Original: C HOICE N ET : R OBUST L EARNING BY R EVEALING O UTPUT C ORRELATIONS
Revised: C HOICE N ET : R OBUST L EARNING BY R EVEALING O UTPUT C ORRELATIONS

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: In this paper, we focus on the supervised learning problem with corrupt training data.
Revised: In this paper, we focus on the supervised learning problem with corrupt training data.

Original: We assume that the training dataset is generated from a mixture of a target distribution and other unknown distributions.
Revised: We assume that the training dataset is generated from a mixture of a target distribution and other unknown distributions.

Original: We estimate the quality of each data by revealing the correlation between the generated distribution and the target distribution.
Revised: We estimate the quality of each data by revealing the correlation between the generated distribution and the target distribution.

Original: To this end, we present a novel framework referred to here as ChoiceNet that can robustly infer the target distribution in the presence of inconsistent data.
Revised: To this end, we present a novel framework referred to here as ChoiceNet that can robustly infer the target distribution in the presence of inconsistent data.

Original: We demonstrate that the proposed framework is applicable to both classiﬁcation and regression tasks.
Revised: We demonstrate that the proposed framework is applicable to both classiﬁcation and regression tasks.

Original: Particularly, ChoiceNet is evaluated in comprehensive experiments, where we show that it constantly outperforms existing baseline methods in the handling of noisy data in synthetic regression tasks as well as behavior cloning problems.
Revised: Particularly, ChoiceNet is evaluated in comprehensive experiments, where we show that it constantly outperforms existing baseline methods in the handling of noisy data in synthetic regression tasks as well as behavior cloning problems.

Original: In the classiﬁcation tasks, we apply the proposed method to the MNIST and CIFAR-10 datasets and it shows superior performances in terms of robustness to different types of noisy labels.
Revised: In the classiﬁcation tasks, we apply the proposed method to the MNIST and CIFAR-10 datasets and it shows superior performances in terms of robustness to different types of noisy labels.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Training a deep neural network requires immense amounts of training data which are often collected using crowdsourcing methods, such as Amazon’s Mechanical Turk (AMT).
Revised: Training a deep neural network requires immense amounts of training data which are often collected using crowdsourcing methods, such as Amazon’s Mechanical Turk (AMT).

Original: However, in practice, the crowd-sourced labels are often noisy (Bi et al., 2014).
Revised: However, in practice, the crowd-sourced labels are often noisy (Bi et al., 2014).

Original: Furthermore, deep neural networks are vulnerable to over-ﬁtting given the noisy training data in that they are capable of memorizing the entire dataset even with inconsistent labels, leading to a poor generalization performance (Zhang et al., 2016).
Revised: Furthermore, deep neural networks are vulnerable to over-ﬁtting given the noisy training data in that they are capable of memorizing the entire dataset even with inconsistent labels, leading to a poor generalization performance (Zhang et al., 2016).

Original: Assuming that a training dataset is generated from a mixture of a target distribution and other distributions, we address this problem through the principled idea of revealing the correlation between the target distribution and the other distributions.
Revised: Assuming that a training dataset is generated from a mixture of a target distribution and other distributions, we address this problem through the principled idea of revealing the correlation between the target distribution and the other distributions.

Original: We present a framework for robust learning which is applicable to arbitrary neural network architectures such as convolutional neural networks (He et al., 2016a) or recurrent neural networks (Chung et al., 2014).
Revised: We present a framework for robust learning which is applicable to arbitrary neural network architectures such as convolutional neural networks (He et al., 2016a) or recurrent neural networks (Chung et al., 2014).

Original: We call this framework ChoiceNet.
Revised: We call this framework ChoiceNet.

Original: Throughout this paper, we aim to address the following questions:
Revised: Throughout this paper, we aim to address the following questions:

Original: How can we measure the quality of training data in a principled manner?
Revised: How can we measure the quality of training data in a principled manner?

Original: In the presence of inconsistent outputs, how can we infer the target distribution in a scalable manner?
Revised: In the presence of inconsistent outputs, how can we infer the target distribution in a scalable manner?

Original: Traditionally, noisy outputs are handled by modeling additive random distributions, often leading to robust loss functions (Hampel et al., 2011).
Revised: Traditionally, noisy outputs are handled by modeling additive random distributions, often leading to robust loss functions (Hampel et al., 2011).

Original: However, we argue that these approaches are too restrictive when handling severe outliers or inconsistencies in the datasets.
Revised: However, we argue that these approaches are too restrictive when handling severe outliers or inconsistencies in the datasets.

Original: To address the ﬁrst question, we leverage the concept of a correlation.
Revised: To address the ﬁrst question, we leverage the concept of a correlation.

Original: Precisely, we measure the quality of training data using the correlation between the target distribution and the data generating distribution.
Revised: Precisely, we measure the quality of training data using the correlation between the target distribution and the data generating distribution.

Original: However, estimating the correct correlation requires an access to a target distribution, whereas learning the correct target distribution requires knowing the correlation between the distributions to be known, making it a chicken-and-egg problem.
Revised: However, estimating the correct correlation requires an access to a target distribution, whereas learning the correct target distribution requires knowing the correlation between the distributions to be known, making it a chicken-and-egg problem.

Original: To address the second question, we simultaneously estimate the target distribution as well as the correlation in an end-to-end-manner using stochastic gradient decent methods, in this case Adam (Kingma & Ba, 2014), to achieve scalability.
Revised: To address the second question, we simultaneously estimate the target distribution as well as the correlation in an end-to-end-manner using stochastic gradient decent methods, in this case Adam (Kingma & Ba, 2014), to achieve scalability.

Original: The cornerstone of the proposed method is a mixture of correlated density network (MCDN) block.
Revised: The cornerstone of the proposed method is a mixture of correlated density network (MCDN) block.

Original: First, we present a Cholesky transform method for sampling the weights of a neural network that enables us to model correlated outputs.
Revised: First, we present a Cholesky transform method for sampling the weights of a neural network that enables us to model correlated outputs.

Original: We also present an effective regularizer to train ChoiceNet.
Revised: We also present an effective regularizer to train ChoiceNet.

Original: To the best of our knowledge, this represents the ﬁrst approach simultaneously to infer the target distribution and the output correlations using a neural network in an end-to-end manner.
Revised: To the best of our knowledge, this represents the ﬁrst approach simultaneously to infer the target distribution and the output correlations using a neural network in an end-to-end manner.

Original: Revealing the output correlations was proposed in earlier work (Bonilla et al., 2008), in which a multi-task Gaussian process prediction (MTGPP) model is proposed.
Revised: Revealing the output correlations was proposed in earlier work (Bonilla et al., 2008), in which a multi-task Gaussian process prediction (MTGPP) model is proposed.

Original: In particular, MTGPP used correlated Gaussian processes to model multiple tasks by learning a free-form cross-covariance matrix.
Revised: In particular, MTGPP used correlated Gaussian processes to model multiple tasks by learning a free-form cross-covariance matrix.

Original: However, due to the multi-task learning setting, it is not suitable for learning a single target function.
Revised: However, due to the multi-task learning setting, it is not suitable for learning a single target function.

Original: In other work (Choi et al., 2016), a leverage optimization method which optimizes the leverage of each demonstrations is proposed.
Revised: In other work (Choi et al., 2016), a leverage optimization method which optimizes the leverage of each demonstrations is proposed.

Original: Unlike to former study (Bonilla et al., 2008), the latter (Choi et al., 2016) focused on inferring a single expert policy by incorporating a sparsity constraint by assuming that the most demonstrations are collected from a skillful consistent expert.
Revised: Unlike a former study (Bonilla et al., 2008), the latter (Choi et al., 2016) focused on inferring a single expert policy by incorporating a sparsity constraint by assuming that the most demonstrations are collected from a skillful consistent expert.

Original: However, these methods suffer from the scalability issues when applying on large-scale tasks.
Revised: However, these methods suffer from the scalability issues when applying on large-scale tasks.

Original: ChoiceNet is initially applied to a synthetic regression task, where we demonstrate its robustness to extreme outliers and ability to distinguish the target distribution and noise distributions.
Revised: ChoiceNet is initially applied to a synthetic regression task, where we demonstrate its robustness to extreme outliers and ability to distinguish the target distribution and noise distributions.

Original: We then apply it to a behavior cloning scenario where the demonstrations are collected from both an expert and an adversarial policies.
Revised: We then apply it to a behavior cloning scenario where the demonstrations are collected from both an expert and an adversarial policies.

Original: Subsequently, we move on to the classiﬁcation tasks using the MNIST and CIFAR-10 datasets.
Revised: Subsequently, we move on to the classiﬁcation tasks using the MNIST and CIFAR-10 datasets.

Original: We show that the proposed method outperforms existing baseline methods in terms of robustness with regard to the handling different types of noisy labels.
Revised: We show that the proposed method outperforms existing baseline methods in terms of robustness with regard to the handling different types of noisy labels.

Original:  R ELATED W ORK
Revised:  R ELATED W ORK

Original: Recently, robustness in deep learning has been actively studied (Fawzi et al., 2017) as deep neural networks are being applied to diverse tasks involving real-world applications such as autonomous driving (Paden et al., 2016) or medical diagnosis (Gulshan et al., 2016) where a simple malfunction can have catastrophic results (AP & REUTERS, 2016).
Revised: Recently, robustness in deep learning has been actively studied (Fawzi et al., 2017) as deep neural networks are being applied to diverse tasks involving real-world applications such as autonomous driving (Paden et al., 2016) or medical diagnosis (Gulshan et al., 2016) where a simple malfunction can have catastrophic results (AP & REUTERS, 2016).

Original: Perhaps, the most actively studied area regarding robustness in deep learning is the modeling and defense against adversarial attacks in the input domain (Aung et al., 2017; Sinha et al., 2017; Carlini & Wagner, 2017; Papernot et al., 2016).
Revised: Existing work for handing noisy training data can be categorized into four groups: small-loss tricks Jiang et al. ; Ren et al.

Original: Adversarial examples are intentionally designed inputs that cause incorrect predictions in learned models by adding a small perturbation that is scarcely recognized by humans (Goodfellow et al., 2014).
Revised: ; Han et al. (2018); Malach & Shalev-Shwartz (2017), estimating label corruptions Patrini et al. ; Goldberger & Ben-Reuven (2017);

Original: While this is a substantially important research direction, we focus on the noise in the outputs, e.g., outliers from different distributions or random labels.
Revised: Sukhbaatar et al.

Original: A number of studies such as Bekker & Goldberger (2016); Patrini et al.
Revised: Bekker & Goldberger (2016); Hendrycks et al. ; Veit et al.

Original:  ; Goldberger & Ben-Reuven (2017); Jindal et al.
Revised: (2017), using robust loss functions Natarajan et al. ; Belagiannis et al.

Original:  ; Liu et al.
Revised: (2015), and explicit and implicit regularization methods Reed et al. (2014); Lee (2013); Goodfellow et al. ; Xie et al. (2016); Tokozume et al.

Original: (2017) deal with the problems which arise when handling noisy labels in the training dataset in that massive datasets such as the ImageNet dataset (Deng et al., 2009) are often mostly from crowdsourcing and which thus may contain inaccurate and inconsistent labels (Bi et al., 2014).
Revised: Zhang et al. ; Miyato et al. ; Tarvainen & Valpola (2017); Laine & Aila (2017). Our proposed method is mostly related to the robust loss function approach but cannot fully categorized into this group in that we present a novel architecture, a mixture of correlated density network block, for achieving robustness based on the correlation estimation.

Original: To deal with noisy labels, an earlier study in Bekker &
Revised: First of all, the small-loss tricks selectively focus on training instances based on a certain criterion such as having small cost values Han et al. (Malach & Shalev-Shwartz, 2017) proposed a meta-algorithm for tackling the noisy label problem by training two networks only when the predictions of the two networks disagree, where selecting a proper network from among the two networks can be done using an additional clean dataset. (Ren et al., 2018) reweighs the weight of each training instance using a small amount of clean validation data. MentorNet Jiang et al.

Revised: (2017) concentrated on the training of an additional neural network, which assigns a weight to each instance of training data to supervise the training of a base network, termed StudentNet, to overcome the over-ﬁtting of corrupt training data.

Revised: Recent work Han et al.

Revised: (2018) presented Co-teaching by maintaining two separate networks where each network is trained with small-loss instances selected from its peer network.

Revised: The second group of estimating label corruption information is mainly presented for classiﬁcation tasks where training labels are assumed to be corrupted with a possibly unknown corruption matrix.

Original:  Goldberger (2016) proposed an extra layer for the modeling of output noises.
Revised: An earlier study in Bekker & Goldberger (2016) proposed an extra layer for the modeling of output noises.

Original: Jindal et al.
Revised: Jindal et al.

Original: (2016) extended the aforementioned approachin Bekker & Goldberger (2016) by adding an additional noise adaptation layer with aggressive dropout regularization.
Revised: (2016) extended the aforementioned approach by adding an additional noise adaptation layer with aggressive dropout regularization.

Original: A similar method was then proposed in
Revised: A similar method was then proposed in

Original: Patrini et al.
Revised: Patrini et al.

Original: (2017) which initially estimated the label corruption matrix with a learned classiﬁer and used the corruption matrix to ﬁne-tune the classiﬁer.
Revised: (2017) which initially estimated the label corruption matrix with a learned classiﬁer and used the corruption matrix to ﬁne-tune the classiﬁer.

Original: Jiang et al. (2017) concentrated on the training of an additional neural network, referred to as MentorNet, which assigns a weight to each instance of training data to supervise the training of a base network, termed StudentNet, to overcome the over-ﬁtting of corrupt training data.
Revised: Other researchers (Goldberger & Ben-Reuven, 2017) presented a robust training method that mimics the EM algorithm to train a neural network, with the label noise modeled as an additional softmax layer, similar to earlier work (Jindal et al., 2016). A self-error-correcting network was also presented (Liu et al., 2017).

Original: One ﬁnal study of note here (Rolnick et al., 2017) analyzed the intrinsic robustness of deep neural network models to massive label noise and empirically showed that a larger batch size with a lower learning rate can be beneﬁcial with regard to the robustness. Motivated by  that work (Rolnick et al., 2017), we train ChoiceNet with a large batch size and  a low learning rate. Unlike previous methods that only require noisy training datasets, some work such as Li et al.
Revised: It switches the training labels based on the learned model at the beginning stages by assuming that the deep model is more accurate during the earlier stage of training. Researchers have also focussed on using robust loss functions; (Natarajan et al., 2013) studied problem of binary classiﬁcation in the presence of random labels and presented a robust surrogate loss function for handling noisy labels.

Original: Malach & Shalev-Shwartz (2017); Hendrycks et al. ; Veit et al. (2017) require a small number of clean datasets. A gold-loss correction method was also presented Hendrycks et al.
Revised: Existing loss functions for classiﬁcation were studied (Ghosh et al., 2017), with the results showing that the mean absolute value of error is inherently robust to label noise.

Original: (2018); it initially learns a label corruption matrix using a small clean dataset and then uses the corruption matrix to retrain a corrected classiﬁer. A label-cleaning network has also been proposed in Veit et al.
Revised: In other work (Belagiannis et al., 2015), a robust loss function for deep regression tasks were proposed using Tukey’s biweight function with median absolute deviation of the residuals.

Original: It corrects noisy labels in the training dataset by leveraging information from a small clean dataset.
Revised: The last group focusses on using implicit or explicit regularization methods while training.

Original: Adding small label noises while training is known to be beneﬁcial to training, as it can be regarded as an effective regularization method (Lee, 2013; Goodfellow et al., 2016).
Revised: Adding small label noises while training is known to be beneﬁcial to training, as it can be regarded as an effective regularization method (Lee, 2013; Goodfellow et al., 2016).

Original: Similar methods have been proposed to tackle noisy outputs.
Revised: Similar methods have been proposed to tackle noisy outputs.

Original: A bootstrapping method (Reed et al., 2014) which train a neural network with a convex combination of the output of the current network and the noisy target was proposed.
Revised: A bootstrapping method (Reed et al., 2014) which train a neural network with a convex combination of the output of the current network and the noisy target was proposed.

Original: Xie et al.
Revised: Xie et al.

Original: (2016) proposed DisturbLabel, a simple method which randomly replaces a percentage of the labels with incorrect values for each iteration.
Revised: (2016) proposed DisturbLabel, a simple method which randomly replaces a percentage of the labels with incorrect values for each iteration.

Original: Mixing both input and output data was also proposed (Tokozume et al., 2018; Zhang et al., 2017).
Revised: Mixing both input and output data was also proposed (Tokozume et al., 2018; Zhang et al., 2017).

Original: One study (Zhang et al., 2017) considered the image recognition problem under label noise and the other (Tokozume et al., 2018) focused on a sound recognition problem.
Revised: One study (Zhang et al., 2017) considered the image recognition problem under label noise and the other (Tokozume et al., 2018) focused on a sound recognition problem.

Revised: Temporal ensemble was proposed in Laine & Aila (2017) where an unsupervised loss term of ﬁtting the output of of an augmented input to the augmented target updated with an exponential moving average.

Revised: (Tarvainen & Valpola, 2017) extends the temporal ensemble in Laine & Aila (2017) by introducing a consistency cost function that minimizes the distance between the weights of the student model and the teacher model.

Revised: (Miyato et al., 2018) presented new regularization method based on virtual adversarial loss which measures the smoothness of conditional label distribution given input.

Revised: Minimizing the virtual adversarial loss has a regularizing effect in that it makes the model smooth at each data point.

Revised: The foundation of the proposed method is the mixture of correlated density network (MCDN) block where the output distribution is modeled using a mixture of correlated distributions.

Original: Modeling correlations of output training data has been actively studied in light of Gaussian processes (Rasmussen, 2006).
Revised: Modeling correlations of output training data has been actively studied in light of Gaussian processes (Rasmussen, 2006).

Original: MTGPP (Bonilla et al., 2008) that models the correlations of multiple tasks via Gaussian process regression was also proposed. Due to the  multi-task setting, however, Bonilla et al. (2008) is not suitable for robust regression tasks.
Revised: MTGPP (Bonilla et al., 2008) that models the correlations of multiple tasks via Gaussian process regression was proposed in the context of multi-task setting.

Original: Choi et al.
Revised: Choi et al.

Original: (2016) proposed a robust learning from demonstration method using a sparse constrained leverage optimization method which estimates the correlation between training outputs.
Revised: (2016) proposed a robust learning from demonstration method using a sparse constrained leverage optimization method which estimates the correlation between training outputs and showed its robustness compared to several baselines.

Original: Unlike the former study in Bonilla et al.
Revised: On the other hand, Platanios et al.

Original: (2008), the latter above in Choi et al. (2016)  can robustly recover the expert policy function.
Revised: (2016) presented a graphical model that can model the structure of the error rates of learned classiﬁers using non-parametric Bayesian methods.

Original: While our problem setting is similar to the latter study (Choi et al., 2016), we propose end-to-end learning of both the target distribution and the correlation of each training data, thus offering, a clear advantage in terms of scalability.
Revised: We would like to note that while our problem setting is similar to the latter study (Choi et al., 2016), we propose end-to-end learning of both the target distribution and the correlation of each training data, thus offering, a clear advantage in terms of scalability.

Original: The aforementioned study (Choi et al., 2016) also requires the design of a proper kernel structure, which is not suitable for high-dimensional inputs and classiﬁcation problems.

Original:  C HOICE N ET
Revised:  C HOICE N ET

Original: In this section, we present the methodology and the model architecture of ChoiceNet.
Revised: In this section, we present the methodology and the model architecture of ChoiceNet.

Original: A main ingredient of ChoiceNet is a mixture of correlated density network (MCDN) block upon the arbitrary base network.
Revised: A main ingredient of ChoiceNet is a mixture of correlated density network (MCDN) block upon the arbitrary base network.

Original: First, we illustrate the motivation of the MCDN block.
Revised: First, we illustrate the motivation of the MCDN block.

Original: Section 3.1 introduces Cholesky transform which enables correlated sampling and legitimates applying the reparametrization trick.
Revised: Section 3.1 introduces Cholesky transform which enables correlated sampling and legitimates applying the reparametrization trick.

Original: Subsequently, we present the mechanism of ChoiceNet in Section 3.2 and loss functions for regression and classiﬁcation tasks in Section 3.3.
Revised: Subsequently, we present the mechanism of ChoiceNet in Section 3.2 and loss functions for regression and classiﬁcation tasks in Section 3.3.

Original: Modeling Corrupt Output As stated in Section 1, we focus on the problem of supervised learning on training data with corrupt outputs.
Revised: Modeling Corrupt Output As stated in Section 1, we focus on the problem of supervised learning on training data with corrupt outputs.

Original: Denote an unknown clean dataset by D clean whose elements ( x
Revised: Denote an unknown clean dataset by D clean whose elements ( x

Original: , y ) ∈
Revised: , y ) ∈

Original: D clean are determined by a relation y = f
Revised: D clean are determined by a relation y = f

Original: ( x ) .
Revised: ( x ) .

Original: For a classiﬁcation task, an accurate label y ∈ { 0 , 1 } exists for each x .
Revised: For a classiﬁcation task, an accurate label y ∈ { 0 , 1 } exists for each x .

Original: We assume a corrupt data ( x , ˆ y
Revised: We assume a corrupt data ( x , ˆ y

Original: ) ∈
Revised: ) ∈

Original: D corrupt is given such that
Revised: D corrupt is given such that

Original: Regression :
Revised: Regression :

Revised: [Equation]

Revised: Classiﬁcation :

Original: [Equation]
Revised: [Equation]

Original: where g ( · ) is an arbitrary function.
Revised: where g ( · ) is an arbitrary function.

Original: Here ε and ξ are additive noise (usually heteroscedastic) and p indicates the corruption (or mixture) probability.
Revised: Here ε and ξ are additive noise (usually heteroscedastic) and p indicates the corruption (or mixture) probability.

Original: The above setting employs the random choice under
Revised: The above setting employs the random choice under

Original: Bernoulli distribution but one can consider a multinoulli distribution instead.
Revised: Bernoulli distribution but one can consider a multinoulli distribution instead.

Original: The corrupt data can be modeled by the conditional density estimation via a mixture of distributions:
Revised: The corrupt data can be modeled by the conditional density estimation via a mixture of distributions:

Original: [Equation]
Revised: [Equation]

Original: where π target and π noise represent the ratio of target and noisy data, respectively.
Revised: where π target and π noise represent the ratio of target and noisy data, respectively.

Original: In this paper, we deal with the target conditional density P ( ·|· ) using a parametrized distribution with expected measurement variance ˆ σ 2 i.e., P ( ·| x ) =
Revised: In this paper, we deal with the target conditional density P ( ·|· ) using a parametrized distribution with expected measurement variance ˆ σ 2 i.e., P ( ·| x ) =

Original: N ( f θ ( x ) , ˆ σ 2 ) where f θ is a neural network and θ is a set of parameters.
Revised: N ( f θ ( x ) , ˆ σ 2 ) where f θ is a neural network and θ is a set of parameters.

Original: Analogous to the mixture density network (Bishop (1994)), we tackle the noise conditional distribution Q ( ·|· ) parametrized also by θ .
Revised: Analogous to the mixture density network (Bishop (1994)), we tackle the noise conditional distribution Q ( ·|· ) parametrized also by θ .

Original: However, one major difference is that, we quantify its irrelevance (or independence) by utilizing the correlation ρ between P and Q .
Revised: However, one major difference is that, we quantify its irrelevance (or independence) by utilizing the correlation ρ between P and Q .

Original: Intuitively speaking, irrelevant noisy data will be modeled to be collected from a class of Q with relatively small or negative ρ .
Revised: Intuitively speaking, irrelevant noisy data will be modeled to be collected from a class of Q with relatively small or negative ρ .

Original: Since we assume that the correlation information is not explicitly given, we model the ρ of each data to be a function of an input x i.e., ρ φ ( x ) , parametrized by φ and jointly optimize φ and θ using a mixture distribution.
Revised: Since we assume that the correlation information is not explicitly given, we model the ρ of each data to be a function of an input x i.e., ρ φ ( x ) , parametrized by φ and jointly optimize φ and θ using a mixture distribution.

Original: The MCDN block in Section 3.2 is proposed for this purpose.
Revised: The MCDN block in Section 3.2 is proposed for this purpose.

Original: 3. C HOLESKY T RANSFORM AND C ORRELATED S AMPLING
Revised: 3. C HOLESKY T RANSFORM AND C ORRELATED S AMPLING

Original: In this section, we present a novel method on how to model dependencies among output distributions.
Revised: In this section, we present a novel method on how to model dependencies among output distributions.

Original: Given parameters Θ = ( ρ, µ, σ 1 , σ 2 ) , Cholesky transform is a mapping from ( w, z ) ∈ R 2 to R which is deﬁned by
Revised: Given parameters Θ = ( ρ, µ, σ 1 , σ 2 ) , Cholesky transform is a mapping from ( w, z ) ∈ R 2 to R which is deﬁned by

Original: [Equation]
Revised: [Equation]

Original: Let W and Z be uncorrelated random variables such that E [ W ] = µ W , V ( W ) =
Revised: Let W and Z be uncorrelated random variables such that E [ W ] = µ W , V ( W ) =

Original: σ 2 W , E
Revised: σ 2 W , E

Original: [ Z ] = 0 , and V ( Z )
Revised: [ Z ] = 0 , and V ( Z )

Original: =
Revised: =

Original: σ 2 Z .
Revised: σ 2 Z .

Original: For − 1 ≤ ρ ≤ 1 , write ˜Θ =
Revised: For − 1 ≤ ρ ≤ 1 , write ˜Θ =

Original: ( ρ, µ W , σ W , σ Z ) and set a new random variable ˜ W by plugging ˜Θ and ( W, Z ) in Cholesky transform i.e. ˜ W :
Revised: ( ρ, µ W , σ W , σ Z ) and set a new random variable ˜ W by plugging ˜Θ and ( W, Z ) in Cholesky transform i.e. ˜ W :

Original: =
Revised: =

Original: T ˜Θ ( W, Z ) .
Revised: T ˜Θ ( W, Z ) .

Original: This transform makes it possible to apply the reparametrization trick (Kingma (2017); Kingma & Welling (2013)) to jointly learn parameters not only µ W , σ W but also ρ .
Revised: This transform makes it possible to apply the reparametrization trick (Kingma (2017); Kingma & Welling (2013)) to jointly learn parameters not only µ W , σ W but also ρ .

Original: Since correlation is invariant to mean translation and variance dilatation (see appendix), it is easy to see
Revised: Since correlation is invariant to mean translation and variance dilatation (see appendix), it is easy to see

Original: that Corr ( W, ˜ W ) = ρ .
Revised: that Corr ( W, ˜ W ) = ρ .

Original: The following theorem further states that a correlation between two random matrices is invariant to an afﬁne transform.
Revised: The following theorem further states that a correlation between two random matrices is invariant to an afﬁne transform.

Original: Theorem.
Revised: Theorem.

Original: Let ρ =
Revised: Let ρ =

Original: ( ρ 1 , . .
Revised: ( ρ 1 , . .

Original: . , ρ K ) ∈ R K .
Revised: . , ρ K ) ∈ R K .

Original: For p ∈ { 1 , 2 } , random matrices W ( p ) ∈ R K × Q are given such that for every k ∈ { 1 , . . .
Revised: For p ∈ { 1 , 2 } , random matrices W ( p ) ∈ R K × Q are given such that for every k ∈ { 1 , . . .

Original: , K } ,
Revised: , K } ,

Original: [Equation]
Revised: [Equation]

Original: Given h
Revised: Given h

Original: =
Revised: =

Original: ( h 1 , . .
Revised: ( h 1 , . .

Original: , h Q ) ∈ R Q , set y ( p ) =
Revised: , h Q ) ∈ R Q , set y ( p ) =

Original: W ( p ) h for each p ∈ { 1 , 2 } .
Revised: W ( p ) h for each p ∈ { 1 , 2 } .

Original: Then an elementwise correlation between y (1) and y (2) equals ρ i.e.
Revised: Then an elementwise correlation between y (1) and y (2) equals ρ i.e.

Original: [Equation]
Revised: [Equation]

Original: Hence ρ is a proper representation of dependencies among the distributions on the feature space.
Revised: Hence ρ is a proper representation of dependencies among the distributions on the feature space.

Original: This theorem allows to use Cholesky transform to generate correlated weight matrices { (cid:102) W k } Kk = upon the feature layer of base network.
Revised: This theorem allows to use Cholesky transform to generate correlated weight matrices { (cid:102) W k } Kk = upon the feature layer of base network.

Original: In the following section, we demonstrate the details of the sampling process in the MCDN block.
Revised: In the following section, we demonstrate the details of the sampling process in the MCDN block.

Original: 3. M ODEL A RCHITECTURE
Revised: 3. M ODEL A RCHITECTURE

Original: In this section, we describe the model architecture and the overall mechanism of ChoiceNet.
Revised: In this section, we describe the model architecture and the overall mechanism of ChoiceNet.

Original: In the followings, τ − 1 > 0 is a constant indicating expected measurement noise and η ( · ) ∈ ( − 1 , 1) is a bounded function, e.g., a hyperbolic tangent.
Revised: In the following, τ − 1 > 0 is a constant indicating expected measurement noise and η ( · ) ∈ ( − 1 , 1) is a bounded function, e.g., a hyperbolic tangent.

Original: W h
Revised: W h

Original: → ρ , W h → π ∈ R K × Q and W h
Revised: → ρ , W h → π ∈ R K × Q and W h

Original: → Σ 

Revised: → Σ

Original: ∈ R D
Revised: ∈ R D

Original: × Q are weight matrices where Q and D denote the dimensions of a feature vector h and output y , respectively, and K is the number of mixtures.
Revised: × Q are weight matrices where Q and D denote the dimensions of a feature vector h and output y , respectively, and K is the number of mixtures.

Original: ρ max is a ﬁxed constant whose value is close to 1 .
Revised: ρ max is a ﬁxed constant whose value is close to 1 .

Original: ChoiceNet is a twofold architecture: (a) a base network and (b) a MCDN block (see Figure 1).
Revised: ChoiceNet is a twofold architecture: (a) a base network and (b) a MCDN block (see Figure 1).

Original: A base network extracts features for a given dataset.
Revised: A base network extracts features for a given dataset.

Original: Then the MCDN block estimates the densities of the data generating distributions through ( µ k , Σ k , π k ) K k =1 .
Revised: Then the MCDN block estimates the densities of the data generating distributions through ( µ k , Σ k , π k ) Kk =1 .

Original: Contrary to the mixture density network (MDN), during the density estimation process, the MCDN block samples correlated weights { (cid:102) W k } Kk =1 using
Revised: Contrary to the mixture density network (MDN), during the density estimation process, the MCDN block samples correlated weights { (cid:102) W k } K k = 1 using

Original: Cholesky transform.
Revised: Cholesky transform.

Original: Consequently, the MCDN block is able to model the correlated outputs i.e. correlated mean vectors µ .
Revised: Consequently, the MCDN block is able to model the correlated outputs i.e. correlated mean vectors µ .

Original: The overall mechanism of ChoiceNet can be elaborated as follows:
Revised: The overall mechanism of ChoiceNet can be elaborated as follows:

Original: [Equation]
Revised: [Equation]

Original: Thanks to the theorem in Section 3.1, for each k ∈ { 1 , . . 
Revised: Thanks to the theorem in Section 3. [Equation] for each k ∈ { 1 , . . .

Original: , K }
Revised: , K }

Original: Corr ( µ k , Wh ) =

Original: Corr(
Revised: Corr

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original: ) = ( ρ k , . . . , ρ k ) ∈ R D and the output density is modeled via mean vectors µ .
Revised: and the output density is modeled via mean vectors µ .

Original: Note that both V ( µ k ) and Σ k are minimized, when ρ k → ± 1 .
Revised: Note that both V ( µ k ) and Σ k are minimized, when ρ k → ± 1 .

Original: Furthermore, as we employ Gaussian distributions in Cholesky transform, the inﬂuences of uninformative or independent data, whose correlations are close to 0, is attenuated as their variances increase (Kendall & Gal (2017)).
Revised: Furthermore, as we employ Gaussian distributions in Cholesky transform, the inﬂuences of uninformative or independent data, whose correlations are close to 0, is attenuated as their variances increase (Kendall & Gal (2017)).

Original: 3. T RAINING O BJECTIVES
Revised: 3. T RAINING O BJECTIVES

Original: Denote a training dataset by D = { ( x i , y i ) :
Revised: Denote a training dataset by D = { ( x i , y i ) :

Original: i = 1 , . .
Revised: i = 1 , . .

Original: . , N } .
Revised: . , N } .

Original: We consider both regression and classiﬁcation tasks.
Revised: We consider both regression and classiﬁcation tasks.

Original: Regression For the regression task, we employ both L 2 -loss and the standard MDN loss (Bishop (1994); Choi et al.
Revised: Regression For the regression task, we employ both L 2 -loss and the standard MDN loss (Bishop (1994); Choi et al.

Original: (2017); Christopher (2016))
Revised: (2017); Christopher (2016))

Original: [Equation]
Revised: [Equation]

Original: where λ 1 and λ 2 are hyper-parameters and N ( ·| µ, Σ) is the density of multivariate Gaussian:
Revised: where λ 1 and λ 2 are hyper-parameters and N ( ·| µ, Σ) is the density of multivariate Gaussian:

Original: [Equation]
Revised: [Equation]

Original: We also add weight decay and the following Kullback-Leibler regularizer to equation
Revised: We also add weight decay and the following Kullback-Leibler regularizer to equation

Original: [Equation]
Revised: [Equation]

Original: The above KL regularizer encourages the mixture components with the strong correlations to have high mixture probabilities.
Revised: The above KL regularizer encourages the mixture components with the strong correlations to have high mixture probabilities.

Original: This guidance is useful since ChoiceNet uses the mean vector
Revised: This guidance is useful since ChoiceNet uses the mean vector

Original: µ 1 ( x i ) of the ﬁrst mixture component at the inference stage.
Revised: µ 1 ( x i ) of the ﬁrst mixture component at the inference stage.

Original: Classiﬁcation In the classiﬁcation task, we suppose each y i is a D -dimensional one-hot vector.
Revised: Classiﬁcation In the classiﬁcation task, we suppose each y i is a D -dimensional one-hot vector.

Original: Unlike the regression task, equation 1 is not appropriate for the classiﬁcation task.
Revised: Unlike the regression task, equation 1 is not appropriate for the classiﬁcation task.

Original: We employ the following loss function:
Revised: We employ the following loss function:

Original: [Equation]
Revised: [Equation]

Original: Here (cid:104)· , ·(cid:105) denotes inner product and for k ∈ { 1 , . .
Revised: Here (cid:104)· , ·(cid:105) denotes inner product and for k ∈ { 1 , . .

Original: , K } , d ∈ { 1 , . . .
Revised: , K } , d ∈ { 1 , . . .

Original: , D }
Revised: , D }

Original: [Equation]
Revised: [Equation]

Original: Similar to the regression task, we use both equation 2 and weight decay.
Revised: Similar to the regression task, we use both equation 2 and weight decay.

Original: [Table]

Original:  E XPERIMENTS 4. R EGRESSION T ASKS
Revised:  E XPERIMENTS 4. R EGRESSION T ASKS

Original: We conduct two regression experiments: 1) a synthetic scenario where the training dataset contains outliers sampled from other distributions and 2) a behavior cloning scenario where the demonstrations are collected from both expert and adversarial policies.
Revised: We conduct two regression experiments: 1) a synthetic scenario where the training dataset contains outliers sampled from other distributions and 2) a behavior cloning scenario where the demonstrations are collected from both expert and adversarial policies.

Original: Synthetic Regression Example We ﬁrst apply ChoiceNet to a simple one-dimensional regression problem of ﬁtting f ( x ) =
Revised: Synthetic Regression Example We ﬁrst apply ChoiceNet to a simple one-dimensional regression problem of ﬁtting f ( x ) =

Original: cos( π 2 x ) exp( − ( x 2 ) 2 ) where x ∈ [ − 3 , +3] as shown in Figure 2.
Revised: cos( π 2 x ) exp( − ( x 2 ) 2 ) where x ∈ [ − 3 , +3] as shown in Figure 2.

Original: ChoiceNet is compared with a naive multilayer perceptron (MLP), a mixture density network (MDN) with ﬁve mixtures where all networks have two hidden layers with 32 nodes with a ReLU activation function.
Revised: ChoiceNet is compared with a naive multilayer perceptron (MLP), a mixture density network (MDN) with ﬁve mixtures where all networks have two hidden layers with 32 nodes with a ReLU activation function.

Original: Gaussian process regression (GPR) (Rasmussen, 2006), leveraged Gaussian process regression (LGPR) with leverage optimization (Choi et al., 2016), and robust Gaussian process regression (RGPR) with an inﬁnite Gaussian process mixture model (Rasmussen & Ghahramani, 2002) are also compared.
Revised: Gaussian process regression (GPR) (Rasmussen, 2006), leveraged Gaussian process regression (LGPR) with leverage optimization (Choi et al., 2016), and robust Gaussian process regression (RGPR) with an inﬁnite Gaussian process mixture model (Rasmussen & Ghahramani, 2002) are also compared.

Original: For the GP based methods, we use a squared-exponential kernel function and the hyper-parameters are determined using a simple median trick (Dai et al., 2014) 1 .
Revised: For the GP based methods, we use a squared-exponential kernel function and the hyper-parameters are determined using a simple median trick (Dai et al., 2014) 1 .

Original: To evaluate its performance in corrupt datasets, we randomly replace the original target values with outliers whose output values are uniformly sampled from − 1 to +3 .
Revised: To evaluate its performance in corrupt datasets, we randomly replace the original target values with outliers whose output values are uniformly sampled from − 1 to +3 .

Original: We vary the outlier rates from 0% (clean) to 80% (extremely noisy).
Revised: We vary the outlier rates from 0% (clean) to 80% (extremely noisy).

Original: Table 1 illustrates the RMSEs (root mean square errors) between the reference target function and the ﬁtted results of ChoiceNet and other compared methods.
Revised: Table 1 illustrates the RMSEs (root mean square errors) between the reference target function and the ﬁtted results of ChoiceNet and other compared methods.

Original: Given an intact training dataset, all the methods show stable performances in that the RMSEs are all below 0 .
Revised: Given an intact training dataset, all the methods show stable performances in that the RMSEs are all below 0 .

Original: Given training datasets whose outlier rates exceed 40% , however, only ChoiceNet successfully ﬁts the target function whereas the other methods fail as shown in Figure 2.
Revised: Given training datasets whose outlier rates exceed 40% , however, only ChoiceNet successfully ﬁts the target function whereas the other methods fail as shown in Figure 2.

Original: To further inspect whether ChoiceNet can distinguish between the target distribution and noise distributions, we train ChoiceNet on two datasets.
Revised: To further inspect whether ChoiceNet can distinguish between the target distribution and noise distributions, we train ChoiceNet on two datasets.

Original: In particular, we use the same target function and replace 50% of the output values whose input values are within 0 to 2 using two different corruptions: one uniformly sampled from − 1 to 3 and the other from a ﬂipped target function.
Revised: In particular, we use the same target function and replace 50% of the output values whose input values are within 0 to 2 using two different corruptions: one uniformly sampled from − 1 to 3 and the other from a ﬂipped target function.

Original: For this experiment, we set K = 2 for better visualization.
Revised: For this experiment, we set K = 2 for better visualization.

Original: As shown in Figure 3(a) and 3(c), ChoiceNet successfully ﬁts the target function.
Revised: As shown in Figure 3(a) and 3(c), ChoiceNet successfully ﬁts the target function.

Original: The correlations of the second component decrease as outliers are introduced as shown in Figure 3(b) and 3(d).
Revised: The correlations of the second component decrease as outliers are introduced as shown in Figure 3(b) and 3(d).

Original: Surprisingly, when the target and noise distribution are negatively correlated (the ﬂipped function case), the correlations of the second component become − 1 as depicted in Figure 3(b).
Revised: Surprisingly, when the target and noise distribution are negatively correlated (the ﬂipped function case), the correlations of the second component become − 1 as depicted in Figure 3(b).

Original: Contrarily, for the uniform corruption case, the correlations of the second[Figure] [Table] component are within 0 and 1 .
Revised: Contrarily, for the uniform corruption case, the correlations of the second component are within 0 and 1 .

Original: We would like to emphasize that this clearly shows the capability of
Revised: We would like to emphasize that this clearly shows the capability of

Original: ChoiceNet to distinguish the target distribution from noisy distributions.
Revised: ChoiceNet to distinguish the target distribution from noisy distributions.

Revised: [Table]

Revised: [Figure]

Revised: [Table]

Original: Behavior Cloning Example
Revised: Behavior Cloning Example

Original: In this experiment, we apply ChoiceNet to behavior cloning tasks when given demonstrations with mixed qualities.
Revised: In this experiment, we apply ChoiceNet to behavior cloning tasks when given demonstrations with mixed qualities.

Original: where the proposed method is compared with
Revised: where the proposed method is compared with

Original: MLP and MDN in two locomotion tasks: HalfCheetah and Walker2d .
Revised: MLP and MDN in two locomotion tasks: HalfCheetah and Walker2d .

Original: The network architectures are identical to those in the synthetic regression example tasks.
Revised: The network architectures are identical to those in the synthetic regression example tasks.

Original: To evaluate the robustness of ChoiceNet, we collect demonstrations from both an expert policy and an adversarial policy where two policies are trained by solving the corresponding reinforcement learning problems using the state-of-the-art proximal policy optimization (PPO) (Schulman et al., 2017).
Revised: To evaluate the robustness of ChoiceNet, we collect demonstrations from both an expert policy and an adversarial policy where two policies are trained by solving the corresponding reinforcement learning problems using the state-of-the-art proximal policy optimization (PPO) (Schulman et al., 2017).

Original: For training adversarial policies for both tasks, we ﬂip the signs of the directional rewards so that the agent gets incentivized by going backward.
Revised: For training adversarial policies for both tasks, we ﬂip the signs of the directional rewards so that the agent gets incentivized by going backward.

Original: We evaluate the performances of the compared methods using 500 state-action pairs with different mixing ratio and measure the average return over 100 consecutive episodes.
Revised: We evaluate the performances of the compared methods using 500 state-action pairs with different mixing ratio and measure the average return over 100 consecutive episodes.

Original: The results are shown in Table 2.
Revised: The results are shown in Table 2.

Original: In both cases, ChoiceNet outperforms compared methods by a signiﬁcant margin.
Revised: In both cases, ChoiceNet outperforms compared methods by a signiﬁcant margin.

Original: 4. C LASSIFICATION T ASKS
Revised: 4. C LASSIFICATION T ASKS

Original: We also conduct classiﬁcation experiments on the MNIST and CIFAR-10 datasets to evaluate the performance of ChoiceNet on corrupt labels.
Revised: We also conduct classiﬁcation experiments on the MNIST and CIFAR-10 datasets to evaluate the performance of ChoiceNet on corrupt labels.

Original: To generate noisy datasets, we follow the setting in (Zhang et al., 2017) which randomly shufﬂes a percentage of the labels in the dataset 2 .
Revised: To generate noisy datasets, we follow the setting in (Zhang et al., 2017) which randomly shufﬂes a percentage of the labels in the dataset 2 .

Original: We vary the corruption probabilities from 50% to 95% for the MNIST dataset and from 20% to 80% for the
Revised: We vary the corruption probabilities from 50% to 95% for the MNIST dataset and from 20% to 80% for the

Original: CIFAR-10 dataset and compare median accuracies after ﬁve runs for each conﬁguration.
Revised: CIFAR-10 dataset and compare median accuracies after ﬁve runs for each conﬁguration.

Original: On both
Revised: On both

Original: MNIST and CIFAR-10 experiments, we also compare ChoiceNet with Mixup (Zhang et al., 2017) which, to the best of our knowledge, shows the state-of-the-art performance on noisy labels.
Revised: MNIST and CIFAR-10 experiments, we also compare ChoiceNet with Mixup (Zhang et al., 2017) which, to the best of our knowledge, shows the state-of-the-art performance on noisy labels.

Original: We set the parameter α of Mixup to be 32 for the baseline network as suggested in the original paper.
Revised: We set the parameter α of Mixup to be 32 for the baseline network as suggested in the original paper.

Original: For
Revised: For

Original: ChoiceNet, we set α to be 1 .
Revised: ChoiceNet, we set α to be 1 .

Original: For the MNIST experiments, we construct two networks: a network with two residual blocks (He et al., 2016b) with 3 × 3 × 64 convolutional layers followed by a fully-connected layer withoutput neurons (ConvNet) and a network with the same two residual blocks followed by a MCDN block (ChoiceNet).
Revised: For the MNIST experiments, we construct two networks: a network with two residual blocks (He et al., 2016b) with 3 × 3 × 64 convolutional layers followed by a fully-connected layer withoutput neurons (ConvNet) and a network with the same two residual blocks followed by a MCDN block (ChoiceNet).

Original: We train each network for 50 epochs with a ﬁxed learning rate of 1 e − 5 .
Revised: We train each network for 50 epochs with a ﬁxed learning rate of 1 e − 5 .

Original: For the CIFAR experiments, we adopt WideResNet (WRN) (Zagoruyko & Komodakis, 2016) withlayers and a widening factor of 4 .
Revised: For the CIFAR experiments, we adopt WideResNet (WRN) (Zagoruyko & Komodakis, 2016) withlayers and a widening factor of 4 .

Original: To construct ChoiceNet, we replace the last layer of WideResNet with a MCDN block.
Revised: To construct ChoiceNet, we replace the last layer of WideResNet with a MCDN block.

Original: We set K = 3 , ρ max = 0 .
Revised: We set K = 3 , ρ max = 0 .

Original: [-95 , λ reg = 0 . 0001 , and ρ k , π k , Σ 0 modules consist 
-]
Revised: [-95 , λ reg = 0 . 0001 , and ρ k , π k , Σ 0 modules consist 
-]

Original: In the corrupt label setting, for a given corruption probability p , the expected ratio of correct labels is (1 − p )

Original: + p × 1 / (number of classes). Additional experiments of replacing the percentage of labels to a random labels and a ﬁxed label can be found in the appendix. of two fully connected layers with 64 hidden units and a ReLU activation function.
Revised: 95 , λ reg = 0 . 0001 , and ρ k , π k , Σ 0 modules consist of two fully connected layers with 64 hidden units and a ReLU activation function.

Original: We train each network for 300 epochs with a minibatch size of 256 .
Revised: We train each network for 300 epochs with a minibatch size of 256 .

Original: We begin with a learning rate of 0 . 1 , and it decays by 1 / 10 after 150 and 225 epochs.
Revised: We begin with a learning rate of 0 . 1 , and it decays by 1 / 10 after 150 and 225 epochs.

Original: We apply random horizontal ﬂip and random crop with 4 − pixel-padding and use a weight decay of 0 . 0001 for the baseline network as (He et al., 2016b).
Revised: We apply random horizontal ﬂip and random crop with 4 − pixel-padding and use a weight decay of 0 . 0001 for the baseline network as (He et al., 2016b).

Original: However, to train ChoiceNet, we reduce the weight decay rate to 1 e − 6 and apply gradient clipping at 1 . 0 .
Revised: However, to train ChoiceNet, we reduce the weight decay rate to 1 e − 6 and apply gradient clipping at 1 . 0 .

Original: We also lower the learning rate to 0 .
Revised: We also lower the learning rate to [Table]

Original: 001 for the ﬁrst epoch to stabilize training.
Revised: 001 for the ﬁrst epoch to stabilize training.

Original: [Table] [Table]
Revised: [Table] [Table]

Original: The classiﬁcation results of the MNIST dataset and the CIFAR dataset are shown in Table 3 and
Revised: The classiﬁcation results of the MNIST dataset and the CIFAR dataset are shown in Table 3 and

Original: Table 4, respectively.
Revised: Table 4, respectively.

Original: In the MNIST experiments, ChoiceNet consistently outperforms ConvNet and
Revised: In the MNIST experiments, ChoiceNet consistently outperforms ConvNet and

Original: ConvNet+Mixup by a signiﬁcant margin, and the difference between the accuracies of ChoiceNet and the others becomes more clear as the corruption probability increases.
Revised: ConvNet+Mixup by a signiﬁcant margin, and the difference between the accuracies of ChoiceNet and the others becomes more clear as the corruption probability increases.

Original: Particularly, the best test accuracy of ChoiceNet reaches 94% even when 90% of the training labels are randomly shufﬂed.
Revised: Particularly, the best test accuracy of ChoiceNet reaches 94% even when 90% of the training labels are randomly shufﬂed.

Original: In the CIFAR-10 experiments, ChoiceNet outperforms WideResNet and achieves its accuracy over 60% even when 80% of the labels are shufﬂed whereas the accuracy of WideResNet drops below 30% .
Revised: In the CIFAR-10 experiments, ChoiceNet outperforms WideResNet and achieves its accuracy over 60% even when 80% of the labels are shufﬂed whereas the accuracy of WideResNet drops below 30% .

Original: When we inspect the training accuracies on the 80% -shufﬂed set, WideResNet tends to overﬁt (memorize) to noisy labels and shows 99 . 8% train accuracy.
Revised: When we inspect the training accuracies on the 80% -shufﬂed set, WideResNet tends to overﬁt (memorize) to noisy labels and shows 99 . 8% train accuracy.

Original: On the contrary, ChoiceNet shows 37 .
Revised: On the contrary, ChoiceNet shows 37 .

Original: When trained with Mixup, both networks become robust to noisy labels to some extent.
Revised: When trained with Mixup, both networks become robust to noisy labels to some extent.

Original: However, the results of the two networks still show signiﬁcant differences except for the 20% corrupt experiments on which both of them show similar accuracies.
Revised: However, the results of the two networks still show signiﬁcant differences except for the 20% corrupt experiments on which both of them show similar accuracies.

Original: Interestingly, when ChoiceNet and
Revised: Interestingly, when ChoiceNet and

Original: Mixup are combined, it achieves a high accuracy of 75% even on the 80% shufﬂed dataset.
Revised: Mixup are combined, it achieves a high accuracy of 75% even on the 80% shufﬂed dataset.

Original: We also note that ChoiceNet (without Mixup) outperforms WideResNet+Mixup when the corruption ratio is over 50% on the last accuracies.
Revised: We also note that ChoiceNet (without Mixup) outperforms WideResNet+Mixup when the corruption ratio is over 50% on the last accuracies.

Original:  C ONCLUSION
Revised:  C ONCLUSION

Original: In this paper, we have presented ChoiceNet that can robustly learn a target distribution given noisy training data.
Revised: In this paper we have presented ChoiceNet that can robustly learn a target distribution given noisy training data.

Original: The keystone of ChoiceNet is the mixture of correlated density network block which can estimate the densities of data distributions using a set of correlated mean functions.
Revised: The keystone of ChoiceNet is the mixture of correlated density network block which can estimate the densities of data distributions using a set of correlated mean functions.

Original: We have demonstrated that ChoiceNet can robustly infer the target distribution on corrupt training data in the following tasks; regression with synthetic data, behavior cloning using demonstrations with mixed qualities, and MNIST and CIFAR-10 image classiﬁcation tasks.
Revised: We have demonstrated that ChoiceNet can robustly infer the target distribution on corrupt training data in the following tasks; regression with synthetic data, behavior cloning using demonstrations with mixed qualities and MNIST and CIFAR-10 image classiﬁcation tasks.

Original: Our experiments verify that
Revised: Our experiments verify that

Original: ChoiceNet outperforms existing methods in the handling of noisy data.
Revised: ChoiceNet outperforms existing methods in the handling of noisy data.

Original: Selecting proper hyper-parameters including the optimal number of mixture components is a compelling topic for the practical usage of ChoiceNet.
Revised: Selecting proper hyper-parameters including the optimal number of mixture components is a compelling topic for the practical usage of ChoiceNet.

Original: Furthermore, one can use ChoiceNet for active learning by evaluating the quality of each training data using through the lens of correlations.
Revised: Furthermore, one can use ChoiceNet for active learning by evaluating the quality of each training data using through the lens of correlations.

Original: We leave these as important questions for future work.
Revised: We leave these as important questions for future work

Original: A P ROOF OF T HEOREM IN S ECTION
Revised: A P ROOF OF T HEOREM IN S ECTION

Original: In this appendix we introduce fundamental theorems which lead to Cholesky transform for given random variables ( W, Z ) .
Revised: In this appendix, we introduce fundamental theorems which lead to Cholesky transform for given random variables ( W, Z ) .

Original: We apply this transform to random matrices W and Z which carry out weight matrices for prediction and a supplementary role, respectively.
Revised: We apply this transform to random matrices W and Z which carry out weight matrices for prediction and a supplementary role, respectively.

Original: We also elaborate the details of conducted experiments with additional illustrative ﬁgures and results.
Revised: We also elaborate the details of conducted experiments with additional illustrative ﬁgures and results.

Original: Particularly, we show additional classiﬁcation experiments with the MNST dataset on different noise conﬁgurations.
Revised: Particularly, we show additional classiﬁcation experiments with the MNST dataset on different noise conﬁgurations [Equation]

Original: Lemma 1 .
Revised: Lemma 1 .

Original: Let W and Z be uncorrelated random variables such that
Revised: Let W and Z be uncorrelated random variables such that

Original: [Equation]
Revised: [Equation]

Original: For a given − 1 ≤ ρ ≤1 , set
Revised: For a given − 1 ≤ ρ ≤

Original: [Equation]
Revised: [Equation]

Original: Then E ˜ Z = 0 , V [Equation] ˜
Revised: set

Original: [Equation]
Revised: [Equation]

Original:  = σ 2 Z , and Corr [Equation] W, ˜ Z )
Revised: Then E ˜ Z = 0 , V ( ˜ Z ) = σ 2 Z , and Corr ( W, ˜ Z )

Original: = ρ .
Revised: = ρ .

Original: Proof.
Revised: Proof.

Original: Since W and Z are uncorrelated, we have
Revised: Since W and Z are uncorrelated, we have

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: By equation 4, we directly obtain
Revised: By equation [Equation] we directly obtain

Original: [Equation]
Revised: [Equation]

Original: Also, by equation 4 and equation 6,
Revised: Also, by equation 4 and equation 6,

Original: [Equation]
Revised: [Equation]

Original: Similarly,
Revised: Similarly,

Original: [Equation]
Revised: [Equation]

Original: Therefore
Revised: Therefore

Original: [Equation]
Revised: [Equation]

Original: The lemma is proved.
Revised: The lemma is proved.

Original: Lemma 2 .
Revised: Lemma 2 .

Original: Assume the same condition in Lemma 1 and deﬁne ˜
Revised: Assume the same condition in Lemma 1 and deﬁne ˜

Original: Z as equation 5.
Revised: Z as equation 5.

Original: For given functions ϕ : R → R and ψ : R → (0 , ∞ ) , set ˜ W := ϕ ( ρ ) + ψ ( ρ ) ˜ Z .
Revised: For given functions ϕ : R → R and ψ : R → (0 , ∞ ) , set ˜ W := ϕ ( ρ ) + ψ ( ρ ) ˜ Z .

Original: Then
Revised: Then

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: Note that
Revised: Note that

Original: [Equation]
Revised: [Equation]

Original: Therefore, by Lemma
Revised: Therefore, by Lemma

Original: [Equation]
Revised: [Equation]

Original: Hence
Revised: Hence

Revised: Corr

Original: [Equation]
Revised: [Equation]

Original: The lemma is proved.
Revised: The lemma is proved.

Original: Now we prove the aforementioned theorem in Section 3.
Revised: Now we prove the aforementioned theorem in Section 3.

Original: Theorem.
Revised: Theorem.

Original: Let ρ =
Revised: Let ρ =

Original: ( ρ 1 , . .
Revised: ( ρ 1 , . .

Original: . , ρ K ) ∈ R K .
Revised: . , ρ K ) ∈ R K .

Original: For p ∈ { 1 , 2 } , random matrices W ( p ) ∈ R K × Q are given such that for every k ∈ { 1 , . . .
Revised: For p ∈ { 1 , 2 } , random matrices W ( p ) ∈ R K × Q are given such that for every k ∈ { 1 , . . .

Original: , K } ,
Revised: , K } ,

Original: [Equation]
Revised: [Equation]

Original: Given h
Revised: Given h

Original: =
Revised: =

Original: ( h 1 , . .
Revised: ( h 1 , . .

Original: , h Q ) ∈ R Q , set y ( p ) =
Revised: , h Q ) ∈ R Q , set y ( p ) =

Original: W ( p ) h for each p ∈ { 1 , 2 } .
Revised: W ( p ) h for each p ∈ { 1 , 2 } .

Original: Then an elementwise correlation between y (1) and y (2) equals ρ i.e.
Revised: Then an elementwise correlation between y (1) and y (2) equals ρ i.e.

Original: [Equation]
Revised: [Equation]

Original: Proof.
Revised: Proof.

Original: First we prove that for p ∈ { 1 , 2 } and k ∈ { 1 , . . .
Revised: First we prove that for p ∈ { 1 , 2 } and k ∈ { 1 , . . .

Original: , K }
Revised: , K }

Original: [Equation]
Revised: [Equation]

Original: Note that
Revised: Note that

Original: [Equation]
Revised: [Equation]

Original: By equation 
Revised: By equation 7,

Original: [Equation]
Revised: [Equation]

Original: so equation 8 is proved.
Revised: so equation 8 is proved.

Original: Next we prove
Revised: Next we prove

Original: [Equation]
Revised: [Equation]

Original: Observe that
Revised: Observe that

Original: [Equation]
Revised: [Equation]

Original: Similarly,
Revised: Similarly

Original: [Equation]
Revised: [Equation]

Original: Hence equation 9 is proved
Revised: Cov ( y (1) k [Equation] y (2) k ) =

Original: [Equation]
Revised: [Equation]

Revised: Hence equation 9 is proved.

Original: Therefore by equation 8 and equation
Revised: Therefore by equation 8 and equation

Original: [Equation]
Revised: [Equation]

Original: The theorem is proved.
Revised: The theorem is proved.

Original: Remark.
Revised: Remark.

Original: Recall the deﬁnition of Cholesky transform: for − 1 < ρ <
Revised: Recall the deﬁnition of Cholesky transform: for − 1 < ρ <

Original: T ( ρ,µ W ,σ W ,σ Z ) ( w, z )

Original: [Equation]
Revised: [Equation]

Original: Note that we do not assume W and Z should follow typical distributions.
Revised: Note that we do not assume W and Z should follow typical distributions.

Original: Hence every above theorems hold for general class of random variables.
Revised: Hence every above theorems hold for general class of random variables.

Original: Additionally, by Theorem 2 and equation 10, ˜ W has the following ρ -dependent behaviors;
Revised: Additionally, by Theorem 2 and equation 10, ˜ W has the following ρ -dependent behaviors;

Original: [Equation]
Revised: [Equation]

Original: Thus strongly correlated weights ˜ W i.e. ρ ≈ 1 , provide prediction with conﬁdence while uncorrelated weights encompass uncertainty.
Revised: Thus strongly correlated weights ˜ W i.e. ρ ≈ 1 , provide prediction with conﬁdence while uncorrelated weights encompass uncertainty.

Original: These different behaviors of weights perform regularization and preclude over-ﬁtting caused by bad data since uncorrelated and negative correlated weights absorb vague and outlier pattern, respectively.
Revised: These different behaviors of weights perform regularization and preclude over-ﬁtting caused by bad data since uncorrelated and negative correlated weights absorb vague and outlier pattern, respectively.

Original: B A DDITIONAL E XPERIMENTS B. R EGRESSION T ASKS B.1. S YNTHETIC E XAMPLE
Revised: B A DDITIONAL E XPERIMENTS B. R EGRESSION T ASKS B.1. S YNTHETIC E XAMPLE

Original: We provide more ﬁtting results for the synthetic example in Figure 4.
Revised: We provide more ﬁtting results for the synthetic example in Figure 4.

Original: Given an intact dataset, all compared methods robustly ﬁt the given training data.
Revised: Given an intact dataset, all compared methods robustly ﬁt the given training data.

Original: However, other methods fail to correctly ﬁt the underlying target function given corrupt data.
Revised: However, other methods fail to correctly ﬁt the underlying target function given corrupt data.

Original: When the outlier rate exceeds 90% all tested methods fail to ﬁt.
Revised: When the outlier rate exceeds 90% all tested methods fail to ﬁt.

Original: B.1. A UTONOMOUS D RIVING E XPERIMENT
Revised: B.1. A UTONOMOUS D RIVING E XPERIMENT

Original: Autonomous Driving Experiment
Revised: Autonomous Driving Experiment

Original: In this experiment, we apply ChoiceNet to a autonomous driving scenario in a simulated environment.
Revised: In this experiment, we apply ChoiceNet to a autonomous driving scenario in a simulated environment.

Original: In particular, the tested methods are asked to learn the  policy from driving demonstrations collected from both safe and careless driving modes.
Revised: In particular, the tested methods are asked to learn the [Table] [Table] policy from driving demonstrations collected from both safe and careless driving modes.

Original: We use the same set of methods used for the previous task.
Revised: We use the same set of methods used for the previous task.

Original: The policy function is deﬁned as a mapping between four dimensional input features consist of three frontal distances to left, center, and right lanes and lane deviation distance from the center of the lane to the desired heading.
Revised: The policy function is deﬁned as a mapping between four dimensional input features consist of three frontal distances to left, center [Table] and right lanes and lane deviation distance from the center of the lane to the desired heading.

Original: Once the desired heading is computed, the angular velocity of a car is computed by 10 ∗ ( θ desired − θ current ) and the directional velocity is ﬁxed to 10 m/s .
Revised: Once the desired heading is computed, the angular velocity of a car is computed by 10 ∗ ( θ desired − θ [Table] ) and the directional velocity is ﬁxed to 10 m/s .

Original: The driving demonstrations are collected from keyboard inputs by human users.
Revised: The driving demonstrations are collected from keyboard inputs by human users.

Original: The objective of this experiment is to assess its performance on a training set generated from[Table] [Table] two different distributions.
Revised: The objective of this experiment is to assess its performance on a training set generated from two different distributions.

Original: We would like to note that this task does not have a reference target function in that all demonstrations are collected manually.
Revised: We would like to note that this task does not have a reference target function in that all demonstrations are collected manually.

Original: Hence, we evaluated the performances of the compared methods by running the trained policies on a straight track by randomly deploying static cars.
Revised: Hence, we evaluated the performances of the compared methods by running the trained policies on a straight track by randomly deploying static cars.

Original: Table 5 and Table 6 indicate collision rates and RMS lane deviation distances of the tested methods, respectively, where the statistics are computed from 50 independent runs on the straight lane by randomly placing static cars as shown in Figure 5.
Revised: Table 5 and Table 6 indicate collision rates and RMS lane deviation distances of the tested methods, respectively, where the statistics are computed from 50 independent runs on the straight lane by randomly placing static cars as shown in Figure 5.

Original: ChoiceNet clearly outperforms compared methods in terms of both safety (low collision rates) and stability (low RMS lane deviation distances).
Revised: ChoiceNet clearly outperforms compared methods in terms of both safety (low collision rates) and stability (low RMS lane deviation distances).

Original: Here, we describe the features used for the autonomous driving experiments.
Revised: Here, we describe the features used for the autonomous driving experiments.

Original: As shown in the manuscript, we use a four dimensional feature, a lane deviation distance of an ego car, and three frontal distances to the closest car at left, center, and right lanes as shown in Figure 6.
Revised: As shown in the manuscript, we use a four dimensional feature, a lane deviation distance of an ego car, and three frontal distances to the closest car at left, center, and right lanes as shown in Figure 6.

Original: We upperbound the frontal distance to 40 m .
Revised: We upperbound the frontal distance to 40 m .

Original: Figure 7(a) and 7(b) illustrate manually collected trajectories of a safe driving mode and a careless driving mode.
Revised: Figure 7(a) and 7(b) illustrate manually collected trajectories of a safe driving mode and a careless driving mode.

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Revised: A BLATION S TUDY ON MNIST

Original: Above ﬁgures show the results of ablation study when varying the number of mixture K and the expected measurement variance 
Revised: Above ﬁgures show the results of ablation study when varying the number of mixture K and the expected measurement variance τ − 1 .

Original: − where 90% of train labels are randomly shufﬂed  right  synthetic one-dimensional regression problem 
Revised: Left two ﬁgures indicate test accuracies using the MNIST dataset where 90% of train labels are randomly shufﬂed and right two ﬁgures are RMSEs using a synthetic one-dimensional regression problem in Section 4.1.

Original: Section We observe that having bigger K is beneﬁcial to  accuracies.
Revised: We observe that having bigger K is beneﬁcial to the classiﬁcation accuracies.

Original: In fact, the results achieved here with K equals  better than  ones reported in  submitted manuscript
Revised: In fact, the results achieved here with K equals 15 andare better than the ones reported in the submitted manuscript.

Original:  1 does not affect much unless  is exceedingly large [Figure]
Revised: τ − 1 does not affect much unless it is exceedingly large.

Original:  D IFFERENT TYPES OF N OISE ON MNIST
Revised: B.2. D IFFERENT TYPES OF N OISE ON MNIST

Original: Here, we present additional experimental results using the MNIST dataset on following three different scenarios:
Revised: Here, we present additional experimental results using the MNIST dataset on following three different scenarios:

Original: Biased label experiments where we randomly assign the percentage of the training labels to label 0 .
Revised: Biased label experiments where we randomly assign the percentage of the training labels to label 0 .

Original: Random shufﬂe experiments where we randomly replace the percentage of the training labels from the uniform multinomial distribution.
Revised: Random shufﬂe experiments where we randomly replace the percentage of the training labels from the uniform multinomial distribution.

Original: Random permutation experiments where we replace the percentage of the labels based on the label permutation matrix where we follow the random permutation in (Reed et al., 2014).
Revised: Random permutation experiments where we replace the percentage of the labels based on the label permutation matrix where we follow the random permutation in (Reed et al., 2014).

Original: The best and ﬁnal accuracies on the intact test dataset for biased label experiments are shown in
Revised: The best and ﬁnal accuracies on the intact test dataset for biased label experiments are shown in

Original: Table 7.
Revised: Table 7.

Original: In all corruption rates, ChoiceNet achieves the best performance compared to two baseline methods.
Revised: In all corruption rates, ChoiceNet achieves the best performance compared to two baseline methods.

Original: The learning curves of the biased label experiments are depicted in Figure 8.
Revised: The learning curves of the biased label experiments are depicted in Figure 8.

Original: Particularly, we observe unstable learning curves regarding the test accuracies of ConvNet and Mixup.
Revised: Particularly, we observe unstable learning curves regarding the test accuracies of ConvNet and Mixup.

Original: As training accuracies of such methods show stable learning behaviors, this can be interpreted as the networks are simply memorizing noisy labels.
Revised: As training accuracies of such methods show stable learning behaviors, this can be interpreted as the networks are simply memorizing noisy labels.

Original: In the contrary, the learning curves of ChoiceNet show stable behaviors which clearly indicates the robustness of the proposed method. 
Revised: In the contrary, the learning curves of ChoiceNet show stable behaviors which clearly indicates the robustness of the proposed method. [Table]

Original:  The experimental results and learning curves of the random shufﬂe experiments are shown in Tableand Figure 9.
Revised: [Table] The experimental results and learning curves of the random shufﬂe experiments are shown in Tableand Figure 9.

Original: The convolutional neural networks trained with Mixup show robust learning behaviors when 80% of the training labels are uniformly shufﬂed.
Revised: The convolutional neural networks trained with Mixup show robust learning behaviors when 80% of the training labels are uniformly shufﬂed.

Original: However, given an extremely noisy dataset ( 90% and 95% ), the test accuracies of baseline methods decrease as the number of epochs increases.
Revised: However, given an extremely noisy dataset ( 90% and 95% ), the test accuracies of baseline methods decrease as the number of epochs increases.

Original: ChoiceNet shows outstanding robustness to the noisy dataset in that the test accuracies do not drop even after 50 epochs for the cases where the corruption rates are below 90% .
Revised: ChoiceNet shows outstanding robustness to the noisy dataset in that the test accuracies do not drop even after 50 epochs for the cases where the corruption rates are below 90% .

Original: For the 95% case, however, over-ﬁtting is occured in all methods.
Revised: For the 95% case, however, over-ﬁtting is occured in all methods.

Original: Table 9 and Figure 10 illustrate the results of the random permutation experiments.
Revised: Table 9 and Figure 10 illustrate the results of the random permutation experiments.

Original: Specifically, we change the labels of randomly selected training data using a permutation rule:

Original: [Table]

Original: [Table]

Original: [Table]

Original: [Table] [Table] [Table] 9) → (7 , 9 , 0 , 4 , 2 , 1 , 3 , 5 , 6 , 8) following (Reed et al., 2014).
Revised: Specifically, we change the labels of randomly selected training data using a permutation rule: (0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9) → (7 , 9 , 0 , 4 , 2 , 1 , 3 , 5 , 6 , 8) following (Reed et al., 2014).

Original: We argue that this setting is more arduous than the random shufﬂe case in that we are intentionally changing the labels based on predeﬁned permutation rules.[Figure] [Figure] [Figure]
Revised: We argue that this setting is more arduous than the random shufﬂe case in that we are intentionally changing the labels based on predeﬁned permutation rules.

Original: B.2. CIFAR- 
Revised: B.2. CIFAR-

Original: Here, we present detailed learning curves of the CIFAR-10 experiments while varying the noise level from 20% to 80% following the conﬁgurations in (Zhang et al., 2017). 
Revised: Here, we present detailed learning curves of the CIFAR-10 experiments while varying the noise level from 20% to 80% following the conﬁgurations in (Zhang et al., 2017). [Table] [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Revised: [Figure]