how to quantify ‘brightness’ and second what kind of actionable insights this level of brightness gives. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
+            823,
+            146
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recent work has also demonstrated that the saliency map these methods produced may be very sensitive to completely uninteresting properties of the data distribution. In particular, (14) showed that simply applying a mean shift to the dataset may cause some saliency methods to result in significant changes in the given explanation. (9) also showed that saliency methods can be easily tricked. ",
+        "bbox": [
+            174,
+            152,
+            825,
+            222
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 DEEPDREAM, NEURON LEVEL INVESTIGATION METHODS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            246,
+            607,
+            258
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "There are techniques, such as DeepDream, which can be used to visualize patterns that maximally activates each neuron of a neural network. The technique starts from an image of random noise and iteratively modifies the image in order to maximally activate a neuron or a linear combination of neurons of interest (17; 18). This technique has offered some insights into the information encoded in a neuron’s activation. This technique also has opened up opportunities for AI-aided art (17). ",
+        "bbox": [
+            174,
+            272,
+            825,
+            342
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, the DeepDream method does not satisfy our desiderata 1) accessibility, 2) customization, and 4) quantification. It does not satisfy 1) because in order to apply it a user must first understand what a neuron is, and second be familiar enough with the the internal architecture in order to choose which neurons to visualize. It does not satisfy 2) because no current method exists to find which neurons correspond to semantically meaningful concepts such as gender, and it is unclear whether or not such a neuron even exists. It does not satisfy 4) because we do not understand these pictures and there is currently no method to quantify how these pictures relate to the output of the network. This method again does not provide actionable insight. ",
+        "bbox": [
+            174,
+            348,
+            825,
+            460
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As we show later, DeepDream may be combined with TCAV in order to identify and visualize interesting directions in a layer. Prior work on DeepDream has typically chosen neurons or linear combinations of neurons at random to visualize. Note that a user of TCAV does need to pick a layer in the network for which to apply TCAV to. However, if only the final prediction is concerned, the last layer can be used by default. ",
+        "bbox": [
+            174,
+            467,
+            825,
+            536
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 WHY WE NEED DESIDERATA 3 - PLUG-IN READINESS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            559,
+            580,
+            573
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To achieve interpretability, we have two options: (1) restrict ourselves to inherently interpretable models or (2) post-process our models in way that yields insights. ",
+        "bbox": [
+            176,
+            587,
+            823,
+            614
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Users may choose option (1) as there are a number of methods to build inherently interpretable models (12; 7; 28; 31; 29; 4). If the users are willing and are able to adapt these models, then this is the gold standard. Although building inherently interpretable machine learning models may be possible for some domains, doing so may result in decreased performance. Furthermore, changing the model may be costly for users who already have a model implemented. ",
+        "bbox": [
+            174,
+            622,
+            825,
+            691
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A method that can be applied without retraining or modifying the network could be instantly used to interpret existing models. Increasingly, attention is turning to the importance of providing explanations for ML decisions (for one example, consider (10)). As a result, there is a growing need for interpretation techniques that can be applied “out of the box,” that is, without rebuilding or retraining existing systems. ",
+        "bbox": [
+            174,
+            699,
+            825,
+            768
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.4 A NOTE ON DESIDERATA 3 - PLUG-IN READINESS AND LOCAL EXPLANATIONS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            791,
+            751,
+            805
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "One of many challenges of building a post-processing interpretion method is to ensure that the explanation is truthful to the model’s behavior. By “truthful” we mean that explanations are roughly consistent with model’s internal state. For example, if the explanations are created completely independently of the model (11), it has high probability of having such inconsistencies. ",
+        "bbox": [
+            174,
+            818,
+            823,
+            875
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The plug-in readiness desiderata poses interesting challenge for explanations to remain consistent with the model’s behavior. Recently, there has been work showing that saliency methods contains such inconsistencies. For instance, (14) show that saliency methods are vulnerable to constant shift in input that does not affect classification. It has also shown that the methods can be easily tricked (9). ",
+        "bbox": [
+            176,
+            882,
+            823,
+            922
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "One way to improve consistency between explanations and the model’s reasoning is to use the generated explanation as an input, and check the network’s output for validation. This is typically used in perturbation-based interpretability methods (16; 20). These methods perturb the input, and use the network’s response to generate explanations. They maintain the consistency either locally or globally1 by construction. TCAV is a type of perturbation method. ",
+        "bbox": [
+            174,
+            138,
+            823,
+            208
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Even a truthful explanation may be misleading if it is only locally truthful (20). For example, since the importance of features only needs to be truthful in the vicinity of the data point of interest, there is no guarantee that the method will not generate two completely conflicting explanations. These inconsistencies may result in decreased user trust at a minimum. On the other hand, making a globally truthful explanation may be difficult as the networks decision boundaries may be complex. TCAV produces globally explanations, and uses model’s output to generate explanations to maintain consistency between explanations and the model’s reasoning. ",
+        "bbox": [
+            174,
+            215,
+            825,
+            314
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 TCAV METHOD ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            343,
+            341,
+            359
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We introduce a method that allows for global linear interpretability of a highly flexible class of models, namely deep feedforward neural networks trained on classification tasks. As a form of explanation, TCAV uses concepts that are provided by users, instead of using predetermined set of input features or input classes. These concepts are tied to real-world data that represents concepts of interest. Users express their concepts of interest using examples — a set of data points exemplifying the concept. These concepts enable testing the relative importance of concepts used in classification. ",
+        "bbox": [
+            174,
+            380,
+            825,
+            464
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Informally, the key idea is the following: although feedforward networks learn highly nonlinear functions there is evidence that they work by gradually disentangling concepts of interest, layer by layer (2; 3). It has also been shown that representations are not necessarily contained in individual neurons but more generally in linear combinations of neurons (19; 27). Thus the space of neuron activations in layers of a neural network may have a meaningful global linear structure. Furthermore, if such a structure exists, we can uncover it by training a linear classifier mapping the representation in a single layer to a human selected set of concepts. ",
+        "bbox": [
+            174,
+            470,
+            825,
+            569
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We now formalize this intuition. First, we formally define a concept activation vector. Let us imagine that an analyst is interested in a given concept $C$ (e.g., striped textures) and has gathered two sets of data points, $P _ { C }$ and $N$ , that represent positive and negative examples of this concept (say, photos of striped objects, versus a set of random photos). There is a lot of flexibility in how to choose $N$ , we often choose a random set of images of the same size as $P _ { C }$ . Here we represent the input to the network as vector in $\\mathbb { R } ^ { n }$ , and $P _ { C } , N \\subset \\mathbb { R } ^ { n }$ . Consider a layer $l$ of the feedforward network consisting of $m$ neurons. Then running inference on an input example and looking at the activations at layer $l$ yields a function $f _ { l } : \\mathbb { R } ^ { n }  \\mathbb { R } ^ { m }$ . For a set of inputs $X \\subseteq \\mathbb { R } ^ { n }$ we denote by $f _ { l } ( X )$ to be the set of layer activations $\\{ f _ { l } ( x ) : x \\in X \\}$ . Note that for convolutional networks we view a layer in it’s flattened form, thus a layer of width $w$ , height $h$ , and $c$ channels becomes a flat vector of $m = w \\times h \\times c$ activations. ",
+        "bbox": [
+            174,
+            575,
+            825,
+            728
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The two sets $P _ { C }$ and $N$ then naturally give rise to two sets of activation vectors in layer $l$ , namely $f _ { l } ( P _ { C } )$ and $f _ { l } ( N )$ . We can then train a linear classifier on the binary classification task of distinguishing between these two sets. The weights of this classifier are an element $\\boldsymbol { v } _ { C } ^ { l } \\in \\mathbb { R } ^ { m }$ . We call this the concept activation vector for the concept $C$ . ",
+        "bbox": [
+            174,
+            734,
+            823,
+            791
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A variation on this idea is a relative concept activation vector. Here the analyst selects two sets of inputs that represent two different concepts, $C$ and $D$ . Training a classifier on $f _ { l } ( P _ { C } )$ and $f _ { l } ( P _ { D } )$ yields a vector $v _ { C , D } ^ { l } \\in \\mathbb { R } ^ { m }$ . The vector $\\bar { v } _ { C , D } ^ { l }$ intuitively defines a $1 - d$ subspace in layer $l$ where the projection of an embedding $f _ { l } ( x )$ along this subspace measures whether $x$ is more relevant to concept $C$ or $D$ . ",
+        "bbox": [
+            174,
+            797,
+            825,
+            869
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A key benefit of this technique is the flexibility allowed in choosing the set $P _ { C }$ . Rather than being tied to the class labels in the training data, an analyst can–after training–select sets that correspond to any concept of interest. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We now describe two ways that this technique be used to interpret a feedforward neural network trained on an image classification task. First, we show relative importance testing between $M$ number of concepts, $P _ { C i }$ , where $i \\in M$ . In this case, each $v _ { C } ^ { l }$ is learned by treating all $P _ { C i } , i \\neq j$ as $N$ . Second, we can also test one concept, $P _ { C }$ , against a random concept, where $N$ are set of random pictures. In the next section we discuss the results of experiments with these methods. ",
+        "bbox": [
+            174,
+            152,
+            825,
+            222
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 APPLICATION: TESTING THE IMPORTANCE OF CONCEPTS IN CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            239,
+            745,
+            253
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The real value of CAVs comes from the fact that they may be used to test the relative importance of concepts. With CAVs, we can formulate generating explanation as a task of performing two-tailed statistical significance test (in particular, z-test). Given samples of class images (e.g., zebra pictures) and two concept vectors A and B, we perform two-tailed $\\mathbf { Z }$ -testing to invalidate the null hypothesis that there is no difference in importance of concepts A and B for the class. We perform this testing for each pair of concepts. For example, an analyst might ask about a photo of a zebra, “was the presence of stripes in the image relevant to the model’s classification of the image as a zebra?” In some cases, with tedious effort in a photo retouching program, it might be possible answer questions like this directly. However, CAVs provide a faster and more general technique. ",
+        "bbox": [
+            173,
+            265,
+            825,
+            390
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Consider an input $x$ and concept $C$ of interest. At inference time, this will give rise to activations $f _ { l } ( x )$ at layer $l$ . We can then modify this vector of activations in various ways using the concept vector $v _ { C } ^ { l }$ . For example, we might create a new vector $w ^ { - } = f _ { l } ( x ) - v _ { C } ^ { l }$ , which might be interpreted as a counterfactual of “ $x$ with less of concept $C ^ { \\mathfrak { n } }$ . Performing inference starting at layer $l + 1$ based on $w ^ { - }$ will give a new classification result $y _ { w }$ . We can also create a new vector $w ^ { + } = f _ { l } ( x ) + v _ { C } ^ { l }$ , which might be interpreted as a counterfactual of $^ { 6 6 } x$ with more of concept $C ^ { \\mathfrak { n } }$ . ",
+        "bbox": [
+            174,
+            397,
+            825,
+            481
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus to test the relevance of stripes to the classification of a zebra, we would either add or subtract a ‘stripe’ concept activation vector from the zebra image embedding, run forward propagation the resulting embedding and examine the output of the network. Large changes in the probability of zebra would indicate that stripes played a key role in the classification. Simply adding the vector is a bit ad-hoc, especially when the norm of the vector may depend on how exactly the CAV was trained. However, we found this naive method to empirically yield results that were consistently semantically meaningful. We also note that this addition is loosely related to directional derivative – saliency maps take the derivative of the logits with respect to each pixel, while our work takes derivatives with respect to a concept direction. Future work should explore more principled methods, to measure relevance of concepts to classification. ",
+        "bbox": [
+            173,
+            487,
+            825,
+            627
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Quantitatively, the metric of the influence of the concept $C$ to class $k$ can be measured by ",
+        "bbox": [
+            171,
+            633,
+            758,
+            648
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/29bffc26b9151075475a0adce823328e5bbb5cc3bd2a4b42e3800ce521ccbe90.jpg",
+        "text": "$$\nI _ { w } ^ { u p } = \\frac { 1 } { N } \\sum _ { i } ^ { N } \\mathbb { 1 } ( p _ { k } ( y ) < p _ { k } ( y _ { w } ) ) \\qquad I _ { w } ^ { d o w n } = \\frac { 1 } { N } \\sum _ { i } ^ { N } \\mathbb { 1 } ( p _ { k } ( y ) > p _ { k } ( y _ { w } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            654,
+            746,
+            696
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $p _ { k } ( y )$ represents the probability of class $k$ of prediction $y$ and $i = \\{ 1 , 2 , \\dots , N \\}$ where $N$ is the number of images of the class for inspection (e.g., zebra). Intuitively speaking, $I _ { w } ^ { u p / d o w n }$ measures the ratio of data points that become ‘more/less like class $k '$ after the modification with concept vector $v _ { C } ^ { l }$ . ",
+        "bbox": [
+            173,
+            702,
+            823,
+            762
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We can perform statistical significance testing in order to quantify the concept importance; we can test a hypothesis that one concept is more/less important than another concept. In other words, the null hypothesis that no color is significant. To test this hypothesis we can do $\\mathbf { z }$ -testing on the measured importance values, and ask the question what is the probability that random concept vectors would observe the measured difference. ",
+        "bbox": [
+            174,
+            768,
+            825,
+            839
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 APPLICATION: WHERE ARE CONCEPTS LEARNED? ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            856,
+            562,
+            869
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As a simple demonstration of the value of CAVs, we describe a technique for localizing where a concept is disentangled in a network. A common view of feedforward networks is that different concepts are represented at different layers. For example, many image classification networks are said to detect textures at low levels, with increasingly complex concepts recognized at later layers. Typically evidence for this view comes from various methods of reverse-engineering (3; 30; 17) the behavior of individual neurons in the network. ",
+        "bbox": [
+            176,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Using concept activation vectors, however, we can approach this question from the opposite direction. We can pick out a particular concept $C$ in advance, and consider the concept vectors $v _ { C } ^ { 1 } , v _ { C } ^ { 2 } , \\ldots , v _ { C } ^ { \\hat { k } }$ for each layer in the network. The accuracies of the linear classifiers, $a _ { C } ^ { 1 } , a _ { C } ^ { 2 } , \\dots , a _ { C } ^ { k }$ then provide a measure of how well disentangled the concept is in each layer. This may be viewed as a generalization of the ideas in (2; 3; 19). In the next section we describe the results of an experiment based on this idea. ",
+        "bbox": [
+            174,
+            152,
+            825,
+            238
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            273,
+            281,
+            290
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we first show evidence that the learned CAV indeed detect the concepts of interest. Then we show the results for hypothesis testing using these CAVs, and the insights one can gain from it. ",
+        "bbox": [
+            176,
+            315,
+            823,
+            357
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "All our experiments are based on the model from (26) using the publicly available model parameters from (1). The pictures used to learn the CAVs are collected from the latest ImageNet Fall 2011 release (21). Each concept vector is learned using between 30-500 pictures as input. We intentionally chose not to use all the available pictures in order to mirror realistic scenarios where users may not be able to collect a large amount of data. The class ‘arms’ only had 30 pictures (all manually collected), since ImageNet does not include this as a part of the dataset. The pictures used to learn texture concepts are taken from the (3) data set. In order to learn concept vectors for colors, we generated 500 pictures synthetically by generating the color channels randomly. We provide experiments both where CAV’s are trained to distinguish between a set of concepts (e.g. red, yellow, green, blue) and when one CAV is trained per concept (with $N$ chosen to be a random collection of images). ",
+        "bbox": [
+            174,
+            364,
+            825,
+            517
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 EVIDENCE OF THE VALIDITY OF LINEARITY ASSUMPTION",
+        "text_level": 1,
+        "bbox": [
+            174,
+            551,
+            612,
+            564
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section we describe experiments that indicate our linearly learned CAVsalign with their intended concepts. First, the linear maps used to construct the CAVsare accurate in predicting the concepts. The point in the network where these concepts in the networks are learned (i.e., when accuracy becomes high) is consistent with general knowledge about the representations learned by neural networks. Low level features such as edges and textures are detected in the early layers and higher level concepts are only reliably detected in the upper layers. Next, we use activation maximization techniques (18) to visualize each of the CAVs and observe that the patterns are consistent with the provided concept. Finally, we show the top $k$ images that are most similar in terms of cosine similarity to each CAV for further qualitative confirmation. ",
+        "bbox": [
+            174,
+            582,
+            825,
+            708
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 WHERE EACH CONCEPT IS LEARNED.",
+        "text_level": 1,
+        "bbox": [
+            174,
+            739,
+            483,
+            753
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows the accuracy of the linear classifiers at each layer in the network for each type of CAV. Each classifier is trained to distinguish one concept from other concepts (e.g., textures1 set contains ‘stripes’, ‘zigzagged’ and ‘dotted’ texture concepts). Overall, we observe high accuracy as measured by a held out test set of size 1/3 that of the training size. This is evidence that the given concepts are linearly separable in many layers of the network. ",
+        "bbox": [
+            174,
+            768,
+            823,
+            839
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Note that the accuracy of more abstract CAV (e.g., objects) increases in higher layers of the network. The accuracy of a simpler CAV, such as color, is high throughout the entire network. This agrees with prior work on visualizing the learned representations at different layers of neural networks (30). ",
+        "bbox": [
+            174,
+            847,
+            825,
+            888
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This experiment does not yet show that these CAVs align with the concepts that makes sense semantically to humans. We demonstrate this with the next set of experiments. ",
+        "bbox": [
+            174,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/1441b37dc50d080387d63509b7c11bc69d89d54a33c77b39193f5bb58be65338.jpg",
+        "image_caption": [
+            "Figure 1: Where each of the CAVs are learned. Accuracy at each layer for CAV. Simple concepts (e.g., colors) achieve high performance in all layers than more abstract concepts (e.g. persons, objects) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            101,
+            745,
+            308
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/49b8d0a897236ccfa31ec4a26fc54cfdd0b00fd124a51d76cf7e67f7a0396894.jpg",
+        "image_caption": [
+            "Figure 2: Deepdreamed fibrous texture, knitted texture, corgis and Siberian huskey CAVs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            397,
+            792,
+            849
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we use the activation maximization technique (17) to visualize the learned representations in the direction of the CAV. We use the same techniques from (18). As typically done, we use a random image as a starting point (17) for the optimization to avoid choosing an arbitrary image as a starting point. Figure 2 shows highly recognizable features, such as knitted textures, and corgi faces. ",
+        "bbox": [
+            174,
+            143,
+            825,
+            213
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/ee49dc3172e84df9b8f15518b7c2ee18146d0eb960a82555b5e0fe4c645a38f9.jpg",
+        "image_caption": [
+            "Figure 3: Deepdreamed CAV texture concepts for each layer (each row) in inception. We can identify textures better in the mid-layer (mixed4d), then later layers. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            229,
+            753,
+            672
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We can also observe how each concept is represented as the depth increases. Images in Fig. 3 show the results for sets of textures. Interestingly, there is a layer (around mixed 4d) where textures are clearly identifiable, after which they become less and less recognizable. ",
+        "bbox": [
+            176,
+            742,
+            821,
+            784
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The left images in Fig. 4 show the results for set of colors — green, blue, yellow and red (each column) and for the set of layers (lower to higher layers from top to bottom). The training set for the color CAVs are generated by simply replacing each RGB channel with randomly sampled values around 0.5, while leaving other channels empty. Note how the color CAVs also contain many random textures, this is perhaps not surprising - there is no reason to expect any direction in a layer to be associated purely with one concept while being orthogonal to all other concepts. ",
+        "bbox": [
+            174,
+            791,
+            825,
+            875
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The right image in Fig. 4 shows higher level concepts. The columns represent zebra, siberian huskies, and corgis. It is interesting to note how the zebra CAVs include textures suggestive of water, trees, and grass. This suggests that the model associates all of these concepts with the classification of zebra and is likely a learned bias that results from the background of zebra pictures in the training data. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/5607ed63116e3a31d0927291a67f547d95acf985633cb45cd990a661de836fc5.jpg",
+        "image_caption": [
+            "Figure 4: Deepdreamed CAV color and animal concepts for each layer (each row) in inception. Zebra CAVs include textures suggestive of water, trees and grass. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            241,
+            101,
+            754,
+            506
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            588,
+            823,
+            617
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "These visualizations provide some qualitative confirmation that the learned directions align with the given concepts. This also shows that DeepDream may be combined with TCAV in order to identify and visualize interesting directions in a layer. Prior work on DeepDream has typically chosen neurons or linear combinations of neurons at random to visualize. The next section provides further evidence that the CAVs are indeed aligned with the concept of interest using real data. ",
+        "bbox": [
+            174,
+            623,
+            825,
+            694
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4.1.3 QUALITATIVE CONFIRMATION: PICTURES THAT ARE SIMILAR TO CAV ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            720,
+            709,
+            734
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In order to qualitatively confirm that the learned CAV aligns with meaningful concepts, we compute cosine similarity between a set of pictures (all from one class) to the CAV. Fig. 5 shows that the top corgi images that aligns with striped CAV selects pictures with striped objects (e.g., a striped tube or vertical blinds in the background). Thus being similar to the striped CAV meant being highly similar to one of the other concepts in the CAV training set. We also show that CAVs for more abstract concepts can be constructed (e.g., CEO). Recall that many of these learned concepts are not classes that the NN learned to classify. ",
+        "bbox": [
+            174,
+            750,
+            825,
+            847
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also observe that if there is not much relation between pictures and the concept, the cosine similarity is low. This experiment qualitatively supports our argument that the alignment between CAVs and the meaningful concepts. Note that the images least similar to striped appear with a checkered or knitted pattern, likely due to the fact that the striped CAV was trained relative to checkered and knitted images. ",
+        "bbox": [
+            176,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/eeca4cd44f32c8fec820df8d34b0f4df61cc3640eaabae50ca682a1e05fdbeb4.jpg",
+        "image_caption": [
+            "Figure 5: Top and bottom 3 similar pictures of each concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            104,
+            815,
+            464
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "When using TCAV, in order to ensure that the provided examples were sufficient to learn CAV, we recommend users to perform this qualitative test by checking cosine similarity with each concepts to the class. ",
+        "bbox": [
+            176,
+            553,
+            825,
+            595
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "4.2 TESTING RELATIVE IMPORTANCE OF CONCEPTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            647,
+            549,
+            661
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this section, we describe how concept activation vectors may be used for quantitative testing of the relative importance of concepts. ",
+        "bbox": [
+            174,
+            686,
+            823,
+            715
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Inof 6, we show . This mea $I _ { w ^ { + } } ^ { u p }$ for eight classes. We do not list at when CAVs are added, the p $I _ { w ^ { - } } ^ { u p }$ , as they always appear to be the inversebility of the class rarely stays the same. $I _ { w ^ { + } } ^ { u p }$ This confirms that adding or subtracting $v _ { C } ^ { l }$ clearly impacts the classification task. In particular the $v _ { C } ^ { l }$ direction is aligned with the direction that measures the probability of the target class. ",
+        "bbox": [
+            174,
+            722,
+            825,
+            784
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Note that adding random directions should be expected to change the probability of the target class in some way. We find, however, that for several CAV’s which are semantically relevant to the class (e.g., red to fire engine) the CAVdirection will consistently increase the probability of class for most images of this class. On the other hand, we show that random directions (e.g., a concept learned from random set of images) tend to be much less consistent. This difference allows us to perform a more rigorous hypothesis test in a following section. ",
+        "bbox": [
+            174,
+            790,
+            825,
+            875
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This testing identifies relationships between concepts and the target class that agree with human intuition (e.g., fire engines are red, cucumbers are bumpy, zebras are striped, CEOs wear suits). In the next section, we describe insights we gained from these tests. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/870201a410f3cfcd791e0af99ea5534afbc501e22d44e82f189663921649aef0.jpg",
+        "image_caption": [
+            "Figure 6: Testing Relative Importance of Concepts "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            97,
+            808,
+            314
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "4.3 GAINING INSIGHTS",
+        "text_level": 1,
+        "bbox": [
+            174,
+            372,
+            348,
+            386
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this section, we show results that confirm common-sense intuitions about training images, as a kind of sanity check that TCAV provides valid results. We then describe results that surprised us, but lead to insights about the dataset that we did not initially have. ",
+        "bbox": [
+            174,
+            400,
+            825,
+            441
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In Fig 6, the yellow color was more influential to cab class than any other colors. For people who have lived in cities where this is the standard color for taxis, this is no shock — and indeed, most ImageNet pictures of cabs are yellow. Similarly, the ‘women’ concept is important to ‘bikini’ class, likely due to the fact that training images have humans wearing the bikini rather than pictures of the clothing. More subtly, we also discovered that the ‘model women’ concept is important to ‘bikini’ This lead us to realize that the most of ‘bikini’ class pictures feature professional models, typically very slim, posing in their bikinis — probably not well representative samples of bikinis in the real world. Note that the network was trained with bikini and cab classes, and the concept ‘yellow’, ‘women’ and ‘model women’ are not part of classes used to train this network2. ",
+        "bbox": [
+            174,
+            448,
+            825,
+            574
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The graph for ‘dumbbell’ class in Fig 6 shows that ‘arms’ concept was more important to predict dumbbell class than other concepts. This finding is consistent with previous qualitative findings from (17), where they discovered that the DeepDream picture of a dumbbell also showed an arm holding it. TCAV allows for quantitative confirmation of this previously qualitative finding. Moreover, unlike all other concepts in this figure, we only collected 30 pictures of each concept (ImageNet did not have arms as a label). Despite the small number of examples, the TCAV method is able to identify this relationship. ",
+        "bbox": [
+            174,
+            580,
+            825,
+            678
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The flexibility of the TCAV method makes it easy to explore a network for other surprising associations. For example, we saw that ‘baby’ concept was important to ‘school bus’ class. It turns out that some school bus pictures include young children riding or standing in front of the bus. Another discovery was to find that ‘red’ concept is important for cucumber class. We believe this due to the fact that a number of cucumber pictures include other red vegetables, such as tomato and carrot. ",
+        "bbox": [
+            174,
+            685,
+            825,
+            755
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We believe that TCAV can be useful in discovering these insights for many applications, including for example, to improve fairness in the model. ",
+        "bbox": [
+            174,
+            762,
+            823,
+            790
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "4.4 RANDOM IMAGE COMPARISONS TO FILTER SPURIOUS RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            811,
+            647,
+            824
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "When constructing the CAV, the choice of negative samples may cause spurious suggestions that a concept is relevant to a class when it actually is not. In other words, the learned CAV may accidentally be aligned with something related to the class, and cause high $I _ { w ^ { + } } ^ { u p }$ . For instance, when we tested unrelated concepts and classes, such as zebra to a set of textures, honeycombed, lace-like and bumpy textures, we found that lace-like concept shows high $I _ { w ^ { + } } ^ { u p }$ . ",
+        "bbox": [
+            174,
+            838,
+            825,
+            880
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/6447cdbc97429574a74d25523ef26a8ac68c7afca3b9608e6f7718e25f29d984.jpg",
+        "image_caption": [
+            "Figure 7: Histogram of the mean of and random concepts. $I _ { w ^ { + } } ^ { u p }$ for the relationship of the zebra class with striped, lace-like "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            338,
+            101,
+            655,
+            277
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            353,
+            823,
+            383
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "One might argue that lace is vaguely related to a zebra’s stripes, but in fact even if we choose both $P _ { C }$ and $N$ to be independent random sets of images we still often observe $I _ { w ^ { + } } ^ { u p }$ to be relatively high. This is not surprising: there are directions in a layer $l$ which are highly aligned with the zebra concept, and randomly chosen directions will typically have small but non-zero projection along these directions. ",
+        "bbox": [
+            174,
+            388,
+            825,
+            458
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "One way to filter out these spurious results is to do statistical testing against random concepts. Using different sets of random images for the negative class and striped images for $P _ { C }$ , we make 50-100 CAVs, all of which represent concept ‘striped’. For each ‘striped’ CAV we can measure $I _ { w ^ { + } } ^ { u p }$ . We can also generate a set of ‘random’ CAVs by choosing random images for both $P _ { C }$ and $\\ddot { N }$ . Then we performed a $z$ -test to see if the mean $I _ { w ^ { + } } ^ { u p }$ s from striped CAVs are statistically different from the mean Iup+ of random CAVs. We can successfully filter out some spurious correlations, including lace-like concepts with zebras by using this method, see the histogram 4.4. ",
+        "bbox": [
+            173,
+            465,
+            825,
+            565
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            589,
+            318,
+            606
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We have introduced the notion of a “concept activation vector,” or CAV, which is a flexible way to probe the internal representation of a concept in a classification network. Since CAVs may be defined via a set of example inputs, rather than custom coding, they are well suited to use by non-experts. We then described a technique (Testing with CAVs, or TCAV) for quantifying the relation between a CAV and a particular class. The TCAV technique allows us to provide quantitative answers to questions such as, “How important are the stripes to the classification of a zebra?” ",
+        "bbox": [
+            173,
+            623,
+            825,
+            707
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To provide evidence for the value of the TCAV technique, we described a series of experiments which supported common-sense intuition, for example, that stripes are indeed important to the identification of zebras. In addition, we used the DeepDream technique to create images whose internal representations approximate certain CAVs. The resulting pictures were strongly evocative of the original concepts. Finally, we described how the TCAV technique may be used to find associations between concepts, both obvious (“yellow” and “taxi”) and non-obvious (“red” and “cucumber”). ",
+        "bbox": [
+            174,
+            714,
+            825,
+            797
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In addition to analyzing a single network, TCAV can be also used to compare and contrast a pair of networks. For example, one can compare the relative importance of concepts to determine how the different choices of training process or architecture influences learning of each concept. Based on the results, users can perform model selection based on the concepts that are more or less important for the task. ",
+        "bbox": [
+            174,
+            804,
+            825,
+            875
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "An interesting direction for future work may be to explore applications of using CAVs to adjust the results of a network during inference time. Adding a scalar multiple of a CAV to the activations of an intermediate layer can, as shown in our experiments, allow us to deemphasize or enhance conceptual aspects of an input. One potential application, for example, might be to reduce bias the network has learned from training data. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            148,
+            328,
+            161
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Chris Olah, Alexander Mordvintsev and Ludwig Schubert for generously allowing us to use their code for DeepDream. ",
+        "bbox": [
+            174,
+            171,
+            823,
+            199
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            220,
+            285,
+            236
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[1] \"http://storage.googleapis.com/download.tensorflow.org/models/ inception5h.zip\", 2017.   \n[2] Guillaume Alain and Yoshua Bengio. Understanding intermediate layers using linear classifier probes. arXiv preprint arXiv:1610.01644, 2016.   \n[3] David Bau, Bolei Zhou, Aditya Khosla, Aude Oliva, and Antonio Torralba. Network dissection: Quantifying interpretability of deep visual representations. In Computer Vision and Pattern Recognition, 2017.   \n[4] Rich Caruana, Yin Lou, Johannes Gehrke, Paul Koch, Marc Sturm, and Noemie Elhadad. Intelligible models for healthcare: Predicting pneumonia risk and hospital 30-day readmission. In KDD, 2015.   \n[5] Piotr Dabkowski and Yarin Gal. Real time image saliency for black box classifiers. arXiv preprint arXiv:1705.07857, 2017.   \n[6] Been Doshi-Velez, Finale; Kim. Towards a rigorous science of interpretable machine learning. In eprint arXiv:1702.08608, 2017.   \n[7] Finale Doshi-Velez, Byron C Wallace, and Ryan Adams. Graph-sparse lda: A topic model with structured sparsity. In Aaai, pages 2575–2581, 2015.   \n[8] Dumitru Erhan, Yoshua Bengio, Aaron Courville, and Pascal Vincent. Visualizing higher-layer features of a deep network. University of Montreal, 1341:3, 2009.   \n[9] Amirata Ghorbani, Abubakar Abid, and James Zou. Interpretation of neural networks is fragile. arXiv preprint arXiv:1710.10547, 2017.   \n[10] Bryce Goodman and Seth Flaxman. European union regulations on algorithmic decisionmaking and a” right to explanation”. arXiv preprint arXiv:1606.08813, 2016.   \n[11] Lisa Anne Hendricks, Zeynep Akata, Marcus Rohrbach, Jeff Donahue, Bernt Schiele, and Trevor Darrell. Generating visual explanations. In European Conference on Computer Vision, pages 3–19. Springer, 2016.   \n[12] Been. Kim, Cynthia Rudin, and Julie Shah. The Bayesian Case Model: A generative approach for case-based reasoning and prototype classification. In NIPS, 2014.   \n[13] Been Kim, Julie Shah, and Finale Doshi-Velez. Mind the gap: A generative approach to interpretable feature selection and extraction. In Advances in Neural Information Processing Systems, 2015.   \n[14] Pieter-Jan Kindermans, Sara Hooker, Julius Adebayo, Maximilian Alber, Kristof T Schutt,¨ Sven Dahne, Dumitru Erhan, and Been Kim. The (un) reliability of saliency methods. ¨ arXiv preprint arXiv:1711.00867, 2017.   \n[15] G.A. Klein. Do decision biases explain too much. HFES, 1989.   \n[16] Pang Wei Koh and Percy Liang. Understanding black-box predictions via influence functions. arXiv preprint arXiv:1703.04730, 2017.   \n[17] Alexander Mordvintsev, Christopher Olah, and Mike Tyka. Inceptionism: Going deeper into neural networks. Google Research Blog. Retrieved June, 20:14, 2015. ",
+        "bbox": [
+            174,
+            242,
+            826,
+            926
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "[18] Chris Olah, Alexander Mordvintsev, and Ludwig Schubert. Feature visualization. Distill, 2017. https://distill.pub/2017/feature-visualization. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "[19] Maithra Raghu, Justin Gilmer, Jason Yosinski, and Jascha Sohl-Dickstein. Svcca: Singular vector canonical correlation analysis for deep understanding and improvement. arXiv preprint arXiv:1706.05806, 2017.   \n[20] Marco Tulio Ribeiro, Sameer Singh, and Carlos Guestrin. “why should i trust you?”: Explaining the predictions of any classifier. arXiv preprint arXiv:1602.04938, 2016.   \n[21] Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, et al. Imagenet large scale visual recognition challenge. International Journal of Computer Vision, 115(3):211–252, 2015.   \n[22] Ramprasaath R Selvaraju, Abhishek Das, Ramakrishna Vedantam, Michael Cogswell, Devi Parikh, and Dhruv Batra. Grad-cam: Why did you say that? arXiv preprint arXiv:1611.07450, 2016.   \n[23] David Silver, Aja Huang, Chris J. Maddison, Arthur Guez, Laurent Sifre, George van den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, Sander Dieleman, Dominik Grewe, John Nham, Nal Kalchbrenner, Ilya Sutskever, Timothy Lillicrap, Madeleine Leach, Koray Kavukcuoglu, Thore Graepel, and Demis Hassabis. Mastering the game of go with deep neural networks and tree search. 2016.   \n[24] Daniel Smilkov, Nikhil Thorat, Been Kim, Fernanda Viegas, and Martin Wattenberg. Smooth- ´ grad: removing noise by adding noise. arXiv preprint arXiv:1706.03825, 2017.   \n[25] Mukund Sundararajan, Ankur Taly, and Qiqi Yan. Axiomatic attribution for deep networks. arXiv preprint arXiv:1703.01365, 2017.   \n[26] Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan, Vincent Vanhoucke, and Andrew Rabinovich. Going deeper with convolutions. In Computer Vision and Pattern Recognition (CVPR), 2015.   \n[27] Christian Szegedy, Wojciech Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow, and Rob Fergus. Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199, 2013.   \n[28] Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58:267–288, 1994.   \n[29] Berk Ustun, Stefano Traca, and Cynthia Rudin. Supersparse linear integer models for inter- \\` pretable classification. arXiv preprint arXiv:1306.6677, 2013.   \n[30] Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In European conference on computer vision, pages 818–833. Springer, 2014.   \n[31] Hui Zou, Trevor Hastie, and Robert Tibshirani. Sparse principal component analysis. Journal of Computational and Graphical Statistics, 15:2006, 2004. ",
+        "bbox": [
+            169,
+            123,
+            826,
+            742
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "top 3 images of corgis similar to knitted concept top 3 images of salmon similar to dotted concept top 3 images of zebra similar to striped concept bottom 3 images of zebra similar to striped concept top 3 images of salmon similar to striped concept top 3 images of corgis similar to porous concept bottom 3 images of salmon similar to porous concept top 3 images of CEO similar to labcoat concept ",
+        "bbox": [
+            279,
+            103,
+            669,
+            119
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/01b3822fb3eea757f6fbb991156dc9e825965ef8eac09c9a03467497b05c3c08.jpg",
+        "image_caption": [
+            "bottom 3 images of corgis similar to knitted concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            122,
+            663,
+            220
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/f5ddb8ddf0f6c759ef7df7ffefc00591c09695ac4900f21db33229d3d1dd6fcb.jpg",
+        "image_caption": [
+            "top 3 images of salmon similar to knitted concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            243,
+            661,
+            343
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/c1b24a920c554f159642c276a82f7cd711be40997195c96e7ef2b3f09899949e.jpg",
+        "image_caption": [
+            "bottom 3 images of salmon similar to knitted concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            369,
+            660,
+            469
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/bb600dc8143c4759f82ec90d2defd3e58f16acd566de3df5a16d3712964e62f7.jpg",
+        "image_caption": [
+            "top 3 images of corgis similar to dotted concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            491,
+            661,
+            590
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/4422827ed9964fad3b546446cb9b3718dfd6836d3abde969a3283b6cd40955fb.jpg",
+        "image_caption": [
+            "bottom 3 images of corgis similar to dotted concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            618,
+            661,
+            717
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/f31229d80a5b9117fe451fb3313d1cde2c55eed737bb5ac8637080bd6239dd33.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            279,
+            739,
+            666,
+            839
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            277,
+            141,
+            676,
+            157
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/eb15b1d1aef2abe5ed47e345643091095d978f926bfaa5ec9944f5a65177c190.jpg",
+        "image_caption": [
+            "bottom 3 images of salmon similar to dotted concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            160,
+            661,
+            258
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/42a7f50c718bd58e8ca7e7321a0752f1b45ba0d35e68a7aa4b3f8ede06d7a8ac.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            281,
+            661,
+            382
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            279,
+            388,
+            666,
+            405
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/f18ecbe9b05f2a322a30aded5ce92211f9cb540bce5a55871d42a1c01724f544.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            409,
+            663,
+            507
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            277,
+            511,
+            697,
+            527
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/af7f653dbbc97cf2f1003ac953309319d1daa94b98f4f810816f256fc210ca55.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            530,
+            661,
+            630
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            277,
+            637,
+            681,
+            652
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/9eb868e11fe2c7c693de4e430d8a0fa4d917206b2d6c1cb6214886ae43acdeac.jpg",
+        "image_caption": [
+            "bottom 3 images of salmon similar to striped concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            655,
+            661,
+            755
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/a9194247c8e9bfef4d529a103e0b2c7ec250d9250a4a246da754d57c7888f300.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            777,
+            651,
+            877
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            279,
+            141,
+            671,
+            157
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/659e698cd8c34a679eae7b4cb7507e3acb93c2395006885442ed31c1082f135c.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            160,
+            661,
+            258
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/22c4eba3ee7b905bc7c45217211f1ff6b0df98129aa23466cfe74fa12b5da9d0.jpg",
+        "image_caption": [
+            "bottom 3 images of corgis similar to porous concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            281,
+            661,
+            382
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/b1bcea7459e07dbcb2fdae25ba0a3c61806985b8d9b3dcbfcb2ce126c0dcf6a4.jpg",
+        "image_caption": [
+            "top 3 images of salmon similar to porous concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            409,
+            661,
+            507
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            272,
+            511,
+            707,
+            526
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/2468eee00877786d08f73a6af3013c317a436a1597b64db2cfc027277286c31b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            274,
+            529,
+            661,
+            630
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            279,
+            645,
+            665,
+            661
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/96b3ed5be1dfa39fc2b8d09d1f2f4cd2aac2baa42026271f35d95c6b108a4b9e.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            665,
+            629,
+            737
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "bottom 3 images of CEO similar to labcoat concept ",
+        "text_level": 1,
+        "bbox": [
+            279,
+            748,
+            696,
+            765
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/416ccbc89685be2f129eb829d141d841dfa7762d0493c809ee76158f28dd70c4.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            290,
+            767,
+            532,
+            848
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "top 3 images of doctor similar to labcoat concept ",
+        "text_level": 1,
+        "bbox": [
+            279,
+            150,
+            676,
+            165
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/632b3bdc7bb382e19cdb36a9e3c36935ecf5e4ccfdc0d382c80af8ad83284abf.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            289,
+            169,
+            545,
+            251
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "bottom 3 images of doctor similar to labcoat concept ",
+        "text_level": 1,
+        "bbox": [
+            279,
+            263,
+            707,
+            280
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/27ecbafc0375cb6b45e2248924df4b7cf103a59f796661d4bff81dbddd5c9ef1.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            285,
+            290,
+            607,
+            358
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "top 3 images of boss similar to labcoat concept ",
+        "bbox": [
+            281,
+            388,
+            663,
+            405
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/22e90a163737e447c938bc9ada629835c39b4027152e744a2f409a698fb682c4.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            285,
+            407,
+            714,
+            481
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "bottom 3 images of boss similar to labcoat concept ",
+        "text_level": 1,
+        "bbox": [
+            281,
+            491,
+            694,
+            507
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/ad6ba7f5ef0a2c4f3856c002247f076228e571e888b8959f44cbc453e550b76b.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            517,
+            650,
+            594
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "top 3 images of model women similar to labcoat concept ",
+        "bbox": [
+            277,
+            636,
+            661,
+            650
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/b78c3c794c3d27365046be17b7257c00c65df1cfc5320f01f3b951427683b284.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            290,
+            654,
+            549,
+            738
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "bottom 3 images of model women similar to labcoat concept ",
+        "text_level": 1,
+        "bbox": [
+            277,
+            748,
+            687,
+            762
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/c4d158efb5dba5094eedba6d313430cbb69614514117962ccee926838234bd71.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            772,
+            616,
+            849
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "top 3 images of CEO similar to suit concept ",
+        "text_level": 1,
+        "bbox": [
+            279,
+            141,
+            632,
+            157
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/9e6c40a4d640ea959afdeac711037aeefc4f45015d875353d98914b0296707bd.jpg",
+        "image_caption": [
+            "bottom 3 images of CEO similar to suit concept "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            160,
+            665,
+            238
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/84b29d7baddfc7416441ffa1c84629941cafa7aeb63ffbc43f1314d56e3295c3.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            289,
+            276,
+            622,
+            361
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "top 3 images of head nurse similar to suit concept ",
+        "bbox": [
+            279,
+            388,
+            686,
+            405
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/7941aece2d284b2d12ae06e37c20b7418489af82cbdddf6678901e3ab7f4bee6.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            289,
+            407,
+            648,
+            487
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "bottom 3 images of head nurse similar to suit concept ",
+        "text_level": 1,
+        "bbox": [
+            279,
+            500,
+            717,
+            515
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/822a296548f1c1a1b0389763ba65f4a8fee539dc7c888616791c3fffb2974b05.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            525,
+            612,
+            607
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "top 3 images of boss similar to suit concept ",
+        "bbox": [
+            281,
+            636,
+            633,
+            652
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a97202b4da32a100ff39dded02c3dc8baa1ed42fc1df81528eef7a0e4522bc4.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            289,
+            656,
+            669,
+            731
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "bottom 3 images of boss similar to suit concept ",
+        "text_level": 1,
+        "bbox": [
+            281,
+            743,
+            665,
+            760
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/5f017e47462b8d08f74f5449d1373eec586ee2a4081e300a7df27e3d123a668d.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            770,
+            673,
+            843
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/ed56d96c321b63e0a6e943406764402efa529b05def270c5c06b1e4dde41c051.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            20,
+            816,
+            928
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/0e1e21e63967bff36b2fb4201ac3e842dfd9a9b3d53c324eda59db1232c79eba.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            252,
+            820,
+            763
+        ],
+        "page_idx": 20
+    }
+]

parse/train/rkaqxm-0b/rkaqxm-0b.md

@@ -0,0 +1,353 @@
+# NEURAL COMPOSITIONAL DENOTATIONAL SEMANTICS FOR QUESTION ANSWERING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Answering compositional questions requiring multi-step reasoning is challenging for current models. We introduce an end-to-end differentiable model for interpreting questions, which is inspired by formal approaches to semantics. Each span of text is represented by a denotation in a knowledge graph, together with a vector that captures ungrounded aspects of meaning. Learned composition modules recursively combine constituents, culminating in a grounding for the complete sentence which is an answer to the question. For example, to interpret not green, the model will represent green as a set of entities, not as a trainable ungrounded vector, and then use this vector to parametrize a composition function to perform a complement operation. For each sentence, we build a parse chart subsuming all possible parses, allowing the model to jointly learn both the composition operators and output structure by gradient descent. We show the model can learn to represent a variety of challenging semantic operators, such as quantifiers, negation, disjunctions and composed relations on a synthetic question answering task. The model also generalizes well to longer sentences than seen in its training data, in contrast to LSTM and RelNet baselines. We will release our code.
+
+# 1 INTRODUCTION
+
+Compositionality is a mechanism by which the meanings of complex expressions are systematically determined from the meanings of their parts, and has been widely assumed in the study of both natural languages (Montague, 1973), as well as programming and logical languages, as a means for allowing speakers to generalize to understanding an infinite number of sentences. Popular neural network approaches to question answering use a restricted form of compositionality, typically encoding a sentence word-by-word from left-to-right, and finally executing the complete sentence encoding against a knowledge source (Perez et al., 2017). Such models can fail to generalize from training sentences in surprising ways. Inspired by linguistic theories of compositional semantics, we instead build a latent tree of interpretable expressions over a sentence, recursively combining constituents using a small set of neural modules. When tested on longer questions than are found in the training data, we find that our model achieves higher performance than baselines using LSTMs and RelNets.
+
+Our approach resembles Montague semantics, in which a tree of interpretable expressions is built over the sentence, with nodes combined by a small set of composition functions. However, both the structure of the sentence and the neural modules that handle composition are learned by end-to-end gradient descent. To achieve this, we define the parametric form of small set of neural modules, and then build a parse chart over each sentence subsuming all possible trees. Each node in the chart represents a span of text with a distribution over groundings (in terms of booleans and knowledge base nodes and edges), as well as a vector representing aspects of the meaning that have not yet been grounded. The representation for a node is built by taking a weighted sum over different ways of building the node (similarly to Maillard et al. (2017)).
+
+Typical neural network approaches to grounded question answering first encode a question from left-to-right with a recurrent neural network (RNNs), and then evaluate the encoding against an encoding of the knowledge source (for example, a knowledge base or image) (Santoro et al., 2017). In contrast to classical approaches to compositionality, constituents of complex expressions are not given explicit interpretations in isolation. For example, in Which cubes are large or green?, an RNN encoder will not explicitly build an interpretation for the expression large or green. We show that such approaches can generalize poorly when tested on more complex sentences than they were trained on. In contrast, our approach imposes strong independence assumptions that give a linguistically motivated inductive bias. In particular, it enforces that phrases are interpreted independently of surrounding words, allowing the model to generalize naturally to interpreting phrases in different contexts. In the previous example, large or green will be represented as a particular set of entities in a knowledge graph, and be intersected with the set of entities represented by the cubes node.
+
+![](images/3e5ee45df3806e70730c7a946cf9d9784259baa6a59de50b2aaa5a49ac02fbfd.jpg)  
+Figure 1: A correct parse for a question given the knowledge graph on the right, using our model. We show the type for each node, and its denotation in terms of the knowledge graph. The words or and not are represented by vectors, which parameterize composition modules. The denotation for the complete question represents the answer to the question. Nodes here have types $E$ for sets of entities, $R$ for relations, $V$ for ungrounded vectors, $E V$ for a combination of entities and a vector, and $\phi$ for semantically vacuous nodes. While we show only one parse tree here, our model builds a parse chart subsuming all trees.
+
+Another perspective on our work is as a method for learning the layouts of Neural Module Networks (NMNs) (Andreas et al., 2016b). Work on NMNs has focused on how to construct the structure of the network, variously using rules, parsers and reinforcement learning (Andreas et al., 2016a; Hu et al., 2017). Our end-to-end differentiable model jointly learns structures and modules by gradient descent.
+
+# 2 MODEL OVERVIEW
+
+Our task is to answer a question $q = w _ { 1 \ldots | q | }$ , with respect to a Knowledge Graph (KG) consisting of nodes $\mathcal { E }$ (representing entities) and labelled directed edges $\mathcal { R }$ (representing relationship between entities). In our task, answers are either booleans, or specific subsets of nodes from the KG.
+
+Our model builds a parse for the sentence, in which phrases are grounded in the KG, and a small set of composition modules are used to combine phrases, resulting in a grounding for the complete question sentence that answers the question. For example, in Figure 1, the phrases not and cylindrical are interpreted as a function word and an entity set, and then not cylindrical is interpreted by computing the complement of the entity set. The node at the root of the parse tree is the answer to the question.
+
+We describe a compositional neural model that answers such questions by:
+
+1. Grounding individual tokens in a Knowledge Graph. Tokens can either be grounded as particular sets of entities and relations in the KG, as ungrounded vectors, or marked as being semantically vacuous. For each word, we learn parameters that are used to compute a distribution over semantic types and corresponding denotations in a KG (§ 4.1).   
+2. Combining representations for adjacent phrases into representations for larger phrases, using trainable neural composition modules $( \ S 3 . 2 )$ . This produces a denotation for the phrase.   
+3. Assigning a binary-tree structure to the question sentence, which determines how words are grounded, and which phrases are combined using which modules. We build a parse chart subsuming all possible structures, and train a parsing model to increase the likelihood of structures leading to the correct answer to questions. Different parses leading to a denotation for a phrase of type $t$ are merged into an expected denotation, allowing dynamic programming $( \ S 4 )$ .   
+4. Answering the question, with the most likely grounding of the phrase spanning the sentence.
+
+# 3 COMPOSITIONAL SEMANTICS
+
+# 3.1 SEMANTIC TYPES
+
+Our model classifies spans of text into different semantic types to represent their meaning as explicit denotations or ungrounded vectors. All phrases are assigned a distribution over semantic types. The semantic type determines how a phrase is grounded, and which composition modules can be used to combine it with other phrases. A phrase spanning $w _ { i \ldots j }$ has a denotation $[ [ w _ { i . . j } ] ] _ { K G } ^ { t }$ for each semantic type $t$ J K. For example, in Figure 1, red thing corresponds to a set of entities, left corresponds to a set of relations, and not is treated as an ungrounded vector.
+
+The semantic types we define can be classified into the three different categories. Below we describe these semantic types and their corresponding representations.
+
+Grounded Semantic Types: Spans of text that can be fully grounded in the KG.
+
+1. Entity $\mathbf { ( E ) }$ : Spans of text that can be grounded to a set of entities in the KG, for example: red sphere or large cube. $\mathbf { E }$ -type span grounding is represented as a soft-attention value for each entity, $[ p _ { e _ { 1 } } , \dotsc , p _ { e _ { | \varepsilon | } } ]$ , where $0 \leq p _ { e _ { i } } \leq 1$ . This can be viewed as a soft version of a logical set-valued denotation, which we refer to as a ‘soft entity set’.
+
+2. Relation $\mathbf { ( R ) }$ : Spans of text that can be grounded to a set of relations from the KG, for example: left of or not right of or above. R-type span grounding is represented by a soft adjacency matrix $A \in \mathbb { R } ^ { | \mathcal { E } | \times | \mathcal { E } | }$ where $A _ { i j } = 1$ denotes a directed edge from $e _ { i } \to e _ { j }$ .
+
+3. Truth (T): Spans of text that can be interpreted as having a True/False denotation, for example: Is anything red?, $I s$ one ball green and are no cubes red? T-type span grounding is represented using a real-value $p _ { t r u e }$ , $0 \leq p _ { t r u e } \leq 1$ , that denotes the probability of the span being True.
+
+Ungrounded Semantic Types: Spans of text whose meaning cannot be grounded in the KG.
+
+1. Vector (V): This type is used for spans representing functions that cannot yet be grounded in the KG, for example words such as and or every. These spans are represented using 4 different real-valued vectors $v _ { 1 } \in \mathbb { R } ^ { 2 } , v _ { 2 } \in \mathbb { R } ^ { 3 } , v _ { 3 } \in \mathbb { R } ^ { 4 } , v _ { 4 } \in \mathbb { R } ^ { 5 }$ that are used to parameterize different composition modules described below in $\ S 3 . 2$ .
+
+2. Vacuous $( \phi )$ : Spans that are considered semantically vacuous, but are necessary syntactically, e.g. of in left of a cube. During composition, these nodes act as identity functions.
+
+Partially-Grounded Semantic Types: Spans of text that can only be partially grounded in the knowledge graph, such as and red or are four spheres. Here, we represent the span by a combination of a grounding and vectors, representing grounded and ungrounded aspects of meaning respectively. The grounded component of the representation will typically combine with another fully grounded representation, and the ungrounded vectors will parameterize the composition module. We define 3 semantic types of this kind: EV, RV and TV, corresponding to the combination of entities, relations and boolean groundings with an ungrounded vector. Here, the word represented by the vectors can be viewed as a binary function, one of whose arguments has been supplied.
+
+# 3.2 COMPOSITION MODULES
+
+Next, we describe how we compose phrase representations (from $\ S 3 . 1 )$ to create representations for larger phrases. We define a small number of generic composition modules, that take as input two constituents of text with their corresponding semantic representations (grounded representations and ungrounded vectors), and outputs the semantic type and corresponding representation of the larger constituent. The composition modules are parameterized by the trainable word vectors.
+
+These can be divided into several categories:
+
+Composition modules resulting in fully grounded denotations: Described in Figure 2.
+
+![](images/ef3ce494ebce8a87057633fea6542442deb0c6193fcdd67293d68026bd3be4a5.jpg)
+
+$$
+\begin{array} { r l } & { \quad \ ' \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \cup _ { e _ { i } } = \sigma \Bigl ( \mathbf { v _ { 1 } } \cdot \left[ \begin{array} { l } { p _ { e _ { i } } ^ { R } } \\ { 1 } \end{array} \right] \Bigr ) } \\ & { \underset { \ @ \ @ } { \mathsf { V } } \quad \mathsf { c y l i n d r i c a l } } \\ & { \underset { \bigodot \updownarrow } { \mathsf { C y l i n d r i c a l } } } \end{array}
+$$
+
+$\mathbf { E } + \mathbf { E }  \mathbf { E }$ : This module performs a function on a pair of soft entity sets, parameterized by the model’s global parameter vector $[ w _ { 1 } , w _ { 2 } , b ]$ to produce a new soft entity set. The composition function for a single entity’s resulting attention value is shown. Such a composition module can be used to interpret compound nouns and entity appositions. For example, the composition module shown above learns to output the intersection of two entity sets.
+
+$$
+\begin{array} { r } { \begin{array} { c } { \bullet _ { \boxtimes \mathbf { E } } ^ { \cup } } \\ { \hfill \phantom { \sum } _ { \begin{array} { c } { \prime } \\ { \prime } \\ { \mathbf { E } } \\ { \mathbf { E } } \end{array} } \setminus \begin{array} { r } { p _ { e _ { i } } = \sigma \left( \mathbf { v _ { 2 } } \cdot \left[ \begin{array} { l } { p _ { e _ { i } } ^ { L } } \\ { p _ { e _ { i } } ^ { R } } \\ { 1 } \end{array} \right] \right.} \\ { \vdots \hfill \mathrm { ~ s u r ~ o r ~ p u r p 1 e } } \\ { \mathbf { \bigoplus } _ { \mathbf { 0 } } \mathbf { E } } \end{array} } \end{array}  } \end{array}
+$$
+
+$\mathbf { V } + \mathbf { E }  \mathbf { E }$ : This module performs a function on a soft entity set, parameterized by a word vector, to produce a new soft entity set. For example, the word not learns to take the complement of a set of entities. The entity attention representation of the resulting span is computed by using the indicated function that takes the $\boldsymbol { v } _ { 1 } \in \mathbb { R } ^ { 2 }$ vector of the $\mathbf { V }$ constituent as a parameter argument and the entity attention vector of the $\mathbf { E }$ constituent as a function argument.
+
+![](images/5ce6a0e45402c1bb7b4dbe583d64a0e14efc9b650262ab7a978f2824679378bf.jpg)
+
+$\mathbf { E V + E  E }$ : This module combines two soft entity sets into a third set, parameterized by the $v _ { 2 }$ word vector. This composition function is similar to a linear threshold unit and is capable of modeling various mathematical operations such as logical conjunctions, disjunctions, differences etc. for different values of $v _ { 2 }$ . For example, the word or learns to model set union.
+
+$$
+\begin{array}{c} \begin{array} { r l } & { \overline { { \mathrm { I } \mathrm { \tiny ~ T r u e } } } } \\ & { \qquad \boldsymbol { \Bigg / } ^ { \mathsf { T } } \setminus \underbrace { p _ { t r u e } } _ { \mathsf { E } } = \sigma ( v _ { 3 } ^ { 1 } \Bigg [ \sum _ { e _ { i } } \sigma ( [ v _ { 3 } ^ { 3 } ] \cdot [ \begin{array} { l } { p _ { e _ { i } } ^ { R } } \\ { 1 } \end{array} ] } \end{array} ) ] + v _ { 3 } ^ { 2 } )  \\ & { \qquad \mathsf { V } } \\ & { \textsf { s } \underset { \ @ \boldsymbol { \Psi } } { \mathsf { a n y t h i n g ~ c y l } } \mathrm { i n d r ~ i c a l } } \\ & { \qquad \boldsymbol { \Theta } \stackrel { \qquad } { \Pi } \overleftrightarrow { \boldsymbol { \Psi } } \big \square } \end{array}
+$$
+
+$\mathbf { R } + \mathbf { E }  \mathbf { E }$ : This module composes a set of relations (represented as a single soft adjacency matrix) and a soft entity set to produce an output soft entity set. The composition function uses the adjacency matrix representation of the $\mathbf { R }$ -span and the soft entity set representation of the E-span.
+
+$\mathbf { V } + \mathbf { E }  \mathbf { T }$ : This module maps a soft entity set onto a soft boolean, parameterized by word vector $( v _ { 3 } )$ . The module counts whether a sufficient number of elements are in (or out) of the set. For example, the word any should test if a set is non-empty.
+
+$$
+\begin{array} { r l } & { \underbrace { \overline { { \left[ \mathsf { F a l s e } \right] } } } _ { \displaystyle { \int } } \mathsf { T } _ { \mathrm { \normalfont ~ \ " ~ \ " ~ } } p _ { t r u e } = \sigma \left( v _ { 4 } ^ { 1 } \left[ \sum _ { e _ { i } } \sigma \left( \begin{array} { c } { \left[ v _ { 4 } ^ { 3 } \right] } \\ { v _ { 4 } ^ { 4 } } \\ { \left[ v _ { 5 } ^ { 5 } \right] } \end{array} \cdot \left[ \begin{array} { c } { p _ { e _ { i } } ^ { L } } \\ { p _ { e _ { i } } ^ { R } } \\ { 1 } \end{array} \right] \right) \right] + v _ { 4 } ^ { 5 } \right) } \\ & { \vdots \begin{array} { l } { \mathsf { E V } } \\ { \mathrm { e v e r y ~ c y ~ l ~ i n d e r ~ \quad ~ b ~ l ~ u e } } \\ { \bigoplus } \end{array} } \end{array}
+$$
+
+$\mathbf { E V + E }  \mathbf { T }$ : This module combines two soft entity sets into a soft boolean, which is useful for modelling generalized quantifiers. For example, in is every cylinder blue, the module can use the inner sigmoid to test if an element $e _ { i }$ is in the set of cylinders $( p _ { e _ { i } } ^ { L } \approx 1 )$ ) but not in the set of blue things $( p _ { e _ { i } } ^ { R } \approx 0 )$ ), and then use the outer sigmoid to return a value close to 1 if the sum of elements matching this property is close to 0.
+
+$$
+\begin{array} { c } { { { \underbrace { \left[ \mathsf { F a l s e } \right] } } _ { \displaystyle { \mathsf { T } } } _ { \displaystyle { \mathrm { s } } _ { t r u e } } = \sigma \left( \mathbf { v } _ { 2 } \cdot \left[ \begin{array} { l } { p _ { t r u e } ^ { L } } \\ { p _ { t r u e } ^ { R } } \\ { 1 } \end{array} \right] \right) } } \\ { { { \mathsf { T } } } } \\ { { { \mathsf { Z } } \mathsf { b a l l s ~ r e d ~ a n d ~ } } } \\ { { { \underbrace { \left[ \mathsf { F a l s e } \right] \ @ { \mathsf { o D } } } } _ { \displaystyle { \left[ \mathsf { T r u e } \right] } } \in { \mathsf { S l u e } } } } \end{array}
+$$
+
+![](images/46ffd5f89b90e1ffa26a1bc73dbbdc87f2868559651ed96abd911bbd6be36748.jpg)
+
+$\mathbf { T V } + \mathbf { T }  \mathbf { T }$ : This module maps a pair of soft booleans into a soft boolean using the $v _ { 2 }$ word vector to parameterize the composition function. Similar to $\mathbf { E V + E }  \mathbf { E }$ , this module facilitates modeling a range of boolean set operations. Using the same functional form for different composition functions, allows our model to use the same ungrounded word vector $\left( v _ { 2 } \right)$ for compositions that are semantically analogous.
+
+$\mathbf { R } \mathbf { V } + \mathbf { R }  \mathbf { R }$ : This module composes a pair of soft set of relations to a produce an output soft set of relations. For example, the relations left and above are composed by the word $o r$ to produce a set of relations such that entities $e _ { i }$ and $e _ { j }$ are related if either of the two relations exists between them. The functional form for this composition is similar to $\mathbf { E V + E }  \mathbf { E }$ and $\mathbf { T V } + \mathbf { T }  \mathbf { T }$ modules.
+
+Figure 2: Composition Modules that compose two constituent span representations into the representation for the combined larger span, using the indicated equations.
+
+Composition with $\phi$ -typed nodes: Phrases with type $\phi$ are treated as being semantically transparent identity functions. Phrases of any other type can combined with these with no change to their type or representation.
+
+Composition modules resulting in partially grounded denotations: We define several simple modules that combine fully grounded phrases with ungrounded phrases, by deterministically taking the union of the representations, giving phrases with partially grounded representations $( \ S \ 3 . 1 )$ . These modules are useful for when words act as binary functions; here they combine with their first argument. For example, in Figure 1, or and not cylindrical combine to make a phrase containing both the vectors for or and the entity set for not cylindrical.
+
+# 4 PARSING MODEL
+
+Here, we describe how our model classifies question tokens into different semantic type spans and compute their representations $( \ S 4 . 1 )$ , recursively uses the composition modules defined above to parse the question appropriately into a soft latent tree that provides the answer $( \ S 4 . 2 )$ . The model is trained end-to-end using only question-answer supervision $( \ S 4 . 3 )$ .
+
+# 4.1 LEXICAL REPRESENTATION ASSIGNMENT
+
+Each token in the question sentence is assigned a distribution over the semantic types, and given a grounding for each type. Tokens can only be assigned the E, R, V, and $\phi$ semantic types. For example, the token cylindrical in the question in Fig. 1 is assigned a distribution over the 4 semantic types (one shown) and for the $\mathbf { E }$ type, the representation computed is the set of cylindrical entities.
+
+Semantic Type Distribution for Tokens: To compute the semantic type distribution, our model represents each word $w$ in the word vocabulary $\nu$ , and each semantic type $t$ using an embedding vector; $v _ { w } , v _ { t } \in \mathbb { R } ^ { d }$ . The semantic type distribution is assigned with a softmax:
+
+$$
+p ( t | w _ { i } ) \propto \exp ( v _ { t } \cdot v _ { w _ { i } } )
+$$
+
+Grounding for Tokens: For each of the four semantic type assignments for question tokens, we need to compute/assign their corresponding representations.
+
+1. E-Type Representation: Each entity $e \in { \mathcal { E } }$ , is represented using an embedding vector $v _ { e } \in \mathbb { R } ^ { d }$ based on the concatenation of vectors for its properties. For each token $w$ , we use its word vector to find the probability of each entity being part of the $\mathbf { E }$ -Type grounding:
+
+$$
+p _ { e _ { i } } ^ { w } = \sigma ( v _ { e _ { i } } \cdot v _ { w } ) ~ \forall e _ { i } \in \mathcal { E }
+$$
+
+For example, in Fig. 1, the word red will be grounded as all the red entities.
+
+2. R-Type Representation: Each relation $r \in \mathcal { R }$ , is represented using an embedding vector $v _ { r } \in \mathbb { R } ^ { d }$ . For each token $w _ { i }$ in the question, we first compute a distribution over relations it could refer to, and then use this distribution to compute the expected adjacency matrix that forms the $\mathbf { R }$ -type representation for this token.
+
+$$
+p ( r | w _ { i } ) \propto \exp ( v _ { r } \cdot v _ { w _ { i } } )
+$$
+
+$$
+A ^ { w _ { i } } = \sum _ { r \in \mathcal { R } } p ( r | w _ { i } ) \cdot A _ { r }
+$$
+
+For example, the word left in Fig. 1 is grounded as the subset of edges with the label ‘left’.
+
+3. V-Type Representation: For each word $w \in \mathcal { V }$ , we learn four vectors $v _ { 1 } \in \mathbb { R } ^ { 2 } , v _ { 2 } \in$ $\mathbb { R } ^ { 3 } , \tilde { v _ { 3 } } \in \mathbb { R } ^ { \bar { 4 } } , v _ { 4 } \in \mathbb { R } ^ { 5 }$ , and use these as the representation for words with the $\mathbf { V }$ -Type.
+
+4. $\phi$ -Type Representation: This type is used for semantically vacuous words, which do not require a representation.
+
+# 4.2 PARSING QUESTIONS
+
+To learn the correct structure for applying composition modules, we use a simple parsing model. We build a parse-chart over the question encompassing all possible trees by applying all composition modules, similar to a standard CRF-based PCFG parser using the CKY algorithm. Each node in the parse-chart, for each span $w _ { i \ldots j }$ of the question, is represented as a distribution over different semantic types with their corresponding representations. This distribution is computed by weighing the different ways of composing the span’s constituents.
+
+Phrase Semantic Type Potential: Each node in the parse-chart is associated with a potential value $\psi ( i , j , t )$ , that is the score assigned by the model to the t semantic type for the $w _ { i \ldots j }$ span. This is computed from all possible ways to form the span $w _ { i \ldots j }$ with type t. For a particular composition of span $w _ { i \dots k }$ of type $\mathbf { t _ { 1 } }$ and $w _ { k + 1 \ldots j }$ of type $\mathbf { t _ { 2 } }$ , using the $\mathbf { t _ { 1 } } + \mathbf { t _ { 2 } }  \mathbf { t }$ module, the score is:
+
+$$
+\psi ( i , j , k , t _ { 1 } + t _ { 2 }  t ) = [ \psi ( i , k , t _ { 1 } ) \cdot \psi ( k + 1 , j , t _ { 2 } ) \cdot \exp ( \sum _ { x } f _ { x } ^ { ( t _ { 1 } + t _ { 2 }  t ) } ( i , j , k | q ) ) ]
+$$
+
+where, $f _ { x } ^ { ( t _ { 1 } + t _ { 2 }  t ) } ( i , j , k | q )$ are six feature functions; a trainable weight for each word per module in the vocabulary, that correspond to: $f _ { 1 }$ : word that appears before the start of the span $w _ { i - 1 }$ ; $f _ { 2 }$ : first word in the span $w _ { i }$ ; $f _ { 3 }$ : last word in the left constituent $w _ { k }$ ; $f _ { 4 }$ : first word in the right constituent $w _ { k + 1 }$ ; $f _ { 5 }$ : last word in the right constituent $w _ { j }$ ; and $f _ { 6 }$ : word that appears after the span $w _ { j + 1 }$ .
+
+The token semantic type potential of $w _ { i }$ , $\psi ( i , i , i , t _ { 1 } + t _ { 2 }  t )$ , is the same as $p ( t | w _ { i } )$ (Eq. 1).
+
+The final t-type potential of $w _ { i \ldots j }$ is computed by summing over scores from all possible compositions:
+
+$$
+\psi ( i , j , t ) = \sum _ { k = i } ^ { j - 1 } \sum _ { \stackrel { \scriptstyle ( t _ { 1 } + t _ { 2 } \to t ) } { \scriptstyle \in \mathrm { M o d u l e s } } } \psi ( i , j , k , t _ { 1 } + t _ { 2 } \to t )
+$$
+
+Combining Phrase Representations: To compute the span $w _ { i \ldots j }$ ’s denotation with type t, $[ [ w _ { i . . j } ] ] _ { K G } ^ { t }$ , we compute an expected output representation from all possible compositions.
+
+$$
+[ w _ { i \cdot . . j } ] _ { K G } ^ { t } = \frac { 1 } { \psi ( i , j , t ) } \sum _ { \stackrel { k = i } { \epsilon \mathrm { \bf ~ ( } t _ { 1 } + t _ { 2 }  t \mathrm { \bf ~ ) } } } ^ { j - 1 } \psi ( i , j , k , t _ { 1 } + t _ { 2 }  t ) \ast [ [ w _ { i \cdot . . k . . j } ] ] _ { K G } ^ { t _ { 1 } + t _ { 2 }  t }
+$$
+
+where resulti $[ [ w _ { i . . j } ] ] _ { K G } ^ { t }$ , is the t-type repre composition of ationwith pan usi $w _ { i \ldots j }$ , e $\mathbb { [ } w _ { i \dots k \dots j } ] _ { K G } ^ { t _ { 1 } + t _ { 2 }  t }$ is the representationcomposition module. $w _ { i , k }$ $w _ { k + 1 \ldots j }$ $\mathbf { t _ { 1 } } ~ + ~ \mathbf { t _ { 2 } } ~  ~ \mathbf { t }$
+
+Answer Grounding: By recursively computing the phrase semantic-type potentials and representations, we can infer the semantic type distribution of the complete question sentence (Eq. 8) and the resulting grounding for different semantic type $t$ , $[ [ w _ { 1 . . | q | } ] ] _ { K G } ^ { t }$ .
+
+$$
+p ( t | q ) \propto \psi ( 1 , | q | , t )
+$$
+
+The answer-type (boolean or subset of entities) for the question is computed using:
+
+$$
+t ^ { * } = \underset { t \in \mathbf { T } , \mathbf { E } } { \mathrm { a r g m a x } } \ p ( t | q )
+$$
+
+The corresponding grounding is $[ [ w _ { 1 . . | q | } ] ] _ { K G } ^ { t ^ { * } }$ , which answers the question.
+
+# 4.3 TRAINING OBJECTIVE
+
+Given a dataset D of (question, answer, knowledge-graph) tuples, {qi, ai, KGi}i=|D|i=1 , we train our model to maximize the log-likelihood of the correct answers. Answers are either booleans, or specific subsets of entities from the KG. We denote the semantic type of the answer as $a _ { t }$ . If the answer is boolean, $a \in \{ 0 , 1 \}$ , otherwise is a subset of entities from the KG, i.e. $a = \{ e _ { j } \}$ . The model’s answer to a question is found by taking its representation of the complete question, containing a distribution over types and the representation for each type. We maximize the following objective:
+
+$$
+\begin{array} { r l } & { \hat { \mathbf { \xi } } = \displaystyle \sum _ { i } \log p ( \boldsymbol { a } ^ { i } | \mathbf { q } ^ { i } , \mathbf { K G } ^ { i } ) \quad \quad \quad \quad ( 1 0 ) } \\ & { = \displaystyle \sum _ { i } \left[ \underbrace { \left( 1 _ { a _ { i } ^ { i } = \mathbf { r } } \left[ \log ( p _ { t r u e } ) ^ { a ^ { i } } ( 1 - p _ { t r u e } ) ^ { ( 1 - a ^ { i } ) } \right] \right) } _ { e _ { j } ^ { i } \in a ^ { i } } + \left( \frac { 1 _ { a _ { i } ^ { i } = \mathbf { E } } } { | \mathcal { E } ^ { i } | } \left[ \log \prod _ { e _ { j } ^ { i } \in a ^ { i } } p _ { e _ { j } ^ { i } } \prod _ { e _ { j } ^ { i } } ( 1 - p _ { e _ { j } ^ { i } } ) \right] \right) \right] } \end{array}
+$$
+
+We also add $L _ { 2 }$ -regularization for the scalar parsing features introduced in $\ S 4 . 2$ .
+
+# 5 DATASET
+
+We generate a dataset of question-answers based on the CLEVR dataset (Johnson et al., 2017), which contains knowledge graphs containing attribute information of objects and relations between them.
+
+We generate a new set of questions for this data, as existing questions contain some biases that can be exploited by models (Johnson et al. (2017) found that many spatial relation questions can be answered only using absolute spatial information and many long questions can be answered correctly without performing all steps of reasoning), and many questions are over 40 words long, which is intractable given that the size of our computation graph is cubic in the question length. Future work should explore scaling our approach to longer questions. We generate 75K questions for training and 37.5K for validation.
+
+Our question set tests various challenging semantic operators. These include conjunctions (e.g. Is anything red or is anything large?), negations (e.g. What is not spherical?), counts (e.g. Are five spheres green?), quantifiers (e.g. Is every red thing cylindrical?), and relations (e.g. What is left of and above a cube?). We employ some simple tests to remove trivial biases from the dataset.
+
+We create two test sets: one drawn from the same distribution as the training data (37.5K), and another containing longer questions than the training data (22.5K).
+
+Our COMPLEX QUESTIONS test set contains the same words and constructions, but chained into longer questions. For example, it contains questions such as What is a cube that is right of a metallic thing that is beneath a blue thing? and Are two red things that are above a sphere metallic?. These questions require more multi-step reasoning to solve.
+
+# 6 EXPERIMENTS
+
+In this section we describe our experimentation setting, the baseline models we compare to, and the various experiments demonstrating the ability of our model to answer compositional questions referring to KG and its ability to generalize to unseen longer questions and new attribute combinations.
+
+# 6.1 EXPERIMENTATION SETTING
+
+Here we describe the training details of our model and the baseline models.
+
+Representing Entities: Each entity in the CLEVR dataset consists of 4 attributes. For each attribute-value, we learn an embedding vector and concatenate the 4-embedding vectors to form the representation for the entity.
+
+Training Details: Training the model is complicated by the large number of poor local minima, as the model needs to learn both good syntactic structures and the complex semantics of neural modules.
+
+To simplify training, we use Curriculum Learning (Bengio et al., 2009) to pre-train the model on an easier subset of questions. We use a 2-step schedule where we first train our model on simple attribute match (What is a red sphere?), attribute existence (Is anything blue?) and boolean composition $I s$ anything green and is anything purple?) questions and in the second step on all questions jointly.
+
+Table 1: Results for Short Questions: Performance of our model compared to baseline models on the Short Questions test set. The LSTM (NO KG) has accuracy close to chance, showing that the questions lack trivial biases.Our model almost perfectly solves all questions showing its ability to learn challenging semantic operators, and parse questions only using weak end-to-end supervision.   
+
+<table><tr><td>Model</td><td>Boolean Questions</td><td>Entity Set Questions</td><td>Relation Questions</td><td>Overall</td></tr><tr><td>LSTM (No KG)</td><td>50.7</td><td>14.4</td><td>17.5</td><td>27.2</td></tr><tr><td>LSTM (NO RELATION)</td><td>88.5</td><td>99.9</td><td>15.7</td><td>84.9</td></tr><tr><td>RELATION NETWORK</td><td>85.6</td><td>89.7</td><td>97.6</td><td>89.4</td></tr><tr><td>Our Model</td><td>99.9</td><td>100</td><td>100.0</td><td>99.9</td></tr></table>
+
+We tune the hyper-parameters using validation accuracy. We train using SGD with learning rate of 0.5 and mini-batch size of 4, regularization constant of 0.3. When assigning the semantic type distribution to the words at the leaves, we add a small positive bias of $+ 1$ for $\phi$ -type and a small negative bias of $- 1$ for the $\mathbf { E }$ -type score before the softmax. Our trainable parameters are: question word embeddings (64-dimensional), relation embeddings (64-dimensional), entity attribute-value embeddings (16-dimensional), four vectors per word for $\mathbf { V }$ -type representations, six scalar feature scores per module per word for the parsing model, and the global parameter vector for the $\mathbf { E + E {  } E }$ module.
+
+Baseline Models: We use three baseline models for comparison. A simple LSTM (NO KG) model that encodes the question using an LSTM network and answers questions without access to the KG. Another LSTM based model, LSTM (NO RELATION), that has access only to the entities of the KG but not the relationship information between them. Finally, we train a RELATION NETWORK (Santoro et al., 2017) augmented model, which achieved state-of-the-art performance on the CLEVR dataset using image state descriptions. Details about the baseline models are given in the Appendix section.
+
+# 6.2 EXPERIMENTS
+
+Short Questions Performance: In Table 1, we see that our model is able to perfectly answer all the questions in the test set. This demonstrates our model can learn challenging semantic operators using composition modules, as well as learn to parse the questions from only using weak endto-end supervision. The RELATION NETWORK also achieves good performance, particularly on questions involving relations, but is weaker than our model on some question types. The LSTM (NO RELATION) model also achieves good performance on questions not involving relations, which are out of scope for the model.
+
+Complex Question Performance: Table 2 shows results on complex questions, which are constructed by combining components of shorter questions. We use the same models as in Table 1, which were trained and developed only on shorter questions. Answering longer questions requires complex multi-hop reasoning, and the ability to generalize from the language seen in its training data to new types of questions. Results show that all baselines achieve close to random performance on this task, despite high accuracy for shorter questions. This shows the challenges in generalizing RNN encoders beyond their training data. In contrast, the strong inductive bias from our model structure allows the model to generalize to complex questions much more easily than RNN encoders.
+
+Generalization to Unseen Attribute Combination: We also measure how well models generalize to unseen attribute combinations in knowledge graphs (using the COGENT subset of CLEVR). For example, the test set contains ‘blue spheres’ that are not found in the training set. None of the models showed a significant reduction in performance in this setting.
+
+Error Analysis: Analyzing the errors of our model, we find that most errors are due to incorrect assignments of structure, rather than semantic errors from the modules. For example, in the question Are four red spheres beneath a metallic thing small?, our model produces a parse where it composes metallic thing small into a single node instead of composing red spheres beneath a metallic thing into a single node. Future work should use more sophisticated parsing models.
+
+<table><tr><td>Model</td><td>Non-Relation Questions</td><td>Relation Questions</td><td>Overall</td></tr><tr><td>LSTM (NO KG)</td><td>46.0</td><td>39.6</td><td>41.4</td></tr><tr><td>LSTM(NO RELATION)</td><td>62.2</td><td>49.2</td><td>52.2</td></tr><tr><td>RELATION NETWORK</td><td>51.1</td><td>38.9</td><td>41.5</td></tr><tr><td>Our Model</td><td>81.8</td><td>85.4</td><td>84.6</td></tr></table>
+
+Table 2: Results for Complex Questions: All baseline models fail to generalize to questions requiring longer chains of reasoning than seen during training. Our model substantially outperforms the baselines, showing its ability to perform complex multi-hop reasoning, and generalize from its training data. Analysis suggests that most errors from our model are due to assigning incorrect structures, not mistakes by the composition modules.
+
+# 7 RELATED WORK
+
+Many approaches have been proposed to perform question-answering against structured knowledge sources. Semantic parsing models have attempted to learn structures over pre-defined discrete operators, to produce logical forms that can be executed to answer the question. Early work trained using gold-standard logical forms (Zettlemoyer & Collins, 2005; Kwiatkowski et al., 2010), whereas later efforts have only used answers to questions (Liang et al., 2011; Krishnamurthy & Kollar, 2013; Pasupat & Liang, 2015). A key difference is that our model must learn semantic operators from data, which may be necessary to model the fuzzy interpretations of some function words like many or few.
+
+Another similar line of work is neural program induction models, such as Neural Programmer (Neelakantan et al., 2016) and Neural Symbolic Machine (Liang et al., 2017). These models learn to produce programs composed of predefined operators using weak supervision to answer questions against semi-structured tables.
+
+Neural module networks have recently been proposed for learning semantic operators (Andreas et al., 2016b) for question answering. This model assumes that the structure of the semantic parse is given, and must only learn a set of operators. Dynamic Neural Module Networks (D-NMN) extend this approach by selecting from a small set of candidate module structures (Andreas et al., 2016a). In contrast, our approach learns a model over all possible structures for interpreting a question.
+
+Our work is most similar to the most recently proposed N2NMN (Hu et al., 2017) model, an end-toend version of D-NMN. This model learns both semantic operators and the layout in which to compose them. However, optimizing the layouts requires reinforcement learning, which is challenging due to the high variance of policy gradients, whereas our approach is end-to-end differentiable.
+
+# 8 CONCLUSION
+
+We have introduced a model for answering questions requiring compositional reasoning that combines ideas from compositional semantics with end-to-end learning of composition operators and structure. We demonstrated that the model is able to learn a number of complex composition operators from end task supervision, and have shown that the linguistically motivated inductive bias imposed by the structure of the model allows it to generalize well beyond its training data. Future work should explore scaling the model to other question answering tasks.
+
+# REFERENCES
+
+Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Learning to compose neural networks for question answering. In HLT-NAACL, 2016a.
+
+Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Neural module networks. In CVPR, pp. 39–48, 2016b.
+
+Yoshua Bengio, Jer´ ome Louradour, Ronan Collobert, and Jason Weston. Curriculum learning. In ˆ Proceedings of the 26th annual international conference on machine learning, pp. 41–48. ACM, 2009.
+
+Ronghang Hu, Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Kate Saenko. Learning to reason: End-to-end module networks for visual question answering. CoRR, abs/1704.05526, 2017.
+
+Justin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C. Lawrence Zitnick, and Ross Girshick. Clevr: A diagnostic dataset for compositional language and elementary visual reasoning. In CVPR, July 2017.
+
+Jayant Krishnamurthy and Thomas Kollar. Jointly learning to parse and perceive: Connecting natural language to the physical world. TACL, 1:193–206, 2013.
+
+Tom Kwiatkowski, Luke S. Zettlemoyer, Sharon Goldwater, and Mark Steedman. Inducing probabilistic ccg grammars from logical form with higher-order unification. In EMNLP, 2010.
+
+Chen Liang, Jonathan Berant, Quoc Le, Kenneth D. Forbus, and Ni Lao. Neural symbolic machines: Learning semantic parsers on freebase with weak supervision. In ACL, 2017.
+
+Percy Liang, Michael I Jordan, and Dan Klein. Learning dependency-based compositional semantics. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies-Volume 1, pp. 590–599. Association for Computational Linguistics, 2011.
+
+Jean Maillard, Stephen Clark, and Dani Yogatama. Jointly learning sentence embeddings and syntax with unsupervised tree-lstms. CoRR, abs/1705.09189, 2017. URL http://arxiv.org/abs/ 1705.09189.
+
+Richard Montague. The proper treatment of quantification in ordinary English. In K. J. J. Hintikka, J. Moravcsic, and P. Suppes (eds.), Approaches to Natural Language, pp. 221–242. Reidel, Dordrecht, 1973.
+
+Arvind Neelakantan, Quoc V. Le, Mart´ın Abadi, Andrew McCallum, and Dario Amodei. Learning a natural language interface with neural programmer. CoRR, abs/1611.08945, 2016.
+
+Panupong Pasupat and Percy Liang. Compositional semantic parsing on semi-structured tables. In ACL, 2015.
+
+Ethan Perez, Harm de Vries, Florian Strub, Vincent Dumoulin, and Aaron C. Courville. Learning visual reasoning without strong priors. CoRR, abs/1707.03017, 2017. URL http://arxiv. org/abs/1707.03017.
+
+Adam Santoro, David Raposo, David G. T. Barrett, Mateusz Malinowski, Razvan Pascanu, Peter Battaglia, and Timothy P. Lillicrap. A simple neural network module for relational reasoning. CoRR, abs/1706.01427, 2017. URL http://arxiv.org/abs/1706.01427.
+
+Luke S Zettlemoyer and Michael Collins. Learning to map sentences to logical form: Structured classification with probabilistic categorial grammars. UAI, 2005.
+
+# APPENDIX
+
+BASELINE MODELS
+
+LSTM (NO KG)
+
+We use a LSTM network to encode the question as a vector $q$ . We also define three other parameter vectors, $t , e$ and $b$ that are used to predict the answer-type $P ( \bar { a } = \mathbf { T } ) = \sigma ( \boldsymbol { q } \cdot \boldsymbol { t } )$ , entity attention value $p _ { e _ { i } } = \sigma ( q \cdot e )$ , and the probability of the answer being True $p _ { t r u e } = \sigma ( q \cdot b )$ .
+
+LSTM (NO RELATION)
+
+Similar to LSTM (NO RELATION), the question is encoded using a LSTM network as vector $q$ . Similar to our model, we learn entity attribute-value embeddings and represent each entity as the concatenation of the 4 attribute-value embeddings, $\boldsymbol { v } _ { e _ { i } }$ . Similar to LSTM (NO RELATION), we also define the $t$ parameter vector to predict the answer-type. The entity-attention values are predicted as $p _ { e _ { i } } = \sigma ( v _ { e _ { i } } \cdot q )$ . To predict the probability of the boolean-type answer being true, we first add the entity representations to form $b { \bar { \mathbf { \theta } } } = \sum _ { e _ { i } } v _ { e _ { i } }$ , then make the prediction as $p _ { t r u e } = \sigma ( q \cdot b )$ .
+
+# RELATION NETWORK AUGMENTED MODEL
+
+The original formulation of the relation network module is as follows:
+
+$$
+R N ( q , K G ) = f _ { \phi } \left( \sum _ { i , j } g _ { \theta } ( e _ { i } , e _ { j } , q ) \right)
+$$
+
+where $e _ { i } , e _ { j }$ are the representations of the entities and $q$ is the question representation from an LSTM network. The output of the Relation Network module is a scalar score value for the elements in the answer vocabulary. Since our dataset contains entity-set valued answers, we modified the module in the following manner.
+
+We concatenate the object pair representations with the representations of the pair of directed relationships between them1. We then use the Relation Network module to produce an output representation for each entity in the KB, in the following manner:
+
+$$
+R N _ { e _ { i } } = f _ { \phi } \Bigg ( \sum _ { j } g _ { \theta } ( e _ { i } , e _ { j } , r _ { i j } ^ { 1 } , r _ { i j } ^ { 2 } , q ) \Bigg )
+$$
+
+Similar to the LSTM baselines, we define a parameter vector $t$ to predict the answer-type as:
+
+$$
+P ( a = \mathbf { T } ) = \sigma ( q \cdot t )
+$$
+
+$$
+P ( a = \mathbf { E } ) = 1 - P ( a = \mathbf { T } )
+$$
+
+To predict the probability of the boolean type answer being true, we define a parameter vector $b$ and predict as following:
+
+$$
+p _ { t r u e } = \sigma \bigg ( b \cdot \sum _ { e _ { i } } R N _ { e _ { i } } \bigg )
+$$
+
+To predict the entity-attention values, we use a separate attribute-embedding matrix to first generate the output representation for each entity, $e _ { i } ^ { o u t }$ , then predict the output attention values as follows:
+
+$$
+p _ { e _ { i } } = \sigma \bigg ( R N _ { e _ { i } } \cdot e _ { i } ^ { o u t } \bigg )
+$$
+
+We tried other architectures as well, but this modification provided the best performance on the validation set. We also tuned the hyper-parameters and found the setting from Santoro et al. (2017) to work the best based on validation accuracy. We used a different 2-step curriculum to train the RELATION NETWORK module, in which we replace the Boolean questions with the relation questions in the first-schedule and jointly train on all questions in the subsequent schedule.

parse/train/rkaqxm-0b/rkaqxm-0b_content_list.json

@@ -0,0 +1,1776 @@
+[
+    {
+        "type": "text",
+        "text": "NEURAL COMPOSITIONAL DENOTATIONAL SEMANTICS FOR QUESTION ANSWERING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            826,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            174,
+            398,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            238,
+            544,
+            253
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Answering compositional questions requiring multi-step reasoning is challenging for current models. We introduce an end-to-end differentiable model for interpreting questions, which is inspired by formal approaches to semantics. Each span of text is represented by a denotation in a knowledge graph, together with a vector that captures ungrounded aspects of meaning. Learned composition modules recursively combine constituents, culminating in a grounding for the complete sentence which is an answer to the question. For example, to interpret not green, the model will represent green as a set of entities, not as a trainable ungrounded vector, and then use this vector to parametrize a composition function to perform a complement operation. For each sentence, we build a parse chart subsuming all possible parses, allowing the model to jointly learn both the composition operators and output structure by gradient descent. We show the model can learn to represent a variety of challenging semantic operators, such as quantifiers, negation, disjunctions and composed relations on a synthetic question answering task. The model also generalizes well to longer sentences than seen in its training data, in contrast to LSTM and RelNet baselines. We will release our code. ",
+        "bbox": [
+            233,
+            270,
+            766,
+            491
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            517,
+            336,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Compositionality is a mechanism by which the meanings of complex expressions are systematically determined from the meanings of their parts, and has been widely assumed in the study of both natural languages (Montague, 1973), as well as programming and logical languages, as a means for allowing speakers to generalize to understanding an infinite number of sentences. Popular neural network approaches to question answering use a restricted form of compositionality, typically encoding a sentence word-by-word from left-to-right, and finally executing the complete sentence encoding against a knowledge source (Perez et al., 2017). Such models can fail to generalize from training sentences in surprising ways. Inspired by linguistic theories of compositional semantics, we instead build a latent tree of interpretable expressions over a sentence, recursively combining constituents using a small set of neural modules. When tested on longer questions than are found in the training data, we find that our model achieves higher performance than baselines using LSTMs and RelNets. ",
+        "bbox": [
+            174,
+            547,
+            825,
+            700
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our approach resembles Montague semantics, in which a tree of interpretable expressions is built over the sentence, with nodes combined by a small set of composition functions. However, both the structure of the sentence and the neural modules that handle composition are learned by end-to-end gradient descent. To achieve this, we define the parametric form of small set of neural modules, and then build a parse chart over each sentence subsuming all possible trees. Each node in the chart represents a span of text with a distribution over groundings (in terms of booleans and knowledge base nodes and edges), as well as a vector representing aspects of the meaning that have not yet been grounded. The representation for a node is built by taking a weighted sum over different ways of building the node (similarly to Maillard et al. (2017)). ",
+        "bbox": [
+            174,
+            708,
+            825,
+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Typical neural network approaches to grounded question answering first encode a question from left-to-right with a recurrent neural network (RNNs), and then evaluate the encoding against an encoding of the knowledge source (for example, a knowledge base or image) (Santoro et al., 2017). In contrast to classical approaches to compositionality, constituents of complex expressions are not given explicit interpretations in isolation. For example, in Which cubes are large or green?, an RNN encoder will not explicitly build an interpretation for the expression large or green. We show that such approaches can generalize poorly when tested on more complex sentences than they were trained on. In contrast, our approach imposes strong independence assumptions that give a linguistically motivated inductive bias. In particular, it enforces that phrases are interpreted independently of surrounding words, allowing the model to generalize naturally to interpreting phrases in different contexts. In the previous example, large or green will be represented as a particular set of entities in a knowledge graph, and be intersected with the set of entities represented by the cubes node. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/3e5ee45df3806e70730c7a946cf9d9784259baa6a59de50b2aaa5a49ac02fbfd.jpg",
+        "image_caption": [
+            "Figure 1: A correct parse for a question given the knowledge graph on the right, using our model. We show the type for each node, and its denotation in terms of the knowledge graph. The words or and not are represented by vectors, which parameterize composition modules. The denotation for the complete question represents the answer to the question. Nodes here have types $E$ for sets of entities, $R$ for relations, $V$ for ungrounded vectors, $E V$ for a combination of entities and a vector, and $\\phi$ for semantically vacuous nodes. While we show only one parse tree here, our model builds a parse chart subsuming all trees. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            98,
+            761,
+            275
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            380,
+            825,
+            463
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Another perspective on our work is as a method for learning the layouts of Neural Module Networks (NMNs) (Andreas et al., 2016b). Work on NMNs has focused on how to construct the structure of the network, variously using rules, parsers and reinforcement learning (Andreas et al., 2016a; Hu et al., 2017). Our end-to-end differentiable model jointly learns structures and modules by gradient descent. ",
+        "bbox": [
+            174,
+            469,
+            825,
+            526
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 MODEL OVERVIEW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            546,
+            366,
+            561
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our task is to answer a question $q = w _ { 1 \\ldots | q | }$ , with respect to a Knowledge Graph (KG) consisting of nodes $\\mathcal { E }$ (representing entities) and labelled directed edges $\\mathcal { R }$ (representing relationship between entities). In our task, answers are either booleans, or specific subsets of nodes from the KG. ",
+        "bbox": [
+            176,
+            577,
+            823,
+            619
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our model builds a parse for the sentence, in which phrases are grounded in the KG, and a small set of composition modules are used to combine phrases, resulting in a grounding for the complete question sentence that answers the question. For example, in Figure 1, the phrases not and cylindrical are interpreted as a function word and an entity set, and then not cylindrical is interpreted by computing the complement of the entity set. The node at the root of the parse tree is the answer to the question. ",
+        "bbox": [
+            174,
+            626,
+            825,
+            695
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We describe a compositional neural model that answers such questions by: ",
+        "bbox": [
+            174,
+            703,
+            661,
+            717
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Grounding individual tokens in a Knowledge Graph. Tokens can either be grounded as particular sets of entities and relations in the KG, as ungrounded vectors, or marked as being semantically vacuous. For each word, we learn parameters that are used to compute a distribution over semantic types and corresponding denotations in a KG (§ 4.1).   \n2. Combining representations for adjacent phrases into representations for larger phrases, using trainable neural composition modules $( \\ S 3 . 2 )$ . This produces a denotation for the phrase.   \n3. Assigning a binary-tree structure to the question sentence, which determines how words are grounded, and which phrases are combined using which modules. We build a parse chart subsuming all possible structures, and train a parsing model to increase the likelihood of structures leading to the correct answer to questions. Different parses leading to a denotation for a phrase of type $t$ are merged into an expected denotation, allowing dynamic programming $( \\ S 4 )$ .   \n4. Answering the question, with the most likely grounding of the phrase spanning the sentence. ",
+        "bbox": [
+            207,
+            729,
+            826,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 COMPOSITIONAL SEMANTICS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            452,
+            117
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 SEMANTIC TYPES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            339,
+            148
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our model classifies spans of text into different semantic types to represent their meaning as explicit denotations or ungrounded vectors. All phrases are assigned a distribution over semantic types. The semantic type determines how a phrase is grounded, and which composition modules can be used to combine it with other phrases. A phrase spanning $w _ { i \\ldots j }$ has a denotation $[ [ w _ { i . . j } ] ] _ { K G } ^ { t }$ for each semantic type $t$ J K. For example, in Figure 1, red thing corresponds to a set of entities, left corresponds to a set of relations, and not is treated as an ungrounded vector. ",
+        "bbox": [
+            174,
+            161,
+            825,
+            244
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The semantic types we define can be classified into the three different categories. Below we describe these semantic types and their corresponding representations. ",
+        "bbox": [
+            173,
+            252,
+            823,
+            280
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Grounded Semantic Types: Spans of text that can be fully grounded in the KG. ",
+        "bbox": [
+            173,
+            296,
+            712,
+            311
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. Entity $\\mathbf { ( E ) }$ : Spans of text that can be grounded to a set of entities in the KG, for example: red sphere or large cube. $\\mathbf { E }$ -type span grounding is represented as a soft-attention value for each entity, $[ p _ { e _ { 1 } } , \\dotsc , p _ { e _ { | \\varepsilon | } } ]$ , where $0 \\leq p _ { e _ { i } } \\leq 1$ . This can be viewed as a soft version of a logical set-valued denotation, which we refer to as a ‘soft entity set’. ",
+        "bbox": [
+            214,
+            324,
+            825,
+            381
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2. Relation $\\mathbf { ( R ) }$ : Spans of text that can be grounded to a set of relations from the KG, for example: left of or not right of or above. R-type span grounding is represented by a soft adjacency matrix $A \\in \\mathbb { R } ^ { | \\mathcal { E } | \\times | \\mathcal { E } | }$ where $A _ { i j } = 1$ denotes a directed edge from $e _ { i } \\to e _ { j }$ . ",
+        "bbox": [
+            212,
+            386,
+            825,
+            430
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3. Truth (T): Spans of text that can be interpreted as having a True/False denotation, for example: Is anything red?, $I s$ one ball green and are no cubes red? T-type span grounding is represented using a real-value $p _ { t r u e }$ , $0 \\leq p _ { t r u e } \\leq 1$ , that denotes the probability of the span being True. ",
+        "bbox": [
+            212,
+            435,
+            825,
+            492
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Ungrounded Semantic Types: Spans of text whose meaning cannot be grounded in the KG. ",
+        "bbox": [
+            176,
+            508,
+            790,
+            523
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "1. Vector (V): This type is used for spans representing functions that cannot yet be grounded in the KG, for example words such as and or every. These spans are represented using 4 different real-valued vectors $v _ { 1 } \\in \\mathbb { R } ^ { 2 } , v _ { 2 } \\in \\mathbb { R } ^ { 3 } , v _ { 3 } \\in \\mathbb { R } ^ { 4 } , v _ { 4 } \\in \\mathbb { R } ^ { 5 }$ that are used to parameterize different composition modules described below in $\\ S 3 . 2$ . ",
+        "bbox": [
+            214,
+            536,
+            825,
+            593
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2. Vacuous $( \\phi )$ : Spans that are considered semantically vacuous, but are necessary syntactically, e.g. of in left of a cube. During composition, these nodes act as identity functions. ",
+        "bbox": [
+            212,
+            598,
+            825,
+            627
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Partially-Grounded Semantic Types: Spans of text that can only be partially grounded in the knowledge graph, such as and red or are four spheres. Here, we represent the span by a combination of a grounding and vectors, representing grounded and ungrounded aspects of meaning respectively. The grounded component of the representation will typically combine with another fully grounded representation, and the ungrounded vectors will parameterize the composition module. We define 3 semantic types of this kind: EV, RV and TV, corresponding to the combination of entities, relations and boolean groundings with an ungrounded vector. Here, the word represented by the vectors can be viewed as a binary function, one of whose arguments has been supplied. ",
+        "bbox": [
+            174,
+            643,
+            825,
+            756
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 COMPOSITION MODULES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            773,
+            390,
+            787
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Next, we describe how we compose phrase representations (from $\\ S 3 . 1 )$ to create representations for larger phrases. We define a small number of generic composition modules, that take as input two constituents of text with their corresponding semantic representations (grounded representations and ungrounded vectors), and outputs the semantic type and corresponding representation of the larger constituent. The composition modules are parameterized by the trainable word vectors. ",
+        "bbox": [
+            174,
+            801,
+            825,
+            871
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "These can be divided into several categories: ",
+        "bbox": [
+            178,
+            877,
+            465,
+            892
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Composition modules resulting in fully grounded denotations: Described in Figure 2. ",
+        "bbox": [
+            171,
+            909,
+            767,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/ef3ce494ebce8a87057633fea6542442deb0c6193fcdd67293d68026bd3be4a5.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            232,
+            107,
+            428,
+            172
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bb632c2beb5be4a908eb6d1c669c0abc1549ebe02d7dadf1196feabfae35a1a6.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\quad \\ ' \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\cup _ { e _ { i } } = \\sigma \\Bigl ( \\mathbf { v _ { 1 } } \\cdot \\left[ \\begin{array} { l } { p _ { e _ { i } } ^ { R } } \\\\ { 1 } \\end{array} \\right] \\Bigr ) } \\\\ & { \\underset { \\ @ \\ @ } { \\mathsf { V } } \\quad \\mathsf { c y l i n d r i c a l } } \\\\ & { \\underset { \\bigodot \\updownarrow } { \\mathsf { C y l i n d r i c a l } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            568,
+            121,
+            764,
+            174
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { E } + \\mathbf { E }  \\mathbf { E }$ : This module performs a function on a pair of soft entity sets, parameterized by the model’s global parameter vector $[ w _ { 1 } , w _ { 2 } , b ]$ to produce a new soft entity set. The composition function for a single entity’s resulting attention value is shown. Such a composition module can be used to interpret compound nouns and entity appositions. For example, the composition module shown above learns to output the intersection of two entity sets. ",
+        "bbox": [
+            173,
+            178,
+            488,
+            291
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dde7bbc535ecd34d70aad49b22e8f7fc24fcf533e1b572bb17c808a624cb96ce.jpg",
+        "text": "$$\n\\begin{array} { r } { \\begin{array} { c } { \\bullet _ { \\boxtimes \\mathbf { E } } ^ { \\cup } } \\\\ { \\hfill \\phantom { \\sum } _ { \\begin{array} { c } { \\prime } \\\\ { \\prime } \\\\ { \\mathbf { E } } \\\\ { \\mathbf { E } } \\end{array} } \\setminus \\begin{array} { r } { p _ { e _ { i } } = \\sigma \\left( \\mathbf { v _ { 2 } } \\cdot \\left[ \\begin{array} { l } { p _ { e _ { i } } ^ { L } } \\\\ { p _ { e _ { i } } ^ { R } } \\\\ { 1 } \\end{array} \\right] \\right.} \\\\ { \\vdots \\hfill \\mathrm { ~ s u r ~ o r ~ p u r p 1 e } } \\\\ { \\mathbf { \\bigoplus } _ { \\mathbf { 0 } } \\mathbf { E } } \\end{array} } \\end{array}  } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            209,
+            300,
+            460,
+            388
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { V } + \\mathbf { E }  \\mathbf { E }$ : This module performs a function on a soft entity set, parameterized by a word vector, to produce a new soft entity set. For example, the word not learns to take the complement of a set of entities. The entity attention representation of the resulting span is computed by using the indicated function that takes the $\\boldsymbol { v } _ { 1 } \\in \\mathbb { R } ^ { 2 }$ vector of the $\\mathbf { V }$ constituent as a parameter argument and the entity attention vector of the $\\mathbf { E }$ constituent as a function argument. ",
+        "bbox": [
+            509,
+            176,
+            825,
+            291
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/5ce6a0e45402c1bb7b4dbe583d64a0e14efc9b650262ab7a978f2824679378bf.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            544,
+            292,
+            795,
+            382
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { E V + E  E }$ : This module combines two soft entity sets into a third set, parameterized by the $v _ { 2 }$ word vector. This composition function is similar to a linear threshold unit and is capable of modeling various mathematical operations such as logical conjunctions, disjunctions, differences etc. for different values of $v _ { 2 }$ . For example, the word or learns to model set union. ",
+        "bbox": [
+            173,
+            392,
+            488,
+            481
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5caf5d5f441dbb43facb84f7fc5c77c148b5d01c4a7e37497b213663d15ee84f.jpg",
+        "text": "$$\n\\begin{array}{c} \\begin{array} { r l } & { \\overline { { \\mathrm { I } \\mathrm { \\tiny ~ T r u e } } } } \\\\ & { \\qquad \\boldsymbol { \\Bigg / } ^ { \\mathsf { T } } \\setminus \\underbrace { p _ { t r u e } } _ { \\mathsf { E } } = \\sigma ( v _ { 3 } ^ { 1 } \\Bigg [ \\sum _ { e _ { i } } \\sigma ( [ v _ { 3 } ^ { 3 } ] \\cdot [ \\begin{array} { l } { p _ { e _ { i } } ^ { R } } \\\\ { 1 } \\end{array} ] } \\end{array} ) ] + v _ { 3 } ^ { 2 } )  \\\\ & { \\qquad \\mathsf { V } } \\\\ & { \\textsf { s } \\underset { \\ @ \\boldsymbol { \\Psi } } { \\mathsf { a n y t h i n g ~ c y l } } \\mathrm { i n d r ~ i c a l } } \\\\ & { \\qquad \\boldsymbol { \\Theta } \\stackrel { \\qquad } { \\Pi } \\overleftrightarrow { \\boldsymbol { \\Psi } } \\big \\square } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            178,
+            483,
+            488,
+            554
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { R } + \\mathbf { E }  \\mathbf { E }$ : This module composes a set of relations (represented as a single soft adjacency matrix) and a soft entity set to produce an output soft entity set. The composition function uses the adjacency matrix representation of the $\\mathbf { R }$ -span and the soft entity set representation of the E-span. ",
+        "bbox": [
+            509,
+            392,
+            825,
+            469
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { V } + \\mathbf { E }  \\mathbf { T }$ : This module maps a soft entity set onto a soft boolean, parameterized by word vector $( v _ { 3 } )$ . The module counts whether a sufficient number of elements are in (or out) of the set. For example, the word any should test if a set is non-empty. ",
+        "bbox": [
+            173,
+            558,
+            488,
+            622
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f9a0b8c32f1c77b1e9370d77f274ccebbefd15564a6c587375f6b5ed67b63d0.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underbrace { \\overline { { \\left[ \\mathsf { F a l s e } \\right] } } } _ { \\displaystyle { \\int } } \\mathsf { T } _ { \\mathrm { \\normalfont ~ \\ \" ~ \\ \" ~ } } p _ { t r u e } = \\sigma \\left( v _ { 4 } ^ { 1 } \\left[ \\sum _ { e _ { i } } \\sigma \\left( \\begin{array} { c } { \\left[ v _ { 4 } ^ { 3 } \\right] } \\\\ { v _ { 4 } ^ { 4 } } \\\\ { \\left[ v _ { 5 } ^ { 5 } \\right] } \\end{array} \\cdot \\left[ \\begin{array} { c } { p _ { e _ { i } } ^ { L } } \\\\ { p _ { e _ { i } } ^ { R } } \\\\ { 1 } \\end{array} \\right] \\right) \\right] + v _ { 4 } ^ { 5 } \\right) } \\\\ & { \\vdots \\begin{array} { l } { \\mathsf { E V } } \\\\ { \\mathrm { e v e r y ~ c y ~ l ~ i n d e r ~ \\quad ~ b ~ l ~ u e } } \\\\ { \\bigoplus } \\end{array} } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            526,
+            481,
+            826,
+            551
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { E V + E }  \\mathbf { T }$ : This module combines two soft entity sets into a soft boolean, which is useful for modelling generalized quantifiers. For example, in is every cylinder blue, the module can use the inner sigmoid to test if an element $e _ { i }$ is in the set of cylinders $( p _ { e _ { i } } ^ { L } \\approx 1 )$ ) but not in the set of blue things $( p _ { e _ { i } } ^ { R } \\approx 0 )$ ), and then use the outer sigmoid to return a value close to 1 if the sum of elements matching this property is close to 0. ",
+        "bbox": [
+            509,
+            558,
+            825,
+            661
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f9995a8ec204d5e90269ab1dd98ecc963dedef62f8bc41c0ea0ebf4016c21f6a.jpg",
+        "text": "$$\n\\begin{array} { c } { { { \\underbrace { \\left[ \\mathsf { F a l s e } \\right] } } _ { \\displaystyle { \\mathsf { T } } } _ { \\displaystyle { \\mathrm { s } } _ { t r u e } } = \\sigma \\left( \\mathbf { v } _ { 2 } \\cdot \\left[ \\begin{array} { l } { p _ { t r u e } ^ { L } } \\\\ { p _ { t r u e } ^ { R } } \\\\ { 1 } \\end{array} \\right] \\right) } } \\\\ { { { \\mathsf { T } } } } \\\\ { { { \\mathsf { Z } } \\mathsf { b a l l s ~ r e d ~ a n d ~ } } } \\\\ { { { \\underbrace { \\left[ \\mathsf { F a l s e } \\right] \\ @ { \\mathsf { o D } } } } _ { \\displaystyle { \\left[ \\mathsf { T r u e } \\right] } } \\in { \\mathsf { S l u e } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            683,
+            480,
+            784
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/46ffd5f89b90e1ffa26a1bc73dbbdc87f2868559651ed96abd911bbd6be36748.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            537,
+            667,
+            803,
+            779
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { T V } + \\mathbf { T }  \\mathbf { T }$ : This module maps a pair of soft booleans into a soft boolean using the $v _ { 2 }$ word vector to parameterize the composition function. Similar to $\\mathbf { E V + E }  \\mathbf { E }$ , this module facilitates modeling a range of boolean set operations. Using the same functional form for different composition functions, allows our model to use the same ungrounded word vector $\\left( v _ { 2 } \\right)$ for compositions that are semantically analogous. ",
+        "bbox": [
+            173,
+            789,
+            488,
+            890
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { R } \\mathbf { V } + \\mathbf { R }  \\mathbf { R }$ : This module composes a pair of soft set of relations to a produce an output soft set of relations. For example, the relations left and above are composed by the word $o r$ to produce a set of relations such that entities $e _ { i }$ and $e _ { j }$ are related if either of the two relations exists between them. The functional form for this composition is similar to $\\mathbf { E V + E }  \\mathbf { E }$ and $\\mathbf { T V } + \\mathbf { T }  \\mathbf { T }$ modules. ",
+        "bbox": [
+            509,
+            787,
+            825,
+            890
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Figure 2: Composition Modules that compose two constituent span representations into the representation for the combined larger span, using the indicated equations. ",
+        "bbox": [
+            173,
+            901,
+            825,
+            930
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Composition with $\\phi$ -typed nodes: Phrases with type $\\phi$ are treated as being semantically transparent identity functions. Phrases of any other type can combined with these with no change to their type or representation. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Composition modules resulting in partially grounded denotations: We define several simple modules that combine fully grounded phrases with ungrounded phrases, by deterministically taking the union of the representations, giving phrases with partially grounded representations $( \\ S \\ 3 . 1 )$ . These modules are useful for when words act as binary functions; here they combine with their first argument. For example, in Figure 1, or and not cylindrical combine to make a phrase containing both the vectors for or and the entity set for not cylindrical. ",
+        "bbox": [
+            173,
+            162,
+            825,
+            247
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 PARSING MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            268,
+            348,
+            285
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here, we describe how our model classifies question tokens into different semantic type spans and compute their representations $( \\ S 4 . 1 )$ , recursively uses the composition modules defined above to parse the question appropriately into a soft latent tree that provides the answer $( \\ S 4 . 2 )$ . The model is trained end-to-end using only question-answer supervision $( \\ S 4 . 3 )$ . ",
+        "bbox": [
+            174,
+            301,
+            825,
+            358
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 LEXICAL REPRESENTATION ASSIGNMENT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            376,
+            506,
+            390
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Each token in the question sentence is assigned a distribution over the semantic types, and given a grounding for each type. Tokens can only be assigned the E, R, V, and $\\phi$ semantic types. For example, the token cylindrical in the question in Fig. 1 is assigned a distribution over the 4 semantic types (one shown) and for the $\\mathbf { E }$ type, the representation computed is the set of cylindrical entities. ",
+        "bbox": [
+            174,
+            402,
+            825,
+            459
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Semantic Type Distribution for Tokens: To compute the semantic type distribution, our model represents each word $w$ in the word vocabulary $\\nu$ , and each semantic type $t$ using an embedding vector; $v _ { w } , v _ { t } \\in \\mathbb { R } ^ { d }$ . The semantic type distribution is assigned with a softmax: ",
+        "bbox": [
+            173,
+            476,
+            825,
+            518
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d4344ce33e7b05e3504c0db8283184e10b34bcb1699b7f90f1b41a247e52c6c.jpg",
+        "text": "$$\np ( t | w _ { i } ) \\propto \\exp ( v _ { t } \\cdot v _ { w _ { i } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            418,
+            527,
+            578,
+            545
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Grounding for Tokens: For each of the four semantic type assignments for question tokens, we need to compute/assign their corresponding representations. ",
+        "bbox": [
+            174,
+            560,
+            826,
+            589
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "1. E-Type Representation: Each entity $e \\in { \\mathcal { E } }$ , is represented using an embedding vector $v _ { e } \\in \\mathbb { R } ^ { d }$ based on the concatenation of vectors for its properties. For each token $w$ , we use its word vector to find the probability of each entity being part of the $\\mathbf { E }$ -Type grounding: ",
+        "bbox": [
+            215,
+            603,
+            825,
+            646
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ad01d43998c7067de4fa3d948b57a97cb1366dc48eb208f53cebf04492d3647.jpg",
+        "text": "$$\np _ { e _ { i } } ^ { w } = \\sigma ( v _ { e _ { i } } \\cdot v _ { w } ) ~ \\forall e _ { i } \\in \\mathcal { E }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            436,
+            654,
+            620,
+            672
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For example, in Fig. 1, the word red will be grounded as all the red entities. ",
+        "bbox": [
+            232,
+            679,
+            727,
+            694
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2. R-Type Representation: Each relation $r \\in \\mathcal { R }$ , is represented using an embedding vector $v _ { r } \\in \\mathbb { R } ^ { d }$ . For each token $w _ { i }$ in the question, we first compute a distribution over relations it could refer to, and then use this distribution to compute the expected adjacency matrix that forms the $\\mathbf { R }$ -type representation for this token. ",
+        "bbox": [
+            212,
+            699,
+            825,
+            756
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a595a11b1aa365fb4e1ba5662c0488c297dee3e4cc3f30c0728a860b38d24e99.jpg",
+        "text": "$$\np ( r | w _ { i } ) \\propto \\exp ( v _ { r } \\cdot v _ { w _ { i } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            446,
+            763,
+            611,
+            781
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffa7844fb113824012ee46eca0585bc9c262b09ea126cf4a8c0c0492080ae226.jpg",
+        "text": "$$\nA ^ { w _ { i } } = \\sum _ { r \\in \\mathcal { R } } p ( r | w _ { i } ) \\cdot A _ { r }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            446,
+            801,
+            611,
+            835
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For example, the word left in Fig. 1 is grounded as the subset of edges with the label ‘left’. ",
+        "bbox": [
+            228,
+            840,
+            823,
+            856
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3. V-Type Representation: For each word $w \\in \\mathcal { V }$ , we learn four vectors $v _ { 1 } \\in \\mathbb { R } ^ { 2 } , v _ { 2 } \\in$ $\\mathbb { R } ^ { 3 } , \\tilde { v _ { 3 } } \\in \\mathbb { R } ^ { \\bar { 4 } } , v _ { 4 } \\in \\mathbb { R } ^ { 5 }$ , and use these as the representation for words with the $\\mathbf { V }$ -Type. ",
+        "bbox": [
+            214,
+            861,
+            821,
+            890
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4. $\\phi$ -Type Representation: This type is used for semantically vacuous words, which do not require a representation. ",
+        "bbox": [
+            210,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 PARSING QUESTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            362,
+            118
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To learn the correct structure for applying composition modules, we use a simple parsing model. We build a parse-chart over the question encompassing all possible trees by applying all composition modules, similar to a standard CRF-based PCFG parser using the CKY algorithm. Each node in the parse-chart, for each span $w _ { i \\ldots j }$ of the question, is represented as a distribution over different semantic types with their corresponding representations. This distribution is computed by weighing the different ways of composing the span’s constituents. ",
+        "bbox": [
+            173,
+            128,
+            825,
+            214
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Phrase Semantic Type Potential: Each node in the parse-chart is associated with a potential value $\\psi ( i , j , t )$ , that is the score assigned by the model to the t semantic type for the $w _ { i \\ldots j }$ span. This is computed from all possible ways to form the span $w _ { i \\ldots j }$ with type t. For a particular composition of span $w _ { i \\dots k }$ of type $\\mathbf { t _ { 1 } }$ and $w _ { k + 1 \\ldots j }$ of type $\\mathbf { t _ { 2 } }$ , using the $\\mathbf { t _ { 1 } } + \\mathbf { t _ { 2 } }  \\mathbf { t }$ module, the score is: ",
+        "bbox": [
+            174,
+            227,
+            825,
+            286
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/06480f453f0c85596444e23614f3c1963afdc7e0fcb6c3098aa49fd17afebbcc.jpg",
+        "text": "$$\n\\psi ( i , j , k , t _ { 1 } + t _ { 2 }  t ) = [ \\psi ( i , k , t _ { 1 } ) \\cdot \\psi ( k + 1 , j , t _ { 2 } ) \\cdot \\exp ( \\sum _ { x } f _ { x } ^ { ( t _ { 1 } + t _ { 2 }  t ) } ( i , j , k | q ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            290,
+            782,
+            333
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where, $f _ { x } ^ { ( t _ { 1 } + t _ { 2 }  t ) } ( i , j , k | q )$ are six feature functions; a trainable weight for each word per module in the vocabulary, that correspond to: $f _ { 1 }$ : word that appears before the start of the span $w _ { i - 1 }$ ; $f _ { 2 }$ : first word in the span $w _ { i }$ ; $f _ { 3 }$ : last word in the left constituent $w _ { k }$ ; $f _ { 4 }$ : first word in the right constituent $w _ { k + 1 }$ ; $f _ { 5 }$ : last word in the right constituent $w _ { j }$ ; and $f _ { 6 }$ : word that appears after the span $w _ { j + 1 }$ . ",
+        "bbox": [
+            173,
+            338,
+            826,
+            398
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The token semantic type potential of $w _ { i }$ , $\\psi ( i , i , i , t _ { 1 } + t _ { 2 }  t )$ , is the same as $p ( t | w _ { i } )$ (Eq. 1). ",
+        "bbox": [
+            173,
+            404,
+            792,
+            420
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The final t-type potential of $w _ { i \\ldots j }$ is computed by summing over scores from all possible compositions: ",
+        "bbox": [
+            171,
+            425,
+            823,
+            441
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d45ba39fc09d3dac74b1f8df488c164797b562e917f23b4dff0b37bec56b139f.jpg",
+        "text": "$$\n\\psi ( i , j , t ) = \\sum _ { k = i } ^ { j - 1 } \\sum _ { \\stackrel { \\scriptstyle ( t _ { 1 } + t _ { 2 } \\to t ) } { \\scriptstyle \\in \\mathrm { M o d u l e s } } } \\psi ( i , j , k , t _ { 1 } + t _ { 2 } \\to t )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            339,
+            448,
+            656,
+            503
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Combining Phrase Representations: To compute the span $w _ { i \\ldots j }$ ’s denotation with type t, $[ [ w _ { i . . j } ] ] _ { K G } ^ { t }$ , we compute an expected output representation from all possible compositions. ",
+        "bbox": [
+            171,
+            516,
+            825,
+            546
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5819b16b6ee6eadacd08dd1a9e04db41d7f4dfa0a3444fd3506b0ea9407f7942.jpg",
+        "text": "$$\n[ w _ { i \\cdot . . j } ] _ { K G } ^ { t } = \\frac { 1 } { \\psi ( i , j , t ) } \\sum _ { \\stackrel { k = i } { \\epsilon \\mathrm { \\bf ~ ( } t _ { 1 } + t _ { 2 }  t \\mathrm { \\bf ~ ) } } } ^ { j - 1 } \\psi ( i , j , k , t _ { 1 } + t _ { 2 }  t ) \\ast [ [ w _ { i \\cdot . . k . . j } ] ] _ { K G } ^ { t _ { 1 } + t _ { 2 }  t }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            551,
+            756,
+            606
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where resulti $[ [ w _ { i . . j } ] ] _ { K G } ^ { t }$ , is the t-type repre composition of ationwith pan usi $w _ { i \\ldots j }$ , e $\\mathbb { [ } w _ { i \\dots k \\dots j } ] _ { K G } ^ { t _ { 1 } + t _ { 2 }  t }$ is the representationcomposition module. $w _ { i , k }$ $w _ { k + 1 \\ldots j }$ $\\mathbf { t _ { 1 } } ~ + ~ \\mathbf { t _ { 2 } } ~  ~ \\mathbf { t }$ ",
+        "bbox": [
+            174,
+            612,
+            826,
+            643
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Answer Grounding: By recursively computing the phrase semantic-type potentials and representations, we can infer the semantic type distribution of the complete question sentence (Eq. 8) and the resulting grounding for different semantic type $t$ , $[ [ w _ { 1 . . | q | } ] ] _ { K G } ^ { t }$ . ",
+        "bbox": [
+            174,
+            656,
+            825,
+            700
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f10a70136f349c15c4d9380be47c7703ce75d3541884fee5f812dc40253ffe33.jpg",
+        "text": "$$\np ( t | q ) \\propto \\psi ( 1 , | q | , t )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            705,
+            566,
+            723
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The answer-type (boolean or subset of entities) for the question is computed using: ",
+        "bbox": [
+            178,
+            728,
+            715,
+            743
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5427cd8b43b1721cbb2b65a016448942e5a4e4cd6386d797951b45cabcc6d660.jpg",
+        "text": "$$\nt ^ { * } = \\underset { t \\in \\mathbf { T } , \\mathbf { E } } { \\mathrm { a r g m a x } } \\ p ( t | q )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            747,
+            568,
+            775
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The corresponding grounding is $[ [ w _ { 1 . . | q | } ] ] _ { K G } ^ { t ^ { * } }$ , which answers the question. ",
+        "bbox": [
+            174,
+            781,
+            656,
+            799
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 TRAINING OBJECTIVE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            814,
+            370,
+            829
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Given a dataset D of (question, answer, knowledge-graph) tuples, {qi, ai, KGi}i=|D|i=1 , we train our model to maximize the log-likelihood of the correct answers. Answers are either booleans, or specific subsets of entities from the KG. We denote the semantic type of the answer as $a _ { t }$ . If the answer is boolean, $a \\in \\{ 0 , 1 \\}$ , otherwise is a subset of entities from the KG, i.e. $a = \\{ e _ { j } \\}$ . The model’s answer to a question is found by taking its representation of the complete question, containing a distribution over types and the representation for each type. We maximize the following objective: ",
+        "bbox": [
+            173,
+            838,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e046895b4070573cef01bad5b055669e8afb8a045a600b1a1ee52d48774690fb.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { \\mathbf { \\xi } } = \\displaystyle \\sum _ { i } \\log p ( \\boldsymbol { a } ^ { i } | \\mathbf { q } ^ { i } , \\mathbf { K G } ^ { i } ) \\quad \\quad \\quad \\quad ( 1 0 ) } \\\\ & { = \\displaystyle \\sum _ { i } \\left[ \\underbrace { \\left( 1 _ { a _ { i } ^ { i } = \\mathbf { r } } \\left[ \\log ( p _ { t r u e } ) ^ { a ^ { i } } ( 1 - p _ { t r u e } ) ^ { ( 1 - a ^ { i } ) } \\right] \\right) } _ { e _ { j } ^ { i } \\in a ^ { i } } + \\left( \\frac { 1 _ { a _ { i } ^ { i } = \\mathbf { E } } } { | \\mathcal { E } ^ { i } | } \\left[ \\log \\prod _ { e _ { j } ^ { i } \\in a ^ { i } } p _ { e _ { j } ^ { i } } \\prod _ { e _ { j } ^ { i } } ( 1 - p _ { e _ { j } ^ { i } } ) \\right] \\right) \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            118,
+            828,
+            204
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also add $L _ { 2 }$ -regularization for the scalar parsing features introduced in $\\ S 4 . 2$ . ",
+        "bbox": [
+            173,
+            246,
+            699,
+            261
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            281,
+            282,
+            297
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We generate a dataset of question-answers based on the CLEVR dataset (Johnson et al., 2017), which contains knowledge graphs containing attribute information of objects and relations between them. ",
+        "bbox": [
+            174,
+            313,
+            823,
+            340
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We generate a new set of questions for this data, as existing questions contain some biases that can be exploited by models (Johnson et al. (2017) found that many spatial relation questions can be answered only using absolute spatial information and many long questions can be answered correctly without performing all steps of reasoning), and many questions are over 40 words long, which is intractable given that the size of our computation graph is cubic in the question length. Future work should explore scaling our approach to longer questions. We generate 75K questions for training and 37.5K for validation. ",
+        "bbox": [
+            174,
+            347,
+            825,
+            445
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our question set tests various challenging semantic operators. These include conjunctions (e.g. Is anything red or is anything large?), negations (e.g. What is not spherical?), counts (e.g. Are five spheres green?), quantifiers (e.g. Is every red thing cylindrical?), and relations (e.g. What is left of and above a cube?). We employ some simple tests to remove trivial biases from the dataset. ",
+        "bbox": [
+            174,
+            452,
+            825,
+            508
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We create two test sets: one drawn from the same distribution as the training data (37.5K), and another containing longer questions than the training data (22.5K). ",
+        "bbox": [
+            171,
+            515,
+            823,
+            544
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our COMPLEX QUESTIONS test set contains the same words and constructions, but chained into longer questions. For example, it contains questions such as What is a cube that is right of a metallic thing that is beneath a blue thing? and Are two red things that are above a sphere metallic?. These questions require more multi-step reasoning to solve. ",
+        "bbox": [
+            174,
+            550,
+            825,
+            606
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            627,
+            326,
+            642
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section we describe our experimentation setting, the baseline models we compare to, and the various experiments demonstrating the ability of our model to answer compositional questions referring to KG and its ability to generalize to unseen longer questions and new attribute combinations. ",
+        "bbox": [
+            174,
+            659,
+            825,
+            700
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.1 EXPERIMENTATION SETTING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            718,
+            415,
+            733
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Here we describe the training details of our model and the baseline models. ",
+        "bbox": [
+            176,
+            744,
+            666,
+            760
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Representing Entities: Each entity in the CLEVR dataset consists of 4 attributes. For each attribute-value, we learn an embedding vector and concatenate the 4-embedding vectors to form the representation for the entity. ",
+        "bbox": [
+            174,
+            773,
+            825,
+            816
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Training Details: Training the model is complicated by the large number of poor local minima, as the model needs to learn both good syntactic structures and the complex semantics of neural modules. ",
+        "bbox": [
+            174,
+            832,
+            821,
+            861
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To simplify training, we use Curriculum Learning (Bengio et al., 2009) to pre-train the model on an easier subset of questions. We use a 2-step schedule where we first train our model on simple attribute match (What is a red sphere?), attribute existence (Is anything blue?) and boolean composition $I s$ anything green and is anything purple?) questions and in the second step on all questions jointly. ",
+        "bbox": [
+            176,
+            867,
+            825,
+            924
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/c7bd77745ea37f93bb26e2e2137ed7d47af414a6053b4581ff3508b00ca391a9.jpg",
+        "table_caption": [
+            "Table 1: Results for Short Questions: Performance of our model compared to baseline models on the Short Questions test set. The LSTM (NO KG) has accuracy close to chance, showing that the questions lack trivial biases.Our model almost perfectly solves all questions showing its ability to learn challenging semantic operators, and parse questions only using weak end-to-end supervision. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Boolean Questions</td><td>Entity Set Questions</td><td>Relation Questions</td><td>Overall</td></tr><tr><td>LSTM (No KG)</td><td>50.7</td><td>14.4</td><td>17.5</td><td>27.2</td></tr><tr><td>LSTM (NO RELATION)</td><td>88.5</td><td>99.9</td><td>15.7</td><td>84.9</td></tr><tr><td>RELATION NETWORK</td><td>85.6</td><td>89.7</td><td>97.6</td><td>89.4</td></tr><tr><td>Our Model</td><td>99.9</td><td>100</td><td>100.0</td><td>99.9</td></tr></table>",
+        "bbox": [
+            173,
+            101,
+            828,
+            184
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We tune the hyper-parameters using validation accuracy. We train using SGD with learning rate of 0.5 and mini-batch size of 4, regularization constant of 0.3. When assigning the semantic type distribution to the words at the leaves, we add a small positive bias of $+ 1$ for $\\phi$ -type and a small negative bias of $- 1$ for the $\\mathbf { E }$ -type score before the softmax. Our trainable parameters are: question word embeddings (64-dimensional), relation embeddings (64-dimensional), entity attribute-value embeddings (16-dimensional), four vectors per word for $\\mathbf { V }$ -type representations, six scalar feature scores per module per word for the parsing model, and the global parameter vector for the $\\mathbf { E + E {  } E }$ module. ",
+        "bbox": [
+            173,
+            279,
+            825,
+            390
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Baseline Models: We use three baseline models for comparison. A simple LSTM (NO KG) model that encodes the question using an LSTM network and answers questions without access to the KG. Another LSTM based model, LSTM (NO RELATION), that has access only to the entities of the KG but not the relationship information between them. Finally, we train a RELATION NETWORK (Santoro et al., 2017) augmented model, which achieved state-of-the-art performance on the CLEVR dataset using image state descriptions. Details about the baseline models are given in the Appendix section. ",
+        "bbox": [
+            174,
+            407,
+            825,
+            491
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.2 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            510,
+            315,
+            523
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Short Questions Performance: In Table 1, we see that our model is able to perfectly answer all the questions in the test set. This demonstrates our model can learn challenging semantic operators using composition modules, as well as learn to parse the questions from only using weak endto-end supervision. The RELATION NETWORK also achieves good performance, particularly on questions involving relations, but is weaker than our model on some question types. The LSTM (NO RELATION) model also achieves good performance on questions not involving relations, which are out of scope for the model. ",
+        "bbox": [
+            174,
+            537,
+            825,
+            633
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Complex Question Performance: Table 2 shows results on complex questions, which are constructed by combining components of shorter questions. We use the same models as in Table 1, which were trained and developed only on shorter questions. Answering longer questions requires complex multi-hop reasoning, and the ability to generalize from the language seen in its training data to new types of questions. Results show that all baselines achieve close to random performance on this task, despite high accuracy for shorter questions. This shows the challenges in generalizing RNN encoders beyond their training data. In contrast, the strong inductive bias from our model structure allows the model to generalize to complex questions much more easily than RNN encoders. ",
+        "bbox": [
+            174,
+            651,
+            825,
+            763
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Generalization to Unseen Attribute Combination: We also measure how well models generalize to unseen attribute combinations in knowledge graphs (using the COGENT subset of CLEVR). For example, the test set contains ‘blue spheres’ that are not found in the training set. None of the models showed a significant reduction in performance in this setting. ",
+        "bbox": [
+            176,
+            780,
+            825,
+            837
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Error Analysis: Analyzing the errors of our model, we find that most errors are due to incorrect assignments of structure, rather than semantic errors from the modules. For example, in the question Are four red spheres beneath a metallic thing small?, our model produces a parse where it composes metallic thing small into a single node instead of composing red spheres beneath a metallic thing into a single node. Future work should use more sophisticated parsing models. ",
+        "bbox": [
+            174,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b7219d0fdd0d0f550ec5594667621bf2abd893b5bb8739c6cf4de07c1c1ede3e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Non-Relation Questions</td><td>Relation Questions</td><td>Overall</td></tr><tr><td>LSTM (NO KG)</td><td>46.0</td><td>39.6</td><td>41.4</td></tr><tr><td>LSTM(NO RELATION)</td><td>62.2</td><td>49.2</td><td>52.2</td></tr><tr><td>RELATION NETWORK</td><td>51.1</td><td>38.9</td><td>41.5</td></tr><tr><td>Our Model</td><td>81.8</td><td>85.4</td><td>84.6</td></tr></table>",
+        "bbox": [
+            223,
+            101,
+            774,
+            184
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Results for Complex Questions: All baseline models fail to generalize to questions requiring longer chains of reasoning than seen during training. Our model substantially outperforms the baselines, showing its ability to perform complex multi-hop reasoning, and generalize from its training data. Analysis suggests that most errors from our model are due to assigning incorrect structures, not mistakes by the composition modules. ",
+        "bbox": [
+            173,
+            193,
+            825,
+            263
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            290,
+            344,
+            306
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Many approaches have been proposed to perform question-answering against structured knowledge sources. Semantic parsing models have attempted to learn structures over pre-defined discrete operators, to produce logical forms that can be executed to answer the question. Early work trained using gold-standard logical forms (Zettlemoyer & Collins, 2005; Kwiatkowski et al., 2010), whereas later efforts have only used answers to questions (Liang et al., 2011; Krishnamurthy & Kollar, 2013; Pasupat & Liang, 2015). A key difference is that our model must learn semantic operators from data, which may be necessary to model the fuzzy interpretations of some function words like many or few. ",
+        "bbox": [
+            174,
+            321,
+            825,
+            420
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Another similar line of work is neural program induction models, such as Neural Programmer (Neelakantan et al., 2016) and Neural Symbolic Machine (Liang et al., 2017). These models learn to produce programs composed of predefined operators using weak supervision to answer questions against semi-structured tables. ",
+        "bbox": [
+            174,
+            426,
+            825,
+            481
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Neural module networks have recently been proposed for learning semantic operators (Andreas et al., 2016b) for question answering. This model assumes that the structure of the semantic parse is given, and must only learn a set of operators. Dynamic Neural Module Networks (D-NMN) extend this approach by selecting from a small set of candidate module structures (Andreas et al., 2016a). In contrast, our approach learns a model over all possible structures for interpreting a question. ",
+        "bbox": [
+            174,
+            489,
+            825,
+            559
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our work is most similar to the most recently proposed N2NMN (Hu et al., 2017) model, an end-toend version of D-NMN. This model learns both semantic operators and the layout in which to compose them. However, optimizing the layouts requires reinforcement learning, which is challenging due to the high variance of policy gradients, whereas our approach is end-to-end differentiable. ",
+        "bbox": [
+            174,
+            565,
+            825,
+            622
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "8 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            642,
+            318,
+            659
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have introduced a model for answering questions requiring compositional reasoning that combines ideas from compositional semantics with end-to-end learning of composition operators and structure. We demonstrated that the model is able to learn a number of complex composition operators from end task supervision, and have shown that the linguistically motivated inductive bias imposed by the structure of the model allows it to generalize well beyond its training data. Future work should explore scaling the model to other question answering tasks. ",
+        "bbox": [
+            174,
+            674,
+            825,
+            758
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            780,
+            287,
+            795
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Learning to compose neural networks for question answering. In HLT-NAACL, 2016a. ",
+        "bbox": [
+            173,
+            803,
+            823,
+            833
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Neural module networks. In CVPR, pp. 39–48, 2016b. ",
+        "bbox": [
+            171,
+            842,
+            825,
+            871
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Yoshua Bengio, Jer´ ome Louradour, Ronan Collobert, and Jason Weston. Curriculum learning. In ˆ Proceedings of the 26th annual international conference on machine learning, pp. 41–48. ACM, 2009. ",
+        "bbox": [
+            176,
+            881,
+            825,
+            922
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Ronghang Hu, Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Kate Saenko. Learning to reason: End-to-end module networks for visual question answering. CoRR, abs/1704.05526, 2017. ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Justin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C. Lawrence Zitnick, and Ross Girshick. Clevr: A diagnostic dataset for compositional language and elementary visual reasoning. In CVPR, July 2017. ",
+        "bbox": [
+            176,
+            140,
+            823,
+            184
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jayant Krishnamurthy and Thomas Kollar. Jointly learning to parse and perceive: Connecting natural language to the physical world. TACL, 1:193–206, 2013. ",
+        "bbox": [
+            171,
+            193,
+            825,
+            222
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Tom Kwiatkowski, Luke S. Zettlemoyer, Sharon Goldwater, and Mark Steedman. Inducing probabilistic ccg grammars from logical form with higher-order unification. In EMNLP, 2010. ",
+        "bbox": [
+            174,
+            231,
+            825,
+            260
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Chen Liang, Jonathan Berant, Quoc Le, Kenneth D. Forbus, and Ni Lao. Neural symbolic machines: Learning semantic parsers on freebase with weak supervision. In ACL, 2017. ",
+        "bbox": [
+            174,
+            267,
+            823,
+            297
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Percy Liang, Michael I Jordan, and Dan Klein. Learning dependency-based compositional semantics. In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Human Language Technologies-Volume 1, pp. 590–599. Association for Computational Linguistics, 2011. ",
+        "bbox": [
+            174,
+            305,
+            828,
+            362
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jean Maillard, Stephen Clark, and Dani Yogatama. Jointly learning sentence embeddings and syntax with unsupervised tree-lstms. CoRR, abs/1705.09189, 2017. URL http://arxiv.org/abs/ 1705.09189. ",
+        "bbox": [
+            174,
+            371,
+            826,
+            414
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Richard Montague. The proper treatment of quantification in ordinary English. In K. J. J. Hintikka, J. Moravcsic, and P. Suppes (eds.), Approaches to Natural Language, pp. 221–242. Reidel, Dordrecht, 1973. ",
+        "bbox": [
+            174,
+            422,
+            826,
+            465
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Arvind Neelakantan, Quoc V. Le, Mart´ın Abadi, Andrew McCallum, and Dario Amodei. Learning a natural language interface with neural programmer. CoRR, abs/1611.08945, 2016. ",
+        "bbox": [
+            171,
+            473,
+            821,
+            503
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Panupong Pasupat and Percy Liang. Compositional semantic parsing on semi-structured tables. In ACL, 2015. ",
+        "bbox": [
+            171,
+            512,
+            823,
+            541
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ethan Perez, Harm de Vries, Florian Strub, Vincent Dumoulin, and Aaron C. Courville. Learning visual reasoning without strong priors. CoRR, abs/1707.03017, 2017. URL http://arxiv. org/abs/1707.03017. ",
+        "bbox": [
+            171,
+            549,
+            825,
+            592
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Adam Santoro, David Raposo, David G. T. Barrett, Mateusz Malinowski, Razvan Pascanu, Peter Battaglia, and Timothy P. Lillicrap. A simple neural network module for relational reasoning. CoRR, abs/1706.01427, 2017. URL http://arxiv.org/abs/1706.01427. ",
+        "bbox": [
+            176,
+            602,
+            823,
+            645
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Luke S Zettlemoyer and Michael Collins. Learning to map sentences to logical form: Structured classification with probabilistic categorial grammars. UAI, 2005. ",
+        "bbox": [
+            169,
+            654,
+            823,
+            683
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            264,
+            118
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "BASELINE MODELS",
+        "bbox": [
+            176,
+            132,
+            312,
+            147
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "LSTM (NO KG) ",
+        "bbox": [
+            174,
+            157,
+            290,
+            172
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We use a LSTM network to encode the question as a vector $q$ . We also define three other parameter vectors, $t , e$ and $b$ that are used to predict the answer-type $P ( \\bar { a } = \\mathbf { T } ) = \\sigma ( \\boldsymbol { q } \\cdot \\boldsymbol { t } )$ , entity attention value $p _ { e _ { i } } = \\sigma ( q \\cdot e )$ , and the probability of the answer being True $p _ { t r u e } = \\sigma ( q \\cdot b )$ . ",
+        "bbox": [
+            174,
+            183,
+            825,
+            226
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "LSTM (NO RELATION) ",
+        "bbox": [
+            174,
+            238,
+            338,
+            253
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Similar to LSTM (NO RELATION), the question is encoded using a LSTM network as vector $q$ . Similar to our model, we learn entity attribute-value embeddings and represent each entity as the concatenation of the 4 attribute-value embeddings, $\\boldsymbol { v } _ { e _ { i } }$ . Similar to LSTM (NO RELATION), we also define the $t$ parameter vector to predict the answer-type. The entity-attention values are predicted as $p _ { e _ { i } } = \\sigma ( v _ { e _ { i } } \\cdot q )$ . To predict the probability of the boolean-type answer being true, we first add the entity representations to form $b { \\bar { \\mathbf { \\theta } } } = \\sum _ { e _ { i } } v _ { e _ { i } }$ , then make the prediction as $p _ { t r u e } = \\sigma ( q \\cdot b )$ . ",
+        "bbox": [
+            173,
+            262,
+            826,
+            356
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "RELATION NETWORK AUGMENTED MODEL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            369,
+            477,
+            385
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The original formulation of the relation network module is as follows: ",
+        "bbox": [
+            174,
+            395,
+            632,
+            410
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e29359df289515509c960c49323684264789fc0af5228b50a42bcc3f7edf96d0.jpg",
+        "text": "$$\nR N ( q , K G ) = f _ { \\phi } \\left( \\sum _ { i , j } g _ { \\theta } ( e _ { i } , e _ { j } , q ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            410,
+            627,
+            454
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $e _ { i } , e _ { j }$ are the representations of the entities and $q$ is the question representation from an LSTM network. The output of the Relation Network module is a scalar score value for the elements in the answer vocabulary. Since our dataset contains entity-set valued answers, we modified the module in the following manner. ",
+        "bbox": [
+            174,
+            454,
+            825,
+            511
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We concatenate the object pair representations with the representations of the pair of directed relationships between them1. We then use the Relation Network module to produce an output representation for each entity in the KB, in the following manner: ",
+        "bbox": [
+            174,
+            517,
+            821,
+            559
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/875ebea8f6525d96cc9a5608f311598009901c166335e9c6519277b5e5ede6cc.jpg",
+        "text": "$$\nR N _ { e _ { i } } = f _ { \\phi } \\Bigg ( \\sum _ { j } g _ { \\theta } ( e _ { i } , e _ { j } , r _ { i j } ^ { 1 } , r _ { i j } ^ { 2 } , q ) \\Bigg )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            560,
+            630,
+            604
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Similar to the LSTM baselines, we define a parameter vector $t$ to predict the answer-type as: ",
+        "bbox": [
+            171,
+            606,
+            781,
+            619
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/405092d619cb68fff4a57c43694cd45362d16cc707073ca361bc611d36dc7c03.jpg",
+        "text": "$$\nP ( a = \\mathbf { T } ) = \\sigma ( q \\cdot t )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            621,
+            568,
+            638
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/04d8e870220d41d7226e7782596058b0e4e4142a1db9e8689360fb531fcbd498.jpg",
+        "text": "$$\nP ( a = \\mathbf { E } ) = 1 - P ( a = \\mathbf { T } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            651,
+            593,
+            669
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To predict the probability of the boolean type answer being true, we define a parameter vector $b$ and predict as following: ",
+        "bbox": [
+            174,
+            675,
+            826,
+            703
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3bac60f81ed58c215dea31ce46eb7e0c34b2f5e208b0077342bc31379f5050db.jpg",
+        "text": "$$\np _ { t r u e } = \\sigma \\bigg ( b \\cdot \\sum _ { e _ { i } } R N _ { e _ { i } } \\bigg )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            410,
+            700,
+            586,
+            741
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "To predict the entity-attention values, we use a separate attribute-embedding matrix to first generate the output representation for each entity, $e _ { i } ^ { o u t }$ , then predict the output attention values as follows: ",
+        "bbox": [
+            173,
+            747,
+            825,
+            776
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d6c91783b5e0ee2a3f8d806cd86e97a06999a3fea68d4df0d3f54ed793e10de8.jpg",
+        "text": "$$\np _ { e _ { i } } = \\sigma \\bigg ( R N _ { e _ { i } } \\cdot e _ { i } ^ { o u t } \\bigg )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            777,
+            576,
+            813
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We tried other architectures as well, but this modification provided the best performance on the validation set. We also tuned the hyper-parameters and found the setting from Santoro et al. (2017) to work the best based on validation accuracy. We used a different 2-step curriculum to train the RELATION NETWORK module, in which we replace the Boolean questions with the relation questions in the first-schedule and jointly train on all questions in the subsequent schedule. ",
+        "bbox": [
+            173,
+            819,
+            825,
+            890
+        ],
+        "page_idx": 10
+    }
+]