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+# IS ATTENTION BETTER THAN MATRIX DECOMPOSITION?
+
+Zhengyang $\mathbf { G e n g ^ { 1 , 2 } }$ , Meng-Hao $\mathbf { G u o ^ { 3 } }$ ∗, Hongxu Chen4, Xia $\mathbf { L i } ^ { 2 }$ , Ke Wei4, Zhouchen $\mathbf { L i n ^ { 2 , 5 } }$ † 1Zhejiang Lab; 2Key Lab. of Machine Perception (MoE), School of EECS, Peking University; 3Tsinghua University; 4School of Data Science, Fudan University; 5Pazhou Lab
+
+# ABSTRACT
+
+As an essential ingredient of modern deep learning, attention mechanism, especially self-attention, plays a vital role in the global correlation discovery. However, is hand-crafted attention irreplaceable when modeling the global context? Our intriguing finding is that self-attention is not better than the matrix decomposition (MD) model developed 20 years ago regarding the performance and computational cost for encoding the long-distance dependencies. We model the global context issue as a low-rank completion problem and show that its optimization algorithms can help design global information blocks. This paper then proposes a series of Hamburgers, in which we employ the optimization algorithms for solving MDs to factorize the input representations into sub-matrices and reconstruct a low-rank embedding. Hamburgers with different MDs can perform favorably against the popular global context module self-attention when carefully coping with gradients back-propagated through MDs. Comprehensive experiments are conducted in the vision tasks where it is crucial to learn the global context, including semantic segmentation and image generation, demonstrating significant improvements over self-attention and its variants. Code is available.
+
+# 1 INTRODUCTION
+
+Since self-attention and transformer (Vaswani et al., 2017) showed significant advantages over recurrent neural networks and convolutional neural networks in capturing long-distance dependencies, attention has been widely adopted by computer vision (Wang et al., 2018; Zhang et al., 2019a) and natural language processing (Devlin et al., 2019) for global information mining. However, is hand-crafted attention irreplaceable when modeling the global context?
+
+This paper focuses on a new approach to design global context modules. The key idea is, if we formulate the inductive bias like the global context into an objective function, the optimization algorithm to minimize the objective function can construct a computational graph, i.e., the architecture we need in the networks. We particularize this idea by developing a counterpart for the most representative global context module, self-attention. Considering extracting global information in the networks as finding a dictionary and the corresponding codes to capture the inherent correlation, we model the context discovery as low-rank completion of the input tensor and solve it via matrix decomposition. This paper then proposes a global correlation block, Hamburger, by employing matrix decomposition to factorize the learned representation into sub-matrices so as to recover the clean low-rank signal subspace. The iterative optimization algorithm to solve matrix decomposition defines the central computational graph, i.e., Hamburger’s architecture.
+
+Our work takes advantage of the matrix decomposition models as the foundation of Hamburger, including Vector Quantization (VQ) (Gray & Neuhoff, 1998), Concept Decomposition (CD) (Dhillon & Modha, 2001), and Non-negative Matrix Factorization (NMF) (Lee & Seung, 1999). Additionally, instead of directly applying Back-Propagation Through Time (BPTT) algorithm (Werbos et al., 1990) to differentiate the iterative optimization, we adopt a truncated BPTT algorithm, i.e., one-step gradient, to back-propagate the gradient effectively. We illustrate the advantages of Hamburger in the fundamental vision tasks where global information has been proven crucial, including semantic segmentation and image generation. The experiments prove that optimization-designed Hamburger can perform competitively with state-of-the-art attention models when avoiding the unstable gradient back-propagated through the iterative computational graph of MD. Hamburger sets new state-ofthe-art records on the PASCAL VOC dataset (Everingham et al., 2010) and PASCAL Context dataset (Mottaghi et al., 2014) for semantic segmentation and surpasses existing attention modules for GANs in the large scale image generation on ImageNet (Deng et al., 2009).
+
+The contributions of this paper are listed as follows:
+
+• We show a white-box approach to design global information blocks, i.e., by turning the optimization algorithm that minimizes an objective function, in which modeling the global correlation is formulated as a low-rank completion problem, into the architecture. We propose Hamburger, a light yet powerful global context module with ${ \mathcal { O } } ( n )$ complexity, surpassing various attention modules on semantic segmentation and image generation. We figure out that the main obstacle of applying MD in the networks is the unstable backward gradient through its iterative optimization algorithm. As a pragmatic solution, the proposed one-step gradient facilitates the training of Hamburger with MDs.
+
+# 2 METHODOLOGY
+
+# 2.1 WARM UP
+
+Since matrix decomposition is pivotal to the proposed Hamburger, we first review the idea of matrix decomposition. A common view is that matrix decomposition factorizes the observed matrix into a product of several sub-matrices, e.g., Singular Value Decomposition. However, a more illuminating perspective is that, by assuming the generation process, matrix decomposition acts as the inverse of the generation, disassembling the atoms that make up the complex data. From the reconstruction of the original matrices, matrix decomposition recovers the latent structure of observed data.
+
+Suppose that the given data are arranged as the columns of a large matrix $\pmb { X } = [ \pmb { x } _ { 1 } , \dots , \pmb { x } _ { n } ] \in \mathbb { R } ^ { d \times n }$ . A general assumption is that there is a low-dimensional subspace, or a union of multiple subspaces hidden in $\boldsymbol { X }$ . That is, there exists a dictionary matrix $D = [ \bar { \bf d } _ { 1 } , \boldsymbol { \cdot } \cdot \cdot , { \bf d } _ { r } ] \in \mathbb { R } ^ { d \times r }$ and corresponding codes $C = [ \mathbf { c } _ { 1 } , \cdot \cdot \cdot , \mathbf { c } _ { n } ] \in \mathbb { R } ^ { r \times n }$ that $\boldsymbol { X }$ can be expressed as
+
+$$
+X = \overbrace { \bar { X } + E = D C } ^ { g e n e r a t i o n } + E ,
+$$
+
+where $\bar { \boldsymbol { X } } \in \mathbb { R } ^ { d \times n }$ is the output low-rank reconstruction, and $\pmb { { \cal E } } \in \mathbb { R } ^ { d \times n }$ is the noise matrix to be discarded. Here we assume that the recovered matrix $\bar { X }$ has the low-rank property, such that
+
+$$
+\operatorname { r a n k } ( { \bar { X } } ) \leq \operatorname* { m i n } ( \operatorname { r a n k } ( D ) , \operatorname { r a n k } ( C ) ) \leq r \ll \operatorname* { m i n } ( d , n ) .
+$$
+
+Different MDs can be derived by assuming structures to matrices $\mathbf { \delta } _ { D , C }$ , and $\pmb { { \cal E } }$ (Kolda & Bader, 2009; Udell et al., 2016). MD is usually formulated as an objective with various constraints and then solved by optimization algorithms, with classic applications to image denoising (Wright et al., 2009; Lu et al., 2014), inpainting (Mairal et al., 2010), and feature extraction (Zhang et al., 2012).
+
+# 2.2 PROPOSED METHOD
+
+We focus on building global context modules for the networks without painstaking hand-crafted design. Before starting our discussion, we review the representative hand-designed context block self-attention pithily.
+
+The attention mechanism aims at finding a group of concepts for further conscious reasoning from massive unconscious context (Xu et al., 2015; Bengio, 2017; Goyal et al., 2019). As a representative, self-attention (Vaswani et al., 2017) is proposed for learning long-range dependencies in machine translation,
+
+$$
+{ \mathrm { A t t e n t i o n } } \left( Q , K , V \right) = { \mathrm { s o f t m a x } } \left( { \frac { Q K ^ { \top } } { \sqrt { d } } } \right) V ,
+$$
+
+where $Q , K , V \in \mathbb { R } ^ { n \times d }$ are features projected by linear transformations from the input. Selfattention extracts global information via attending all tokens at a time rather than the typical one-byone processing of recurrent neural networks.
+
+![](images/f4b8507f6011e66a2d30ba95d9361b6126ed0a6cc21d7971535b8e125c610ec2.jpg)  
+Figure 1: Overview of Hamburger
+
+Though self-attention and its variants achieved great success, researchers are confronted with (1) developing new global context modules based on self-attention, typically via hand-crafted engineering, and (2) explaining why current attention models work. This paper bypasses both issues and finds a method to easily design global context modules via a well-defined white-box toolkit. We try to formulate the human inductive bias, like the global context, as an objective function and use the optimization algorithm to solve such a problem to design the module’s architecture. The optimization algorithm creates a computational graph, takes some input, and finally outputs the solution. We apply the computational graph of optimization algorithms for the central part of our context module.
+
+Based on this approach, we need to model the networks’ global information issue as an optimization problem. Take the convolutional neural networks (CNN) as an example for further discussion. The networks output a tensor $\mathcal { X } \in \mathbb { R } ^ { C \times H \times W }$ after we feed into an image. Since the tensor can be seen as a set of $H W C$ -dimensional hyper-pixels, we unfold the tensor into a matrix $\pmb { X } \in \mathbb { R } ^ { C \times H W }$ . When the module learns the long-range dependencies or the global context, the hidden assumption is that the hyper-pixels are inherently correlated. For the sake of simplicity, we assume that hyper-pixels are linearly dependent, which means that each hyper-pixel in $\boldsymbol { X }$ can be expressed as the linear combination of bases whose elements are typically much less than $H W$ . In the ideal situation, the global information hidden in $\boldsymbol { X }$ can be low-rank. However, due to vanilla CNN’s poor ability to model the global context (Wang et al., 2018; Zhang et al., 2019a), the learned $\boldsymbol { X }$ is usually corrupted with redundant information or incompleteness. The above analysis suggests a potential method to model the global context, i.e., by completing the low-rank part $\bar { X }$ in the unfolded matrix $\boldsymbol { X }$ and discarding the noise part $\pmb { \cal E }$ , using the classic matrix decomposition models described in Eq. (1), which filters out the redundancy and incompleteness at the same time. We thus model learning the global context as a low-rank completion problem with matrix decomposition as its solution. Using the notion of Sec. 2.1, the general objective function of matrix decomposition is
+
+$$
+\operatorname* { m i n } _ { D , C } \mathcal { L } ( X , D C ) + \mathcal { R } _ { 1 } ( D ) + \mathcal { R } _ { 2 } ( C )
+$$
+
+where $\mathcal { L }$ is the reconstruction loss, $\mathcal { R } _ { 1 }$ and $\mathcal { R } _ { 2 }$ are regularization terms for the dictionary $_ D$ and the codes $C$ . Denote the optimization algorithm to minimize Eq. (4) as $\mathcal { M } . \mathcal { M }$ is the core architecture we deploy in our global context module. To help readers further understand this modeling, We also provide a more intuitive illustration in Appendix G.
+
+In the later sections, we introduce our global context block, Hamburger, and then discuss detailed MD models and optimization algorithms for $\mathcal { M }$ . Finally, we handle the gradient issue for back-propagation through matrix decomposition.
+
+# 2.2.1 HAMBURGER
+
+Hamburger consists of one slice of “ham” (matrix decomposition) and two slices of “bread” (linear transformation). As the name implies, Hamburger first maps the input $\boldsymbol { Z } \in \mathbb { R } ^ { d _ { z } \times n }$ into feature space with a linear transformation $W _ { l } ^ { ' } \in \mathbb { R } ^ { d \times d _ { z } }$ , namely “lower bread”, then uses matrix decomposition $\mathcal { M }$ to solve a low-rank signal subspace, corresponding to the “ham”, and finally transforms extracted signals into the output with another linear transformation $W _ { u } \in \mathbb { R } ^ { d _ { z } \times d }$ , called “upper bread”,
+
+$$
+\begin{array} { r } { \mathcal { H } ( Z ) = W _ { u } \mathcal { M } ( W _ { l } Z ) , } \end{array}
+$$
+
+where $\mathcal { M }$ is matrix decomposition to recover the clear latent structure, functioning as a global nonlinearity. Detailed architectures of $\mathcal { M }$ , i.e., optimization algorithms to factorize $\boldsymbol { X }$ , are discussed in Sec. 2.2.2. Fig. 1 describes the architecture of Hamburger, where it collaborates with the networks via Batch Normalization (BN) (Ioffe & Szegedy, 2015), a skip connection, and finally outputs $\mathbf { Y }$ ,
+
+$$
+\begin{array} { r } { Y = Z + \mathrm { B N } ( \mathcal { H } ( Z ) ) . } \end{array}
+$$
+
+# 2.2.2 HAMS
+
+This section describes the structure of “ham”, i.e., $\mathcal { M }$ in Eq. (5). As discussed in the previous section, by formulating the global information discovery as an optimization problem of MD, algorithms to solve MD naturally compose $\mathcal { M } , \mathcal { M }$ takes the output of “lower bread” as its input and computes a low-rank reconstruction as its output, denoted as $\boldsymbol { X }$ and $\bar { X }$ , respectively.
+
+$$
+\mathcal { M } ( X ) = \bar { X } = D C .
+$$
+
+We investigate two MD models for $\mathcal { M }$ , Vector Quantization (VQ), and Non-negative Matrix Factorization (NMF) to solve $_ D$ and $C$ and reconstruct $\bar { X }$ , while leaving Concept Decomposition (CD) to Appendix B. The selected MD models are introduced briefly because we endeavor to illustrate the importance of the low-rank inductive bias and the optimization-driven designing method for global context modules rather than any specific MD models. It is preferred to abstract the MD part as a whole, i.e., $\mathcal { M }$ in the context of this paper, and focus on how Hamburger can show the superiority in its entirety.
+
+Vector Quantization Vector Quantization (VQ) (Gray & Neuhoff, 1998), a classic data compression algorithm, can be formulated as an optimization problem in term of matrix decomposition:
+
+$$
+\operatorname* { m i n } _ { D , C } \| X - D C \| _ { F } \quad { \mathrm { s . t . ~ } } \mathbf { c } _ { i } \in \{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } , \cdot \cdot \cdot , \mathbf { e } _ { r } \} ,
+$$
+
+where $e _ { i }$ is the canonical basis vector, $\mathbf { e } _ { i } = [ 0 , \cdots , 1 , \cdots , 0 ] ^ { \top }$ . The solution to minimize the ith   
+objective in Eq. (8) is K-means (Gray & Neuhoff, 1998). However, to ensure that VQ is differentiable, we replace the hard arg min and Euclidean distance with sof tmax and cosine similarity, leading to Alg. 1, where cosine(D, X) is a similarity matrix whose entries satisfy cosine(D, X)ij = d>i xjkdkkxk , and sof tmax is applied column-wise and $T$ is the temperature. Further we can obtain a hard assignment by a one-hot vector when $T  0$ .
+
+<table><tr><td>Algorithm 1 Ham: Soft VQ</td></tr><tr><td>Input X. Initialize D, C.</td></tr><tr><td>for k from 1 to K do</td></tr><tr><td>C ← softmax(⊥cosine(D,X))</td></tr><tr><td>D ← XCTdiag(C1n)-1</td></tr><tr><td>end for</td></tr><tr><td>Output X = DC.</td></tr></table>
+
+<table><tr><td>Algorithm2Ham:NMF with MU</td></tr><tr><td>Input X. Initialize non-negative D, C</td></tr><tr><td>for k from 1 to K do (DTX）ij</td></tr><tr><td>Cij←Cij (DT DC)ij</td></tr><tr><td>(xCT）ij Dij←Dij</td></tr><tr><td>(DCCT)ij end for</td></tr><tr><td>Output X = DC.</td></tr></table>
+
+Non-negative Matrix Factorization If we impose non-negative constraints on the dictionary $_ { D }$ and the codes $C$ , it leads to Non-negative Matrix Factorization (NMF) (Lee & Seung, 1999):
+
+$$
+\operatorname* { m i n } _ { D , C } \| X - D C \| _ { F } \quad \mathrm { s . t . } D _ { i j } \geq 0 , C _ { j k } \geq 0 .
+$$
+
+To satisfy the non-negative constraints, we add a ReLU non-linearity before putting $\boldsymbol { X }$ into NMF. We apply the Multiplicative Update (MU) rules (Lee & Seung, 2001) in Alg. 2 to solve NMF, which guarantees the convergence.
+
+As white-box global context modules, VQ, CD, and NMF are straightforward and light, showing remarkable efficiency. They are formulated into optimization algorithms that mainly consist of matrix multiplications with the complexity $\mathcal { O } ( n d r )$ , much cheaper than complexity $\mathcal { O } ( n ^ { 2 } \bar { d } )$ in self-attention as $r \ll n$ . All three MDs are memory-friendly since they avoid generating a large $n \times n$ matrix as an intermediate variable, like the product of $Q$ and $\kappa$ of self-attention in Eq. (3). In the later section, our experiments prove MDs are at least on par with self-attention, though the architectures of $\mathcal { M }$ are created by optimization and look different from classic dot product self-attention.
+
+# 2.3 ONE-STEP GRADIENT
+
+Since $\mathcal { M }$ involves an optimization algorithm as its computational graph, a crux to fuse it into the networks is how the iterative algorithm back-propagates gradient. The RNN-like behavior of optimization suggests Back-Propagation Through Time (BPTT) algorithm (Werbos et al., 1990) as the standard choice to differentiate the iterative process. We first review the BPTT algorithm below. However, in practice, the unstable gradient from BPTT does harm Hamburger’s performances. Hence we build an abstract model to analyze the drawbacks of BPTT and try to find a pragmatic solution while considering MD’s nature as an optimization algorithm.
+
+As shown in Fig. 2, x, y and $\mathbf { h } ^ { t }$ denote input, output and intermediate result at time step $t$ , respectively, while $\mathcal { F }$ and $\mathcal { G }$ are operators. At each time step, the model receives the same input $\mathbf { x }$ processed by the underlying networks.
+
+$$
+\begin{array} { r } { \mathbf { h } ^ { t + 1 } = \mathcal { F } ( \mathbf { h } ^ { t } , \mathbf { x } ) . } \end{array}
+$$
+
+The intermediate results $\mathbf { h } ^ { i }$ are all discarded. Only the output of the last step $\mathbf { h } ^ { t }$ is passed through $\mathcal { G }$ for output $\mathbf { y }$ ,
+
+$$
+\mathbf { y } = \mathcal { G } ( \mathbf { h } ^ { t } ) .
+$$
+
+In the BPPT algorithm, the gradient from output $\mathbf { y }$ to input $\mathbf { x }$ is given, according to the Chain rule:
+
+$$
+\frac { \partial \mathbf { y } } { \partial \mathbf { x } } = \sum _ { i = 0 } ^ { t - 1 } \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { t } } \left( \prod _ { j = t - i } ^ { t - 1 } \frac { \partial \mathbf { h } ^ { j + 1 } } { \partial \mathbf { h } ^ { j } } \right) \frac { \partial \mathbf { h } ^ { t - i } } { \partial \mathbf { x } } .
+$$
+
+![](images/5ab6d727f0cd57f41588f1cd83d1aa6d9c71c62cfac4892ef03dfa8bdeb4c6b9.jpg)  
+Figure 2: One-step Gradient
+
+A thought experiment is to consider $t \to \infty$ , leading to a fully converged result $\mathbf { h } ^ { * }$ and infinite terms in Eq. (12). We suppose
+
+that both $\mathcal { F }$ and $\mathcal { G }$ are Lipschitz with constants $L _ { h }$ w.r.t. h, $L _ { x } w . r . t . \textbf { x }$ , and $L _ { \mathcal { G } }$ , and $L _ { h } < 1$ . Note that these assumptions apply to a large number of optimization or numerical methods. Then we have:
+
+Proposition 1 $\{ \mathbf { h } ^ { i } \} _ { t }$ has linear convergence.
+
+Proposition 2
+
+Table 1: One-step Gradient & BPTT   
+
+<table><tr><td>Method</td><td>One-step</td><td>BPTT</td></tr><tr><td>VQ</td><td>77.7(77.4)</td><td>76.6(76.3)</td></tr><tr><td>CD</td><td>78.1(77.5)</td><td>75.0(74.6)</td></tr><tr><td>NMF</td><td>78.3(77.8)</td><td>77.4(77.0)</td></tr></table>
+
+$$
+\begin{array} { r l } & { \underset { t  \infty } { \operatorname* { l i m } } \frac { \partial \mathbf { y } } { \partial \mathbf { x } } = \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { * } } ( I - \frac { \partial \mathcal { F } } { \partial \mathbf { h } ^ { * } } ) ^ { - 1 } \frac { \partial \mathcal { F } } { \partial \mathbf { x } } . } \\ & { \underset { t  \infty } { \operatorname* { l i m } } \Vert \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { 0 } } \Vert = 0 , \underset { t  \infty } { \operatorname* { l i m } } \Vert \frac { \partial \mathbf { y } } { \partial \mathbf { x } } \Vert \leq \frac { L _ { \mathcal { G } } L _ { x } } { 1 - L _ { h } } . } \end{array}
+$$
+
+Proposition 3
+
+It is easy to incur gradient vanishing w.r.t. $\mathbf { h } ^ { 0 }$ when $L _ { h }$ is close to 0 and gradient explosion w.r.t. $\mathbf { x }$ $\begin{array} { r } { ( { I - \frac { \partial \mathcal { F } } { \partial { \bf h ^ { * } } } } ) ^ { - 1 } } \end{array}$ when $L _ { h }$ is close to 1. The Jacobian matrix when the largest eigenvalue of $\textstyle { \frac { \partial { \mathcal { F } } } { \partial \mathbf { h } } }$ , i.e., the Lipschitz constant of ∂x , moreover, suffers from an ill-conditioned term $\mathcal { F }$ w.r.t. h, approaches 1 and its minimal eigenvalue typically stays near 0, thus restricts the capability of the gradient to search the well-generalized solution in the parameter space. The erratic scale and spectrum of the gradient back through the optimization algorithm indicate the infeasibility to apply BPTT to Hamburger directly, corroborated by the experiments in Tab. 1, using the same ablation settings as Sec. 3.1.
+
+The analysis inspires us a possible solution. Note that there are a multiplication of multiple Jacobian matrices ∂hj∂hj−1 and a summation of an infinite series in BPTT algorithm, leading to uncontrollable scales of gradients. It enlightens us to drop some minor terms in the gradient while preserving its dominant terms to ensure the direction is approximately right. Considering terms of Eq. (12) as a series, i.e., $\begin{array} { r } { \{ \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { t } } \left( \prod _ { j = t - i } ^ { t - 1 } \frac { \partial \mathbf { h } ^ { j + 1 } } { \partial \mathbf { h } ^ { j } } \right) \frac { \partial \mathbf { h } ^ { t - i } } { \partial \mathbf { x } } \} _ { i } } \end{array}$ , it makes sense to use the first term of this series to approximate the gradient if the scale of its terms decays exponentially measured by the operator norm. The first term of the gradient is from the last step of optimization, leading to the one-step gradient,
+
+$$
+\widehat { \frac { \partial \mathbf { y } } { \partial \mathbf { x } } } = \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { t } } \frac { \partial \mathbf { h } ^ { t } } { \partial \mathbf { x } } .
+$$
+
+The one-step gradient is a linear approximation of the BPTT algorithm when $t \to \infty$ according to the Proposition 2. It is easy to implement, requiring a no_grad operation in PyTorch (Paszke et al., 2019) or stop_gradient operation in TensorFlow (Abadi et al., 2016) and reducing the time and space complexity from $\mathcal { O } ( t )$ in BPTT to $\mathcal { O } ( 1 )$ . We test adding more terms to the gradient but its performance is worse than using one step. According to experimental results, one-step gradient is acceptable to back-propagate gradient through MDs.
+
+Table 2: Ablation on components of Hamburger with NMF Ham.   
+
+<table><tr><td>Method</td><td>mIoU(%)</td><td>Params</td></tr><tr><td>baseline</td><td>75.9(75.7)</td><td>32.67M</td></tr><tr><td>basic</td><td>78.3(77.8)</td><td>+0.50M</td></tr><tr><td> - ham</td><td>75.8(75.6)</td><td>+0.50M</td></tr><tr><td>- upper bread</td><td>77.0(76.8)</td><td>+0.25M</td></tr><tr><td>- lower bread</td><td>77.3(77.2)</td><td>+0.25M</td></tr><tr><td>only ham</td><td>77.0(76.8)</td><td>+0M</td></tr></table>
+
+# 3 EXPERIMENTS
+
+In this section we present experimental results demonstrating the techniques described above. Two vision tasks that benefit a lot from global information and attention mechanism attract us, including semantic segmentation (over 50 papers using attention) and deep generative models like GANs (most state-of-the-art GANs adopt self-attention since SAGAN (Zhang et al., 2019a)). Both tasks are highly competitive and thus enough for comparing Hamburger with self-attention. Ablation studies show the importance of MD in Hamburger as well as the necessity of the one-step gradient. We emphasize the superiority of Hamburger on modeling global context over self-attention regarding both performance and computational cost.
+
+# 3.1 ABLATION EXPERIMENTS
+
+We choose to conduct all ablation experiments on the PASCAL VOC dataset (Everingham et al., 2010) for semantic segmentation, and report mIoU of 5 runs on the validation set in the form of best(mean). ResNet-50 (He et al., 2016) with output stride 16 is the backbone for all ablation experiments. We employ a $3 \times 3$ conv with BN (Ioffe & Szegedy, 2015) and ReLU to reduce channels from 2048 to 512 and then add Hamburger, the same location as popular attentions in semantic segmentation. For detailed training settings, please see Appendix E.1.
+
+![](images/24419391ec90442015c0b287b0277f4cc7ec68ebb6455c8e2d42b50f8207a791.jpg)  
+Figure 3: Ablation on $d$ and $r$
+
+Breads and Hams We ablate each part of the Hamburger. Removing MD (ham) causes the most severe decay in performance, attesting to the importance of MD. Even if only the parameter-free MD is added (only ham), the performance can visibly improve. Parameterization also helps the Hamburger process the extracted features. Bread, especially upper bread, contributes considerable performance.
+
+Latent Dimension $d$ and $r$ It is worth noting that there is no simple linear relation between $d$ and $r$ with performances measured by mIoU, though $d = 8 r$ is a satisfactory choice. Experiments show that even $r = 8$ performs well, revealing that it can be very cheap for modeling the global context.
+
+![](images/56fcdfe0681b4e23dc01a7fc1b5dd2f92c91af7f0ab029fc8d3476e531ab88aa.jpg)  
+Figure 4: Ablation on $K$
+
+Iterations $K$ We test more optimization steps in the evaluation stage. In general, the same $K$ for training and test is recommended. $K = 6$ is enough for CD and NMF, while even $K = 1$ is acceptable for VQ. Typically $3 { \sim } 6$ steps are enough since simple MD’s prior is still biased, and full convergence can overfit it. The few iterations are cheap and act as early stopping.
+
+# 3.2 A CLOSE LOOK AT HAMBURGER
+
+To understand the behavior of Hamburger in the networks, we visualize the spectrums of representations before and after Hamburger on the PASCAL VOC validation set. The input and output tensors are unfolded to $\mathbb { R } ^ { C \times H W }$ . The accumulative ratio of squared largest $r$ singular values over total squared singular values of the unfolded matrix has been shown in Fig. 5. A truncated spectrum is usually observed in classic matrix decomposition models’ results due to the low-rank reconstruction. In the networks, Hamburger also promotes energy concentration while preserving informative details via the skip connection. Additionally, we visualize the feature maps before and after Hamburger in Fig. 6. MD helps Hamburger learn interpretable global information by zeroing out uninformative channels, removing irregular noises, and completing details according to the context.
+
+![](images/f95a516bc0b67b123fc10b829a93b9a88723b677beefb69a3c882a275803e995.jpg)  
+Figure 5: Accumulative Ratio
+
+![](images/de1972c49d4bb1cb288ed917e599d9f10e60ec02432255c0fb5ea6f5af8b179a.jpg)  
+Figure 6: Visualization of feature maps
+
+# 3.3 A COMPARISON WITH ATTENTION
+
+This section shows the advantages of MD-based Hamburger over attention-related context modules in computational cost, memory consumption, and inference time. We compare Hamburger (Ham) with self-attention (SA) (Vaswani et al., 2017), Dual Attention (DA) module from DANet (Fu et al., 2019), Double Attention module from $A ^ { 2 }$ Net (Chen et al., 2018b), APC module from APCNet (He et al., 2019b), DM module from DMNet (He et al., 2019a), ACF module from CFNet (Zhang et al., 2019b), reporting parameters and costs of processing a tensor $\mathcal { Z } \in \mathbb { R } ^ { 1 \times 5 1 2 \times 1 2 8 \times 1 2 8 }$ in Tab. 3. Excessive memory usage is the key bottleneck of cooperating with attention in real applications. Hence we also provide the GPU load and inference time on NVIDIA TITAN Xp. In general, Hamburger is light in computation and memory compared with attention-related global context modules.
+
+Table 3: Comparisons between Hamburger and context modules.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Params</td><td rowspan="2">MACs</td><td colspan="2">GPU Load</td><td colspan="2">GPU Time</td></tr><tr><td>Train</td><td>Infer</td><td>Train</td><td>Infer</td></tr><tr><td>SA</td><td>1.00M</td><td>292G</td><td>5253MB</td><td>2148MB</td><td>242.0ms</td><td>82.2ms</td></tr><tr><td>DA</td><td>4.82M</td><td>79.5G</td><td>2395MB</td><td>2203MB</td><td>72.6ms</td><td>64.4ms</td></tr><tr><td>A2</td><td>1.01M</td><td>25.7G</td><td>326MB</td><td>165MB</td><td>22.9ms</td><td>8.0ms</td></tr><tr><td>APC</td><td>2.03M</td><td>17.6G</td><td>458MB</td><td>264MB</td><td>26.5ms</td><td>11.6ms</td></tr><tr><td>DM</td><td>3.00M</td><td>35.1G</td><td>557MB</td><td>268MB</td><td>65.7ms</td><td>23.3ms</td></tr><tr><td>ACF</td><td>0.75M</td><td>79.5G</td><td>1380MB</td><td>627MB</td><td>71.0ms</td><td>22.6ms</td></tr><tr><td>Ham (CD)</td><td>0.50M</td><td>16.2G</td><td>162MB</td><td>102MB</td><td>20.0ms</td><td>13.0ms</td></tr><tr><td>Ham (NMF)</td><td>0.50M</td><td>17.6G</td><td>202MB</td><td>98MB</td><td>15.6ms</td><td>7.7ms</td></tr></table>
+
+# 3.4 SEMANTIC SEGMENTATION
+
+We benchmark Hamburger on the PASCAL VOC dataset (Everingham et al., 2010), and the PASCAL Context dataset (Mottaghi et al., 2014), against state-of-the-art attentions. We use ResNet-101 (He et al., 2016) as our backbone. The output stride of the backbone is 8. The segmentation head is the same as ablation experiments. NMF is usually better than CD and VQ in ablation studies (see Tab. 1). Therefore, we mainly test NMF in further experiments. We use HamNet to represent ResNet with Hamburger in the following section.
+
+Results on the PASCAL VOC test set, and the PASCAL Context validation set, are illustrated in Tab. 4, and Tab. 5, respectively. We mark all attention-based models with ∗ in which diverse attentions compose the segmentation heads. Though semantic segmentation is a saturated task, and most contemporary published works have approximate performances, Hamburger shows considerable improvements over previous state-of-the-art attention modules.
+
+Table 4: Comparisons with state-of-the-art on the PASCAL VOC test set w/o COCO pretraining.   
+
+<table><tr><td>Method</td><td>mIoU(%)</td></tr><tr><td>PSPNet (Zhao et al., 2017)</td><td>82.6</td></tr><tr><td>DFN* (Yu et al., 2018)</td><td>82.7</td></tr><tr><td>EncNet (Zhang et al., 2018)</td><td>82.9</td></tr><tr><td>DANet* (Fu et al., 2019)</td><td>82.6</td></tr><tr><td>DMNet* (He et al., 2019a)</td><td>84.4</td></tr><tr><td>APCNet* (He et al., 2019b)</td><td>84.2</td></tr><tr><td>CFNet* (Zhang et al., 2019b)</td><td>84.2</td></tr><tr><td>SpyGR* (Li et al., 2020)</td><td>84.2</td></tr><tr><td>SANet* (Zhong et al., 2020)</td><td>83.2</td></tr><tr><td>OCR* (Yuan et al., 2020)</td><td>84.3</td></tr><tr><td>HamNet</td><td>85.9</td></tr></table>
+
+Table 5: Results on the PASCAL-Context Val set.   
+
+<table><tr><td>Method</td><td>mIoU(%)</td></tr><tr><td>PSPNet (Zhao et al., 2017)</td><td>47.8</td></tr><tr><td>SGR* (Liang et al., 2018)</td><td>50.8</td></tr><tr><td>EncNet (Zhang et al., 2018)</td><td>51.7</td></tr><tr><td>DANet* (Fu et al., 2019)</td><td>52.6</td></tr><tr><td>EMANet* (Li et al., 2019)</td><td>53.1</td></tr><tr><td>DMNet* (He et al., 2019a)</td><td>54.4</td></tr><tr><td>APCNet* (He et al., 2019b)</td><td>54.7</td></tr><tr><td>CFNet* (Zhang et al., 2019b)</td><td>54.0</td></tr><tr><td>SpyGR* (Li et al., 2020)</td><td>52.8</td></tr><tr><td>SANet* (Zhong et al., 2020)</td><td>53.0</td></tr><tr><td>OCR*(Yuan et al., 2020)</td><td>54.8</td></tr><tr><td>HamNet</td><td>55.2</td></tr></table>
+
+# 3.5 IMAGE GENERATION
+
+Attention presents as the global context description block in deep generative models like GANs. Most state-of-the-art GANs for conditional image generation integrate self-attention into their architectures since SAGAN (Zhang et al., 2019a), e.g., BigGAN (Brock et al., 2018), $\mathrm { S ^ { 3 } G A N }$ (Luciˇ c et al.´ , 2019), and LOGAN (Wu et al., 2019). It is convincing to benchmark MDbased Hamburger in the challenging conditional image generation task on ImageNet (Deng et al., 2009).
+
+Table 6: Results on ImageNet $1 2 8 \times 1 2 8$ . ∗ are from Tab. 1 and Tab. 2 of Zhang et al. (2019a).   
+
+<table><tr><td>Method</td><td>FID↓</td></tr><tr><td>SNGAN-projection*</td><td>27.62</td></tr><tr><td>SAGAN*</td><td>18.28</td></tr><tr><td>HamGAN-baby</td><td>16.05</td></tr><tr><td>YLG</td><td>15.94</td></tr><tr><td>HamGAN-strong</td><td>14.77</td></tr></table>
+
+Experiments are conducted to compare Hamburger with self-attention on ImageNet $1 2 8 \times 1 2 8$ . Selfattention is replaced by Hamburger with NMF ham in both generator and discriminator at feature resolution $3 2 \times 3 2$ , named as HamGAN-baby. HamGAN achieves an appreciable improvement in Fr´echet Inception Distance (FID) (Heusel et al., 2017) over SAGAN. Additionally, we compare Hamburger with a recently developed attention variant Your Local GAN (YLG) (Daras et al., 2020) using their codebase and the same training settings, named HamGAN-strong. HamGAN-strong offers over $5 \%$ improvement in FID while being $15 \%$ faster for the total training time and $3 . 6 \mathbf { x }$ faster for the module time (1.54 iters/sec of HamGAN, 1.31 iters/sec of YLG, and 1.65 iters/sec without both context modules, averaged from 1000 iterations) on the same TPUv3 training platform.
+
+# 4 RELATED WORK
+
+General Survey for Attention The last five years have witnessed a roaring success of attention mechanisms (Bahdanau et al., 2015; Mnih et al., 2014; Xu et al., 2015; Luong et al., 2015) in deep learning. Roughly speaking, the attention mechanism is a term of adaptively generating the targets’ weights to be attended according to the requests. Its architectures are diverse, and the most well-known one is dot product self-attention (Vaswani et al., 2017). The attention mechanism has a wide range of applications, from a single source (Lin et al., 2017) to multi-source inputs (Luong et al., 2015; Parikh et al., 2016), from global information discovery (Wang et al., 2018; Zhang et al., 2019a) to local feature extraction (Dai et al., 2017; Parmar et al., 2019).
+
+Previous researchers attempt to explain the effectiveness of attention mechanisms from numerous aspects. Capturing long-range dependencies (Wang et al., 2018), sequentially decomposing visual scenes (Eslami et al., 2016; Kosiorek et al., 2018), inferring relationships between the part and the whole (Sabour et al., 2017; Hinton et al., 2018), simulating interactions between objects (Greff et al., 2017; van Steenkiste et al., 2018), and learning the dynamics of environments (Goyal et al., 2019) are often considered as the underlying mechanisms of attention.
+
+One common idea from biology is that attention simulates the emergence of concerns in many unconscious contexts (Xu et al., 2015). Some work tries to interpret the attention mechanism by visualizing or attacking attention weights (Serrano & Smith, 2019; Jain & Wallace, 2019; Wiegreffe & Pinter, 2019), while others formulate attention into non-local operation (Wang et al., 2018) or diffusion models (Tao et al., 2018; Lu et al., 2019) or build attention-like models via Expectation Maximization (Greff et al., 2017; Hinton et al., 2018; Li et al., 2019) or Variational Inference (Eslami et al., 2016) on a mixture model. A connection between transformer and graph neural network is discussed as well (Liang et al., 2018; Zhang et al., 2019c). Overall, discussions towards attention are still far from reaching agreements or consistent conclusions.
+
+Efficient Attention Recent works develop efficient attention modules via low-rank approximation in both computer vision (Chen et al., 2018b; Zhu et al., 2019; Chen et al., 2019; Li et al., 2019) and natural language processing (Mehta et al., 2019; Katharopoulos et al., 2020; Wang et al., 2020; Song et al., 2020). Technically, the low-rank approximation usually targets at the correlation matrix, i.e., the product of $Q$ and $\kappa$ after the sof tmax operation, using a product of two smaller matrices to approximate the correlation matrix and applying the associative law to save the memory cost and computation, where the approximation involves kernel functions or other similarity functions. Other works (Babiloni et al., 2020; Ma et al., 2019) make efforts to formulate attention into tensor form but may generate large intermediate variables. In this paper, we do not approximate attention or make it efficient. This paper formulates modeling the global context as a low-rank completion problem. The computation and memory efficiency is a by-product of the low-rank assumption on the clean signal subspace and optimization algorithms as architectures.
+
+Matrix Decomposition in Deep Learning There is a long history of combining MD with deep learning. Researchers focus on reducing the parameters in the networks via factorization on the weights, including the softmax layer (Sainath et al., 2013), the convolutional layer (Zhong et al., 2019), and the embedding layer (Lan et al., 2019). Tariyal et al. (2016) attempts to construct deep dictionary learning for feature extraction and trains the model greedily. This paper tries to factorize the representations to recover a clean signal subspace as the global context and provide a new formulation for modeling the long-range dependencies via matrix decomposition.
+
+# 5 CONCLUSION
+
+This paper studies modeling long-range dependencies in the networks. We formulate learning the global context as a low-rank completion problem. Inspired by such a low-rank formulation, we develop the Hamburger module based on well-studied matrix decomposition models. By specializing matrix decomposition’s objective function, the computational graph created by its optimization algorithm naturally defines ham, Hamburger’s core architecture. Hamburger learns interpretable global context via denoising and completing its input and improves the spectrum’s concentration. It is startling that, when prudently coped with the backward gradient, even simple matrix decomposition proposed 20 years ago is as powerful as self-attention in challenging vision tasks semantic segmentation and image generation, as well as light, fast, and memory-efficient. We plan to extend Hamburger to natural language processing by integrating positional information and designing a decoder like Transformer, build a theoretical foundation for the one-step gradient trick or find a better method to differentiate MDs, and integrate advanced MDs in the future.
+
+# ACKNOWLEDGMENTS
+
+Zhouchen Lin is supported by NSF China (grant no.s 61625301 and 61731018), Major Scientific Research Project of Zhejiang Lab (grant no.s 2019KB0AC01 and 2019KB0AB02), Beijing Academy of Artificial Intelligence, and Qualcomm. We thank Google’s Tensorflow Research Cloud (TFRC) for providing us Cloud TPUs.
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+
+# A TABLE OF NOTION
+
+Table 7: Summary of notations in this paper   
+
+<table><tr><td rowspan=1 colspan=2>βXXZ</td><td rowspan=1 colspan=1>A scalar.A vector.A matrix.A tensor.</td></tr><tr><td rowspan=1 colspan=2>1nXiht8</td><td rowspan=1 colspan=1>A vector whose n elements are all 1.i-th column of matrix X.Vector h at time step t.Jacobian matrix of y W.r.t. X.</td></tr><tr><td rowspan=1 colspan=1>XX||F</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Operator norm.Frobenius norm.</td></tr><tr><td rowspan=1 colspan=2>diagcosinesoftmaxnormalize</td><td rowspan=1 colspan=1>Map a vector to a diagonal matrix.Cosine similarity used in Alg.1.Column-wise softmax function.Column-wise normalization by L2 norm.</td></tr></table>
+
+# B HAMS
+
+Additionally, we introduce another type of ham adopted by Hamburger, Concept Decomposition.
+
+Concept Decomposition We first enhance Concept Decomposition (Dhillon & Modha, 2001) to the following form:
+
+$$
+\begin{array} { l } { \displaystyle \operatorname* { m i n } _ { D , C } \| X - D C \| _ { F } ^ { 2 } + \beta \| C \| _ { F } ^ { 2 } } \\ { \displaystyle \mathrm { s . t . } D \in \arg \operatorname* { m a x } _ { D } \mathcal { Q } \left( D , X \right) . } \end{array}
+$$
+
+This problem has a closed solution w.r.t. $C$ under a given $_ { D }$ , i.e., $\pmb { C } = ( \pmb { D } ^ { \top } \pmb { D } + \beta \pmb { I } ) ^ { - 1 } \pmb { D } ^ { \top } \pmb { X }$ . Since $D ^ { \top } D + \beta I$ is a positive definite matrix with a regularized conditional number, the inverse can be more numerically stable than the original one where a semi-positive definite matrix $D ^ { \top } D$ is given under $\beta = 0$ . In practice, 0.01 or 0.1 makes no difference for $\beta$ .
+
+<table><tr><td>Algorithm 3 Ham: Soft CD</td></tr><tr><td>Input X. Initialize D, C</td></tr><tr><td>for k from 1 to K do</td></tr><tr><td>C ← softmax(Tcosine(D,X))</td></tr><tr><td>D ← normalize(XCT)</td></tr><tr><td>end for</td></tr><tr><td>C ← (DD+ βI)-1DTX</td></tr><tr><td>Output X = DC.</td></tr></table>
+
+The dictionary in CD is given by spherical $\mathbf { K }$ -means (Dhillon & Modha, 2001) with objective $\mathcal { Q } \left( D , X \right)$ , as mentioned in Eq. (14).
+
+$$
+\begin{array} { r l } { \arg \operatorname* { m a x } } & { \sum _ { j = 1 } ^ { r } \sum _ { \mathbf { x } \in \pi _ { j } } c o s i n e \left( \mathbf { x } , \mathbf { d } _ { j } \right) } \\ { D , \{ \pi _ { j } \} _ { r } } & { \phantom { \sum _ { j = 1 } ^ { r } \sum _ { \mathbf { x } \in \pi _ { j } } c o s i n e \left( \mathbf { x } , \mathbf { d } _ { j } \right) } } \\ { \mathrm { s . t . } } & { \| \mathbf { d } _ { j } \| = 1 . } \end{array}
+$$
+
+The same strategy as VQ is adopted to make the whole algorithm differentiable, however, in which each column of $_ { D }$ is normalized to be a unit vector and thus differs from VQ.
+
+# C PROOF OF PROPOSITIONS
+
+We investigate an abstract RNN model inspired by numerical methods to understand the drawbacks of BPTT algorithm in differentiating the optimization algorithm of MDs, $\mathcal { M }$ . We show the propositions in Sec. 2.3 to illustrate the unstable gradient from $\mathcal { M }$ when using BPTT algorithm, considering MDs’ nature as optimization algorithms.
+
+Proposition 1 The iterations of $\mathcal { F }$ have linear convergence.
+
+Proof. It is obvious that $\mathcal { F }$ is a contraction mapping w.r.t. h under arbitrary given $\mathbf { x }$ . We can then conclude $\{ \mathbf { h } ^ { t } \}$ is a Cauthy sequence and $\mathcal { F } ( * , \mathbf { x } )$ admits a unique fixed point $\mathbf { h } ^ { * }$ due to Banach Fixed Point Theorem.
+
+$$
+\begin{array} { r l } & { \| \mathbf { h } ^ { t + 1 } - \mathbf { h } ^ { * } \| = \| \mathcal { F } ( \mathbf { h } ^ { t } , \mathbf { x } ) - \mathcal { F } ( \mathbf { h } ^ { * } , \mathbf { x } ) \| } \\ & { \qquad \leq L _ { h } \| \mathbf { h } ^ { t } - \mathbf { h } ^ { * } \| } \end{array}
+$$
+
+Eq. (16) shows the linear convergence.
+
+Proposition 2 $\begin{array} { r } { \operatorname* { l i m } _ { t  \infty } \frac { \partial \mathbf { y } } { \partial \mathbf { x } } = \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { \ast } } ( \pmb { I } - \frac { \partial \mathcal { F } } { \partial \mathbf { h } ^ { \ast } } ) ^ { - 1 } \frac { \partial \mathcal { F } } { \partial \mathbf { x } } . } \end{array}$
+
+Proof. Note that $\mathcal { F } ( \ast ,  { \mathbf { x } } )$ admits a unique fixed point $\mathbf { h } ^ { * }$ under arbitrary given $\mathbf { x }$ , i.e.,
+
+$$
+\mathbf { h } ^ { * } = \mathcal { F } ( \mathbf { h } ^ { * } , \mathbf { x } ) \quad \Longrightarrow \quad \mathbf { h } ^ { * } - \mathcal { F } ( \mathbf { h } ^ { * } , \mathbf { x } ) = \mathbf { 0 }
+$$
+
+By differentiating the above equation, we can obtain
+
+$$
+( I - \frac { \partial \mathcal { F } } { \partial \mathbf { h } ^ { * } } ) \frac { \partial \mathbf { h } ^ { * } } { \partial \mathbf { x } } = \frac { \partial \mathcal { F } } { \partial \mathbf { x } }
+$$
+
+The Jacobian matrix $\begin{array} { r } { I - \frac { \partial \mathcal { F } } { \partial \mathbf { h } ^ { * } } } \end{array}$ is invertible, which implies the existence of the implicit function $\mathbf { h } ^ { * } ( \mathbf { x } )$ . Immediately, we have
+
+$$
+\operatorname* { l i m } _ { t \to \infty } \frac { \partial \mathbf { y } } { \partial \mathbf { x } } = \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { * } } \frac { \partial \mathbf { h } ^ { * } } { \partial \mathbf { x } } = \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { * } } ( \pmb { I } - \frac { \partial \mathcal { F } } { \partial \mathbf { h } ^ { * } } ) ^ { - 1 } \frac { \partial \mathcal { F } } { \partial \mathbf { x } } ,
+$$
+
+which completes the proof.
+
+Proposition 3 $\begin{array} { r } { \underset { t  \infty } { \operatorname* { l i m } } \Vert \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { 0 } } \Vert = 0 , \underset { t  \infty } { \operatorname* { l i m } } \Vert \frac { \partial \mathbf { y } } { \partial \mathbf { x } } \Vert \leq \frac { L _ { \mathcal { G } } L _ { x } } { 1 - L _ { h } } . } \end{array}$
+
+Proof.
+
+$$
+\| \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { 0 } } \| = \| \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { t } } \prod _ { i = 1 } ^ { t } \frac { \partial \mathbf { h } ^ { i } } { \partial \mathbf { h } ^ { i - 1 } } \| \leq \| \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { t } } \| \prod _ { i = 1 } ^ { t } \| \frac { \partial \mathbf { h } ^ { i } } { \partial \mathbf { h } ^ { i - 1 } } \| \leq L _ { \mathcal { G } } L _ { h } ^ { t }
+$$
+
+Then we have:
+
+$$
+\operatorname* { l i m } _ { t  \infty } \lVert \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { 0 } } \rVert = 0 .
+$$
+
+$$
+\begin{array} { r l } & { \| \displaystyle \frac { \partial \mathbf { y } } { \partial \mathbf { x } } \| = \| \sum _ { i = 0 } ^ { t } \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { t } } \| \prod _ { j = i + 1 } ^ { t } \frac { \partial \mathbf { h } ^ { j } } { \partial \mathbf { h } ^ { j - 1 } } \frac { \partial \mathbf { h } ^ { i } } { \partial \mathbf { x } } \| } \\ & { \qquad \le \displaystyle \sum _ { i = 0 } ^ { t } \| \frac { \partial \mathbf { y } } { \partial \mathbf { h } ^ { t } } \| \prod _ { j = i + 1 } ^ { t } \| \frac { \partial \mathbf { h } ^ { j } } { \partial \mathbf { h } ^ { j - 1 } } \| \| \frac { \partial \mathbf { h } ^ { i } } { \partial \mathbf { x } } \| } \\ & { \qquad \le L  { \mathcal { C } } ( \displaystyle \sum _ { i = 0 } ^ { t - 1 } L _ { n } ^ { i } ) L _ { x } } \\ & { \qquad = \displaystyle \frac { L _ { G } L _ { x } ( 1 - L _ { h } ^ { t } ) } { 1 - L _ { h } } } \end{array}
+$$
+
+Then we have:
+
+$$
+\operatorname* { l i m } _ { t \to \infty } \| \frac { \partial \mathbf { y } } { \partial \mathbf { x } } \| \leq \frac { L _ { \mathcal { G } } L _ { x } } { 1 - L _ { h } } .
+$$
+
+# D DATASETS
+
+PASCAL VOC The PASCAL VOC dataset (Everingham et al., 2010) is a widely used dataset in both semantic segmentation and detection. For segmentation, it contains 10,582 images for training, 1,449 images for validation and 1,456 images for testing. PASCAL VOC dataset involves 20 foreground object classes and a background class for segmentation and detection.
+
+PASCAL Context The PASCAL Context dataset (Mottaghi et al., 2014) is a challenging dataset in semantic segmentation, which provides detailed labels and involves 59 foreground object classes and a background class for segmentation. It consists of 4,998 and 5,105 images in training and validation set, respectively.
+
+ILSVRC 2012 The ILSVRC 2012 (ImageNet) (Deng et al., 2009) dataset contains 1.3M training samples and $5 0 \mathrm { k }$ test images, categorized into 1000 object classes. We resize images to resolution $1 2 8 \times 1 2 8$ , as done in SNGAN with projection (Miyato & Koyama, 2018) and SAGAN (Zhang et al., 2019a).
+
+# E DETAILS OF EXPERIMENTS
+
+# E.1 ABALATION EXPERIMENTS
+
+We use dilated ResNet-50 (He et al., 2016) with the output stride 16 as the backbone. The backbone is pre-trained on ImageNet (Deng et al., 2009). We apply a poly-learning rate policy under batch size 12 and 30k iterations (about 35 epochs) for fast experiments (less than 12 hours using 1 NVIDIA TITAN Xp GPU). The initial learning rate is set to 0.009, multiplied by $\begin{array} { r } { ( 1 - \frac { i t e r } { i t e r _ { m a x } } ) ^ { \tilde { 0 . } 9 } } \end{array}$ )0.9 after each iteration, with momentum 0.9 and weight decay 0.0001. Hyperparameters of Hamburger are the same as Appendix E.3.
+
+# E.2 A COMPARISON WITH ATTENTION MECHANISM
+
+We report MACs according to Molchanov et al. (2016), using torchprofile1, a more accurate profiler for Pytorch. Real-time cost is measured by built-in Pytorch memory tools on NVIDIA TITAN Xp GPU with a input tensor $\mathcal { Z } \in \mathbb { R } ^ { 1 \times 5 1 2 \times 1 2 8 \times 1 2 8 }$ . Inference times are averaged results from 20 repeats of 100 runs.
+
+# E.3 SEMANTIC SEGMENTATION
+
+Architectures We use ResNet-101 (He et al., 2016) with the ouptput strid 8 as our backbone. We adopt dilated convolution (Chen et al., 2018a) to preserve more detail spatial information and enlarge receptive field as done in the backbone of state-of-the-art attention models ( $\mathrm { F u }$ et al., 2019; Li et al., 2019; Zhang et al., 2019b). We employ a $3 \times 3$ convolution layer with BN and ReLU to reduce channels from 2048 to 512 and then add Hamburger on the top of the backbone. Note that the input of Hamburger is a tensor $\mathcal { Z } \in \mathbb { R } ^ { C \times H \times W }$ . We unfold $\mathcal { Z }$ to a matrix $\boldsymbol { Z } \in \mathbb { R } ^ { C \times H W }$ and set $d _ { z } = C$ and $n = H W$ for Hamburger. Latent dimension $d$ and $r$ , i.e., the column vectors’ dimension of the input matrix $\pmb { X } \in \mathbb { R } ^ { d \times n }$ to $\mathcal { M }$ and the number of atoms in the dictionary $\pmb { D } \in \mathbb { R } ^ { r \times d }$ , are set to 512 and 64. The iterations of MD’s optimization algorithm, $K$ , are set to 6. Non-negative Matrix Factorization (NMF) is our default ham for semantic segmentation.
+
+Data augmentation In the training stage, we apply random left-right flipping, random scaling (from 0.5 to 2), and cropping to augment the training data. Images are resized to $5 1 3 \times 5 1 3$ for the PASCAL VOC dataset and the PASCAL Context dataset. In the test stage, the multi-scale and flipping strategy is applied as other state-of-the-art attention-based models ( $\mathrm { F u }$ et al., 2019; Yuan & Wang, 2018; Yuan et al., 2020).
+
+Optimization We use mini-batch SGD with momentum 0.9 to train HamNet. Synchronized Batch Normalization is adopted in experiments on semantic segmentation. All backbones are fine-tuned from ImageNet (Deng et al., 2009) pre-training. Following previous works (Zhao et al., 2017; Chen et al., 2018a), we apply a poly-learning rate policy. The initial learning rate is multiplied by $\begin{array} { r } { ( 1 - \frac { i t e r } { i t e r _ { m a x } } ) ^ { 0 . 9 } } \end{array}$ )0.9. For the PASCAL VOC dataset, learning rate, weight decay, batch size, iterations are set to 0.009, 0.0001, 16, and $6 0 \mathrm { k }$ , respectively. We fine-tune HamNet on the PASCAL VOC trainval set with the learning rate down to a tenth. The learning rate, weight decay, batch size, iterations are 0.002, 0.0001, 16, and 25k for the PASCAL-Context dataset.
+
+# E.4 IMAGE GENERATION
+
+We use the official GAN codebase2 from Tensorflow (Abadi et al., 2016) and TF-GAN to train HamGAN and evaluate FID.
+
+Architectures Experiments on ImageNet are conducted using the same architecture as SAGAN (Zhang et al., 2019a), and YLG (Daras et al., 2020), including Spectral Normalization (Miyato et al., 2018) in both the generator and the discriminator, conditional Batch Normalization in the generator, and class projection in the discriminator (Miyato & Koyama, 2018). Hamburger with NMF ham is placed at feature resolution $3 2 \times 3 2$ in both the generator and the discriminator where self-attention can obtain the best FID according to Zhang et al. (2019a). We use $d = 8 r$ for Hamburger, and $d$ is the same as the input channels, while the optimization steps $K$ are 6. Restricted to expenditures of training GANs on ImageNet, $d , r$ , and $K$ are decided according to the ablation experiments on semantic segmentation without new ablation experiments.
+
+Optimization For all models, we use Adam (Kingma & Ba, 2015) optimizer with TTUR (Heusel et al., 2017). HamGAN employs the same training settings as SAGAN (Miyato et al., 2018) and YLG (Daras et al., 2020), respectively.
+
+Evaluation metrics The quality of images generated by GANs are evaluated by Fr´echet Inception Distance (FID) (Heusel et al., 2017). Lower FID indicates that the model can generate higher-fidelity images. In our experiments, $5 0 \mathrm { k }$ images are sampled from the generator to compute FID. We evaluate HamGAN for 6 runs and report the best FID to approximately match the convention in the modern GAN research like Kurach et al. (2019) and CR-GAN (Zhang et al., 2020), reporting top $5 \% / 1 5 \%$ results in the experiments.
+
+# F FURTHER RESULTS FROM ABLATION EXPERIMENTS
+
+Table 8: Ablation on initializations.   
+
+<table><tr><td>Init</td><td>NMF</td><td>CD</td><td>VQ</td></tr><tr><td>fixed</td><td>77.4(77.3)</td><td>77.7(77.4)</td><td>77.3(76.9)</td></tr><tr><td>learned</td><td>76.8(76.5)</td><td>75.0(73.7)</td><td>75.9(75.8)</td></tr><tr><td>random</td><td>78.3(77.8)</td><td>77.9(77.3)</td><td>77.7(77.4)</td></tr><tr><td>online</td><td>77.8(77.5)</td><td>78.1(77.5)</td><td>78.0(77.2)</td></tr></table>
+
+Initialization We test four types of initialization for the dictionary $_ { D }$ , including fixed initialization, learned initialization, random initialization, and warm start with online update. Usually, random initialization is the best choice that means we can sample each entry of $_ { D }$ from a given distribution like Uniform $( 0 , 1 )$ as the initialization of the optimization algorithm $\mathcal { M }$ . For NMF, after initializing $_ { D }$ , we initialize $\begin{array} { r } { \dot { \mathbf { C } } = s o f t m a x ( \frac { 1 } { T } c o s i n e ( \mathbf { D } , \mathbf { \dot { X } } ) ) } \end{array}$ since $\mathbf { K }$ -means is usually applied for initializing NMF and this initialization for $\bar { C }$ is equivalent to a single update in Spherical K-means. A special reminder is that it is not suitable to initialize either $_ D$ or $C$ to values too close to 0 due to the property of the MU rule. So the temperature $T$ is recommended to be a higher value like 1 in this initialization for $C$ . Random initialization also works for $C$ in NMF with scores 77.8(77.6) when sampling $C _ { i j } \sim \mathrm { U n i f o r m } ( 0 , 1 )$ . Note that learned initialization is always the worst one since the BPTT algorithm is employed to learn the initialization that the gradient from $\mathcal { M }$ may impede the training of the backbone, instead of the one-step gradient. Warm start benefits MD with unit vectors in the dictionary $_ { D }$ like CD. In general, random initialization is good enough for all three selected MD models. A possible reason is that it can enforce the network to adapt to the results solved by different initializations during the training process, acting like an inner augmentation.
+
+Temperature $T$ As we have claimed, when $T$ approaches 0, we can get a solution close to the original problem in both VQ and CD. In VQ and CD experiments, a relatively low temperature $T$ is more recommended to solve a better $_ { D }$ for MD. However, it will not receive more gains but increase the variance during training if we further lower $T$ .
+
+Table 9: Influence of temperature $T$ with CD ham.   
+
+<table><tr><td>Temperature T</td><td>mIoU(%)</td></tr><tr><td>1</td><td>77.1(77.0)</td></tr><tr><td>0.1</td><td>78.2(77.5)</td></tr><tr><td>0.01</td><td>78.1(77.5)</td></tr></table>
+
+Iterations $K$ We take the iterations $K$ of optimization algorithms $\mathcal { M }$ for all three MD models, NMF, CD, and VQ, into our consideration. More iterations and even fully converged results for $\mathcal { M }$ are tested in the evaluation stage but worse than little optimization steps. The smaller $K$ , ranging from 1 to 8, can be treated as early stopping for the optimization algorithm $\mathcal { M }$ , obtaining satisfactory performances. For a detailed visualization, see Fig. 7, Fig. 8, Fig. 9.
+
+![](images/ee93b30769d5d4762b9066b623172a18833ce1847f0d1ae8d6a1d369d1da547a.jpg)  
+Figure 7: Impacts of $K$ on NMF
+
+![](images/914e4316a63ddbfbb8243122ca075bd60a7f09d19f7433ae36ae010d149f63ee.jpg)  
+Figure 8: Impacts of $K$ on CD
+
+![](images/f53bad29ad9f09cc2f2d4a29c77f9f785dc4c10e87d76b8c8c8480e7eb9930bb.jpg)  
+Figure 9: Impacts of $K$ on VQ
+
+# G AN INTUITIVE ILLUSTRATION
+
+In this section, we hope to give an example to help our readers develop insight into why the low-rank assumption is useful for modeling the representations’ global context.
+
+The low-rank assumption helps because it represents the inductive bias that the low-level representations contain limited and much less high-level concepts than the scale of the representations themselves. Imagine an image in which a person works on the road. Many hyper-pixels extracted by the backbone CNN will describe the road. Note that the road can be considered as repeats of small road patches, which means that we can represent the road via modeling the basic road patches and repeat them. Mathematically, it is equivalent to finding a small set of bases $_ D$ corresponding to different road patches and a coefficient matrix $C$ that captures the relation between the elementary road patches and the hyper-pixels. This example illustrates that the high-level concepts, i.e., the global context, can be low-rank in the ideal situation.
+
+The hyper-pixels describing the road patches have close semantic attributes. However, due to the vanilla CNN’s inefficiency for modeling the long-range dependencies, the learned representation contains too many local details and incorrect information, lacking global guidance. Imagine that the person in the image wears gloves. When we see the gloves patch locally, we think that this patch describes gloves. When we consider the global context, we can understand that this patch is a part of a person. The semantic information is hierarchical, depending on at which level we hope to comprehend. This work aims at enabling the networks to understand the context globally via the low-rank completion formulation. We thus model the incorrect information, namely the redundancies and incompleteness, as a noise matrix. To emphasize the global context, we decompose the representations into two parts, a low-rank global information matrix and a noise matrix, by employing the optimization algorithm to recover the clean signal subspace, discard the noises, and enhance the global information via the skip connection. It could be learned from the data on how much global information the networks need for a specific task.

parse/train/4G2dEuRZ7eO/4G2dEuRZ7eO.md

@@ -0,0 +1,290 @@
+# Progressive Coordinate Transforms for Monocular 3D Object Detection
+
+Li Wang1∗ Li Zhang1† Yi Zhu2 Zhi Zhang2 Tong He2 Mu Li2 Xiangyang Xue1 1Fudan University 2Amazon Inc.
+
+# Abstract
+
+Recognizing and localizing objects in the 3D space is a crucial ability for an AI agent to perceive its surrounding environment. While significant progress has been achieved with expensive LiDAR point clouds, it poses a great challenge for 3D object detection given only a monocular image. While there exist different alternatives for tackling this problem, it is found that they are either equipped with heavy networks to fuse RGB and depth information or empirically ineffective to process millions of pseudo-LiDAR points. With in-depth examination, we realize that these limitations are rooted in inaccurate object localization. In this paper, we propose a novel and lightweight approach, dubbed Progressive Coordinate Transforms (PCT) to facilitate learning coordinate representations. Specifically, a localization boosting mechanism with confidence-aware loss is introduced to progressively refine the localization prediction. In addition, semantic image representation is also exploited to compensate for the usage of patch proposals. Despite being lightweight and simple, our strategy leads to superior improvements on the KITTI and Waymo Open Dataset monocular 3D detection benchmarks. At the same time, our proposed PCT shows great generalization to most coordinatebased 3D detection frameworks. The code is available at: https://github.com/ amazon-research/progressive-coordinate-transforms.
+
+# 1 Introduction
+
+Object detection is a fundamental and challenging task in scene understanding applications. Recently, 3D object detection has received increasing attention and found applications in a wide range of scenarios such as autonomous driving, robotics, visual navigation and mixed reality. Despite the great progress from the area of 2D object detection [34, 49, 40, 18, 4], 3D object detection remains a largely unsolved problem as it aims to predict the object location in the 3D space alongside 3D object dimension and orientation.
+
+Existing prevalent approaches [50, 44, 35, 10, 11] for 3D object detection largely rely on LiDAR sensors, which provide accurate 3D point clouds of the scene. Although these approaches achieve superior performance, the dependence on expensive equipment severely limits their applicability to generic 3D perception. There also exists a cheaper alternative that takes a single-view RGB image as input, termed as monocular 3D object detection. However, its performance is far from satisfactory as itself is an ill-posed problem due to the loss of depth information in 2D image planes. Hence, several recent attempts introduce depth information to help monocular 3D detection. Such attempts can be roughly categorized into two directions, pixel-based and coordinate-based. Pixel-based approaches [9, 36, 31, 41] turn to use estimated depth map as additional input for improved detection performance. But at the same time, this leads to heavy computational burden and large memory footprint since they often operate on the entire image. Coordinated-based approaches [42, 29, 46, 27] pursue the coordinate representations as in LiDAR-based methods. They use the predicted depth map to convert the monocular image pixels to 3D coordinate representations, then apply a 3D detector on the converted coordinates. In particular, they are often lightweight since their network inputs are object proposals generated by 2D detectors [29, 27]. However, the performance of coordinated-based methods lags far behind LiDAR-based methods. So we ask, can we identify the bottleneck that holds back the 3D detection accuracy of coordinate-based methods and how can we improve them?
+
+Table 1: Probing investigation on coordinate-based methods, PatchNet [27] and Pseudo-LiDAR [42]. We examine the potential improvement by replacing the predicted factor with the corresponding ground truth. $^ *$ indicates our reproduced performance. We can see that coordinate-based methods mostly suffer from inaccurate localization.   
+
+<table><tr><td rowspan="2">Factor</td><td colspan="3">PatchNet*[AP3D/APBEV]</td><td colspan="3">Pseudo-LiDAR*[AP3D/APBEv]</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Baseline</td><td>26.31/34.14</td><td>36.40/46.80</td><td>21.07/28.04</td><td>23.04/31.06</td><td>32.27/42.45</td><td>19.67/25.67</td></tr><tr><td>dimension</td><td>27.26/34.62</td><td>40.32/47.24</td><td>24.29/28.38</td><td>25.88/31.97</td><td>36.09/44.35</td><td>20.88/26.60</td></tr><tr><td>rotation</td><td>26.25/34.04</td><td>36.09/46.25</td><td>23.49/27.99</td><td>23.88/31.31</td><td>32.42/42.74</td><td>19.85/26.07</td></tr><tr><td>X</td><td>32.80/41.43</td><td>45.60/56.22</td><td>27.38/34.63</td><td>28.36/36.77</td><td>39.69/50.78</td><td>25.08/29.92</td></tr><tr><td>y</td><td>30.16/34.14</td><td>40.94/46.80</td><td>24.58/28.04</td><td>25.53/31.06</td><td>35.19/42.45</td><td>20.69/25.67</td></tr><tr><td>Z</td><td>42.42/53.48</td><td>55.42/68.29</td><td>35.54/45.60</td><td>38.37/50.81</td><td>50.04/63.96</td><td>32.24/43.32</td></tr><tr><td>location(xyz)</td><td>72.58/75.27</td><td>81.41/85.14</td><td>57.69/66.10</td><td>64.36/73.37</td><td>79.13/83.77</td><td>55.64/58.27</td></tr></table>
+
+In order to determine the bottleneck, we conduct an investigation on two widely adopted coordinatebased methods, PatchNet [27] and Pseudo-LiDAR [42]. Specifically, for each prediction target, we examine the potential improvement by replacing its value with the corresponding ground truth, and then re-compute the 3D detection accuracy. As shown in Table 1, using ground truth dimension and rotation do not bring significant improvements over the baseline. But using ground truth location (i.e., x/y/z values of the objects) almost triples detection accuracy. This indicates that coordinate-based methods mostly suffer from inaccurate localization even with the assistance of estimated depth maps.
+
+Based on this observation, we focus on improving the accuracy of 3D center localization. In this work, we propose a lightweight and generalized approach, called Progressive Coordinate Transforms (PCT), to enhance the localization capability for coordinate-based methods. First of all, since the localization regression network in most coordinate-based methods is less accurate but lightweight, we propose to progressively refine its prediction similar to gradient boosting [12, 13]. To be specific, a localization regression network can be seen as a weak learner, and we progressively train multiple consecutive networks such that each network fits the regression residual from the previous networks. These networks share the same lightweight structure so that the computation overhead is negligible. We also predict a confidence score for each network to help stabilize the end-to-end training. We term this progressive refining strategy as confidence-aware localization boosting (CLB). Compared to image-only or pixel-based methods, coordinated-based methods suffer from the problem of missing global context information due to the use of patched input. In order to further improve the localization accuracy, we exploit semantic image representations from 2D detector. We term this module as global context encoding (GCE). We find that GCE can not only improve center localization accuracy, but also contribute to the final 3D box estimation.
+
+Through extensive experiments, our progressive coordinate transforms, consisting of CLB and GCE, is shown to improve popular coordinate-based models [42, 27] by generating more accurate localization. Without bells and whistles, we achieve state-of-the-art monocular 3D detection performance on KITTI [16, 17, 15] with a strong base method [27]. Additionally, this also leads to superior improvements on Waymo Open Dataset [38] compared with the base method PatchNet.
+
+# 2 Related work
+
+# 2.1 Monocular 3D object detection
+
+Existing paradigms for monocular 3D object detection can be categorized into two types: image-only methods and depth-assisted methods.
+
+For image-only methods, they often adapt architectures and good practices from popular 2D detectors [34, 49, 40]. However, locating objects in 3D space is much more challenging without depth information. Hence, several works [30, 2, 25, 6, 49] integrate geometry consistency into the training strategy to constrain the localization prediction. Deep3DBox [30] divides orientation into multi-bins to stably regress them, and combines the 2D-3D box constraint to recover accurate 3D object pose. M3D-RPN [2] utilizes the geometric relationship between 2D and 3D perspectives by sharing the prior anchors and classification targets. MonoPair [6] leverages the spatial relationships between paired objects to improve accuracy on occluded objects. To further improve the performance of truncated objects, MonoFlex [48] decouples the features learning and prediction of truncated objects, and formulates an depth estimation to adaptively combine independent estimator based on uncertainty. [33] designs CaDDN as a fully differentiable end-to-end approach for joint depth estimation and object detection.
+
+Depth-assisted methods often estimate a depth map given a input image, and use it in different ways. Some pixel-based approaches [9, 26] directly feed images and estimated depth maps into networks to generate depth-aware features and enhance the 3D detection performance. Some other coordinatebased approaches first transform the pixels of input images to 3D coordinates by leveraging the depth and camera information, then feed the coordinate proposals to a 3D detector. Pioneering work Pseudo-LiDAR [42] imitates the process of LiDAR-based approaches, which uses LiDAR-based 3D detector upon coordinates proposals. AM3D [29] explores the multi-modal input fusion to embed the complementary RGB cue into the network. Recently, PatchNet [27] points out that the efficacy of pseudo-LiDAR representation comes from the coordinate transform, instead of sophisticated LiDAR-based networks. Hence, they design a simple 2D CNN to perform 3D detection. In this work, we follow the research of coordinate-based methods [42, 27]. Instead of regressing 3D localization directly with a single lightweight network, we propose to progressively refine the prediction inspired by gradient boosting. We also incorporate RGB image information to complement patch proposals and enhance global context modeling. Different from AM3D [29], we utilize the RGB features from the 2D detector directly which can share the same context, and we do not need to train another RGB network from scratch.
+
+# 2.2 Gradient boosting
+
+Gradient boosting is a well-known greedy algorithm proposed in [7], which trains a sequence of learners and progressively improves the prediction results. It is a general learning framework, and has been verified to be a formidable force when applied with lightweight learners. Meanwhile, when each learner in the sequence is heavy, the computation cost becomes high and the performance is not beneficial [24]. Early works in 2D detection area [19, 21, 22] also adopt the boosting mechanism following a standard cascade paradigm, and achieve improved performance. Li et al. [21] treat face detection as an image retrieval task and improve it with a boosted exemplar-based face detector. Karianakis et al. [19] and Li et al. [22] feed convolutional features of proposals instead of hand-crafted features to boosted classifiers and distinguish objects from backgrounds. We can also find the usage of gradient boosting in other computer vision tasks [37, 51].
+
+To our best knowledge, we are the first to explore the boosting mechanism in coordinate-based methods for 3D object detection. We perform this mechanism in two folds. First, instead of the entire 3D detection pipeline, we only progressively boost the localization regression network as its computational cost is insignificant comparing to the entire pipeline. Second, we refine the localization with an additional confidence score in the boosting procedure, such that the loss is balanced. These choices greatly improves the performance with small extra parameters.
+
+# 3 Background
+
+Before diving into the details, we first revisit recent coordinate-based monocular 3D detection methods and present a visual depiction of its common pipeline in Figure 1. The framework usually consists of four main components: 2D bounding box generation, depth map estimation, data transformation and 3D box estimation. Specifically, given an image $I$ , the process can be described as:
+
+2D bounding box generation. An off-the-shelf 2D object detector $F _ { 2 d }$ such as Faster R-CNN [34] is employed on image $I$ to generate region of interests (RoIs), $\mathscr { R } = F _ { 2 d } ( I )$ .
+
+![](images/7300362b1ae21126e252898156e0cc2a3864d52b7ebabd1472089dd56a29c819.jpg)  
+Figure 1: A common pipeline of coordinate-based monocular 3D detectors. It consists of four steps to predict the final 3D boxes: 2D bounding box generation, depth map estimation, data transformation and 3D box estimation. In this work, we focus on improving the last step: 3D box estimation.
+
+Depth map estimation. An off-the-shelf depth estimator $F _ { z }$ such as DORN [14] is applied on image $I$ to predict its depth map, $\mathcal { Z } = F _ { z } ( I )$ .
+
+Data transformation. To convert a pixel $( u , v )$ within a RoI to 3D space, the associated depth $z = \mathcal { Z } ( u , v )$ is used to transform it into its 3D coordinates $\left( c _ { x } , c _ { y } , c _ { z } \right)$ by
+
+$$
+c _ { x } = { \frac { ( u - u ^ { \prime } ) \times z } { f _ { u } } } ; \quad c _ { y } = { \frac { ( v - v ^ { \prime } ) \times z } { f _ { v } } } ; \quad c _ { z } = z
+$$
+
+Here, $\left( c _ { x } , c _ { y } , c _ { z } \right)$ is a pixel in the generated 3D coordinate patch $c$ . $( u ^ { \prime } , v ^ { \prime } )$ is the camera principal point. $f _ { u }$ and $f _ { v }$ are the focal length along horizontal and vertical axis, respectively. $u ^ { \prime } , v ^ { \prime } , \bar { f } _ { u } , f _ { v }$ are usually provided by the datasets.
+
+3D box estimation. Once the 3D coordinates for each RoI are available, the final step is to predict 3D boxes with their center location, rotation and dimension. Different networks $F _ { 3 d }$ such as Frustum PointNet [32] can be employed to conduct 3D box prediction, $\boldsymbol { B } = F _ { 3 d } ( \boldsymbol { c } )$ . Here, $\boldsymbol { B }$ includes the center location $( x , y , z )$ , rotation $( \theta )$ , and dimensions $( w , h , l )$ of the 3D box.
+
+Since the first two steps use off-the-shelf models and the third step can be computed analytically, in this paper, we focus on improving the last step of coordinated-based methods. In particular, our goal is to improve the accuracy of localization prediction motivated by the observation in Table 1.
+
+# 4 Method
+
+In this section, we present our progressive coordinate transforms (PCT) for improved 3D detection. In order to obtain more accurate localization predictions, we introduce a confidence-aware localization boosting mechanism (CLB) in Sec. 4.1 to progressively refine the prediction. Then in Sec. 4.2, we incorporate RGB image information by a global context encoding (GCE) strategy to compensate for the drawbacks of using patch proposals. In the end, we illustrate the overall framework of PCT in Figure 2 (a).
+
+# 4.1 Confidence-aware localization boosting
+
+Following Frustum PointNet [32], most coordinate-based methods [42, 45, 27, 43] divide the last step of 3D box estimation into two major components. The first component is a lightweight 3D localization regressor $F$ , whose input is 3D coordinate proposals generated from data transformation. The second component is a relatively heavy network $G$ used to regress the final 3D box $\boldsymbol { B }$ . Recalling the results in Table 1, 3D localization performance is the weakest point of a coordinate-based model, accounting for up to 50 AP loss when all other modules keep intact. Therefore, can we find an efficient way to improve the accuracy of localization prediction and also generalizes to other coordinated-based methods?
+
+Gradient boosting [12, 13] is a general learning framework that combines multiple weak learners into a single strong one in an iterative fashion. Let $\mathcal { L } ( x )$ be the risk of ensemble models, the algorithm devotes to seek an approximation $h ( x )$ to minimize $\mathcal { L } ( y ^ { \ast } , x ) = \Psi ( y ^ { \ast } , h ( x ) )$ , where $y ^ { * }$ is a target value, $\Psi ( \cdot )$ is the loss function. $h ( x )$ is a linear combination of a set of weak (base) learners $f _ { t } ( x )$ from some class $\mathcal { F }$ , i.e., $\begin{array} { r } { h ( x ) = \sum _ { t = 1 } ^ { t = T } \gamma _ { t } f _ { t } ( x ) + c o n s t } \end{array}$ . Here, $T$ is the total training iterations and $\gamma _ { t }$ is the corresponding weight for each weak learner. To minimize the empirical risk, the algorithm starts with a model $h _ { 0 } ( x )$ , and then incrementally expands it in a greedy manner. This process manages to fit a new weak learner to the residual errors made by the previous set of learners. Mathematically, the
+
+![](images/9523a01cc930e55907dce22cf2a7d99cd0709e1b67369f64a9b03e5aa9e5b4bd.jpg)  
+Figure 2: (a) Schematic illustration of proposed Coordinate Transforms (PCT). We treat $F$ as a weaker learner, and perform coordinate transform $T$ steps via confidence-aware localization boosting (CLB module) to obtain a better localization. Then the refined coordinate proposals combined with corresponding encoded RGB features (GCE module) are fed into network $\mathbf { G }$ to generate final 3D bounding boxes. (b) The data flow of CLB mechanism in detail. For step $t$ , it takes refined coordinate patches $c _ { t - 1 }$ as input, which is transformed based on predicted $\Delta ( x _ { t - 1 } , y _ { t - 1 } , z _ { t - 1 } )$ . Then network $F _ { t }$ generate the residual localization for the next step, and confidence $s _ { t }$ is also generated during the training process.
+
+optimization can be formulated as
+
+$$
+\begin{array} { r l } & { h _ { 0 } ( x ) = \underset { \gamma _ { 0 } } { \arg \operatorname* { m i n } } \mathcal { L } _ { 0 } ( y ^ { * } , x ) ; } \\ & { ~ \quad \quad \cdots } \\ & { h _ { t } ( x ) = h _ { t - 1 } ( x ) + \underset { f _ { t } \in \mathcal { F } } { \arg \operatorname* { m i n } } \mathcal { L } _ { t } ( y ^ { * } , h _ { t - 1 } ( x ) + f _ { t } ( x ) ) . } \end{array}
+$$
+
+Inspired by gradient boosting, we imitate its optimization procedure to progressively adapt localization prediction by multiple localization regressors instead of a single one used in previous works [42, 29, 46, 27]. To be specific, we treat localization network $F$ as a weak learner and stack multiple of them as shown in Figure 2 (b). Given $F$ is a lightweight network, the extra computational cost brought by gradient boosting is insignificant. After the data transformation step, each 2D bounding box obtains its corresponding coordinate patch $c$ . We take the coordinate patch as the input of weak learner $F$ to regress the center localization residual $\Delta ( x , y , z )$ based on the prediction from previous stage,
+
+$$
+\Delta ( x _ { t } , y _ { t } , z _ { t } ) = F _ { t } ( c _ { t - 1 } ) ; { \mathrm { ~ w h e r e ~ } } c _ { t - 1 } = c _ { 0 } - \gamma _ { 0 } ( x _ { 0 } , y _ { 0 } , z _ { 0 } ) - \sum _ { i = 1 } ^ { t - 1 } \gamma _ { i } \Delta ( x _ { i } , y _ { i } , z _ { i } ) .
+$$
+
+We denote $c _ { 0 }$ and $( x _ { 0 } , y _ { 0 } , z _ { 0 } )$ to be the initial coordinate input patch and object location, respectively. Coordinate input patch $c _ { t - 1 }$ is then transformed according to the localization residual prediction, and fed into the next weak learner.
+
+Thus the risk at stage $t$ can be written as,
+
+$$
+\mathcal { L } _ { F _ { t } } ( ( x ^ { * } , y ^ { * } , z ^ { * } ) , c _ { t - 1 } ) = \Psi ( ( x ^ { * } , y ^ { * } , z ^ { * } ) , \gamma _ { 0 } ( x _ { 0 } , y _ { 0 } , z _ { 0 } ) + \sum _ { i = 1 } ^ { t - 1 } \gamma _ { i } \Delta ( x _ { i } , y _ { i } , z _ { i } ) )
+$$
+
+where $( x ^ { * } , y ^ { * } , z ^ { * } )$ indicates the ground truth location. At this point, we can easily see that $\mathrm { E q 4 }$ is a natural derivation from $\operatorname { E q } 2$ . After $T$ iterations, the final adjusted prediction $c _ { T }$ is fed into the network $G$ to estimate the 3D box $\boldsymbol { B }$ $3 , i . e . B = G ( c _ { T } )$ .
+
+Confidence-aware network loss In the case that the target of weak learner $f$ is differentiable, gradient boosting solves the optimization problem in a forward greedy manner as shown in Eq 2. For each iteration, it first fits the weak learner $f$ to the residual error, and then the optimal value of the coefficient weight $\gamma$ is determined for this weak learner. The optimization procedures train iteratively for $T$ iterations.
+
+However, for our center regression task, we would like to train $T$ weak localization networks $F _ { t } ( c _ { t - 1 } ) , t \in { 1 , . . . , T }$ and a 3D box prediction network $G$ in an end-to-end manner instead of bootstrapping. This is challenging in terms of both computational cost and optimization stability, given the simultaneous training of a set of weak learners and their coefficient weights. Therefore, we first simplify the problem by treating all $\gamma _ { t }$ the same and only focus on optimizing the localization networks. However, the contribution from each weak learner $F$ may not be the same during end-toend training, which leads to unstable optimization. Hence, we tailor a confidence-aware boosting loss to facilitate network training, by learning confidence score $s _ { t }$ for each localization loss function $\mathcal { L } _ { F _ { t } }$ . The confidence score $s _ { t }$ is learned from a small decoder and followed a self-balancing formulation closely coupled to the network loss. The overall loss function is defined as
+
+$$
+\mathcal { L } ( \boldsymbol { B } ^ { * } , \boldsymbol { c } _ { 0 } ) = \sum _ { t = 1 } ^ { T } s _ { t } \ast \mathcal { L } _ { \boldsymbol { F } _ { t } } ( ( \boldsymbol { x } ^ { * } , \boldsymbol { y } ^ { * } , \boldsymbol { z } ^ { * } ) , \boldsymbol { c } _ { t - 1 } ) + \lambda _ { s } \prod _ { t = 1 } ^ { T } ( 1 - s _ { t } ) + \mathcal { L } _ { G } ( \boldsymbol { B } ^ { * } , \boldsymbol { c } _ { T } ) ,
+$$
+
+where $B ^ { * }$ is the 3D box ground truth, $( x ^ { \ast } , y ^ { \ast } , z ^ { \ast } ) \in B ^ { \ast }$ and $\lambda _ { s }$ is the balance weight. $s _ { t }$ is the prediction after sigmoid, which represents the confidence of the localization regression loss at $t ^ { t h }$ stage, and $\textstyle \prod _ { t = 1 } ^ { T } ( 1 - s _ { t } )$ is the penalty on the network uncertainty. In other words, if $s _ { t }$ is approaching 1, which means the network is confident about localization refinement at $t ^ { t h }$ stage, then no penalty will be applied. Otherwise, the uncertainty of regression loss is high, thus triggers a higher penalty.
+
+# 4.2 Global context encoding
+
+Typical coordinate-based methods [42, 45, 27, 43] perform 3D detection based on 2D RoIs, which is similar to two-stage 2D detection frameworks, such as Faster R-CNN [34]. In a two-stage 2D object detection framework, the second stage reuses the features from the first stage via RoIPooling [34] or RoIAlign operators [18] guided by ROI proposals, and then a small decoder is used for localization refinement. However, in 3D detection, only cropped patches with coordinates information are fed to the network for 3D box regression. Neither RGB information nor global context is included.
+
+Considering that the RGB information is a vital visual clue, we explore its aggregation in the last 3D box estimation step. Similar to two-stage 2D detectors, we obtain the RGB information by directly cropping the corresponding features from a 2D detector $F _ { 2 d }$ . Then the input to 3D box estimator $G$ can be formulated as $\bar { c \mathbf { \eta } } = \{ [ \mathcal { D } ( { \boldsymbol u } , { \boldsymbol v } ) , \mathcal { A } ( { \boldsymbol u } , { \boldsymbol v } ) ] , \forall ( { \boldsymbol u } , { \boldsymbol v } ) \in \mathcal { R } \}$ . Here, $\mathcal { D } ( \cdot )$ represents the data transformation function and $\boldsymbol { \mathcal { A } } ( \cdot )$ represents the RoIAlign operation. Both operations are performed upon the generated regions of interest from $\mathcal { R }$ .
+
+After RoIAlign operation, features are of size $C \times K \times K$ , where $C$ is the number of channels and $K \times K$ is the corresponding width and height, respectively. A small feature encoder is then used to encode cropped features into vectors with the dimension of $C$ . A feature fusion is followed to integrate coordinate representations with the obtained image representations. Benefiting from the large receptive fields of image feature representations, 3D box estimator can now have access to global context. Besides, directly cropping on RGB features also avoids learning image representations of RoIs from scratch and reduces the overall network parameters.
+
+# 5 Experiments
+
+Two monocular 3D detection benchmarks are introduced in Sec. 5.1 and Sec. 5.2, while experimental implementation details are described in Sec. 5.3. In Sec. 5.4, we conduct main analysis on KITTI dataset [16, 17, 15] with base method PatchNet [27] given its current best performance. More experiments on Waymo Open Dataset [38] are also demonstrated to further verify the generality of our proposed PCT in Sec. 5.5.
+
+# 5.1 KITTI setup
+
+We first evaluate our method on the KITTI benchmark [16, 17, 15], which contains 7,481 and 7,518 images for training and testing respectively. We follow [5] to split the 7,481 training images into 3712 for training and 3,769 for validation.
+
+Table 2: Ablative analysis on KITTI validation set for $\mathrm { A P _ { 3 D } }$ and $\mathrm { A P _ { B E V } }$ at $\mathrm { I o U } = 0 . 7$ . Experiment Group (I) is the baseline method. Different experiment settings are explored: (II) applying localization boosting without confidence constraint, (III) performing confidence-aware localization boosting algorithm, (IV) adding global context encoding on Group (II), (V) our full approach.   
+
+<table><tr><td rowspan="2">Group</td><td rowspan="2">Localization Boosting</td><td rowspan="2">Uncertainty</td><td rowspan="2">GCE</td><td colspan="3">AP3D</td><td colspan="3">APBEV</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>I</td><td>-</td><td>-</td><td>1</td><td>25.88</td><td>36.07</td><td>20.99</td><td>33.34</td><td>46.39</td><td>27.54</td></tr><tr><td>Ⅱ</td><td>√</td><td></td><td>1</td><td>26.78</td><td>37.29</td><td>24.11</td><td>34.39</td><td>47.08</td><td>28.28</td></tr><tr><td>Ⅲ</td><td>√</td><td>√</td><td>=</td><td>27.24</td><td>38.32</td><td>24.39</td><td>33.92</td><td>46.77</td><td>27.98</td></tr><tr><td>IV</td><td>√</td><td></td><td></td><td>27.12</td><td>37.38</td><td>24.11</td><td>34.46</td><td>46.70</td><td>28.32</td></tr><tr><td>V</td><td>√</td><td>√</td><td></td><td>27.53</td><td>38.39</td><td>24.44</td><td>34.65</td><td>47.16</td><td>28.47</td></tr></table>
+
+Precision-recall curves are adopted for evaluation, and we report the average precision (AP) results of 3D and Bird’s eye view (BEV) object detection on KITTI validation and test set. For fair comparison to previous literature, the 40 recall positions-based metric $A P | _ { R 4 0 }$ is reported on test set while $\bar { A } P | _ { R 1 1 }$ is reported on validation set. Three levels of difficulty are defined in the benchmark according to the 2D bounding box height, occlusion, and truncation degree, namely, “Easy”, “Mod.”, and “Hard”. The KITTI benchmark ranks all methods based on the $\mathrm { A P _ { 3 D } }$ of “Mod.”. In particular, we focus on the “Car” category as in [42, 27], and we adopt $\mathrm { I o U } = 0 . 7$ as threshold for evaluation.
+
+# 5.2 Waymo setup
+
+We also carry out experiments on large-scale, high quality and diverse dataset, Waymo open dataset [38]. It provides pre-defined 798 training sequences and 202 validation sequences from different scenes, and another 150 test sequences without labels. The dataset contains camera images from five high-resolution pinhole cameras, and we only consider images with their 3D labels from front camera for monocular 3D detection task. We sample every third frame from the training sequences (total 52,386 images) as in CaDDN [33] to form the training set due to its large scale. And validation set contains all the 39,848 images from 202 different scenes.
+
+For evaluation, we adopt the officially released evaluation [39] to calculate the mean average precision (mAP) and the mean average precision weighted by heading (mAPH). Two levels are included according to difficulty rating, which are defined by LiDAR points. 3D labels without any points are ignored, LEVEL_2 is assigned to examples when it contains equal or lesser than 5 points, while the rest of the examples are assigned to LEVEL_1. Additionally, three distances (0 - 30m, 30 - 50m, $5 0 \mathrm { m }$ $- \infty )$ ) to sensor are considered during evaluation.
+
+# 5.3 Implementation details
+
+Our overall framework of PCT can be visualized in Figure 2. In terms of implementation details, we instantiate our algorithm on two widely adopted coordinate-based methods with public released code [28], PatchNet and Pseudo LiDAR. Bearing efficiency in mind, we use a real-time 2D detector RTM3D [23] with DLA-34 [47] as backbone. For the sake of fair comparison, we adopt depth predictor DORN [14] on KITTI dataset as in most depth-assisted literature. Since there is no published depth results on Waymo open dataset, we adopt a most recent monocular depth estimator AdaBins [1] trained on Waymo training set. For the CLB mechanism, we inherit the original localization regression framework in each method. Each $F _ { t }$ shares the same structure. The corresponding confidence is generated following the last convolutional layers of $F _ { t }$ with three linear layers and a Sigmoid function. $T = 3$ and $\lambda _ { s } = 1$ are set for the following experiments except for the ablation study on boosting iterations. For GCE module, we get the corresponding input image features by performing RoIAlign on the features from last convolutional layer of 2D detector. We set the output of RoIAlign as $1 6 \times 1 6$ . As 2D detector use DLA-34 as backbone, the obtained image feature representations have the size of $6 4 \times 1 6 \times 1 6$ and entitle arbitrary sized receptive field theoretically due to the embedded deformable convolution [8]. The structure of feature encoder in global context module is two common $6 4 \times 3 \times 3$ convolutional layers (stride $^ { = 4 }$ ) and a $6 4 \times 1 \times 1$ convolutional layer (stride ${ \mathop : } = 1$ ). Hence, image feature representations are encoded to a vector with 64-dim. The obtained features are then concatenated with the coordinate feature vectors from the final global pooling of box prediction network $G$ . With the lightweight structure, we are able to optimize the network end-to-end on a single Nvidia V100
+
+![](images/e1d29ed86083cfdcecf02e52932d516313e169150f98e7200986c3955e7caf75.jpg)  
+Figure 3: The statistic analysis and comparison on different Localization boosting stage when $T = 3$ . The vertical axis of the chart represents the number of samples after normalization. “loc. $1 / 2 / 3 ^ { \circ }$ denotes the $1 / 2 / 3 ^ { t h }$ step of localization errors in $F$ and “loc. $. 4 \ "$ is the final localization errors in $G$ . Note that when the curve is more thin, tall, and closer to zeros, the localization is more accurate.
+
+GPU with 16G memory. The training criterion for network $F$ and $G$ and other training settings follow the corresponding base methods for fair comparison.
+
+# 5.4 Method analysis on KITTI dataset
+
+Main ablative analysis In Table 2, we conduct ablation studies to analyze the effectiveness of our contributions: I) Without any localization regression network $F$ , network $G$ generates 3D box prediction directly. II) This configuration only contains localization boosting part without confidence constraint. III) The entire CLB mechanism is included to progressively regress center localization. IV) GCE module is added to the network based on the localization boosting block without confidence constraint since the feature fusion can be performed on either the localization regression networks $F$ or 3D box prediction network $G$ . V) Our full method with all the components.
+
+As depicted in Table 2, we can observe that the performance continues to grow with the addition of every component. From group II, localization boosting brings a noticeable improvement on all settings especially “Hard” set, which confirms its effectiveness in increasing localization accuracy. Group III shows that balancing training loss by adding confidence leads to better and more stable optimization. Group IV reveals that the proposed GCE module can effectively equip RGB information and global context with 3D coordinate representations. In the end, Group V demonstrates the complementarity of the proposed CLB mechanism and GCE module, leading to an improvement from 25.88/36.07/20.99 to 27.53/38.39/24.44 compared with Group I.
+
+Table 3: Comparison of different boosting iteration settings on KITTI validation split set.   
+
+<table><tr><td rowspan="2">Localization Boost (T)</td><td colspan="3">AP3D/APBEV</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>1</td><td>25.88/33.34</td><td>36.07/46.39</td><td>20.99/27.54</td></tr><tr><td>1</td><td>26.31/34.14</td><td>36.40/46.80</td><td>21.07/28.04</td></tr><tr><td>2</td><td>26.69/34.06</td><td>37.17/46.42</td><td>24.04/28.03</td></tr><tr><td>3</td><td>26.78/34.39</td><td>37.29/47.08</td><td>24.11/28.28</td></tr><tr><td>4</td><td>26.77/34.21</td><td>37.12/47.00</td><td>23.48/28.23</td></tr><tr><td>5</td><td>26.64/34.43</td><td>37.24/47.04</td><td>23.89/28.27</td></tr></table>
+
+Table 4: Evaluation of different coordinate feature fusion with GCE on KITTI validation set. Baseline is the Group (II) in Table 2.   
+
+<table><tr><td>Method</td><td colspan="3">AP3D</td></tr><tr><td>Baseline</td><td>Mod. 26.78</td><td>Easy 37.29</td><td>Hard 24.11</td></tr><tr><td>F+GCE</td><td>27.08</td><td>37.33</td><td>24.07</td></tr><tr><td>G+GCE</td><td>27.12</td><td>37.38</td><td>24.11</td></tr><tr><td>All + GCE</td><td>27.07</td><td>37.43</td><td>24.18</td></tr></table>
+
+Localization boosting iteration settings We explore the effect of different localization boosting iteration settings in this part. For a fair comparison, we do not perform the confidence constraint on regression loss. As illustrated in Table 3, when boosting iteration $T = 3$ , we achieve the best 3D detection performance. More iterations of boosting do not bring improvements, which might be caused by overfitting with the increasing of network parameters.
+
+To verify the improvement of each step in boosting procedure, we conduct the comparison of localization errors at iteration $T = 3$ on the specific metrics (location “xyz”) of the ground truth. In particular, three stacked localization networks $F$ generate intermediate localization $^ { \bullet \bullet } \mathrm { l o c . } 1 / 2 / 3 ^ { \bullet \bullet }$ and $G$ output the final localization “loc.4”. As shown in Figure 3, we can see that the distributions of $\mathbf { \ddot { x } } ^ { , 5 }$ , “y” and “z” errors tend to distributed to zero with localization boosting iterating. For instance, the red line in left chart is narrow and tall near zero along horizontal axis compared with other lines, which means that the corresponding x coordinate is more accurate than others. This further suggests that localization boosting is useful for object localization. The corresponding BEV visualization will be shown in Supplementary Material.
+
+![](images/c1f84afe258a114f12040dbc85530e9d69baa6299bc5ce42f2def65beeebaa14.jpg)  
+Figure 4: Qualitative comparison of ground truth (green), base method PatchNet (blue), and our method (red) on KITTI val set. The first and second rows show RGB and BEV images respectively.
+
+Table 5: Comparison of generalization on KITTI validation set. $^ *$ denotes that the method is reproduced by ourselves.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">AP3D</td><td colspan="3">APBEV</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Pseudo-LiDAR*[42]</td><td>23.04</td><td>32.27</td><td>19.67</td><td>31.06</td><td>42.45</td><td>25.67</td></tr><tr><td>Pseudo-LiDAR+ CLB</td><td>24.14</td><td>34.46</td><td>20.16</td><td>32.41</td><td>44.98</td><td>26.82</td></tr><tr><td>Pseudo-LiDAR + CLB +GCE</td><td>24.43</td><td>34.34</td><td>20.18</td><td>32.50</td><td>45.35</td><td>26.91</td></tr></table>
+
+Impact of global context encoding We also explore where the global context representation fusion operates. We take Group II as the baseline, and perform feature fusion on localization regression network $F$ (row one), 3D estimation network $G$ (row two) or on both (final row). RoI features are encoded into a vector with GCE and then concatenate with the feature vectors from network ( $F$ or $G$ ) global pooling. As shown in Table 4, the operation on $G$ outperforms it on $F$ , which indicates that image representation is more suitable for the overall box prediction rather than only localization as it contains the additional semantic appearance information. Although operation on all networks achieves a lightly higher than it on $G$ on the “Easy” and “Hard” set, introducing parameters is much larger due to operation on stacked localization networks. Hence, we only apply GCE on network $G$ in our approach for a lightweight network and avoid overfitting during training.
+
+Generally applicable to other coordinate-based algorithm In this section, we demonstrate the generalization capability of our algorithm to classic coordinate-based methods Pseudo-LiDAR [42]. As shown in Table 5, each component of our algorithm improves the original methods a lot. Specially, our approach improves Pseudo-LiDAR by 1.43/2.06/0.51 while 1.22/1.99/3.36 gains on PatchNet.
+
+Comparison with state-of-the-arts We build our test detector on the current state-of-the-art coordinate-based method PatchNet, and results are shown in Table 6. Quantitatively, our method achieves the highest performance on “Mod.” set with 22 FPS on NvidiaTesla v100 including 2D detector inference time, which is the main setting for ranking on the benchmark. Specially, large margins, 2.25/5.32/1.14 on 3D detection and 2.17/6.68/0.95 on BEV, are observed over the base method PatchNet with only additional 3.41M parameters. Besides, our methods also outperforms the pixel-based state-of-the-arts methods Liu et al. [26] especially on “Hard” set.
+
+Qualitative comparisons are shown in Figure 4. The ground truth, base method (PatchNet), and our method are colored in green, blue, and red, respectively. For better visualization, the first and second rows show RGB images and BEV images, respectively. Compared with the base method, our algorithm can produce higher-quality 3D bounding boxes in different kinds of scenes.
+
+Table 6: Comparison with SoTA methods on the KITTI test set at $\mathrm { I o U } = 0 . 7$ . Our algorithm achieves new SoTA performance. “Depth” means if the method belongs to depth-assisted methods or not. “Type” indicates the method input pattern, “Pixel” denotes the methods with image as inputs directly while “Coordinate” means the coordinate-based methods with 3D coordinates as inputs.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Depth</td><td rowspan="2">Type</td><td colspan="3">AP3D</td><td colspan="3">APBEV</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>AM3D [29]</td><td>yes</td><td>Coordinate</td><td>10.74</td><td>16.50</td><td>9.52</td><td>17.32</td><td>25.30</td><td>14.91</td></tr><tr><td>PatchNet [27]</td><td>yes</td><td>Coordinate</td><td>11.12</td><td>15.68</td><td>10.17</td><td>16.86</td><td>22.97</td><td>14.97</td></tr><tr><td>GrooMeD-NMS[20]</td><td>no</td><td>Pixel</td><td>12.32</td><td>18.10</td><td>9.65</td><td>18.27</td><td>26.19</td><td>14.05</td></tr><tr><td>Kinematic3D[3]</td><td>yes</td><td>Pixel</td><td>12.72</td><td>19.07</td><td>9.17</td><td>17.52</td><td>26.69</td><td>13.10</td></tr><tr><td>DDMP-3D[41]</td><td>yes</td><td>Pixel</td><td>12.78</td><td>19.71</td><td>9.80</td><td>17.89</td><td>28.08</td><td>13.44</td></tr><tr><td>Liu et al. [26]</td><td>yes</td><td>Pixel</td><td>13.25</td><td>21.65</td><td>9.91</td><td>17.98</td><td>29.81</td><td>13.08</td></tr><tr><td>PCT</td><td>yes</td><td>Coordinate</td><td>13.37</td><td>21.00</td><td>11.31</td><td>19.03</td><td>29.65</td><td>15.92</td></tr></table>
+
+# 5.5 Results on Waymo Open Dataset
+
+Table 7 shows the results of base method PatchNet [27] and our proposed PCT. It can be observed that our method consistently outperforms the base method on mAP/mAPH of $0 . 5 0 \% / 0 . 5 1 \%$ and $0 . 2 8 \% / 0 . 3 0 \%$ on the LEVEL_1 and LEVEL_2 difficulties respectively under $\mathrm { I o U } = 0 . 7$ . Again, our method is efficient, e.g, it takes 5 days to complete training on large scale Waymo dataset with a 8-GPU node. More qualitative results can be seen at Supplementary Material.
+
+Table 7: 3D performance on Waymo validation set. We demonstrate results of base method PatchNet [27] and corresponding PCT at $\mathrm { I o U } = 0 . 7$ and $\mathrm { I 0 U } = 0 . 5$ . Our proposed PCT achieves consistent improvements on all settings.   
+
+<table><tr><td rowspan="2">Difficulty</td><td rowspan="2">Threshold</td><td rowspan="2">Method</td><td colspan="4">3D mAP/3D mAPH</td></tr><tr><td>Overall</td><td>0-30m</td><td>30-50m</td><td>50-8</td></tr><tr><td rowspan="3">LEVEL_1</td><td>IoU=0.7</td><td>PatchNet PCT</td><td>0.39/0.37 0.89 / 0.88</td><td>1.67 / 1.63 3.18 / 3.15</td><td>0.13/0.12</td><td>0.03/0.03 0.07 /0.07</td></tr><tr><td rowspan="2">IoU=0.5</td><td>PatchNet</td><td>2.92/2.74</td><td>10.03/9.75</td><td>0.27 / 0.27 1.09/ 0.96</td><td>0.23/0.18</td></tr><tr><td>PCT</td><td>4.20 /4.15</td><td>14.70 / 14.54</td><td>1.78 / 1.75</td><td>0.39 / 0.39</td></tr><tr><td rowspan="3">LEVEL_2</td><td>IoU=0.7</td><td>PatchNet</td><td>0.38/0.36</td><td>1.67 / 1.63</td><td>0.13/0.11</td><td>0.03/0.03</td></tr><tr><td rowspan="2">IoU=0.5</td><td>PCT PatchNet</td><td>0.66 / 0.66</td><td>3.18 /3.15</td><td>0.27 /0.26</td><td>0.07 /0.07</td></tr><tr><td></td><td>2.42/2.28</td><td>10.01/9.73</td><td>1.07 /0.94</td><td>0.22/0.16</td></tr><tr><td></td><td></td><td>PCT</td><td>4.03 /3.99</td><td>14.67 / 14.51</td><td>1.74 / 1.71</td><td>0.36 / 0.35</td></tr></table>
+
+# 6 Conclusions
+
+In this paper, we have introduced a novel approach PCT to address the inaccurate localization problem for monocular 3D object detection. This is achieved by iteratively transforming the coordinate representation with a confidence-aware booting mechanism. Meanwhile, global context is introduced to compensate for the missing of semantic image representation in coordinated-based methods. Through extensive experiments, we have shown that our proposed PCT substantially improve the performance of the coordinate-based model by a large margin, and achieve state-of-the-art monocular 3D detection performance on KITTI test set. Moreover, we also show consistent improvements compared to the strong baseline on the large-scale Waymo Open dataset.
+
+There are several limitations that could indicate the possible directions for future work. First, the performance of off-the-shelf 2D detector directly influences the accuracy of coordinate-based methods, hence how to effectively design the 3D box estimation algorithm to fit with existing 2D detectors is important. Second, we only concentrate on the lightweight coordinate-based methods. It requires further exploration to extend our approach to pixel-based methods. Finally, our proposed global context encoding is a simple module. Despite working well, a more tailored feature fusion strategy between coordinate representation and RGB image representation is worth exploring.
+
+Potential impacts. Our method focuses on the monocular 3D detection which can be applied in autonomous driving field. One potential social problem of our work is that it may aggravate the employment crisis of human servants and drivers, which replaces human with autonomous robots and intelligent systems.
+
+# Acknowledgments
+
+This work was supported by Shanghai Municipal Science and Technology Major Projects (No.2021SHZDZX0103 and No.2018SHZDZX01).
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We have described exactly in abstract section and Sec. 1.   
+(b) Did you describe the limitations of your work? [Yes] See Sec. 6.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Sec. 6.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We demonstrate the codes in Github website. The dataset is public and URL [15, 39] is also attached in Sec. 5.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Sec. 5.3.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We report experiments results at the fixed seed.
+
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] The type of resources can be seen at Sec. 5.3. And each GPU can run two experiments, thus, our included experiments (total number of 15) in paper require about 8 Nvidia Tesla v100 GPUs (16G).
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See Sec. 5   
+(b) Did you mention the license of the assets? [Yes] We conduct experiments on KITTI (license: CC BY-NC-SA 3.0) and Waymo (license: Custom (non-commercial)).   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] We conduct experiments only on public datasets.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/4G2dEuRZ7eO/4G2dEuRZ7eO_content_list.json

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+        "text": "Progressive Coordinate Transforms for Monocular 3D Object Detection ",
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+    {
+        "type": "text",
+        "text": "Li Wang1∗ Li Zhang1† Yi Zhu2 Zhi Zhang2 Tong He2 Mu Li2 Xiangyang Xue1 1Fudan University 2Amazon Inc. ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
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+    },
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+        "text": "Recognizing and localizing objects in the 3D space is a crucial ability for an AI agent to perceive its surrounding environment. While significant progress has been achieved with expensive LiDAR point clouds, it poses a great challenge for 3D object detection given only a monocular image. While there exist different alternatives for tackling this problem, it is found that they are either equipped with heavy networks to fuse RGB and depth information or empirically ineffective to process millions of pseudo-LiDAR points. With in-depth examination, we realize that these limitations are rooted in inaccurate object localization. In this paper, we propose a novel and lightweight approach, dubbed Progressive Coordinate Transforms (PCT) to facilitate learning coordinate representations. Specifically, a localization boosting mechanism with confidence-aware loss is introduced to progressively refine the localization prediction. In addition, semantic image representation is also exploited to compensate for the usage of patch proposals. Despite being lightweight and simple, our strategy leads to superior improvements on the KITTI and Waymo Open Dataset monocular 3D detection benchmarks. At the same time, our proposed PCT shows great generalization to most coordinatebased 3D detection frameworks. The code is available at: https://github.com/ amazon-research/progressive-coordinate-transforms. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Object detection is a fundamental and challenging task in scene understanding applications. Recently, 3D object detection has received increasing attention and found applications in a wide range of scenarios such as autonomous driving, robotics, visual navigation and mixed reality. Despite the great progress from the area of 2D object detection [34, 49, 40, 18, 4], 3D object detection remains a largely unsolved problem as it aims to predict the object location in the 3D space alongside 3D object dimension and orientation. ",
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+        "text": "Existing prevalent approaches [50, 44, 35, 10, 11] for 3D object detection largely rely on LiDAR sensors, which provide accurate 3D point clouds of the scene. Although these approaches achieve superior performance, the dependence on expensive equipment severely limits their applicability to generic 3D perception. There also exists a cheaper alternative that takes a single-view RGB image as input, termed as monocular 3D object detection. However, its performance is far from satisfactory as itself is an ill-posed problem due to the loss of depth information in 2D image planes. Hence, several recent attempts introduce depth information to help monocular 3D detection. Such attempts can be roughly categorized into two directions, pixel-based and coordinate-based. Pixel-based approaches [9, 36, 31, 41] turn to use estimated depth map as additional input for improved detection performance. But at the same time, this leads to heavy computational burden and large memory footprint since they often operate on the entire image. Coordinated-based approaches [42, 29, 46, 27] pursue the coordinate representations as in LiDAR-based methods. They use the predicted depth map to convert the monocular image pixels to 3D coordinate representations, then apply a 3D detector on the converted coordinates. In particular, they are often lightweight since their network inputs are object proposals generated by 2D detectors [29, 27]. However, the performance of coordinated-based methods lags far behind LiDAR-based methods. So we ask, can we identify the bottleneck that holds back the 3D detection accuracy of coordinate-based methods and how can we improve them? ",
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+    {
+        "type": "table",
+        "img_path": "images/4737389cf59a150a31599b624a8e493a4cf5799866976f2476b32a23fe07ca0e.jpg",
+        "table_caption": [
+            "Table 1: Probing investigation on coordinate-based methods, PatchNet [27] and Pseudo-LiDAR [42]. We examine the potential improvement by replacing the predicted factor with the corresponding ground truth. $^ *$ indicates our reproduced performance. We can see that coordinate-based methods mostly suffer from inaccurate localization. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Factor</td><td colspan=\"3\">PatchNet*[AP3D/APBEV]</td><td colspan=\"3\">Pseudo-LiDAR*[AP3D/APBEv]</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Baseline</td><td>26.31/34.14</td><td>36.40/46.80</td><td>21.07/28.04</td><td>23.04/31.06</td><td>32.27/42.45</td><td>19.67/25.67</td></tr><tr><td>dimension</td><td>27.26/34.62</td><td>40.32/47.24</td><td>24.29/28.38</td><td>25.88/31.97</td><td>36.09/44.35</td><td>20.88/26.60</td></tr><tr><td>rotation</td><td>26.25/34.04</td><td>36.09/46.25</td><td>23.49/27.99</td><td>23.88/31.31</td><td>32.42/42.74</td><td>19.85/26.07</td></tr><tr><td>X</td><td>32.80/41.43</td><td>45.60/56.22</td><td>27.38/34.63</td><td>28.36/36.77</td><td>39.69/50.78</td><td>25.08/29.92</td></tr><tr><td>y</td><td>30.16/34.14</td><td>40.94/46.80</td><td>24.58/28.04</td><td>25.53/31.06</td><td>35.19/42.45</td><td>20.69/25.67</td></tr><tr><td>Z</td><td>42.42/53.48</td><td>55.42/68.29</td><td>35.54/45.60</td><td>38.37/50.81</td><td>50.04/63.96</td><td>32.24/43.32</td></tr><tr><td>location(xyz)</td><td>72.58/75.27</td><td>81.41/85.14</td><td>57.69/66.10</td><td>64.36/73.37</td><td>79.13/83.77</td><td>55.64/58.27</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In order to determine the bottleneck, we conduct an investigation on two widely adopted coordinatebased methods, PatchNet [27] and Pseudo-LiDAR [42]. Specifically, for each prediction target, we examine the potential improvement by replacing its value with the corresponding ground truth, and then re-compute the 3D detection accuracy. As shown in Table 1, using ground truth dimension and rotation do not bring significant improvements over the baseline. But using ground truth location (i.e., x/y/z values of the objects) almost triples detection accuracy. This indicates that coordinate-based methods mostly suffer from inaccurate localization even with the assistance of estimated depth maps. ",
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+        "type": "text",
+        "text": "Based on this observation, we focus on improving the accuracy of 3D center localization. In this work, we propose a lightweight and generalized approach, called Progressive Coordinate Transforms (PCT), to enhance the localization capability for coordinate-based methods. First of all, since the localization regression network in most coordinate-based methods is less accurate but lightweight, we propose to progressively refine its prediction similar to gradient boosting [12, 13]. To be specific, a localization regression network can be seen as a weak learner, and we progressively train multiple consecutive networks such that each network fits the regression residual from the previous networks. These networks share the same lightweight structure so that the computation overhead is negligible. We also predict a confidence score for each network to help stabilize the end-to-end training. We term this progressive refining strategy as confidence-aware localization boosting (CLB). Compared to image-only or pixel-based methods, coordinated-based methods suffer from the problem of missing global context information due to the use of patched input. In order to further improve the localization accuracy, we exploit semantic image representations from 2D detector. We term this module as global context encoding (GCE). We find that GCE can not only improve center localization accuracy, but also contribute to the final 3D box estimation. ",
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+        "type": "text",
+        "text": "Through extensive experiments, our progressive coordinate transforms, consisting of CLB and GCE, is shown to improve popular coordinate-based models [42, 27] by generating more accurate localization. Without bells and whistles, we achieve state-of-the-art monocular 3D detection performance on KITTI [16, 17, 15] with a strong base method [27]. Additionally, this also leads to superior improvements on Waymo Open Dataset [38] compared with the base method PatchNet. ",
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+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "2.1 Monocular 3D object detection ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Existing paradigms for monocular 3D object detection can be categorized into two types: image-only methods and depth-assisted methods. ",
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+        "type": "text",
+        "text": "For image-only methods, they often adapt architectures and good practices from popular 2D detectors [34, 49, 40]. However, locating objects in 3D space is much more challenging without depth information. Hence, several works [30, 2, 25, 6, 49] integrate geometry consistency into the training strategy to constrain the localization prediction. Deep3DBox [30] divides orientation into multi-bins to stably regress them, and combines the 2D-3D box constraint to recover accurate 3D object pose. M3D-RPN [2] utilizes the geometric relationship between 2D and 3D perspectives by sharing the prior anchors and classification targets. MonoPair [6] leverages the spatial relationships between paired objects to improve accuracy on occluded objects. To further improve the performance of truncated objects, MonoFlex [48] decouples the features learning and prediction of truncated objects, and formulates an depth estimation to adaptively combine independent estimator based on uncertainty. [33] designs CaDDN as a fully differentiable end-to-end approach for joint depth estimation and object detection. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Depth-assisted methods often estimate a depth map given a input image, and use it in different ways. Some pixel-based approaches [9, 26] directly feed images and estimated depth maps into networks to generate depth-aware features and enhance the 3D detection performance. Some other coordinatebased approaches first transform the pixels of input images to 3D coordinates by leveraging the depth and camera information, then feed the coordinate proposals to a 3D detector. Pioneering work Pseudo-LiDAR [42] imitates the process of LiDAR-based approaches, which uses LiDAR-based 3D detector upon coordinates proposals. AM3D [29] explores the multi-modal input fusion to embed the complementary RGB cue into the network. Recently, PatchNet [27] points out that the efficacy of pseudo-LiDAR representation comes from the coordinate transform, instead of sophisticated LiDAR-based networks. Hence, they design a simple 2D CNN to perform 3D detection. In this work, we follow the research of coordinate-based methods [42, 27]. Instead of regressing 3D localization directly with a single lightweight network, we propose to progressively refine the prediction inspired by gradient boosting. We also incorporate RGB image information to complement patch proposals and enhance global context modeling. Different from AM3D [29], we utilize the RGB features from the 2D detector directly which can share the same context, and we do not need to train another RGB network from scratch. ",
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+    {
+        "type": "text",
+        "text": "2.2 Gradient boosting ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Gradient boosting is a well-known greedy algorithm proposed in [7], which trains a sequence of learners and progressively improves the prediction results. It is a general learning framework, and has been verified to be a formidable force when applied with lightweight learners. Meanwhile, when each learner in the sequence is heavy, the computation cost becomes high and the performance is not beneficial [24]. Early works in 2D detection area [19, 21, 22] also adopt the boosting mechanism following a standard cascade paradigm, and achieve improved performance. Li et al. [21] treat face detection as an image retrieval task and improve it with a boosted exemplar-based face detector. Karianakis et al. [19] and Li et al. [22] feed convolutional features of proposals instead of hand-crafted features to boosted classifiers and distinguish objects from backgrounds. We can also find the usage of gradient boosting in other computer vision tasks [37, 51]. ",
+        "bbox": [
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+        "type": "text",
+        "text": "To our best knowledge, we are the first to explore the boosting mechanism in coordinate-based methods for 3D object detection. We perform this mechanism in two folds. First, instead of the entire 3D detection pipeline, we only progressively boost the localization regression network as its computational cost is insignificant comparing to the entire pipeline. Second, we refine the localization with an additional confidence score in the boosting procedure, such that the loss is balanced. These choices greatly improves the performance with small extra parameters. ",
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+    {
+        "type": "text",
+        "text": "3 Background ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Before diving into the details, we first revisit recent coordinate-based monocular 3D detection methods and present a visual depiction of its common pipeline in Figure 1. The framework usually consists of four main components: 2D bounding box generation, depth map estimation, data transformation and 3D box estimation. Specifically, given an image $I$ , the process can be described as: ",
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+    {
+        "type": "text",
+        "text": "2D bounding box generation. An off-the-shelf 2D object detector $F _ { 2 d }$ such as Faster R-CNN [34] is employed on image $I$ to generate region of interests (RoIs), $\\mathscr { R } = F _ { 2 d } ( I )$ . ",
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+        "img_path": "images/7300362b1ae21126e252898156e0cc2a3864d52b7ebabd1472089dd56a29c819.jpg",
+        "image_caption": [
+            "Figure 1: A common pipeline of coordinate-based monocular 3D detectors. It consists of four steps to predict the final 3D boxes: 2D bounding box generation, depth map estimation, data transformation and 3D box estimation. In this work, we focus on improving the last step: 3D box estimation. "
+        ],
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+        "type": "text",
+        "text": "Depth map estimation. An off-the-shelf depth estimator $F _ { z }$ such as DORN [14] is applied on image $I$ to predict its depth map, $\\mathcal { Z } = F _ { z } ( I )$ . ",
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+        "type": "text",
+        "text": "Data transformation. To convert a pixel $( u , v )$ within a RoI to 3D space, the associated depth $z = \\mathcal { Z } ( u , v )$ is used to transform it into its 3D coordinates $\\left( c _ { x } , c _ { y } , c _ { z } \\right)$ by ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/14a9d072d611d93be41fbce212a03d4aebb6c15be68918003b3b1b4f24dc2f51.jpg",
+        "text": "$$\nc _ { x } = { \\frac { ( u - u ^ { \\prime } ) \\times z } { f _ { u } } } ; \\quad c _ { y } = { \\frac { ( v - v ^ { \\prime } ) \\times z } { f _ { v } } } ; \\quad c _ { z } = z\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Here, $\\left( c _ { x } , c _ { y } , c _ { z } \\right)$ is a pixel in the generated 3D coordinate patch $c$ . $( u ^ { \\prime } , v ^ { \\prime } )$ is the camera principal point. $f _ { u }$ and $f _ { v }$ are the focal length along horizontal and vertical axis, respectively. $u ^ { \\prime } , v ^ { \\prime } , \\bar { f } _ { u } , f _ { v }$ are usually provided by the datasets. ",
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+    {
+        "type": "text",
+        "text": "3D box estimation. Once the 3D coordinates for each RoI are available, the final step is to predict 3D boxes with their center location, rotation and dimension. Different networks $F _ { 3 d }$ such as Frustum PointNet [32] can be employed to conduct 3D box prediction, $\\boldsymbol { B } = F _ { 3 d } ( \\boldsymbol { c } )$ . Here, $\\boldsymbol { B }$ includes the center location $( x , y , z )$ , rotation $( \\theta )$ , and dimensions $( w , h , l )$ of the 3D box. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Since the first two steps use off-the-shelf models and the third step can be computed analytically, in this paper, we focus on improving the last step of coordinated-based methods. In particular, our goal is to improve the accuracy of localization prediction motivated by the observation in Table 1. ",
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+    {
+        "type": "text",
+        "text": "4 Method ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we present our progressive coordinate transforms (PCT) for improved 3D detection. In order to obtain more accurate localization predictions, we introduce a confidence-aware localization boosting mechanism (CLB) in Sec. 4.1 to progressively refine the prediction. Then in Sec. 4.2, we incorporate RGB image information by a global context encoding (GCE) strategy to compensate for the drawbacks of using patch proposals. In the end, we illustrate the overall framework of PCT in Figure 2 (a). ",
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+    {
+        "type": "text",
+        "text": "4.1 Confidence-aware localization boosting ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Following Frustum PointNet [32], most coordinate-based methods [42, 45, 27, 43] divide the last step of 3D box estimation into two major components. The first component is a lightweight 3D localization regressor $F$ , whose input is 3D coordinate proposals generated from data transformation. The second component is a relatively heavy network $G$ used to regress the final 3D box $\\boldsymbol { B }$ . Recalling the results in Table 1, 3D localization performance is the weakest point of a coordinate-based model, accounting for up to 50 AP loss when all other modules keep intact. Therefore, can we find an efficient way to improve the accuracy of localization prediction and also generalizes to other coordinated-based methods? ",
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+        "type": "text",
+        "text": "Gradient boosting [12, 13] is a general learning framework that combines multiple weak learners into a single strong one in an iterative fashion. Let $\\mathcal { L } ( x )$ be the risk of ensemble models, the algorithm devotes to seek an approximation $h ( x )$ to minimize $\\mathcal { L } ( y ^ { \\ast } , x ) = \\Psi ( y ^ { \\ast } , h ( x ) )$ , where $y ^ { * }$ is a target value, $\\Psi ( \\cdot )$ is the loss function. $h ( x )$ is a linear combination of a set of weak (base) learners $f _ { t } ( x )$ from some class $\\mathcal { F }$ , i.e., $\\begin{array} { r } { h ( x ) = \\sum _ { t = 1 } ^ { t = T } \\gamma _ { t } f _ { t } ( x ) + c o n s t } \\end{array}$ . Here, $T$ is the total training iterations and $\\gamma _ { t }$ is the corresponding weight for each weak learner. To minimize the empirical risk, the algorithm starts with a model $h _ { 0 } ( x )$ , and then incrementally expands it in a greedy manner. This process manages to fit a new weak learner to the residual errors made by the previous set of learners. Mathematically, the ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/9523a01cc930e55907dce22cf2a7d99cd0709e1b67369f64a9b03e5aa9e5b4bd.jpg",
+        "image_caption": [
+            "Figure 2: (a) Schematic illustration of proposed Coordinate Transforms (PCT). We treat $F$ as a weaker learner, and perform coordinate transform $T$ steps via confidence-aware localization boosting (CLB module) to obtain a better localization. Then the refined coordinate proposals combined with corresponding encoded RGB features (GCE module) are fed into network $\\mathbf { G }$ to generate final 3D bounding boxes. (b) The data flow of CLB mechanism in detail. For step $t$ , it takes refined coordinate patches $c _ { t - 1 }$ as input, which is transformed based on predicted $\\Delta ( x _ { t - 1 } , y _ { t - 1 } , z _ { t - 1 } )$ . Then network $F _ { t }$ generate the residual localization for the next step, and confidence $s _ { t }$ is also generated during the training process. "
+        ],
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "optimization can be formulated as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/5ab34280d4187cd864ac3803d97d914adc2f9a15a48103362d16d5d743052146.jpg",
+        "text": "$$\n\\begin{array} { r l } & { h _ { 0 } ( x ) = \\underset { \\gamma _ { 0 } } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { 0 } ( y ^ { * } , x ) ; } \\\\ & { ~ \\quad \\quad \\cdots } \\\\ & { h _ { t } ( x ) = h _ { t - 1 } ( x ) + \\underset { f _ { t } \\in \\mathcal { F } } { \\arg \\operatorname* { m i n } } \\mathcal { L } _ { t } ( y ^ { * } , h _ { t - 1 } ( x ) + f _ { t } ( x ) ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Inspired by gradient boosting, we imitate its optimization procedure to progressively adapt localization prediction by multiple localization regressors instead of a single one used in previous works [42, 29, 46, 27]. To be specific, we treat localization network $F$ as a weak learner and stack multiple of them as shown in Figure 2 (b). Given $F$ is a lightweight network, the extra computational cost brought by gradient boosting is insignificant. After the data transformation step, each 2D bounding box obtains its corresponding coordinate patch $c$ . We take the coordinate patch as the input of weak learner $F$ to regress the center localization residual $\\Delta ( x , y , z )$ based on the prediction from previous stage, ",
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+    {
+        "type": "equation",
+        "img_path": "images/5227bdc64906cb8cf1a3d4af428388bcc5f4335bcb869d84e75ac23e8ecdeb79.jpg",
+        "text": "$$\n\\Delta ( x _ { t } , y _ { t } , z _ { t } ) = F _ { t } ( c _ { t - 1 } ) ; { \\mathrm { ~ w h e r e ~ } } c _ { t - 1 } = c _ { 0 } - \\gamma _ { 0 } ( x _ { 0 } , y _ { 0 } , z _ { 0 } ) - \\sum _ { i = 1 } ^ { t - 1 } \\gamma _ { i } \\Delta ( x _ { i } , y _ { i } , z _ { i } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "We denote $c _ { 0 }$ and $( x _ { 0 } , y _ { 0 } , z _ { 0 } )$ to be the initial coordinate input patch and object location, respectively. Coordinate input patch $c _ { t - 1 }$ is then transformed according to the localization residual prediction, and fed into the next weak learner. ",
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+    {
+        "type": "text",
+        "text": "Thus the risk at stage $t$ can be written as, ",
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+        "type": "equation",
+        "img_path": "images/c0c428b59c5e734a011fd61e6570f80f4a32c0a9c85ad8208e7147ca4dfca147.jpg",
+        "text": "$$\n\\mathcal { L } _ { F _ { t } } ( ( x ^ { * } , y ^ { * } , z ^ { * } ) , c _ { t - 1 } ) = \\Psi ( ( x ^ { * } , y ^ { * } , z ^ { * } ) , \\gamma _ { 0 } ( x _ { 0 } , y _ { 0 } , z _ { 0 } ) + \\sum _ { i = 1 } ^ { t - 1 } \\gamma _ { i } \\Delta ( x _ { i } , y _ { i } , z _ { i } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $( x ^ { * } , y ^ { * } , z ^ { * } )$ indicates the ground truth location. At this point, we can easily see that $\\mathrm { E q 4 }$ is a natural derivation from $\\operatorname { E q } 2$ . After $T$ iterations, the final adjusted prediction $c _ { T }$ is fed into the network $G$ to estimate the 3D box $\\boldsymbol { B }$ $3 , i . e . B = G ( c _ { T } )$ . ",
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+    },
+    {
+        "type": "text",
+        "text": "Confidence-aware network loss In the case that the target of weak learner $f$ is differentiable, gradient boosting solves the optimization problem in a forward greedy manner as shown in Eq 2. For each iteration, it first fits the weak learner $f$ to the residual error, and then the optimal value of the coefficient weight $\\gamma$ is determined for this weak learner. The optimization procedures train iteratively for $T$ iterations. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "However, for our center regression task, we would like to train $T$ weak localization networks $F _ { t } ( c _ { t - 1 } ) , t \\in { 1 , . . . , T }$ and a 3D box prediction network $G$ in an end-to-end manner instead of bootstrapping. This is challenging in terms of both computational cost and optimization stability, given the simultaneous training of a set of weak learners and their coefficient weights. Therefore, we first simplify the problem by treating all $\\gamma _ { t }$ the same and only focus on optimizing the localization networks. However, the contribution from each weak learner $F$ may not be the same during end-toend training, which leads to unstable optimization. Hence, we tailor a confidence-aware boosting loss to facilitate network training, by learning confidence score $s _ { t }$ for each localization loss function $\\mathcal { L } _ { F _ { t } }$ . The confidence score $s _ { t }$ is learned from a small decoder and followed a self-balancing formulation closely coupled to the network loss. The overall loss function is defined as ",
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+    {
+        "type": "equation",
+        "img_path": "images/ab37a8e43592edd65a8b898e2029bb744cd460f514ab885ccf296fa4eb18c719.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { B } ^ { * } , \\boldsymbol { c } _ { 0 } ) = \\sum _ { t = 1 } ^ { T } s _ { t } \\ast \\mathcal { L } _ { \\boldsymbol { F } _ { t } } ( ( \\boldsymbol { x } ^ { * } , \\boldsymbol { y } ^ { * } , \\boldsymbol { z } ^ { * } ) , \\boldsymbol { c } _ { t - 1 } ) + \\lambda _ { s } \\prod _ { t = 1 } ^ { T } ( 1 - s _ { t } ) + \\mathcal { L } _ { G } ( \\boldsymbol { B } ^ { * } , \\boldsymbol { c } _ { T } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $B ^ { * }$ is the 3D box ground truth, $( x ^ { \\ast } , y ^ { \\ast } , z ^ { \\ast } ) \\in B ^ { \\ast }$ and $\\lambda _ { s }$ is the balance weight. $s _ { t }$ is the prediction after sigmoid, which represents the confidence of the localization regression loss at $t ^ { t h }$ stage, and $\\textstyle \\prod _ { t = 1 } ^ { T } ( 1 - s _ { t } )$ is the penalty on the network uncertainty. In other words, if $s _ { t }$ is approaching 1, which means the network is confident about localization refinement at $t ^ { t h }$ stage, then no penalty will be applied. Otherwise, the uncertainty of regression loss is high, thus triggers a higher penalty. ",
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+    {
+        "type": "text",
+        "text": "4.2 Global context encoding ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Typical coordinate-based methods [42, 45, 27, 43] perform 3D detection based on 2D RoIs, which is similar to two-stage 2D detection frameworks, such as Faster R-CNN [34]. In a two-stage 2D object detection framework, the second stage reuses the features from the first stage via RoIPooling [34] or RoIAlign operators [18] guided by ROI proposals, and then a small decoder is used for localization refinement. However, in 3D detection, only cropped patches with coordinates information are fed to the network for 3D box regression. Neither RGB information nor global context is included. ",
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+    {
+        "type": "text",
+        "text": "Considering that the RGB information is a vital visual clue, we explore its aggregation in the last 3D box estimation step. Similar to two-stage 2D detectors, we obtain the RGB information by directly cropping the corresponding features from a 2D detector $F _ { 2 d }$ . Then the input to 3D box estimator $G$ can be formulated as $\\bar { c \\mathbf { \\eta } } = \\{ [ \\mathcal { D } ( { \\boldsymbol u } , { \\boldsymbol v } ) , \\mathcal { A } ( { \\boldsymbol u } , { \\boldsymbol v } ) ] , \\forall ( { \\boldsymbol u } , { \\boldsymbol v } ) \\in \\mathcal { R } \\}$ . Here, $\\mathcal { D } ( \\cdot )$ represents the data transformation function and $\\boldsymbol { \\mathcal { A } } ( \\cdot )$ represents the RoIAlign operation. Both operations are performed upon the generated regions of interest from $\\mathcal { R }$ . ",
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+    {
+        "type": "text",
+        "text": "After RoIAlign operation, features are of size $C \\times K \\times K$ , where $C$ is the number of channels and $K \\times K$ is the corresponding width and height, respectively. A small feature encoder is then used to encode cropped features into vectors with the dimension of $C$ . A feature fusion is followed to integrate coordinate representations with the obtained image representations. Benefiting from the large receptive fields of image feature representations, 3D box estimator can now have access to global context. Besides, directly cropping on RGB features also avoids learning image representations of RoIs from scratch and reduces the overall network parameters. ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Two monocular 3D detection benchmarks are introduced in Sec. 5.1 and Sec. 5.2, while experimental implementation details are described in Sec. 5.3. In Sec. 5.4, we conduct main analysis on KITTI dataset [16, 17, 15] with base method PatchNet [27] given its current best performance. More experiments on Waymo Open Dataset [38] are also demonstrated to further verify the generality of our proposed PCT in Sec. 5.5. ",
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+    {
+        "type": "text",
+        "text": "5.1 KITTI setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We first evaluate our method on the KITTI benchmark [16, 17, 15], which contains 7,481 and 7,518 images for training and testing respectively. We follow [5] to split the 7,481 training images into 3712 for training and 3,769 for validation. ",
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+    {
+        "type": "table",
+        "img_path": "images/f370de2688269b944ce974de911d8e853d117ac31909683ff90d3fd447e02d7b.jpg",
+        "table_caption": [
+            "Table 2: Ablative analysis on KITTI validation set for $\\mathrm { A P _ { 3 D } }$ and $\\mathrm { A P _ { B E V } }$ at $\\mathrm { I o U } = 0 . 7$ . Experiment Group (I) is the baseline method. Different experiment settings are explored: (II) applying localization boosting without confidence constraint, (III) performing confidence-aware localization boosting algorithm, (IV) adding global context encoding on Group (II), (V) our full approach. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Group</td><td rowspan=\"2\">Localization Boosting</td><td rowspan=\"2\">Uncertainty</td><td rowspan=\"2\">GCE</td><td colspan=\"3\">AP3D</td><td colspan=\"3\">APBEV</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>I</td><td>-</td><td>-</td><td>1</td><td>25.88</td><td>36.07</td><td>20.99</td><td>33.34</td><td>46.39</td><td>27.54</td></tr><tr><td>Ⅱ</td><td>√</td><td></td><td>1</td><td>26.78</td><td>37.29</td><td>24.11</td><td>34.39</td><td>47.08</td><td>28.28</td></tr><tr><td>Ⅲ</td><td>√</td><td>√</td><td>=</td><td>27.24</td><td>38.32</td><td>24.39</td><td>33.92</td><td>46.77</td><td>27.98</td></tr><tr><td>IV</td><td>√</td><td></td><td></td><td>27.12</td><td>37.38</td><td>24.11</td><td>34.46</td><td>46.70</td><td>28.32</td></tr><tr><td>V</td><td>√</td><td>√</td><td></td><td>27.53</td><td>38.39</td><td>24.44</td><td>34.65</td><td>47.16</td><td>28.47</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Precision-recall curves are adopted for evaluation, and we report the average precision (AP) results of 3D and Bird’s eye view (BEV) object detection on KITTI validation and test set. For fair comparison to previous literature, the 40 recall positions-based metric $A P | _ { R 4 0 }$ is reported on test set while $\\bar { A } P | _ { R 1 1 }$ is reported on validation set. Three levels of difficulty are defined in the benchmark according to the 2D bounding box height, occlusion, and truncation degree, namely, “Easy”, “Mod.”, and “Hard”. The KITTI benchmark ranks all methods based on the $\\mathrm { A P _ { 3 D } }$ of “Mod.”. In particular, we focus on the “Car” category as in [42, 27], and we adopt $\\mathrm { I o U } = 0 . 7$ as threshold for evaluation. ",
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+    {
+        "type": "text",
+        "text": "5.2 Waymo setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We also carry out experiments on large-scale, high quality and diverse dataset, Waymo open dataset [38]. It provides pre-defined 798 training sequences and 202 validation sequences from different scenes, and another 150 test sequences without labels. The dataset contains camera images from five high-resolution pinhole cameras, and we only consider images with their 3D labels from front camera for monocular 3D detection task. We sample every third frame from the training sequences (total 52,386 images) as in CaDDN [33] to form the training set due to its large scale. And validation set contains all the 39,848 images from 202 different scenes. ",
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+    {
+        "type": "text",
+        "text": "For evaluation, we adopt the officially released evaluation [39] to calculate the mean average precision (mAP) and the mean average precision weighted by heading (mAPH). Two levels are included according to difficulty rating, which are defined by LiDAR points. 3D labels without any points are ignored, LEVEL_2 is assigned to examples when it contains equal or lesser than 5 points, while the rest of the examples are assigned to LEVEL_1. Additionally, three distances (0 - 30m, 30 - 50m, $5 0 \\mathrm { m }$ $- \\infty )$ ) to sensor are considered during evaluation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.3 Implementation details ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Our overall framework of PCT can be visualized in Figure 2. In terms of implementation details, we instantiate our algorithm on two widely adopted coordinate-based methods with public released code [28], PatchNet and Pseudo LiDAR. Bearing efficiency in mind, we use a real-time 2D detector RTM3D [23] with DLA-34 [47] as backbone. For the sake of fair comparison, we adopt depth predictor DORN [14] on KITTI dataset as in most depth-assisted literature. Since there is no published depth results on Waymo open dataset, we adopt a most recent monocular depth estimator AdaBins [1] trained on Waymo training set. For the CLB mechanism, we inherit the original localization regression framework in each method. Each $F _ { t }$ shares the same structure. The corresponding confidence is generated following the last convolutional layers of $F _ { t }$ with three linear layers and a Sigmoid function. $T = 3$ and $\\lambda _ { s } = 1$ are set for the following experiments except for the ablation study on boosting iterations. For GCE module, we get the corresponding input image features by performing RoIAlign on the features from last convolutional layer of 2D detector. We set the output of RoIAlign as $1 6 \\times 1 6$ . As 2D detector use DLA-34 as backbone, the obtained image feature representations have the size of $6 4 \\times 1 6 \\times 1 6$ and entitle arbitrary sized receptive field theoretically due to the embedded deformable convolution [8]. The structure of feature encoder in global context module is two common $6 4 \\times 3 \\times 3$ convolutional layers (stride $^ { = 4 }$ ) and a $6 4 \\times 1 \\times 1$ convolutional layer (stride ${ \\mathop : } = 1$ ). Hence, image feature representations are encoded to a vector with 64-dim. The obtained features are then concatenated with the coordinate feature vectors from the final global pooling of box prediction network $G$ . With the lightweight structure, we are able to optimize the network end-to-end on a single Nvidia V100 ",
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+    {
+        "type": "image",
+        "img_path": "images/e1d29ed86083cfdcecf02e52932d516313e169150f98e7200986c3955e7caf75.jpg",
+        "image_caption": [
+            "Figure 3: The statistic analysis and comparison on different Localization boosting stage when $T = 3$ . The vertical axis of the chart represents the number of samples after normalization. “loc. $1 / 2 / 3 ^ { \\circ }$ denotes the $1 / 2 / 3 ^ { t h }$ step of localization errors in $F$ and “loc. $. 4 \\ \"$ is the final localization errors in $G$ . Note that when the curve is more thin, tall, and closer to zeros, the localization is more accurate. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "GPU with 16G memory. The training criterion for network $F$ and $G$ and other training settings follow the corresponding base methods for fair comparison. ",
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+    {
+        "type": "text",
+        "text": "5.4 Method analysis on KITTI dataset ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Main ablative analysis In Table 2, we conduct ablation studies to analyze the effectiveness of our contributions: I) Without any localization regression network $F$ , network $G$ generates 3D box prediction directly. II) This configuration only contains localization boosting part without confidence constraint. III) The entire CLB mechanism is included to progressively regress center localization. IV) GCE module is added to the network based on the localization boosting block without confidence constraint since the feature fusion can be performed on either the localization regression networks $F$ or 3D box prediction network $G$ . V) Our full method with all the components. ",
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+    {
+        "type": "text",
+        "text": "As depicted in Table 2, we can observe that the performance continues to grow with the addition of every component. From group II, localization boosting brings a noticeable improvement on all settings especially “Hard” set, which confirms its effectiveness in increasing localization accuracy. Group III shows that balancing training loss by adding confidence leads to better and more stable optimization. Group IV reveals that the proposed GCE module can effectively equip RGB information and global context with 3D coordinate representations. In the end, Group V demonstrates the complementarity of the proposed CLB mechanism and GCE module, leading to an improvement from 25.88/36.07/20.99 to 27.53/38.39/24.44 compared with Group I. ",
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+    {
+        "type": "table",
+        "img_path": "images/26211e7c811f6262880294138835cc0c4276008ead17df8fe481ee5b57f31cd2.jpg",
+        "table_caption": [
+            "Table 3: Comparison of different boosting iteration settings on KITTI validation split set. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Localization Boost (T)</td><td colspan=\"3\">AP3D/APBEV</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>1</td><td>25.88/33.34</td><td>36.07/46.39</td><td>20.99/27.54</td></tr><tr><td>1</td><td>26.31/34.14</td><td>36.40/46.80</td><td>21.07/28.04</td></tr><tr><td>2</td><td>26.69/34.06</td><td>37.17/46.42</td><td>24.04/28.03</td></tr><tr><td>3</td><td>26.78/34.39</td><td>37.29/47.08</td><td>24.11/28.28</td></tr><tr><td>4</td><td>26.77/34.21</td><td>37.12/47.00</td><td>23.48/28.23</td></tr><tr><td>5</td><td>26.64/34.43</td><td>37.24/47.04</td><td>23.89/28.27</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/49d656a84388a9be6c8b3a0d8ef0c606f283323843b8f17dcbba777e30cc8033.jpg",
+        "table_caption": [
+            "Table 4: Evaluation of different coordinate feature fusion with GCE on KITTI validation set. Baseline is the Group (II) in Table 2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"3\">AP3D</td></tr><tr><td>Baseline</td><td>Mod. 26.78</td><td>Easy 37.29</td><td>Hard 24.11</td></tr><tr><td>F+GCE</td><td>27.08</td><td>37.33</td><td>24.07</td></tr><tr><td>G+GCE</td><td>27.12</td><td>37.38</td><td>24.11</td></tr><tr><td>All + GCE</td><td>27.07</td><td>37.43</td><td>24.18</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Localization boosting iteration settings We explore the effect of different localization boosting iteration settings in this part. For a fair comparison, we do not perform the confidence constraint on regression loss. As illustrated in Table 3, when boosting iteration $T = 3$ , we achieve the best 3D detection performance. More iterations of boosting do not bring improvements, which might be caused by overfitting with the increasing of network parameters. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To verify the improvement of each step in boosting procedure, we conduct the comparison of localization errors at iteration $T = 3$ on the specific metrics (location “xyz”) of the ground truth. In particular, three stacked localization networks $F$ generate intermediate localization $^ { \\bullet \\bullet } \\mathrm { l o c . } 1 / 2 / 3 ^ { \\bullet \\bullet }$ and $G$ output the final localization “loc.4”. As shown in Figure 3, we can see that the distributions of $\\mathbf { \\ddot { x } } ^ { , 5 }$ , “y” and “z” errors tend to distributed to zero with localization boosting iterating. For instance, the red line in left chart is narrow and tall near zero along horizontal axis compared with other lines, which means that the corresponding x coordinate is more accurate than others. This further suggests that localization boosting is useful for object localization. The corresponding BEV visualization will be shown in Supplementary Material. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/c1f84afe258a114f12040dbc85530e9d69baa6299bc5ce42f2def65beeebaa14.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative comparison of ground truth (green), base method PatchNet (blue), and our method (red) on KITTI val set. The first and second rows show RGB and BEV images respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/46e887bd10e5ffc09d3241262afdfc5b064b579ea8c733880e1f83b560e26699.jpg",
+        "table_caption": [
+            "Table 5: Comparison of generalization on KITTI validation set. $^ *$ denotes that the method is reproduced by ourselves. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">AP3D</td><td colspan=\"3\">APBEV</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>Pseudo-LiDAR*[42]</td><td>23.04</td><td>32.27</td><td>19.67</td><td>31.06</td><td>42.45</td><td>25.67</td></tr><tr><td>Pseudo-LiDAR+ CLB</td><td>24.14</td><td>34.46</td><td>20.16</td><td>32.41</td><td>44.98</td><td>26.82</td></tr><tr><td>Pseudo-LiDAR + CLB +GCE</td><td>24.43</td><td>34.34</td><td>20.18</td><td>32.50</td><td>45.35</td><td>26.91</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Impact of global context encoding We also explore where the global context representation fusion operates. We take Group II as the baseline, and perform feature fusion on localization regression network $F$ (row one), 3D estimation network $G$ (row two) or on both (final row). RoI features are encoded into a vector with GCE and then concatenate with the feature vectors from network ( $F$ or $G$ ) global pooling. As shown in Table 4, the operation on $G$ outperforms it on $F$ , which indicates that image representation is more suitable for the overall box prediction rather than only localization as it contains the additional semantic appearance information. Although operation on all networks achieves a lightly higher than it on $G$ on the “Easy” and “Hard” set, introducing parameters is much larger due to operation on stacked localization networks. Hence, we only apply GCE on network $G$ in our approach for a lightweight network and avoid overfitting during training. ",
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+        "text": "Generally applicable to other coordinate-based algorithm In this section, we demonstrate the generalization capability of our algorithm to classic coordinate-based methods Pseudo-LiDAR [42]. As shown in Table 5, each component of our algorithm improves the original methods a lot. Specially, our approach improves Pseudo-LiDAR by 1.43/2.06/0.51 while 1.22/1.99/3.36 gains on PatchNet. ",
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+        "text": "Comparison with state-of-the-arts We build our test detector on the current state-of-the-art coordinate-based method PatchNet, and results are shown in Table 6. Quantitatively, our method achieves the highest performance on “Mod.” set with 22 FPS on NvidiaTesla v100 including 2D detector inference time, which is the main setting for ranking on the benchmark. Specially, large margins, 2.25/5.32/1.14 on 3D detection and 2.17/6.68/0.95 on BEV, are observed over the base method PatchNet with only additional 3.41M parameters. Besides, our methods also outperforms the pixel-based state-of-the-arts methods Liu et al. [26] especially on “Hard” set. ",
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+        "text": "Qualitative comparisons are shown in Figure 4. The ground truth, base method (PatchNet), and our method are colored in green, blue, and red, respectively. For better visualization, the first and second rows show RGB images and BEV images, respectively. Compared with the base method, our algorithm can produce higher-quality 3D bounding boxes in different kinds of scenes. ",
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+            "Table 6: Comparison with SoTA methods on the KITTI test set at $\\mathrm { I o U } = 0 . 7$ . Our algorithm achieves new SoTA performance. “Depth” means if the method belongs to depth-assisted methods or not. “Type” indicates the method input pattern, “Pixel” denotes the methods with image as inputs directly while “Coordinate” means the coordinate-based methods with 3D coordinates as inputs. "
+        ],
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+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td rowspan=\"2\">Depth</td><td rowspan=\"2\">Type</td><td colspan=\"3\">AP3D</td><td colspan=\"3\">APBEV</td></tr><tr><td>Mod.</td><td>Easy</td><td>Hard</td><td>Mod.</td><td>Easy</td><td>Hard</td></tr><tr><td>AM3D [29]</td><td>yes</td><td>Coordinate</td><td>10.74</td><td>16.50</td><td>9.52</td><td>17.32</td><td>25.30</td><td>14.91</td></tr><tr><td>PatchNet [27]</td><td>yes</td><td>Coordinate</td><td>11.12</td><td>15.68</td><td>10.17</td><td>16.86</td><td>22.97</td><td>14.97</td></tr><tr><td>GrooMeD-NMS[20]</td><td>no</td><td>Pixel</td><td>12.32</td><td>18.10</td><td>9.65</td><td>18.27</td><td>26.19</td><td>14.05</td></tr><tr><td>Kinematic3D[3]</td><td>yes</td><td>Pixel</td><td>12.72</td><td>19.07</td><td>9.17</td><td>17.52</td><td>26.69</td><td>13.10</td></tr><tr><td>DDMP-3D[41]</td><td>yes</td><td>Pixel</td><td>12.78</td><td>19.71</td><td>9.80</td><td>17.89</td><td>28.08</td><td>13.44</td></tr><tr><td>Liu et al. [26]</td><td>yes</td><td>Pixel</td><td>13.25</td><td>21.65</td><td>9.91</td><td>17.98</td><td>29.81</td><td>13.08</td></tr><tr><td>PCT</td><td>yes</td><td>Coordinate</td><td>13.37</td><td>21.00</td><td>11.31</td><td>19.03</td><td>29.65</td><td>15.92</td></tr></table>",
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+        "text": "5.5 Results on Waymo Open Dataset ",
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+        "text": "Table 7 shows the results of base method PatchNet [27] and our proposed PCT. It can be observed that our method consistently outperforms the base method on mAP/mAPH of $0 . 5 0 \\% / 0 . 5 1 \\%$ and $0 . 2 8 \\% / 0 . 3 0 \\%$ on the LEVEL_1 and LEVEL_2 difficulties respectively under $\\mathrm { I o U } = 0 . 7$ . Again, our method is efficient, e.g, it takes 5 days to complete training on large scale Waymo dataset with a 8-GPU node. More qualitative results can be seen at Supplementary Material. ",
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+            "Table 7: 3D performance on Waymo validation set. We demonstrate results of base method PatchNet [27] and corresponding PCT at $\\mathrm { I o U } = 0 . 7$ and $\\mathrm { I 0 U } = 0 . 5$ . Our proposed PCT achieves consistent improvements on all settings. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Difficulty</td><td rowspan=\"2\">Threshold</td><td rowspan=\"2\">Method</td><td colspan=\"4\">3D mAP/3D mAPH</td></tr><tr><td>Overall</td><td>0-30m</td><td>30-50m</td><td>50-8</td></tr><tr><td rowspan=\"3\">LEVEL_1</td><td>IoU=0.7</td><td>PatchNet PCT</td><td>0.39/0.37 0.89 / 0.88</td><td>1.67 / 1.63 3.18 / 3.15</td><td>0.13/0.12</td><td>0.03/0.03 0.07 /0.07</td></tr><tr><td rowspan=\"2\">IoU=0.5</td><td>PatchNet</td><td>2.92/2.74</td><td>10.03/9.75</td><td>0.27 / 0.27 1.09/ 0.96</td><td>0.23/0.18</td></tr><tr><td>PCT</td><td>4.20 /4.15</td><td>14.70 / 14.54</td><td>1.78 / 1.75</td><td>0.39 / 0.39</td></tr><tr><td rowspan=\"3\">LEVEL_2</td><td>IoU=0.7</td><td>PatchNet</td><td>0.38/0.36</td><td>1.67 / 1.63</td><td>0.13/0.11</td><td>0.03/0.03</td></tr><tr><td rowspan=\"2\">IoU=0.5</td><td>PCT PatchNet</td><td>0.66 / 0.66</td><td>3.18 /3.15</td><td>0.27 /0.26</td><td>0.07 /0.07</td></tr><tr><td></td><td>2.42/2.28</td><td>10.01/9.73</td><td>1.07 /0.94</td><td>0.22/0.16</td></tr><tr><td></td><td></td><td>PCT</td><td>4.03 /3.99</td><td>14.67 / 14.51</td><td>1.74 / 1.71</td><td>0.36 / 0.35</td></tr></table>",
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+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
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+        "text": "In this paper, we have introduced a novel approach PCT to address the inaccurate localization problem for monocular 3D object detection. This is achieved by iteratively transforming the coordinate representation with a confidence-aware booting mechanism. Meanwhile, global context is introduced to compensate for the missing of semantic image representation in coordinated-based methods. Through extensive experiments, we have shown that our proposed PCT substantially improve the performance of the coordinate-based model by a large margin, and achieve state-of-the-art monocular 3D detection performance on KITTI test set. Moreover, we also show consistent improvements compared to the strong baseline on the large-scale Waymo Open dataset. ",
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+        "text": "There are several limitations that could indicate the possible directions for future work. First, the performance of off-the-shelf 2D detector directly influences the accuracy of coordinate-based methods, hence how to effectively design the 3D box estimation algorithm to fit with existing 2D detectors is important. Second, we only concentrate on the lightweight coordinate-based methods. It requires further exploration to extend our approach to pixel-based methods. Finally, our proposed global context encoding is a simple module. Despite working well, a more tailored feature fusion strategy between coordinate representation and RGB image representation is worth exploring. ",
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+        "type": "text",
+        "text": "Potential impacts. Our method focuses on the monocular 3D detection which can be applied in autonomous driving field. One potential social problem of our work is that it may aggravate the employment crisis of human servants and drivers, which replaces human with autonomous robots and intelligent systems. ",
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+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
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+        "text": "This work was supported by Shanghai Municipal Science and Technology Major Projects (No.2021SHZDZX0103 and No.2018SHZDZX01). ",
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+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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In ICCV workshops, 2019.   \n[44] Yan Yan, Yuxing Mao, and Bo Li. Second: Sparsely embedded convolutional detection. Sensors, 2018.   \n[45] Xiaoqing Ye, Liang Du, Yifeng Shi, Yingying Li, Xiao Tan, Jianfeng Feng, Errui Ding, and Shilei Wen. Monocular 3d object detection via feature domain adaptation. In ECCV, 2020.   \n[46] Yurong You, Yan Wang, Wei-Lun Chao, Divyansh Garg, Geoff Pleiss, Bharath Hariharan, Mark Campbell, and Kilian Q Weinberger. Pseudo-lidar++: Accurate depth for 3d object detection in autonomous driving. arXiv preprint, 2019.   \n[47] Fisher Yu, Dequan Wang, Evan Shelhamer, and Trevor Darrell. Deep layer aggregation. In CVPR, 2018.   \n[48] Yunpeng Zhang, Jiwen Lu, and Jie Zhou. Objects are different: Flexible monocular 3d object detection. In CVPR, 2021.   \n[49] Xingyi Zhou, Dequan Wang, and Philipp Krähenbühl. Objects as points. arXiv preprint, 2019.   \n[50] Yin Zhou and Oncel Tuzel. Voxelnet: End-to-end learning for point cloud based 3d object detection. In CVPR, 2018.   \n[51] Yi Zhu, Zhongyue Zhang, Chongruo Wu, Zhi Zhang, Tong He, Hang Zhang, R. Manmatha, Mu Li, and Alexander Smola. Improving semantic segmentation via self-training. arXiv preprint arXiv:2004.14960, 2020. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We have described exactly in abstract section and Sec. 1.   \n(b) Did you describe the limitations of your work? [Yes] See Sec. 6.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Sec. 6.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We demonstrate the codes in Github website. The dataset is public and URL [15, 39] is also attached in Sec. 5.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Sec. 5.3.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We report experiments results at the fixed seed. ",
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parse/train/HJgXsjA5tQ/HJgXsjA5tQ.md

@@ -0,0 +1,549 @@
+# ON THE LOSS LANDSCAPE OF A CLASS OF DEEP NEURAL NETWORKS WITH NO BAD LOCAL VALLEYS
+
+Quynh Nguyen Saarland University, Germany
+
+Mahesh Chandra Mukkamala Saarland University, Germany
+
+Matthias Hein University of Tübingen, Germany
+
+# ABSTRACT
+
+We identify a class of over-parameterized deep neural networks with standard activation functions and cross-entropy loss which provably have no bad local valley, in the sense that from any point in parameter space there exists a continuous path on which the cross-entropy loss is non-increasing and gets arbitrarily close to zero. This implies that these networks have no sub-optimal strict local minima.
+
+# 1 INTRODUCTION
+
+It has been empirically observed in deep learning (Dauphin et al., 2014; Goodfellow et al., 2015) that the training problem of over-parameterized1 deep CNNs (LeCun et al., 1990; Krizhevsky et al., 2012) does not seem to have a problem with bad local minima. In many cases, local search algorithms like stochastic gradient descent (SGD) frequently converge to a solution with zero training error even though the training objective is known to be non-convex and potentially has many distinct local minima (Auer et al., 1996; Safran & Shamir, 2018). This indicates that the problem of training practical over-parameterized neural networks is still far from the worst-case scenario where the problem is known to be NP-hard (Blum & Rivest., 1989; Sima, 2002; Livni et al., 2014; ShalevShwartz et al., 2017). A possible hypothesis is that the loss landscape of these networks is“wellbehaved” so that it becomes amenable to local search algorithms like SGD and its variants. As not all neural networks have a well-behaved loss landscape, it is interesting to identify sufficient conditions on their architecture so that this is guaranteed. In this paper our motivation is to come up with such a class of networks in a practically relevant setting, that is we study multi-class problems with the usual empirical cross-entropy loss and deep (convolutional) networks and almost no assumptions on the training data, in particular no distributional assumptions. Thus our results directly apply to the networks which we use in the experiments.
+
+![](images/39469606f77237c568a7d4dcbaf77f29d8956c9eac32274d06f09e18edc54d4d.jpg)  
+Figure 1: An example loss landscape with bad local valleys (left) and without bad local valley (right).
+
+Our contributions. We identify a family of deep networks with skip connections to the output layer whose loss landscape has no bad local valleys (see Figure 1 for an illustration). Our setting is for the empirical loss and there are no distributional assumptions on the training data. Moreover, we study directly the standard cross-entropy loss for multi-class problems. There are little assumptions on the network structure which can be arbitrarily deep and can have convolutional layers (weight sharing) and skip-connections between hidden layers. From a practical perspective, one can generate an architecture which fulfills our conditions by taking an existing CNN architecture and then adding skip-connections from a random subset of $N$ neurons ( $N$ is the number of training samples), possibly from multiple hidden layers, to the output layer (see Figure 2 for an illustration). For these networks we show that there always exists a continuous path from any point in parameter space on which the loss is non-increasing and gets arbitrarily close to zero. We note that this implies the loss landscape has no strict local minima, but theoretically non-strict local minima can still exist. Beside that, we show that the loss has also no local maxima.
+
+Beside the theoretical analysis, we show in experiments that despite achieving zero training error, the aforementioned class of neural networks generalize well in practice when trained with SGD whereas an alternative training procedure guaranteed to achieve zero training error has significantly worse generalization performance and is overfitting. Thus we think that the presented class of neural networks offer an interesting test bed for future work to study the implicit bias/regularization of SGD.
+
+# 2 DESCRIPTION OF NETWORK ARCHITECTURE
+
+We consider a family of deep neural networks which have $d$ input units, $H$ hidden units, $m$ output units and satisfy the following conditions:
+
+1. Every hidden unit of the first layer can be connected to an arbitrary subset of input units.   
+2. Every hidden unit at higher layers can take as input an arbitrary subset of hidden units from (multiple) lower hidden layers.   
+3. Any subgroup of hidden units lying on the same layer can have non-shared or shared weights, in the later case their number of incoming units have to be equal.   
+4. There exist $N$ hidden units which are connected to the output nodes with independent weights ( $N$ denotes the number of training samples).   
+5. The output of every hidden unit $j$ in the network, denoted as $f _ { j } : \mathbb { R } ^ { d }  \mathbb { R }$ , is given as
+
+$$
+f _ { j } ( x ) = \sigma _ { j } \Bigl ( b _ { j } + \sum _ { k : k \to j } f _ { k } ( x ) u _ { k \to j } \Bigr )
+$$
+
+where $x \in \mathbb { R } ^ { d }$ is an input vector of the network, $\sigma _ { j } : \mathbb { R }  \mathbb { R }$ is the activation function of unit $j$ , $b _ { j } \in \mathbb { R }$ is the bias of unit $j$ , and $u _ { k \to j } \in \mathbb { R }$ the weight from unit $k$ to unit $j$ .
+
+This definition covers a class of deep fully connected and convolutional neural networks with an additional condition on the number of connections to the output layer. In particular, while conventional architectures have just connections from the last hidden layer to the output, we require in our setting that there must exist at least $N$ neurons, “regardless” of their hidden layer, that are connected to the output layer. Essentially, this means that if the last hidden layer of a traditional network has just $L < N$ neurons then one can add connections from $N - L$ neurons in the hidden layers below it to the output layer so that the network fulfills our conditions.
+
+Similar skip-connections have been used in DenseNet (Huang et al., 2017) which are different from identity skip-connections as used in ResNets (He et al., 2016). In Figure 2 we illustrate a network with and without skip connections to the output layer which is analyzed in this paper. We note that several architectures like DenseNets Huang et al. (2017) already have skip-connections between hidden layers in their original architecture, whereas our special skip-connections go from hidden layers directly to the output layer. As our framework allow both kinds to exist in the same network (see Figure 2 for an example), we would like to separate them from each other by making the convention that in the following skip-connections, if not stated otherwise, always refer to ones which connect hidden neurons to output neurons.
+
+We denote by $d$ the dimension of the input and index all neurons in the network from the input layer to the output layer as $1 , 2 , \ldots , d , d + 1 , \ldots , d + H , d + H + 1 , \ldots , d + H + m$ which correspond to $d$ input units, $H$ hidden units and $m$ output units respectively. As we only allow directed arcs from lower layers to upper layers, it follows that $k \ < \ j$ for every $k  j$ . Let $N$ be the number of training samples. Suppose that there are $M$ hidden neurons which are directly connected to the output with independent weights where it holds $N \leq M \leq H$ . Let $\{ p _ { 1 } , \hdots , p _ { M } \}$ with $p _ { j } \in \{ d + 1 , \ldots , d + H \}$ be the set of hidden units which are directly connected to the output units. Let $\mathrm { i n } ( j )$ be the set of incoming nodes to unit $j$ and $u _ { j } = [ u _ { k  j } ] _ { k \in \mathrm { i n } ( j ) }$ the weight vector of the $j$ -th unit. Let $U = ( u _ { d + 1 } , \dots , u _ { d + H } , b _ { d + 1 } , \dots , b _ { d + H } )$ denote the set of all weights and biases of all hidden units in the network. Let $V \in \mathbb { R } ^ { M \times m }$ be the weight matrix which connects the $M$ hidden neurons to the $m$ output units of the network. An important quantity in the following is the matrix $\Psi \in \mathbb { R } ^ { N \times M }$ defined as
+
+![](images/d8b5942633c28d84dac3472b71378ffc0e0fa2ecfac8051002037caa31e2bc80.jpg)  
+Figure 2: Left: An example neural network represented as directed acyclic graph. Right: The same network with skip connections added from a subset of hidden neurons to the output layer. All neurons with the same color can have shared or non-shared weights.
+
+$$
+\Psi = \left[ \begin{array} { c c c } { { f _ { p _ { 1 } } ( x _ { 1 } ) } } & { { \ldots } } & { { f _ { p _ { M } } ( x _ { 1 } ) } } \\ { { \vdots } } & { { } } & { { \vdots } } \\ { { f _ { p _ { 1 } } ( x _ { N } ) } } & { { \ldots } } & { { f _ { p _ { M } } ( x _ { N } ) } } \end{array} \right]
+$$
+
+As $\Psi$ depends on $U$ , we write $\Psi _ { U }$ or $\Psi ( U )$ as a function of $U$ . Let $G \in \mathbb { R } ^ { N \times m }$ be the output of the network for all training samples. In particular, $G _ { i j }$ is the value of the $j$ -th output neuron for training sample $x _ { i }$ . It follows from our definition that
+
+$$
+G _ { i j } = \langle \Psi _ { i : } , V _ { : j } \rangle = \sum _ { k = 1 } ^ { M } f _ { p _ { k } } ( x _ { i } ) V _ { k j } , \quad \forall i \in [ N ] , j \in [ m ]
+$$
+
+Let $( x _ { i } , y _ { i } ) _ { i = 1 } ^ { N }$ be the training set where $y _ { i }$ denotes the target class for sample $x _ { i }$ . In the following we analyze the commonly used cross-entropy loss given as
+
+$$
+\Phi ( U , V ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } - \log \Big ( \frac { e ^ { G _ { i y _ { i } } } } { \sum _ { k = 1 } ^ { m } e ^ { G _ { i k } } } \Big )
+$$
+
+We refer to Section $\textrm { C }$ in the appendix for extension of our results to general convex losses. The cross-entropy loss is bounded from below by zero but this value is not attained. In fact the global minimum of the cross-entropy loss need not exist e.g. if a classifier achieves zero training error then by upscaling the function to infinity one can drive the loss arbitrarily close to zero. Due to this property, we do not study the global minima of the cross-entropy loss but the question if and how one can achieve zero training error. Moreover, we note that sufficiently small cross-entropy loss implies zero training error as shown in the following lemma.
+
+Lemma 2.1 If $\begin{array} { r } { \Phi ( U , V ) < \frac { \log ( 2 ) } { N } } \end{array}$ , then the training error is zero.
+
+Proof: We note that if $\begin{array} { r } { \Phi ( U , V ) < \frac { \log ( 2 ) } { N } } \end{array}$ < log(2)N , then it holds due to the positivity of the loss,
+
+$$
+\operatorname* { m a x } _ { i = 1 , \dots , N } - \log \Big ( \frac { e ^ { G _ { i y _ { i } } } } { \sum _ { k = 1 } ^ { m } e ^ { G _ { i k } } } \Big ) \leq \sum _ { i = 1 } ^ { N } - \log \Big ( \frac { e ^ { G _ { i y _ { i } } } } { \sum _ { k = 1 } ^ { m } e ^ { G _ { i k } } } \Big ) < \log ( 2 ) .
+$$
+
+This implies that for all $i = 1 , \ldots , N$ ,
+
+$$
+\log \left( 1 + \sum _ { k \neq y _ { i } } e ^ { G _ { i k } - G _ { i y _ { i } } } \right) < \log ( 2 ) \quad \Longrightarrow \quad \sum _ { k \neq y _ { i } } e ^ { G _ { i k } - G _ { i y _ { i } } } < 1 .
+$$
+
+In particular: $\begin{array} { r } { \operatorname* { m a x } _ { k \neq y _ { i } } e ^ { G _ { i k } - G _ { i y _ { i } } } < 1 } \end{array}$ and thus $\begin{array} { r } { \operatorname* { m a x } _ { k \neq y _ { i } } G _ { i k } - G _ { i y _ { i } } < 0 } \end{array}$ for all $i = 1 , \ldots , N$ which implies the result. 
+
+# 3 MAIN RESULT
+
+The following conditions are required for the main result to hold.
+
+Assumption 3.1 increasing
+
+1. All activation functions $\{ \sigma _ { d + 1 } , \ldots , \sigma _ { d + H } \}$ are real analytic and strictly
+
+2. Among $M$ neurons $\{ p _ { 1 } , \hdots , p _ { M } \}$ which are connected to the output units, there exist $N \leq M$ neurons, say w.l.o.g. $\{ p _ { 1 } , \dotsc , p _ { N } \}$ , such that one of the following conditions hold:
+
+• For every $1 \le j \le N : \sigma _ { p _ { j } }$ is bounded and $\begin{array} { r } { \operatorname* { l i m } _ { t  - \infty } \sigma _ { p _ { j } } ( t ) = 0 } \end{array}$ • For every $1 \le j \le N : \sigma _ { p _ { j } }$ is the softplus activation (3), and there exists a backward path from $p _ { j }$ to the first hidden layer s.t. on this path there is no neuron which has skip-connections to the output or shared weights with other skip-connection neurons.
+
+3. The input patches of different training samples are distinct. In particular, let $n _ { 1 }$ be the number of units in the first hidden layer and denote by $S _ { i }$ for $i \in [ d + 1 , d + n _ { 1 } ]$ their input support, then for all $r \neq s \in [ N ]$ , and $i \in [ d + 1 , d + n _ { 1 } ]$ , it holds $x _ { r } | _ { S _ { i } } \neq x _ { s } | _ { S _ { i } }$ .
+
+The first condition of Assumption 3.1 is satisfied for softplus, sigmoid, tanh, etc, whereas the second condition is fulfilled for sigmoid and softplus. For softplus activation function (smooth approximation of ReLU),
+
+$$
+\sigma _ { \gamma } ( t ) = \frac { 1 } { \gamma } \log ( 1 + e ^ { \gamma t } ) , ~ \mathrm { f o r ~ s o m e } \gamma > 0 ,
+$$
+
+we require an additional assumption on the network architecture. The third condition is always satisfied for fully connected networks if the training samples are distinct. For CNNs, this condition means that the corresponding input patches across different training samples are distinct. This could be potentially violated if the first convolutional layer has very small receptive fields. However, if this condition is violated for the given training set then after an arbitrarily small random perturbation of all training inputs it will be satisfied with probability 1. Note that the $M$ neurons which are directly connected to the output units can lie on different hidden layers in the network. Also there is no condition on the width of every individual hidden layer as long as the total number of hidden neurons in the network is larger than $N$ so that our condition $M \geq N$ is feasible.
+
+Overall, we would like to stress that Assumption 3.1 covers a quite large class of interesting network architectures but nevertheless allows us to show quite strong results on their empirical loss landscape.
+
+The following key lemma shows that for almost all $U$ , the matrix $\Psi ( U )$ has full rank.
+
+Lemma 3.2 Under Assumption 3.1, the set of $U$ such that $\Psi ( U )$ has not full rank N has Lebesgue measure zero.
+
+Proof: (Proof sketch) Due to space limitation, we can only present below a proof sketch. We refer the reader to the appendix for the detailed proof. The proof consists of two main steps. First, we show that there exists $U$ s.t. the submatrix $\Psi _ { 1 : N , 1 : N }$ has full rank. By Assumption 3.1, all activation functions are real analytic, thus the determinant of $\Psi _ { 1 : N , 1 : N }$ is a real analytic function of the network parameters which $\Psi$ depends on. By the first result, this determinant function is not identically zero, thus Lemma A.1 shows that the set of $U$ for which $\Psi$ has not full rank has Lebesgue measure zero.
+
+Now we sketch the proof for the first step, that is to find a $U$ s.t. $\Psi$ has full rank. By Assumption 3.1.3, one can always choose the weight vectors of the first hidden layer so that every neuron at this layer has distinct values for different training samples. For higher neurons, we set their initial weight vectors to be unit vectors with exactly one 1 and 0 elsewhere. Note that the above construction of weights can be easily done so that all the neurons from the same layer and with the same number of incoming units can have shared/unshared weights according to our description of network architecture in Section 2. Let $c ( j )$ be the neuron below $j$ s.t. $u _ { c ( j )  j } = 1$ . To find $U$ , we are going to scale up each weight vector $u _ { j }$ by a positive scalar $\alpha _ { j }$ . The idea is to show that the determinant of $\Psi _ { 1 : N , 1 : N }$ is non-zero for some positive value of $\{ \alpha _ { j } \}$ . The biases can be chosen in such a way that the following holds for some $\beta \in \mathbb { R }$ ,
+
+$$
+\Psi _ { i j } = f _ { p _ { j } } ( x _ { i } ) = \sigma _ { p _ { j } } \Big ( \beta + \alpha _ { p _ { j } } \big ( f _ { c ( p _ { j } ) } ( x _ { i } ) - f _ { c ( p _ { j } ) } ( x _ { j } ) \big ) \Big ) \quad \forall j \in [ N ]
+$$
+
+where $f _ { c ( p _ { j } ) } ( \boldsymbol { x } ) = \sigma _ { c ( p _ { j } ) } ( \alpha _ { c ( p _ { j } ) } \sigma _ { c ( c ( p _ { j } ) ) } ( \ldots f _ { q _ { j } } ( \boldsymbol { x } ) \ldots ) )$ with $q _ { j }$ being the index of some neuron in the first hidden layer. Note that by our construction the value of unit $q _ { j }$ is distinct at different training samples, and thus it follows from the strict monotonic property of activation functions from Assumption 3.1 and the positivity of $\big \{ \alpha _ { c ( p _ { j } ) } , . . . \big \}$ that $f _ { c ( p _ { j } ) } ( x _ { i } ) \ne f _ { c ( p _ { j } ) } ( x _ { j } )$ for every $i \neq j$ . Next, we show that the set of training samples can be re-ordered in such a way that it holds $f _ { c ( p _ { j } ) } ( x _ { i } ) < f _ { c ( p _ { j } ) } ( x _ { j } )$ for every $i > j$ . Note that this re-ordering does not affect the rank of $\Psi$ Now, the intuition is that if one let $\alpha _ { p _ { j } }$ go to infinity then $\Psi _ { i j }$ converges to zero for $i > j$ because it holds for all activations from Assumption 3.1 that $\begin{array} { r } { \operatorname* { l i m } _ { t  - \infty } \sigma _ { p _ { j } } ( t ) = 0 } \end{array}$ . Thus the determinant of $\Psi _ { 1 : 1 , 1 : N }$ converges to $\begin{array} { r } { \prod _ { i = 1 } ^ { N } \sigma _ { p _ { j } } ( \beta ) } \end{array}$ which can be chosen to be non-zero by a predefined value of $\beta$ in which case $\Psi$ will have full rank. The detailed proof basically will show how to choose the specific values of $\{ \alpha _ { j } \}$ so that all the above criteria are met. In particular, it is important to make sure that the weight vectors of two neurons ${ j , j ^ { \prime } }$ from the same layer will be scaled by the same factor $\alpha _ { j } = \alpha _ { j \prime }$ as we want to maintain any potential weight-sharing conditions. The choice of activation functions from the second condition of Assumption 3.1 basically determines how the values of $\alpha$ should be chosen. 
+
+While we conjecture that the result of Lemma 3.2 holds for softplus activation function without the additional condition as mentioned in Assumption 3.1, the proof of this is considerably harder for such a general class of neural networks since one has to control the output of neurons with skip connection from different layers which depend on each other. However, please note that the condition is also not too restrictive as it just might require more connections from lower layers to upper layers but it does not require that the network is wide. Before presenting our main result, we first need a formal definition of bad local valleys.
+
+Definition 3.3 The $\alpha$ -sublevel set of $\Phi$ is defined as $L _ { \alpha } = \{ ( U , V ) \mid \Phi ( U , V ) < \alpha \}$ . A local valley is defined as a connected component of some sublevel set $L _ { \alpha }$ . A bad local valley is a local valley on which the loss function $\Phi$ cannot be made “arbitrarily small”.
+
+Intuitively, a typical example of a bad local valley is a small neighborhood around a sub-optimal strict local minimum. We are now ready to state our main result.
+
+Theorem 3.4 The following holds under Assumption 3.1:
+
+1. There exist uncountably many solutions with zero training error.
+
+2. The loss landscape of $\Phi$ does not have any bad local valley.
+
+3. There exists no suboptimal strict local minimum.
+
+4. There exists no local maximum.
+
+# Proof:
+
+1. By Lemma 3.2 the set of $U$ such that $\Psi ( U )$ has not full rank $N$ has Lebesgue measure zero. Given $U$ such that $\Psi$ has full rank, the linear system $\Psi ( U ) V = Y$ has for every possible target output matrix $Y \in \mathbb { R } ^ { N \times m }$ at least one solution $V$ . As this is possible for almost all $U$ , there exist uncountably many solutions achieving zero training error.
+
+2. Let $C$ be a non-empty, connected component of some $\alpha$ -sublevel set $L _ { \alpha }$ for $\alpha > 0$ . Suppose by contradiction that the loss on $C$ cannot be made arbitrarily small, that is there exists an $\epsilon > 0$ such that $\Phi ( U , V ) ~ \ge ~ \epsilon$ for all $( U , V ) \in C$ , where $\epsilon \ < \ \alpha$ . By definition, $L _ { \alpha }$ can be written as the pre-image of an open set under a continuous function, that is $L _ { \alpha } = \Phi ^ { - 1 } ( \{ a \mid a < \alpha \} )$ , and thus $L _ { \alpha }$ must be an open set (see Proposition A.2). Since $C$ is a non-empty connected component of $L _ { \alpha }$ , $C$ must be an open set as well, and thus $C$ has non-zero Lebesgue measure. By Lemma 3.2 the set of $U$ where $\Psi ( U )$ has not full rank has measure zero and thus $C$ must contain a point $( U , V )$ such that $\Psi ( U )$ has full rank. Let $Y$ be the usual zero-one one-hot encoding of the target network output. As $\Psi ( U )$ has full rank, there always exist $V ^ { * }$ such that $\Psi ( U ) V ^ { * } = Y t ^ { * }$ , where $\begin{array} { r } { t ^ { * } = \log \left( \frac { m - 1 } { e ^ { \frac { \epsilon } { 2 } } - 1 } \right) } \end{array}$ Note that the loss of $( U , V ^ { * } )$ is
+
+$$
+\Phi ( U , V ^ { * } ) = - \log \Big ( \frac { e ^ { t ^ { * } } } { e ^ { t ^ { * } } + ( m - 1 ) } \Big ) = \log ( 1 + ( m - 1 ) e ^ { - t ^ { * } } ) = \frac { \epsilon } { 2 } .
+$$
+
+As the cross-entropy loss $\Phi ( U , V )$ is convex in $V$ and $\Phi ( U , V ) < \alpha$ we have for the line segment $V ( \lambda ) = \lambda V + ( 1 - \lambda ) V ^ { * }$ for $\lambda \in [ 0 , 1 ]$ ,
+
+$$
+\Phi ( U , V ( \lambda ) ) \le \lambda \Phi ( U , V ) + ( 1 - \lambda ) \Phi ( U , V ^ { * } ) < \lambda \alpha + ( 1 - \lambda ) \frac { \epsilon } { 2 } < \alpha .
+$$
+
+Thus the whole line segment is contained in $L _ { \alpha }$ and as $C$ is a connected component it has to be contained in $C$ . However, this contradicts the assumption that for all $( U , V ) \in C$ it holds $\Phi ( U , V ) \ge \epsilon$ . Thus on every connected component $C$ of $L _ { \alpha }$ the training loss can be made arbitrarily close to zero and thus the loss landscape has no bad valleys.
+
+3. Let $( U _ { 0 } , V _ { 0 } )$ be a strict suboptimal local minimum, then there exists $r > 0$ such that $\Phi ( U , V ) > \Phi ( U _ { 0 } , V _ { 0 } ) > 0$ for all $( U , V ) \in { \cal B } ( ( U _ { 0 } , V _ { 0 } ) , r ) \ : \backslash \ : \{ ( U _ { 0 } , V _ { 0 } ) \}$ where $B ( \cdot , r )$ denotes a closed ball of radius $r$ . Let $\begin{array} { r } { \alpha = \operatorname* { m i n } _ { ( U , V ) \in \partial B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big ) } \Phi ( U , V ) } \end{array}$ which exists as $\Phi$ is continuous and the boundary $\partial B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ of $B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ is compact. Note that $\alpha > \Phi ( U _ { 0 } , V _ { 0 } )$ as $( U _ { 0 } , V _ { 0 } )$ is a strict local minimum. Consider the sub-level set $D = L _ { \frac { \alpha + \Phi ( U _ { 0 } , V _ { 0 } ) } { 2 } }$ . As ent o $\begin{array} { r } { \Phi ( U _ { 0 } , V _ { 0 } ) < \frac { \alpha + \Phi ( U _ { 0 } , V _ { 0 } ) } { 2 } } \end{array}$ α+Φ(U0,V0) it holds (U0, V0) ∈ D. Let E be the , that i t holds $D$ $( U _ { 0 } , V _ { 0 } )$ $( U _ { 0 } , V _ { 0 } ) \in E \subseteq D$ $E \subset B { \big ( } ( U _ { 0 } , V _ { 0 } ) , r { \big ) }$ as $\begin{array} { r } { \Phi ( U , V ) < \frac { \alpha + \Phi ( U _ { 0 } , V _ { 0 } ) } { 2 } < \alpha } \end{array}$ α+Φ(U0,V0) < α for all (U, V ) ∈ E. Moreover, $\Phi ( U , V ) \ge \Phi ( U _ { 0 } , V _ { 0 } ) > 0$ for all $( U , V ) \in E$ and thus $\Phi$ can not be made arbitrarily small on a connected component of a sublevel set of $\Phi$ and thus $E$ would be a bad local valley which contradicts 3.3.2.
+
+4. Suppose by contradiction that $( U , V )$ is a local maximum. Then the Hessian of $\Phi$ is negative semi-definite. However, as principal submatrices of negative semi-definite matrices are again negative semi-definite, then also the Hessian of $\Phi$ w.r.t $V$ must be negative semidefinite. However, $\Phi$ is always convex in $V$ and thus its Hessian restricted to $V$ is positive semi-definite. The only matrix which is both p.s.d. and n.s.d. is the zero matrix. It follows that $\nabla _ { V } ^ { 2 } \Phi ( U , V ) = 0$ . One can easily show that
+
+$$
+\nabla _ { V _ { : j } } ^ { 2 } \Phi = \sum _ { i = 1 } ^ { N } \frac { e ^ { G _ { i j } } } { \sum _ { k = 1 } ^ { m } e ^ { G _ { i k } } } \Bigl ( 1 - \frac { e ^ { G _ { i j } } } { \sum _ { k = 1 } ^ { m } e ^ { G _ { i k } } } \Bigr ) \Psi _ { i : } \Psi _ { i : } ^ { T }
+$$
+
+From Assumption 3.1 it holds that there exists $j \in [ N ]$ s.t. $\sigma _ { p _ { j } }$ is strictly positive, and thus some entries of $\Psi _ { i }$ : must be strictly positive. Moreover, one has ${ \frac { e ^ { G _ { i j } } } { \sum _ { k = 1 } ^ { m } e ^ { G _ { i k } } } } \in ( 0 , 1 )$ . It follows that some entries of $\nabla _ { V : j } ^ { 2 } \Phi$ must be strictly positive. Thus $\nabla _ { V : j } ^ { 2 } \Phi$ cannot be identically zero, leading to a contradiction. Therefore $\Phi$ has no local maximum.
+
+Theorem 3.4 shows that there are infinitely many solutions which achieve zero training error, and the loss landscape is nice in the sense that from any point in the parameter space there exists a continuous path that drives the loss arbitrarily close to zero (and thus a solution with zero training error) on which the loss is non-increasing.
+
+While the networks are over-parameterized, we show in the next Section 4 that the modification of standard networks so that they fulfill our conditions leads nevertheless to good generalization performance, often even better than the original network. We would like to note that the proof of Theorem 3.4 also suggests a different algorithm to achieve zero training error: one initializes all weights, except the weights to the output layer, randomly (e.g. Gaussian weights), denoted as $U$ , and then just solves the linear system $\Psi ( U ) V = Y$ to obtain the weights $V$ to the output layer. Basically, this algorithm uses the network as a random feature generator and fits the last layer directly to achieve zero training error. The algorithm is successful with probability 1 due to Lemma 3.2. Note that from a solution with zero training error one can drive the cross-entropy loss to zero by upscaling to infinity but this does not change the classifier. We will see, that this simple algorithm shows bad generalization performance and overfitting, whereas training the full network with SGD leads to good generalization performance. This might seem counter-intuitive as our networks have more parameters than the original networks but is inline with recent observations in Zhang et al. (2017) that state-of-the art networks, also heavily over-parameterized, can fit even random labels but still generalize well on the original problem. Due to this qualitative difference of SGD and the simple algorithm which both are able to find solutions with zero training error, we think that our class of networks is an ideal test bed to study the implicit regularization/bias of SGD, see e.g. Soudry et al. (2018).
+
+# 4 EXPERIMENTS
+
+The main purpose of this section is to investigate the generalization ability of practical neural networks with skip-connections added to the output layer to fulfill Assumption 3.1.
+
+Datasets. We consider MNIST and CIFAR10 datasets. MNIST contains $5 . 5 \times 1 0 ^ { 4 }$ training samples and $1 0 ^ { 4 }$ test samples, and CIFAR10 has $5 \times 1 0 ^ { 4 }$ training samples and $1 0 ^ { 4 }$ test samples. We do not use any data pre-processing nor data-augmentation in all of our experiments.
+
+Network architectures. For MNIST, we use a plain CNN architecture with 13 layers, denoted as CNN13 (see Table 3 in the appendix for more details about this architecture). For CIFAR10 we use VGG11, VGG13, VGG16 (Simonyan & Zisserman, 2015) and DenseNet121 (Huang et al., 2017). As the VGG models were originally proposed for ImageNet and have very large fully connected layers, we adapted these layers for CIFAR10 by reducing their width from 4096 to 128. For each given network, we create the corresponding skip-networks by adding skip-connections to the output so that our condition $M \geq N$ from the main theorem is satisfied. In particular, we aggregate all neurons of all the hidden layers in a pool and randomly choose from there a subset of $N$ neurons to be connected to the output layer (see e.g. Figure 2 for an illustration). As existing network architectures have a large number of feature maps per layer, the total number of neurons is often very large compared to number of training samples, thus it is easy to choose from there a subset of $N$ neurons to connect to the output. In the following, we test both sigmoid and softplus activation function $( \gamma = 2 0 )$ ) for each network architecture and their skip-variants. We use the standard cross-entropy loss and train all models with SGD+Nesterov momentum for 300 epochs. The initial learning rate is set to 0.1 for Densenet121 and 0.01 for the other architectures. Following Huang et al. (2017), we also divide the learning rate by 10 after $5 0 \%$ and $7 5 \%$ of the total number of training epochs. Note that we do not use any explicit regularization like weight decay or dropout.
+
+The main goal of our experiments is to investigate the influence of the additional skip-connections to the output layer on the generalization performance. We report the test accuracy for the original models and the ones with skip-connections to the output layer. For the latter one we have two different algorithms: standard SGD for training the full network as described above (SGD) and the randomized procedure (rand). The latter one uses a slight variant of the simple algorithm described at the end of the last section: randomly initialize the weights of the network $U$ up to the output layer by drawing each of them from a truncated Gaussian distribution with zero mean and variance $\textstyle { \frac { 2 } { d } }$ where $d$ is the number of weight parameters and the truncation is done after $\pm 2$ standard deviations (standard keras initialization), then use SGD to optimize the weights $V$ for a linear classifier with fixed features $\Psi ( U )$ which is a convex optimization problem.
+
+Our experimental results are summarized in Table 1 for MNIST and Table 2 for CIFAR10. For skip-models, we report mean and standard deviation over 8 random choices of the subset of $N$ neurons connected to the output.
+
+Discussion of results. First of all, we note that adding skip connections to the output improves the test accuracy in almost all networks (with the exception of Densenet121) when the full network is
+
+<table><tr><td></td><td>Sigmoid activation function</td><td>Softplus activation function</td></tr><tr><td>CNN13</td><td>11.35</td><td>99.20</td></tr><tr><td>CNN13-skip (SGD)</td><td>98.40 ± 0.07</td><td>99.14 ± 0.04</td></tr></table>
+
+Table 1: Test accuracy $( \% )$ of CNN13 on MNIST dataset. CNN13 denotes the original architecture from Table 3 while CNN13-skip denotes the corresponding skip-model. There are in total 179, 840 hidden neurons from the original CNN13 (see Table 3), out of which we choose a random subset of $N = 5 5$ , 000 neurons to connect to the output layer to obtain CNN13-skip.
+
+Table 2: Traning and test accuracy of several CNN architectures with/without skip-connections on CIFAR10 (no data-augmentation). For each original model A, A-skip denotes the corresponding skip-model in which a subset of $N$ hidden neurons “randomly selected” from the hidden layers are connected to the output units. For Densenet121, these neurons are randomly chosen from the first dense block. The names in open brackets (rand/SGD) specify how the networks are trained: rand $U$ is randomized and fixed while $V$ is learned with SGD), SGD (both $U$ and $V$ are optimized with SGD). Additional experimental results with data-augmentation are shown in Table 5 in the appendix.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Sigmoid activation function</td><td rowspan=1 colspan=2>Softplus activation function</td></tr><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>Test acc (%)</td><td rowspan=1 colspan=1>Train acc (%)</td><td rowspan=1 colspan=1>Test acc (%)</td><td rowspan=1 colspan=1>Train acc (%)</td></tr><tr><td rowspan=1 colspan=1>VGG11VGG11-skip (rand)VGG11-skip (SGD)</td><td rowspan=1 colspan=1>1062.81 ± 0.3972.51 ± 0.35</td><td rowspan=1 colspan=1>10100100</td><td rowspan=1 colspan=1>78.9264.49 ± 0.3880.57 ± 0.40</td><td rowspan=1 colspan=1>100100100</td></tr><tr><td rowspan=1 colspan=1>VGG13VGG13-skip (rand)VGG13-skip (SGD)</td><td rowspan=1 colspan=1>1061.50 ± 0.3470.24 ± 0.39</td><td rowspan=1 colspan=1>10100100</td><td rowspan=1 colspan=1>80.8461.42 ± 0.4081.94 ± 0.40</td><td rowspan=1 colspan=1>100100100</td></tr><tr><td rowspan=1 colspan=1>VGG16VGG16-skip (rand)VGG16-skip (SGD)</td><td rowspan=1 colspan=1>1061.57 ± 0.4170.61 ± 0.36</td><td rowspan=1 colspan=1>10100100</td><td rowspan=1 colspan=1>81.3361.46 ± 0.3481.91 ± 0.24</td><td rowspan=1 colspan=1>100100100</td></tr><tr><td rowspan=1 colspan=1>Densenet121Densenet121-skip (rand)Densenet121-skip (SGD)</td><td rowspan=1 colspan=1>86.4152.07 ± 0.4881.47 ± 1.03</td><td rowspan=1 colspan=1>100100100</td><td rowspan=1 colspan=1>89.3155.39 ± 0.4886.76 ± 0.49</td><td rowspan=1 colspan=1>100100100</td></tr></table>
+
+trained with SGD. In particular, for the sigmoid activation function the skip connections allow for all models except Densenet121 to get reasonable performance whereas training the original model fails. This effect can be directly related to our result of Theorem 3.4 that the loss landscape of skip-networks has no bad local valley and thus it is not difficult to reach a solution with zero training error (see Section F in the appendix for more detailed discussions on this issue, as well as Section E for a visual example which shows why the skip-models can succeed while the original models fail). The exception is Densenet121 which gets already good performance for the sigmoid activation function for the original model. We think that the reason is that the original Densenet121 architecture has already quite a lot of skip-connections between the hidden layers which thus improves the loss surface already so that the additional connections added to the output units are not necessary anymore.
+
+The second interesting observation is that we do not see any sign of overfitting for the SGD version even though we have increased for all models the number of parameters by adding skip connections to the output layer and we know from Theorem 3.4 that for all the skip-models one can easily achieve zero training error. This is in line with the recent observation of Zhang et al. (2017) that modern heavily over-parameterized networks can fit everything (random labels, random input) but nevertheless generalize well on the original training data when trained with SGD. This is currently an active research area to show that SGD has some implicit bias (Neyshabur et al., 2017; Brutzkus et al., 2018; Soudry et al., 2018) which leads to a kind of regularization effect similar to the linear least squares problem where SGD converges to the minimum norm solution. Our results confirm that there is an implicit bias as we see a strong contrast to the (skip-rand) results obtained by using the network as a random feature generator and just fitting the connections to the output units (i.e. $V$ ) which also leads to solutions with zero training error with probability 1 as shown in Lemma 3.2 and the proof of Theorem 3.4. For this version we see that the test accuracy gets worse as one is moving from simpler networks (VGG11) to more complex ones (VGG16 and Densenet121) which is a sign of overfitting. Thus we think that our class of networks is also an interesting test bed to understand the implicit regularization effect of SGD. It seems that SGD selects from the infinite pool of solutions with zero training error one which generalizes well, whereas the randomized feature generator selects one with much worse generalization performance.
+
+# 5 RELATED WORK
+
+In the literature, many interesting theoretical results have been developed on the loss surface of neural networks Yu & Chen (1995); Haeffele & Vidal (2017); Choromanska et al. (2015); Kawaguchi (2016); Safran & Shamir (2016); Hardt & Ma (2017); Yun et al. (2017); Lu & Kawaguchi (2017); Venturi et al. (2018); Liang et al. (2018b); Zhang et al. (2018); Nouiehed & Razaviyayn (2018). The behavior of SGD for the minimization of training objective has been also analyzed for various settings (Andoni et al., 2014; Sedghi & Anandkumar, 2015; Janzamin et al., 2016; Gautier et al., 2016; Brutzkus & Globerson, 2017; Soltanolkotabi, 2017; Soudry & Hoffer, 2017; Zhong et al., 2017; Tian, 2017; Du et al., 2018; Wang et al., 2018) to name a few. Most of current results are however limited to shallow networks (one hidden layer), deep linear networks and/or making simplifying assumptions on the architecture or the distribution of training data. An interesting recent exception is Liang et al. (2018a) where they show that for binary classification one neuron with a skip-connection to the output layer and exponential activation function is enough to eliminate all bad local minima under mild conditions on the loss function. More closely related in terms of the setting are (Nguyen & Hein, 2017; 2018) where they study the loss surface of fully connected and convolutional networks if one of the layers has more neurons than the number of training samples for the standard multi-class problem. However, the presented results are stronger as we show that our networks do not have any suboptimal local valley or strict local minima and there is less over-parameterization if the number of classes is small.
+
+# 6 CONCLUSION
+
+We have identified a class of deep neural networks whose loss landscape has no bad local valleys. While our networks are over-parameterized and can easily achieve zero training error, they generalize well in practice when trained with SGD. Interestingly, a simple different algorithm using the network as random feature generator also achieves zero training error but has significantly worse generalization performance. Thus we think that our class of models is an interesting test bed for studying the implicit regularization effect of SGD.
+
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+
+# A MATHEMATICAL TOOLS
+
+In the proof of Lemma 3.2 we make use of the following property of analytic functions.
+
+Lemma A.1 (Nguyen, 2015; Mityagin, 2015) If $f : \mathbb { R } ^ { n }  \mathbb { R }$ is a real analytic function which is not identically zero then the set $\{ x \in \mathbb { R } ^ { n } \mid f ( x ) = 0 \}$ has Lebesgue measure zero.
+
+We recall the following standard result from topology (see e.g. Apostol (1974), Theorem 4.23, p. 82), which is used in the proof of Theorem 3.4.
+
+Proposition A.2 Let $f : \mathbb { R } ^ { m }  \mathbb { R } ^ { n }$ be a continuous function. If $U \subseteq \mathbb { R } ^ { n }$ is an open set then $f ^ { - 1 } ( U )$ is also open.
+
+# B PROOF OF LEMMA 3.2
+
+Proof: We assume w.l.o.g. that $\{ p _ { 1 } , \dotsc , p _ { N } \}$ is a subset of the neurons with skip connections to the output layer and satisfy Assumption 3.1. In the following, we will show that there exists a weight configuration $U$ such that the submatrix $\Psi _ { 1 : N , 1 : N }$ has full rank. Using then that the determinant is an analytic function together with Lemma A.1, we will conclude that the set of weight configurations $U$ such that $\Psi$ has not full rank has Lebesgue measure zero.
+
+We remind that all the hidden units in the network are indexed from the first hidden layer till the higher layers as $d + 1 , \dotsc , d + H$ . For every hidden neuron $j \in [ d + 1 , d + H ]$ , $u _ { j }$ denotes the associated weight vector
+
+$$
+u _ { j } = [ u _ { k  j } ] _ { k \in \mathrm { i n } ( j ) } \in \mathbb { R } ^ { | \mathrm { i n } ( j ) | } , \quad \mathrm { w h e r e ~ i n } ( j ) = \mathrm { t h e ~ s e t ~ o f ~ i n c o m i n g ~ u n i t s ~ t o ~ u n i t ~ } j .
+$$
+
+Let $n _ { 1 }$ be the number of units of the first hidden layer. For every neuron $j$ from the first hidden layer, let us define the pre-activation output gj,
+
+$$
+g _ { j } ( x _ { i } ) = \sum _ { k  j } ( x _ { i } ) _ { k } u _ { k } {  } j .
+$$
+
+Due to Assumptions 3.1 (condition 3), we can always choose the weights $\{ u _ { d + 1 } , \ldots , u _ { d + n _ { 1 } } \}$ so that the output of every neuron in the first layer is distinct for different training samples, that is $g _ { j } ( x _ { i } ) \neq g _ { j } ( x _ { i ^ { \prime } } )$ for every $j \in [ d + 1 , d + n _ { 1 } ]$ and $i \neq i ^ { \prime }$ . For every neuron $j \in [ d + n _ { 1 } + 1 , d + H ]$ in the higher layers we choose the weight vector $u _ { j }$ such that it has exactly one 1 and 0 elsewhere. According to our definition of network in Section 2, the weight vectors of neurons of the same layer need not have the same dimension, but any subgroup of these neurons can still have shared weights as long as the dimensions among them agree. Thus the above choice of $u$ is always possible. In the following, let $c ( j )$ denote the neuron below $j$ such that $u _ { c ( j )  j } = 1$ . This leads to
+
+$$
+\sum _ { k  j } f _ { k } ( x ) u _ { k  j } = f _ { c ( j ) } ( x ) .
+$$
+
+Let $\alpha : = ( \alpha _ { d + 1 } , \dots , \alpha _ { d + H } )$ be a tuple of positive scalars. Let $\beta \in \mathbb { R }$ such that $\sigma _ { p _ { j } } ( \beta ) \neq 0$ for every $j \in [ N ]$ . We consider a family of configurations of network parameters of the form $( \alpha _ { j } u _ { j } , b _ { j } ) _ { j = d + 1 } ^ { d + H }$ where the biases are chosen as
+
+$$
+\begin{array} { l } { b _ { p _ { j } } = \beta - \alpha _ { p _ { j } } g _ { p _ { j } } ( x _ { j } ) \quad \forall j \in [ N ] , p _ { j } \in [ d + 1 , d + n _ { 1 } ] } \\ { b _ { p _ { j } } = \beta - \alpha _ { p _ { j } } f _ { c ( p _ { j } ) } ( x _ { j } ) \quad \forall j \in [ N ] , p _ { j } \notin [ d + 1 , d + n _ { 1 } ] } \\ { b _ { j } = 0 \quad \forall j \in \{ d + 1 , \ldots , d + H \} \setminus \{ p _ { 1 } , \ldots , p _ { N } \} } \end{array}
+$$
+
+Note that the assignment of biases can be done via a forward pass through the network. By the above choice of biases and our definition of neurons in Section 2, we have
+
+$$
+\begin{array} { r l } & { f _ { p _ { j } } ( x _ { i } ) = \sigma _ { p _ { j } } \Big ( \beta + \alpha _ { p _ { j } } \big ( g _ { p _ { j } } ( x _ { i } ) - g _ { p _ { j } } ( x _ { j } ) \big ) \Big ) , \quad \forall j \in [ N ] , p _ { j } \in [ d + 1 , d + n _ { 1 } ] , } \\ & { f _ { p _ { j } } ( x _ { i } ) = \sigma _ { p _ { j } } \Big ( \beta + \alpha _ { p _ { j } } \big ( f _ { c ( p _ { j } ) } ( x _ { i } ) - f _ { c ( p _ { j } ) } ( x _ { j } ) \big ) \Big ) \quad \forall j \in [ N ] , p _ { j } \notin [ d + 1 , d + n _ { 1 } ] , } \\ & { f _ { j } ( x _ { i } ) = \sigma _ { j } \Big ( \alpha _ { j } f _ { c ( j ) } ( x _ { i } ) \Big ) \quad \forall j \in \{ d + n _ { 1 } + 1 , \ldots , d + H \} \setminus \{ p _ { 1 } , \ldots , p _ { N } \} , } \\ & { f _ { j } ( x _ { i } ) = \sigma _ { j } \Big ( \alpha _ { j } g _ { j } ( x _ { i } ) \Big ) \quad \forall j \in \{ d + 1 , \ldots , d + n _ { 1 } \} \setminus \{ p _ { 1 } , \ldots , p _ { N } \} . } \end{array}
+$$
+
+One notes that the output of every skip-connection neuron $p _ { j }$ is given by the first equation if $p _ { j }$ lies on the first layer and by the second equation if $p _ { j }$ lies on higher layers. In the following, to reduce notational complexity we make a convention that: $f _ { c ( p _ { j } ) } = g _ { p _ { j } }$ for every $p _ { j }$ lies on the first layer. This allows us to use the second equation for every skip-connection neuron, that is,
+
+$$
+f _ { p _ { j } } ( x _ { i } ) = \sigma _ { p _ { j } } \left( \beta + \alpha _ { p _ { j } } \left( f _ { c ( p _ { j } ) } ( x _ { i } ) - f _ { c ( p _ { j } ) } ( x _ { j } ) \right) \right) \forall j \in [ N ] .
+$$
+
+Now, since $\alpha > 0$ and all activation functions are strictly increasing by Assumption 3.1, one can easily show from the above recursive definitions that if $p _ { j }$ is a skip-connection neuron which does not lie on the first hidden layer then one has the relation: $f _ { c ( p _ { j } ) } ( x _ { i } ) < f _ { c ( p _ { j } ) } ( x _ { j } )$ if and only if $g _ { q _ { j } } ( x _ { i } ) < g _ { q _ { j } } ( x _ { j } )$ , where $q _ { j }$ is some neuron in the first hidden layer. This means if one sorts the elements of the set $\left\{ f _ { c ( p _ { j } ) } ( x _ { 1 } ) , \ldots , f _ { c ( p _ { j } ) } ( x _ { N } ) \right\}$ in increasing order then for every positive tuple $\alpha$ , the order is fully determined by the corresponding order of $\left\{ g _ { q _ { j } } ( x _ { 1 } ) , \ldots , g _ { q _ { j } } ( x _ { N } ) \right\}$ for some neuron $q _ { j }$ in the first layer. Note that this order can be different for different neurons $q _ { j }$ in the first layer, and thus can be different for different skip-connection neurons $p _ { j }$ . Let $\pi$ be a permutation such that it holds for every $j = 1 , 2 , \dots , N$ that
+
+$$
+\pi ( j ) = \underset { i \in \{ 1 , \ldots , N \} \setminus \{ \pi ( 1 ) , \ldots , \pi ( j - 1 ) \} } { \arg \operatorname* { m a x } } f _ { c ( p _ { j } ) } ( x _ { i } )
+$$
+
+It follows from above that $\pi$ is fully determined by the values of $g$ at the first layer. By definition one has $f _ { c ( p _ { j } ) } ( x _ { \pi _ { i } } ) \mathop { < } _ { c ( p _ { j } ) } ( x _ { \pi _ { j } } )$ for every $i > j$ . Since $\pi$ is independent of every positive tuple $\alpha$ and fully determined by the values of $g$ , it can be fixed in the beginning. One can assume w.l.o.g. that $\pi$ is the identity permutation as otherwise one can reorder the training samples according to $\pi$ so that the rank of $\Psi$ does not change. Thus it holds for every $\alpha > 0$ that
+
+$$
+\delta _ { i j } : = f _ { c ( p _ { j } ) } ( x _ { i } ) - f _ { c ( p _ { j } ) } ( x _ { j } ) < 0 \quad \forall i , j \in [ N ] , i > j
+$$
+
+Now, we are ready to show that there exists a positive tuple $\alpha$ for which $\Psi$ has full rank. We consider two cases of the activation functions of skip-connection neurons as stated in Assumption 3.1:
+
+• In the first case, the activation functions $\sigma _ { p _ { j } } : \mathbb { R }  \mathbb { R }$ for every $j \in [ N ]$ are strictly increasing, bounded and $\begin{array} { r } { \operatorname* { l i m } _ { t  - \infty } \sigma _ { p _ { j } } ( t ) = \mathbf { \bar { 0 } } } \end{array}$ . In the following, let $l ( j )$ denote the layer index of the hidden unit $j$ . For every hidden unit $j \in \{ d + 1 , \ldots , d + H \}$ we set $\alpha _ { j }$ to be the maximum of certain bounds (explained later in (10)) associated to all skip-connection neurons $p _ { k }$ lying on the same layer, that is,
+
+$$
+\alpha _ { j } = \operatorname* { m a x } \left\{ 1 , \operatorname* { m a x } _ { \substack { k \in [ N ] | l ( p _ { k } ) = l ( j ) } } \operatorname* { m a x } _ { i > k } \frac { \sigma _ { p _ { k } } ^ { - 1 } ( \epsilon ) - \beta } { f _ { c ( p _ { k } ) } ( x _ { i } ) - f _ { c ( p _ { k } ) } ( x _ { k } ) } \right\}
+$$
+
+where $\epsilon > 0$ is an arbitrarily small constant which will be specified later. There are a few remarks we want to make for Eq. (9) before proceeding with our proof. First, the second term in (9) can be empty if there is no skip-connection unit $p _ { k }$ which lies on the same layer as unit $j$ , in which case $\alpha _ { j }$ is simply set to 1. Second, $\alpha _ { j }$ ’s are well-defined by constructing the values $f _ { c ( p _ { k } ) } ( x _ { r } )$ , $\dot { r } = 1 , \ldots , N$ by a forward pass through the network (note that the network is a directed, acyclic graph; in particular, in the formula of $\alpha _ { j }$ , one has $l ( c ( p _ { k } ) ) < l ( p _ { k } ) = l ( j )$ and thus the computation of $\alpha _ { j }$ is feasible given the values of hidden units lying below the layer of unit $j$ , namely $f _ { c ( p _ { k } ) } .$ ). Third, if $j$ and $j ^ { \prime }$ are two neurons from the same layer, i.e. $l ( j ) = l ( j ^ { \prime } )$ , then it follows from (9) that $\alpha _ { j } = \alpha _ { j \prime }$ , meaning that their corresponding weight vectors are scaled by the same factor, thus any potential weight sharing conditions imposed on these neurons can still be satisfied.
+
+The main idea of choosing the above values of $\alpha$ is to obtain
+
+$$
+\begin{array} { r } { \Psi _ { i j } = f _ { p _ { j } } ( x _ { i } ) \le \epsilon \quad \forall i , j \in [ N ] , i > j . } \end{array}
+$$
+
+To see this, one first observes that the inequality (8) holds for the constructed values of $\alpha$ since they are all positive. From (9) it holds for every skip-connection unit $p _ { j }$ that
+
+$$
+\alpha _ { p _ { j } } > \operatorname* { m a x } _ { i > j } \frac { \sigma _ { p _ { j } } ^ { - 1 } ( \epsilon ) - \beta } { f _ { c ( p _ { j } ) } ( x _ { i } ) - f _ { c ( p _ { j } ) } ( x _ { j } ) } \quad \forall j \in [ N ]
+$$
+
+which combined with (8) leads to
+
+$$
+\begin{array} { r } { \alpha _ { p _ { j } } \big ( f _ { c ( p _ { j } ) } ( x _ { i } ) - f _ { c ( p _ { j } ) } ( x _ { j } ) \big ) \le \sigma _ { p _ { j } } ^ { - 1 } ( \epsilon ) - \beta \quad \forall i , j \in [ N ] , i > j . } \end{array}
+$$
+
+and thus using (6) we obtain (10).
+
+Coming back to the main proof of the lemma, since $\sigma _ { p _ { j } } ( j \in [ N ] )$ are bounded there exists a finite positive constant $C$ such that it holds that
+
+$$
+\vert \Psi _ { i j } \vert \le C \quad \forall i , j \in [ N ]
+$$
+
+By the Leibniz-formula one has
+
+$$
+\operatorname* { d e t } ( \Psi _ { 1 : N , 1 : N } ) = \prod _ { j = 1 } ^ { N } \sigma _ { p _ { j } } ( \beta ) + \sum _ { \pi \in S _ { N } \backslash \{ \gamma \} } \mathrm { s i g n } ( \pi ) \prod _ { j = 1 } ^ { N } \Psi _ { \pi ( j ) j }
+$$
+
+where $S _ { N }$ is the set of all $N !$ permutations of the set $\{ 1 , \ldots , N \}$ and $\gamma$ is the identity permutation. Now, one observes that for every permutation $\pi \neq \gamma$ , there always exists at least one component $j$ where $\pi ( j ) > j$ in which case it follows from (10) and (11) that
+
+$$
+\Bigl | \sum _ { \pi \in S _ { N } \backslash \{ \gamma \} } \mathrm { s i g n } ( \pi ) \prod _ { j = 1 } ^ { N } \Psi _ { \pi ( j ) j } \Bigr | \ \le N ! C ^ { N - 1 } \epsilon
+$$
+
+By choosing $\begin{array} { r } { \epsilon = \frac { \bigg | \prod _ { j = 1 } ^ { N } \sigma _ { p _ { j } } ( \beta ) \bigg | } { 2 N ! C ^ { N - 1 } } } \end{array}$ , we get that
+
+$$
+\operatorname* { d e t } ( \Psi _ { 1 : N , 1 : N } ) \geq \prod _ { j = 1 } ^ { N } \sigma _ { p _ { j } } ( \beta ) - \frac { 1 } { 2 } \prod _ { j = 1 } ^ { N } \sigma _ { p _ { j } } ( \beta ) = \frac { 1 } { 2 } \prod _ { j = 1 } ^ { N } \sigma _ { p _ { j } } ( \beta ) \neq 0
+$$
+
+and thus $\Psi$ has full rank.
+
+• In the second case we consider the softplus activation function which satisfies our Assumption 3.1 that there exists a backward path from every skip-connection neuron $p _ { j }$ to the first hidden layer s.t. on this path there is no neuron which has skip-connections to the output or shared weights with other skip-connection neurons.
+
+We choose all the weights and biases similarly to the first case. The only difference is that for every skip-connection neuron $p _ { j } ( 1 \leq j \leq N )$ , the position of 1 in its weight vector $\boldsymbol { \underline { { u } } _ { p _ { j } } }$ is chosen s.t. the value of neuron $p _ { j }$ is determined by the first neuron on the corresponding backward path as stated in Assumption 3.1, that is,
+
+$$
+\sum _ { k  p _ { j } } f _ { k } ( x _ { i } ) u _ { k  p _ { j } } = f _ { c ( p _ { j } ) } ( x _ { i } ) .
+$$
+
+For skip-connection neurons we set all $\{ \alpha _ { p 1 } , \dotsc , \alpha _ { p _ { N } } \}$ to some scalar variable $\alpha$ , and for non-skip connection neurons $j$ we set $\alpha _ { j } = 1$ . From (6) and equations of (5) we have
+
+$$
+\begin{array} { r l } & { f _ { p _ { j } } ( x _ { i } ) = \sigma _ { p _ { j } } \Big ( \beta + \alpha \big ( f _ { c ( p _ { j } ) } ( x _ { i } ) - f _ { c ( p _ { j } ) } ( x _ { j } ) \big ) \Big ) \quad \forall j \in [ N ] , } \\ & { \quad f _ { j } ( x _ { i } ) = \sigma _ { j } \big ( f _ { c ( j ) } ( x _ { i } ) \big ) \quad \forall j \in \{ d + 1 , \ldots , d + H \} \setminus \{ p _ { 1 } , \ldots , p _ { N } \} . } \end{array}
+$$
+
+Note that with above construction of $u$ and $\alpha$ , the only case where our weight sharing conditions can be potentially violated is between a skip-connection neuron ( $( \alpha _ { j } = \alpha )$ ) with a neuron on a backward path ${ \bf \Phi } _ { \cdot } ( x _ { j } = 1$ ). However, this is not possible because our assumption in this case states that there is no weight sharing between a skip-connection neuron and a neuron on one of the backward paths.
+
+Next, by our assumption the recursive backward path $c ^ { ( k ) } ( p _ { j } )$ does not contain any skipconnection unit and thus will eventually end up at some neuron $q _ { j } \in [ d + 1 , d + n _ { 1 } ]$ in the first hidden layer after some finite number of steps. Thus we can write for every $j \in [ N ]$
+
+$$
+f _ { p _ { j } } ( x _ { i } ) = \sigma _ { p _ { j } } \Bigl ( \beta + \alpha \bigl ( f _ { c ( p _ { j } ) } ( x _ { i } ) - f _ { c ( p _ { j } ) } ( x _ { j } ) \bigr ) \Bigr ) ,
+$$
+
+where
+
+$$
+f _ { c ( p _ { j } ) } ( x _ { i } ) = \sigma _ { c ( p _ { j } ) } ( \sigma _ { c ( c ( p _ { j } ) ) } ( \hdots ( g _ { q _ { j } } ( x _ { i } ) ) \hdots ) ) \quad \forall i \in [ N ] .
+$$
+
+Moreover, we have from (8) that $f _ { c ( p _ { j } ) } ( x _ { i } ) < f _ { c ( p _ { j } ) } ( x _ { j } )$ for every $i > j$ . Note that softplus fulfills for $t < 0$ , $\begin{array} { r } { \sigma _ { \gamma } ( t ) \leq \frac { 1 } { \gamma } e ^ { \gamma t } } \end{array}$ , whereas for $t > 0$ one has $\begin{array} { r } { \sigma _ { \gamma } ( t ) \leq \frac { 1 } { \gamma } + t } \end{array}$ . The latter property implies $\begin{array} { r } { \sigma ^ { ( K ) } ( t ) \le \frac { K } { \gamma } + t } \end{array}$ . Finally, this together implies that there exist positive constants $c _ { 1 } , c _ { 2 } , c _ { 3 } , c _ { 4 }$ such that it hods
+
+$$
+| \prod _ { j = 1 } ^ { N } \Psi _ { \pi ( j ) j } | \leq c _ { 1 } e ^ { - \alpha c _ { 2 } } ( c _ { 3 } + \alpha ) ^ { N - 1 } .
+$$
+
+This can be made arbitrarily small by increasing $\alpha$ . Thus we get
+
+$$
+\operatorname* { l i m } _ { \alpha \to \infty } \operatorname* { d e t } ( \Psi _ { 1 : N , 1 : N } ) = \prod _ { j = 1 } ^ { N } \sigma _ { p _ { j } } ( \beta ) \neq 0
+$$
+
+So far, we have shown that there always exist $U$ such that $\Psi$ has full rank. Since every activation function is real analytic by Assumption 3.1, every entry of $\Psi$ is also a real analytic function of the network parameters where $\Psi$ depends on. The set of low rank matrices $\Psi$ can be characterized by a system of equations such that all the $\textstyle { \binom { M } { N } }$ determinants of all $N \times N$ sub-matrices of $\Psi$ are zero. As the determinant is a polynomial in the entries of the matrix and thus an analytic function of the entries and composition of analytic functions are again analytic, we conclude that each determinant is an analytic function of $U$ . As shown above, there exists at least one $U$ such that one of these determinant functions is not identically zero and thus by Lemma A.1, the set of $U$ where this determinant is zero has measure zero. But as all submatrices need to have low rank in order that $\Psi$ has low rank, it follows that the set of $U$ where $\Psi$ has low rank has just measure zero. $\boxed { \begin{array} { r l } \end{array} }$
+
+# C EXTENSION OF THEOREM 3.4 TO GENERAL CONVEX LOSSES
+
+In this section, we consider a more general training objective, defined as
+
+$$
+\Phi ( U , V ) = \varphi ( G ( U , V ) )
+$$
+
+where $G ( U , V ) = \Psi ( U ) V \in \mathbb { R } ^ { N \times m }$ is the output of the network for all training samples at some given parameters $( U , V )$ , and $\varphi : \mathbb { R } ^ { N \times m }  \mathbb { R }$ the loss function applied on the network output.
+
+Assumption C.1 The loss function $\varphi : \mathbb { R } ^ { N \times m }  \mathbb { R }$ is convex and bounded from below.
+
+One can easily check that the following loss functions satisfy Assumption C.1 as they are all convex and bounded from below by zero:
+
+1. The cross-entropy loss from Equation 2, in particular:
+
+$$
+\varphi ( G ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } - \log \Big ( \frac { e ^ { G _ { i y _ { i } } } } { \sum _ { k = 1 } ^ { m } e ^ { G _ { i k } } } \Big ) ,
+$$
+
+where $( x _ { i } , y _ { i } ) _ { i = 1 } ^ { N }$ is the training data with $y _ { i }$ being the ground-truth class of $x _ { i }$
+
+2. The standard square loss (for classification/regression tasks)
+
+$$
+\varphi ( G ) = \frac { 1 } { 2 } \left. G - Y \right. _ { F } ^ { 2 } ,
+$$
+
+where $Y \in \mathbb { R } ^ { N \times m }$ is the ground-truth matrix.
+
+3. The multi-class Hinge-loss (for classification tasks)
+
+$$
+\varphi ( G ) = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \operatorname* { m a x } _ { j \neq y _ { i } } \operatorname* { m a x } ( 0 , 1 - ( G _ { i y _ { i } } - G _ { i j } ) ) ,
+$$
+
+where $( x _ { i } , y _ { i } ) _ { i = 1 } ^ { N }$ is the training data with $y _ { i }$ being the ground-truth class of $x _ { i }$ .
+
+By Assumption C.1, $\varphi$ is bounded from below, thus it attains a finite infimum:
+
+$$
+p ^ { * } : = \operatorname* { i n f } _ { G \in \mathbb { R } ^ { N \times m } } \varphi ( G ) < \infty .
+$$
+
+Basically, $p ^ { * }$ serves as a lower bound on our training objective $\Phi$ . For the above examples, it holds $p ^ { * } = 0$ . Next, we adapt the definition of “bad local valleys” from Definition 3.3 to the current setting.
+
+Definition C.2 The $\alpha$ -sublevel set of $\Phi$ is defined as $L _ { \alpha } = \{ ( U , V ) \mid \Phi ( U , V ) < \alpha \} \ : ,$ . A local valley is defined as a connected component of some sublevel set $L _ { \alpha }$ . A bad local valley is a local valley on which the training objective $\Phi$ cannot be made arbitrarily close to $p ^ { * }$ .
+
+The following result extends Theorem 3.4 to general convex losses. The proofs are mostly similar as before, but we present them below for the completeness and convenience of the reader.
+
+Theorem C.3 The following holds under Assumption 3.1 and Assumption $C . I$ :
+
+1. There exist uncountably many solutions with zero training error.
+
+2. The loss landscape of $\Phi$ does not have any bad local valley.
+
+3. There exists no suboptimal strict local minimum.
+
+4. For cross-entropy loss (2) and square loss (14) there exists no local maximum.
+
+# Proof:
+
+1. By Lemma 3.2 the set of $U$ such that $\Psi ( U )$ has not full rank $N$ has Lebesgue measure zero. Given $U$ such that $\Psi$ has full rank, the linear system $\Psi ( U ) V = Y$ has for every possible target output matrix $Y \in \mathbb { R } ^ { N \times m }$ at least one solution $V$ . As this is possible for almost all $U$ , there exist uncountably many solutions achieving zero training error.
+
+2. Let $C$ be a non-empty, connected component of some sub-level set $L _ { \alpha }$ where $\alpha > p ^ { * }$ . Note that $L _ { \alpha } = \mathcal { O }$ if $\alpha \leq p ^ { * }$ by Definition C.2. Given any $\epsilon \in ( p ^ { * } , \alpha )$ , we will show that $C$ always contains a point $( U , V )$ s.t. $\Phi ( U , V ) \leq \epsilon$ as this would imply that the loss $\Phi$ restricted to $C$ can always attain arbitrarily small value close to $p ^ { * }$ .
+
+We note that $L _ { \alpha } = \Phi ^ { - 1 } ( ( - \infty , \alpha ) )$ is an open set according to Proposition A.2. Since $C$ is a non-empty connected component of $L _ { \alpha }$ , $C$ must also be an open set with non-zero Lebesgue measure. By Lemma 3.2 the set of $U$ where $\Psi ( U )$ has not full rank has measure zero and thus $C$ must contain a point $( U , V )$ such that $\Psi ( U )$ has full rank. By Assumption C.1, $\varphi$ attains its infimum at $p ^ { * } < \epsilon$ , and thus by continuity of $\varphi$ , there exists $\dot { G } ^ { * } \in \mathbb { R } ^ { N \times m }$ such that $p ^ { * } \leq \varphi ( G ^ { * } ) \leq \epsilon$ . As $\Psi ( U )$ has full rank, there always exist $V ^ { * }$ such that $\Psi ( U ) V ^ { * } = G ^ { * }$ Now, one notes that the loss $\Phi ( U , V ) = \varphi ( \Psi ( U ) V )$ is convex in $V$ , and that $\Phi ( U , V ) < \alpha$ thus we have for the line segment $V ( \lambda ) = \lambda V + ( 1 - \lambda ) V ^ { * }$ for $\lambda \in [ 0 , 1 ]$ ,
+
+$$
+\Phi ( U , V ( \lambda ) ) \le \lambda \Phi ( U , V ) + ( 1 - \lambda ) \Phi ( U , V ^ { * } ) < \lambda \alpha + ( 1 - \lambda ) \epsilon < \alpha .
+$$
+
+Thus the whole line segment from $( U , V )$ to $( U , V ^ { * } )$ is contained in $L _ { \alpha }$ . Since $C$ is a connected component of $L _ { \alpha }$ which contains $( U , V )$ , it follows that $( U , V ^ { * } ) \in C$ . Moreover, one has $\Phi ( U , \bar { V } ^ { * } ) = \varphi ( G ^ { * } ) \leq \epsilon .$ , which thus implies that the loss can always be made $\epsilon$ -small inside the set $C$ for every $\epsilon \in ( p ^ { * } , \alpha )$ .
+
+3. Let $( U _ { 0 } , V _ { 0 } )$ be a strict suboptimal local minimum, then there exists $r > 0$ such that $\Phi ( U , V ) > \Phi ( U _ { 0 } , V _ { 0 } ) > p ^ { * }$ for all $( U , V ) \in { \cal B } ( ( U _ { 0 } , V _ { 0 } ) , r ) \ : \backslash \ : \{ ( U _ { 0 } , V _ { 0 } ) \}$ where $B ( \cdot , r )$ denotes a closed ball of radius r. Let α = min(U,V )∈∂B(U0,V0),r
 which exists as $\Phi$ is continuous and the boundary $\partial B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ of $B \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ is compact. Note that $\Phi ( U _ { 0 } , V _ { 0 } ) < \alpha$ as $( U _ { 0 } , V _ { 0 } )$ is a strict local minimum, and thus $( U _ { 0 } , \dot { V } _ { 0 } ) \in L _ { \alpha }$ . Let $E$ be the connected component of $L _ { \alpha }$ which contains $( U _ { 0 } , V _ { 0 } )$ , that is, $( U _ { 0 } , V _ { 0 } ) \in E \subseteq L _ { \alpha }$ . Since the loss of every point inside $E$ is strictly smaller than $\alpha$ , whereas the loss of every point on the boundary $\mathrm { \widehat { \partial } } D \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ is greater than or equal to $\alpha$ , $E$ must be contained in the interior of the ball, that is $E \subset \dot { B } \big ( ( U _ { 0 } , V _ { 0 } ) , r \big )$ . Moreover, $\Phi ( U , V ) \ge \Phi ( U _ { 0 } , V _ { 0 } ) > p ^ { * }$ for every $( U , V ) \in E$ and thus the values of $\Phi$ restricted to $E$ can not be arbitrarily close to $p ^ { * }$ , which means that $E$ is a bad local valley, which contradicts G.3.2.
+
+4. The proof for cross-entropy loss is similar to Theorem 3.4. For square loss, one has
+
+$$
+\Phi ( U , V ) = \frac { 1 } { 2 } \left\| \Psi ( U ) V - Y \right\| _ { F } ^ { 2 } = \frac { 1 } { 2 } \left\| \left( \mathbb { I } _ { m } \otimes \Psi ( U ) \right) \nu e c ( V ) - \nu e c ( Y ) \right\| _ { 2 } ^ { 2 }
+$$
+
+w.r.t. where $V$ $\otimes$ is denotes Kronecker product, and $\nabla _ { \nu e c ( V ) } ^ { 2 } \Phi = ( \mathbb { I } _ { m } \diamondsuit \Psi ( U ) ) ^ { T } ( \mathbb { I } _ { m } \otimes \Psi ( U ) )$ $\mathbb { I } _ { m }$ an $m \times m$ . identity matrix. The hessian of $\Phi$
+
+Suppose by contradiction that $( U , V )$ is a local maximum. Then the Hessian of $\Phi$ is negative semi-definite. As principal submatrices of negative semi-definite matrices are again negative semi-definite, the Hessian of $\Phi$ w.r.t $V$ must be also negative semi-definite. However, $\Phi$ is which is both p.s.d. and n.s.d. is the zero matrix. It follows that convex in $V$ thus its Hessian restricted to $V$ must be positive semi-definite. The only matrix $\nabla _ { \nu e c ( V ) } ^ { 2 } \Phi ( U , V ) \stackrel { \cdot } { = } 0$ and thus $\Psi ( U ) = 0$ . By Assumption 3.1, there exists $j \in [ M ]$ s.t. $\sigma _ { p _ { j } }$ is strictly positive, thus some entries of $\Psi ( U )$ must be strictly positive, and so $\Psi ( U )$ cannot be identically zero, leading to a contradiction. Therefore $\Phi$ has no local maximum.
+
+D THE ARCHITECTURE OF CNN13 FROM TABLE 1: SEE TABLE 3   
+
+<table><tr><td rowspan=1 colspan=1>Layer</td><td rowspan=1 colspan=1>Output size</td><td rowspan=1 colspan=1>#neurons</td></tr><tr><td rowspan=1 colspan=1>Input: 28 × 28</td><td rowspan=1 colspan=1>28×28×1</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>3×3conv-64,stride1</td><td rowspan=1 colspan=1>28×28×64</td><td rowspan=1 colspan=1>50176</td></tr><tr><td rowspan=1 colspan=1>3×3conv-64,stride1</td><td rowspan=1 colspan=1>28×28×64</td><td rowspan=1 colspan=1>50176</td></tr><tr><td rowspan=1 colspan=1>3×3conv-64，stride 2</td><td rowspan=1 colspan=1>14×14×64</td><td rowspan=1 colspan=1>12544</td></tr><tr><td rowspan=1 colspan=1>3 ×3conv -128,stride1</td><td rowspan=1 colspan=1>14×14×128</td><td rowspan=1 colspan=1>25088</td></tr><tr><td rowspan=1 colspan=1>3 ×3conv-128,stride1</td><td rowspan=1 colspan=1>14×14×128</td><td rowspan=1 colspan=1>25088</td></tr><tr><td rowspan=1 colspan=1>3 ×3conv-128,stride2</td><td rowspan=1 colspan=1>7×7×128</td><td rowspan=1 colspan=1>6272</td></tr><tr><td rowspan=1 colspan=1>3 ×3conv -256,stride1</td><td rowspan=1 colspan=1>7×7×256</td><td rowspan=1 colspan=1>12544</td></tr><tr><td rowspan=1 colspan=1>1 × 1conv - 256,stride1</td><td rowspan=1 colspan=1>7×7×256</td><td rowspan=1 colspan=1>12544</td></tr><tr><td rowspan=1 colspan=1>3×3conv-256,stride2</td><td rowspan=1 colspan=1>4×4×256</td><td rowspan=1 colspan=1>4096</td></tr><tr><td rowspan=1 colspan=1>3×3conv-256,stride1</td><td rowspan=1 colspan=1>4×4×256</td><td rowspan=1 colspan=1>4096</td></tr><tr><td rowspan=1 colspan=1>3×3conv-256,stride2</td><td rowspan=1 colspan=1>2×2×256</td><td rowspan=1 colspan=1>1024</td></tr><tr><td rowspan=1 colspan=1>3 × 3conv- 256,stride1</td><td rowspan=1 colspan=1>2×2×256</td><td rowspan=1 colspan=1>1024</td></tr><tr><td rowspan=1 colspan=1>3×3conv-256,stride2</td><td rowspan=1 colspan=1>1×1×256</td><td rowspan=1 colspan=1>256</td></tr><tr><td rowspan=1 colspan=3>Fully connected,10 output units</td></tr></table>
+
+Table 3: The architecture of CNN13 for MNIST dataset. There are in total 179, 840 hidden neurons.
+
+# E VISUALIZATION OF THE LOSS LANDSCAPE BEFORE AND AFTER ADDING SKIP-CONNECTIONS TO THE OUTPUT LAYER
+
+Similar to Li et al. (2018); Goodfellow et al. (2015), we visualize the loss surface restricted to a two dimensional subspace of the parameter space. The subspace is chosen to go through some point $( U _ { 0 } , V _ { 0 } )$ learned by SGD and spanned by two random directions $( U _ { 1 } , V _ { 1 } )$ and $( U _ { 2 } , V _ { 2 } )$ .
+
+For the purpose of illustration, we train with SGD a two-hidden-layer fully connected network with 784 and 300 hidden units respectively, followed by a 10-way softmax. The training set consists of 1024 images, which are randomly selected from MNIST dataset. After adding skip-connections to the output, the network fulfills $M = N = 1 0 2 4$ . Figure 3 shows the heat map of the loss surface before and after adding skip-connections. One can see a visible effect that skip-connections have helped to smooth the loss landscape near a small sub-optimal region and allows gradient descent to flow directly from there to the bottom of the landscape with smaller objective value.
+
+# F DISCUSSION OF TRAINING ERROR IN TABLE 2
+
+Training error for the experiment in Table 2. As shown in Table 2, the training error is zero in all cases, except when the original VGG models are used with sigmoid activation function. The reason, as noticed in our experiments, is because the learning of these sigmoidal networks converges quickly to a constant zero classifier (i.e. the output of the last hidden layer converges to zero), which makes both training and test accuracy converge to $1 0 \%$ and the loss in Equation (2) converges to $- \log ( 1 / 1 0 )$ . While we are not aware of a theoretical explanation for this behavior, it is not restricted to the specific architecture of VGGs but hold in general for plain sigmoidal networks with depth ${ > } 5$ as pointed out earlier by Glorot & Bengio (2010). As shown in Table 2, Densenets however do not suffer from this phenomenon, probably because they already have skip-connections between all the hidden layers of a dense block, thus gradients can easily flow from the output to every layer of a dense block, which makes the training of this network with sigmoid activation function become feasible.
+
+![](images/67f0ad64a205e40f6d91db078bdced8be23d0dd2fb47417fc8a78564b9679d34.jpg)  
+Figure 3: Loss surface of a two-hidden-layer network on a small MNIST dataset.
+
+![](images/99cc486e2c89f05108ef52cdbb45384c550c70c8a158dc518a8efb2ebf7127e8.jpg)
+
+Discussion of convergence speed. For sigmoid activation, we noticed that skip-models when trained with the random sampling approach (skip-rand) often converge much slower than when trained with full SGD (skip-SGD). In our experiments, to be sure that one gets absolute zero training error, we set the number of training epochs to 5000 for the former case and 1000 for the later. Perhaps a better learning rate schedule might help to reduce this number, or maybe not, but this is beyond the scope of this paper. For softplus activation, we noticed a much faster convergence – all models often converge within 300 epochs to absolute zero training error.
+
+Skip-connections are also helpful for training very deep networks with softplus activation. Previously we have shown that skip-connections are helpful for training deep sigmoidal networks. In this part, we show a similar result for softplus activation function. For the purpose of illustration, we create a small dataset with $N = 1 0 0 0$ training images randomly chosen from CIFAR10 dataset. We use a very deep network with 150 fully connected layers, each of width 10, and softplus activation. A skip-model is created by adding skip-connections from $N$ randomly chosen neurons to the output units. We train both networks with SGD. The best learning rate for each model is empirically chosen from $\left\{ 1 0 ^ { - 2 } , 1 0 ^ { - 3 } , 1 0 ^ { - 4 } , 1 0 ^ { - 5 } \right\}$ . We report the training loss and training error of both models in Figure 5. One can see that the skip-network easily converge to zero training error within 200 epochs, whereas the original network has stronger fluctuations and fails to converge after 1000 epochs. This is directly related to our result of Theorem 3.4 in the sense that skip-connections can help to smooth the loss landscape and enable effective training of very deep networks.
+
+![](images/e2f8aa5f93b0ade6b31678fc1a12d924e7b7ca999478079dcb0075a6b677c85d.jpg)  
+Figure 5: Training progress of a 150-layer neural network with and without skip-connections.
+
+# G ADDITIONAL EXPERIMENTS: MAX-POOLING OF VGGS ARE REPLACED BY 2X2 CONVOLUTIONAL LAYERS OF STRIDE 2
+
+The original VGG Simonyan & Zisserman (2015) and Densenet Huang et al. (2017) contain pooling layers in their architecture. In particular, original VGGs have max-pooling layers, and original Densenets have averaging pooling layers. In the following, we will clarify how/if these pooling layers have been used in our experiments in Table 2, and whether and how our theretical results are appicable to this case, as well as presenting additional experimental results in this regard.
+
+First of all, we note that Densenets Huang et al. (2017) contain pooling layers only after the first dense block. Meanwhile, as noted in Table 2, our experiments with Densenets only use skip-connections from hidden units of the first dense block, and thus Lemma 3.2 and Theorem 3.4 are applicable. The reason is that one can restrict the full-rank analysis of matrix $\Psi$ in Lemma 3.2 to the hidden units of the first dense block, so that it follows that the set of parameters of the first dense block where $\Psi$ has not full rank has Lebesgue measure zero, from which our results of Theorem 3.4 follow immediately.
+
+However for VGGs in Table 2, we kept their max-pooling layers similar to the original architecture as we wanted to have a fair comparison between our skip-models and the original models. In this setting, our results are not directly applicable because we lose the analytic property of the entries of $\Psi$ w.r.t. its dependent parameters, which is crucial to prove Lemma 3.2. Therefore in this section, we would like to present additional results to Table 2 in which we replace all max-pooling layers of all VGG models from Table 2 with $2 \mathbf { x } 2$ convolutional layers of stride 2. In this case, the whole network consists of only convolutional and fully connected layers, hence our theoretical results are applicable.
+
+The experimental results are presented in Table 4. Overall, our main observations are similar as before. The performance gap between original models and their corresponding skip-variants are approximately the same as in Table 2 or slightly more pronounced in some cases. A one-to-one comparison with Table 2 also shows that the performance of skip-models themselves have decreased by $4 - 7 \%$ after the replacement of max-pooling layers with $2 \mathbf { x } 2$ convolutional layers. This is perhaps not so surprising because the problem gets potentially harder when the network has more layers to be learned, especially in case of sigmoid activation where the decrease is sharper. Similar to Table 2, adding skip-connections to the output units still prove to be very helpful – it improves the result for softplus while making the training of deep networks with sigmoid activation become possible at all. Finally, the training of full network with SGD still yields significantly better solutions in terms of generalization error than the random feature approach. This confirms once again the implicit bias of SGD towards high quality solutions among infinitely many solutions with zero training error.
+
+Table 4: Test accuracy $( \% )$ of VGG networks from Table 2 where max-pooling layers are replaced by $2 \mathbf { x } 2$ convolutional layers of stride 2 (denoted as mp2conv). Other notations are similar to Table 2.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Sigmoid activation function</td><td rowspan=1 colspan=2>Softplus activation function</td></tr><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>C-10</td><td rowspan=1 colspan=1>C-10+</td><td rowspan=1 colspan=1>C-10</td><td rowspan=1 colspan=1>C-10+</td></tr><tr><td rowspan=1 colspan=1>VGG11-mp2convVGG11-mp2conv-skip (rand)VGG11-mp2conv-skip (SGD)</td><td rowspan=1 colspan=1>1053.65 ± 0.6664.45 ± 0.31</td><td rowspan=1 colspan=1>10-80.15 ± 0.59</td><td rowspan=1 colspan=1>74.5355.51 ± 0.4376.18 ± 0.58</td><td rowspan=1 colspan=1>88.80-89.93 ± 0.19</td></tr><tr><td rowspan=1 colspan=1>VGG13-mp2convVGG13-mp2conv-skip (rand)</td><td rowspan=1 colspan=1>1053.45 ± 0.23</td><td rowspan=2 colspan=1>10-82.40 ± 0.23</td><td rowspan=2 colspan=1>74.0453.33 ± 0.6775.58 ± 0.77</td><td rowspan=2 colspan=1>90.37-91.04 ± 0.20</td></tr><tr><td rowspan=1 colspan=1>VGG13-mp2conv-skip (SGD)</td><td rowspan=1 colspan=1>63.53 ± 0.37</td></tr><tr><td rowspan=1 colspan=1>VGG16-mp2convVGG16-mp2conv-skip (rand)VGG16-mp2conv-skip (SGD)</td><td rowspan=1 colspan=1>1053.83 ± 0.3065.77 ± 0.67</td><td rowspan=1 colspan=1>10-83.06 ± 0.34</td><td rowspan=1 colspan=1>74.0055.34 ± 0.6476.52 ± 0.78</td><td rowspan=1 colspan=1>90.38-91.00 ± 0.23</td></tr></table>
+
+# H DATA-AUGMENTATION RESULTS FOR TABLE 2
+
+The following Table 5 shows additional results to Table 2 where data-augmentation is used now. For data-augmentation, we follow the procedure as described in (Zagoruyko & Komodakis, 2016) by considering random crops of size $3 2 \times 3 2$ after 4 pixel padding on each side of the training images and random horizontal flips with probability 0.5. For the convenience of the reader, we also repeat the results of Table 2 in the new table 5.
+
+Table 5: Test accuracy $( \% )$ of several CNN architectures with/without skip-connections on CIFAR10 $^ +$ denotes data augmentation). For each model A, A-skip denotes the corresponding skip-model in which a subset of $N$ hidden neurons “randomly selected” from the hidden layers are connected to the output units. For Densenet121, these neurons are randomly chosen from the first dense block. The names in open brackets (rand/SGD) specify how the networks are trained: rand ( $U$ is randomized and fixed while $V$ is learned with SGD), SGD (both $U$ and $V$ are optimized with SGD).   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=2>Sigmoid activation function</td><td rowspan=1 colspan=2>Softplus activation function</td></tr><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>C-10</td><td rowspan=1 colspan=1>C-10+</td><td rowspan=1 colspan=1>C-10</td><td rowspan=1 colspan=1>C-10+</td></tr><tr><td rowspan=2 colspan=1>VGG11VGG11-skip (rand)VGG11-skip (SGD)</td><td rowspan=2 colspan=1>1062.81 ± 0.3972.51 ± 0.35</td><td rowspan=2 colspan=1>10-85.55 ± 0.09</td><td rowspan=1 colspan=1>78.92</td><td rowspan=2 colspan=1>88.62-89.32 ± 0.16</td></tr><tr><td rowspan=1 colspan=1>64.49 ± 0.3880.57 ± 0.40</td></tr><tr><td rowspan=1 colspan=1>VGG13VGG13-skip (rand)VGG13-skip (SGD)</td><td rowspan=1 colspan=1>1061.50 ± 0.3470.24 ± 0.39</td><td rowspan=1 colspan=1>10-86.48 ± 0.32</td><td rowspan=1 colspan=1>80.8461.42 ± 0.4081.94 ± 0.40</td><td rowspan=1 colspan=1>90.58-91.06 ± 0.12</td></tr><tr><td rowspan=1 colspan=1>VGG16VGG16-skip (rand)VGG16-skip (SGD)</td><td rowspan=1 colspan=1>1061.57 ± 0.4170.61 ± 0.36</td><td rowspan=1 colspan=1>10-86.42 ± 0.31</td><td rowspan=1 colspan=1>81.3361.46 ± 0.3481.91 ± 0.24</td><td rowspan=1 colspan=1>90.68-91.00 ± 0.22</td></tr><tr><td rowspan=1 colspan=1>Densenet121Densenet121-skip (rand)Densenet121-skip (SGD)</td><td rowspan=1 colspan=1>86.4152.07 ± 0.4881.47 ± 1.03</td><td rowspan=1 colspan=1>90.93-90.32 ± 0.50</td><td rowspan=1 colspan=1>89.3155.39 ± 0.4886.76 ± 0.49</td><td rowspan=1 colspan=1>94.20-93.23 ± 0.42</td></tr></table>

parse/train/HJguLo0cKQ/HJguLo0cKQ.md

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+# STRENGTH IN NUMBERS: TRADING-OFF ROBUSTNESS AND COMPUTATION VIA ADVERSARIALLY-TRAINED ENSEMBLES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+While deep learning has led to remarkable results on a number of challenging problems, researchers have discovered a vulnerability of neural networks in adversarial settings, where small but carefully chosen perturbations to the input can make the models produce extremely inaccurate outputs. This makes these models particularly unsuitable for safety-critical application domains (e.g. self-driving cars) where robustness is extremely important. Recent work has shown that augmenting training with adversarially generated data provides some degree of robustness against test-time attacks. In this paper we investigate how this approach scales as we increase the computational budget given to the defender. We show that increasing the number of parameters in adversarially-trained models increases their robustness, and in particular that ensembling smaller models while adversarially training the entire ensemble as a single model is a more efficient way of spending said budget than simply using a larger single model. Crucially, we show that it is the adversarial training of the ensemble, rather than the ensembling of adversarially trained models, which provides robustness.
+
+# 1 INTRODUCTION
+
+Deep neural networks have demonstrated state-of-the-art performance in a wide range of application domains Krizhevsky et al. (2012). However, researchers have discovered that deep networks are in some sense ‘brittle’, in that small changes to their inputs can result in wildly different outputs (Huang et al., 2017; Jia & Liang, 2017; Szegedy et al., 2013). For instance, practically imperceptible (to human) modifications to images can result in misclassification of the image with high confidence. Not only are networks susceptible to these ‘attacks’, but these attacks are also relatively easy to compute using standard optimization techniques (Carlini & Wagner, 2017b; Goodfellow et al., 2014). These changes are often referred to as adversarial perturbations, in the sense that an adversary could craft a very small change to the input in order to create an undesirable outcome. This phenomenon is not unique to image classification, nor to particular network architectures, nor to particular training algorithms (Papernot et al., 2016; 2017).
+
+Adversarial attacks can be broken into different categories depending on how much knowledge of the underlying model the adversary has access to. In ‘white-box’ attacks the adversary has full access to the model, and can perform both forward and backwards passes (though not change the weights or logic of the network) (Carlini & Wagner, 2017a; Goodfellow et al., 2014). In the ‘black-box’ setting the adversary has no access to the model, but perhaps knows the dataset that the model was trained on (Papernot et al., 2016; 2017). Despite several recent papers demonstrating new defences against adversarial attacks (Akhtar & Mian, 2018; Guo et al., 2017; Liao et al., 2017; Song et al., 2017; Tramer et al., 2018; Warde-Farley & Goodfellow, 2016; Xie et al., 2017; Yuan et al., 2017), recent \` papers have demonstrated that most of these new defences are still susceptible to attacks and largely just obfuscate the gradients that the attacker can follow, and that non-gradient based attacks are still effective Uesato et al. (2018); Athalye et al. (2018).
+
+Exploring Tradeoff of Computation and Robustness In many safety-critical application domains (e.g. self-driving cars), robustness is extremely important even if it comes at the cost of increased computation. This motivated the central question considered by this paper: Is it possible to increase adversarial robustness of a classifier at the cost of increased computation?
+
+There are a number of possibilities to employ extra computation available at runtime. We can use a much larger model that requires more time to run, execute the original model multiple times and aggregate the predictions, or instead of using a single model, make predictions from a portfolio or ensemble of models. While researchers have proposed the use of portfolios and ensembles as a mechanism to improve adversarial robustness Abbasi & Gagne (2017); Thilo Strauss (2017), our ´ experimental results indicate that stronger adversaries are able to attack the ensembles successfully.
+
+Contributions In this paper, we study and analyze the trade-off of adversarial robustness and computation (memory and runtime). We propose the use of adversarial training of ensemble of models and through an exhaustive ablative analysis make the following empirical findings:
+
+• increased computation and/or model size can be used to increase robustness,   
+• ensembles on their own are not very robust, but can be made robust through adversarial training where the ensemble is treated as a single model,   
+• adversarially trained ensembles are more robust than adversarially trained individual models requiring the same amount of parameters/computation
+
+Related Work Recently, Tramer et al. (2018) investigated the use of ensembles for adversarial \` robustness. However, their goal and approach was quite different from the technique we are investigating. In Tramer et al. (2018), the authors generated adversarial perturbations using an ensemble of \` pre-trained models in order to transfer the example to another model during training. This procedure decouples adversarial example generation from the current model, and consequently the model being trained cannot simply ‘overfit’ to the procedure for generating adversarial examples, which they generally took to be single-step attack methods. The authors demonstrated strong robustness of the resulting trained model to black-box attacks. By contrast, in this paper we investigate using an ensemble of models as our predictive model, and we train the models using multi-step adversarial training. We show increased robustness to both black-box and white-box adversarial attacks using this strategy.
+
+# 2 PRELIMINARIES
+
+Here we lay out the basics of attacking a neural network by the generation of adversarial examples. Denote an input to the network as $\boldsymbol { x } \in \mathbb { R } ^ { d }$ with correct label $\hat { y } \in \mathcal { V } \subset \mathbb { N }$ , and let $m _ { \theta } : \mathbb { R } ^ { d }  \mathbb { R } ^ { | \bar { y } | }$ be the mapping performed by the neural network which is parameterized by $\theta \in \mathbb { R } ^ { p }$ . Let $L : \mathcal { V } \times \mathbb { R } ^ { | \mathcal { V } | } \to$ $\mathbb { R }$ denote the loss we are trying to minimize (e.g., the cross-entropy). When training a neural network we seek to solve
+
+$$
+\begin{array} { r l } { \mathrm { m i n i m i z e } } & { { } \mathbb { E } _ { ( x , y ) \sim \mathcal { D } } L ( \hat { y } , m _ { \theta } ( x ) ) } \end{array}
+$$
+
+over variable $\theta$ , where $\mathcal { D }$ is the data distribution. Given any fixed $\theta$ we can generate (untargeted) adversarial inputs by perturbing the input $x$ so as to maximize the loss. We restrict ourselves to small perturbations around a nominal input, and we denote by $\boldsymbol { B }$ this set of allowable inputs. For example, if we restrict ourselves to small perturbations in $\ell _ { \infty }$ norm around a nominal input $x ^ { \mathrm { n o m } }$ then we could set $\mathcal { B } = \left\{ x \vert \| x - x ^ { \mathrm { n o m } } \| _ { \infty } \leq \epsilon \right\}$ where $\epsilon > 0$ is the tolerance. A common approach for generating adversarial examples is projected gradient descent Carlini $\&$ Wagner (2016), i.e., to iteratively update the input $x$ by
+
+$$
+\tilde { { \boldsymbol { x } } } ^ { k + 1 } = \Pi _ { { \boldsymbol { \mathcal { B } } } } ( \tilde { { \boldsymbol { x } } } ^ { k } + \eta \nabla _ { x } L ( y , m _ { \theta } ( \tilde { { \boldsymbol { x } } } ^ { k } ) ) ) ,
+$$
+
+where typically $x ^ { 0 } = x + \epsilon$ for some noise $\epsilon$ , $\eta > 0$ is a step-size parameter and $\Pi _ { B }$ denotes the Euclidean projection on $\boldsymbol { B }$ . We add noise to the initial point so that the network can’t memorize the training dataset and mask or obfuscate the gradients at that point Uesato et al. (2018); Athalye et al. (2018), in other words the added noise encourages generalization of adversarial robustness to the test dataset. If instead of using the gradient we just use the sign of the gradient then this is the fast-gradient-sign method Goodfellow et al. (2014). Empirically speaking, for most networks just a few steps of either of these procedures is sufficient to generate an $\tilde { x }$ that is close to $x ^ { \mathrm { n o m } }$ but has a different label with high confidence.
+
+In this paper we are primarily concerned with the performance of ensembles of models when trained with adversarial training Madry et al. (2017). In adversarial training we train a network to minimize a weighted sum of two losses (where the relative weighting is a hyper-parameter). The first loss is the standard loss of the problem we are trying to solve on the normal training data, e.g., the cross-entropy for a classification task. The second loss is the same function as the first loss, except evaluated on adversarially generated data, where typically the adversarial data is generated by attacking the network at that time-step. In other words we replace the problem in eq. (1) with
+
+$$
+\mathbb { E } _ { ( x , y ) \sim \mathcal { D } } ( L ( \hat { y } , m _ { \theta } ( x ) ) + \rho L ( \hat { y } , m _ { \theta } ( \tilde { x } ) ) )
+$$
+
+where $\rho \geq 0$ is the weighting parameter and $\tilde { x }$ is an adversarial example generated from $x$ at model parameters $\theta$ using, for example, the update in eq. (2). This problem is usually approximated by sampling and minimizing the empirical expectation.
+
+# 3 ADVERSARIALLY-TRAINED ENSEMBLES
+
+In this section we lay out the basic strategy of using ensembles of models to increase robustness to adversarial attacks. The notion of ensemble used here simply involves taking $k$ separatelyparameterized models and averaging their predictions. If the output of network $i$ as a function of input $x$ and with network parameters $\theta _ { i }$ is given by $p ( \cdot | x , \theta _ { i } ) = m _ { \theta _ { i } } ( \bar { x } )$ , then the output of the ensemble is
+
+$$
+p ( \boldsymbol { y } | \boldsymbol { x } ) = \frac { 1 } { k } \sum _ { i = 1 } ^ { k } p ( \boldsymbol { y } | \boldsymbol { x } , \boldsymbol { \theta } _ { i } ) .
+$$
+
+Alternatively, we could consider using a ‘gating network’ to generate data-dependent weights for each model rather than a simple average, though we found the performance to be similar.
+
+Using ensembles to improve the performance of statistical models is a very old idea; see, e.g. Opitz & Maclin (1999) for a survey. The basic intuition is that several weak models can be combined in such a way that the ensemble performs better than any individual, and is sometimes explained as being caused by the errors of the models ‘cancelling’ with one another.
+
+In order to ensure that the models are actually producing different outputs the diversity of the models must be maintained. This can be done in several ways, such as bootstrapping the data, whereby each model gets a slightly different copy of the data, or using totally different model types or architectures. In the case that the model training procedure is convex, and if all models architectures are the same and are getting the same data, then the models in the ensemble would be expected to converge on the same parameters. In the case of neural networks however, the model training procedure is not convex and so our strategy for maintaining diversity is very simple—initialize each model differently. Due to the nature of training neural networks it is likely that differently initialized networks will converge (assuming they do, in fact, converge) to different points of the parameter space. The insight that only different initialization is required is not new, previous papers have observed that different initialization is sufficient for uncertainty estimation Lakshminarayanan et al. (2016); Osband et al. (2016).
+
+Different initialization for networks has an appealing interpretation. If we take a Bayesian approach to the classification problem, then we have a prior over possible model parameters, $p ( \theta )$ , a likelihood of the data, $p ( D | \theta ) ^ { \overline { { } } }$ , and a probability of a label $y$ given an input and a model, $\overset { \cdot } { p ( \boldsymbol { y } | \boldsymbol { x } , \boldsymbol { \theta } ) }$ . The ‘Bayes-optimal’ classification of a new data point $x$ is given by
+
+$$
+y ^ { \star } = \mathrm { a r g m a x } _ { y } \int _ { \theta } p ( y | x , \theta ) p ( D | \theta ) p ( \theta ) .
+$$
+
+This classifier is optimal in the sense that no other classifier can outperform it on average, given the same model class and knowledge of the prior and likelihood; however, the formulation is intractable for all but small problems. We can consider approximating it by the following approach, sample initial parameters from the prior $p ( \theta )$ and run an iterative procedure to (approximately) maximize the likelihood $p ( D | \theta )$ . Very loosely speaking, we can consider this procedure as approximately sampling from the posterior over models $p ( \theta | D ) \propto p ( D | \theta ) p ( \theta )$ . Consequently, we output the classification
+
+$$
+y ^ { \star } = \operatorname { a r g m a x } _ { y } \sum _ { i = 1 } ^ { k } p ( y | x , \theta _ { i } ) ,
+$$
+
+Algorithm 1 Adversarial ensemble training using PGD under $\ell _ { \infty }$ norm constraint input: $k$ neural networks $m _ { \theta _ { i } }$ , $i = 1 , \ldots , k$ ; attack steps $N$ ; step sizes $\eta , \hat { \eta }$ ; initial variance $\sigma$ adversarial loss weighting $\rho$ ; perturbation width $\delta$
+
+initialize: neural network parameters $\theta _ { i } ^ { 0 }$ randomly, $i = 1 , \ldots , k$   
+for time-step $t = 0 , 1 , \ldots , \mathbf { i }$ o sample input minibatch $( x , \hat { y } ) \sim \mathcal { D }$ initialize $\bar { \tilde { x } } { } ^ { 0 } = x + \epsilon$ where $\epsilon \sim \mathcal { N } ( 0 , \sigma ^ { 2 } I )$ define $\boldsymbol { \mathcal { B } } = \{ x ^ { \prime } \ | \ \| x - x ^ { \prime } \| _ { \infty } \leq \delta \}$ for $k = 0 , \ldots , N - 1$ do
+
+$$
+\tilde { { \boldsymbol { x } } } ^ { k + 1 } = \Pi _ { \mathcal { B } } \big ( \tilde { { \boldsymbol { x } } } ^ { k } + \hat { \eta } \nabla _ { x } L \big ( \hat { y } , \frac { 1 } { k } \sum _ { i = 1 } ^ { k } m _ { \theta _ { i } } ( \tilde { { \boldsymbol { x } } } ^ { k } ) \big ) \big )
+$$
+
+# end for
+
+update parameters for each $i = 1 , \ldots , k$ :
+
+$$
+\theta _ { i } ^ { t + 1 } = \theta _ { i } ^ { t } - \eta \nabla _ { \theta _ { i } } \left( L ( \hat { y } , \frac { 1 } { k } \sum _ { j = 1 } ^ { k } m _ { \theta _ { j } ^ { t } } ( x ) ) + \rho L ( \hat { y } , \frac { 1 } { k } \sum _ { j = 1 } ^ { k } m _ { \theta _ { j } ^ { t } } ( \tilde { x } ^ { N } ) ) \right)
+$$
+
+# end for
+
+i.e., the best guess of the ensemble. The role of initialization therefore is that of sampling from our prior over possible model parameters.
+
+Adversarial training of Ensembles Up to this point we have discussed the use of ensembles for improving classification performance and approximating the Bayes optimal classifier. Typically speaking neural networks appear to not benefit much from ensembling in terms of nominal performance. Here, however, we make the claim that adversarially trained ensembles of networks provide a level of robustness to adversarial attacks. When using ensembles the loss function for adversarial training in (3) is replaced by the mean of the loss over the $k$ models, i.e., now we want to solve
+
+$$
+\begin{array} { r l } { \mathrm { m i n i m i z e } } & { \mathbb { E } _ { ( x , y ) \sim \mathcal { D } } \left( L \big ( \hat { y } , \frac { 1 } { k } \sum _ { i = 1 } ^ { k } m _ { \theta _ { i } } ( x ) \big ) + \rho L \big ( \hat { y } , \frac { 1 } { k } \sum _ { i = 1 } ^ { k } m _ { \theta _ { i } } ( \tilde { x } ) \big ) \right) } \end{array}
+$$
+
+over variables $\theta _ { i }$ , $i = 1 , \ldots , k .$ , and where $\tilde { x }$ is an adversarial example generating by attacking the entire ensemble. The exact procedure is outlined in Algorithm 1. We demonstrate empirically in the numerical results section that this procedure increases robustness to adversarial inputs. Following these results, we offer an analysis and hypothesis why ensembles outperform single models, even when controlling for number of parameters.
+
+# 4 EXPERIMENTAL SETUP
+
+# 4.1 MODELS COMPARED
+
+Non-Adversarial Benchmarks The Baseline model for our investigation is a Wide ResNet (Zagoruyko & Komodakis, 2016) consisting of a $3 \times 3$ convolution layer, followed by three layers containing 28 ResNet blocks of width factor 10, followed by batch normalization Ioffe & Szegedy (2015) layer, followed by a ReLU (Nair & Hinton, 2010), and by a final linear layer projecting into the logits of the CIFAR-10 classes. All models we experimented with here are variations on this architecture, and where hyperparameters are not explicitly referenced, they are assumed to be the same as this base model. Ensemble2 contains two copies of the baseline architecture. This has twice the number of parameters of the baseline. Together with the base model, these constitute our non-adversarially trained benchmarks.
+
+Adversarial Models When adding adversarial training to the baseline architecture, we obtain our SingleAdv benchmark, which has the same number of parameters as the baseline. When trained with adversarial training, whereby the whole ensemble is attacked by Iterated Fast Gradient Sign Method (IFGSM) (Kurakin et al., 2016) at each training step to obtain adversarial inputs, we refer to the ensemble as Ensemble2Adv. This ensemble has as many parameters as its non-adversarially-trained counterparts.
+
+![](images/d0b32f5f21f442b831718d3024e33f4917dfb7e7b43a58ba01d3feaeb82919d0.jpg)  
+Figure 1: Schematic depiction of classes of models compared in this paper. Here, $n$ indicates the number of parameters in the base model, $\hat { y }$ indicates the ground trouth label, $x$ is a clean input from the dataset, $\tilde { x }$ is that input after a number of steps of the chosen adversarial training attack (7 steps of IFGSM in our experiments), $y$ is the output distribution according to the network based on clean input $x$ , and $\tilde { y }$ is the output based on adversarial input $\tilde { x }$ . Adversarially trained networks are shown to have two inputs (and two losses) for compactness, but in practice two parameter-sharing copies of the network will be instantiated, with one taking clean input, the other taking adversarial input, and their losses will be computed separately and averaged before optimisation.
+
+Comparisons to Ensemble2Adv In order to compare Ensemble2Adv to the SingleAdv benchmark while controlling for number of parameters, we introduce a variant DoubleAdv of this benchmark with ResNet blocks of width 15, which yields roughly the same number of parameters as Ensemble2Adv. Finally, we train two separately parameterised instances of SingleAdv and ensemble them at test time for the purpose of evaluating the hypothesis that it is adversarial training of ensembles that provides and advantage, and call this test-time model SeparateEnsemble2Adv.
+
+The model variations described here are illustrated in Figure 1, which can serve as a basis for repeating these experiments with a different base model architecture.
+
+# 4.2 TRAINING PROCEDURE
+
+We train and evaluate our models on CIFAR-10 (Krizhevsky & Hinton, 2009). We use similar hyperparameters to Zagoruyko & Komodakis (2016), with additional iterations to account for the fact that minimizing the adversarial objective requires more training steps. We train all models for 500,000 iterations using a momentum optimiser with minibatches of size 128, with an initial learning rate of 0.1, a momentum of 0.9, and a learning rate factor (decay) of 0.2 after $\{ \mathrm { 3 0 k , 6 0 k , 9 0 k } \}$ steps. When doing adversarial training, we train both on “clean” versions of the minibatch images, and on adversarial examples produced by 7 steps of IFGSM, following Madry et al. (2017). The cross-entropy losses with regard to the ground truth labels for both the adversarial and clean images are averaged to obtain gradients for the model (i.e. $\rho = 1$ ).
+
+# 4.3 EVALUATION PROCEDURE
+
+During training, we run an evaluation job which evaluates the accuracy of the model on the entire CIFAR-10 test set. We consider two white-box adversaries, both with a maximum $L _ { \infty }$ perturbation of 8 (out of 255): IFGSM which performs the iterated fast gradient sign method update, which is equivalent to steepest descent with respect to the $L _ { \mathrm { i n f } }$ norm Madry et al. (2017); Kurakin et al. (2016) and PGD which performs projected gradient descent using the Adam Kingma & Ba (2014) update rule. During training, we evaluate using IFGSM7, the training adversary which performs 7 iterations of the IFGSM update, also used in Madry et al. (2017), as well as PGD5 and PGD20, the 5 and 20-step versions of our PGD attack. Additionally, for the best model, we run these attacks 500 steps in order to estimate the strongest possible attacks.
+
+We further include a black-box adversary in our evaluation procedure. We use a dataset of precomputed adversarial examples, following the procedure in Liu et al. (2016) against an ensemble of a Wide ResNet Zagoruyko & Komodakis (2016) and VGG-like Simonyan & Zisserman (2014) architectures. The two models are trained with standard training procedures and achieve $9 6 . 0 \%$ and $9 4 . 5 \%$ accuracy respectively on the CIFAR-10 clean test set, and are ensembled by an arithmetic mean of their logits. The adversary is the PGD20 adversary which fools all members of the ensemble on $100 \%$ of the evaluation set. We note that the exact values for robustness of networks to black box attacks can be highly contingent on the similarity between the original and attacked networks Uesato et al. (2018), rather than the true adversarial robustness of the attacked network. However, we include black box accuracies for best practice, as a check against models which achieve illusory robustness through obscured gradients Goodfellow et al. (2014).
+
+We trained and evaluated each model with three separate random seeds. Evaluation outliers, caused by occasional crashes of evaluation jobs, are removed according to the following procedure. We compute a smoothed version of each time series by using a centered rolling median window of width 50. We take the absolute difference of each original time series and its smoothed form, compute the mean of the difference, and replace points in the original time series with their smoothed version only when the absolute difference exceeds three standard deviations with this mean. This removes at most two outlier points per model per evaluation in our runs. Evaluation time series for different seeds are then interpolated to obtain results on the same 1000 time-steps, which are then averaged across seeds, per model class.
+
+# 5 RESULTS AND ANALYSIS
+
+We give a numerical break down of evaluation accuracies for the metrics described above, during and at the end training, in Table 1: in Table 1a, we report the average of the last 10 evaluation steps for all models, and in Table 1b, we report the evaluation metrics at the time step where each model obtained the best evaluation score on FGSM5. In Figures 2a and 2b, we show the evolution of evaluation accuracies for selected metrics. To more thoroughly evaluate the models compared here, we show in Figure 2c how the accuracy of our models drops as the number of PGD attack steps increases. We report the evaluation results for 500 steps of PGD of the model snapshots used for Table 1b in Table 1c.
+
+Figures 2a and 2b show that adversarially trained models uniformly outperform non-adversarially trained ones. Especially with weaker attacks, such as IFGSM5 and PGD5, non-adversarially trained models exhibit some recovery of robustness to attacks after 2–300,000 steps of training, but this is not stable and decays with further training. We further confirm that even such models which achieve some robustness against weak adversaries have true adversarial robustness close to $0 \%$ when the adversarial optimization is run for longer. In contrast, the robustness of adversarially trained models is stable throughout training. We read, in Table 1b, that all models incorporating adversarial training do slightly worse on the CIFAR-10 test, suffering a drop of roughly 10 points in accuracy, a phenomenon which was also observed in other work Madry et al. (2017). On PGD20, the smallest gap between an adversarially trained model and a baseline is $22 \%$ . Ensemble2Adv yields an improvement of $7 \%$ over a SingleAdv, of $5 \%$ over the parameterically equivalent DoubleAdv, and of $29 \%$ over the non-adversarially trained Ensemble2Adv.
+
+In Figure 2c, we see that while the accuracies of the Ensemble2Adv drop more readily as the number of attack steps increases, they preserve a gap 7 accuracy points over the SingleAdv benchmark. Here, we also compare to an ensemble, Separate2Adv, where the individual models in the ensemble were separately adversarially trained. We observe that this ensemble produces a robustness to adversarial attacks which is closer to the SingleAdv results than to Ensemble2Adv, despite having the exact same structure and number of parameters. We present the evaluation accuracies after 500 steps of PGD in Table 1c, which maintains the relative ordering and rough gaps between models seen in Table 1b, thereby helping validate our results.
+
+Table 1: Evaluation Results   
+(a) Average of last 10 evaluation steps   
+
+<table><tr><td></td><td>clean accuracy</td><td>IFGSM5 accuracy</td><td>PGD5 accuracy</td><td>PGD20 accuracy</td><td>black box accuracy</td></tr><tr><td>Baseline</td><td>0.94</td><td>0.34</td><td>0.15</td><td>0.01</td><td>0.27</td></tr><tr><td>Ensemble2</td><td>0.94</td><td>0.59</td><td>0.44</td><td>0.30</td><td>0.22</td></tr><tr><td>Ensemble4</td><td>0.91</td><td>0.50</td><td>0.40</td><td>0.34</td><td>0.26</td></tr><tr><td>SingleAdv</td><td>0.82</td><td>0.55</td><td>0.44</td><td>0.43</td><td>0.80</td></tr><tr><td>DoubleAdv</td><td>0.83</td><td>0.57</td><td>0.46</td><td>0.44</td><td>0.82</td></tr><tr><td>Ensemble2Adv</td><td>0.85</td><td>0.62</td><td>0.55</td><td>0.52</td><td>0.83</td></tr><tr><td>Ensemble4Adv</td><td>0.87</td><td>0.66</td><td>0.58</td><td>0.53</td><td>0.85</td></tr></table>
+
+(b) Evaluation results for model at best IFGSM5 training step
+
+<table><tr><td></td><td>clean accuracy</td><td>FGSM5 accuracy</td><td>PGD5 accuracy</td><td>PGD20 accuracy</td><td>black box accuracy</td></tr><tr><td>Baseline</td><td>0.95</td><td>0.57</td><td>0.29</td><td>0.09</td><td>0.26</td></tr><tr><td>Ensemble2</td><td>0.95</td><td>0.65</td><td>0.52</td><td>0.38</td><td>0.22</td></tr><tr><td>Ensemble4</td><td>0.93</td><td>0.60</td><td>0.48</td><td>0.43</td><td>0.24</td></tr><tr><td>SingleAdv</td><td>0.84</td><td>0.57</td><td>0.46</td><td>0.45</td><td>0.81</td></tr><tr><td>DoubleAdv</td><td>0.85</td><td>0.60</td><td>0.48</td><td>0.47</td><td>0.84</td></tr><tr><td>Ensemble2Adv</td><td>0.87</td><td>0.64</td><td>0.56</td><td>0.52</td><td>0.85</td></tr><tr><td>Ensemble4Adv</td><td>0.88</td><td>0.67</td><td>0.58</td><td>0.52</td><td>0.86</td></tr></table>
+
+(c) Model accuracy after 500 attack steps.
+
+<table><tr><td></td><td></td><td></td><td colspan="4">Ensemble</td></tr><tr><td></td><td>Baseline</td><td>SingleAdv</td><td>DoubleAdv</td><td>-2</td><td>-2Adv</td><td>Separate2Adv</td></tr><tr><td>IFGSM</td><td>0.16</td><td>0.46</td><td>0.47</td><td>0.13</td><td>0.55</td><td>0.49</td></tr><tr><td>PGD</td><td>0.04</td><td>0.44</td><td>0.47</td><td>0.02</td><td>0.52</td><td>0.47</td></tr></table>
+
+# 6 DISCUSSION
+
+In this section we briefly discuss the possible reasons for the behaviours observed. As we saw, an ensemble of models trained adversarially outperforms the other setups at test time. We suspect, that this might be happening due to a mechanism described below.
+
+When the model is being trained, it is exposed to pairs of images, both “clean” and adversarially modified. The adversarial training exploits the fact that the original image is close to the decision boundary of the model. The model then, when provided with both clean and adversarial image would attempt to modify the decision boundary in order to engulf them both. It is relatively easy to imagine why SingleAdv would be weaker then the other models—it simply has less parameters than the competition. In order to accommodate the adversarial example it has to compromise the decision boundary somewhere else, pulling it close to other clean images, making it vulnerable to subsequent attack. This is illustrated in Figure 3a.
+
+The possible reason why Ensemble2Adv outperforms DoubleAdv is more elusive. Both models have the same number of parameters, so one could expect them to display a similar performance. As Ensemble2Adv is more robust to white box attack during test time we argue, that this might be due to the fact that in abundance of flexibility DoubleAdv tends often to spread out thin “tentacles”
+
+![](images/5c4b090b6ca456a5dcfd4f927d29c66331eca4de1a9b116061e75e6ef1649367.jpg)  
+Figure 2: Evaluation Curves
+
+(c) Accuracy under PGD attack as a function of the number of attack steps.
+
+![](images/4055ccb2906a1d043215a15cebe4524d3bf6c21f32ba5a13b0e698208f4bc5dc.jpg)  
+Figure 3: Different responses of various architectures to adversarial training. Solid lines represent decision boundary of the models that see only “clean” images. Dashed lines are the boundaries modified due to the presence of adversarial training. The black dot is a clean image, the red dot is its adversarial modification. In the presence of two models $\mathrm { M o d e l } _ { 1 }$ is blue and $\mathrm { M o d e l _ { 2 } }$ is green. In case it needs to be specified with respect to which model the adversarial example is constructed the red dot has a circle in an appropriate color around it.
+
+(Figure 3b) which do not cover up too much of a space. On the other hand, given that Ensemble2Adv is comprised of two separate models they are both subject to lesser ability to overfit following from the smaller number of parameters available in them. Thus we argue, that in most cases the modification of the model with adversarial training covers the adversarial example by modifying one model more than the other. This way the decision boundary of the model modified to a lesser degree still “provides protection” for “clean” images, while at the same time the “tentacle” generated by the model modified more is thicker than the one DoubleAdv creates. We illustrate that with Figure 3c.
+
+Finally, it was shown that SeparateEnsemble2Adv is outperformed by a Ensemble2Adv trained “jointly”. We think that this is due to the fact that the adversarial training has to weaken both of the submodels simultaneously. Figure 3d illustrates that.
+
+For a further illustration of the effects of adversarial training we plotted actual images of the decision boundaries for non-adversarially and adversarially trained Baseline and 2-Ensemble models (Figure 4).
+
+![](images/06b72f963fb9ff1deb22648a64125a6f8bf11253c105f734c872dc3a70301e2e.jpg)  
+Figure 4: Decision boundaries for various architectures/training methods. Each column shows the decision regions of two models on the same 2-dimensional plane in the space of all images. On every picture the black dot corresponds to the datapoint—an unaltered (ship) image from the test dataset. The light rectangle superimposed over the dot represents the bounds of the permitted attack region within the region. The two red arrows are the two vectors—attack directions on the base image, with respect to respectively the first and second tested model. The red dots are the images resulting from the attack. The plane presented is then the (unique) 2-dimensional plane containing those 3 points. Dark grey is void (outside the slice boundaries), and all other pixels are generated by a forward pass of the model at those coordinates, with the colour used representing the majority class.
+
+Decision regions of the models are 3072-dimensional sets, so visualizing them itself poses a challenge. What we present are color-coded values of the models restricted to 2-dimensional planes in the space of all images, chosen so that the original image and the closest adversarial example (or attempt to find one) for both models in a pair being compared are co-planar. We observe, amongst other things, some support for the hypothesis put forward in Figure 3: adversarial training adds “thickness” around the natural image points, pushing the boundary further away from them, and in doing so, making adversarial examples harder to find (even within the test set); ensembling makes some classes more “consistent” within the decision plane, but introduce small “pockets” or “tentacles” of other classes; and the combinator thereof removes said pockets to create large regions of the correct class around images. We believe that such an approach of choosing a good plane and plotting the values of models on it is a more informative way of visualizing phenomena taking place in the universe of robustness and adversarial examples than more traditional approaches like t-SNE plots (Maaten & Hinton, 2008).
+
+# 7 CONCLUSIONS AND FURTHER WORK
+
+In this paper, we provide an empirical study of the effect of increasing the number of parameters in a model trained with adversarial training methods, with regard to its robustness to test-time adversarial attacks. We showed that while increasing parameters improves robustness, it is better to do so by ensembling smaller models than by producing one larger model. Through our experiments, we show that this result is not only due to ensembling alone, or to the implicit robustness of an ensemble of adversarially trained models, but specifically to due to the adversarial training of an ensemble as if it were a single model. We proposed a high level interpretation of why this phenomenon might occur. Further work should seek to determine whether scaling the number of models in the ensemble while controlling for number of parameters produces significant improvements over the minimal ensembles studied here in an attempt to draw conclusions about why such architectures are generally more robust than larger single models, even under adversarial training.
+
+# REFERENCES
+
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+
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+        "type": "text",
+        "text": "STRENGTH IN NUMBERS: TRADING-OFF ROBUSTNESS AND COMPUTATION VIA ADVERSARIALLY-TRAINED ENSEMBLES ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While deep learning has led to remarkable results on a number of challenging problems, researchers have discovered a vulnerability of neural networks in adversarial settings, where small but carefully chosen perturbations to the input can make the models produce extremely inaccurate outputs. This makes these models particularly unsuitable for safety-critical application domains (e.g. self-driving cars) where robustness is extremely important. Recent work has shown that augmenting training with adversarially generated data provides some degree of robustness against test-time attacks. In this paper we investigate how this approach scales as we increase the computational budget given to the defender. We show that increasing the number of parameters in adversarially-trained models increases their robustness, and in particular that ensembling smaller models while adversarially training the entire ensemble as a single model is a more efficient way of spending said budget than simply using a larger single model. Crucially, we show that it is the adversarial training of the ensemble, rather than the ensembling of adversarially trained models, which provides robustness. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep neural networks have demonstrated state-of-the-art performance in a wide range of application domains Krizhevsky et al. (2012). However, researchers have discovered that deep networks are in some sense ‘brittle’, in that small changes to their inputs can result in wildly different outputs (Huang et al., 2017; Jia & Liang, 2017; Szegedy et al., 2013). For instance, practically imperceptible (to human) modifications to images can result in misclassification of the image with high confidence. Not only are networks susceptible to these ‘attacks’, but these attacks are also relatively easy to compute using standard optimization techniques (Carlini & Wagner, 2017b; Goodfellow et al., 2014). These changes are often referred to as adversarial perturbations, in the sense that an adversary could craft a very small change to the input in order to create an undesirable outcome. This phenomenon is not unique to image classification, nor to particular network architectures, nor to particular training algorithms (Papernot et al., 2016; 2017). ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Adversarial attacks can be broken into different categories depending on how much knowledge of the underlying model the adversary has access to. In ‘white-box’ attacks the adversary has full access to the model, and can perform both forward and backwards passes (though not change the weights or logic of the network) (Carlini & Wagner, 2017a; Goodfellow et al., 2014). In the ‘black-box’ setting the adversary has no access to the model, but perhaps knows the dataset that the model was trained on (Papernot et al., 2016; 2017). Despite several recent papers demonstrating new defences against adversarial attacks (Akhtar & Mian, 2018; Guo et al., 2017; Liao et al., 2017; Song et al., 2017; Tramer et al., 2018; Warde-Farley & Goodfellow, 2016; Xie et al., 2017; Yuan et al., 2017), recent \\` papers have demonstrated that most of these new defences are still susceptible to attacks and largely just obfuscate the gradients that the attacker can follow, and that non-gradient based attacks are still effective Uesato et al. (2018); Athalye et al. (2018). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Exploring Tradeoff of Computation and Robustness In many safety-critical application domains (e.g. self-driving cars), robustness is extremely important even if it comes at the cost of increased computation. This motivated the central question considered by this paper: Is it possible to increase adversarial robustness of a classifier at the cost of increased computation? ",
+        "bbox": [
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+        "page_idx": 0
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+    {
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+    {
+        "type": "text",
+        "text": "There are a number of possibilities to employ extra computation available at runtime. We can use a much larger model that requires more time to run, execute the original model multiple times and aggregate the predictions, or instead of using a single model, make predictions from a portfolio or ensemble of models. While researchers have proposed the use of portfolios and ensembles as a mechanism to improve adversarial robustness Abbasi & Gagne (2017); Thilo Strauss (2017), our ´ experimental results indicate that stronger adversaries are able to attack the ensembles successfully. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Contributions In this paper, we study and analyze the trade-off of adversarial robustness and computation (memory and runtime). We propose the use of adversarial training of ensemble of models and through an exhaustive ablative analysis make the following empirical findings: ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• increased computation and/or model size can be used to increase robustness,   \n• ensembles on their own are not very robust, but can be made robust through adversarial training where the ensemble is treated as a single model,   \n• adversarially trained ensembles are more robust than adversarially trained individual models requiring the same amount of parameters/computation ",
+        "bbox": [
+            215,
+            292,
+            825,
+            376
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Related Work Recently, Tramer et al. (2018) investigated the use of ensembles for adversarial \\` robustness. However, their goal and approach was quite different from the technique we are investigating. In Tramer et al. (2018), the authors generated adversarial perturbations using an ensemble of \\` pre-trained models in order to transfer the example to another model during training. This procedure decouples adversarial example generation from the current model, and consequently the model being trained cannot simply ‘overfit’ to the procedure for generating adversarial examples, which they generally took to be single-step attack methods. The authors demonstrated strong robustness of the resulting trained model to black-box attacks. By contrast, in this paper we investigate using an ensemble of models as our predictive model, and we train the models using multi-step adversarial training. We show increased robustness to both black-box and white-box adversarial attacks using this strategy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here we lay out the basics of attacking a neural network by the generation of adversarial examples. Denote an input to the network as $\\boldsymbol { x } \\in \\mathbb { R } ^ { d }$ with correct label $\\hat { y } \\in \\mathcal { V } \\subset \\mathbb { N }$ , and let $m _ { \\theta } : \\mathbb { R } ^ { d }  \\mathbb { R } ^ { | \\bar { y } | }$ be the mapping performed by the neural network which is parameterized by $\\theta \\in \\mathbb { R } ^ { p }$ . Let $L : \\mathcal { V } \\times \\mathbb { R } ^ { | \\mathcal { V } | } \\to$ $\\mathbb { R }$ denote the loss we are trying to minimize (e.g., the cross-entropy). When training a neural network we seek to solve ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b8fe346b942fb8ab4f188bc857a254f5bf015df189ee744642145ad7c90fb56.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathrm { m i n i m i z e } } & { { } \\mathbb { E } _ { ( x , y ) \\sim \\mathcal { D } } L ( \\hat { y } , m _ { \\theta } ( x ) ) } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "over variable $\\theta$ , where $\\mathcal { D }$ is the data distribution. Given any fixed $\\theta$ we can generate (untargeted) adversarial inputs by perturbing the input $x$ so as to maximize the loss. We restrict ourselves to small perturbations around a nominal input, and we denote by $\\boldsymbol { B }$ this set of allowable inputs. For example, if we restrict ourselves to small perturbations in $\\ell _ { \\infty }$ norm around a nominal input $x ^ { \\mathrm { n o m } }$ then we could set $\\mathcal { B } = \\left\\{ x \\vert \\| x - x ^ { \\mathrm { n o m } } \\| _ { \\infty } \\leq \\epsilon \\right\\}$ where $\\epsilon > 0$ is the tolerance. A common approach for generating adversarial examples is projected gradient descent Carlini $\\&$ Wagner (2016), i.e., to iteratively update the input $x$ by ",
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+    {
+        "type": "equation",
+        "img_path": "images/c9fbd5edf2aa5dafb555e62e99d6a09eeda4f47f33136a11ca0303ed96fd15b0.jpg",
+        "text": "$$\n\\tilde { { \\boldsymbol { x } } } ^ { k + 1 } = \\Pi _ { { \\boldsymbol { \\mathcal { B } } } } ( \\tilde { { \\boldsymbol { x } } } ^ { k } + \\eta \\nabla _ { x } L ( y , m _ { \\theta } ( \\tilde { { \\boldsymbol { x } } } ^ { k } ) ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where typically $x ^ { 0 } = x + \\epsilon$ for some noise $\\epsilon$ , $\\eta > 0$ is a step-size parameter and $\\Pi _ { B }$ denotes the Euclidean projection on $\\boldsymbol { B }$ . We add noise to the initial point so that the network can’t memorize the training dataset and mask or obfuscate the gradients at that point Uesato et al. (2018); Athalye et al. (2018), in other words the added noise encourages generalization of adversarial robustness to the test dataset. If instead of using the gradient we just use the sign of the gradient then this is the fast-gradient-sign method Goodfellow et al. (2014). Empirically speaking, for most networks just a few steps of either of these procedures is sufficient to generate an $\\tilde { x }$ that is close to $x ^ { \\mathrm { n o m } }$ but has a different label with high confidence. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper we are primarily concerned with the performance of ensembles of models when trained with adversarial training Madry et al. (2017). In adversarial training we train a network to minimize a weighted sum of two losses (where the relative weighting is a hyper-parameter). The first loss is the standard loss of the problem we are trying to solve on the normal training data, e.g., the cross-entropy for a classification task. The second loss is the same function as the first loss, except evaluated on adversarially generated data, where typically the adversarial data is generated by attacking the network at that time-step. In other words we replace the problem in eq. (1) with ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/f197ef5806c345e30c754e32c4975131fc10d294b050fd8dabda5b53adcf6ebf.jpg",
+        "text": "$$\n\\mathbb { E } _ { ( x , y ) \\sim \\mathcal { D } } ( L ( \\hat { y } , m _ { \\theta } ( x ) ) + \\rho L ( \\hat { y } , m _ { \\theta } ( \\tilde { x } ) ) )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\rho \\geq 0$ is the weighting parameter and $\\tilde { x }$ is an adversarial example generated from $x$ at model parameters $\\theta$ using, for example, the update in eq. (2). This problem is usually approximated by sampling and minimizing the empirical expectation. ",
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+    {
+        "type": "text",
+        "text": "3 ADVERSARIALLY-TRAINED ENSEMBLES ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we lay out the basic strategy of using ensembles of models to increase robustness to adversarial attacks. The notion of ensemble used here simply involves taking $k$ separatelyparameterized models and averaging their predictions. If the output of network $i$ as a function of input $x$ and with network parameters $\\theta _ { i }$ is given by $p ( \\cdot | x , \\theta _ { i } ) = m _ { \\theta _ { i } } ( \\bar { x } )$ , then the output of the ensemble is ",
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+        "type": "equation",
+        "img_path": "images/8ff044190cc6bd96a68f565e0fd27ce5900cc7650a07a5211a8918d4dd2c97e4.jpg",
+        "text": "$$\np ( \\boldsymbol { y } | \\boldsymbol { x } ) = \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { \\theta } _ { i } ) .\n$$",
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+    {
+        "type": "text",
+        "text": "Alternatively, we could consider using a ‘gating network’ to generate data-dependent weights for each model rather than a simple average, though we found the performance to be similar. ",
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+        "type": "text",
+        "text": "Using ensembles to improve the performance of statistical models is a very old idea; see, e.g. Opitz & Maclin (1999) for a survey. The basic intuition is that several weak models can be combined in such a way that the ensemble performs better than any individual, and is sometimes explained as being caused by the errors of the models ‘cancelling’ with one another. ",
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+    {
+        "type": "text",
+        "text": "In order to ensure that the models are actually producing different outputs the diversity of the models must be maintained. This can be done in several ways, such as bootstrapping the data, whereby each model gets a slightly different copy of the data, or using totally different model types or architectures. In the case that the model training procedure is convex, and if all models architectures are the same and are getting the same data, then the models in the ensemble would be expected to converge on the same parameters. In the case of neural networks however, the model training procedure is not convex and so our strategy for maintaining diversity is very simple—initialize each model differently. Due to the nature of training neural networks it is likely that differently initialized networks will converge (assuming they do, in fact, converge) to different points of the parameter space. The insight that only different initialization is required is not new, previous papers have observed that different initialization is sufficient for uncertainty estimation Lakshminarayanan et al. (2016); Osband et al. (2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Different initialization for networks has an appealing interpretation. If we take a Bayesian approach to the classification problem, then we have a prior over possible model parameters, $p ( \\theta )$ , a likelihood of the data, $p ( D | \\theta ) ^ { \\overline { { } } }$ , and a probability of a label $y$ given an input and a model, $\\overset { \\cdot } { p ( \\boldsymbol { y } | \\boldsymbol { x } , \\boldsymbol { \\theta } ) }$ . The ‘Bayes-optimal’ classification of a new data point $x$ is given by ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/228a3851e4f7fde453c442f6fd4f2a0fe3c3e48f761b3673be38d61ef97ef45b.jpg",
+        "text": "$$\ny ^ { \\star } = \\mathrm { a r g m a x } _ { y } \\int _ { \\theta } p ( y | x , \\theta ) p ( D | \\theta ) p ( \\theta ) .\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "This classifier is optimal in the sense that no other classifier can outperform it on average, given the same model class and knowledge of the prior and likelihood; however, the formulation is intractable for all but small problems. We can consider approximating it by the following approach, sample initial parameters from the prior $p ( \\theta )$ and run an iterative procedure to (approximately) maximize the likelihood $p ( D | \\theta )$ . Very loosely speaking, we can consider this procedure as approximately sampling from the posterior over models $p ( \\theta | D ) \\propto p ( D | \\theta ) p ( \\theta )$ . Consequently, we output the classification ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/05915fb5b861975ff9c258344dae5397600a4ac4647845094e921dd701d37f8c.jpg",
+        "text": "$$\ny ^ { \\star } = \\operatorname { a r g m a x } _ { y } \\sum _ { i = 1 } ^ { k } p ( y | x , \\theta _ { i } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Adversarial ensemble training using PGD under $\\ell _ { \\infty }$ norm constraint input: $k$ neural networks $m _ { \\theta _ { i } }$ , $i = 1 , \\ldots , k$ ; attack steps $N$ ; step sizes $\\eta , \\hat { \\eta }$ ; initial variance $\\sigma$ adversarial loss weighting $\\rho$ ; perturbation width $\\delta$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "initialize: neural network parameters $\\theta _ { i } ^ { 0 }$ randomly, $i = 1 , \\ldots , k$   \nfor time-step $t = 0 , 1 , \\ldots , \\mathbf { i }$ o sample input minibatch $( x , \\hat { y } ) \\sim \\mathcal { D }$ initialize $\\bar { \\tilde { x } } { } ^ { 0 } = x + \\epsilon$ where $\\epsilon \\sim \\mathcal { N } ( 0 , \\sigma ^ { 2 } I )$ define $\\boldsymbol { \\mathcal { B } } = \\{ x ^ { \\prime } \\ | \\ \\| x - x ^ { \\prime } \\| _ { \\infty } \\leq \\delta \\}$ for $k = 0 , \\ldots , N - 1$ do ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/6b313c9e80d7c33ecfbf99d09b0bc64dcc563fac7984cb71579d966b83e80912.jpg",
+        "text": "$$\n\\tilde { { \\boldsymbol { x } } } ^ { k + 1 } = \\Pi _ { \\mathcal { B } } \\big ( \\tilde { { \\boldsymbol { x } } } ^ { k } + \\hat { \\eta } \\nabla _ { x } L \\big ( \\hat { y } , \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } m _ { \\theta _ { i } } ( \\tilde { { \\boldsymbol { x } } } ^ { k } ) \\big ) \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "end for ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "update parameters for each $i = 1 , \\ldots , k$ : ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/fdb92fd55519aff0b13a4724a636ca992f6c6d42702223e6054b575f2140ded6.jpg",
+        "text": "$$\n\\theta _ { i } ^ { t + 1 } = \\theta _ { i } ^ { t } - \\eta \\nabla _ { \\theta _ { i } } \\left( L ( \\hat { y } , \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } m _ { \\theta _ { j } ^ { t } } ( x ) ) + \\rho L ( \\hat { y } , \\frac { 1 } { k } \\sum _ { j = 1 } ^ { k } m _ { \\theta _ { j } ^ { t } } ( \\tilde { x } ^ { N } ) ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "end for ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "i.e., the best guess of the ensemble. The role of initialization therefore is that of sampling from our prior over possible model parameters. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Adversarial training of Ensembles Up to this point we have discussed the use of ensembles for improving classification performance and approximating the Bayes optimal classifier. Typically speaking neural networks appear to not benefit much from ensembling in terms of nominal performance. Here, however, we make the claim that adversarially trained ensembles of networks provide a level of robustness to adversarial attacks. When using ensembles the loss function for adversarial training in (3) is replaced by the mean of the loss over the $k$ models, i.e., now we want to solve ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/702a2f34296e9b38b44185023d747c23244907dc343f1ea5fbaf35335ad1db7d.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathrm { m i n i m i z e } } & { \\mathbb { E } _ { ( x , y ) \\sim \\mathcal { D } } \\left( L \\big ( \\hat { y } , \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } m _ { \\theta _ { i } } ( x ) \\big ) + \\rho L \\big ( \\hat { y } , \\frac { 1 } { k } \\sum _ { i = 1 } ^ { k } m _ { \\theta _ { i } } ( \\tilde { x } ) \\big ) \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "over variables $\\theta _ { i }$ , $i = 1 , \\ldots , k .$ , and where $\\tilde { x }$ is an adversarial example generating by attacking the entire ensemble. The exact procedure is outlined in Algorithm 1. We demonstrate empirically in the numerical results section that this procedure increases robustness to adversarial inputs. Following these results, we offer an analysis and hypothesis why ensembles outperform single models, even when controlling for number of parameters. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 MODELS COMPARED ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Non-Adversarial Benchmarks The Baseline model for our investigation is a Wide ResNet (Zagoruyko & Komodakis, 2016) consisting of a $3 \\times 3$ convolution layer, followed by three layers containing 28 ResNet blocks of width factor 10, followed by batch normalization Ioffe & Szegedy (2015) layer, followed by a ReLU (Nair & Hinton, 2010), and by a final linear layer projecting into the logits of the CIFAR-10 classes. All models we experimented with here are variations on this architecture, and where hyperparameters are not explicitly referenced, they are assumed to be the same as this base model. Ensemble2 contains two copies of the baseline architecture. This has twice the number of parameters of the baseline. Together with the base model, these constitute our non-adversarially trained benchmarks. ",
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+        "type": "text",
+        "text": "Adversarial Models When adding adversarial training to the baseline architecture, we obtain our SingleAdv benchmark, which has the same number of parameters as the baseline. When trained with adversarial training, whereby the whole ensemble is attacked by Iterated Fast Gradient Sign Method (IFGSM) (Kurakin et al., 2016) at each training step to obtain adversarial inputs, we refer to the ensemble as Ensemble2Adv. This ensemble has as many parameters as its non-adversarially-trained counterparts. ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/d0b32f5f21f442b831718d3024e33f4917dfb7e7b43a58ba01d3feaeb82919d0.jpg",
+        "image_caption": [
+            "Figure 1: Schematic depiction of classes of models compared in this paper. Here, $n$ indicates the number of parameters in the base model, $\\hat { y }$ indicates the ground trouth label, $x$ is a clean input from the dataset, $\\tilde { x }$ is that input after a number of steps of the chosen adversarial training attack (7 steps of IFGSM in our experiments), $y$ is the output distribution according to the network based on clean input $x$ , and $\\tilde { y }$ is the output based on adversarial input $\\tilde { x }$ . Adversarially trained networks are shown to have two inputs (and two losses) for compactness, but in practice two parameter-sharing copies of the network will be instantiated, with one taking clean input, the other taking adversarial input, and their losses will be computed separately and averaged before optimisation. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "Comparisons to Ensemble2Adv In order to compare Ensemble2Adv to the SingleAdv benchmark while controlling for number of parameters, we introduce a variant DoubleAdv of this benchmark with ResNet blocks of width 15, which yields roughly the same number of parameters as Ensemble2Adv. Finally, we train two separately parameterised instances of SingleAdv and ensemble them at test time for the purpose of evaluating the hypothesis that it is adversarial training of ensembles that provides and advantage, and call this test-time model SeparateEnsemble2Adv. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The model variations described here are illustrated in Figure 1, which can serve as a basis for repeating these experiments with a different base model architecture. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4.2 TRAINING PROCEDURE ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We train and evaluate our models on CIFAR-10 (Krizhevsky & Hinton, 2009). We use similar hyperparameters to Zagoruyko & Komodakis (2016), with additional iterations to account for the fact that minimizing the adversarial objective requires more training steps. We train all models for 500,000 iterations using a momentum optimiser with minibatches of size 128, with an initial learning rate of 0.1, a momentum of 0.9, and a learning rate factor (decay) of 0.2 after $\\{ \\mathrm { 3 0 k , 6 0 k , 9 0 k } \\}$ steps. When doing adversarial training, we train both on “clean” versions of the minibatch images, and on adversarial examples produced by 7 steps of IFGSM, following Madry et al. (2017). The cross-entropy losses with regard to the ground truth labels for both the adversarial and clean images are averaged to obtain gradients for the model (i.e. $\\rho = 1$ ). ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4.3 EVALUATION PROCEDURE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "During training, we run an evaluation job which evaluates the accuracy of the model on the entire CIFAR-10 test set. We consider two white-box adversaries, both with a maximum $L _ { \\infty }$ perturbation of 8 (out of 255): IFGSM which performs the iterated fast gradient sign method update, which is equivalent to steepest descent with respect to the $L _ { \\mathrm { i n f } }$ norm Madry et al. (2017); Kurakin et al. (2016) and PGD which performs projected gradient descent using the Adam Kingma & Ba (2014) update rule. During training, we evaluate using IFGSM7, the training adversary which performs 7 iterations of the IFGSM update, also used in Madry et al. (2017), as well as PGD5 and PGD20, the 5 and 20-step versions of our PGD attack. Additionally, for the best model, we run these attacks 500 steps in order to estimate the strongest possible attacks. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "We further include a black-box adversary in our evaluation procedure. We use a dataset of precomputed adversarial examples, following the procedure in Liu et al. (2016) against an ensemble of a Wide ResNet Zagoruyko & Komodakis (2016) and VGG-like Simonyan & Zisserman (2014) architectures. The two models are trained with standard training procedures and achieve $9 6 . 0 \\%$ and $9 4 . 5 \\%$ accuracy respectively on the CIFAR-10 clean test set, and are ensembled by an arithmetic mean of their logits. The adversary is the PGD20 adversary which fools all members of the ensemble on $100 \\%$ of the evaluation set. We note that the exact values for robustness of networks to black box attacks can be highly contingent on the similarity between the original and attacked networks Uesato et al. (2018), rather than the true adversarial robustness of the attacked network. However, we include black box accuracies for best practice, as a check against models which achieve illusory robustness through obscured gradients Goodfellow et al. (2014). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We trained and evaluated each model with three separate random seeds. Evaluation outliers, caused by occasional crashes of evaluation jobs, are removed according to the following procedure. We compute a smoothed version of each time series by using a centered rolling median window of width 50. We take the absolute difference of each original time series and its smoothed form, compute the mean of the difference, and replace points in the original time series with their smoothed version only when the absolute difference exceeds three standard deviations with this mean. This removes at most two outlier points per model per evaluation in our runs. Evaluation time series for different seeds are then interpolated to obtain results on the same 1000 time-steps, which are then averaged across seeds, per model class. ",
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+    {
+        "type": "text",
+        "text": "5 RESULTS AND ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We give a numerical break down of evaluation accuracies for the metrics described above, during and at the end training, in Table 1: in Table 1a, we report the average of the last 10 evaluation steps for all models, and in Table 1b, we report the evaluation metrics at the time step where each model obtained the best evaluation score on FGSM5. In Figures 2a and 2b, we show the evolution of evaluation accuracies for selected metrics. To more thoroughly evaluate the models compared here, we show in Figure 2c how the accuracy of our models drops as the number of PGD attack steps increases. We report the evaluation results for 500 steps of PGD of the model snapshots used for Table 1b in Table 1c. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Figures 2a and 2b show that adversarially trained models uniformly outperform non-adversarially trained ones. Especially with weaker attacks, such as IFGSM5 and PGD5, non-adversarially trained models exhibit some recovery of robustness to attacks after 2–300,000 steps of training, but this is not stable and decays with further training. We further confirm that even such models which achieve some robustness against weak adversaries have true adversarial robustness close to $0 \\%$ when the adversarial optimization is run for longer. In contrast, the robustness of adversarially trained models is stable throughout training. We read, in Table 1b, that all models incorporating adversarial training do slightly worse on the CIFAR-10 test, suffering a drop of roughly 10 points in accuracy, a phenomenon which was also observed in other work Madry et al. (2017). On PGD20, the smallest gap between an adversarially trained model and a baseline is $22 \\%$ . Ensemble2Adv yields an improvement of $7 \\%$ over a SingleAdv, of $5 \\%$ over the parameterically equivalent DoubleAdv, and of $29 \\%$ over the non-adversarially trained Ensemble2Adv. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "In Figure 2c, we see that while the accuracies of the Ensemble2Adv drop more readily as the number of attack steps increases, they preserve a gap 7 accuracy points over the SingleAdv benchmark. Here, we also compare to an ensemble, Separate2Adv, where the individual models in the ensemble were separately adversarially trained. We observe that this ensemble produces a robustness to adversarial attacks which is closer to the SingleAdv results than to Ensemble2Adv, despite having the exact same structure and number of parameters. We present the evaluation accuracies after 500 steps of PGD in Table 1c, which maintains the relative ordering and rough gaps between models seen in Table 1b, thereby helping validate our results. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 1: Evaluation Results ",
+            "(a) Average of last 10 evaluation steps "
+        ],
+        "table_footnote": [
+            "(b) Evaluation results for model at best IFGSM5 training step "
+        ],
+        "table_body": "<table><tr><td></td><td>clean accuracy</td><td>IFGSM5 accuracy</td><td>PGD5 accuracy</td><td>PGD20 accuracy</td><td>black box accuracy</td></tr><tr><td>Baseline</td><td>0.94</td><td>0.34</td><td>0.15</td><td>0.01</td><td>0.27</td></tr><tr><td>Ensemble2</td><td>0.94</td><td>0.59</td><td>0.44</td><td>0.30</td><td>0.22</td></tr><tr><td>Ensemble4</td><td>0.91</td><td>0.50</td><td>0.40</td><td>0.34</td><td>0.26</td></tr><tr><td>SingleAdv</td><td>0.82</td><td>0.55</td><td>0.44</td><td>0.43</td><td>0.80</td></tr><tr><td>DoubleAdv</td><td>0.83</td><td>0.57</td><td>0.46</td><td>0.44</td><td>0.82</td></tr><tr><td>Ensemble2Adv</td><td>0.85</td><td>0.62</td><td>0.55</td><td>0.52</td><td>0.83</td></tr><tr><td>Ensemble4Adv</td><td>0.87</td><td>0.66</td><td>0.58</td><td>0.53</td><td>0.85</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "(c) Model accuracy after 500 attack steps. "
+        ],
+        "table_body": "<table><tr><td></td><td>clean accuracy</td><td>FGSM5 accuracy</td><td>PGD5 accuracy</td><td>PGD20 accuracy</td><td>black box accuracy</td></tr><tr><td>Baseline</td><td>0.95</td><td>0.57</td><td>0.29</td><td>0.09</td><td>0.26</td></tr><tr><td>Ensemble2</td><td>0.95</td><td>0.65</td><td>0.52</td><td>0.38</td><td>0.22</td></tr><tr><td>Ensemble4</td><td>0.93</td><td>0.60</td><td>0.48</td><td>0.43</td><td>0.24</td></tr><tr><td>SingleAdv</td><td>0.84</td><td>0.57</td><td>0.46</td><td>0.45</td><td>0.81</td></tr><tr><td>DoubleAdv</td><td>0.85</td><td>0.60</td><td>0.48</td><td>0.47</td><td>0.84</td></tr><tr><td>Ensemble2Adv</td><td>0.87</td><td>0.64</td><td>0.56</td><td>0.52</td><td>0.85</td></tr><tr><td>Ensemble4Adv</td><td>0.88</td><td>0.67</td><td>0.58</td><td>0.52</td><td>0.86</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/a982ee2343680d25dab313daf8de311056f9a88836c6096b7f577b9a904603a5.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td></td><td colspan=\"4\">Ensemble</td></tr><tr><td></td><td>Baseline</td><td>SingleAdv</td><td>DoubleAdv</td><td>-2</td><td>-2Adv</td><td>Separate2Adv</td></tr><tr><td>IFGSM</td><td>0.16</td><td>0.46</td><td>0.47</td><td>0.13</td><td>0.55</td><td>0.49</td></tr><tr><td>PGD</td><td>0.04</td><td>0.44</td><td>0.47</td><td>0.02</td><td>0.52</td><td>0.47</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section we briefly discuss the possible reasons for the behaviours observed. As we saw, an ensemble of models trained adversarially outperforms the other setups at test time. We suspect, that this might be happening due to a mechanism described below. ",
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+    {
+        "type": "text",
+        "text": "When the model is being trained, it is exposed to pairs of images, both “clean” and adversarially modified. The adversarial training exploits the fact that the original image is close to the decision boundary of the model. The model then, when provided with both clean and adversarial image would attempt to modify the decision boundary in order to engulf them both. It is relatively easy to imagine why SingleAdv would be weaker then the other models—it simply has less parameters than the competition. In order to accommodate the adversarial example it has to compromise the decision boundary somewhere else, pulling it close to other clean images, making it vulnerable to subsequent attack. This is illustrated in Figure 3a. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The possible reason why Ensemble2Adv outperforms DoubleAdv is more elusive. Both models have the same number of parameters, so one could expect them to display a similar performance. As Ensemble2Adv is more robust to white box attack during test time we argue, that this might be due to the fact that in abundance of flexibility DoubleAdv tends often to spread out thin “tentacles” ",
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+    {
+        "type": "image",
+        "img_path": "images/5c4b090b6ca456a5dcfd4f927d29c66331eca4de1a9b116061e75e6ef1649367.jpg",
+        "image_caption": [
+            "Figure 2: Evaluation Curves "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(c) Accuracy under PGD attack as a function of the number of attack steps. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4055ccb2906a1d043215a15cebe4524d3bf6c21f32ba5a13b0e698208f4bc5dc.jpg",
+        "image_caption": [
+            "Figure 3: Different responses of various architectures to adversarial training. Solid lines represent decision boundary of the models that see only “clean” images. Dashed lines are the boundaries modified due to the presence of adversarial training. The black dot is a clean image, the red dot is its adversarial modification. In the presence of two models $\\mathrm { M o d e l } _ { 1 }$ is blue and $\\mathrm { M o d e l _ { 2 } }$ is green. In case it needs to be specified with respect to which model the adversarial example is constructed the red dot has a circle in an appropriate color around it. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(Figure 3b) which do not cover up too much of a space. On the other hand, given that Ensemble2Adv is comprised of two separate models they are both subject to lesser ability to overfit following from the smaller number of parameters available in them. Thus we argue, that in most cases the modification of the model with adversarial training covers the adversarial example by modifying one model more than the other. This way the decision boundary of the model modified to a lesser degree still “provides protection” for “clean” images, while at the same time the “tentacle” generated by the model modified more is thicker than the one DoubleAdv creates. We illustrate that with Figure 3c. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Finally, it was shown that SeparateEnsemble2Adv is outperformed by a Ensemble2Adv trained “jointly”. We think that this is due to the fact that the adversarial training has to weaken both of the submodels simultaneously. Figure 3d illustrates that. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For a further illustration of the effects of adversarial training we plotted actual images of the decision boundaries for non-adversarially and adversarially trained Baseline and 2-Ensemble models (Figure 4). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/06b72f963fb9ff1deb22648a64125a6f8bf11253c105f734c872dc3a70301e2e.jpg",
+        "image_caption": [
+            "Figure 4: Decision boundaries for various architectures/training methods. Each column shows the decision regions of two models on the same 2-dimensional plane in the space of all images. On every picture the black dot corresponds to the datapoint—an unaltered (ship) image from the test dataset. The light rectangle superimposed over the dot represents the bounds of the permitted attack region within the region. The two red arrows are the two vectors—attack directions on the base image, with respect to respectively the first and second tested model. The red dots are the images resulting from the attack. The plane presented is then the (unique) 2-dimensional plane containing those 3 points. Dark grey is void (outside the slice boundaries), and all other pixels are generated by a forward pass of the model at those coordinates, with the colour used representing the majority class. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            184,
+            101,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Decision regions of the models are 3072-dimensional sets, so visualizing them itself poses a challenge. What we present are color-coded values of the models restricted to 2-dimensional planes in the space of all images, chosen so that the original image and the closest adversarial example (or attempt to find one) for both models in a pair being compared are co-planar. We observe, amongst other things, some support for the hypothesis put forward in Figure 3: adversarial training adds “thickness” around the natural image points, pushing the boundary further away from them, and in doing so, making adversarial examples harder to find (even within the test set); ensembling makes some classes more “consistent” within the decision plane, but introduce small “pockets” or “tentacles” of other classes; and the combinator thereof removes said pockets to create large regions of the correct class around images. We believe that such an approach of choosing a good plane and plotting the values of models on it is a more informative way of visualizing phenomena taking place in the universe of robustness and adversarial examples than more traditional approaches like t-SNE plots (Maaten & Hinton, 2008). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSIONS AND FURTHER WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we provide an empirical study of the effect of increasing the number of parameters in a model trained with adversarial training methods, with regard to its robustness to test-time adversarial attacks. We showed that while increasing parameters improves robustness, it is better to do so by ensembling smaller models than by producing one larger model. Through our experiments, we show that this result is not only due to ensembling alone, or to the implicit robustness of an ensemble of adversarially trained models, but specifically to due to the adversarial training of an ensemble as if it were a single model. We proposed a high level interpretation of why this phenomenon might occur. Further work should seek to determine whether scaling the number of models in the ensemble while controlling for number of parameters produces significant improvements over the minimal ensembles studied here in an attempt to draw conclusions about why such architectures are generally more robust than larger single models, even under adversarial training. ",
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+    {
+        "type": "text",
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parse/train/MmCRswl1UYl/MmCRswl1UYl.md

@@ -0,0 +1,325 @@
+# OPEN QUESTION ANSWERING OVER TABLES AND TEXT
+
+Wenhu Chen1 ∗, Ming-Wei Chang2, Eva Schlinger2, William Wang1, William W. Cohen2
+
+1University of California, Santa Barbara 2Google Research {wenhuchen, william}@cs.ucsb.edu {mingweichang, eschling, wcohen}@google.com
+
+# ABSTRACT
+
+In open question answering (QA), the answer to a question is produced by retrieving and then analyzing documents that might contain answers to the question. Most open QA systems have considered only retrieving information from unstructured text. Here we consider for the first time open QA over both tabular and textual data and present a new large-scale dataset Open Table-and-Text Question Answering (OTT-QA) to evaluate performance on this task1. Most questions in OTT-QA require multi-hop inference across tabular data and unstructured text, and the evidence required to answer a question can be distributed in different ways over these two types of input, making evidence retrieval challenging—our baseline model using an iterative retriever and BERT-based reader achieves an exact match score less than $10 \%$ . We then propose two novel techniques to address the challenge of retrieving and aggregating evidence for OTT-QA. The first technique is to use “early fusion” to group multiple highly relevant tabular and textual units into a fused block, which provides more context for the retriever to search for. The second technique is to use a cross-block reader to model the cross-dependency between multiple retrieved evidence with global-local sparse attention. Combining these two techniques improves the score significantly, to above $27 \%$ .
+
+# 1 INTRODUCTION
+
+Open question answering considers the problem of retrieving documents from a fixed corpus with a retriever, and then analyzes retrieved evidence to provide answers to a given question with a reader. Prior open question answering systems focused only on retrieving and reading free-form passages or documents. However, a significant amount of real-world information is stored in other forms, such as semi-structured web tables due to its compact representation to aggregate related information. For example, tables are often used to hold large quantities of related facts, especially numeric facts, such as ‘Career Statistics for Lebron James’. This type of detailed information is found much less frequently in unstructured text. Tables are also commonly used for collections of homogeneous entities or recurring events, like ‘List of Periodic Comets’ or ‘List of Champions League Winners since $\exists \in 6 6 ^ { \prime }$ . Hence tabular information serves as an excellent complement to textual data, especially in the open setting. Despite these advantages, no previous studies have exploited the millions of web tables to augment their open QA system.
+
+In this paper, we describe the first study to jointly exploit tables and text for open-domain question answering. For this purpose, we construct a new dataset, Open Table-and-Text Question Answering (OTT-QA). OTT-QA is built on the HybridQA dataset (Chen et al., 2020), and like HybridQA, OTTQA questions are multi-hop questions which require aggregating information from both tables and text to answer. However, unlike HybridQA, OTT-QA requires the system to retrieve relevant tables and text — in contrast, in HybridQA, the ground truth tables and textual passages required for each question are given. To produce OTT-QA’s questions, we begin by re-annotating the questions from HybridQA to ‘decontextualize’ them—i.e., we make questions suitable for the open-domain setting so that unique answers can be determined from the question alone, without needing context from the provided text and tables. We then add new questions to remove potential biases. After these steps, OTT-QA contains $4 5 K$ human-annotated questions that require retrieving and aggregating information over tables and text from the whole Wikipedia. Examples from OTT-QA are depicted in Figure 1. Note the table and passages contain non-overlapping information, and both of them must be understood to answer the question. For example, the question has a low lexical overlap with the passage about the ‘Lakers’, and it needs the table as the bridge to retrieve this passage. Such cross-modality multi-hop retrieval features OTT-QA. More examples are displayed in Appendix.
+
+![](images/1a81c8d90729bc68de9a95ff23ef30b585c5155dba20f18ce01577242c537f78.jpg)  
+Figure 1: The problem setting: A OTT-QA model needs to retrieve from two candidate pools and then perform multi-hop reasoning to find answers.
+
+OTT-QA is distinguished from the existing QA datasets in two aspects. Existing table-based QA datasets (Pasupat & Liang, 2015; Yu et al., 2018; Chen et al., 2020) operates in the closed setting without requiring any retrieval, whereas most existing open QA datasets (Joshi et al., 2017; Yang et al., 2018) require only text retrieval, not table retrieval. One dataset, Natural Questions (NQ) (Kwiatkowski et al., 2019) includes some tabular information in its corpus, but the tables are nearly always of a restricted type (infobox tables with only a single row). In contrast, OTT-QA models require retrieving both tabular data and text, and unlike the NQ dataset, requires information fusion from text and tables in non-trivial ways. OTT-QA poses novel and realistic challenges to both the retriever and reader in open QA though the questions are less natural than the real queries from NQ (Kwiatkowski et al., 2019). Retrievers for OTT-QA need to consider two information formats, making the search space larger. Even worse, as questions in OTT-QA often require multi-hop inference, one round of retrieval is often not enough. Readers for OTT-QA also need to aggregate a significant amount of knowledge-intensive information, compared to other reader models: a single table in OTT-QA has an average length of over 300 words. Moreover, readers are often expected to process multiple retrieved units due to the uncertainty in retrieval, which makes it difficult to design strong reader models (Devlin et al., 2019; Liu et al., 2019) with a length limit of 512 tokens.
+
+The baseline system that we propose to address these challenges uses an iterative retriever (Sun et al., 2019; Qi et al., 2019; Min et al., 2019; Ding et al., 2019; Asai et al., 2019) and a BERT reader (Devlin et al., 2019). The iterative retriever explores multiple evidence documents iteratively, interacting with the candidate pool to gradually reformulate the query. Beam search is used to find multiple subsets of documents that may contain all the required evidence, and each subset is then fed to the BERT reader to predict the answer span. The highest-scored prediction is chosen as the answer. The iterative retriever needs to re-encode the query with a big transformer and re-search over the candidate pool, such a procedure (especially dense) can be computationally expensive. Furthermore, the BERT reader fails to capture a global overview of the retrieved documents, which leads to bad local optimum in the model prediction.
+
+We propose a more sophisticated system that addresses these challenges with two novel strategies: namely fusion retrieval and cross-block reading. The fusion retriever first pre-aligns the table segments to their highly related passages, using entity linking. Then, the aligned table segments and passages are grouped as a fused block, which contains aggregated information from two modalities; hence, compared to the previous documents, it contains richer context to benefit the following retrieval. We view the fused block as the basic unit to be retrieved, and instead of performing multiple runs of retrieval iteratively, the fusion retriever is used once to retrieve the top $K$ fused blocks; however, due to errors in fusion and retrieval, the retrieved top-1 fused block might not contain the necessary information. We thus also propose a cross-block reader based on a sparse-attention based transformer architecture (Ainslie et al., 2020; Zaheer et al., 2020), which can process extremely long sequences efficiently. We use the cross-block reader to read all the top-K retrieved fused blocks jointly. Both strategies have proven effective compared to the baseline system: the best model combining the two strategies improves the accuracy of the baseline system by a huge margin.
+
+# 2 BACKGROUND
+
+The aim of an open QA system is to extract an answer to a question $q$ from a given large corpus. Most open QA models are retriever-reader models, which extract answers in two steps: retrieval and reading. In the retrieval step, a retrieval model $f$ is used to retrieve a set of passages from the text corpus. In the reading step, the reader is then used to extract the answer from them.
+
+Retrieval Function There are two commonly-used types of retrieval function $f$ : sparse retrievers and dense retrievers. Our sparse retriever uses a unigram-based BM-25 score to retrieve an evidence unit $b$ from the candidate pool $\mathbb { B }$ . Our dense retrieval function is a dual-encoder model (Bromley et al., 1994), and we follow (Lee et al., 2019; Guu et al., 2020) for the dual encoder design. The query and the passage are encoded with separate Transformers. As in (Devlin et al., 2019), the vector corresponding to the first token, [CLS], is used as a “pooled” representation of the sequence. The dense retrieval function is the dot product between $h _ { q } = \mathtt { B E R T } _ { \mathbb { Q } } ( q ) [ \mathrm { C L S } ]$ and $h _ { b } = \mathtt { B E R T _ { B } } ( b ) [ \mathrm { C L S } ]$ for each evidence block $b$ in the candidate corpus—i.e., the scoring function is $f ( q , b ) = h _ { q } ^ { T } h _ { b }$ , which can viewed as finding the nearest neighbor in vector space. In the multi-hop open QA setting (Yang et al., 2018), an iterative retrieval function (Sun et al., 2019; Min et al., 2019; Ding et al., 2019) is proposed, which defines the retrieval process as an auto-regressive formula. Our iterative retriever function is denoted as $f ( [ q , b _ { 1 } , \cdots , b _ { j - 1 } ] , b _ { j } )$ , which appends the previous $j - 1$ rounds of retrieval to the original $q$ in in the $j$ -th round of retrieval. Beam search is used in test time.
+
+Single-Block Reader Due to the uncertainty in retrieval, the top-1 document might not always contain the answer. Existing models normally retrieve the top- $k$ documents and feed them to the reader for span selection. The standard reader (Chen et al., 2017; Joshi et al., 2017) aims to extract a span from each of the retrieved blocks $b _ { i }$ and assign a confidence $f ( q , b _ { i } ) f _ { r e a d } ( a | q , b _ { i } )$ to it, with $f ( \boldsymbol { q } , \boldsymbol { b } _ { i } )$ indicating the retrieval probability and $f _ { r e a d } ( a | q , b _ { i } )$ denoting the span selection probability by reader. Multiple answers $\{ \bar { a } _ { 1 } , \cdots , \bar { a } _ { k } \}$ are ranked with this confidence score and the highest scored answer span $\hat { a }$ is the final answer. Note that the reader needs to run $k$ times, once for each of the top- $k$ retrievals. We refer to this model as the single-block reader and use it as our baseline.
+
+HybridQA HybridQA (Chen et al., 2020), a closed-domain QA dataset, is the most related to ours. During the annotation of HybridQA, a table $T$ and its relevant passages $\{ P _ { 1 } , \cdots , P _ { N } \}$ (surrounding text and hyperlinked passage) are given to a crowd worker to write questions which necessarily require both the passage and table information to answer. The original dataset contains 72K multi-hop questions paired with 13K tables with their paired passages. During training/testing time, the ground-truth tables and passages are given to a model, HYBRIDER, to find the final answer. HYBRIDER also serves as an important baseline in our paper.
+
+# 3 TASK AND DATASET
+
+In OTT-QA, the retrieval corpus consists of a set of table candidates $\mathbb { B } _ { T }$ and a set of passage candidates $\mathbb { B } _ { P }$ . The task is to answer question $q$ by extracting answer strings from blocks $b \in \mathbb { B } _ { T } \cup \mathbb { B } _ { P }$ , where $b$ can be either textual and tabular data. We adopt the standard exact match (EM) and F1 scores (Yang et al., 2018) for evaluation. Different from HybridQA, OTT-QA’s table candidates are web tables without hyperlinks provided. This decision was made to make the problem setting more general, as otherwise systems that solve OTT-QA could only be applied to high-quality data in Wikipedia. However, in OTT-QA, we provide hyperlinks in the training subset, but not dev/test set. Removing hyperlinks in tables makes the overall task much more challenging, but makes the final systems applicable to more general domains. Thus, an OTT-QA model needs to jointly retrieve both tables and text, without abusing gold hyperlinks, and then aggregate them to find the answer.
+
+Candidate Pool For our table collection $\mathbb { B } _ { T }$ , we extracted all Wikipedia regular tables with their metadata including page title, page section title, and section text. The metadata, denoted $T _ { M }$ , is essential for de-contextualization. We obtain a table corpus containing over $4 0 0 k$ high-quality tables with an average length of 320 words including metadata. For the text passage collection $\mathbb { B } _ { P }$ , we crawl English Wikipedia dump pages and filter out noisy pages. We follow HybridQA (Chen et al., 2020) and only keep a maximum of 12 sentences in the introduction section as the passage. We obtain a corpus containing over 5 million passages, with an average of 94 words.
+
+Notation We define each table as a matrix $T$ , which consists of cells $T _ { i , j }$ with $i$ specifying the row, and $j$ specifying the column. Each cell $T _ { i , j }$ could be a number, date, phrase or even sentence due to its semi-structured nature. However, a single complete table with structured representation (Herzig et al., 2020) can easily exceed the 512-token limit, which poses great challenges to the downstream reader to process top- $K$ retrieval. Hence we propose to decompose each table $T$ into multiple rows $R _ { i }$ , which are combined with the headers, metadata, and global max/min information from the original table as a table segment. The table segment is used as the basic retrieval block in our paper. This decomposition procedure increases candidate $\mathbb { B } _ { T }$ from $4 0 0 k$ to 5 million, making the retrieval problems even more fine-grained and more challenging. Our table segment representation is described in Appendix subsection B.1. In summary, we build a candidate pool of 5 million table segments $\mathbb { B } _ { T }$ and a pool of 5 million passages $\mathbb { B } _ { P }$ . We denote as $\mathbb { B }$ as our full candidate pool, which our model needs to find the block $b$ (a table segment or a passage) containing the answer span.
+
+# 3.1 QUESTION AND ANSWER ANNOTATIONS
+
+Our question and answer pairs are built upon the existing HybridQA (Chen et al., 2020) dataset, with several significant changes. First, crowd workers ‘decontextualize’ the questions so that they are not under-specified or context-dependent, and thus suitable for the open setting. Second, we add more questions to the development/test set to remove possible annotation bias. During annotation, we adopt strict quality control2 and more details are described in Appendix section A.1.
+
+Decontextualization Most questions in HybridQA are contextualized with a given table and several passages, with corresponding questions written by crowd workers. Often, the crowdsourced questions assume the context. For example, the questions might contain the words "the players" because the given table is about "Netherlands players". We thus needed ‘decontextualize’ (Parikh et al., 2020) the original context-dependent questions, so they could serve as standalone questions, specific enough to imply a unique answer relative to the corpus. To discourage excessive unwanted modification, we enforce a two-step annotation procedure, as depicted in Figure 2. In the first phase, the worker is only allowed to insert minimum words or phrases (or replace pronouns) into the questions based on the information presented by Wikipedia Title, Section Title, and Section Text to make the question have a unique answer. After this step, we often potentially obtain overly-complicated questions that are artificial and unnatural. Therefore, we manually selected the worst $2 5 \%$ questions and sent them back to make them more concise and natural.
+
+OTT-QA Annotation   
+
+<table><tr><td colspan="6">Page Title:Netherlandsatthe European Track Championships Section Title: European Track Championships (elite) 2010-current</td><td rowspan="2"></td></tr><tr><td>schema</td><td>Medal</td><td>Championship</td><td>Name</td><td>Event</td><td>Ranking</td></tr><tr><td rowspan="2">content</td><td>Silver</td><td>2010 Pruszk6w</td><td>Tim Veldt</td><td>Men&#x27;somnium</td><td>2nd</td><td rowspan="2"></td></tr><tr><td>Bronze</td><td>2011 Apeldoorn</td><td>Kirsten Wild</td><td>Women&#x27;somnium</td><td>3rd</td></tr></table>
+
+Figure 2: The ‘de-contextualization’ annotation phase of OTT-QA. In the first step, the annotator is restricted to add phrases from the context. In the second step, the annotator is specifically requested to make the sentence more concise and natural.   
+
+<table><tr><td rowspan=1 colspan=1>0. Original</td><td rowspan=1 colspan=1>Which city does the player winning the silver medal in Men&#x27;s Omnium come from?</td></tr><tr><td rowspan=1 colspan=1>1. Insertion</td><td rowspan=1 colspan=1>Which citydoes the Netherlands player winning the Men&#x27;s Omnium silver medalin ETCafter 2010 come from?</td></tr><tr><td rowspan=1 colspan=1>2. Naturalize</td><td rowspan=1 colspan=1>Which city does the Netherlands Men&#x27;s Omnium silver medalist after 2010 in ETC come from?</td></tr></table>
+
+Additional Evaluation Examples As all the questions from HybridQA are based on the $1 3 k$ tables from the HybridQA set, no questions are asked about the newly crawled $4 0 0 k$ tables. This potentially generates unwanted statistical biases or artifacts for the model to exploit, and potentially biases the final evaluation results. Therefore, we randomly sampled another 1100 tables from the newly crawled tables, and follow the original annotation process used by HybridQA to re-collect 2200 new questions. These new questions were mainly used in the dev/test set. Below we refer to the subset of tables used by original HybridQA as the in-domain tables.
+
+Distant Supervision Signals For the in-domain tables $( \approx 8 k )$ , the cell-wise hyperlinks are provided in OTT-QA as a potential signal for supervision. We use $H _ { i , j } = \left\{ b _ { 1 } , b _ { 2 } , . . . \in \mathbb { B } _ { P } \right\}$ to denote the hyperlinks in cell $T _ { i , j }$ . Since in HybridQA the oracle fine-grained answer span is not explicitly annotated, we approximate this by traversing the table and hyperlinked pasasages to find all exact matches. This process contains some noise—a manual study reveals that it roughly contains $15 \%$ error. We use this ‘weakly-supervised’ fine-grained information to train our models. We denote the ‘approximate’ block of the answer span for answer $a$ as $b _ { a }$ , and use it to train our model.
+
+# 3.2 DATASET STATISTICS
+
+After annotation, we sampled roughly 2K questions from the in-domain HybridQA dataset, and then mix them with the newly collected out-domain questions to construct our dev and test sets. Finally, we have 41,469 questions in the training set, 2,214 questions in the dev set, and 2,158 questions in the test set. We conduct more in-detailed analysis over the reasoning types and show them in the Appendix A.3, a remarkable difference from original HybridQA is that a proportion of questions actually have multiple plausible inference chains in the open-domain setting.
+
+# 4 MODEL
+
+Our model for OTT-QA is a retriever-reader model with new designs for both retriever and reader. As discussed briefly above, we propose to use a fusion retriever instead of using a standard iterative retrieve, and we also propose to use cross-block readers to replace a standard single-block reader.
+
+![](images/2acc0185ed2ffbf5a53701e3e2f86f8902e28a7779fb79c3aa1d5ba55ee781e0.jpg)  
+Figure 3: Left: Iterative 3-step retrieval over individual blocks (baseline). Right: Fusion 1-step retrieval over fused groups, which greatly lowers the cost of iterative encoding and retrieving.
+
+# 4.1 FUSION RETRIEVER
+
+Iterative retrieval (Figure 3, Left) has the following issues. First, iterative retrieval training often requires having supervision signals for every retrieval step to reach good performance, which is not available in OTT-QA. The iterative retrieval also suffers from the problem of error propagation, as early mistakes can propagate to later retrieval stages. Finally, the computation cost for applying a dual-encoder for iterative retrieval is very high, as for every stage, the query embedding has to be re-encoded to include the entire retrieval history.
+
+We propose an alternative strategy to replace multi-step retrieval, namely fusion retrieval (Figure 3, Right). In the fusion retriever, we first use an ‘early fusion’ strategy to group relevant heterogeneous data before retrieval. The fusion procedure groups several highly-relevant blocks from different modalities as a self-contained group (fused block), which provides more clues for the retriever to utilize. Early fusion is very important for retrieving table segments, which often have incomplete context by themselves. The early fusion process aims to fuse a table segment and relevant passages into a group. Here we propose to fuse entities mentioned in a table segment to the appropriate passages for those entities; this is similar to document expansion based on a traditional entity linking step. The problem is challenging due to the mismatch between the lexical forms from the
+
+How many points per game did Lebron James get in the NBA Season suspended by COVID?
+
+![](images/168a1ec45d5773c47ac87b7f4d18326eed75dfa9f6ab5d6dd61725e5ad956428.jpg)  
+Figure 4: Left: Single-block reader with input shorter than 512 tokens (baseline). Right: Crossblock reader with length over 4K tokens, and $\bar { A }$ denotes the global state assigned to local block A. The single-block reader is stuck at local optimum, while cross-block reader outputs global optimum.
+
+table (which for brevity are often abbreviated) and the relevant passage titles. For example, a cell in the table of "NCAA Division I Men’s Football Tournament" contains the term "Penn State". Directly matching "Penn State" against the passage corpus will lead to "Penn State University" rather than the ground-truth hyperlinked entity, named "Penn State Nittany Lions football". Therefore, we propose an additional augmentation step, which takes in a table segment block $b _ { T }$ and generates a sequence of augmented queries $q _ { 1 } , q _ { 2 } , \cdots , q _ { n }$ token by token to make the queries more similar to the passage title. The augmented queries are then used to search for nearest neighbors in the passage corpus $\mathbb { B } _ { P }$ using BM25 as the final entity linking step, which is depicted in Appendix. The query augmentation is implemented with a GPT-2 model (Radford et al., 2019), fine-tuned on the supervised pairs of (table segment, hyperlink) from the in-domain tables. Each $b _ { T }$ is fed to find its companions $b _ { P } ^ { 1 } , \cdots , b _ { P } ^ { n }$ , they are collectively called $b _ { F }$ .
+
+We follow the standard dual-encoder setting (section 2) and the only difference is that we replace the input of the block encoder with $h _ { b } = \mathtt { B E R T _ B } ( [ b _ { T } , b _ { P } ^ { 1 } , \cdot \cdot \cdot , b _ { P } ^ { n } ] )$ ., which captures the cross-attention between the table and the text within a block. The fused embedding contains richer context from both modalities to complement each other. The retriever only needs to retrieve once from the candidate pool, which dramatically decreases the complexity compared to the existing iterative retrievers.
+
+To enhance the neural retrieval system to retrieve fused blocks, we apply the Inverse Cloze Task (ICT) (Lee et al., 2019) pretraining task on the corpus of fused blocks. ICT is a way to generate pseudo-training data for dense retrieval. Unlike standard document-wise ICT, our fused block contains both table segments and multiple passages. Given a fused block $b _ { F }$ , we generate the pseudoquery in the following way: 1) we first corrupt the table segment by randomly dropping half of the words from the table metadata and cells to obtain a partial table segment $\hat { b } _ { T }$ . 2) We then randomly sample a sentence $\hat { b } _ { P }$ from the fused passage. We combine $\hat { b } _ { T }$ and $\bar { \boldsymbol { b } } _ { P }$ as a pseudo query $\hat { q }$ and pair it with the original fused block $b _ { F }$ as pre-training data. The pre-training data is applied to enhance the dual encoder’s ability to select lexically matched documents. After pre-training, the retriever is fine-tuned on OTT-QA. Finally, at inference time, the retriever is used to retrieve the top $K$ fused blocks for a question, which are then passed to the reader for answer prediction.
+
+# 4.2 CROSS-BLOCK READER
+
+The reader typically needs to process the top- $k$ retrieved blocks returned by the retriever to extract the best answer, as the top-1 block might not contain enough evidence to answer the question. As demonstrated in Figure 4, the cross-block reader aims to address this issue by using cross attention between different blocks to model their dependencies. To obtain the cross-block reader, we take the pre-trained long-range sparse attention transformer (ETC) (Ainslie et al., 2020), which can accept up to 4096 tokens as input, and then fine-tune the model on the distant supervision data. During training, the ground truth (fused) blocks are mixed with hard negative blocks from the retriever. We take the top- $k$ retrieval results to fill the 4096 token space (roughly 15 fused blocks).
+
+Cross-attention between blocks allows a much more powerful way to aggregate information across the $k$ retrieved blocks compared to the single-block reader, especially when the blocks are fused. This is feasible because of the design of the sparse attention structure in ETC, which can constrain the attention of each token to its neighboring tokens within a local radius in its local block. Such sparse attention can decrease the attention computation complexity from quadratic $\mathcal { O } ( N ^ { 2 } )$ to linear $\mathcal { O } ( N | R | )$ , where $| R |$ is the local radius (where $N = 4 0 9 6$ and $| R | = 8 4$ in our experiments). To allow cross-block interaction, ETC assigns a global state for each local block in the long sequence, and blocks can attention to each other through multiple layers of such global-local structures.
+
+Table 1: Main Results. We conduct experiments with both sparse and dense retrievers using the dev set, and then select the best setting to report the test set results (as indicated by the word ”Best”). Fusion-Retriever and Cross-Block Reader are combined to obtain the highest score. $\dagger$ are ablations.   
+
+<table><tr><td>Retriever</td><td colspan="2">Dev-Sparse</td><td colspan="2">Dev-Dense</td><td colspan="2">Test-Best</td></tr><tr><td>Model</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>HYBRIDER(Top-1) (Chen et al., 2020)</td><td>8.7</td><td>10.9</td><td>8.9</td><td>11.3</td><td>8.4</td><td>10.6</td></tr><tr><td>HYBRIDER (best Top-K) (Chen et al.,2020)</td><td>9.9</td><td>12.2</td><td>10.3</td><td>13.0</td><td>9.7</td><td>12.8</td></tr><tr><td>Iterative-Retrieval + Single-Block Reader</td><td>9.8</td><td>13.3</td><td>7.9</td><td>11.1</td><td>9.6</td><td>13.1</td></tr><tr><td>Fusion-Retrieval + Single-Block Reader</td><td>14.3</td><td>17.8</td><td>13.8</td><td>17.2</td><td>13.4</td><td>16.9</td></tr><tr><td>Iterative-Retrieval + Cross-Block Reader</td><td>17.1</td><td>20.7</td><td>14.4</td><td>18.5</td><td>16.9</td><td>20.9</td></tr><tr><td>Fusion-Retrieval + Cross-Block Reader</td><td>27.7</td><td>31.8</td><td>28.1</td><td>32.5</td><td>27.2</td><td>31.5</td></tr><tr><td>† Table-only Retrieval + Cross-Block Reader</td><td>4.6</td><td>6.9</td><td>4.9</td><td>7.2</td><td>4.4</td><td>7.0</td></tr><tr><td>† Text-only Retrieval + Cross-Block Reader</td><td>8.2</td><td>12.4</td><td>8.9</td><td>12.8</td><td>8.8</td><td>12.1</td></tr><tr><td>† Oracle Link + Fusion-Retrieval + Cross-Block Reader</td><td>35.8</td><td>40.1</td><td>35.2</td><td>39.9</td><td>35.0</td><td>39.5</td></tr><tr><td>† Oracle Table + Link (w/o Retrieval) + HYBRIDER</td><td>44.1</td><td>50.8</td><td>44.1</td><td>50.8</td><td>43.0</td><td>49.8</td></tr></table>
+
+# 5 EXPERIMENTS
+
+All of our code is based on Tensorflow (Abadi et al., 2016). For the retriever part, the sparse retriever is built on top of DrQA (Chen et al., 2017) with unigram features, and the dense retriever is built with BERT. The single-block retriever is based on BERT-uncased, and the cross-block reader is based on ETC (Ainslie et al., 2020). Both of them consist of 12 layers with a hidden size of 768, the minor differences in the relative positional embedding used in ETC. All the models are trained with a learning rate of 1e-5 optimized by AdamW (Loshchilov & Hutter, 2019). We use in-batch negatives (Lee et al., 2019) to train our dense retrievers. A more detailed implementation of the baseline iterative retriever is described in Appendix subsection B.2. In fusion retriever, we use the ‘fused’ block containing the ‘approximate’ answer block $b _ { a }$ as the positive instance. In iterative retriever, since the auto-regressive model $f ( b _ { j } | q , b _ { 1 } , \cdot \cdot \cdot , b _ { j - 1 } )$ requires fine-grained inference chain for step-wise supervision, which is not given in OTT-QA. We apply lexical match based heuristics to synthesize inference chains as weakly supervised training data (described in Appendix). For all the dense retrievers, we pre-train with 10K steps using the generated pseudo query and then fine-tune them another 10K step using a batch size of 2048. For the cross-block reader, we fine-tune with a batch size of 64. Both are using 16 cloud TPUs.
+
+Main Results In our experiments, we experiment with different types of retriever and reader models under both sparse and dense setting, the details are described as follows:
+
+• HYBRIDER: this model, designed for closed domain HybridQA questions, is one baseline. Since this model requires a ground truth table with its hyperlinks to do modularized reasoning, we use BM25 to retrieve the most relevant table and passages to reconstruct an ‘approximated’ input for this model. We experiment with top-1,2,3,4 cases where we use the answer with the highest confidence as the final result. We also directly feed the ground-truth table and hyperlinks to HYBRIDER, which roughly estimates an upper limit of this task.
+
+• Iterative-Retriever (Sparse): We use a 2-step iterative retriever: in the first step, we apply the question to retrieve the top-10 table segments and top-10 passages. In the second step, we use each retrieved table segment to retrieve its related top-5 passages and concatenate each retrieved passage title with the original question to retrieve the top-5 table segments. We merge and calculate the retrieval score of each unique block and rank them by their score. For the single-block reader, we split the retrieved blocks into 512-token chunks and feed them to the BERT reader. For the cross-block reader, we truncate the top 4096 subword tokens and only feed these tokens to reader.
+
+• Iterative-Retriever (Dense): We use a 3-step iterative retriever. In the first step, we encode the question and retrieve the top-8 blocks (either table segment or passage); in the second step, we concatenate the previous retrieved block and the question to re-encode the query vector to further retrieve top-4 blocks; similarly, the last step retrieves top-2 blocks.
+
+• Fusion Retriever (Iterative): We use a sparse retriever to directly retrieve the top-15 fused blocks based on bag-of-words BM25 score, and then split it into individual table segments and passage blocks. Since passage could be associated with multiple fused blocks, we merge duplicate blocks and use their summed score. Finally, we rank each block based on its merged retrieval score and truncate the first 4096 subword tokens for the next step.
+
+• Fusion Retriever (Iterative): We use a dual-encoder dense retriever to directly retrieve the top-15 fused blocks, and then follow the same procedure as above. Without specifying the dense retriever uses ICT for pre-training by default.
+
+• Fusion Retriever w/o ICT and w/o GPT-2: these two ablation studies are aimed to show the effectiveness of our proposed ICT pre-training and query augmentation.
+
+The main results are presented in Table 1. First, we can observe that best HYBRIDER top-2 can only achieve a comprised exact match of $9 . 9 \%$ while the oracle HYBRIDER can obtain a score of $44 \%$ , which reflects the difficulties of the hybrid retrieval in our dataset. We restrain the retriever to only retrieve table and text to answer the questions and report their results in Table 1, even with the strong cross-block reader, the model only obtains $10 \%$ EM. These experiments demonstrate the necessity to integrate information from both forms in OTT-QA.
+
+By combining the standard iterative retriever and single-block reader, we can slightly improve the score can to roughly $10 \%$ . By replacing the iterative retriever with the proposed sparse fusion retriever, the EM score can reach $14 \%$ , a $4 . 5 \%$ absolute improvement. By replacing the single-block with the proposed cross-block reader, the EM score can reach $17 \%$ , a $7 \%$ absolute improvement. However, by combining the two strategies, the final EM score can reach $28 \%$ , with an $18 \%$ absolute improvement, which is greater than the sum of individual improvements. The observation suggests the two components can affect each other in a positive way. We conjecture that the fusion retriever is more likely to retrieve mutually-supportive blocks in a group, which makes the multi-hop reasoning across different blocks easier for the following cross-block reader. In comparison, the iterative retriever retrieves isolated table segments and passages separately, which can easily miss out on the bridging evidence for building the complete reasoning chain. Thus, the cross-block reader cannot maximize its advantage in reasoning across blocks.
+
+By removing the ICT pre-training and query augmentation, we observe that the Dev-EM score drops to $2 4 . 6 \%$ . By removing the GPT-2 query augmentation, the Dev-EM performance drops to $2 2 . 1 \%$ . These two results indicate the effectiveness of the proposed two strategies. By replacing the predicted hyperlinks with the oracle links, the fusion model performance can increase by $7 \%$ EM. This indicates that there is still plenty of room to improve for the table-passage fusion model.
+
+Linker/Retriever Results To understand the results more, we evaluate the standalone tablepassage entity linking accuracy and retriever recall.
+
+![](images/f37485a0990a6823b4c0f884eec011731f701b3d6821fc45ee7559680313f918.jpg)  
+Figure 5: Entity linker performance (F1).
+
+![](images/17edf79954540b2f232608d739bf1038e6ce1ebf55c8d67513b924256f783432.jpg)  
+Figure 6: Retriever performance $( \mathrm { H I T S } @ 4 \mathrm { K } )$
+
+We consider the following linking models: a) BM25 model, which directly uses the cell value to retrieve passages based on their titles without query augmentation, b) a Dual-Encoder model, which encodes the cell value and meta information into a query vector to compute dot-product over all the passage candidate to retrieve, c) a GPT-2 model, which first augments the cell value by the context and then uses BM25. We demonstrate our findings in Figure 5, and evaluate with table-segment-wise
+
+F1 score. We observe that directly using BM25 leads to compromised precision of $3 0 . 3 \%$ , which is mainly due to the lack of context information. By using a dual-encoder retriever, the precision can be improved to $42 \%$ . However, many table segments have either zero or multiple linked passages and can be better modeled by an auto-regressive retrieval process.
+
+We use $\mathrm { H I T S } @ 4 \mathrm { K }$ is used to measure the retriever performance, which indicates the chance of ground truth block existing in the retrieved 4096 subword tokens. The results are reported in Figure 6. We vary the steps of iterative retrievers to show the necessity of multi-hop retrieval in OTTQA. We observe that the 1-step retrieval has the lowest recall because the answer block in OTT-QA normally has a lower lexical overlap with the query. Adding the second retrieval step can greatly improve the recall, but adding the third retrieval hop has very little impact. In contrast to the iterative retriever, the fusion retriever can consistently improve the performance over the iterative setting for both sparse and dense setting. The sparse setting can rise from $3 5 . 8 \%$ to $4 8 . 1 \%$ indicating the advantage of ‘early’ fusion. The dense retriever’s improvement is more dramatic (from $2 7 . 2 \%$ to $5 2 . 4 \% )$ . We believe this is because the iterative retriever heavily relies on noisy synthetic inference chain data, while the fusion retriever does not require such a fine-grained supervision signal, thus less prone to noise. To better understand the retriever, we conduct detailed error analysis in Appendix C.
+
+# 6 RELATED WORK
+
+Table Retrieval: Tables are pervasive on the Web, there have been some studies on mining web tables to answer open-domain questions (Sun et al., 2016; Chakrabarti et al., 2020). In Sun et al. (2016), the authors have proposed a pipeline framework to first detect the topic entity and then generate a candidate chain, finally ranking chains to predict the answer cell. In Chakrabarti et al. (2020), the authors investigate different similarity matching features to retrieve tables from the web. Our paper is significantly different from these two studies in two aspects: 1) the previous papers use private small-scale datasets while we collect a large-scale dataset and release it for public use, 2) the previous studies are restricted to only using tables as evidence, while our paper considers a more realistic and challenging setting with both table and text corpus. Tables have been a ubiquitous information representation form to express semi-structured information. There has been a long-standing effort to utilize tables in natural language processing applications (Pasupat & Liang, 2015; Zhong et al., 2017; Yu et al., 2018; Parikh et al., 2020; Chen et al., 2019). However, these existing tasks are restricted to in-domain cases without requiring any retrieval, and our paper is the first to investigate retrieving web tables for downstream tasks. Another pair of related works are TAPAS (Herzig et al., 2020) and TABERT (Yin et al., 2020), which investigate joint pre-training over textual and tabular data. Our method draws inspiration from these models, and also uses special tokens and embeddings to encode spatial and logical operations inside tables.
+
+Long Range Transformer: Recently, many transformer variants to resolve the $\mathcal { O } ( n ^ { 2 } )$ attention cost have been proposed including Sparse Attention (Child et al., 2019), Reformer (Kitaev et al., 2020), Routing Transformer (Roy et al., 2020), Longformer (Beltagy et al., 2020) and ETC (Ainslie et al., 2020). These different transformer models apply hierarchical architecture, local-sensitive hashing, global-local state to decrease the attention complexity to nearly linear. Our cross-block reader is based on ETC (Ainslie et al., 2020), but unlike prior works that process one long document for QA, our task requires reading multiple blocks containing both structured and unstructured data. To handle the long sequence of retrieved documents in open-domain question answering, Fusionin-Decoder (Izacard & Grave, 2020) has been proposed to replace the extractive model with an encoder-decoder generative model. The long sequence of passages are split and encoded independently to decrease the computation complexity, but the decoder still uses full attention over the tens of thousands of encoded vectors to generate the answer token by token. Such full-attention can decrease the decoding speed by an order of magnitude, while our sparse-attention-based cross-block reader can still maintain the same speed as the standard BERT model.
+
+# 7 CONCLUSION
+
+We focus on the problem of performing open question answering over tables and text in this paper. By proposing the fusion retriever and sparse reader, we manage the increase the model’s effectiveness and efficiency by a large margin. One interesting question we would like to ask in the future is: can we extend open question answering system to more modalities like images or audios, etc?
+
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+
+# A DATASET COLLECTION
+
+# A.1 DATASET ANNOTATION
+
+Filtering The original HybridQA dataset contains over $7 2 k$ questions paired with $1 3 k$ hyperlinked tables. We adopt two filtering heuristics to make the decontextualization easier. First, we filter out tables without enough meta-information or containing too much non-textual information3. Second, we filter out overly-long questions, i.e., questions longer than 30 words. These two filtering heuristics result in a cleaner subset of $4 6 k$ questions paired with $9 k$ in-domain tables.
+
+Quality Control During annotation, we conduct strict manual quality evaluation over the decontextualized questions, with the following criteria: 1) the annotated question retains the same semantics and answer as before, 2) the annotated question still requires multi-hop reasoning over both table and passages, and 3) the annotated question is concise and fluent. The manual quality checking was performed over batches distributed to the same annotator. Each batch consists of six questions, one of which will be sampled to decide the acceptance/rejection of the whole batch. The overall acceptance rate for the crowd-sourcing job is $71 \%$ , and a rejected job was re-distributed until it was accepted.
+
+# A.2 DATASET EXAMPLES
+
+We demonstrate more examples in Figure 7, which includes more diverse inference chains, like table text; text table text; text $^ +$ text comparative table. Our model is able to perform these reasoning types quite well by jointly matching a query against a fused table-text block.
+
+![](images/238b0ff545cec87ff7962e1ffe943a76b51472e23e4894efbed8226134617f79.jpg)  
+Figure 7: More examples from OTT-QA
+
+# A.3 QUESTION TYPES
+
+We randomly sampled 100 questions from the dataset to manually analyze the kinds of inference chains seen in OTT-QA and divide the major types into the following categories:
+
+1. Single hop questions $( 1 3 \% )$ require reading one table or one passage to answer. 2. Two hop questions $( 5 7 \% )$ require reading one passage and one table to answer. These can be subclassified as ‘table bridge’ ‘answer text’4 or ‘text bridge’ ‘answer table’.
+
+3. Multi-hop questions $( 3 0 \% )$ require reading two passages and one table to answer. These mainly following the reasoning chain of ‘text bridge’ ‘table bridge’ ‘answer text’.
+
+4. Questions with multiple reasoning paths: Due to information redundancy in Wikipedia, similar information can appear in both tables and text. We find that $9 \%$ of questions are answerable by reading one text passage, $18 \%$ of questions are answerable by reading two text passages and $4 \%$ of questions are answerable by reading two tables.
+
+# B MODEL DETAILS
+
+# B.1 RETRIEVAL BLOCK REPRESENTATION
+
+The table decomposition is visualized in Figure 8. The title/section title are prefixed to the table segment. We add the row position token ‘1st’ and a max/min special token over the column to infuse global table information into the segmented unit. The column embedding is added as another vector to the representation. The table segment representation is relatively small and easy to deal with in the following reader model. After the table-passage alignment, we group the highly related units together and represent them as the lower part demonstrated in Figure 8. We add [SEP] tokens to separate different passages and set their type id to 0. Such a flattened representation for fused block $b _ { F }$ will be used throughout our experiments for both sparse/dense retriever and ETC reader.
+
+![](images/ff906d672aec3f62692c5486b5abdefcae9c9c5b32a591e69086ad4bcf25402a.jpg)  
+Figure 8: The decomposition of the original table into segments.
+
+# B.2 ITERATIVE RETRIEVER
+
+Iterative retrieval has been used in recent graph-based multi-hop retrieval models to gradually retrieve documents to find the correct supporting evidence. Specifically, the retriever conditions the $i$ -th round retrieval on the previous round of retrieval results.
+
+Sparse Retriever The sparse retriever uses uni-gram lexical feature to compute the BM-25 score between the $q , . . , b _ { 1 . . j - 1 }$ over $b _ { j } \in \mathbb { B }$ to get the top candidates. Here we describe a two-step retrieval procedure, called here the LxM procedure. In the first step, the model calculates the BM25 score between question over all the candidates in $\mathbb { B }$ to select top $_ { \textrm { L } / 2 }$ table segments $b _ { T }$ , and $_ { \textrm { L } / 2 }$ passages $b _ { P }$ . In the second step, the question is concatenated with the retrieved table segment to form $_ { \textrm { L } / 2 }$ new queries $[ q ; b _ { T } ]$ which are used to retrieve LM/2 passages from $\mathbb { B }$ . The question is also concatenated with the retrieved passage titles to form another $\mathrm { \ K } / 2$ queries $[ q ; b _ { P } ]$ to retrieve LM/2 table segments. The retrieval procedure results in at most $_ { \mathrm { L M + L } }$ unique blocks. Each unique block aggregates its score from two rounds, denoted as $f ( \boldsymbol { q } , \boldsymbol { b } )$ , which is used to rank the top-K candidates for the next step. We truncate the top-K candidate by thresholding their combined length.
+
+Dual-Encoder Retriever The dual encoder uses a BERT-based encoder to compress each question, table segment, and passage into a fixed-length vector and then computes the dot product between fixed vectors to obtain the highest scored candidates from pool $\mathbb { B }$ . However, since the dataset does not provide an explicit supervision signal for the iterative retrieval, we heuristically synthesize some noisy retrieval chains using lexical matching. The retrieval inference chain is depicted as $b _ { 1 }  b _ { 2 }  b _ { K }$ , which is used to train the model $f ( b _ { k } | q , b _ { 1 \dots k - 1 } )$ in a supervised manner. At inference time, the dual encoder retriever will encode a query $q$ into a fixed vector and retrieve the first $L$ blocks from $\mathbb { B }$ . The blocks are appended to query $q$ to form $L$ new queries $[ q ; b _ { i } ]$ , which is re-encoded and search for $L M$ new neighbors. We experiment with a maximum of 3-step retrieval of $\mathbf { L x M x N }$ to obtain a maximum of $_ \mathrm { L + L x M + L M N }$ unique blocks. Similarly, each unique block aggregates its score from different rounds to select the top-K candidates for the next step.
+
+# B.3 SPARSE FUSED RETRIEVER
+
+The sparse fused retriever uses the uni-gram lexical feature to compute the BM-25 score between $q$ over $b _ { F } \in \mathbb { B } _ { F }$ . The uni-gram feature of $b _ { F }$ is based on the representation depicted in subsection B.1. Note that this BM25 feature will be much more abundant than the BM25 feature in iterative sparse retriever because it encloses more uni-grams. Instead of doing multiple rounds of retrieval, the fused retrieval once retrieve once over the candidate pool and treat all the units inside the block as the same retrieval score. Finally, We truncate the top- $\mathbf { \nabla } \cdot \mathbf { K }$ candidate by thresholding their combined length.
+
+# B.4 QUERY AUGMENTATION
+
+The query augmentation procedure is depicted in Figure 9.
+
+![](images/e3ba7b02b6601e568357a58b66f0283761608109d5c97af1bbe51e579b2ae779.jpg)  
+Figure 9: Fusion: 1) GPT-2 query augmentation, 2) nearest neighbor search over passages.
+
+# B.5 DENSE RETRIEVAL/IN-BATCH NEGATIVE
+
+Recently, different dense-retrieval methods (Lee et al., 2019; Guu et al., 2020; Karpukhin et al., 2020) based on dual-encoders (Bromley et al., 1994) have been shown to surpass traditional sparse retrieval in open-QA models. The query and the passage are both encoded using a Transformer, which produces a vector for every token. As in (Devlin et al., 2019), the vector corresponding to the first token, [CLS], which is used as a “pooled” representation of the sequence (denoted $\mathtt { B E R T } _ { \mathtt { C L S } } ,$ ). The dense retrieval function can be represented as the dot product between $\mathtt { B E R T } _ { \mathtt { C L S } } ( q )$ and $\mathtt { B E R T } _ { \mathtt { C L S } } ( p )$ for each document in the text collection, much like TF-IDF (Chen et al., 2017) and BM25 (Robertson & Zaragoza, 2009) on some Open QA datasets. To train the dual-encoder, the in-batch negative trick (Yih et al., 2011; Karpukhin et al., 2020) plays an important role, which uses B training instances in each batch and views the other B-1 instances inside the batch as the negatives. In this way, the model reuses computation and effectively trains on $B ^ { 2 }$ question/document pairs in each batch.
+
+# C PERFORMANCE ANALYSIS
+
+# C.1 QUESTION TYPE BREAKDOWN PERFORMANCE
+
+We measure our best model’s performance (dense fusion retriever $^ +$ cross-block reader) and baseline model (dense Iterative-Retriever $^ +$ single-block reader) on different question types (subsection A.3) to show the breakdown statistics in Figure 10 and Figure 11. As we can observe, the gap between our model vs. baseline in 1-hop question is less significant as 2-hop and 3-hop questions. The iterative retriever’s performance is sensitive to the number of hops in the question, which is the largely due to the error propagation in the beam search stage. If the retriever fails to include the golden block in the earlier stage beam, the retrieval in later stage cannot recover from such failure. In contrast, our fusion retriever can group the related information prior to retrieval to retrieve all the blocks at once, which makes the model less prone to the error propagation issue. Another reason is due to the cross-block reader, which can reason over different blocks in the latent space, such implicit reasoning can also decrease the error propagation issue. To sum up, our model is more powerful to deal with complex multi-hop open questions with much less performance drop.
+
+![](images/8b6b5290ce661d3c29bf208f02e8d077c6fe5ebaaa030ab421db42c69127e048.jpg)  
+Figure 10: Breakdown for iterative retriever Figure 11: Breakdown for fusion retriever $^ +$ $^ +$ sing-block reader. cross-block reader.   
+Figure 12: The main error types in the retriever.
+
+# C.2 RETRIEVER ERROR ANALYSIS
+
+We conduct error analysis to see what are the major issues with the retriever and conclude the following types in Figure 12. The major issues causing the system to retrieve unrelated evidence are low lexical overlap, fusion errors, numerical reasoning and distracting passages or tables. In the
+
+<table><tr><td>Reason</td><td>Example</td><td>Groundtruth Block</td></tr><tr><td>Low Lexical Overlap</td><td>Where is the NYU Alumni from 1980 who now serves as.</td><td>Table: New York University Alumni in Politics and Science....</td></tr><tr><td rowspan="2">Error in Fusion</td><td rowspan="2">Who is the coach of Pittsburgh team in NCAA division I football team in 1980?</td><td>Table: List of NCAA Division I Cell: Pittsburgh</td></tr><tr><td>Fail to link to the passage: Pitsburgh Panthers</td></tr><tr><td rowspan="2">Numerical Reasoning</td><td rowspan="2">Who achieves the highest score in the Grand Prix 1980s Men&#x27;s..</td><td>Table: List of Grand Prix 1980s Men’&#x27;s.</td></tr><tr><td>The time is not in regular format, SQL operation</td></tr><tr><td rowspan="2">Distraction</td><td rowspan="2">Where of 1990 Grammy Award winner for ... .come from in?</td><td>cannot select the max/min row. Table: Grammy Lifetime Achievement Award</td></tr><tr><td>Many tables from are about general &quot;Grammy Awards&quot;,the groundtruth table only differentiate</td></tr><tr><td colspan="2">Retriever Error Breakdown</td><td>from the others a little bit. Breakdown of Errors</td></tr><tr><td colspan="2">40% 36% 32%</td><td></td></tr><tr><td colspan="2">30% 24%</td><td></td></tr><tr><td colspan="2">20%</td><td>28% 49%</td></tr><tr><td colspan="2">10% 8%</td><td>23%</td></tr><tr><td colspan="2">0% Retriever</td><td></td></tr></table>
+
+Low-Lexical-Overlap case, the errors are mainly coming from the abbreviation, rephrasing of the table metadata, for example, ‘New York University’ is shortened as ‘NYU’, etc. In the Fusion-Error case, the issue is mainly because the entity-linking model fails to fuse all the hyperlinked passages, the error $( \mathrm { F 1 } { = } 5 0 \%$ ) is quantitatively reflected in the entity-linker-performance figure. NumericalReasoning error is mainly related to the failure to find max/min/earliest/latest row in the table. The distraction error is mainly caused by some distracting passages or tables having very similar information. We sample 50 error samples from the dev-set and attribute their errors to the above categories. As shown in the left part, we found that the numerical reasoning error is not as severe as the other three types because the proportion of questions requiring it is relatively small. Besides the low-lexical overlap error, which is general across other open QA datasets like NQ and HoptpotQA, we found the fusion and distraction errors quite specific in our dataset.
+
+• (Fusion) Questions which ask about tables which are linked to too many linked passages. For example, a question over table “Team Record” in https://en.wikipedia.org/ wiki/Sevens_Grand_Prix_Series is hard because some table rows associate with over 10 passages, it’s hard to link them and fuse all of them into a fused block. (Distraction) Questions which ask about topics which are contained by too many similar tables, it’s hard to differentiate the true one. For example, there are over ten tables in https://en.wikipedia.org/wiki/List_of_RMIT_University_ people, these similar tables can easily distract the attention of the retriever to select the wrong one from the same page.
+
+From our quantitative results, we can attribute the errors to retriever and reader, among all the examples, $49 \%$ of examples cannot find the correct supporting block. For the rest $51 \%$ examples with correct block retrieved, the reader fails to select the correct span for $23 \%$ of them.
+
+# C.3 LENGTH SENSITIVITY ANALYSIS OF RETRIEVAL/READER
+
+![](images/11cfe553441603a5e1f65463ef1c5ca2fb1bc5ebf77c3196dccfc26979e706c3.jpg)  
+We perform sensitivity analysis for both retriever and reader in Figure 13. We gradually increase   
+Figure 13: Analyzing retriever performance.
+
+the length limit of retrieved evidence from 400 to 4096 to first visualize its impact on the sparse iterative and dense fusion retriever. For both fusion and iterative retriever, we can observe that both of their recall $@ \mathrm { K }$ significantly improves as the length limit increases. With a low budget of token limit, their performance is gap is smaller because its performance is dominated by the singlehop questions in the dataset. As the length limit increases, the improvement for fusion retriever is steeper than iterative retriever because the contextualized fusion block becomes easier to retrieve than standalone table segment or passage.
+
+We also visualize the input length’s impact on the single-block The performance of single vs crossblock reader. With a low budget of token limit, both single-block and cross-block readers are comparable. However, as the limit increases to 4000, the cross-block reader can digest long input with its sparse attention mechanism to achieve better scores, while the single-block reader needs to truncate the information to read independently, which leads to a even lower EM score due to introduced noise. This observation reveals the importance of modeling cross-attention between different retrieved evidence units to reach a consistent answer. In single-block reader, dealing with different blocks independently can lead to suboptimal prediction in our dataset.
+
+# D CONNECTION TO EXISTING WORK
+
+KB and Text The problem combining structured and unstructured data has been studied in question answering. The previous approaches are mainly divided into two categories: 1) FusionNet and PullNet (Sun et al., 2018; 2019) simulate a KB-incomplete setting by masking out some triples from a knowledge graph and use textual information to complete the masked KB triples; these experiments are conducted on KB-based QA datasets. 2) DrKIT (Dhingra et al., 2019) and KnowledgeGuided Retrieval (Min et al., 2019) propose to use entity mentions and relations to guide the retrieval from the web. However, the KB is mainly used as an assisting tool, rather than a necessary information source. In OTT-QA, the structured data is used as necessary information in a realistic setting. The two information forms are combined in a non-trivial way, which makes the problem much harder than the other structure-unstructured QA settings.
+
+Entity Linking Our generative entity linker is related to knowledge-enhanced language understanding (Petroni et al., 2020), which proposes a seq2seq model to deal with different knowledge-intensive tasks like slot filling, entity linking, etc. There is a concurrent related work on auto-regressive entity linking (De Cao et al., 2020), which also demonstrates the advantages of using an autoregressive generation model for entity retrieval.

parse/train/Ovp8dvB8IBH/Ovp8dvB8IBH.md

@@ -0,0 +1,479 @@
+# NEGATIVE DATA AUGMENTATION
+
+Abhishek Sinha1∗ Kumar Ayush1∗ Jiaming Song1∗ Burak Uzkent1 Hongxia Jin2
+
+# Stefano Ermon1
+
+Department of Computer Science1   
+Stanford University   
+{a7b23, kayush, tsong, buzkent, ermon}@stanford.edu
+
+Samsung Research America2
+
+# ABSTRACT
+
+Data augmentation is often used to enlarge datasets with synthetic samples generated in accordance with the underlying data distribution. To enable a wider range of augmentations, we explore negative data augmentation strategies (NDA) that intentionally create out-of-distribution samples. We show that such negative out-of-distribution samples provide information on the support of the data distribution, and can be leveraged for generative modeling and representation learning. We introduce a new GAN training objective where we use NDA as an additional source of synthetic data for the discriminator. We prove that under suitable conditions, optimizing the resulting objective still recovers the true data distribution but can directly bias the generator towards avoiding samples that lack the desired structure. Empirically, models trained with our method achieve improved conditional/unconditional image generation along with improved anomaly detection capabilities. Further, we incorporate the same negative data augmentation strategy in a contrastive learning framework for self-supervised representation learning on images and videos, achieving improved performance on downstream image classification, object detection, and action recognition tasks. These results suggest that prior knowledge on what does not constitute valid data is an effective form of weak supervision across a range of unsupervised learning tasks.
+
+# 1 INTRODUCTION
+
+Data augmentation strategies for synthesizing new data in a way that is consistent with an underlying task are extremely effective in both supervised and unsupervised learning (Oord et al., 2018; Zhang et al., 2016; Noroozi & Favaro, 2016; Asano et al., 2019). Because they operate at the level of samples, they can be combined with most learning algorithms. They allow for the incorporation of prior knowledge (inductive bias) about properties of typical samples from the underlying data distribution (Jaiswal et al., 2018; Antoniou et al., 2017), e.g., by leveraging invariances to produce additional “positive” examples of how a task should be solved.
+
+To enable users to specify an even wider range of inductive biases, we propose to leverage an alternative and complementary source of prior knowledge that specifies how a task should not be solved. We formalize this intuition by assuming access to a way of generating samples that are guaranteed to be out-of-support for the data distribution, which we call a Negative Data Augmentation (NDA). Intuitively, negative out-of-distribution (OOD) samples can be leveraged as a useful inductive bias because they provide information about the support of the data distribution to be learned by the model. For example, in a density estimation problem we can bias the model to avoid putting any probability mass in regions which we know a-priori should have zero probability. This can be an effective prior if the negative samples cover a sufficiently large area. The best NDA candidates are ones that expose common pitfalls of existing models, such as prioritizing local structure over global structure (Geirhos et al., 2018); this motivates us to consider known transformations from the literature that intentionally destroy the spatial coherence of an image (Noroozi & Favaro, 2016; DeVries & Taylor, 2017; Yun et al., 2019), such as Jigsaw transforms.
+
+Building on this intuition, we introduce a new GAN training objective where we use NDA as an additional source of fake data for the discriminator as shown in Fig. 1. Theoretically, we can show that if the NDA assumption is valid, optimizing this objective will still recover the data distribution in the limit of infinite data. However, in the finite data regime, there is a need to generalize beyond the empirical distribution (Zhao et al., 2018). By explicitly providing the discriminator with samples we want to avoid, we are able to bias the generator towards avoiding undesirable samples thus improving generation quality.
+
+Furthermore, we propose a way of leveraging NDA for unsupervised representation learning. We propose a new contrastive predictive coding (He et al., 2019; Han et al., 2019) (CPC) objective that encourages the distribution of representations corresponding to in-support data to become disjoint from that of NDA data. Empirically, we show that applying NDA with our proposed transformations (e.g., forcing the representation of normal and jigsaw images to be disjoint) improves performance in downstream tasks.
+
+With appropriately chosen NDA strategies, we obtain superior empirical performance on a variety of tasks, with almost no cost in computation. For generative modeling, models trained with NDA achieve better image generation, image translation and anomaly detection performance compared with the same model trained without NDA. Similar gains are observed on representation learning for images and videos over downstream tasks such as image classification, object detection and action recognition.
+
+![](images/839636c254c2fc5d8ab0051fdcb57f96bd382e429beee902224044cf0e07643c.jpg)  
+Figure 1: Negative Data Augmentation for GANs.
+
+These results suggest that NDA has much potential to improve a variety of self-supervised learning techniques.
+
+# 2 NEGATIVE DATA AUGMENTATION
+
+The input to most learning algorithms is a dataset of samples from an underlying data distribution $p _ { d a t a }$ . While $p _ { d a t a }$ is unknown, learning algorithms always rely on prior knowledge about its properties (inductive biases (Wolpert & Macready, 1997)), e.g., by using specific functional forms such as neural networks. Similarly, data augmentation strategies exploit known invariances of $p _ { d a t a }$ , such as the conditional label distribution being invariant to semantic-preserving transformations.
+
+While typical data augmentation strategies exploit prior knowledge about what is in support of $p _ { \mathrm { d a t a } }$ , in this paper, we propose to exploit prior knowledge about what is not in the support of $p _ { \mathrm { d a t a } }$ . This information is often available for common data modalities (e.g., natural images and videos) and is under-exploited by existing approaches. Specifically, we assume: (1) there exists an alternative distribution $\overline { { p } }$ such that its support is disjoint from that of $p _ { d a t a }$ ; and (2) access to a procedure to efficiently sample from $\overline { { p } }$ . We emphasize $\overline { { p } }$ need not be explicitly defined (e.g., through an explicit density) – it may be implicitly defined by a dataset or by a procedure that transforms samples from $p _ { \mathrm { d a t a } }$ into ones from $\overline { { p } }$ by suitably altering their structure.
+
+![](images/fbeba05957d446edd2c4a9643e443666ad3df3272642786ba7b17a7ec6414623.jpg)  
+Figure 2: Negative augmentations produce out-of-distribution samples lacking the typical structure of natural images; these negative samples can be used to inform a model on what it should not learn.
+
+Analogous to typical data augmentations, NDA strategies are by definition domain and task specific. In this paper, we focus on natural images and videos, and leave the application to other domains (such as natural language processing) as future work. How do we select a good NDA strategy? According to the manifold hypothesis (Fefferman et al., 2016), natural images lie on low-dimensional manifolds: $p _ { d a t a }$ is supported on a low-dimensional manifold of the ambient (pixel) space. This suggests that many negative data augmentation strategies exist. Indeed, sampling random noise is in most cases a valid NDA. However, while this prior is generic, it is not very informative, and this NDA will likely be ineffective for most learning problems. Intuitively, NDA is informative if its support is close (in a suitable metric) to that of $p _ { d a t a }$ , while being disjoint. These negative samples will provide information on the “boundary” of the support of $p _ { d a t a }$ , which we will show is helpful in several learning problems. In most of our tasks, the images are processed by convolutional neural networks (CNNs) that are good at processing local features but not necessarily global features (Geirhos et al., 2018). Therefore, we may consider NDA examples to be ones that preserve local features (“informative”) and break global features, so that it forces the CNNs to learn global features (by realizing NDAs are different from real data).
+
+Leveraging this intuition, we show several image transformations from the literature that can be viewed as generic NDAs over natural images in Figure 2, that we will use for generative modeling and representation learning in the following sections. Details about these transformations can be found in Appendix B.
+
+# 3 NDA FOR GENERATIVE ADVERSARIAL NETWORKS
+
+![](images/e708ddd2aa35f8aa54beac8042a0759af0d8cdfb0b1cfe59150666fca3040521.jpg)  
+Figure 3: Schematic overview of our NDA framework. Left: In the absence of NDA, the support of a generative model $P _ { \theta }$ (blue oval) learned from samples (green dots) may “over-generalize” and include samples from $\overline { { P _ { 1 } } }$ or $\overline { { P _ { 2 } } }$ . Right: With NDA, the learned distribution $P _ { \theta }$ becomes disjoint from NDA distributions $\overline { { P _ { 1 } } }$ and $\overline { { P _ { 2 } } }$ , thus pushing $P _ { \theta }$ closer to the true data distribution $p _ { d a t a }$ (green oval). As long as the prior is consistent, i.e. the supports of $\overline { { P _ { 1 } } }$ and $\overline { { P _ { 2 } } }$ are truly disjoint from $p _ { d a t a }$ , the best fit distribution in the infinite data regime does not change.
+
+In GANs, we are interested in learning a generative model $G _ { \theta }$ from samples drawn from some data distribution $p _ { \mathrm { d a t a } }$ (Goodfellow et al., 2014). GANs use a binary classifier, the so-called discriminator $D _ { \phi }$ , to distinguish real data from generated (fake) samples. The generator $G _ { \theta }$ is trained via the following mini-max objective that performs variational Jensen-Shannon divergence minimization:
+
+$$
+\begin{array} { c } { { \displaystyle \operatorname* { m i n } _ { G _ { \theta } \in \mathcal { P } ( \mathcal { X } ) } \operatorname* { m a x } _ { D _ { \phi } } L _ { \mathrm { J S } } ( G _ { \theta } , D _ { \phi } ) \quad \mathrm { w h e r e } } } \\ { { \displaystyle L _ { \mathrm { J S } } ( G _ { \theta } , D _ { \phi } ) = \mathbb { E } _ { \mathbf { x } \sim p _ { \mathrm { d a t a } } } \left[ \log ( D _ { \phi } ( \pmb { x } ) ) \right] + \mathbb { E } _ { \mathbf { x } \sim G _ { \theta } } \left[ \log ( 1 - D _ { \phi } ( \pmb { x } ) ) \right] } } \end{array}
+$$
+
+This is a special case to the more general variational $f$ -divergence minimization objective (Nowozin et al., 2016). The optimal $D _ { \phi }$ for any $G _ { \theta }$ is $( p _ { \mathrm { d a t a } } / G _ { \theta } ) / ( 1 + p _ { \mathrm { d a t a } } / G _ { \theta } )$ , so the discriminator can serve as a density ratio estimator between $p _ { \mathrm { d a t a } }$ and $G _ { \theta }$ .
+
+With sufficiently expressive models and infinite capacity, $G _ { \theta }$ will match $p _ { \mathrm { d a t a } }$ . In practice, however, we have access to finite datasets and limited model capacity. This means that the generator needs to generalize beyond the empirical distribution, which is challenging because the number of possible discrete distributions scale doubly exponentially w.r.t. to the data dimension. Hence, as studied in (Zhao et al., 2018), the role of the inductive bias is critical. For example, Zhao et al. (2018) report that when trained on images containing 2 objects only, GANs and other generative models can sometimes “generalize” by generating images with 1 or 3 objects (which were never seen in the training set). The generalization behavior – which may or may not be desirable – is determined by factors such as network architectures, hyperparameters, etc., and is difficult to characterize analytically.
+
+Here we propose to bias the learning process by directly specifying what the generator should not generate through NDA. We consider an adversarial game based on the following objective:
+
+$$
+\operatorname* { m i n } _ { G _ { \theta } \in \mathcal { P } ( \mathcal { X } ) } \operatorname* { m a x } _ { D _ { \phi } } L _ { \mathrm { J S } } ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { P } } , D _ { \phi } )
+$$
+
+where the negative samples are generated from a mixture of $G _ { \theta }$ (the generator distribution) and $\overline { { P } }$ (the NDA distribution); the mixture weights are controlled by the hyperparameter $\lambda$ . Intuitively, this can help addresses the above “over-generalization” issue, as we can directly provide supervision on what should not be generated and thus guide the support of $G _ { \theta }$ (see Figure 3) . For instance, in the object count example above, we can empirically prevent the model from generating images with an undesired number of objects (see Appendix Section A for experimental results on this task).
+
+In addition, the introduction of NDA samples will not affect the solution of the original GAN objective in the limit. In the following theorem, we show that given infinite training data and infinite capacity discriminators and generators, using NDA will not affect the optimal solution to the generator, i.e. the generator will still recover the true data distribution.
+
+Theorem 1. Let $\overline { { P } } \in \mathcal { P } ( \mathcal { X } )$ be any distribution over $\mathcal { X }$ with disjoint support than $p _ { \mathrm { d a t a } } ,$ , i.e., such that $\operatorname { s u p p } ( p _ { \mathrm { d a t a } } ) \cap \operatorname { s u p p } ( { \overline { { P } } } ) = \emptyset$ . Let $D _ { \phi } : \mathcal { X }  \mathbb { R }$ be the set of all discriminators over $\mathcal { X }$ , $f : \mathbb { R } _ { \geq 0 } \to \mathbb { R }$ be a convex, semi-continuous function such that $f ( 1 ) = { \dot { 0 } }$ , $f ^ { \star }$ be the convex conjugate of $f$ , ${ \overline { { f } } } ^ { \prime }$ its derivative, and $G _ { \theta }$ be a distribution with sample space $\mathcal { X }$ . Then $\forall \lambda \in ( 0 , 1 ]$ , we have:
+
+$$
+\underset { G _ { \theta } \in \mathcal { P } ( \mathcal { X } ) } { \arg \operatorname* { m i n } } \ \underset { D _ { \phi } : \mathcal { X } \to \mathbb { R } } { \operatorname* { m a x } } L _ { f } \big ( G _ { \theta } , D _ { \phi } \big ) = \underset { G _ { \theta } \in \mathcal { P } ( \mathcal { X } ) } { \arg \operatorname* { m i n } } \ \underset { D _ { \phi } : \mathcal { X } \to \mathbb { R } } { \operatorname* { m a x } } L _ { f } \big ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { P } } , D _ { \phi } \big ) = p _ { \mathrm { d a t a } }
+$$
+
+where $L _ { f } ( Q , D _ { \phi } ) = \mathbb { E } _ { \pmb { x } \sim p _ { \mathrm { d a t a } } } [ D _ { \phi } ( \pmb { x } ) ] - \mathbb { E } _ { \pmb { x } \sim Q } [ f ^ { \star } ( D _ { \phi } ( \pmb { x } ) ) ]$ is the objective for $f$ -GAN (Nowozin et al., 2016). However, the optimal discriminators are different for the two objectives:
+
+$$
+\begin{array} { c } { \underset { D _ { \phi } : \mathcal { X }  \mathbb { R } } { \arg \operatorname* { m a x } } L _ { f } ( G _ { \theta } , D _ { \phi } ) = f ^ { \prime } \big ( p _ { \mathrm { d a t a } } / G _ { \theta } \big ) } \\ { \underset { \mathrm { a r g } \operatorname* { m a x } } { \arg \operatorname* { m a x } } L _ { f } \big ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { P } } , D _ { \phi } \big ) = f ^ { \prime } \big ( p _ { \mathrm { d a t a } } / \big ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { P } } \big ) \big ) } \end{array}
+$$
+
+Proof. See Appendix C.
+
+The above theorem shows that in the limit of infinite data and computation, adding NDA changes the optimal discriminator solution but not the optimal generator. In practice, when dealing with finite data, existing regularization techniques such as weight decay and spectral normalization (Miyato et al., 2018) allow potentially many solutions that achieve the same objective value. The introduction of NDA samples allows us to filter out certain solutions by providing additional inductive bias through OOD samples. In fact, the optimal discriminator will reflect the density ratio between $p _ { \mathrm { d a t a } }$ and $\bar { \lambda ( \bar { G } _ { \theta } + ( 1 - \lambda ) \bar { P } }$ (see Eq.(6)), and its values will be higher for samples from $p _ { \mathrm { d a t a } }$ compared to those from $\overline { { P } }$ . As we will show in Section 5, a discriminator trained with this objective and suitable NDA performs better than relevant baselines for other downstream tasks such as anomaly detection.
+
+# 4 NDA FOR CONSTRASTIVE REPRESENTATION LEARNING
+
+Using a classifier to estimate a density ratio is useful not only for estimating $f$ -divergences (as in the previous section) but also for estimating mutual information between two random variables. In representation learning, mutual information (MI) maximization is often employed to learn compact yet useful representations of the data, allowing one to perform downstream tasks efficiently (Tishby & Zaslavsky, 2015; Nguyen et al., 2008; Poole et al., 2019b; Oord et al., 2018). Here, we show that NDA samples are also beneficial for representation learning.
+
+In contrastive representation learning (such as CPC (Oord et al., 2018)), the goal is to learn a mapping $h _ { \theta } ( \pmb { x } ) : \mathcal { X } \overset { \mathbf { \bar { \alpha } } } {  } \mathcal { P } ( \mathcal { Z } )$ that maps a datapoint $_ { x }$ to some distribution over the representation space $\mathcal { Z }$ ; once the network $h _ { \theta }$ is learned, representations are obtained by sampling from $z \sim h _ { \theta } ( x )$ . CPC maximizes the following objective:
+
+$$
+I _ { \mathrm { C P C } } ( h _ { \theta } , g _ { \phi } ) : = \mathbb { E } _ { x \sim p _ { \mathrm { d a t a } } ( x ) , z \sim h _ { \theta } ( x ) , \widehat { z } _ { i } \sim p _ { \theta } ( z ) } \left[ \log \frac { n g _ { \phi } ( x , z ) } { g _ { \phi } ( x , z ) + \sum _ { j = 1 } ^ { n - 1 } g _ { \phi } ( x , \widehat { z _ { j } } ) } \right]
+$$
+
+where $\begin{array} { r } { p _ { \theta } ( z ) = \int h _ { \theta } ( z | \boldsymbol x ) p _ { \mathrm { d a t a } } ( \boldsymbol x ) \mathrm d \boldsymbol x } \end{array}$ is the marginal distribution of the representations associated with $p _ { \mathrm { d a t a } }$ . Intuitively, the CPC objective involves an $n$ -class classification problem where $g _ { \phi }$ attempts to identify a matching pair (i.e. $( { \pmb x } , z ) )$ sampled from the joint distribution from the $( n - 1 )$ non-matching pairs (i.e. $( \pmb { x } , \widehat { \pmb { z } } _ { j } ) )$ sampled from the product of marginals distribution. Note that $g _ { \phi }$ bplays the role of a discriminator/critic, and is implicitly estimating a density ratio. As $n  \infty$ , the optimal $g _ { \phi }$ corresponds to an un-normalized density ratio between the joint distribution and the product of marginals, and the CPC objective matches its upper bound which is the mutual information between $X$ and $Z$ (Poole et al., 2019a; Song & Ermon, 2019). However, this objective is no longer able to control the representations for data that are out of support of $p _ { \mathrm { d a t a } }$ , so there is a risk that the representations are similar between $p _ { \mathrm { d a t a } }$ samples and out-of-distribution ones.
+
+To mitigate this issue, we propose to use NDA in the CPC objective, where we additionally introduce a batch of NDA samples, for each positive sample:
+
+$$
+\overline { { I _ { \mathrm { C P C } } } } ( h _ { \theta } , g _ { \phi } ) : = { \mathbb E } \left[ \log \frac { ( n + m ) g _ { \phi } ( { \pmb x } , { \pmb z } ) } { g _ { \phi } ( { \pmb x } , { \pmb z } ) + \sum _ { j = 1 } ^ { n - 1 } g _ { \phi } ( { \pmb x } , \widehat { { \pmb z _ { j } } } ) + \sum _ { k = 1 } ^ { m } g _ { \phi } ( { \pmb x } , \overline { { { \pmb z _ { k } } } } ) } \right]
+$$
+
+where the expectation is taken over $\pmb { x } \sim p _ { \mathrm { d a t a } } ( \pmb { x } ) , z \sim h _ { \theta } ( \pmb { x } ) , \widehat { z } _ { i } \sim p _ { \theta } ( z )$ , $\overline { { \mathbf { x } } } _ { k } \sim \overline { { p } }$ (NDA distribution), $\overline { { \boldsymbol { z } } } _ { k } \sim h _ { \theta } ( \overline { { \boldsymbol { x } } } _ { k } )$ for all $k \in [ m ]$ b. Here, the behavior of $h _ { \theta } ( { \pmb x } )$ when $_ { \textbf { \em x } }$ is NDA is optimized explicitly, allowing us to impose additional constraints to the NDA representations. This corresponds to a more challenging classification problem (compared to basic CPC) that encourages learning more informative representations. In the following theorem, we show that the proposed objective encourages the representations for NDA samples to become disjoint from the representations for $p _ { \mathrm { d a t a } }$ samples, i.e. NDA samples and $p _ { \mathrm { d a t a } }$ samples do not map to the same representation.
+
+Theorem 2. (Informal) The optimal solution to $h _ { \theta }$ in the NDA-CPC objective maps the representations of data samples and NDA samples to disjoint regions.
+
+Proof. See Appendix D for a detailed statement and proof.
+
+# 5 NDA-GAN EXPERIMENTS
+
+In this section we report experiments with different types of NDA for image generation. Additional details about the network architectures and hyperparameters can be found in Appendix K.
+
+Unconditional Image Generation. We conduct experiments on various datasets using the BigGAN architecture (Brock et al., 2018) for unconditional image generation1. We first explore various image transformations from the literature to evaluate which ones are effective as NDA. For each transformation, we evaluate its performance as NDA (training as in Eq. 3) and as a traditional data augmentation strategy, where we enlarge the training set by applying the transformation to real images (denoted PDA for positive data augmentation). Table 1 shows the FID scores for different types of transformations as PDA/NDA. The results suggest that transformations that spatially corrupt the image are strong NDA candidates. It can be seen that Random Horizontal Flip is not effective as an NDA; this is because flipping does not spatially corrupt the image but is rather a semantic preserving transformation, hence the NDA distribution $\overline { { P } }$ is not disjoint from $p _ { d a t a }$ . On the contrary, it is reasonable to assume that if an image is likely under $p _ { d a t a }$ , its flipped variant should also be likely. This is confirmed by the effectiveness of this strategy as PDA.
+
+![](images/5faf6e60edf5bdb943e0bd868b7e918aa4afdf8d393375b9df61dd8ae83ac6b8.jpg)  
+Figure 4: Histogram of difference in the discriminator output for a real image and it’s Jigsaw version.
+
+Table 1: FID scores over CIFAR-10 using different transformations as PDA and NDA in BigGAN. The results indicate that some transformations yield better results when used as NDA. The common feature of such transformations is they all spatially corrupt the images.   
+
+<table><tr><td rowspan="2">w/o Aug.</td><td colspan="2">Jigsaw</td><td colspan="2">Cutout</td><td colspan="2">Stitch</td><td colspan="2">Mixup</td><td colspan="2">Cutmix</td><td colspan="2">Random Crop</td><td colspan="2">Random Flip</td><td colspan="2">Gaussian</td></tr><tr><td>PDA</td><td>NDA</td><td>PDA</td><td>NDA</td><td>PDA</td><td>NDA</td><td>PDA</td><td>NDA</td><td>PDA</td><td>NDA</td><td>PDA</td><td>NDA</td><td>PDA</td><td>NDA</td><td>PDA</td><td>NDA</td></tr><tr><td>18.64</td><td>98.09</td><td>12.61</td><td>79.72</td><td>14.69</td><td>108.69</td><td>13.97</td><td>70.64</td><td>17.29</td><td>90.81</td><td>15.01</td><td>20.02</td><td>15.05</td><td>16.65</td><td>124.32</td><td>44.41 </td><td>18.72</td></tr></table>
+
+Table 2: Comparison of FID scores of different types of NDA for unconditional image generation on various datasets. The numbers in bracket represent the corresponding image resolution in pixels. Jigsaw consistently achieves the best or second best result.   
+
+<table><tr><td></td><td>BigGAN</td><td> Jigsaw</td><td>Stitching</td><td>Mixup</td><td>Cutout</td><td>Cutmix</td><td>CR-BigGAN</td></tr><tr><td>CIFAR-10 (32)</td><td>18.64</td><td>12.61</td><td>13.97</td><td>17.29</td><td>14.69</td><td>15.01</td><td>14.56</td></tr><tr><td>CIFAR-100 (32)</td><td>22.19</td><td>19.72</td><td>20.99</td><td>22.21</td><td>22.08</td><td>20.78</td><td>1</td></tr><tr><td>CelebA (64)</td><td>38.14</td><td>37.24</td><td>37.17</td><td>37.51</td><td>37.39</td><td>37.46</td><td>1</td></tr><tr><td>STL10 (32)</td><td>26.80</td><td>23.94</td><td>26.08</td><td>24.45</td><td>24.91</td><td>25.34</td><td>1</td></tr></table>
+
+We believe spatially corrupted negatives perform well as NDA in that they push the discriminator to focus on global features instead of local ones (e.g., texture). We confirm this by plotting the histogram of differences in the discriminator output for a real image and it’s Jigsaw version as shown in Fig. 4. We show that the difference is (a) centered close to zero for normal BigGAN (so without NDA training, the discriminator cannot distinguish real and Jigsaw samples well), and (b) centered at a positive number (logit 10) for our method (NDA-BigGAN). Following our findings, in our remaining experiments we use Jigsaw, Cutout, Stitch, Mixup and Cutmix as they achieve significant improvements when used as NDA for unconditional image generation on CIFAR-10.
+
+Table 2 shows the FID scores for BigGAN when trained with five types of negative data augmentation on four different benchmarks. Almost all the NDA augmentations improve the baseline across datasets. For all the datasets except CIFAR-100, $\lambda = 0 . 2 5$ , whereas for CIFAR-100 it is 0.5. We show the effect of $\lambda$ on CIFAR-10 performance in Appendix H. We additionally performed an experiment using a mixture of augmentation policy. The results (FID 16.24) were better than the baseline method (18.64) but not as good as using a single strategy.
+
+Conditional Image Generation. We also investigate the benefits of NDA in conditional image generation using BigGAN. The results are shown in Table 3. In this setting as well, NDA gives a significant boost over the baseline model. We again use $\lambda = 0 . 2 5$ for CIFAR-10 and $\lambda = 0 . 5$ for CIFAR-100. For both unconditional and conditional setups we find the Jigsaw and Stitching augmentations to achieve a better FID score than the other augmentations.
+
+Table 3: FID scores for conditional image generation using different NDAs.2   
+
+<table><tr><td></td><td>BigGAN</td><td> Jigsaw</td><td>Stitching</td><td>Mixup</td><td>Cutout</td><td>Cutmix</td><td>CR-BigGAN</td></tr><tr><td>C-10</td><td>11.51</td><td>9.42</td><td>9.47</td><td>13.87</td><td>10.52</td><td>10.3</td><td>11.48</td></tr><tr><td>C-100</td><td>15.04</td><td>14.12</td><td>13.90</td><td>15.27</td><td>14.21</td><td>13.99</td><td>1</td></tr></table>
+
+Image Translation. Next, we apply the NDA method to image translation. In particular, we use the Pix2Pix model (Isola et al., 2017) that can perform image-to-image translation using GANs provided paired training data. Here, the generator is conditioned on an image $\mathcal { T }$ , and the discriminator takes as input the concatenation of generated/real image and $\mathcal { T }$ . We use $\mathrm { P i x 2 P i x }$ for semantic segmentation on Cityscapes dataset (Cordts et al., 2016) (i.e. photos labels). Table 4 shows the quantitative gains obtained by using Jigsaw NDA3 while Figure 7 in Appendix F highlights the qualitative improvements. The NDA-Pix2Pix model avoids noisy segmentation on objects including buildings and trees.
+
+Table 4: Results on CityScapes, using per pixel accuracy $( { \mathrm { P p . } } )$ , per class accuracy (Pc.) and mean Intersection over Union (mIOU). We compare Pix2Pix and its NDA version.   
+
+<table><tr><td>Metric</td><td>Pp.</td><td>Pc.</td><td>mIOU</td></tr><tr><td>Pix2Pix (cGAN)</td><td>0.80</td><td>0.24</td><td>0.27</td></tr><tr><td>NDA (cGAN)</td><td>0.84</td><td>0.34</td><td>0.28</td></tr><tr><td>Pix2Pix (L1+cGAN)</td><td>0.72</td><td>0.23</td><td>0.18</td></tr><tr><td>NDA (L1+cGAN)</td><td>0.75</td><td>0.28</td><td>0.22</td></tr></table>
+
+Table 5: AUROC scores for different OOD datasets. OOD-1 contains different datasets, while OOD-2 contains the set of 19 different corruptions in CIFAR-10-C (Hendrycks & Dietterich, 2018) (the average score is reported).   
+
+<table><tr><td></td><td></td><td>BigGAN</td><td>Jigsaw</td><td>EBM</td></tr><tr><td rowspan="6">00D-1</td><td>DTD</td><td>0.70</td><td>0.69</td><td>0.48</td></tr><tr><td>SVHN</td><td>0.75</td><td>0.61</td><td>0.63</td></tr><tr><td>Places-365</td><td>0.35</td><td>0.58</td><td>0.68</td></tr><tr><td>TinyImageNet</td><td>0.40</td><td>0.62</td><td>0.67</td></tr><tr><td>CIFAR-100</td><td>0.63</td><td>0.64</td><td>0.50</td></tr><tr><td>Average</td><td>0.57</td><td>0.63</td><td>0.59</td></tr><tr><td>00D-2</td><td>CIFAR-10-C</td><td>0.56</td><td>0.63</td><td>0.60</td></tr></table>
+
+Anomaly Detection. As another added benefit of NDA for GANs, we utilize the output scores of the BigGAN discriminator for anomaly detection. We experiment with 2 different types of OOD datasets. The first set consists of SVHN (Netzer et al., 2011), DTD (Cimpoi et al., 2014), Places365 (Zhou et al., 2017), TinyImageNet, and CIFAR-100 as the OOD datapoints following the protocol in (Du & Mordatch, 2019; Hendrycks et al., 2018). We train BigGAN w/ and w/o Jigsaw NDA on the train set of CIFAR-10 and then use the output value of discriminator to classify the test set of CIFAR-10 (not anomalous) and different OOD datapoints (anomalous) as anomalous or not. We use the AUROC metric as proposed in (Hendrycks & Gimpel, 2016) to evaluate the anomaly detection performance. Table 5 compares the performance of NDA with a likelihood based model (Energy Based Models (EBM (Du & Mordatch, 2019)). Results show that Jigsaw NDA performs much better than baseline BigGAN and other generative models. We did not include other NDAs as Jigsaw achieved the best results.
+
+We consider the extreme corruptions in CIFAR-10-C (Hendrycks & Dietterich, 2018) as the second set of OOD datasets. It consists of 19 different corruptions, each having 5 different levels of severity. We only consider the corruption of highest severity for our experiment, as these constitute a significant shift from the true data distribution. Averaged over all the 19 different corruptions, the AUROC score for the normal BigGAN is 0.56, whereas the BigGAN trained with Jigsaw NDA achieves 0.63. The histogram of difference in discriminator’s output for clean and OOD samples are shown in Figure 8 in the appendix. High difference values imply that the Jigsaw NDA is better at distinguishing OOD samples than the normal BigGAN.
+
+# 6 REPRESENTATION LEARNING USING CONTRASTIVE LOSS AND NDA
+
+Unsupervised Learning on Images. In this section, we perform experiments on three benchmarks: (a) CIFAR10 (C10), (b) CIFAR100 (C100), and (c) ImageNet-100 (Deng et al., 2009) to show the benefits of NDA on representation learning with the contrastive loss function. In our experiments, we use the momentum contrast method (He et al., 2019), MoCo-V2, as it is currently the state-of-theart model on unsupervised learning on ImageNet. For C10 and C100, we train the MoCo-V2 model for unsupervised learning (w/ and w/o NDA) for 1000 epochs. On the other hand, for ImageNet-100, we train the MoCo-V2 model (w/ and w/o NDA) for 200 epochs. Additional hyperparameter details can be found in the appendix. To evaluate the representations, we train a linear classifier on the representations on the same dataset with labels. Table 6 shows the top-1 accuracy of the classifier. We find that across all the three datasets, different NDA approaches outperform MoCo-V2. While Cutout NDA performs the best for C10, the best performing NDA for C100 and ImageNet-100 are Jigsaw and Mixup respectively. Figure 9 compares the cosine distance of the representations learned w/ and w/o NDA (jigsaw) and shows that jigsaw and normal images are projected far apart from each other when trained using NDA whereas with original MoCo-v2 they are projected close to each other.
+
+Transfer Learning for Object Detection. We transfer the network pre-trained over ImageNet-100 for the task of Pascal-VOC object detection using a Faster R-CNN detector (C4 backbone) Ren et al. (2015). We fine-tune the network on Pascal VOC $2 0 0 7 + 2 0 1 2$ trainval set and test it on the 2007 test set. The baseline MoCo achieves 38.47 AP, 65.99 AP50, 38.81 AP75 whereas the MoCo trained with mixup NDA gets 38.72 AP, 66.23 AP50, 39.16 AP75 (an improvement of $\approx 0 . 3$ ).
+
+Table 6: Top-1 accuracy results on image recognition w/ and w/o NDA on MoCo-V2.   
+
+<table><tr><td></td><td>MoCo-V2</td><td>Jigsaw</td><td>Stitching</td><td>Cutout</td><td>Cutmix</td><td>Mixup</td></tr><tr><td>CIFAR-10</td><td>91.20</td><td>91.66</td><td>91.59</td><td>92.26</td><td>91.51</td><td>91.36</td></tr><tr><td>CIFAR-100</td><td>69.63</td><td>70.17</td><td>69.21</td><td>69.81</td><td>69.83</td><td>69.99</td></tr><tr><td>ImageNet-100</td><td>69.41</td><td>69.95</td><td>69.54</td><td>69.77</td><td>69.61</td><td>70.01</td></tr></table>
+
+Unsupervised Learning on Videos. In this section, we investigate the benefits of NDA in selfsupervised learning of spatio-temporal embeddings from video, suitable for human action recognition. We apply NDA to Dense Predictive Coding (Han et al., 2019), which is a single stream (RGB only) method for self-supervised representation learning on videos. For videos, we create NDA samples by performing the same transformation on all frames of the video (e.g. the same jigsaw permutation is applied to all the frames of a video). We evaluate the approach by first training the DPC model with NDA on a large-scale dataset (UCF101), and then evaluate the representations by training a supervised action classifier on UCF101 and HMDB51 datasets. As shown in Table 7, Jigsaw and Cutmix NDA improve downstream task accuracy on UCF-101 and HMDB-51, achieving new state-of-the-art performance among single stream (RGB only) methods for self-supervised representation learning (when pre-trained using UCF-101).
+
+Table 7: Top-1 accuracy results on action recognition in videos w/ and w/o NDA in DPC.   
+
+<table><tr><td></td><td>DPC</td><td>Jigsaw</td><td>Stitching</td><td>Cutout</td><td>Cutmix</td><td>Mixup</td></tr><tr><td>UCF-101 (Pre-trained on UCF-101)</td><td>61.35</td><td>64.54</td><td>66.07</td><td>64.52</td><td>63.52</td><td>63.65</td></tr><tr><td>HMDB51 (Pre-trained on UCF-101)</td><td>45.31</td><td>46.88</td><td>45.31</td><td>45.31</td><td>48.43</td><td>43.75</td></tr></table>
+
+# 7 RELATED WORK
+
+In several machine learning settings, negative samples are produced from a statistical generative model. Sung et al. (2019) aim to generate negative data using GANs for semi-supervised learning and novelty detection while we are concerned with efficiently creating negative data to improve generative models and self-supervised representation learning. Hanneke et al. (2018) also propose an alternative theoretical framework that relies on access to an oracle which classifies a sample as valid or not, but do not provide any practical implementation. Bose et al. (2018) use adversarial training to generate hard negatives that fool the discriminator for NLP tasks whereas we obtain NDA data from positive data to improve image generation and representation learning. Hou et al. (2018) use a GAN to learn the negative data distribution with the aim of classifying positive-unlabeled (PU) data whereas we do not have access to a mixture data but rather generate negatives by transforming the positive data.
+
+In contrastive unsupervised learning, common negative examples are ones that are assumed to be further than the positive samples semantically. Word2Vec (Mikolov et al., 2013) considers negative samples to be ones from a different context and CPC-based methods (Oord et al., 2018) such as momentum contrast (He et al., 2019), the negative samples are data augmentations from a different image. Our work considers a new aspect of “negative samples” that are neither generated from some model, nor samples from the data distribution. Instead, by applying negative data augmentation (NDA) to existing samples, we are able to incorporate useful inductive biases that might be difficult to capture otherwise (Zhao et al., 2018).
+
+# 8 CONCLUSION
+
+We proposed negative data augmentation as a method to incorporate prior knowledge through out-ofdistribution (OOD) samples. NDAs are complementary to traditional data augmentation strategies, which are typically focused on in-distribution samples. Using the NDA framework, we interpret existing image transformations (e.g., jigsaw) as producing OOD samples and develop new learning algorithms to leverage them. Owing to rigorous mathematical characterization of the NDA assumption, we are able to theoretically analyze their properties. As an example, we bias the generator of a GAN to avoid the support of negative samples, improving results on conditional/unconditional image generation tasks. Finally, we leverage NDA for unsupervised representation learning in images and videos. By integrating NDA into MoCo-v2 and DPC, we improve results on image and action recognition on CIFAR10, CIFAR100, ImageNet-100, UCF-101, and HMDB-51 datasets. Future work include exploring other augmentation strategies as well as NDAs for other modalities.
+
+# 9 ACKNOWLEDGEMENT
+
+The authors would like to thank Shengjia Zhao and Kristy Choi for reviewing an earlier draft of the paper. This research was supprted by NSF (#1651565, #1522054, #1733686), ONR (N00014-19-1- 2145), AFOSR (FA9550-19-1-0024), ARO, and Amazon AWS.
+
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+Ben Poole, Sherjil Ozair, Aaron van den Oord, Alexander A Alemi, and George Tucker. On variational bounds of mutual information. arXiv preprint arXiv:1905.06922, 2019a.
+
+Ben Poole, Sherjil Ozair, Aaron van den Oord, Alexander A Alemi, and George Tucker. On variational bounds of mutual information. arXiv preprint arXiv:1905.06922, May 2019b.
+
+Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In Advances in neural information processing systems, pp. 91–99, 2015.
+
+Jiaming Song and Stefano Ermon. Understanding the limitations of variational mutual information estimators. arXiv preprint arXiv:1910.06222, October 2019.
+
+Yi Lin Sung, Sung-Hsien Hsieh, Soo-Chang Pei, and Chun-Shien Lu. Difference-seeking generative adversarial network–unseen sample generation. In International Conference on Learning Representations, 2019.
+
+Naftali Tishby and Noga Zaslavsky. Deep learning and the information bottleneck principle. arXiv preprint arXiv:1503.02406, March 2015.
+
+David H Wolpert and William G Macready. No free lunch theorems for optimization. IEEE transactions on evolutionary computation, 1(1):67–82, 1997.
+
+Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo. Cutmix: Regularization strategy to train strong classifiers with localizable features. In Proceedings of the IEEE International Conference on Computer Vision, pp. 6023–6032, 2019.
+
+Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017.
+
+Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In European conference on computer vision, pp. 649–666. Springer, 2016.
+
+Shengjia Zhao, Hongyu Ren, Arianna Yuan, Jiaming Song, Noah Goodman, and Stefano Ermon. Bias and generalization in deep generative models: An empirical study. In Advances in Neural Information Processing Systems, pp. 10792–10801, 2018.
+
+Bolei Zhou, Agata Lapedriza, Aditya Khosla, Aude Oliva, and Antonio Torralba. Places: A 10 million image database for scene recognition. IEEE transactions on pattern analysis and machine intelligence, 40(6):1452–1464, 2017.
+
+# A NUMEROSITY CONTAINMENT
+
+Zhao et al. (2018) systematically investigate generalization in deep generative models using two different datasets: (a) a toy dataset where there are $k$ non-overlapping dots (with random color and location) in the image (see Figure 5a), and (b) the CLEVR dataset where ther are $k$ objects (with random shape, color, location, and size) in the images (see Figure 5b). They train a GAN model (WGAN-GP Gulrajani et al. (2017)) with (either) dataset and observe that the learned distribution does not produce the same number of objects as in the dataset it was trained on. The distribution of the numerosity in the generated images is centered at the numerosity from the dataset, with a slightbias towards over-estimation. For, example when trained on images with six dots, the generated images contain anywhere from two to eight dots (see Figure 6a). The observation is similar when trained on images with two CLEVR objects. The generated images contain anywhere from one to three dots (see Figure 6b).
+
+In order to remove samples with numerosity different from the train dataset, we use such samples as negative data during training. For example, while training on images with six dots we use images with four, five and seven dots as negative data for the GAN. The resulting distribution of the numerosity in the generated images is constrained to six. We observe similar behaviour when training a GAN with images containing two CLEVR objects as positive data and images with one or three objects as negative data.
+
+# B IMAGE TRANSFORMATIONS
+
+Given an image of size $H \times W$ , the different image transformations that we used are described below.
+
+![](images/f8d1a32535fc09525b3ba48a8a768d897ade95a7c0b9d558da68d1fba6da558c.jpg)  
+Figure 5: Toy Datasets used in Numerosity experiments.
+
+![](images/f6d8631628ecda40011e2c2f70a853c25074399ee7e2d1e5eee88e5e20491e78.jpg)  
+Figure 6: Left: Distribution over number of dots. The arrows are the number of dots the learning algorithm is trained on, and the solid line is the distribution over the number of dots the model generates. Right: Distribution over number of CLEVR objects the model generates. Generating CLEVR is harder so we explore only one, but the behaviour with NDA is similar to dots.
+
+Jigsaw- $K$ (Noroozi & Favaro, 2016) We partition the image into a grid of $K \times K$ patches of size $( H / K ) \times ( W / K )$ , indexed by $[ 1 , \ldots , K \times K ]$ . Then we shuffle the image patches according to a random permutation (different from the original order) to produce the NDA image. Empirically, we find $K = 2$ to work the best for Jigsaw- $K$ NDA.
+
+Stitching We stitch two equal-sized patches of two different images, either horizontally $( H / 2 \times W )$ or vertically $( H \times W / 2 )$ , chosen uniformly at random, to produce the NDA image.
+
+Cutout / Cutmix We select a random patch in the image with its height and width lying between one-third and one-half of the image height and width respectively. To construct NDA images, this patch is replaced with the mean pixel value of the patch (like cutout (DeVries & Taylor, 2017) with the only difference that they use zero-masking), or the pixel values of another image at the same location (cutmix (Yun et al., 2019)).
+
+Mixup- $\alpha$ NDA image is constructed from a linear interpolation between two images $_ { \textbf { \em x } }$ and $\textbf {  { y } }$ (Zhang et al., 2017), $\gamma { \pmb x } + ( 1 - \gamma ) { \pmb y }$ ; $\gamma \sim \mathrm { B e t a } ( \alpha , \alpha )$ . $\alpha$ is chosen so that the distribution has high density at 0.5.
+
+Other classes NDA images are sampled from other classes in the same dataset. See Appendix A.
+
+# C NDA FOR GANS
+
+Theorem 1. Let $\overline { { P } } \in \mathcal { P } ( \mathcal { X } )$ be any distribution over $\mathcal { X }$ with disjoint support than $p _ { \mathrm { d a t a } } ,$ , i.e., such that $\operatorname { s u p p } ( p _ { \mathrm { d a t a } } ) \cap \operatorname { s u p p } ( { \overline { { P } } } ) = \emptyset$ . Let $D _ { \phi } : \mathcal { X }  \mathbb { R }$ be the set of all discriminators over $\mathcal { X }$ ,
+
+$f : \mathbb { R } _ { \geq 0 } \to \mathbb { R }$ be a convex, semi-continuous function such that $f ( 1 ) = 0 { \mathrm { ; } }$ , $f ^ { \star }$ be the convex conjugate of $\cdot _ { f } , \bar { f } ^ { \prime }$ its derivative, and $G _ { \theta }$ be a distribution with sample space $\mathcal { X }$ . Then $\forall \lambda \in ( 0 , 1 ]$ , we have:
+
+$$
+\underset { G _ { \theta } \in \mathcal { P } ( \mathcal { X } ) } { \arg \operatorname* { m i n } } \ \underset { D _ { \phi } : \mathcal { X } \to \mathbb { R } } { \operatorname* { m a x } } L _ { f } \big ( G _ { \theta } , D _ { \phi } \big ) = \underset { G _ { \theta } \in \mathcal { P } ( \mathcal { X } ) } { \arg \operatorname* { m i n } } \ \underset { D _ { \phi } : \mathcal { X } \to \mathbb { R } } { \operatorname* { m a x } } L _ { f } \big ( \lambda G _ { \theta } + \big ( 1 - \lambda \big ) \overline { { P } } , D _ { \phi } \big ) = p _ { \mathrm { d a t a } }
+$$
+
+where $L _ { f } ( Q , D _ { \phi } ) = \mathbb { E } _ { \pmb { x } \sim p _ { \mathrm { d a t a } } } [ D _ { \phi } ( \pmb { x } ) ] - \mathbb { E } _ { \pmb { x } \sim Q } [ f ^ { \star } ( D _ { \phi } ( \pmb { x } ) ) ]$ is the objective for $f$ -GAN (Nowozin et al., 2016). However, the optimal discriminators are different for the two objectives:
+
+$$
+\begin{array} { c } { \underset { D _ { \phi } : \mathcal { X }  \mathbb { R } } { \arg \operatorname* { m a x } } L _ { f } ( G _ { \theta } , D _ { \phi } ) = f ^ { \prime } \big ( p _ { \mathrm { d a t a } } / G _ { \theta } \big ) } \\ { \underset { \mathrm { a r g } \operatorname* { m a x } } { \arg \operatorname* { m a x } } L _ { f } \big ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { P } } , D _ { \phi } \big ) = f ^ { \prime } \big ( p _ { \mathrm { d a t a } } / \big ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { P } } \big ) \big ) } \end{array}
+$$
+
+Proof. Let us use $p ( x ) , { \bar { p } } ( x ) , q ( x )$ to denote the density functions of $p _ { \mathrm { d a t a } } , \overline { { P } }$ and $G _ { \theta }$ respectively (and $P , { \overline { { P } } } , Q$ for the respective distributions). First, from Lemma 1 in Nguyen et al. (2008), we have that
+
+$$
+\begin{array} { c } { { \displaystyle \operatorname* { m a x } _ { D _ { \phi } : \mathcal { X } \to \mathbb { R } } L _ { f } ( G _ { \theta } , D _ { \phi } ) = D _ { f } ( P \| G _ { \theta } ) } } \\ { { \displaystyle \operatorname* { m a x } _ { D _ { \phi } : \mathcal { X } \to \mathbb { R } } L _ { f } ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { { P } } } , D _ { \phi } ) = D _ { f } ( P \| \lambda Q + ( 1 - \lambda ) \overline { { { P } } } ) } } \end{array}
+$$
+
+where $D _ { f }$ refers to the $f$ -divergence. Then, we have
+
+$$
+\begin{array} { r l } & { \quad B _ { 2 } ( F ) [ M \otimes { \bf { j } } + ( 1 - \lambda ) F ] } \\ & { = \int _ { \gamma } \langle \partial _ { t } ( x ) + ( 1 - \lambda ) \widetilde { \partial } _ { t } ( x ) \rangle \qquad \left( \frac { p ( x ) } { M ( \sigma ) } \right) [ 1 \quad \lambda ] \widehat { \partial } _ { t } ( x ) \rangle } \\ & { = \int _ { \gamma } \lambda \widehat { \mu } _ { 0 } ( x ) \Big { \Big [ \displaystyle \frac { p ( x ) } { M ( \sigma ) + ( 1 - \lambda ) \widehat { \partial } _ { t } ( x ) } \Big ] } + ( 1 - \lambda ) f ( 0 ) 1 } \\ & { \quad \mathrm { s . a . j ~ } \left( \int _ { \gamma } \psi ( \sigma ) \frac { p ( x ) } { M ( \sigma ) + ( 1 - \lambda ) \widehat { \partial } _ { t } ( x ) } \right) - ( 1 - \lambda ) f ( 0 ) } \\ & { \quad \mathrm { ~ } } \\ & { \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad }  \\ & { \quad \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \quad \mathrm { ~ } \quad \quad \mathrm { ~ } \quad \quad \mathrm { ~ } \quad \quad \mathrm { ~ } \quad p _ { G } ^ { ( 1 ) } [ M \otimes { \bf { j } } ] } \\ & { = \mathrm { ~ \mathcal { A } ~ } \left( \frac { 1 } { \lambda } \int _ { x } \lambda \widehat { \mu } ( x ) \right) \ \ \mathrm { ~ ( ~ \widehat { \mu } _ { 0 } ( x ) ) ~ } \ \mathrm { ~ ( ~ \widehat { \mu } _ { 0 } ( x ) ) ~ } \ \mathrm { ~ ( ~ \widehat { \mu } _ { 0 } ( x ) ) ~ } \ \mathrm { ~ } \mathrm { ~ } \quad \mathrm { ~ } \mathrm { ~ } \quad \mathrm { ~ } \mathrm { ~ } \quad \mathrm { ~ } \mathrm { ~ } \quad \mathrm { ~ } \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \mathrm { ~ } \quad \mathrm { ~ } \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \mathrm { ~ } \quad p _ { G } ^ { ( 2 ) } [ x ] } \\ &  \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \mathrm { ~ } \quad \ \end{array}
+$$
+
+where we use the fact that $f$ is convex with Jensen’s inequality in Eq.(11) and the fact that $p ( x ) { \overline { { p } } } ( x ) = 0 , \forall x \in \mathcal { X }$ in Eq.(12) since $P$ and $\overline { { P } }$ has disjoint support.
+
+We also have
+
+$$
+\begin{array} { l } { \displaystyle D _ { f } ( P | | \lambda P + ( 1 - \lambda ) \overline { { P } } ) = \int _ { \chi } \left( \lambda p ( x ) + ( 1 - \lambda ) \overline { { p } } ( x ) \right) f \left( \frac { p ( x ) } { \lambda p ( x ) + ( 1 - \lambda ) \overline { { p } } ( x ) } \right) } \\ { \displaystyle \qquad = \int _ { \chi } \left( \lambda p ( x ) \right) f \left( \frac { p ( x ) } { \lambda p ( x ) + ( 1 - \lambda ) \overline { { p } } ( x ) } \right) + ( 1 - \lambda ) f ( 0 ) } \\ { \displaystyle \qquad = \int _ { \chi } \left( \lambda p ( x ) \right) f \left( \frac { p ( x ) } { \lambda p ( x ) + 0 } \right) + ( 1 - \lambda ) f ( 0 ) } \\ { \displaystyle \qquad = \lambda f \left( \frac { 1 } { \lambda } \right) + ( 1 - \lambda ) f ( 0 ) } \end{array}
+$$
+
+Therefore, in order for the inequality in Equation 11 to be an equality, we must have that $q ( { \pmb x } ) =$ $p ( { \pmb x } )$ for all $\textbf { \textit { x } } \in \textbf { \textit { X } }$ . Therefore, the generator distribution recovers the data distribution at the equlibrium posed by the NDA-GAN objective, which is also the case for the original GAN objective.
+
+Moreover, from Lemma 1 in Nguyen et al. (2008), we have that:
+
+$$
+\underset { D _ { \phi } } { \arg \operatorname* { m a x } } L _ { f } ( Q , D _ { \phi } ) = f ^ { \prime } ( p _ { \mathrm { d a t a } } / Q )
+$$
+
+Therefore, by replacing $Q$ with $G _ { \theta }$ and $( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { { P } } } )$ , we have:
+
+$$
+\begin{array} { c } { \underset { D _ { \phi } : \mathcal { X }  \mathbb { R } } { \arg \operatorname* { m a x } } L _ { f } ( G _ { \theta } , D _ { \phi } ) = f ^ { \prime } \big ( p _ { \mathrm { d a t a } } / G _ { \theta } \big ) } \\ { \underset { \mathrm { a r g } \ \mathrm { m a x } } { \arg \operatorname* { m a x } } L _ { f } \big ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { P } } , D _ { \phi } \big ) = f ^ { \prime } \big ( p _ { \mathrm { d a t a } } / \big ( \lambda G _ { \theta } + ( 1 - \lambda ) \overline { { P } } \big ) \big ) } \end{array}
+$$
+
+which shows that the optimal discriminators are indeed different for the two objectives.
+
+# D NDA FOR CONTRASTIVE REPRESENTATION LEARNING
+
+We describe the detailed statement of Theorem 2 and proof as follows.
+
+Theorem 3. For some distribution $\overline { { p } }$ over $\mathcal { X }$ such that $\mathrm { s u p p } ( \overline { { p } } ) \cap \mathrm { s u p p } ( p _ { \mathrm { d a t a } } ) = \emptyset$ , and for any maximizer of the NDA-CPC objective
+
+$$
+\hat { h } \in \underset { h _ { \theta } } { \arg \operatorname* { m a x } } \underset { g _ { \phi } } { \operatorname* { m a x } } \overline { { I _ { \mathrm { C P C } } } } ( h _ { \theta } , g _ { \phi } )
+$$
+
+the representations of negative samples are disjoint from that of positive samples for $\hat { h }$ ; i.e., $\forall { \mathbf { } } x \in$ $\operatorname { s u p p } ( p _ { \mathrm { d a t a } } ) , \bar { \pmb { x } } \in \operatorname { s u p p } ( \overline { { p } } )$ ,
+
+$$
+\mathrm { s u p p } ( \hat { h } ( \bar { \pmb x } ) ) \cap \mathrm { s u p p } ( \hat { h } ( \pmb x ) ) = \emptyset
+$$
+
+Proof. We use a contradiction argument to establish the proof. For any representation mapping that maximizes the NDA-CPC objective,
+
+$$
+\hat { h } \in \underset { h _ { \theta } } { \arg \operatorname* { m a x } } \underset { g _ { \phi } } { \operatorname* { m a x } } \overline { { I _ { \mathrm { C P C } } } } ( h _ { \theta } , g _ { \phi } )
+$$
+
+suppose that the positive and NDA samples share some support, i.e., $\exists x \ \in \ \operatorname { s u p p } ( p _ { \mathrm { d a t a } } ) , { \bar { x } } \ \in$ $\operatorname { s u p p } ( { \overline { { p } } } )$ ,
+
+$$
+\operatorname { s u p p } ( { \hat { h } } ( { \bar { x } } ) ) \cap \operatorname { s u p p } ( { \hat { h } } ( \mathbf { x } ) ) \neq \emptyset
+$$
+
+We can always construct $\hat { h } ^ { \prime }$ that shares the same representation with $\hat { h }$ for $p _ { \mathrm { d a t a } }$ but have disjoint representations for NDA samples; i.e., $\forall x \in \mathrm { s u p p } ( \bar { p _ { \mathrm { d a t a } } } ) , \bar { x } \in \mathrm { s u p p } ( \bar { p } )$ , the following two statements are true:
+
+1. $\hat { h } ( \pmb { x } ) = \hat { h } ^ { \prime } ( \pmb { x } )$ ;$2 . \ \operatorname { s u p p } ( { \hat { h } } ^ { \prime } ( { \bar { x } } ) ) \cap \operatorname { s u p p } ( { \hat { h } } ^ { \prime } ( x ) ) = \varnothing .$
+
+Our goal is to prove that:
+
+$$
+\operatorname* { m a x } _ { g _ { \phi } } \overline { { I _ { \mathrm { C P C } } } } ( \hat { h } ^ { \prime } , g _ { \phi } ) > \operatorname* { m a x } _ { g _ { \phi } } \overline { { I _ { \mathrm { C P C } } } } ( \hat { h } , g _ { \phi } )
+$$
+
+which shows a contradiction.
+
+For ease of exposition, let us allow zero values for the output of $g$ , and define $0 / 0 = 0$ (in this case, if $g$ assigns zero to positive values, then the CPC objective becomes $- \infty$ , so it cannot be a maximizer to the objective).
+
+Let $\hat { g } \in \arg \operatorname* { m a x } _ { } \overline { { I _ { \mathrm { C P C } } } } ( \hat { h } , g _ { \phi } )$ be an optimal critic to the representation model $\hat { h _ { \theta } }$ . We then define a following critic function:
+
+$$
+\hat { g } ^ { \prime } ( \pmb { x } , z ) = \left\{ \begin{array} { l l } { \hat { g } ( \pmb { x } , z ) } & { \mathrm { i f } \ \exists \pmb { x } \in \mathrm { s u p p } ( p _ { \mathrm { d a t a } } ) \quad s . t . \quad z \in \mathrm { s u p p } ( \hat { h } ^ { \prime } ( \pmb { x } ) ) } \\ { 0 } & { \mathrm { o t h e r w i s e } } \end{array} \right.
+$$
+
+In other words, the critic assigns the same value for data-representation pairs over the support of $p _ { \mathrm { d a t a } }$ and zero otherwise. From the assumption over $\hat { h }$ $, \exists x \in \mathrm { s u p p } ( p _ { \mathrm { d a t a } } ) , \bar { x } \in \mathrm { s u p p } ( \bar { p } )$ , and $\overline { { z } } \in \operatorname { s u p p } ( \hat { h } ( \bar { \pmb x } ) )$ ,
+
+$$
+\overline { { z } } \in \operatorname { s u p p } ( \hat { h } ( \pmb { x } ) )
+$$
+
+so $( { \pmb x } , \overline { { { \pmb z } } } )$ can be sampled as a positive pair and $\hat { g } ( \pmb { x } , \overline { { \pmb { z } } } ) > 0$ .
+
+Therefore,
+
+$$
+\begin{array} { r l } & { \quad \underset { g _ { \phi } } { \operatorname* { m a x } } \bar { I } _ { \mathrm { C P C } } ( \hat { h } ^ { \prime } , g _ { \phi } ) \geq \bar { I } _ { \mathrm { C P C } } ( \hat { h } ^ { \prime } , \hat { g } ^ { \prime } ) } \\ & { = \mathbb { E } \Bigg [ \log \frac { ( n + m ) \hat { g } ^ { \prime } ( { \pmb x } , z ) } { \hat { g } ^ { \prime } ( { \pmb x } , z ) + \sum _ { j = 1 } ^ { n - 1 } \hat { g } ^ { \prime } ( { \pmb x } , \hat { z _ { j } } ) + \sum _ { k = 1 } ^ { m } \underbrace { \hat { g } ^ { \prime } ( { \pmb x } , \overline { { z _ { k } } } ) } _ { = 0 } } \Bigg ] } \\ & { \geq \mathbb { E } \Bigg [ \log \frac { ( n + m ) \hat { g } ( { \pmb x } , z ) } { \hat { g } ( { \pmb x } , z ) + \sum _ { j = 1 } ^ { n - 1 } \hat { g } ( { \pmb x } , \hat { z _ { j } } ) + \sum _ { k = 1 } ^ { m } \hat { g } ( { \pmb x } , \overline { { z _ { k } } } ) } \Bigg ] } \\ & { = \underset { g _ { \phi } } { \operatorname* { m a x } } \bar { I } _ { \mathrm { C P C } } ( \hat { h } , g _ { \phi } ) } \end{array}
+$$
+
+which proves the theorem via contradiction.
+
+(plug in definition for NDA-CPC)
+
+(Assumption that $\hat { g }$ is optimal critic)
+
+# E WHAT DOES THE THEORY OVER GANS ENTAIL?
+
+Our goal is to show that NDA GAN objectives are principled in the sense that with infinite computation, data, and modeling capacity, NDA GAN will recover the same optimal generator as a regular GAN. In other words, under these assumptions, NDA will not bias the solution in an undesirable way. We note that the NDA GAN objective is as stable as regular GAN in practice since both methods estimate a lower bound to the divergence with the discriminator, and then minimize that lower bound w.r.t. the generator. The estimated divergences are slightly different, but they have the same minimizer (which is the ground truth data distribution). Intuitively, while GAN and NDA GAN will give the same solution asymptotically, NDA GAN might get there faster (with less data) because it leverages a stronger prior over what the support should (not) be.
+
+# F PIX2PIX
+
+![](images/e3eded4084244a3aea0c49c7b05fc1f65f3c1fa3a81f2f96e289a39ec6f8b9e2.jpg)  
+Figure 7 highlights the qualitative improvements when we apply the NDA method to Pix2Pix model (Isola et al., 2017).   
+Figure 7: Qualitative results on Cityscapes.
+
+# G ANOMALY DETECTION
+
+Here, we show the histogram of difference in discriminator’s output for clean and OOD samples in Figure 8. High difference values imply that the Jigsaw NDA is better at distinguishing OOD samples than the normal BigGAN.
+
+![](images/21dfbc8b37ed4f88d3fc8bdb6b58d59414721fb82886cf57fe8e65aadd0c5e9b.jpg)  
+Figure 8: Histogram of D(clean) - D(corrupt) for 3 different corruptions.
+
+H EFFECT OF HYPERPARAMETER ON UNCONDITIONAL IMAGE GENERATION
+
+Here, we show the effect of $\lambda$ for unconditional image generation on CIFAR-10 dataset.
+
+Table 8: Effect of $\lambda$ on the FID score for unconditional image generation on CIFAR-10 using Jigsaw as NDA.   
+
+<table><tr><td>入</td><td>1.0</td><td>0.75</td><td>0.5</td><td>0.25</td><td>0.15</td></tr><tr><td>FID</td><td>18.64</td><td>16.61</td><td>14.95</td><td>12.61</td><td>13.01</td></tr></table>
+
+# I UNSUPERVISED LEARNING ON IMAGES
+
+![](images/a72123d97a87aede345b4de75427adc26031d72957b9992b5033a5af64e678e3.jpg)  
+Figure 9: Comparing the cosine distance of the representations learned with Jigsaw NDA and MocoV2 (shaded blue), and original Moco-V2 (white). With NDA, we project normal and its jigsaw image representations further away from each other than the one without NDA.
+
+# J DATASET PREPARATION FOR FID EVALUATION
+
+For dataset preparation, we follow the the following procedures: (a) CIFAR-10 contains 60K $3 2 \times 3 2$ images with 10 labels, out of which 50K are used for training and 10K are used for testing, (b)
+
+CIFAR-100 contains 60K $3 2 \times 3 2$ images with 100 labels, out of which 50K are used for training and 10K are used for testing, (c) CelebA contains 162,770 train images and 19,962 test images (we resize the images to $6 4 \times 6 4 \mathrm { p x } ,$ ), (d) STL-10 contains 100K (unlabeled) train images and 8K (labeled) test images (we resize the images to $3 2 \times 3 2 \mathrm { p x }$ ). In our experiments the FID is calculated on the test dataset. In particular, we use 10K generated images vs. 10K test images for CIFAR-10, 10K vs. 10K for CIFAR-100, 19,962 vs. 19,962 for CelebA, and 8K vs 8K for STL-10.
+
+# K HYPERPARAMETERS AND NETWORK ARCHITECTURE
+
+Generative Modeling. We use the same network architecture in BigGAN Brock et al. (2018) for our experiments. The code used for our experiments is based over the author’s PyTorch code. For CIFAR-10, CIFAR-100, and CelebA we train for 500 epochs whereas for STL-10 we train for 300 epochs. For all the datasets we use the following hyperparameters: batch-size $= 6 4$ , generator learning rate $= 2 \mathrm { e } { - } 4$ , discriminator learning rate $= 2 \mathrm { e } { - 4 }$ , discriminator update steps per generator update step $= 4$ . The best model was selected on the basis of FID scores on the test set (as explained above).
+
+Momentum Contrastive Learning. We use the official PyTorch implementation for our experiments. For CIFAR-10 and CIFAR-100, we perform unsupervised pre-training for 1000 epochs and supervised training (linear classifier) for 100 epochs. For Imagenet-100, we perform unsupervised pre-training for 200 epochs and supervised training (linear classifier) for 100 epochs. For CIFAR10 and CIFAR-100, we use the following hyperparameters during pre-training: batch-size $= 2 5 6$ , learning-date $= 0 . 3$ , temperature $= 0 . 0 7$ , feature dimensionality $= 2 0 4 8$ . For ImageNet-100 pretraining we have the following: batch-size $= 1 2 8$ , learning-date $= 0 . 0 1 5$ , temperature $= 0 . 2$ , feature dimensionality $= 1 2 8$ . During linear classification we use a batch size of 256 for all the datasets and learning rate of 10 for CIFAR-10, CIFAR-100, whereas for ImageNet-100 we use learning rate of 30.
+
+Dense Predictive Coding. We use the same network architecture and hyper-parameters in DPC Han et al. (2019) for our experiments and use the official PyTorch implementation. We perform self-supervised training on UCF-101 for 200 epochs and supervised training (action classifier) for 200 epochs on both UCF-101 and HMDB51 datasets.
+
+# L CODE
+
+The code to reproduce our experiments is given here.
+
+# M IMPLEMENTATION DETAILS
+
+For our experiment over GAN, we augment the batch of real samples with a negative augmentation of the same batch, and we treat the augmented images as fake images for the discriminator. Similarly, for the contrastive learning experiments, we consider negative augmentation of the query image batch as negatives for that batch.
+
+For all our experiments we used existing open-source models. For experiments over GAN, we use the open-source implementations of BigGAN and Pix2Pix models, and for contrastive learning, we use the open-source implementation of the MoCo-v2 model and Dense Predictive Coding. Hence, we did not explain in detail each of the models. Implementing NDA is quite simple as we only need to generate NDA samples from the images in a mini-batch which only takes several lines of code.
+
+# N DOES THE GAIN OF NDA FOR REPRESENTATION LEARNING COME FROM THE FACT THAT MORE NEGATIVE SAMPLES ARE USED?
+
+We perform the experiments over MoCo-v2 which maintains a queue of negative samples. The number of negatives is around 65,536. With our approach, we use the augmented versions of images in the same batch as negative. We transform both the key and query images to create NDA samples.
+
+Thus, the number of negatives for our approach is $^ { 6 5 , 5 3 6 + 2 }$ (one NDA sample created using query image and other using key image), only 0.00003051664 times more than the original number of negatives samples in MoCo-v2. Thus our experiments are comparable to the baseline MoCo-v2. In terms of computation, we need an additional forward pass in each batch to get the representations of the NDA samples. The normal MoCo-v2 requires 1.09 secs for entire forward computation, which includes forward pass through the network, momentum update of the key encoder and dot product between the positive and negative samples. With NDA, 1 forward computation requires 1.36 secs.
+
+O WHAT HAPPENS WHEN NEGATIVE DATA AUGMENTATIONS ARE NOISY?
+
+Regarding the performance of negative data augmentation, we perform 2 different experiments:
+
+a) When the noise is low - When using jigsaw as our NDA strategy with a $2 \times 2$ grid, one out of the 24 permutations will be the original image. We find that when this special permutation is not removed, or there is $4 \%$ “noise”, the FID score is 12.61, but when it is removed the FID score is 12.59. So, we find that when the noise is low, the performance of our approach is not greatly affected and is robust in such scenarios.
+
+b) When the noise is large - We use random vertical flipping as our NDA strategy, where with $50 \%$ probability the image is vertically flipped during NDA. In this case, the “noise” is large, as $50 \%$ of the time, the negative sample is actually the original image. We contrast this with the “noisefree” NDA strategy where the NDA image is always vertically flipped. We find that for the random vertical flipping NDA, the FID score of BigGAN is 15.84, whereas, with vertical flipping NDA, the FID score of BigGAN is 14.74. So performance degrades with larger amounts of noise.

parse/train/S1xjdoC9Fm/S1xjdoC9Fm.md

@@ -0,0 +1,284 @@
+# OFFLINE DEEP MODELS CALIBRATION WITH BAYESIAN NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In this work the authors show that Bayesian Neural Networks (BNNs) can be efficiently applied to calibrate state-of-the-art Deep Neural Networks (DNN). Our approach acts offline, i.e., it is decoupled from the training of the DNN to be calibrated. This offline approach allow us to apply our BNN calibration to any model regardless of the limitations that the model may present during training. Note that this offline setting is also appropriate in order to deal with privacy concerns of DNN training data or implementation, among others. We show that our approach clearly outperforms other simple maximum likelihood based solutions that have recently shown very good performance, as temperature scaling (Guo et al., 2017). As an example, we reduce the Expected Calibration Error $( \mathrm { E C E } \% )$ from 0.52 to 0.24 on CIFAR-10 and from 4.28 to 2.46 on CIFAR-100 on two Wide ResNet with $9 6 . 1 3 \%$ and $8 0 . 3 9 \%$ accuracy respectively, which are among the best results published for these tasks. Moreover, we show that our approach improves the performance of online methods directly applied to the DNN, e.g. Gaussian processes or Bayesian Convolutional Neural Networks. Finally, this decoupled approach allows us to apply any further improvement to the BNN without considering the computational restrictions imposed by the deep model. In this sense, this offline setting is a practical application where BNNs can be considered, which is one of the main criticisms to these techniques. In terms of reproducibility, we provide all the implementation details in https://github.com/2019submission/bnn.2019.
+
+# 1 INTRODUCTION
+
+Deep Neural Networks (DNNs) have achieved state of art performance in many task such as Image Recognition (Huang et al., 2017; Szegedy et al., 2017; Zagoruyko & Komodakis, 2016), language modeling (Mikolov et al., 2013a;b), machine translation (Vaswani et al., 2017) or speech (Hinton et al., 2012). For that reason, neural networks are now used in many applications. However, this state-of-theart performance is measured in terms of accuracy, but there are many tasks in which the probabilistic information must be also reliable. For example, a probabilistic classifier can be incorporated into a more complex model considering multiple sources of information, by the use of e.g., probabilistic graphical models (Koller & Friedman, 2009), or by combining neural networks with language models in natural language processing tasks (Gulcehre et al., 2017). In addition, probabilistic outputs of classifiers have proven to be useful in many areas apart from classical machine learning tasks, such as language recognition (Brümmer & van Leeuwen, 2006), language models for speech recognition (Tüske et al., 2018) or medical diagnosis (Caruana et al., 2015).
+
+In Bayesian statistics, the reliability of probabilities is measured by their calibration. As a consequence, the machine learning community has been exploring methods to calibrate the output of classifiers to achieve the many beneficial properties of well-calibrated probabilities (Zadrozny & Elkan, 2002a; Cohen & Goldszmidt, 2004; Niculescu-Mizil & Caruana, 2005). Nowadays, there is an increasing interest in obtaining reliable probabilities in the deep learning community. In the past, neural networks trained with a cross-entropy criterion tended to present relatively good calibration. However, a relevant recent work in Guo et al. (2017) has evidenced that modern state-of-the-art neural networks are badly calibrated in general. Moreover, the same work shows that calibration can be dramatically improved by very simple maximum-likelihood parametric techniques, among which Temperature Scaling (TS) is highlighted as the preferred choice, due to its extreme simplicity, very good behavior in general and computational efficiency. In fact, TS outperforms more complex techniques in most cases, leading to the conclusion that good calibration can be better achieved with simpler techniques. This conclusion follows the hypothesis that the space configured by the outputs of a deep model is relatively simple, and therefore good performance, measured as Expected Calibration Error (ECE), can be achieved with very simple models. In fact, TS is a technique that performs nicely in complex multiclass tasks (Guo et al., 2017).
+
+![](images/06621e565922dc9459a398e6539aab8fce8f5c8812f3dd555f33807197667e37.jpg)  
+ResNet-101: $9 3 . 4 6 \%$ accuracy (CIFAR10)   
+Figure 1: Reliability diagrams (Guo et al., 2017) for two networks trained on CIFAR-10 and CIFAR100. The red line represents perfect calibration. We plot the Expected Calibration Error (ECE $\%$ ) for 15 bins (see section 5 for a description). The lower the better.
+
+In general, there are two main approaches to reduce overconfidence: implicit or online and explicit or offline. An implicit method aims at obtaining calibrated distributions directly at the output of the model, while explicit methods post-process the output of the model to be calibrated. In this paper we propose an offline method based on a Bayesian Neural Networks (BNNs) to obtain calibrated probabilities, see figure 2. We use BNNs as we aim at being benefited from two key properties of Bayesian statistics and neural networks: the high expressiveness of neural network models and the capabilities of the Bayesian statistics to model the uncertainty. We assume the hypothesis that as long as the uncertainty is correctly modelled, we can use high expressive function approximators for the task of calibration. These high expressive models are required since we assume that the calibration space is not simple. However, the generalization capability of these models are achieved through proper uncertainty consideration. Figure 1 shows reliability diagrams comparing our BNN method to TS.
+
+This work is organized as follows. We first provide an insight on why Bayesian Statistics and neural networks are suitable to adjust confidence in output probabilities. We then describe our offline calibration approach based on BNNs. We end up comparing our method to TS and reporting clear performance improvements. Finally, we discuss our approach against recent published techniques, enumerate some beneficial properties and propose possible improvements over this contribution.
+
+# 2 BAYESIAN MODELLING AND CALIBRATION
+
+In a classification scenario, calibration can be interpreted as the agreement between the probabilities of a class assigned by a model to a set of samples, and the proportion of those samples where that class is actually the true one.
+
+One way of achieving calibration is to reliably modelling the probabilistic distributions of the data from the classes involved. This is the main strength of Bayesian models, which manage uncertainty properly, in contrast to point-estimate approaches (i.e. Maximum Likelihood or Maximum Posterior). In the former, the uncertainty is incorporated by taking an average of all the likelihood models under the posterior distribution on the parameters, given an observed set of data:
+
+$$
+p ( t | x ) = \mathbb { E } _ { p ( \theta | \mathcal { O } ) } \{ p ( t | x , \theta ) \} ,
+$$
+
+where $\theta$ are model parameters, $x$ represent a sample for which we want to predict a label $t$ and $\mathcal { O } = \{ ( x _ { i } , t _ { i } ) \} _ { i = 1 } ^ { N }$ is the set of observed samples1. In Bayesian approaches, it is indeed the observed data what model how representative a likelihood model is for a particular task, and thus how it influences the predictions. On the other hand, in point-estimate models all the decision is based on a choice of the parameter once the model is trained.
+
+For instance, consider the case of a MAP network (e.g., a typical deep convolutional model trained with cross-entropy loss and $L _ { 2 }$ regularization). This model explains the data based on a point-estimate training, i.e., by representing what is more likely to appear. Nowadays, this gives outstanding accuracy in classification tasks, but it is easy to train an over/under-confident model, i.e., the one that outputs too extreme probabilities, even for unfavourable cases like e.g. when data that has conditions not unseen in the training set, or not explained by the expressiveness of the model itself. This could happen if the true distribution does not lie in the family of parametric models $p ( t | x , \theta )$ . We will refer to both conditions as mismatch. This is very harmful for the calibration, because in those mismatch cases, the model should yield more moderate probabilities, otherwise the classification errors will be more catastrophic. In other words, in tasks where calibration matters, a classification error has unequal consequences if the probabilities are moderate or extreme. Thus, for example, if there exists such mismatching conditions, what is likely to happen, a point estimate will not represent the data (i.e. the probability assigned) in the way it should, possibly leading to over-or under-estimation of probabilities. On the other hand, in the Bayesian framework, a posterior distribution on the parameters could consider networks explaining these mismatching conditions. By averaging different contributions, the model ideally moderates probabilities on unfavourable data. For the sake of illustration, we provide a simple example in appendix A. We encourage unfamiliar readers with Bayesian statistics and calibration to read this appendix.
+
+Regarding the accuracy of Bayesian models, in Bayesian decision theory, if the model used to generate the data is known, the optimal error can be achieved, which also means optimal accuracy if all decision costs are equal. This suggests that a proper way of assigning probabilities is also paramount for the accuracy. Thus, by choosing appropriate densities for the class-conditional probabilities $p ( t | x )$ , such as factorized multivariate Bernoulli distributions like in point estimate models, the accuracy will also be correctly modelled. Moreover, it is well known that Bayesian models asymptotically tend to point estimate models as the data increases in size, see Duda et al. (2000) section 3.4. Therefore, it is expected that good accuracy performance achieved by point-estimate models should be also achievable by Bayesian models, at least for sufficient amounts of data.
+
+# 3 OFFLINE CALIBRATION WITH BAYESIAN NEURAL NETWORKS
+
+The architecture proposed in this work is shown in Figure 2. We apply a BNN to the potentially uncalibrated outputs of a DNN model. The goal is to improve the calibration minimizing the accuracy degradation of the original DNN model. Our approach works offline, meaning that given a DNN model, we project the available data to the space defined by the outputs of the model in the form of logit, i.e. pre-softmax values. This new representation is then the input to our BNN. The BNN aims at taking this uncalibrated so-called logit space and project it to a new calibrated one. The same procedure is applied for the TS method.
+
+![](images/7d26d0fed45dcf9dab9259ca9ea552334e4f35ca5924d1e83b3b311d694ee0c9.jpg)  
+Figure 2: Exemplary representation of the architecture of our proposed model. On the left, an expensive pretrained DNN on ImageNet is trained on a specific dataset (transfer learning). Then, the (uncalibrated) output of such DNN is the input to the BNN calibration stage. This stage is trained by the maximization of the Expected Lower Bound (ELBO) and predictions are done using Monte Carlo integration. The inputs and outputs of the Bayesian stage have same dimension (given by the number of classes), and lie in the so-called logit space. Orange Gaussians on each arrow represent variational distributions on parameters. We do not plot all the arrows for clarity. This Bayesian stage is independent of the previous one as we only require access to the logits of an already trained model.
+
+This off-line set-up presents clear advantages. First, the approach is efficient, since the DNN model does not need to be re-trained for re-calibration. Furthermore, we can incorporate future improvements to the BNN calibration stage without affecting the previous DNN model. Second, our proposal is extremely flexible, as the proposed BNN calibration stage will work with any probabilistic model, even implicitly-calibrated models, with potential benefits on calibration performance. And finally, our proposal preserves privacy, because there is no need to access the original data used to train the DNN model, or even the DNN topology: to be trained, the BNN only needs the data projected to the outputs of the DNN on the logit space, and the original targets $t$ .
+
+For these Bayesian approaches one has to compute the posterior distribution $p ( \theta | \mathcal { O } )$ and the expectation in equation 1. Using configurations that yield to analytic solutions to both problems result in low-expression models for this task. We solve this problem by choosing Neural Networks to parameterize the likelihood $p ( t | x , \theta )$ of our BNN, and therefore taking advantage of the high expressiveness of these models. In this case, several intractabilities arise that must be addressed.
+
+In this work we approximate expectations on equation 1 with Monte Carlo integration, and the posterior is approximated by a variational distribution in terms of the Kulback-Lieber Divergence. This is done by the maximization the Evidence Lower Bound (ELBO). We use stochastic optimization based on the reparameterization trick (Kingma & Welling, 2014; Rezende et al., 2014) to approximate the expectation under the variational distribution, and also mini-batch stochastic optimization for expectations under data distribution. Thus, our training criteria is given by:
+
+$$
+E L B O = \frac { 1 } { N } \sum _ { ( x , t ) \sim p _ { d } ( x , t ) } \Bigl [ \frac { 1 } { K } \sum _ { \theta \sim q _ { \phi } ( \theta ) } [ \log p ( t | x , \theta ) ] - \beta \cdot D _ { K L } \{ q _ { \phi } ( \theta ) / / p ( \theta ) \} \Bigr ] ,
+$$
+
+where we introduce $\beta$ following Blundell et al. (2015).
+
+Our variational distribution is a factorized Gaussian distribution and for that reason we refer to our BNN approach as a basic approach, as we do not incorporate any improvement recently proposed for BNN models, such as normalizing flows, local reparameterization, and so on. Also, we use a standard normal density for the prior. We choose this simple approximation to demonstrate our starting hypothesis: that BNNs can be applied to improve the calibration of state-of-the-art DNN in a very efficient way. Our basic BNN model can be improved by using normalizing flows (Rezende & Mohamed, 2015; Kingma et al., 2016; Huang et al., 2018; van den Berg et al., 2018), auxiliary variables (Agakov & Barber, 2004; Ranganath et al., 2016; Maaløe et al., 2016), local reparameterization (Kingma et al., 2015), combinations of all of them (Louizos & Welling, 2017) or deterministic models (Wu et al., 2018). Also, Cremer et al. (2018) has recently pointed out that amortized inference leads to an additional gap in the bound, in addition to the $D _ { K L }$ gap between the true and variational posteriors; and we can also use other proposals to mitigate this effect (Shu et al., 2018; Kim et al., 2018).
+
+Finally, class predictions are assigned by first computing the logits of a test sample using the first DNN stage $\boldsymbol { B }$ , and then using them as inputs of our BNN to yield calibrated probabilities, which can be formalized as follows:
+
+$$
+\begin{array} { c } { l = { } } \\ { { } } \\ { { p ( t | x , \mathcal { O } ) \approx \displaystyle \frac { 1 } { M } \sum _ { i = 1 } ^ { M } p ( t | l , \theta _ { i } ) ; \theta _ { i } \sim q _ { \phi } ( \theta ) , } } \end{array}
+$$
+
+where $M$ is a value chosen on validation. Note that our proposed BNN is not as efficient as TS for calibration. However, the contributions to the weighted average can be fully parallelized computationally, as predictions do not depend on each other. Thus, we can use modern GPU libraries such as CUBLAS and batch-based operations to dramatically speed-up the process.
+
+# 4 RELATED WORK
+
+To our knowledge, TS (Guo et al., 2017) has been consistently reported as the best technique to improve calibration over a list of classical ways of improving calibration, such as histogram binning (Zadrozny & Elkan, 2001), isotonic regression (Zadrozny & Elkan, 2002b), Platt scaling (Platt, 1999) or Bayesian binning into quantiles (Naeini et al., 2015) among others. For a recent description and performance comparison with modern neural networks, see Guo et al. (2017). On the other hand, there are several works that study overconfident predictions and model uncertainty in different contexts, but without reporting an explicit measurement of calibration performance in deep neural models. For instance, Gal & Ghahramani (2015) connect Bernoulli dropout with BNNs, and Gal & Ghahramani (2016) links Gaussian processes with classical dropout regularized networks, showing how uncertainty estimates can be obtained from this networks. In the latter, the authors state that these Bayesian outputs are not calibrated. In Pereyra et al. (2017), an entropy term is added to the log-likelihood to relax overconfidence. Lakshminarayanan et al. (2017) propose training network ensembles with adversarial noise samples to output confident scores. Chen et al. (2018) propose a model that uses probes of the individual layers of the neural network classifier to create a confidence score for the network output. DeVries & Taylor (2018) train a second output obtained from the penultimate layer of the classifier to be confident by interpolation of the softmax output and the true value, scaled by this score. Lee et al. (2018) proposed a generative approach for detecting outof-distribution samples but evaluates calibration performance comparing their method with normal cross-entropy minimization, using TS as the calibration technique.
+
+On the side of BNNs, Kingma et al. (2015) formalize Gaussian dropout as a Bayesian approach. In Louizos & Welling (2017), novel BNNs are proposed, mixing inverse autoregressive flows (Kingma et al., 2016), auxiliary variables (Maaløe et al., 2016) and local reparameretization (Kingma et al., 2015). None of these approaches measure calibration explicitly on deep neural models, as we do. For instance, Louizos & Welling (2017) and Lakshminarayanan et al. (2017) evaluate uncertainty by training on one dataset and use it on another, expecting a maximum entropy output distribution. More recently, Zhang et al. (2018) propose an scalable inference algorithm that is also asymptotically accurate as MCMC algorithms.
+
+We compare our proposed BNN approach against TS, as to our knowledge it is the state of the art in calibration tasks involving deep neural models, according to the reviewed previous work. TS widely improves calibration without affecting the accuracy, and can be efficiently applied to any model. Formally, given $\mathcal { O }$ , TS maximizes the log-likelihood of the conditional distribution $p ( t | x / T )$ w.r.t. the parameter $T$ . In this case, $x$ also represents the output logits of the deep convolutional model.
+
+In Section 6, we will comparatively discuss the properties and results of our proposal with other works recently found in the literature for calibration on deep models, some of them based on implicit and explicit models.
+
+# 5 EXPERIMENTS
+
+We demonstrate calibration performance on several computer vision models on several datasets. We have used CIFAR10 and CIFAR100 databases (Krizhevsky et al., a;b); SVHN (Netzer et al., 2011); and a GENDER recognition task (Eidinger et al., 2014). We use a validation set randomly taken from the training set with 5000 samples for CIFAR10 and CIFAR100, 10000 samples for SVHN and 4005 for GENDER. This validation set is used to train the TS parameter and to choose the number of Monte Carlo samples, $M$ in equation 2. We report results for the best model on validation for all the tested configurations. We optimize the ELBO using adam optimization (Kingma & Ba, 2014), since it performed better than stochastic gradient descent on our previous experiments. We used $\beta = 0 . 1$ from the set $\lbrace 1 , 0 . 1 , 0 . 0 1 \rbrace$ as it behaves better on validation.
+
+In order to compare our experiments with uncalibrated and TS calibrated probabilities, we used an unbiased estimator of the Expected Calibration Error (ECE) computed as in Guo et al. (2017), with 15 bins. The ECE measures the expected value of the difference between accuracy and confidence:
+
+$$
+\mathrm { E C E } = \sum _ { i = 1 } ^ { 1 5 } \frac { | B _ { i } | } { N } | \mathrm { a c c } ( B _ { i } ) - \mathrm { c o n f } ( B _ { i } ) |
+$$
+
+where $N$ is the number of total samples; $B _ { i }$ represents the set of samples whose predictions $t$ confidence lie in bin $i$ ; $\mathrm { c o n f } ( B _ { i } )$ is the average confidence and $\operatorname { a c c } ( B _ { i } )$ is the accuracy of that bin. We also report the accuracy of our models. This is because a classifier can be perfectly calibrated, but useless from a classification point of view.
+
+Note that we evaluate our proposed method on several state of art configurations of computer vision neural networks over the mentioned datasets: Wide Residual Networks (Zagoruyko & Komodakis, 2016), Residual Networks (He et al., 2016b), Densely Connected Neural Networks (Huang et al., 2017), Pre-Activation Residual Networks (He et al., 2016a), Dual Path Networks (Chen et al., 2017), VGG (Simonyan & Zisserman, 2014) and ResNext (Xie et al., 2017). The results reported in this work are obtained from some pretrained neural networks.
+
+# 5.1 RESULTS
+
+Table 1 shows the results for the different datasets, original DNNs and calibration approaches. We presents results in $\%$ , i.e. multiplying by 100 the result obtained in equation 4. The most important point is that calibration is improved by a wide margin in every model except for two models in CIFAR100 and one model in SVHN. Table 1 shows that in average our proposed calibration method outperforms TS with an insignificant accuracy loss. This means that high expressive models can cope with the calibration task as long as uncertainty is correctly modelled. Therefore, we propose an alternative hypothesis to the one given in Guo et al. (2017) where the authors argued that the calibration space is simple. We argue that if highly complex models outperform simple ones is because the distribution of the calibration space is also complex but the low dimensionality of the logit space makes high expressive models overfit.
+
+We realized that more expressive models are needed by more complex tasks, like CIFAR100. For instance, ResNet-18 GENDER uses BNNs of two layers with two neurons per layer, while WideResNet 40x10 on CIFAR100 uses two layers of 2000 neurons. This reflects that, when dimensionality increases, more expressiveness is needed. Another important point observed in SVHN (see Densenet169, ResNet-50 and WideResNet 16x8) is that TS has degraded calibration by a factor of three in the worst case. In general BNNs do not degrade the calibration.
+
+One drawback of our basic approach is that in some cases we obtain slight accuracy degradation. Accuracy degradation is more relevant only for CIFAR100, however our BNN method reduces ECE15 by a factor of two in some experiments in this task. Moreover, in some cases we are able to improve both, accuracy and calibration, see WideResNet $4 0 \mathrm { x } 1 0$ for CIFAR10 or ResNet-18 for GENDER dataset. Thus, we cannot conclude that BNNs are calibrating at the cost of losing accuracy. This motivates us towards further research on this accuracy degradation, as we expect to solve it with more sophisticated approximations in future work. Possible hypothesis for this degradation are that either the gap between the variational and the true posterior is still large, the variance of the ELBO estimator is large and does not allow us to converge to a better optimal, or the expressiveness of the likelihood model is not enough to deal with the particular logit space distribution. We found that for more complex logit space distribution (100 dimensional in CIFAR100) we could get better accuracy and better ECE increasing the expressiveness of the model. On the other hand in simpler logit space distribution (2 dimensional in GENDER) we found that the expressiveness must be reduced.
+
+Table 1: ECE $1 5 ( \% )$ and Accuracy $( \% )$ comparing model uncalibrated, calibrated with TS and with BNN   
+
+<table><tr><td rowspan="2"></td><td colspan="2">uncalibrated</td><td colspan="2">CIFAR10 Temp Scal</td><td colspan="2">BNN</td></tr><tr><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td></tr><tr><td>WideResNet28x10</td><td>96.13</td><td>1.835</td><td>96.13</td><td>0.518</td><td>96.08</td><td>0.243</td></tr><tr><td>DenseNet 121</td><td>95.49</td><td>2.643</td><td>95.49</td><td>1.011</td><td>95.26</td><td>0.600</td></tr><tr><td>DenseNet 169</td><td>95.49</td><td>2.664</td><td>95.49</td><td>0.826</td><td>95.29</td><td>0.511</td></tr><tr><td>Dual Path Network 92</td><td>95.18</td><td>2.995</td><td>95.18</td><td>1.072</td><td>95.03</td><td>0.730</td></tr><tr><td>ResNet 101</td><td>93.46</td><td>4.268</td><td>93.46</td><td>1.196</td><td>93.38</td><td>0.776</td></tr><tr><td>VGG 19</td><td>93.68</td><td>4.412</td><td>93.68</td><td>1.708</td><td>93.67</td><td>0.843</td></tr><tr><td>Preactivation ResNet 18</td><td>94.93</td><td>3.155</td><td>94.93</td><td>0.570</td><td>94.73</td><td>0.455</td></tr><tr><td>Preactivation ResNet 164</td><td>93.91</td><td>4.102</td><td>93.91</td><td>0.437</td><td>93.82</td><td>0.331</td></tr><tr><td>ResNext29_8x16</td><td>94.79</td><td>2.833</td><td>94.79</td><td>0.741</td><td>94.61</td><td>0.728</td></tr><tr><td>Wide ResNet 40x10</td><td>95.01</td><td>3.001</td><td>95.01</td><td>0.921</td><td>95.08</td><td>0.594</td></tr><tr><td>average</td><td>94.81</td><td>3.191</td><td>94.81</td><td>0.9</td><td>94.70</td><td>0.581</td></tr></table>
+
+<table><tr><td rowspan="2"></td><td colspan="2">uncalibrated</td><td colspan="2">SVHN Temp Scal</td><td colspan="2">BNN</td></tr><tr><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td></tr><tr><td>WideResNet 40x10</td><td>96.95</td><td>1.26</td><td>96.95</td><td>1.17</td><td>96.90</td><td>1.15</td></tr><tr><td>Densenet-121</td><td>96.76</td><td>2.021</td><td>96.76</td><td>1.092</td><td>96.69</td><td>0.716</td></tr><tr><td>Densenet-169</td><td>96.70</td><td>0.363</td><td>96.70</td><td>1.016</td><td>96.59</td><td>0.453</td></tr><tr><td>ResNet 50</td><td>96.47</td><td>0.886</td><td>96.47</td><td>1.030</td><td>96.33</td><td>0.857</td></tr><tr><td>Preactivation ResNet 164</td><td>96.20</td><td>2.539</td><td>96.20</td><td>1.079</td><td>96.08</td><td>0.921</td></tr><tr><td>Wide ResNet 16x8</td><td>96.88</td><td>0.710</td><td>96.88</td><td>1.318</td><td>96.82</td><td>0.739</td></tr><tr><td>Preactivation ResNet 18</td><td>96.15</td><td>1.574</td><td>96.15</td><td>0.645</td><td>96.05</td><td>1.096</td></tr><tr><td>average</td><td>96,587</td><td>1,336</td><td>96,587</td><td>1.05</td><td>96,494</td><td>0.847</td></tr></table>
+
+CIFAR100   
+
+<table><tr><td rowspan="2"></td><td colspan="2">uncalibrated</td><td colspan="2">Temp Scal</td><td colspan="2">BNN</td></tr><tr><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td></tr><tr><td>WideResNet 28x10</td><td>80.39</td><td>4.853</td><td>80.39</td><td>4.276</td><td>77.59</td><td>2.456</td></tr><tr><td>DenseNet 121</td><td>78.8</td><td>8.724</td><td>78.8</td><td>3.476</td><td>75.9</td><td>2.534</td></tr><tr><td>ResNet 101</td><td>72</td><td>11.413</td><td>72</td><td>1.533</td><td>68.7</td><td>1.612</td></tr><tr><td>VGG 19</td><td>72.7</td><td>17.631</td><td>72.7</td><td>4.798</td><td>71.94</td><td>6</td></tr><tr><td>Preactivation ResNet 18</td><td>76.6</td><td>10.780</td><td>76.9</td><td>3.152</td><td>74.3</td><td>1.763</td></tr><tr><td>Preactivation ResNet 164</td><td>73.28</td><td>15.754</td><td>73.28</td><td>2.046</td><td>70.77</td><td>1.461</td></tr><tr><td>ResNext29_8x16</td><td>77.88</td><td>9.678</td><td>77.88</td><td>2.811</td><td>73.97</td><td>2.581</td></tr><tr><td>DenseNet 169</td><td>79.05</td><td>8.883</td><td>79.05</td><td>3.758</td><td>75.58</td><td>2.393</td></tr><tr><td>WideResNet 40x10</td><td>76.74</td><td>14.767</td><td>76.74</td><td>3.765</td><td>76.17</td><td>1.876</td></tr><tr><td>average</td><td>76,36</td><td>11,387</td><td>76,36</td><td>3,291</td><td>73,88</td><td>2,520</td></tr></table>
+
+<table><tr><td rowspan="2"></td><td colspan="2">uncalibrated</td><td colspan="2">GENDER Temp Scal</td><td colspan="2">BNN</td></tr><tr><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td></tr><tr><td>VGG-19</td><td>90.60</td><td>8.08</td><td>90.60</td><td>3.96</td><td>90.50</td><td>2.70</td></tr><tr><td>DenseNet 121</td><td>90.035</td><td>8.803</td><td>90.035</td><td>3.077</td><td>89.961</td><td>1.547</td></tr><tr><td>ResNet 18</td><td>90.42</td><td>8.45</td><td>90.42</td><td>3.8</td><td>90.44</td><td>3.082</td></tr><tr><td>average</td><td>90.352</td><td>8.444</td><td>90.352</td><td>3.612</td><td>90.30</td><td>2.443</td></tr></table>
+
+![](images/d5f2dc77003d6ed18061b37b0766445346960f5546444890d6f27d5782722284.jpg)  
+Figure 3: This figures compares the ECE performance for TS and BNN in test and validation. On the left (CIFAR10) we show the performance of different training parameters. For example 30MC_500 means that the ELBO was optimized using $3 0 \mathrm { M C }$ steps to estimate the expectation and 500 epochs of Adam optimization. On the right (CIFAR100) we show the performance of a BNN trained with different number of epochs up to 2000, showing the robustness against the course of learning.
+
+Finally, we realized that the BNNs are suitable and robust in the experiments carried out. In many experiments we found that all the tested configurations clearly outperform TS, as an example see figure 3.
+
+# 6 DISCUSSION AND CONTRIBUTIONS
+
+There is an increasing interest in adjusting confidence in deep learning, as this models are now becoming part of complex decision systems and critical applications. In the machine learning community there are two main approaches for reducing over-confidence, each one with its own pros and cons: implicit/online and explicit/offline. An implicit method aim at obtaining calibrated distributions directly at the output of the model, while explicit methods post-process the output of the model to be calibrated. Moreover both approaches can use either point estimate or Bayesian probabilistic models.
+
+Explicit approaches have several advantages. First, one can calibrate pre-softmax values provided by other practitioners. Therefore, privacy concerns regarding the model or the original data used to train that model are considered. Second, explicit approaches can be combined with implicit ones (Lakshminarayanan et al., 2017; Seo et al., 2018; Kumar et al., 2018; Chen et al., 2018; DeVries & Taylor, 2018) to further improve calibration, as example see (Kumar et al., 2018; Lee et al., 2018), where TS is used within implicit approaches. Third, one can use impractical models applied directly to a deep model in this offline stage, e.g Bayesian Neural or Gaussian Processed. Fourth, we do not need deep architectures for the Bayesian stage as the input includes the already learned representation of the DNN. This stage only focus on adjusting probabilities. A two layer BNN is unable to reach the same accuracy as a deep convolutional model by its own, however, combined with it, can yield to state-of-the-art accuracy and calibration results, as we showed. Fifth, models to be calibrated does not need to be retrained. This easily let us calibrate, as example, models that make use of pretrained DNN (transfer learning applications). Sixth, any probabilistic model can be calibrated: CNN, LSTM-RNN, BLSTM-RNN, SVM, network ensembles... Seventh, some implicit methods, such as Gal & Ghahramani (2016); Seo et al. (2018) require us to train our deep models with Dropout or stochastic depth, respectively, while ours is totally independent on how the deep model is trained. On the other hand, implicit approaches are less sensible to overfitting. Guo et al. (2017) shows that more complex models yield worse calibration performance. However, we have demonstrated that correctly managing uncertainty allow us using complex models for post-processing, improving state-of-the-art explicit approaches, and getting competitive results with the most recent published implicit ones.
+
+Regarding the calibration performance applied to deep learning models. Our method reaches competitive results with state-of-the-art implicit approaches on deep learning models (Seo et al., 2018; Kumar et al., 2018), and outperforms other proposed explicit (Guo et al., 2017) and implicit (Tran et al., 2018) techniques. In fact, Kumar et al. (2018) obtain competitive results when combining their implicit method with TS, which again shows that offline calibration is a desirable and flexible choice to be combined with implicit models. In fact, we have been able to apply BNNs to a task of interest for the machine learning community, which is the main criticism to these techniques. Kuleshov et al. (2018), which propose a procedure for calibrating Bayesian algorithms only for regression problems, and Lakshminarayanan et al. (2017) argue that a Bayesian treatment do not output calibrated distributions, as the Bayesian deep learning has several restrictions that the machine learning community is trying to overcome. However, this work demonstrates that if we let the major complexity of the task to a deep model, a simple Bayesian approach can adjust probabilities in an efficient way. As shown in our github, 2-layers Bayesian neural nets are enough to adjust probabilities.
+
+In terms of efficiency our BNN method presents several benefits in comparison to other Bayesian or point estimate methods. Although making predictions is more expensive than with TS, this predictions can be fully parallelized, as noted above, computing predictions in only one step. Moreover, a forward through a deep model and a shallow BNN is less computational expensive than a forward through a deep Bayesian convolutional model that requires several forward (and backward) for test (and training). For instance we have models based on BNNs (Gal & Ghahramani, 2015) and based on Gaussian processes (Tran et al., 2018; Milios et al., 2018). Network ensembles (Lakshminarayanan et al., 2017) reduce overconfidence and output calibrated distributions, but it is not measure in a deep model application, and only compared to Monte Carlo Dropout and to the number of ensembles. Ensembles can be also paralellized but in case of deep learning models, which is our field of study, the performance is compromised by the deepness of the different ensembles. As the authors state, computation restriction arises when evaluation of ensembles is done on ImageNet with the Inception network. Our model not only uses shallow neural nets but is only compromised by the number of classes of the task to be performed, and not by the complexity of the task at hand, as once we are able to reach a good accuracy we only focus on adjusting probabilities. Other implicit approaches such as Seo et al. (2018), that compute the cost to be optimized based on several predictions of the model, require to perform as many forwards per training samples as samples we want to estimate the cost parameter. This also compromise performance in deep models.
+
+Finally, Lakshminarayanan et al. (2017) propose to train models with proper scoring rules, such as negative log-likelihood. However, as demonstrated by Guo et al. (2017) it is not clear if deep generative models trained with this criteria presents uncalibrated distributions, at least in implicit approaches.
+
+# 7 CONCLUSION AND FUTURE WORK
+
+This work has shown the many beneficial properties of offline calibration with a Bayesian reasoning. We open future perspectives which include: incorporate Bayesian improvements on the variational posterior with the objective of reducing topologies (efficiency in training and test time), better calibration and accuracy; be able to analyze how the logit dimension influences the expressiveness needed by the likelihood model and which key factors of Bayesian algorithms are critical for good performance; how can we model prior information on the parameters to yield better results; other offline approaches based on Gaussian processes, as example; incorporate training based on different proper scoring rules; measure robustness against adversarial examples; and implement these models in task where having good calibration is critical.
+
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+Christian Szegedy, Sergey Ioffe, Vincent Vanhoucke, and Alexander A. Alemi. Inception-v4, inception-resnet and the impact of residual connections on learning. In Proceedings of the Thirty-First AAAI Conference on Artificial Intelligence, February 4-9, 2017, San Francisco, California, USA., pp. 4278–4284, 2017. URL http://aaai.org/ocs/index.php/AAAI/ AAAI17/paper/view/14806.
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+Gia-Lac Tran, Edwin V. Bonilla, John P. Cunningham, Pietro Michiardi, and Maurizio Filippone. Calibrating deep convolutional gaussian processes, 2018.
+
+Zoltán Tüske, Ralf Schlüter, and Hermann Ney. Investigation on LSTM recurrent n-gram language models for speech recognition. In Interspeech, pp. 3358–3362. ISCA, 2018.
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+Rianne van den Berg, Leonard Hasenclever, Jakub M. Tomczak, and Max Welling. Sylvester normalizing flows for variational inference. CoRR, abs/1803.05649, 2018.
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+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 5998–6008. Curran Associates, Inc., 2017. URL http: //papers.nips.cc/paper/7181-attention-is-all-you-need.pdf.
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+Anqi Wu, Sebastian Nowozin, Edward Meeds, Richard E. Turner, José Miguel Hernández-Lobato, and Alexander L. Gaunt. Fixing variational bayes: Deterministic variational inference for bayesian neural networks, 2018.
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+B. Zadrozny and C. Elkan. Transforming classifier scores into accurate multiclass probability estimates. Proceeding of the Eight International Conference on Knowledge Discovery and Data Mining (KDD’02), 2002a. doi: 10.1145/775047.775151.
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+Bianca Zadrozny and Charles Elkan. Obtaining calibrated probability estimates from decision trees and naive bayesian classifiers. In Proceedings of the Eighteenth International Conference on Machine Learning, ICML ’01, pp. 609–616, San Francisco, CA, USA, 2001. Morgan Kaufmann Publishers Inc. ISBN 1-55860-778-1. URL http://dl.acm.org/citation.cfm?id= 645530.655658.
+
+Bianca Zadrozny and Charles Elkan. Transforming classifier scores into accurate multiclass probability estimates. In Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’02, pp. 694–699, New York, NY, USA, 2002b. ACM. ISBN 1-58113-567-X. doi: 10.1145/775047.775151. URL http://doi.acm.org/10.1145/ 775047.775151.
+
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+
+# APPENDIX A: TOY EXAMPLE ON BAYESIAN CALIBRATION
+
+In this case we consider a problem of density assignment. Formally, we have a set of data $\mathcal { O } =$ $\{ x _ { i } \} _ { i = 1 } ^ { N } , x \in \mathbb { R } ^ { 2 }$ belonging to class $c _ { 1 }$ , where its true distribution belongs to the family of bimodal Gaussian distributions. We want to assign a unimodal Gaussian parametric model $p ( x | \theta )$ where $\theta =$ $( \mu , \Sigma )$ . Note that in this case the parametric model would be unable to recover the true distribution, which is likely to happen in deep learning models due to the complexity of the distributions these models cope with.
+
+In the Bayesian framework the density is computed by averaging each possible model parameterized by $\theta$ , using the posterior distribution computed from the observed data as the distribution over which we take the expectation:
+
+$$
+p ( x | \mathcal { O } , c _ { 1 } ) = \int d \theta _ { 1 } p ( x | \theta _ { 1 } ) \cdot p ( \theta _ { 1 } | \mathcal { O } )
+$$
+
+On the other hand, maximum likelihood would represent the data using the ML estimator $\theta ^ { M L }$ . In this setting, it can be computed uniquely as the loss function has a global optimum.
+
+Figure 4 shows an example of how model averaging improves the assignments of probabilities, and therefore model calibration, contrary to ML. The figure shows some data points generated by our training distribution, where the color of each point is different for each of the Gaussian mixtures. A point-estimate maximum likelihood (ML) model $p ( x | \theta _ { 1 } ^ { M L } )$ (we use $\theta _ { 1 } ^ { M L }$ to refer to the model assigned to $c _ { 1 }$ datapoints) is fitted and represented as red contour lines. It can be seen that the ML model fails to accurately represent the true data distribution, although it can represent one of the two clusters of data moderately well. Two samples not observed in the training set are shown as a black and a gray dot. Also, different plausible likelihood models are represented in dashed contour plots.
+
+We now assume that we have another set of data belonging to class $c _ { 2 }$ , but not represented in this figure as it is far in the data space. We fit $p ( x | \theta _ { 2 } ^ { M L } )$ for this dataset. We assume the prior distribution over the classes to be equal for both classes, and based on Bayes theory decision, our decision rule is given by:
+
+![](images/3bebab5d15da9371e366f0490da0224ba223f37c2f0846d36f5162b8658b88ce.jpg)  
+Figure 4: Bimodal distribution of 2-dimensional training data (orange and blue points) with Maximum Likelihood estimation of a Gaussian distribution (red contour) and other possible likelihood models explaining the data (dashed plots). Green and black dots represent data not seen in the training data, for which densities are to be assigned. Best viewed in color.
+
+$$
+{ \frac { P ( c _ { 1 } | x ) } { P ( c _ { 2 } | x ) } } = { \frac { p ( x | c _ { 1 } ) } { p ( x | c _ { 2 } ) } } ,
+$$
+
+where for generality we do not explicitly indicate if the model $p ( x | c )$ is computed in the ML or Bayesian setting. As long as $p ( x | c _ { 1 } ) > \dot { p } ( x | c _ { 2 } )$ we will assign $c _ { 1 }$ to a given sample $x$ . It is clearly seen that for both the black and gray test samples, the density assigned by $p ( x | c _ { 1 } )$ is greater and thus these samples are assigned to class $c _ { 1 }$ . In fact, if both samples belong to this class we will have a perfect performance in terms of accuracy.
+
+However, although the black dot is correctly assigned to $c _ { 1 }$ , the red ML model assigns extremely low density to it which is undesirable, as it has been actually generated like the rest of the data. This is not a desired behaviour, since it is in fact likely to belong to the blue component of the distribution as it is close to blue training samples in the data space. For that reason, if we compute probabilities under this model, the ultimate confidence would not reflect the true underlying process, and the calibration of the model will be affected. This effect is what we argue is happening in a classification framework: although we correctly choose the class (the cluster $c _ { 1 }$ ) the ML model does not assign a correct probability.
+
+On the other hand, if we take average of all the different models parameterized by $\theta = ( \mu , \Sigma ) ; \mu \in$ $\mathbb { R } ^ { 2 } , \Sigma \in \mathbb { R } ^ { 2 \mathrm { x 2 } }$ (see dashed plots in the figure), the density assigned to the black dot would be raised by some of the models that explain the blue data points. The importance given to each likelihood is given by the posterior $p ( \theta | \mathcal { O } )$ . Therefore, other possibilities apart from the ML density will be considered, and thus we will be better modelling the probabilistic information. For an exact theoretical example on this same density estimation problem see (Minka, 2001). There you can find the exact posterior distribution using non-informative priors on the parameters.

parse/train/S1xjdoC9Fm/S1xjdoC9Fm_content_list.json

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+    {
+        "type": "text",
+        "text": "OFFLINE DEEP MODELS CALIBRATION WITH BAYESIAN NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            452,
+            246,
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+            261
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work the authors show that Bayesian Neural Networks (BNNs) can be efficiently applied to calibrate state-of-the-art Deep Neural Networks (DNN). Our approach acts offline, i.e., it is decoupled from the training of the DNN to be calibrated. This offline approach allow us to apply our BNN calibration to any model regardless of the limitations that the model may present during training. Note that this offline setting is also appropriate in order to deal with privacy concerns of DNN training data or implementation, among others. We show that our approach clearly outperforms other simple maximum likelihood based solutions that have recently shown very good performance, as temperature scaling (Guo et al., 2017). As an example, we reduce the Expected Calibration Error $( \\mathrm { E C E } \\% )$ from 0.52 to 0.24 on CIFAR-10 and from 4.28 to 2.46 on CIFAR-100 on two Wide ResNet with $9 6 . 1 3 \\%$ and $8 0 . 3 9 \\%$ accuracy respectively, which are among the best results published for these tasks. Moreover, we show that our approach improves the performance of online methods directly applied to the DNN, e.g. Gaussian processes or Bayesian Convolutional Neural Networks. Finally, this decoupled approach allows us to apply any further improvement to the BNN without considering the computational restrictions imposed by the deep model. In this sense, this offline setting is a practical application where BNNs can be considered, which is one of the main criticisms to these techniques. In terms of reproducibility, we provide all the implementation details in https://github.com/2019submission/bnn.2019. ",
+        "bbox": [
+            232,
+            276,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            580,
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+            595
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep Neural Networks (DNNs) have achieved state of art performance in many task such as Image Recognition (Huang et al., 2017; Szegedy et al., 2017; Zagoruyko & Komodakis, 2016), language modeling (Mikolov et al., 2013a;b), machine translation (Vaswani et al., 2017) or speech (Hinton et al., 2012). For that reason, neural networks are now used in many applications. However, this state-of-theart performance is measured in terms of accuracy, but there are many tasks in which the probabilistic information must be also reliable. For example, a probabilistic classifier can be incorporated into a more complex model considering multiple sources of information, by the use of e.g., probabilistic graphical models (Koller & Friedman, 2009), or by combining neural networks with language models in natural language processing tasks (Gulcehre et al., 2017). In addition, probabilistic outputs of classifiers have proven to be useful in many areas apart from classical machine learning tasks, such as language recognition (Brümmer & van Leeuwen, 2006), language models for speech recognition (Tüske et al., 2018) or medical diagnosis (Caruana et al., 2015). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In Bayesian statistics, the reliability of probabilities is measured by their calibration. As a consequence, the machine learning community has been exploring methods to calibrate the output of classifiers to achieve the many beneficial properties of well-calibrated probabilities (Zadrozny & Elkan, 2002a; Cohen & Goldszmidt, 2004; Niculescu-Mizil & Caruana, 2005). Nowadays, there is an increasing interest in obtaining reliable probabilities in the deep learning community. In the past, neural networks trained with a cross-entropy criterion tended to present relatively good calibration. However, a relevant recent work in Guo et al. (2017) has evidenced that modern state-of-the-art neural networks are badly calibrated in general. Moreover, the same work shows that calibration can be dramatically improved by very simple maximum-likelihood parametric techniques, among which Temperature Scaling (TS) is highlighted as the preferred choice, due to its extreme simplicity, very good behavior in general and computational efficiency. In fact, TS outperforms more complex techniques in most cases, leading to the conclusion that good calibration can be better achieved with simpler techniques. This conclusion follows the hypothesis that the space configured by the outputs of a deep model is relatively simple, and therefore good performance, measured as Expected Calibration Error (ECE), can be achieved with very simple models. In fact, TS is a technique that performs nicely in complex multiclass tasks (Guo et al., 2017). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/06621e565922dc9459a398e6539aab8fce8f5c8812f3dd555f33807197667e37.jpg",
+        "image_caption": [
+            "ResNet-101: $9 3 . 4 6 \\%$ accuracy (CIFAR10) ",
+            "Figure 1: Reliability diagrams (Guo et al., 2017) for two networks trained on CIFAR-10 and CIFAR100. The red line represents perfect calibration. We plot the Expected Calibration Error (ECE $\\%$ ) for 15 bins (see section 5 for a description). The lower the better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 1
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+    {
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In general, there are two main approaches to reduce overconfidence: implicit or online and explicit or offline. An implicit method aims at obtaining calibrated distributions directly at the output of the model, while explicit methods post-process the output of the model to be calibrated. In this paper we propose an offline method based on a Bayesian Neural Networks (BNNs) to obtain calibrated probabilities, see figure 2. We use BNNs as we aim at being benefited from two key properties of Bayesian statistics and neural networks: the high expressiveness of neural network models and the capabilities of the Bayesian statistics to model the uncertainty. We assume the hypothesis that as long as the uncertainty is correctly modelled, we can use high expressive function approximators for the task of calibration. These high expressive models are required since we assume that the calibration space is not simple. However, the generalization capability of these models are achieved through proper uncertainty consideration. Figure 1 shows reliability diagrams comparing our BNN method to TS. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This work is organized as follows. We first provide an insight on why Bayesian Statistics and neural networks are suitable to adjust confidence in output probabilities. We then describe our offline calibration approach based on BNNs. We end up comparing our method to TS and reporting clear performance improvements. Finally, we discuss our approach against recent published techniques, enumerate some beneficial properties and propose possible improvements over this contribution. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BAYESIAN MODELLING AND CALIBRATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In a classification scenario, calibration can be interpreted as the agreement between the probabilities of a class assigned by a model to a set of samples, and the proportion of those samples where that class is actually the true one. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One way of achieving calibration is to reliably modelling the probabilistic distributions of the data from the classes involved. This is the main strength of Bayesian models, which manage uncertainty properly, in contrast to point-estimate approaches (i.e. Maximum Likelihood or Maximum Posterior). In the former, the uncertainty is incorporated by taking an average of all the likelihood models under the posterior distribution on the parameters, given an observed set of data: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/02b2d89481fa7f7301c102324054435f6c84d4162ba24f1b1d2f17754d70b655.jpg",
+        "text": "$$\np ( t | x ) = \\mathbb { E } _ { p ( \\theta | \\mathcal { O } ) } \\{ p ( t | x , \\theta ) \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\theta$ are model parameters, $x$ represent a sample for which we want to predict a label $t$ and $\\mathcal { O } = \\{ ( x _ { i } , t _ { i } ) \\} _ { i = 1 } ^ { N }$ is the set of observed samples1. In Bayesian approaches, it is indeed the observed data what model how representative a likelihood model is for a particular task, and thus how it influences the predictions. On the other hand, in point-estimate models all the decision is based on a choice of the parameter once the model is trained. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For instance, consider the case of a MAP network (e.g., a typical deep convolutional model trained with cross-entropy loss and $L _ { 2 }$ regularization). This model explains the data based on a point-estimate training, i.e., by representing what is more likely to appear. Nowadays, this gives outstanding accuracy in classification tasks, but it is easy to train an over/under-confident model, i.e., the one that outputs too extreme probabilities, even for unfavourable cases like e.g. when data that has conditions not unseen in the training set, or not explained by the expressiveness of the model itself. This could happen if the true distribution does not lie in the family of parametric models $p ( t | x , \\theta )$ . We will refer to both conditions as mismatch. This is very harmful for the calibration, because in those mismatch cases, the model should yield more moderate probabilities, otherwise the classification errors will be more catastrophic. In other words, in tasks where calibration matters, a classification error has unequal consequences if the probabilities are moderate or extreme. Thus, for example, if there exists such mismatching conditions, what is likely to happen, a point estimate will not represent the data (i.e. the probability assigned) in the way it should, possibly leading to over-or under-estimation of probabilities. On the other hand, in the Bayesian framework, a posterior distribution on the parameters could consider networks explaining these mismatching conditions. By averaging different contributions, the model ideally moderates probabilities on unfavourable data. For the sake of illustration, we provide a simple example in appendix A. We encourage unfamiliar readers with Bayesian statistics and calibration to read this appendix. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Regarding the accuracy of Bayesian models, in Bayesian decision theory, if the model used to generate the data is known, the optimal error can be achieved, which also means optimal accuracy if all decision costs are equal. This suggests that a proper way of assigning probabilities is also paramount for the accuracy. Thus, by choosing appropriate densities for the class-conditional probabilities $p ( t | x )$ , such as factorized multivariate Bernoulli distributions like in point estimate models, the accuracy will also be correctly modelled. Moreover, it is well known that Bayesian models asymptotically tend to point estimate models as the data increases in size, see Duda et al. (2000) section 3.4. Therefore, it is expected that good accuracy performance achieved by point-estimate models should be also achievable by Bayesian models, at least for sufficient amounts of data. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 OFFLINE CALIBRATION WITH BAYESIAN NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The architecture proposed in this work is shown in Figure 2. We apply a BNN to the potentially uncalibrated outputs of a DNN model. The goal is to improve the calibration minimizing the accuracy degradation of the original DNN model. Our approach works offline, meaning that given a DNN model, we project the available data to the space defined by the outputs of the model in the form of logit, i.e. pre-softmax values. This new representation is then the input to our BNN. The BNN aims at taking this uncalibrated so-called logit space and project it to a new calibrated one. The same procedure is applied for the TS method. ",
+        "bbox": [
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+    {
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+        "img_path": "images/7d26d0fed45dcf9dab9259ca9ea552334e4f35ca5924d1e83b3b311d694ee0c9.jpg",
+        "image_caption": [
+            "Figure 2: Exemplary representation of the architecture of our proposed model. On the left, an expensive pretrained DNN on ImageNet is trained on a specific dataset (transfer learning). Then, the (uncalibrated) output of such DNN is the input to the BNN calibration stage. This stage is trained by the maximization of the Expected Lower Bound (ELBO) and predictions are done using Monte Carlo integration. The inputs and outputs of the Bayesian stage have same dimension (given by the number of classes), and lie in the so-called logit space. Orange Gaussians on each arrow represent variational distributions on parameters. We do not plot all the arrows for clarity. This Bayesian stage is independent of the previous one as we only require access to the logits of an already trained model. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This off-line set-up presents clear advantages. First, the approach is efficient, since the DNN model does not need to be re-trained for re-calibration. Furthermore, we can incorporate future improvements to the BNN calibration stage without affecting the previous DNN model. Second, our proposal is extremely flexible, as the proposed BNN calibration stage will work with any probabilistic model, even implicitly-calibrated models, with potential benefits on calibration performance. And finally, our proposal preserves privacy, because there is no need to access the original data used to train the DNN model, or even the DNN topology: to be trained, the BNN only needs the data projected to the outputs of the DNN on the logit space, and the original targets $t$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For these Bayesian approaches one has to compute the posterior distribution $p ( \\theta | \\mathcal { O } )$ and the expectation in equation 1. Using configurations that yield to analytic solutions to both problems result in low-expression models for this task. We solve this problem by choosing Neural Networks to parameterize the likelihood $p ( t | x , \\theta )$ of our BNN, and therefore taking advantage of the high expressiveness of these models. In this case, several intractabilities arise that must be addressed. ",
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+    {
+        "type": "text",
+        "text": "In this work we approximate expectations on equation 1 with Monte Carlo integration, and the posterior is approximated by a variational distribution in terms of the Kulback-Lieber Divergence. This is done by the maximization the Evidence Lower Bound (ELBO). We use stochastic optimization based on the reparameterization trick (Kingma & Welling, 2014; Rezende et al., 2014) to approximate the expectation under the variational distribution, and also mini-batch stochastic optimization for expectations under data distribution. Thus, our training criteria is given by: ",
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+        "img_path": "images/c0a2894a12f857290e27e9045effd3e22919128a4bc155a9a5733d0f174a0e6f.jpg",
+        "text": "$$\nE L B O = \\frac { 1 } { N } \\sum _ { ( x , t ) \\sim p _ { d } ( x , t ) } \\Bigl [ \\frac { 1 } { K } \\sum _ { \\theta \\sim q _ { \\phi } ( \\theta ) } [ \\log p ( t | x , \\theta ) ] - \\beta \\cdot D _ { K L } \\{ q _ { \\phi } ( \\theta ) / / p ( \\theta ) \\} \\Bigr ] ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where we introduce $\\beta$ following Blundell et al. (2015). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our variational distribution is a factorized Gaussian distribution and for that reason we refer to our BNN approach as a basic approach, as we do not incorporate any improvement recently proposed for BNN models, such as normalizing flows, local reparameterization, and so on. Also, we use a standard normal density for the prior. We choose this simple approximation to demonstrate our starting hypothesis: that BNNs can be applied to improve the calibration of state-of-the-art DNN in a very efficient way. Our basic BNN model can be improved by using normalizing flows (Rezende & Mohamed, 2015; Kingma et al., 2016; Huang et al., 2018; van den Berg et al., 2018), auxiliary variables (Agakov & Barber, 2004; Ranganath et al., 2016; Maaløe et al., 2016), local reparameterization (Kingma et al., 2015), combinations of all of them (Louizos & Welling, 2017) or deterministic models (Wu et al., 2018). Also, Cremer et al. (2018) has recently pointed out that amortized inference leads to an additional gap in the bound, in addition to the $D _ { K L }$ gap between the true and variational posteriors; and we can also use other proposals to mitigate this effect (Shu et al., 2018; Kim et al., 2018). ",
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+        "text": "Finally, class predictions are assigned by first computing the logits of a test sample using the first DNN stage $\\boldsymbol { B }$ , and then using them as inputs of our BNN to yield calibrated probabilities, which can be formalized as follows: ",
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+    {
+        "type": "equation",
+        "img_path": "images/edfa99b96d0a49da1a70793188add84e383cdf84a4e24ae46e0f1942a27554ca.jpg",
+        "text": "$$\n\\begin{array} { c } { l = { } } \\\\ { { } } \\\\ { { p ( t | x , \\mathcal { O } ) \\approx \\displaystyle \\frac { 1 } { M } \\sum _ { i = 1 } ^ { M } p ( t | l , \\theta _ { i } ) ; \\theta _ { i } \\sim q _ { \\phi } ( \\theta ) , } } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $M$ is a value chosen on validation. Note that our proposed BNN is not as efficient as TS for calibration. However, the contributions to the weighted average can be fully parallelized computationally, as predictions do not depend on each other. Thus, we can use modern GPU libraries such as CUBLAS and batch-based operations to dramatically speed-up the process. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To our knowledge, TS (Guo et al., 2017) has been consistently reported as the best technique to improve calibration over a list of classical ways of improving calibration, such as histogram binning (Zadrozny & Elkan, 2001), isotonic regression (Zadrozny & Elkan, 2002b), Platt scaling (Platt, 1999) or Bayesian binning into quantiles (Naeini et al., 2015) among others. For a recent description and performance comparison with modern neural networks, see Guo et al. (2017). On the other hand, there are several works that study overconfident predictions and model uncertainty in different contexts, but without reporting an explicit measurement of calibration performance in deep neural models. For instance, Gal & Ghahramani (2015) connect Bernoulli dropout with BNNs, and Gal & Ghahramani (2016) links Gaussian processes with classical dropout regularized networks, showing how uncertainty estimates can be obtained from this networks. In the latter, the authors state that these Bayesian outputs are not calibrated. In Pereyra et al. (2017), an entropy term is added to the log-likelihood to relax overconfidence. Lakshminarayanan et al. (2017) propose training network ensembles with adversarial noise samples to output confident scores. Chen et al. (2018) propose a model that uses probes of the individual layers of the neural network classifier to create a confidence score for the network output. DeVries & Taylor (2018) train a second output obtained from the penultimate layer of the classifier to be confident by interpolation of the softmax output and the true value, scaled by this score. Lee et al. (2018) proposed a generative approach for detecting outof-distribution samples but evaluates calibration performance comparing their method with normal cross-entropy minimization, using TS as the calibration technique. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "On the side of BNNs, Kingma et al. (2015) formalize Gaussian dropout as a Bayesian approach. In Louizos & Welling (2017), novel BNNs are proposed, mixing inverse autoregressive flows (Kingma et al., 2016), auxiliary variables (Maaløe et al., 2016) and local reparameretization (Kingma et al., 2015). None of these approaches measure calibration explicitly on deep neural models, as we do. For instance, Louizos & Welling (2017) and Lakshminarayanan et al. (2017) evaluate uncertainty by training on one dataset and use it on another, expecting a maximum entropy output distribution. More recently, Zhang et al. (2018) propose an scalable inference algorithm that is also asymptotically accurate as MCMC algorithms. ",
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+        "type": "text",
+        "text": "We compare our proposed BNN approach against TS, as to our knowledge it is the state of the art in calibration tasks involving deep neural models, according to the reviewed previous work. TS widely improves calibration without affecting the accuracy, and can be efficiently applied to any model. Formally, given $\\mathcal { O }$ , TS maximizes the log-likelihood of the conditional distribution $p ( t | x / T )$ w.r.t. the parameter $T$ . In this case, $x$ also represents the output logits of the deep convolutional model. ",
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+        "type": "text",
+        "text": "In Section 6, we will comparatively discuss the properties and results of our proposal with other works recently found in the literature for calibration on deep models, some of them based on implicit and explicit models. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We demonstrate calibration performance on several computer vision models on several datasets. We have used CIFAR10 and CIFAR100 databases (Krizhevsky et al., a;b); SVHN (Netzer et al., 2011); and a GENDER recognition task (Eidinger et al., 2014). We use a validation set randomly taken from the training set with 5000 samples for CIFAR10 and CIFAR100, 10000 samples for SVHN and 4005 for GENDER. This validation set is used to train the TS parameter and to choose the number of Monte Carlo samples, $M$ in equation 2. We report results for the best model on validation for all the tested configurations. We optimize the ELBO using adam optimization (Kingma & Ba, 2014), since it performed better than stochastic gradient descent on our previous experiments. We used $\\beta = 0 . 1$ from the set $\\lbrace 1 , 0 . 1 , 0 . 0 1 \\rbrace$ as it behaves better on validation. ",
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+        "type": "text",
+        "text": "In order to compare our experiments with uncalibrated and TS calibrated probabilities, we used an unbiased estimator of the Expected Calibration Error (ECE) computed as in Guo et al. (2017), with 15 bins. The ECE measures the expected value of the difference between accuracy and confidence: ",
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+        "img_path": "images/79c381d76d62a41dfa4a7a1712930a0dfebcb764726e5f2c7de9ca4fc274e5a2.jpg",
+        "text": "$$\n\\mathrm { E C E } = \\sum _ { i = 1 } ^ { 1 5 } \\frac { | B _ { i } | } { N } | \\mathrm { a c c } ( B _ { i } ) - \\mathrm { c o n f } ( B _ { i } ) |\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $N$ is the number of total samples; $B _ { i }$ represents the set of samples whose predictions $t$ confidence lie in bin $i$ ; $\\mathrm { c o n f } ( B _ { i } )$ is the average confidence and $\\operatorname { a c c } ( B _ { i } )$ is the accuracy of that bin. We also report the accuracy of our models. This is because a classifier can be perfectly calibrated, but useless from a classification point of view. ",
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+        "type": "text",
+        "text": "Note that we evaluate our proposed method on several state of art configurations of computer vision neural networks over the mentioned datasets: Wide Residual Networks (Zagoruyko & Komodakis, 2016), Residual Networks (He et al., 2016b), Densely Connected Neural Networks (Huang et al., 2017), Pre-Activation Residual Networks (He et al., 2016a), Dual Path Networks (Chen et al., 2017), VGG (Simonyan & Zisserman, 2014) and ResNext (Xie et al., 2017). The results reported in this work are obtained from some pretrained neural networks. ",
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+    {
+        "type": "text",
+        "text": "5.1 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Table 1 shows the results for the different datasets, original DNNs and calibration approaches. We presents results in $\\%$ , i.e. multiplying by 100 the result obtained in equation 4. The most important point is that calibration is improved by a wide margin in every model except for two models in CIFAR100 and one model in SVHN. Table 1 shows that in average our proposed calibration method outperforms TS with an insignificant accuracy loss. This means that high expressive models can cope with the calibration task as long as uncertainty is correctly modelled. Therefore, we propose an alternative hypothesis to the one given in Guo et al. (2017) where the authors argued that the calibration space is simple. We argue that if highly complex models outperform simple ones is because the distribution of the calibration space is also complex but the low dimensionality of the logit space makes high expressive models overfit. ",
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+        "type": "text",
+        "text": "We realized that more expressive models are needed by more complex tasks, like CIFAR100. For instance, ResNet-18 GENDER uses BNNs of two layers with two neurons per layer, while WideResNet 40x10 on CIFAR100 uses two layers of 2000 neurons. This reflects that, when dimensionality increases, more expressiveness is needed. Another important point observed in SVHN (see Densenet169, ResNet-50 and WideResNet 16x8) is that TS has degraded calibration by a factor of three in the worst case. In general BNNs do not degrade the calibration. ",
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+        "type": "text",
+        "text": "One drawback of our basic approach is that in some cases we obtain slight accuracy degradation. Accuracy degradation is more relevant only for CIFAR100, however our BNN method reduces ECE15 by a factor of two in some experiments in this task. Moreover, in some cases we are able to improve both, accuracy and calibration, see WideResNet $4 0 \\mathrm { x } 1 0$ for CIFAR10 or ResNet-18 for GENDER dataset. Thus, we cannot conclude that BNNs are calibrating at the cost of losing accuracy. This motivates us towards further research on this accuracy degradation, as we expect to solve it with more sophisticated approximations in future work. Possible hypothesis for this degradation are that either the gap between the variational and the true posterior is still large, the variance of the ELBO estimator is large and does not allow us to converge to a better optimal, or the expressiveness of the likelihood model is not enough to deal with the particular logit space distribution. We found that for more complex logit space distribution (100 dimensional in CIFAR100) we could get better accuracy and better ECE increasing the expressiveness of the model. On the other hand in simpler logit space distribution (2 dimensional in GENDER) we found that the expressiveness must be reduced. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 1: ECE $1 5 ( \\% )$ and Accuracy $( \\% )$ comparing model uncalibrated, calibrated with TS and with BNN "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">uncalibrated</td><td colspan=\"2\">CIFAR10 Temp Scal</td><td colspan=\"2\">BNN</td></tr><tr><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td></tr><tr><td>WideResNet28x10</td><td>96.13</td><td>1.835</td><td>96.13</td><td>0.518</td><td>96.08</td><td>0.243</td></tr><tr><td>DenseNet 121</td><td>95.49</td><td>2.643</td><td>95.49</td><td>1.011</td><td>95.26</td><td>0.600</td></tr><tr><td>DenseNet 169</td><td>95.49</td><td>2.664</td><td>95.49</td><td>0.826</td><td>95.29</td><td>0.511</td></tr><tr><td>Dual Path Network 92</td><td>95.18</td><td>2.995</td><td>95.18</td><td>1.072</td><td>95.03</td><td>0.730</td></tr><tr><td>ResNet 101</td><td>93.46</td><td>4.268</td><td>93.46</td><td>1.196</td><td>93.38</td><td>0.776</td></tr><tr><td>VGG 19</td><td>93.68</td><td>4.412</td><td>93.68</td><td>1.708</td><td>93.67</td><td>0.843</td></tr><tr><td>Preactivation ResNet 18</td><td>94.93</td><td>3.155</td><td>94.93</td><td>0.570</td><td>94.73</td><td>0.455</td></tr><tr><td>Preactivation ResNet 164</td><td>93.91</td><td>4.102</td><td>93.91</td><td>0.437</td><td>93.82</td><td>0.331</td></tr><tr><td>ResNext29_8x16</td><td>94.79</td><td>2.833</td><td>94.79</td><td>0.741</td><td>94.61</td><td>0.728</td></tr><tr><td>Wide ResNet 40x10</td><td>95.01</td><td>3.001</td><td>95.01</td><td>0.921</td><td>95.08</td><td>0.594</td></tr><tr><td>average</td><td>94.81</td><td>3.191</td><td>94.81</td><td>0.9</td><td>94.70</td><td>0.581</td></tr></table>",
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+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">uncalibrated</td><td colspan=\"2\">SVHN Temp Scal</td><td colspan=\"2\">BNN</td></tr><tr><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td></tr><tr><td>WideResNet 40x10</td><td>96.95</td><td>1.26</td><td>96.95</td><td>1.17</td><td>96.90</td><td>1.15</td></tr><tr><td>Densenet-121</td><td>96.76</td><td>2.021</td><td>96.76</td><td>1.092</td><td>96.69</td><td>0.716</td></tr><tr><td>Densenet-169</td><td>96.70</td><td>0.363</td><td>96.70</td><td>1.016</td><td>96.59</td><td>0.453</td></tr><tr><td>ResNet 50</td><td>96.47</td><td>0.886</td><td>96.47</td><td>1.030</td><td>96.33</td><td>0.857</td></tr><tr><td>Preactivation ResNet 164</td><td>96.20</td><td>2.539</td><td>96.20</td><td>1.079</td><td>96.08</td><td>0.921</td></tr><tr><td>Wide ResNet 16x8</td><td>96.88</td><td>0.710</td><td>96.88</td><td>1.318</td><td>96.82</td><td>0.739</td></tr><tr><td>Preactivation ResNet 18</td><td>96.15</td><td>1.574</td><td>96.15</td><td>0.645</td><td>96.05</td><td>1.096</td></tr><tr><td>average</td><td>96,587</td><td>1,336</td><td>96,587</td><td>1.05</td><td>96,494</td><td>0.847</td></tr></table>",
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+        "type": "table",
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+            "CIFAR100 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">uncalibrated</td><td colspan=\"2\">Temp Scal</td><td colspan=\"2\">BNN</td></tr><tr><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td></tr><tr><td>WideResNet 28x10</td><td>80.39</td><td>4.853</td><td>80.39</td><td>4.276</td><td>77.59</td><td>2.456</td></tr><tr><td>DenseNet 121</td><td>78.8</td><td>8.724</td><td>78.8</td><td>3.476</td><td>75.9</td><td>2.534</td></tr><tr><td>ResNet 101</td><td>72</td><td>11.413</td><td>72</td><td>1.533</td><td>68.7</td><td>1.612</td></tr><tr><td>VGG 19</td><td>72.7</td><td>17.631</td><td>72.7</td><td>4.798</td><td>71.94</td><td>6</td></tr><tr><td>Preactivation ResNet 18</td><td>76.6</td><td>10.780</td><td>76.9</td><td>3.152</td><td>74.3</td><td>1.763</td></tr><tr><td>Preactivation ResNet 164</td><td>73.28</td><td>15.754</td><td>73.28</td><td>2.046</td><td>70.77</td><td>1.461</td></tr><tr><td>ResNext29_8x16</td><td>77.88</td><td>9.678</td><td>77.88</td><td>2.811</td><td>73.97</td><td>2.581</td></tr><tr><td>DenseNet 169</td><td>79.05</td><td>8.883</td><td>79.05</td><td>3.758</td><td>75.58</td><td>2.393</td></tr><tr><td>WideResNet 40x10</td><td>76.74</td><td>14.767</td><td>76.74</td><td>3.765</td><td>76.17</td><td>1.876</td></tr><tr><td>average</td><td>76,36</td><td>11,387</td><td>76,36</td><td>3,291</td><td>73,88</td><td>2,520</td></tr></table>",
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+        "table_caption": [
+            ""
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">uncalibrated</td><td colspan=\"2\">GENDER Temp Scal</td><td colspan=\"2\">BNN</td></tr><tr><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td><td>Acc</td><td>ECE</td></tr><tr><td>VGG-19</td><td>90.60</td><td>8.08</td><td>90.60</td><td>3.96</td><td>90.50</td><td>2.70</td></tr><tr><td>DenseNet 121</td><td>90.035</td><td>8.803</td><td>90.035</td><td>3.077</td><td>89.961</td><td>1.547</td></tr><tr><td>ResNet 18</td><td>90.42</td><td>8.45</td><td>90.42</td><td>3.8</td><td>90.44</td><td>3.082</td></tr><tr><td>average</td><td>90.352</td><td>8.444</td><td>90.352</td><td>3.612</td><td>90.30</td><td>2.443</td></tr></table>",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/d5f2dc77003d6ed18061b37b0766445346960f5546444890d6f27d5782722284.jpg",
+        "image_caption": [
+            "Figure 3: This figures compares the ECE performance for TS and BNN in test and validation. On the left (CIFAR10) we show the performance of different training parameters. For example 30MC_500 means that the ELBO was optimized using $3 0 \\mathrm { M C }$ steps to estimate the expectation and 500 epochs of Adam optimization. On the right (CIFAR100) we show the performance of a BNN trained with different number of epochs up to 2000, showing the robustness against the course of learning. "
+        ],
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+        "text": "",
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+        "type": "text",
+        "text": "Finally, we realized that the BNNs are suitable and robust in the experiments carried out. In many experiments we found that all the tested configurations clearly outperform TS, as an example see figure 3. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONTRIBUTIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "There is an increasing interest in adjusting confidence in deep learning, as this models are now becoming part of complex decision systems and critical applications. In the machine learning community there are two main approaches for reducing over-confidence, each one with its own pros and cons: implicit/online and explicit/offline. An implicit method aim at obtaining calibrated distributions directly at the output of the model, while explicit methods post-process the output of the model to be calibrated. Moreover both approaches can use either point estimate or Bayesian probabilistic models. ",
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+        "type": "text",
+        "text": "Explicit approaches have several advantages. First, one can calibrate pre-softmax values provided by other practitioners. Therefore, privacy concerns regarding the model or the original data used to train that model are considered. Second, explicit approaches can be combined with implicit ones (Lakshminarayanan et al., 2017; Seo et al., 2018; Kumar et al., 2018; Chen et al., 2018; DeVries & Taylor, 2018) to further improve calibration, as example see (Kumar et al., 2018; Lee et al., 2018), where TS is used within implicit approaches. Third, one can use impractical models applied directly to a deep model in this offline stage, e.g Bayesian Neural or Gaussian Processed. Fourth, we do not need deep architectures for the Bayesian stage as the input includes the already learned representation of the DNN. This stage only focus on adjusting probabilities. A two layer BNN is unable to reach the same accuracy as a deep convolutional model by its own, however, combined with it, can yield to state-of-the-art accuracy and calibration results, as we showed. Fifth, models to be calibrated does not need to be retrained. This easily let us calibrate, as example, models that make use of pretrained DNN (transfer learning applications). Sixth, any probabilistic model can be calibrated: CNN, LSTM-RNN, BLSTM-RNN, SVM, network ensembles... Seventh, some implicit methods, such as Gal & Ghahramani (2016); Seo et al. (2018) require us to train our deep models with Dropout or stochastic depth, respectively, while ours is totally independent on how the deep model is trained. On the other hand, implicit approaches are less sensible to overfitting. Guo et al. (2017) shows that more complex models yield worse calibration performance. However, we have demonstrated that correctly managing uncertainty allow us using complex models for post-processing, improving state-of-the-art explicit approaches, and getting competitive results with the most recent published implicit ones. ",
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+    {
+        "type": "text",
+        "text": "Regarding the calibration performance applied to deep learning models. Our method reaches competitive results with state-of-the-art implicit approaches on deep learning models (Seo et al., 2018; Kumar et al., 2018), and outperforms other proposed explicit (Guo et al., 2017) and implicit (Tran et al., 2018) techniques. In fact, Kumar et al. (2018) obtain competitive results when combining their implicit method with TS, which again shows that offline calibration is a desirable and flexible choice to be combined with implicit models. In fact, we have been able to apply BNNs to a task of interest for the machine learning community, which is the main criticism to these techniques. Kuleshov et al. (2018), which propose a procedure for calibrating Bayesian algorithms only for regression problems, and Lakshminarayanan et al. (2017) argue that a Bayesian treatment do not output calibrated distributions, as the Bayesian deep learning has several restrictions that the machine learning community is trying to overcome. However, this work demonstrates that if we let the major complexity of the task to a deep model, a simple Bayesian approach can adjust probabilities in an efficient way. As shown in our github, 2-layers Bayesian neural nets are enough to adjust probabilities. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In terms of efficiency our BNN method presents several benefits in comparison to other Bayesian or point estimate methods. Although making predictions is more expensive than with TS, this predictions can be fully parallelized, as noted above, computing predictions in only one step. Moreover, a forward through a deep model and a shallow BNN is less computational expensive than a forward through a deep Bayesian convolutional model that requires several forward (and backward) for test (and training). For instance we have models based on BNNs (Gal & Ghahramani, 2015) and based on Gaussian processes (Tran et al., 2018; Milios et al., 2018). Network ensembles (Lakshminarayanan et al., 2017) reduce overconfidence and output calibrated distributions, but it is not measure in a deep model application, and only compared to Monte Carlo Dropout and to the number of ensembles. Ensembles can be also paralellized but in case of deep learning models, which is our field of study, the performance is compromised by the deepness of the different ensembles. As the authors state, computation restriction arises when evaluation of ensembles is done on ImageNet with the Inception network. Our model not only uses shallow neural nets but is only compromised by the number of classes of the task to be performed, and not by the complexity of the task at hand, as once we are able to reach a good accuracy we only focus on adjusting probabilities. Other implicit approaches such as Seo et al. (2018), that compute the cost to be optimized based on several predictions of the model, require to perform as many forwards per training samples as samples we want to estimate the cost parameter. This also compromise performance in deep models. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, Lakshminarayanan et al. (2017) propose to train models with proper scoring rules, such as negative log-likelihood. However, as demonstrated by Guo et al. (2017) it is not clear if deep generative models trained with this criteria presents uncalibrated distributions, at least in implicit approaches. ",
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+    {
+        "type": "text",
+        "text": "7 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "This work has shown the many beneficial properties of offline calibration with a Bayesian reasoning. We open future perspectives which include: incorporate Bayesian improvements on the variational posterior with the objective of reducing topologies (efficiency in training and test time), better calibration and accuracy; be able to analyze how the logit dimension influences the expressiveness needed by the likelihood model and which key factors of Bayesian algorithms are critical for good performance; how can we model prior information on the parameters to yield better results; other offline approaches based on Gaussian processes, as example; incorporate training based on different proper scoring rules; measure robustness against adversarial examples; and implement these models in task where having good calibration is critical. ",
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+    },
+    {
+        "type": "text",
+        "text": "APPENDIX A: TOY EXAMPLE ON BAYESIAN CALIBRATION",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            660,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In this case we consider a problem of density assignment. Formally, we have a set of data $\\mathcal { O } =$ $\\{ x _ { i } \\} _ { i = 1 } ^ { N } , x \\in \\mathbb { R } ^ { 2 }$ belonging to class $c _ { 1 }$ , where its true distribution belongs to the family of bimodal Gaussian distributions. We want to assign a unimodal Gaussian parametric model $p ( x | \\theta )$ where $\\theta =$ $( \\mu , \\Sigma )$ . Note that in this case the parametric model would be unable to recover the true distribution, which is likely to happen in deep learning models due to the complexity of the distributions these models cope with. ",
+        "bbox": [
+            173,
+            137,
+            825,
+            222
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In the Bayesian framework the density is computed by averaging each possible model parameterized by $\\theta$ , using the posterior distribution computed from the observed data as the distribution over which we take the expectation: ",
+        "bbox": [
+            174,
+            228,
+            823,
+            270
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1e2d283fb480a97b485efc37364fec1c23a602a8b652dbc36cc558c6b48bd5bb.jpg",
+        "text": "$$\np ( x | \\mathcal { O } , c _ { 1 } ) = \\int d \\theta _ { 1 } p ( x | \\theta _ { 1 } ) \\cdot p ( \\theta _ { 1 } | \\mathcal { O } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            296,
+            627,
+            329
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, maximum likelihood would represent the data using the ML estimator $\\theta ^ { M L }$ . In this setting, it can be computed uniquely as the loss function has a global optimum. ",
+        "bbox": [
+            174,
+            349,
+            825,
+            378
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 shows an example of how model averaging improves the assignments of probabilities, and therefore model calibration, contrary to ML. The figure shows some data points generated by our training distribution, where the color of each point is different for each of the Gaussian mixtures. A point-estimate maximum likelihood (ML) model $p ( x | \\theta _ { 1 } ^ { M L } )$ (we use $\\theta _ { 1 } ^ { M L }$ to refer to the model assigned to $c _ { 1 }$ datapoints) is fitted and represented as red contour lines. It can be seen that the ML model fails to accurately represent the true data distribution, although it can represent one of the two clusters of data moderately well. Two samples not observed in the training set are shown as a black and a gray dot. Also, different plausible likelihood models are represented in dashed contour plots. ",
+        "bbox": [
+            173,
+            385,
+            825,
+            497
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We now assume that we have another set of data belonging to class $c _ { 2 }$ , but not represented in this figure as it is far in the data space. We fit $p ( x | \\theta _ { 2 } ^ { M L } )$ for this dataset. We assume the prior distribution over the classes to be equal for both classes, and based on Bayes theory decision, our decision rule is given by: ",
+        "bbox": [
+            174,
+            503,
+            825,
+            560
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/3bebab5d15da9371e366f0490da0224ba223f37c2f0846d36f5162b8658b88ce.jpg",
+        "image_caption": [
+            "Figure 4: Bimodal distribution of 2-dimensional training data (orange and blue points) with Maximum Likelihood estimation of a Gaussian distribution (red contour) and other possible likelihood models explaining the data (dashed plots). Green and black dots represent data not seen in the training data, for which densities are to be assigned. Best viewed in color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            281,
+            617,
+            681,
+            853
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8e02371f91dcf976412240d8237356a2313538eddd4a26487a1464c933ee94c.jpg",
+        "text": "$$\n{ \\frac { P ( c _ { 1 } | x ) } { P ( c _ { 2 } | x ) } } = { \\frac { p ( x | c _ { 1 } ) } { p ( x | c _ { 2 } ) } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            113,
+            568,
+            147
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "where for generality we do not explicitly indicate if the model $p ( x | c )$ is computed in the ML or Bayesian setting. As long as $p ( x | c _ { 1 } ) > \\dot { p } ( x | c _ { 2 } )$ we will assign $c _ { 1 }$ to a given sample $x$ . It is clearly seen that for both the black and gray test samples, the density assigned by $p ( x | c _ { 1 } )$ is greater and thus these samples are assigned to class $c _ { 1 }$ . In fact, if both samples belong to this class we will have a perfect performance in terms of accuracy. ",
+        "bbox": [
+            173,
+            150,
+            825,
+            220
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "However, although the black dot is correctly assigned to $c _ { 1 }$ , the red ML model assigns extremely low density to it which is undesirable, as it has been actually generated like the rest of the data. This is not a desired behaviour, since it is in fact likely to belong to the blue component of the distribution as it is close to blue training samples in the data space. For that reason, if we compute probabilities under this model, the ultimate confidence would not reflect the true underlying process, and the calibration of the model will be affected. This effect is what we argue is happening in a classification framework: although we correctly choose the class (the cluster $c _ { 1 }$ ) the ML model does not assign a correct probability. ",
+        "bbox": [
+            174,
+            227,
+            825,
+            338
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "On the other hand, if we take average of all the different models parameterized by $\\theta = ( \\mu , \\Sigma ) ; \\mu \\in$ $\\mathbb { R } ^ { 2 } , \\Sigma \\in \\mathbb { R } ^ { 2 \\mathrm { x 2 } }$ (see dashed plots in the figure), the density assigned to the black dot would be raised by some of the models that explain the blue data points. The importance given to each likelihood is given by the posterior $p ( \\theta | \\mathcal { O } )$ . Therefore, other possibilities apart from the ML density will be considered, and thus we will be better modelling the probabilistic information. For an exact theoretical example on this same density estimation problem see (Minka, 2001). There you can find the exact posterior distribution using non-informative priors on the parameters. ",
+        "bbox": [
+            173,
+            344,
+            825,
+            443
+        ],
+        "page_idx": 15
+    }
+]

parse/train/SkFqf0lAZ/SkFqf0lAZ.md

@@ -0,0 +1,222 @@
+# MEMORY ARCHITECTURES IN RECURRENT NEURAL NETWORK LANGUAGE MODELS
+
+Dani Yogatama♣, Yishu Miao♠, Gabor Melis♣, Wang Ling♣, Adhiguna Kuncoro♣♠   
+Chris Dyer♣, Phil Blunsom♣♠   
+♣DeepMind and ♠University of Oxford   
+dyogatama@google.com, yishu.miao@cs.ox.ac.uk   
+{melisgl,lingwang,akuncoro,cdyer,pblunsom}@google.com
+
+# ABSTRACT
+
+We compare and analyze sequential, random access, and stack memory architectures for recurrent neural network language models. Our experiments on the Penn Treebank and Wikitext-2 datasets show that stack-based memory architectures consistently achieve the best performance in terms of held out perplexity. We also propose a generalization to existing continuous stack models (Joulin & Mikolov, 2015; Grefenstette et al., 2015) to allow a variable number of pop operations more naturally that further improves performance. We further evaluate these language models in terms of their ability to capture non-local syntactic dependencies on a subject-verb agreement dataset (Linzen et al., 2016) and establish new state of the art results using memory augmented language models. Our results demonstrate the value of stack-structured memory for explaining the distribution of words in natural language, in line with linguistic theories claiming a context-free backbone for natural language.
+
+# 1 INTRODUCTION
+
+Sequential recurrent neural networks such as LSTMs (Hochreiter & Schmidhuber, 1997) are the basis of state-of-the-art models of natural language in various tasks. They effectively learn to capture dependencies between events separated in time by learning to store and retrieve information in a hidden state. However, the ability of these methods to discover long-term dependencies is limited by the capacity of their hidden state and the difficulty of propagating reliable gradients. For example, LSTM language models have been shown to struggle to capture non-sequential syntactic dependencies in complex sentences without explicit supervision (Linzen et al., 2016). As an illustration of the kind of dependencies they have difficulty learning, in the sentence, the loss of basic needs providers emigrating from impoverished countries has a damaging effect, correctly predicting singular has rather than its plural form have requires that the LSTM have learned that it depends on the subject, in this case the first noun (loss) rather than any of the intervening non-subject nouns, such as countries. Linzen et al. (2016) show that LSTM language models fail to capture this kind of dependencies, especially as the number of attractors (underlined) increases.
+
+Attempts to improve language models’ ability to capture non-local dependencies have recently been undertaken by introducing an external memory components. These include (i) a soft attention mechanism (Daniluk et al., 2017) and (ii) an explicit memory block or cache model (Tran et al., 2016; Grave et al., 2017). However, since very local context is often most highly informative for predicting the next word, existing memory-augmented RNN LMs use memory just to store information about local context (Daniluk et al., 2017).
+
+In this work, we compare several memory architectures for recurrent neural network language models. Since our goal is to evaluate how well these types of memory architectures learn long term and syntactic dependencies, we focus on language models that are static as opposed to non-static models such as neural cache (Grave et al., 2017) and dynamic evaluation (Krause et al., 2017) that can update their distribution at test time. We consider increasing the capacity of a purely sequential memory model by increasing the capacity of an LSTM, a random access memory model as typified by an attention-based LSTM, and a new variant of a stack augmented recurrent neural network.
+
+Unlike random access memory models, a stack has a built-in bias to discover hierarchical structures that are important in language. A continuous stack memory has been proposed to improve recurrent neural networks (Joulin & Mikolov, 2015; Grefenstette et al., 2015), although it has never been carefully evaluated in benchmark language modeling experiments. In the only set of results for a stack augmented recurrent language model, Joulin & Mikolov (2015) show that a stack augmented vanilla RNN outperforms a standard RNN and is comparable to an LSTM. We augment an LSTM with a stack memory and perform thorough comparisons to evaluate its efficacy as a language model. In contrast to prior work, our continuous stack allows for push, stay, and a variable number of pop operations at each time step (multiple pop operations are useful in modeling natural language sentences since while only a single new word is presented at each time step, multiple syntactic units may come to an end concurrently).
+
+Our motivating hypothesis is that allowing the memory to dynamically store and retrieve contextual information with a stack will drive the model to use the memory to learn dependencies that are difficult to capture by a sequential model. Sequential memory has an easier time learning local dependencies, but it often fails to capture long term dependencies. Random access memory models capture longer range dependencies (i.e., proportional to the window size), but the learner has to infer these from data without any informative structural bias. We hypothesize that introducing a more appropriate inductive bias will make it easier for the model to learn long range and structurally meaningful dependencies, given that the variance in learning such dependencies can be high. Linguistic insights reveal that one possible inductive bias is in the form of a hierarchical nested structure that captures syntactic dependencies. Stack memory models provide a natural way for capturing hierarchical structures, providing an easier path for gradients to flow to particular locations in the past.
+
+Our main contributions in this paper are as follows:
+
+• We thoroughly evaluate the efficacy of a stack augmented RNN as a language model and propose a more expressive extension to existing stack models (§2.1). We compare how a recurrent neural network uses a stack memory, a sequential memory cell (i.e., an LSTM memory cell), and a random access memory (i.e., an attention mechanism) for language modeling. Experiments on the Penn Treebank and Wikitext-2 datasets (§3.2) show that both the stack model and the attention-based model outperform the LSTM model with a comparable (or even larger) number of parameters, and that the stack model eliminates the need to tune window size to achieve the best perplexity.   
+• We assess the ability of these memory models to discover long range structural dependencies commonly encountered in natural language using the subject-verb agreement dataset (Linzen et al., 2016). We achieve new state of the art results and show that the gap in accuracy between a sequential or random access memory model with a stack model gets bigger as the dependencies become more complex (i.e., number of attractors increases; $\ S 3 . 3 )$ . We also analyze the stack and find that the model tends to use it to enhance its sequential memory component in high entropy prediction contexts (§3.4).
+
+# 2 MODEL
+
+We consider a language modeling problem where the goal is to predict the next word $x _ { t }$ given previously seen context words $x _ { 0 } , \ldots , x _ { t - 1 }$ . We represent each input word $x$ by its $D$ -dimensional embedding vector $\mathbf { x } \in \mathbb { R } ^ { D }$ .
+
+Our base model is an LSTM that computes a hidden state at timestep $t$ as follows:
+
+$$
+\begin{array} { r l r l } & { \mathbf i _ { t } = \sigma \big ( \mathbf W _ { i , x } \mathbf x _ { t } + \mathbf W _ { i , h } \mathbf h _ { t - 1 } + \mathbf b _ { i } \big ) \quad } & & { \mathbf f _ { t } = \sigma \big ( \mathbf W _ { f , x } \mathbf x _ { t } + \mathbf W _ { f , h } \mathbf h _ { t - 1 } + \mathbf b _ { f } \big ) } \\ & { \mathbf o _ { t } = \sigma \big ( \mathbf W _ { o , x } \mathbf x _ { t } + \mathbf W _ { o , h } \mathbf h _ { t - 1 } + \mathbf b _ { o } \big ) \quad } & & { \mathbf g _ { t } = \mathrm { t a n h } \big ( \mathbf W _ { g , x } \mathbf x _ { t } + \mathbf W _ { g , h } \mathbf h _ { t - 1 } + \mathbf b _ { g } \big ) } \\ & { \mathbf c _ { t } = \mathbf f _ { t } \odot \mathbf c _ { t - 1 } + \mathbf i _ { t } \odot \mathbf g _ { t } \quad } & & { \mathbf h _ { t } = \mathbf o _ { t } \odot \mathrm { t a n h } \big ( \mathbf c _ { t } \big ) } \end{array}
+$$
+
+Sequential memory. An LSTM has a sequential memory cell c to store and retrieve information that is regulated by the input, output, and forget gates. In order for long-term contextual information to be used in the future, it has to pass through these gates for multiple timesteps.
+
+Random access memory. One common approach to retrieve information from the distant past more reliably is to augment the model with a random access memory block via an attention based
+
+![](images/8f898c37d3bb5705ab491d7152de39c1031529481009bfc5be9228436d0ba168.jpg)  
+Figure 1: Multipop Adaptive Computation Stack Recurrent Neural Network.
+
+method. In this model, we consider the previous $K$ states as the memory block, and construct a memory vector $\mathbf { m } _ { t }$ by a weighted combination of these states:
+
+$$
+\mathbf { m } _ { t } = \sum _ { i = t - K } ^ { t - 1 } a _ { i } \mathbf { h } _ { i } , { \mathrm { w h e r e ~ } } a _ { i } \propto \exp ( \mathbf { w } _ { m , i } \mathbf { h } _ { i } + \mathbf { w } _ { m , h } \mathbf { h } _ { t } )
+$$
+
+Such method can be improved further by partitioning $\mathbf { h }$ into a key, value, and predict subvectors (Daniluk et al., 2017).
+
+Given the LSTM hidden state $\mathbf { h } _ { t }$ and the memory state $\mathbf { m } _ { t }$ , we combine them using a simple function:
+
+$$
+\tilde { \mathbf { h } } _ { t } = \mathbf { W } _ { h , h } \mathbf { h } _ { t } + \mathbf { W } _ { h , m } \mathbf { m } _ { t }
+$$
+
+to get the final representation $\tilde { \mathbf { h } } _ { t }$ . We compute the probability of predicting the next word as $p ( x _ { t } \mid \pmb { x } _ { < t } ) \propto \exp ( \mathbf { x } _ { t } ^ { \top } \tilde { \mathbf { h } } _ { t } + b _ { y , x _ { t } } )$ , where we follow Inan et al. (2017) and reuse the word embedding matrix $\mathbf { X }$ as the softmax parameters.
+
+Stack memory. In this work, we propose to augment a recurrent LSTM language model with a stack memory M that has three basic operations:
+
+• PUSH: Push the current hidden state $\mathbf { h } _ { t }$ onto the stack.   
+• POP: Remove the top element of the stack.   
+• STAY: Keep the stack unchanged.
+
+Figure 1 shows an illustration of our stack augmented RNN. We describe the stack memory in details in the followings.
+
+# 2.1 MULTIPOP ADAPTIVE COMPUTATION STACK
+
+Our stack is a multipop adaptive computation stack—it learns how many POP operations need to be performed before predicting an output. In previous work (Joulin & Mikolov, 2015; Grefenstette et al., 2015), at every timestep $t$ , the job of the memory (stack) controller is to decide whether (i) to push the current state (either $\mathbf { h } _ { t }$ or $\mathbf { x } _ { t }$ ) onto the stack, (ii) to pop the top element of the stack $\mathbf { m } _ { 0 }$ , or (iii) to stay and keep the stack state unchanged. The stack of Joulin & Mikolov (2015) is primarily designed as a single computation stack that performs one of the available operations at every timestep. In order to capture long-term dependencies, the stack learns to carry the information across multiple timesteps by mainly relying on the LSTM hidden states for predictions in between and keeping the state of the stack the same (i.e., by choosing to stay), or by pushing and popping the same number of times in between these timesteps. While this promotes discoveries of hierarchical dependencies, the kind of hierarchical dependencies that it can discover is limited. A multipop stack, on the other hand, has greater flexibility since there are more ways to manipulate its state at each timestep. The stack of Grefenstette et al. (2015) implicitly allows multiple pop operations in a single timestep by setting the pop weights to be greater than one. However, the controller makes this decision based only on the element at the top of the stack (along with the input and the current hidden state), making it less plausible to know whether more than one pop operations are needed since it does not look at other elements of the stack. Our formulation of the multipop operations is more intuitive and takes inspirations from adaptive computation time (Graves, 2017).
+
+Concretely, consider a stack memory with $K$ elements. In all our experiments, we limit the size of the stack to $K = 1 0$ for computational considerations. If the stack requires more than $K$ elements, the bottom element of the stack is removed to make space for the new element, which is added on the top of the stack. In a single computation stack, a feedforward policy network is used to compute the probability of choosing an action $a \in \{ \mathrm { S T A Y , P U S H , P O P } \}$ .
+
+In our stack, we also use a feedforward policy network, but the number of possible POPs is $k \in$ $\{ 0 , 1 , \ldots , K \}$ . Denote the current top two elements of the stack after performing $k$ pops by $\mathbf { m } _ { k , 0 }$ $\mathbf { m } _ { k , 1 }$ , and the state of the stack after $k$ pops by $\operatorname { S T A Y } _ { k }$ (i.e., do $k$ pops and stay) or $\mathrm { P U S H } _ { k }$ (i.e., do $k$ pops and push the current hidden state $\mathbf { h } _ { t }$ ). We compute the probability of choosing an action recursively:
+
+$$
+\begin{array} { r l } & { p ( { \mathrm { S T A Y } } _ { k } \mid \mathbf { x } _ { t } , \mathbf { M } ) = p ( { \mathrm { P O P } } _ { k - 1 } \mid \mathbf { x } _ { t } , \mathbf { m } _ { k - 1 , 0 } , \mathbf { m } _ { k - 1 , 1 } ) \times p ( { \mathrm { S T A Y } } _ { k } \mid \mathbf { x } _ { t } , \mathbf { m } _ { k , 0 } , \mathbf { m } _ { k , 1 } ) } \\ & { p ( { \mathrm { P U S H } } _ { k } \mid \mathbf { x } _ { t } , \mathbf { M } ) = p ( { \mathrm { P O P } } _ { k - 1 } \mid \mathbf { x } _ { t } , \mathbf { m } _ { k - 1 , 0 } , \mathbf { m } _ { k - 1 , 1 } ) \times p ( { \mathrm { P U S H } } _ { k } \mid \mathbf { x } _ { t } , \mathbf { m } _ { k , 0 } , \mathbf { m } _ { k , 1 } ) } \\ & { p ( { \mathrm { P O P } } _ { k } \mid \mathbf { x } _ { t } , \mathbf { M } ) = p ( { \mathrm { P O P } } _ { k - 1 } \mid \mathbf { x } _ { t } , \mathbf { m } _ { k - 1 , 0 } , \mathbf { m } _ { k - 1 , 1 } ) \times p ( { \mathrm { P O P } } _ { k } \mid \mathbf { x } _ { t } , \mathbf { m } _ { k , 0 } , \mathbf { m } _ { k , 1 } ) . } \end{array}
+$$
+
+As a base case, we have $p ( \mathrm { P O P } _ { - 1 } \mid \mathbf { x } _ { t } , \mathbf { m } _ { k - 1 , 0 } , \mathbf { m } _ { k - 1 , 1 } ) = 1$ . To ensure that the probability sums to one, we set $p \big ( \mathrm { P O P } _ { K + 1 } \mid \mathbf { x } _ { t } , \mathbf { m } _ { K , 0 } , \mathbf { m } _ { K , 1 } \big ) = 0$ .
+
+The final stack state is then computed as:
+
+$$
+\mathbf { M } = \sum _ { k = 0 } ^ { K } p ( { \operatorname { S T A Y } } _ { k } \mid \mathbf { x } _ { t } , \mathbf { M } ) \mathbf { M } _ { \operatorname { S T A Y } _ { k } } + p ( { \operatorname { P U S H } } _ { k } \mid \mathbf { x } _ { t } , \mathbf { M } ) \mathbf { M } _ { \operatorname { P U S H } _ { k } } ,
+$$
+
+where $\mathbf { M } _ { * _ { k } }$ is the stack state after performing $k$ POP and PUSH or STAY. Denote the top of the final stack at timestep $t$ as $\mathbf { m } _ { t }$ . The final representation is
+
+$$
+\tilde { \mathbf { h } } _ { t } = \mathbf { W } _ { h , h } \mathbf { h } _ { t } + \mathbf { W } _ { h , m } \mathbf { m } _ { t } .
+$$
+
+We propose to treat this stack as a fully differentiable continuous stack. Alternatively, our stack can also be treated as a discrete stack and trained with reinforcement learning (e.g., with REINFORCE; Williams, 1992). In this case, instead of summing over all possible stack states, we sample according to the probabilities. However, such methods can have slow convergence due to high variance. We include comparisons to discrete and continuous single computation stacks in our experiments (§3).
+
+Adaptive and Variable Computation Networks Previous work on adaptive computation time (Graves, 2017) consider the number of computations as “thinking time”, where they show that their models use more computation time for more difficult predictions. In our work, the number of computations is related to how further back we need to look back when making a prediction at a given timestep. Note that when we decide to push, we push the current hidden state $\mathbf { h } _ { t }$ onto the stack. Since this operation is performed before making a prediction at every timestep, it is possible to use the stack to increase the number of parameters for some predictions (i.e., by pushing $\mathbf { h } _ { t }$ and immediately use it to compute $\tilde { \mathbf { h } } _ { t } = \mathbf { W } _ { h , h } \mathbf { h } _ { t } + \mathbf { W } _ { h , m } \mathbf { m } _ { t }$ , because immediately after a push $\mathbf { m } _ { t } = \mathbf { h } _ { t }$ ). As a result, our stack is also related to variable computation recurrent networks Jernite et al. (2017) that decide the number of dimensions to be used at each timestep.
+
+# 3 EXPERIMENTS
+
+# 3.1 SETUP
+
+We compare the following methods in our experiments:
+
+• Sequential memory: 650-dimension and 920-dimension vanilla LSTMs (650 or 920 for both the word embedding and the LSTM hidden size).   
+• Random access memory: a 650-dimension attention-based LSTM with attention size $K =$ $\{ 1 , 3 , 5 , 1 0 , 1 5 \}$ .
+
+• Stack memory: a single computation discrete or continuous stack, or a multipop adaptive computation continuous stack on top of a 650-dimension LSTM.
+
+Following Inan et al. (2017), we tie word embedding and word classifier layers and apply dropout to these layers with probability 0.6 (value chosen based on preliminary experiment results). We also use recurrent dropout (Semeniuta et al., 2016) and set it to 0.1. We perform non-episodic training with batch size 32 using RMSprop (Hinton, 2012) as our optimization method. We tune the RMSprop learning rate and $\ell _ { 2 }$ regularization parameter for all models on a development set by random search from [0.004, 0.009] and [0.0001, 0.0005] respectively, and use perplexity on the development set to choose the best model.
+
+# 3.2 PERPLEXITY
+
+We use standard language modeling datasets, the Penn TreeBank (PTB) and Wikitext-2 (Wik-2) corpora to evaluate perplexity. Our main results are summarized in Table 1, where we also show comparisons with previous work on these datasets.
+
+Our basic LSTM model is comparable to some of the best LSTM models. The results show that increasing the sequential memory capacity by increasing the hidden size improves performance. However, the improvement is not as significant as adding random access or stack memory. The best attention model is the one with $K = 1 0$ and $K = 1 5$ on PTB and Wik-2 respectively, highlighting the necessity to tune to get the optimal window size. Our results generally agree with Daniluk et al. (2017) that show that increasing the attention size generally improves performance up to a certain threshold.
+
+While both the discrete and continuous single computation stack models perform reasonably well, the discrete model underperforms the continuous model on Wik-2. Recall that we use REINFORCE to learn the optimal discrete stack operations. We leave it to future work to investigate whether better techniques can be used to improve the performance of the discrete stack. The best model on both datasets is consistently the multipop stack.
+
+Overall perplexity on these datasets is strongly dominated by words that require little to no long term dependencies, making it difficult to assess when memory helps. In the next section, we look into a specifically designed linguistic task to get a better understanding of these memory models.
+
+# 3.3 SYNTACTIC DEPENDENCIES
+
+We evaluate these memory models for learning syntax-sensitive dependencies on the number prediction dataset from Linzen et al. (2016). In this dataset, the model is given a sentence up to—but not including—its verb, and the goal is to predict the number of the following verb (singular or plural). For example, given a sentence prefix with different numbers of intervening nouns:
+
+• this robot {is, are}   
+• the users he mentioned $\left\{ i s , a r e \right\}$   
+• many systems , in addition to VBG a page of free text for each knowledge element , also {permit, permits}
+
+the goal is to predict the correct verb form out of the possible answers in the brackets. In total, there are approximately 1.4 million test examples in this dataset with varying degrees of difficulty. One proxy to assess the difficulty of a test example is through the number of attractors (underlined)— which are defined as intervening nouns of the opposite singular/plural form to the subject. Each of the example above has zero, one, and four attractors, respectively. Naturally, examples with fewer numbers of attractors between the head of the syntactic subject and the predicted verb are easier than those with more. We follow the experimental setup in Linzen et al. (2016) and only use test examples where all the attractors are of contrasting form to the main subject (i.e., all intervening nouns between the subject and the verb must be plural if the subject is singular, and vice versa).
+
+One way to do this task is to train a binary classifier that takes the context and predicts an answer. We approach this task from a language modeling perspective, where we simply train a language model and take the word with the higher probability between the two possible answers as the prediction. Success on this task requires a language model that understands syntactic—and in some cases long term—dependencies in natural language. For a purely sequential memory model to do well on this task, it has to be able to carry dependencies over multiple timesteps and attractors. On the other hand, our memory augmented recurrent models need to use the random access or stack memory component in conjunction with the sequential memory of their LSTM core to capture these dependencies.
+
+Table 1: Perplexity on PTB and Wikitext-2 datasets. The two numbers $( { } ^ { * } { \bf M } / { } ^ { * } { \bf M } )$ in the # of params. column for models that we implemented denote the number of parameters for PTB and Wik-2 respectively.   
+
+<table><tr><td rowspan=2 colspan=1>Model</td><td rowspan=2 colspan=1>LSTMhidden size</td><td rowspan=2 colspan=1>#ofparams.</td><td rowspan=1 colspan=2>PTB</td><td rowspan=1 colspan=2>Wik-2</td></tr><tr><td rowspan=1 colspan=1>Dev</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Dev</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=9 colspan=1>Var LSTM(Gal &amp; Ghahramani, 2016)Var LSTM+REAL (Inan et al.,2017)Pointer LSTM (Merity et al., 2017b)Neural Cache (Grave et al., 2017)Neural Cache (Grave et al., 2017)NAS (Zoph &amp; Le,2017)Optimized LSTM (Melis et al., 2017)AWD LSTM (Merity et al., 2017a)AWD LSTM + Cache (Merity et al., 2017a)</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>20M</td><td rowspan=1 colspan=1>81.9</td><td rowspan=1 colspan=1>79.7</td><td rowspan=1 colspan=1>101.7</td><td rowspan=1 colspan=1>96.3</td></tr><tr><td rowspan=1 colspan=1>1500</td><td rowspan=1 colspan=1>51M</td><td rowspan=1 colspan=1>71.1</td><td rowspan=1 colspan=1>68.5</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=7 colspan=1>1=1</td><td rowspan=1 colspan=1>21M</td><td rowspan=1 colspan=1>72.4</td><td rowspan=1 colspan=1>70.9</td><td rowspan=1 colspan=1>84.8</td><td rowspan=4 colspan=1>80.881.668.91</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>72.1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=1>162.4</td><td rowspan=2 colspan=1>==</td></tr><tr><td rowspan=1 colspan=1>54M</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>24M</td><td rowspan=1 colspan=1>60.9</td><td rowspan=1 colspan=1>58.3</td><td rowspan=1 colspan=1>69.1</td><td rowspan=1 colspan=1>65.9</td></tr><tr><td rowspan=1 colspan=1>24M</td><td rowspan=1 colspan=1>60.0</td><td rowspan=1 colspan=1>57.3</td><td rowspan=1 colspan=1>68.6</td><td rowspan=1 colspan=1>65.8</td></tr><tr><td rowspan=1 colspan=1>24M</td><td rowspan=1 colspan=1>53.9</td><td rowspan=1 colspan=1>52.8</td><td rowspan=1 colspan=1>53.8</td><td rowspan=1 colspan=1>52.0</td></tr><tr><td rowspan=2 colspan=1>LSTMLSTM</td><td rowspan=2 colspan=1>650920</td><td rowspan=2 colspan=1>10M/25M16M/40M</td><td rowspan=2 colspan=1>69.267.8</td><td rowspan=1 colspan=1>67.2</td><td rowspan=1 colspan=1>83.9</td><td rowspan=1 colspan=1>80.8</td></tr><tr><td rowspan=1 colspan=1>65.4</td><td rowspan=1 colspan=1>79.8</td><td rowspan=1 colspan=1>77.4</td></tr><tr><td rowspan=1 colspan=1>Attention-1Attention-3Attention-5Attention-10Attention-15</td><td rowspan=1 colspan=1>650</td><td rowspan=1 colspan=1>12M/28M</td><td rowspan=1 colspan=1>68.667.967.567.266.6</td><td rowspan=1 colspan=1>66.165.465.264.763.6</td><td rowspan=1 colspan=1>80.478.778.277.677.7</td><td rowspan=1 colspan=1>76.374.674.673.774.3</td></tr><tr><td rowspan=1 colspan=1>Single Comp.Discrete StackSingle Comp. Continuous StackMultipop Adaptive Continuous Stack</td><td rowspan=1 colspan=1>650</td><td rowspan=1 colspan=1>11M/26M</td><td rowspan=1 colspan=1>66.165.865.9</td><td rowspan=1 colspan=1>63.563.863.5</td><td rowspan=1 colspan=1>78.176.775.9</td><td rowspan=1 colspan=1>74.773.072.4</td></tr></table>
+
+Linzen et al. (2016) concluded that a vanilla LSTM trained only with a language modeling signal is insufficient for capturing such dependencies. They reported an overall accuracy of 93.22 with a language modeling objective (using a 50 dimension LSTM), and 99.17 with a supervised binary classifier objective.
+
+We train the best vanilla LSTM (920 dimensions), the best attention-based LSTM ( $K = 1 0$ , since the Linzen dataset is derived from Wikipedia articles similar to Wik-2), and the best stack LSTM (multipop stack) on the provided training set that contains sentences of similar structures to the test set. There are approximately 3 million tokens on the training set $( \sim 1 4 0 , 0 0 0$ sentences). We tune the learning rate and $\ell _ { 2 }$ hyperparameter on the development set using perplexity as the tuning criterion.
+
+Our results are shown in Table 2. We report accuracies per number of attractors, as well as the overall accuracy and perplexity. Contrary to the Linzen et al. (2016) results, all of our language models perform surprisingly well on this dataset. Our vanilla LSTM model outperforms Linzen’s best LSTM by a significant margin (99.11 vs. 93.22). One possible reason is that we are able to train a much bigger LSTM than Linzen—almost 20 times bigger in hidden size. The results clearly demonstrate the improvements from adding random access and stack memory. The performance of the attention model slowly degrades to the performance of a vanilla LSTM model as the number of attractors increases, since it becomes more difficult for a random access memory mechanism to attend to the syntactic head in the presence of multiple attractors. For example, when there are five attractors, our attention model performs just as well as our vanilla LSTM model.
+
+<table><tr><td rowspan="2">Model</td><td colspan="6">Number of attractors</td><td rowspan="2">Acc.</td><td rowspan="2">Ppx.</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>Best LSTM</td><td>99.3</td><td>97.2</td><td>95.0</td><td>92.2</td><td>90.0</td><td>84.2</td><td>99.11</td><td>23.8</td></tr><tr><td>Best attention</td><td>99.4</td><td>97.7</td><td>95.9</td><td>92.9</td><td>90.7</td><td>84.2</td><td>99.18</td><td>22.7</td></tr><tr><td>Best stack</td><td>99.4</td><td>97.9</td><td>96.5</td><td>93.5</td><td>91.6</td><td>88.0</td><td>99.23</td><td>22.2</td></tr></table>
+
+Table 2: Accuracies on the Linzen number prediction dataset. 0, 1, 2, 3, 4, and 5 refer to the number of attractors between the subject and the predicted verb (see text for details).
+
+Table 3: Examples of mistakes made by competing models on the Linzen number prediction dataset. $\pmb { \chi }$ indicates an incorrect prediction, whereas $\checkmark$ indicates a correct prediction. In general, we observe that the mistakes made by both the LSTM and attention models that are correctly predicted by the stack model (row 2) typically involve longer sentences regardless of the number of attractors.   
+
+<table><tr><td colspan="3">Model</td><td rowspan="2">Example</td></tr><tr><td>LSTM</td><td>attention</td><td>stack</td></tr><tr><td>X</td><td>X</td><td>X</td><td>the NN notes and front cover title {is,are}</td></tr><tr><td>X</td><td>X</td><td>√</td><td>other NNS that in the recent past were part of the JJ parish {is,are}</td></tr><tr><td>X</td><td>√</td><td>X</td><td>the class ofall VBN sets with JJ functions as NNS {form,forms}</td></tr><tr><td>√</td><td>X</td><td>X</td><td>various brands of JJ compound or NN NN {helps,helpl</td></tr><tr><td>X</td><td>√</td><td>√</td><td>score based on penalties forfallen bars,NNS，{falls,fall}</td></tr><tr><td>√</td><td>X</td><td></td><td>the loss of basic needs providers VBG from VBN countries {has,havel</td></tr><tr><td>√</td><td>√</td><td>X</td><td>the construction of the JJwalls,floors,and VBGwalls {is,are}</td></tr></table>
+
+The stack model performs best on this dataset, across all numbers of attractors (except zero, tie with attention), Notably, the advantage of the stack model becomes more pronounced as the number of attractors increases. In $\ S 3 . 4$ , we analyze how the model uses its stack. We take this collection of results as evidence that a hierarchical bias introduced by a stack-like data structure helps the language model to learn better syntactic natural language dependencies.
+
+We also investigate whether mistakes are made on the same test examples. Figure 2 shows a Venn diagram of mistakes made by each of the models. Most mistakes (6115) are the same across all three models. It is clear that adding a stack or an attention mechanism improves a vanilla LSTM model, as shown by the significant decrease in the number of mistakes that are made only by LSTM (3942) to 1938 and 2534 respectively. Nonetheless, since there are still a large number of mistakes that are complementary, an interesting future direction is to combine all three kinds of memory models efficiently in a single language model. Table 3 shows examples of mistakes made by each of these models.
+
+![](images/46f0957461fd509a24ea960e566c7874cc11226280178219366d32a6ac07ee39.jpg)  
+Figure 2: A Venn diagram of mistakes made on the Linzen dataset.
+
+# 3.4 ANALYSIS
+
+In this section, we analyze how our stack model uses its memory to improve predictions. Recall that our adaptive continuous stack has 22 possible stack states at each timestep— $\mathrm { { S T A Y _ { 0 } } }$ , $\mathrm { P U S H _ { \mathrm { 0 } } }$ , $\mathrm { S T A Y _ { 1 } }$ , $\mathrm { P U S H _ { 1 } , \ldots , S T A Y _ { 1 0 } }$ , $\mathrm { P U S H _ { 1 0 } }$ —where the subscript indicates the number of pops that are performed, limited to $K = 1 0$ in our experiments. In Figure 3, we show the number of times each stack state has the highest probability on the Wik-2 test set (left figure, red denotes STAY and blue denotes PUSH) and the maximum action probability (right figure). The LSTM hidden states are not pushed onto the stack most of the times. We inspect what is pushed and find that the model uses the stack to mostly store hidden states for difficult predictions (more details below). While the overall distribution of the maximum action probability is not very peaky since the stack is not used most of the times, we also observe that the action probability is relatively peaky when the stack is activated.
+
+![](images/8de8028aa1b0101b2923084c9d384926dd4a433d64f8f60263369a5a35c478d2.jpg)
+
+![](images/4edfd539dc9daa067c4c014d9f96ffc9939418f47f416a04273bbe2cc565fabe.jpg)  
+Figure 3: Statistics of the stack actions on Wik-2 test dataset. The left plot shows the number of times (in log space) each stack state has the highest probability. The $x$ -axis represents the number of pops, and the color represents the last action taken after POP (red represents STAY, blue represents PUSH). We can see that most words are not pushed onto the stack and that the model takes advantage of the flexibility of the stack to pop multiple times. The right plot shows the value of the maximum action probability for all timesteps. We observe that for most words the maximum action probabilities are not peaky (the distribution is roughly uniform across all stack states). They tend to be peaky only when the model wants to use the stack.   
+Figure 4: An illustration of how the stack memory is written and read for a correct prediction. We follow Linzen et al. (2016) and convert some words (e.g., emigrating, impoverished) to their part of speech tags–given inside the brackets in the example above—to limit the vocabulary size.
+
+We next investigate how the stack improves accuracy on the Linzen dataset by looking into how it operates when making both a correct prediction and a wrong prediction. In Figure 4, we show a randomly selected test sentence the loss of basic needs providers VBG from VBN countries {has, have}. Similar to Figure 3, the red and blue bars represent STAY and PUSH, while the $x$ -axis denotes the number of POPs. The $y$ -axis, on the other hand, denotes the probability of each action. The green dots above each word shows the magnitude of the norm of the contribution of the memory vector $\mathbf { W } _ { h , m } \mathbf { m } _ { t }$ to the final hidden state $\tilde { \mathbf { h } } _ { t }$ (bigger dots represent higher magnitudes). Interestingly, the model seems to use and push onto the stack when the next word prediction has a high entropy. For example, after the word the, of, or from, the model decides to increase its capacity by pushing the current hidden state onto the stack and activates its memory component (as illustrated by bigger green dots). In $\ S 2 . 1$ , we discuss connections of our stack model to adaptive computation time and variable computation RNN, which are designed to explicitly increase their capacity for difficult predictions. Our analysis shows that our stack model also exhibits this kind of behavior. For the Linzen dataset, we conjecture that the stack model is able to perform the best because it uses the stack as a controller to allow the sequential memory component from its base LSTM to carry longer term dependencies (e.g., tracking the subject of the sentence).
+
+For comparison, we also show how the stack operates when it makes an incorrect prediction in Figure 5. Here, the stack behaves similarly, being mostly active for higher entropy predictions, although the model was unable to predict the correct verb right after walls.
+
+![](images/7baddf218e2ada04c49ac6de9edfa4c4b977e95bd0c02141e8c7443e5d6b909d.jpg)  
+Figure 5: An illustration of how the stack memory is written and read for an incorrect prediction.
+
+# 4 CONCLUSION
+
+We proposed a generalization of the continuous stack model that allows a variable number of pop operations, and compared sequential, random access, and stack memory architectures for recurrent neural network language models. Our experiments on PTB and Wik-2 showed that adding the stack memory eliminates the need to tune window size in the random access attention model to achieve the best perplexity. We also evaluated these models on the Linzen syntactic dependencies dataset and demonstrated that the stack augmented model outperforms other methods in terms of both accuracy and perplexity, especially as the number of syntactic attractors increases.
+
+# ACKNOWLEDGEMENTS
+
+The authors thank Edward Grefenstette for valuable feedback on an earlier draft of this paper, Tal Linzen for his assistance with the Linzen dataset, and the DeepMind language group for helpful discussions.
+
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+Tran, Ke, Bisazza, Arianna, and Monz, Christof. Recurrent memory networks for language modeling. In Proc. of NAACL-HLT, 2016.
+
+Williams, Ronald J. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229–256, 1992.
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+Zoph, Barret and Le, Quoc V. Neural architecture search with reinforcement learning. In Proc. of ICLR, 2017.

parse/train/SkFqf0lAZ/SkFqf0lAZ_content_list.json

@@ -0,0 +1,1139 @@
+[
+    {
+        "type": "text",
+        "text": "MEMORY ARCHITECTURES IN RECURRENT NEURAL NETWORK LANGUAGE MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dani Yogatama♣, Yishu Miao♠, Gabor Melis♣, Wang Ling♣, Adhiguna Kuncoro♣♠   \nChris Dyer♣, Phil Blunsom♣♠   \n♣DeepMind and ♠University of Oxford   \ndyogatama@google.com, yishu.miao@cs.ox.ac.uk   \n{melisgl,lingwang,akuncoro,cdyer,pblunsom}@google.com ",
+        "bbox": [
+            184,
+            167,
+            767,
+            242
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            279,
+            544,
+            294
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We compare and analyze sequential, random access, and stack memory architectures for recurrent neural network language models. Our experiments on the Penn Treebank and Wikitext-2 datasets show that stack-based memory architectures consistently achieve the best performance in terms of held out perplexity. We also propose a generalization to existing continuous stack models (Joulin & Mikolov, 2015; Grefenstette et al., 2015) to allow a variable number of pop operations more naturally that further improves performance. We further evaluate these language models in terms of their ability to capture non-local syntactic dependencies on a subject-verb agreement dataset (Linzen et al., 2016) and establish new state of the art results using memory augmented language models. Our results demonstrate the value of stack-structured memory for explaining the distribution of words in natural language, in line with linguistic theories claiming a context-free backbone for natural language. ",
+        "bbox": [
+            233,
+            310,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            517,
+            336,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sequential recurrent neural networks such as LSTMs (Hochreiter & Schmidhuber, 1997) are the basis of state-of-the-art models of natural language in various tasks. They effectively learn to capture dependencies between events separated in time by learning to store and retrieve information in a hidden state. However, the ability of these methods to discover long-term dependencies is limited by the capacity of their hidden state and the difficulty of propagating reliable gradients. For example, LSTM language models have been shown to struggle to capture non-sequential syntactic dependencies in complex sentences without explicit supervision (Linzen et al., 2016). As an illustration of the kind of dependencies they have difficulty learning, in the sentence, the loss of basic needs providers emigrating from impoverished countries has a damaging effect, correctly predicting singular has rather than its plural form have requires that the LSTM have learned that it depends on the subject, in this case the first noun (loss) rather than any of the intervening non-subject nouns, such as countries. Linzen et al. (2016) show that LSTM language models fail to capture this kind of dependencies, especially as the number of attractors (underlined) increases. ",
+        "bbox": [
+            174,
+            547,
+            825,
+            728
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Attempts to improve language models’ ability to capture non-local dependencies have recently been undertaken by introducing an external memory components. These include (i) a soft attention mechanism (Daniluk et al., 2017) and (ii) an explicit memory block or cache model (Tran et al., 2016; Grave et al., 2017). However, since very local context is often most highly informative for predicting the next word, existing memory-augmented RNN LMs use memory just to store information about local context (Daniluk et al., 2017). ",
+        "bbox": [
+            174,
+            736,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we compare several memory architectures for recurrent neural network language models. Since our goal is to evaluate how well these types of memory architectures learn long term and syntactic dependencies, we focus on language models that are static as opposed to non-static models such as neural cache (Grave et al., 2017) and dynamic evaluation (Krause et al., 2017) that can update their distribution at test time. We consider increasing the capacity of a purely sequential memory model by increasing the capacity of an LSTM, a random access memory model as typified by an attention-based LSTM, and a new variant of a stack augmented recurrent neural network. ",
+        "bbox": [
+            174,
+            827,
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+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Unlike random access memory models, a stack has a built-in bias to discover hierarchical structures that are important in language. A continuous stack memory has been proposed to improve recurrent neural networks (Joulin & Mikolov, 2015; Grefenstette et al., 2015), although it has never been carefully evaluated in benchmark language modeling experiments. In the only set of results for a stack augmented recurrent language model, Joulin & Mikolov (2015) show that a stack augmented vanilla RNN outperforms a standard RNN and is comparable to an LSTM. We augment an LSTM with a stack memory and perform thorough comparisons to evaluate its efficacy as a language model. In contrast to prior work, our continuous stack allows for push, stay, and a variable number of pop operations at each time step (multiple pop operations are useful in modeling natural language sentences since while only a single new word is presented at each time step, multiple syntactic units may come to an end concurrently). ",
+        "bbox": [
+            174,
+            103,
+            825,
+            256
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our motivating hypothesis is that allowing the memory to dynamically store and retrieve contextual information with a stack will drive the model to use the memory to learn dependencies that are difficult to capture by a sequential model. Sequential memory has an easier time learning local dependencies, but it often fails to capture long term dependencies. Random access memory models capture longer range dependencies (i.e., proportional to the window size), but the learner has to infer these from data without any informative structural bias. We hypothesize that introducing a more appropriate inductive bias will make it easier for the model to learn long range and structurally meaningful dependencies, given that the variance in learning such dependencies can be high. Linguistic insights reveal that one possible inductive bias is in the form of a hierarchical nested structure that captures syntactic dependencies. Stack memory models provide a natural way for capturing hierarchical structures, providing an easier path for gradients to flow to particular locations in the past. ",
+        "bbox": [
+            174,
+            263,
+            825,
+            416
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our main contributions in this paper are as follows: ",
+        "bbox": [
+            174,
+            422,
+            511,
+            438
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We thoroughly evaluate the efficacy of a stack augmented RNN as a language model and propose a more expressive extension to existing stack models (§2.1). We compare how a recurrent neural network uses a stack memory, a sequential memory cell (i.e., an LSTM memory cell), and a random access memory (i.e., an attention mechanism) for language modeling. Experiments on the Penn Treebank and Wikitext-2 datasets (§3.2) show that both the stack model and the attention-based model outperform the LSTM model with a comparable (or even larger) number of parameters, and that the stack model eliminates the need to tune window size to achieve the best perplexity.   \n• We assess the ability of these memory models to discover long range structural dependencies commonly encountered in natural language using the subject-verb agreement dataset (Linzen et al., 2016). We achieve new state of the art results and show that the gap in accuracy between a sequential or random access memory model with a stack model gets bigger as the dependencies become more complex (i.e., number of attractors increases; $\\ S 3 . 3 )$ . We also analyze the stack and find that the model tends to use it to enhance its sequential memory component in high entropy prediction contexts (§3.4). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 MODEL ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            681,
+            269,
+            698
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We consider a language modeling problem where the goal is to predict the next word $x _ { t }$ given previously seen context words $x _ { 0 } , \\ldots , x _ { t - 1 }$ . We represent each input word $x$ by its $D$ -dimensional embedding vector $\\mathbf { x } \\in \\mathbb { R } ^ { D }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our base model is an LSTM that computes a hidden state at timestep $t$ as follows: ",
+        "bbox": [
+            173,
+            762,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58ee6ca8fa2685034bdeba9e151a9927689f5456f22a812b166bd70c950acbce.jpg",
+        "text": "$$\n\\begin{array} { r l r l } & { \\mathbf i _ { t } = \\sigma \\big ( \\mathbf W _ { i , x } \\mathbf x _ { t } + \\mathbf W _ { i , h } \\mathbf h _ { t - 1 } + \\mathbf b _ { i } \\big ) \\quad } & & { \\mathbf f _ { t } = \\sigma \\big ( \\mathbf W _ { f , x } \\mathbf x _ { t } + \\mathbf W _ { f , h } \\mathbf h _ { t - 1 } + \\mathbf b _ { f } \\big ) } \\\\ & { \\mathbf o _ { t } = \\sigma \\big ( \\mathbf W _ { o , x } \\mathbf x _ { t } + \\mathbf W _ { o , h } \\mathbf h _ { t - 1 } + \\mathbf b _ { o } \\big ) \\quad } & & { \\mathbf g _ { t } = \\mathrm { t a n h } \\big ( \\mathbf W _ { g , x } \\mathbf x _ { t } + \\mathbf W _ { g , h } \\mathbf h _ { t - 1 } + \\mathbf b _ { g } \\big ) } \\\\ & { \\mathbf c _ { t } = \\mathbf f _ { t } \\odot \\mathbf c _ { t - 1 } + \\mathbf i _ { t } \\odot \\mathbf g _ { t } \\quad } & & { \\mathbf h _ { t } = \\mathbf o _ { t } \\odot \\mathrm { t a n h } \\big ( \\mathbf c _ { t } \\big ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Sequential memory. An LSTM has a sequential memory cell c to store and retrieve information that is regulated by the input, output, and forget gates. In order for long-term contextual information to be used in the future, it has to pass through these gates for multiple timesteps. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Random access memory. One common approach to retrieve information from the distant past more reliably is to augment the model with a random access memory block via an attention based ",
+        "bbox": [
+            173,
+            895,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/8f898c37d3bb5705ab491d7152de39c1031529481009bfc5be9228436d0ba168.jpg",
+        "image_caption": [
+            "Figure 1: Multipop Adaptive Computation Stack Recurrent Neural Network. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            254,
+            97,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "method. In this model, we consider the previous $K$ states as the memory block, and construct a memory vector $\\mathbf { m } _ { t }$ by a weighted combination of these states: ",
+        "bbox": [
+            168,
+            305,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/764bea1452fb85a1fee66d831bd40a5b5e98c685daf87f77ba609d8bfa118558.jpg",
+        "text": "$$\n\\mathbf { m } _ { t } = \\sum _ { i = t - K } ^ { t - 1 } a _ { i } \\mathbf { h } _ { i } , { \\mathrm { w h e r e ~ } } a _ { i } \\propto \\exp ( \\mathbf { w } _ { m , i } \\mathbf { h } _ { i } + \\mathbf { w } _ { m , h } \\mathbf { h } _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Such method can be improved further by partitioning $\\mathbf { h }$ into a key, value, and predict subvectors (Daniluk et al., 2017). ",
+        "bbox": [
+            173,
+            390,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given the LSTM hidden state $\\mathbf { h } _ { t }$ and the memory state $\\mathbf { m } _ { t }$ , we combine them using a simple function: ",
+        "bbox": [
+            174,
+            426,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/048affadf22bd83c70ecf8b23fe2e6d3a0b08c1e883811dc19eac7be5ce9811b.jpg",
+        "text": "$$\n\\tilde { \\mathbf { h } } _ { t } = \\mathbf { W } _ { h , h } \\mathbf { h } _ { t } + \\mathbf { W } _ { h , m } \\mathbf { m } _ { t }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            448,
+            589,
+            468
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "to get the final representation $\\tilde { \\mathbf { h } } _ { t }$ . We compute the probability of predicting the next word as $p ( x _ { t } \\mid \\pmb { x } _ { < t } ) \\propto \\exp ( \\mathbf { x } _ { t } ^ { \\top } \\tilde { \\mathbf { h } } _ { t } + b _ { y , x _ { t } } )$ , where we follow Inan et al. (2017) and reuse the word embedding matrix $\\mathbf { X }$ as the softmax parameters. ",
+        "bbox": [
+            173,
+            477,
+            826,
+            522
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Stack memory. In this work, we propose to augment a recurrent LSTM language model with a stack memory M that has three basic operations: ",
+        "bbox": [
+            173,
+            537,
+            823,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• PUSH: Push the current hidden state $\\mathbf { h } _ { t }$ onto the stack.   \n• POP: Remove the top element of the stack.   \n• STAY: Keep the stack unchanged. ",
+        "bbox": [
+            215,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows an illustration of our stack augmented RNN. We describe the stack memory in details in the followings. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 MULTIPOP ADAPTIVE COMPUTATION STACK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            689,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our stack is a multipop adaptive computation stack—it learns how many POP operations need to be performed before predicting an output. In previous work (Joulin & Mikolov, 2015; Grefenstette et al., 2015), at every timestep $t$ , the job of the memory (stack) controller is to decide whether (i) to push the current state (either $\\mathbf { h } _ { t }$ or $\\mathbf { x } _ { t }$ ) onto the stack, (ii) to pop the top element of the stack $\\mathbf { m } _ { 0 }$ , or (iii) to stay and keep the stack state unchanged. The stack of Joulin & Mikolov (2015) is primarily designed as a single computation stack that performs one of the available operations at every timestep. In order to capture long-term dependencies, the stack learns to carry the information across multiple timesteps by mainly relying on the LSTM hidden states for predictions in between and keeping the state of the stack the same (i.e., by choosing to stay), or by pushing and popping the same number of times in between these timesteps. While this promotes discoveries of hierarchical dependencies, the kind of hierarchical dependencies that it can discover is limited. A multipop stack, on the other hand, has greater flexibility since there are more ways to manipulate its state at each timestep. The stack of Grefenstette et al. (2015) implicitly allows multiple pop operations in a single timestep by setting the pop weights to be greater than one. However, the controller makes this decision based only on the element at the top of the stack (along with the input and the current hidden state), making it less plausible to know whether more than one pop operations are needed since it does not look at other elements of the stack. Our formulation of the multipop operations is more intuitive and takes inspirations from adaptive computation time (Graves, 2017). ",
+        "bbox": [
+            173,
+            715,
+            826,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
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+            146
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Concretely, consider a stack memory with $K$ elements. In all our experiments, we limit the size of the stack to $K = 1 0$ for computational considerations. If the stack requires more than $K$ elements, the bottom element of the stack is removed to make space for the new element, which is added on the top of the stack. In a single computation stack, a feedforward policy network is used to compute the probability of choosing an action $a \\in \\{ \\mathrm { S T A Y , P U S H , P O P } \\}$ . ",
+        "bbox": [
+            173,
+            152,
+            825,
+            223
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In our stack, we also use a feedforward policy network, but the number of possible POPs is $k \\in$ $\\{ 0 , 1 , \\ldots , K \\}$ . Denote the current top two elements of the stack after performing $k$ pops by $\\mathbf { m } _ { k , 0 }$ $\\mathbf { m } _ { k , 1 }$ , and the state of the stack after $k$ pops by $\\operatorname { S T A Y } _ { k }$ (i.e., do $k$ pops and stay) or $\\mathrm { P U S H } _ { k }$ (i.e., do $k$ pops and push the current hidden state $\\mathbf { h } _ { t }$ ). We compute the probability of choosing an action recursively: ",
+        "bbox": [
+            176,
+            229,
+            823,
+            300
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0a6b916ba4b26210efb0aefaf7a7de572ead50e17c5f9dfc1d5c3146396a604.jpg",
+        "text": "$$\n\\begin{array} { r l } & { p ( { \\mathrm { S T A Y } } _ { k } \\mid \\mathbf { x } _ { t } , \\mathbf { M } ) = p ( { \\mathrm { P O P } } _ { k - 1 } \\mid \\mathbf { x } _ { t } , \\mathbf { m } _ { k - 1 , 0 } , \\mathbf { m } _ { k - 1 , 1 } ) \\times p ( { \\mathrm { S T A Y } } _ { k } \\mid \\mathbf { x } _ { t } , \\mathbf { m } _ { k , 0 } , \\mathbf { m } _ { k , 1 } ) } \\\\ & { p ( { \\mathrm { P U S H } } _ { k } \\mid \\mathbf { x } _ { t } , \\mathbf { M } ) = p ( { \\mathrm { P O P } } _ { k - 1 } \\mid \\mathbf { x } _ { t } , \\mathbf { m } _ { k - 1 , 0 } , \\mathbf { m } _ { k - 1 , 1 } ) \\times p ( { \\mathrm { P U S H } } _ { k } \\mid \\mathbf { x } _ { t } , \\mathbf { m } _ { k , 0 } , \\mathbf { m } _ { k , 1 } ) } \\\\ & { p ( { \\mathrm { P O P } } _ { k } \\mid \\mathbf { x } _ { t } , \\mathbf { M } ) = p ( { \\mathrm { P O P } } _ { k - 1 } \\mid \\mathbf { x } _ { t } , \\mathbf { m } _ { k - 1 , 0 } , \\mathbf { m } _ { k - 1 , 1 } ) \\times p ( { \\mathrm { P O P } } _ { k } \\mid \\mathbf { x } _ { t } , \\mathbf { m } _ { k , 0 } , \\mathbf { m } _ { k , 1 } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            304,
+            779,
+            359
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As a base case, we have $p ( \\mathrm { P O P } _ { - 1 } \\mid \\mathbf { x } _ { t } , \\mathbf { m } _ { k - 1 , 0 } , \\mathbf { m } _ { k - 1 , 1 } ) = 1$ . To ensure that the probability sums to one, we set $p \\big ( \\mathrm { P O P } _ { K + 1 } \\mid \\mathbf { x } _ { t } , \\mathbf { m } _ { K , 0 } , \\mathbf { m } _ { K , 1 } \\big ) = 0$ . ",
+        "bbox": [
+            174,
+            363,
+            825,
+            392
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The final stack state is then computed as: ",
+        "bbox": [
+            174,
+            398,
+            442,
+            412
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/96ae7df6d373af50f4f2cbe3d74fd5e3448411dfa7d987a446dded7e02b51d36.jpg",
+        "text": "$$\n\\mathbf { M } = \\sum _ { k = 0 } ^ { K } p ( { \\operatorname { S T A Y } } _ { k } \\mid \\mathbf { x } _ { t } , \\mathbf { M } ) \\mathbf { M } _ { \\operatorname { S T A Y } _ { k } } + p ( { \\operatorname { P U S H } } _ { k } \\mid \\mathbf { x } _ { t } , \\mathbf { M } ) \\mathbf { M } _ { \\operatorname { P U S H } _ { k } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            417,
+            715,
+            462
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { M } _ { * _ { k } }$ is the stack state after performing $k$ POP and PUSH or STAY. Denote the top of the final stack at timestep $t$ as $\\mathbf { m } _ { t }$ . The final representation is ",
+        "bbox": [
+            173,
+            467,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37a5896496d46bf4bceb7807cdd8f6452d3f97a786cc7047b9bca267ee8b98e9.jpg",
+        "text": "$$\n\\tilde { \\mathbf { h } } _ { t } = \\mathbf { W } _ { h , h } \\mathbf { h } _ { t } + \\mathbf { W } _ { h , m } \\mathbf { m } _ { t } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            405,
+            501,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We propose to treat this stack as a fully differentiable continuous stack. Alternatively, our stack can also be treated as a discrete stack and trained with reinforcement learning (e.g., with REINFORCE; Williams, 1992). In this case, instead of summing over all possible stack states, we sample according to the probabilities. However, such methods can have slow convergence due to high variance. We include comparisons to discrete and continuous single computation stacks in our experiments (§3). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Adaptive and Variable Computation Networks Previous work on adaptive computation time (Graves, 2017) consider the number of computations as “thinking time”, where they show that their models use more computation time for more difficult predictions. In our work, the number of computations is related to how further back we need to look back when making a prediction at a given timestep. Note that when we decide to push, we push the current hidden state $\\mathbf { h } _ { t }$ onto the stack. Since this operation is performed before making a prediction at every timestep, it is possible to use the stack to increase the number of parameters for some predictions (i.e., by pushing $\\mathbf { h } _ { t }$ and immediately use it to compute $\\tilde { \\mathbf { h } } _ { t } = \\mathbf { W } _ { h , h } \\mathbf { h } _ { t } + \\mathbf { W } _ { h , m } \\mathbf { m } _ { t }$ , because immediately after a push $\\mathbf { m } _ { t } = \\mathbf { h } _ { t }$ ). As a result, our stack is also related to variable computation recurrent networks Jernite et al. (2017) that decide the number of dimensions to be used at each timestep. ",
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+        "text": "3 EXPERIMENTS ",
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+        "text": "3.1 SETUP ",
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+        "text": "We compare the following methods in our experiments: ",
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+        "text": "• Sequential memory: 650-dimension and 920-dimension vanilla LSTMs (650 or 920 for both the word embedding and the LSTM hidden size).   \n• Random access memory: a 650-dimension attention-based LSTM with attention size $K =$ $\\{ 1 , 3 , 5 , 1 0 , 1 5 \\}$ . ",
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+        "text": "• Stack memory: a single computation discrete or continuous stack, or a multipop adaptive computation continuous stack on top of a 650-dimension LSTM. ",
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+        "text": "Following Inan et al. (2017), we tie word embedding and word classifier layers and apply dropout to these layers with probability 0.6 (value chosen based on preliminary experiment results). We also use recurrent dropout (Semeniuta et al., 2016) and set it to 0.1. We perform non-episodic training with batch size 32 using RMSprop (Hinton, 2012) as our optimization method. We tune the RMSprop learning rate and $\\ell _ { 2 }$ regularization parameter for all models on a development set by random search from [0.004, 0.009] and [0.0001, 0.0005] respectively, and use perplexity on the development set to choose the best model. ",
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+        "text": "3.2 PERPLEXITY ",
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+        "type": "text",
+        "text": "We use standard language modeling datasets, the Penn TreeBank (PTB) and Wikitext-2 (Wik-2) corpora to evaluate perplexity. Our main results are summarized in Table 1, where we also show comparisons with previous work on these datasets. ",
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+        "text": "Our basic LSTM model is comparable to some of the best LSTM models. The results show that increasing the sequential memory capacity by increasing the hidden size improves performance. However, the improvement is not as significant as adding random access or stack memory. The best attention model is the one with $K = 1 0$ and $K = 1 5$ on PTB and Wik-2 respectively, highlighting the necessity to tune to get the optimal window size. Our results generally agree with Daniluk et al. (2017) that show that increasing the attention size generally improves performance up to a certain threshold. ",
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+        "type": "text",
+        "text": "While both the discrete and continuous single computation stack models perform reasonably well, the discrete model underperforms the continuous model on Wik-2. Recall that we use REINFORCE to learn the optimal discrete stack operations. We leave it to future work to investigate whether better techniques can be used to improve the performance of the discrete stack. The best model on both datasets is consistently the multipop stack. ",
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+        "text": "Overall perplexity on these datasets is strongly dominated by words that require little to no long term dependencies, making it difficult to assess when memory helps. In the next section, we look into a specifically designed linguistic task to get a better understanding of these memory models. ",
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+        "type": "text",
+        "text": "3.3 SYNTACTIC DEPENDENCIES ",
+        "text_level": 1,
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+        "text": "We evaluate these memory models for learning syntax-sensitive dependencies on the number prediction dataset from Linzen et al. (2016). In this dataset, the model is given a sentence up to—but not including—its verb, and the goal is to predict the number of the following verb (singular or plural). For example, given a sentence prefix with different numbers of intervening nouns: ",
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+        "text": "• this robot {is, are}   \n• the users he mentioned $\\left\\{ i s , a r e \\right\\}$   \n• many systems , in addition to VBG a page of free text for each knowledge element , also {permit, permits} ",
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+        "text": "the goal is to predict the correct verb form out of the possible answers in the brackets. In total, there are approximately 1.4 million test examples in this dataset with varying degrees of difficulty. One proxy to assess the difficulty of a test example is through the number of attractors (underlined)— which are defined as intervening nouns of the opposite singular/plural form to the subject. Each of the example above has zero, one, and four attractors, respectively. Naturally, examples with fewer numbers of attractors between the head of the syntactic subject and the predicted verb are easier than those with more. We follow the experimental setup in Linzen et al. (2016) and only use test examples where all the attractors are of contrasting form to the main subject (i.e., all intervening nouns between the subject and the verb must be plural if the subject is singular, and vice versa). ",
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+        "text": "One way to do this task is to train a binary classifier that takes the context and predicts an answer. We approach this task from a language modeling perspective, where we simply train a language model and take the word with the higher probability between the two possible answers as the prediction. Success on this task requires a language model that understands syntactic—and in some cases long term—dependencies in natural language. For a purely sequential memory model to do well on this task, it has to be able to carry dependencies over multiple timesteps and attractors. On the other hand, our memory augmented recurrent models need to use the random access or stack memory component in conjunction with the sequential memory of their LSTM core to capture these dependencies. ",
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+        "table_caption": [
+            "Table 1: Perplexity on PTB and Wikitext-2 datasets. The two numbers $( { } ^ { * } { \\bf M } / { } ^ { * } { \\bf M } )$ in the # of params. column for models that we implemented denote the number of parameters for PTB and Wik-2 respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=2 colspan=1>Model</td><td rowspan=2 colspan=1>LSTMhidden size</td><td rowspan=2 colspan=1>#ofparams.</td><td rowspan=1 colspan=2>PTB</td><td rowspan=1 colspan=2>Wik-2</td></tr><tr><td rowspan=1 colspan=1>Dev</td><td rowspan=1 colspan=1>Test</td><td rowspan=1 colspan=1>Dev</td><td rowspan=1 colspan=1>Test</td></tr><tr><td rowspan=9 colspan=1>Var LSTM(Gal &amp; Ghahramani, 2016)Var LSTM+REAL (Inan et al.,2017)Pointer LSTM (Merity et al., 2017b)Neural Cache (Grave et al., 2017)Neural Cache (Grave et al., 2017)NAS (Zoph &amp; Le,2017)Optimized LSTM (Melis et al., 2017)AWD LSTM (Merity et al., 2017a)AWD LSTM + Cache (Merity et al., 2017a)</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>20M</td><td rowspan=1 colspan=1>81.9</td><td rowspan=1 colspan=1>79.7</td><td rowspan=1 colspan=1>101.7</td><td rowspan=1 colspan=1>96.3</td></tr><tr><td rowspan=1 colspan=1>1500</td><td rowspan=1 colspan=1>51M</td><td rowspan=1 colspan=1>71.1</td><td rowspan=1 colspan=1>68.5</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=7 colspan=1>1=1</td><td rowspan=1 colspan=1>21M</td><td rowspan=1 colspan=1>72.4</td><td rowspan=1 colspan=1>70.9</td><td rowspan=1 colspan=1>84.8</td><td rowspan=4 colspan=1>80.881.668.91</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>72.1</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=2 colspan=1>162.4</td><td rowspan=2 colspan=1>==</td></tr><tr><td rowspan=1 colspan=1>54M</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>24M</td><td rowspan=1 colspan=1>60.9</td><td rowspan=1 colspan=1>58.3</td><td rowspan=1 colspan=1>69.1</td><td rowspan=1 colspan=1>65.9</td></tr><tr><td rowspan=1 colspan=1>24M</td><td rowspan=1 colspan=1>60.0</td><td rowspan=1 colspan=1>57.3</td><td rowspan=1 colspan=1>68.6</td><td rowspan=1 colspan=1>65.8</td></tr><tr><td rowspan=1 colspan=1>24M</td><td rowspan=1 colspan=1>53.9</td><td rowspan=1 colspan=1>52.8</td><td rowspan=1 colspan=1>53.8</td><td rowspan=1 colspan=1>52.0</td></tr><tr><td rowspan=2 colspan=1>LSTMLSTM</td><td rowspan=2 colspan=1>650920</td><td rowspan=2 colspan=1>10M/25M16M/40M</td><td rowspan=2 colspan=1>69.267.8</td><td rowspan=1 colspan=1>67.2</td><td rowspan=1 colspan=1>83.9</td><td rowspan=1 colspan=1>80.8</td></tr><tr><td rowspan=1 colspan=1>65.4</td><td rowspan=1 colspan=1>79.8</td><td rowspan=1 colspan=1>77.4</td></tr><tr><td rowspan=1 colspan=1>Attention-1Attention-3Attention-5Attention-10Attention-15</td><td rowspan=1 colspan=1>650</td><td rowspan=1 colspan=1>12M/28M</td><td rowspan=1 colspan=1>68.667.967.567.266.6</td><td rowspan=1 colspan=1>66.165.465.264.763.6</td><td rowspan=1 colspan=1>80.478.778.277.677.7</td><td rowspan=1 colspan=1>76.374.674.673.774.3</td></tr><tr><td rowspan=1 colspan=1>Single Comp.Discrete StackSingle Comp. Continuous StackMultipop Adaptive Continuous Stack</td><td rowspan=1 colspan=1>650</td><td rowspan=1 colspan=1>11M/26M</td><td rowspan=1 colspan=1>66.165.865.9</td><td rowspan=1 colspan=1>63.563.863.5</td><td rowspan=1 colspan=1>78.176.775.9</td><td rowspan=1 colspan=1>74.773.072.4</td></tr></table>",
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+        "text": "Linzen et al. (2016) concluded that a vanilla LSTM trained only with a language modeling signal is insufficient for capturing such dependencies. They reported an overall accuracy of 93.22 with a language modeling objective (using a 50 dimension LSTM), and 99.17 with a supervised binary classifier objective. ",
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+        "type": "text",
+        "text": "We train the best vanilla LSTM (920 dimensions), the best attention-based LSTM ( $K = 1 0$ , since the Linzen dataset is derived from Wikipedia articles similar to Wik-2), and the best stack LSTM (multipop stack) on the provided training set that contains sentences of similar structures to the test set. There are approximately 3 million tokens on the training set $( \\sim 1 4 0 , 0 0 0$ sentences). We tune the learning rate and $\\ell _ { 2 }$ hyperparameter on the development set using perplexity as the tuning criterion. ",
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+        "text": "Our results are shown in Table 2. We report accuracies per number of attractors, as well as the overall accuracy and perplexity. Contrary to the Linzen et al. (2016) results, all of our language models perform surprisingly well on this dataset. Our vanilla LSTM model outperforms Linzen’s best LSTM by a significant margin (99.11 vs. 93.22). One possible reason is that we are able to train a much bigger LSTM than Linzen—almost 20 times bigger in hidden size. The results clearly demonstrate the improvements from adding random access and stack memory. The performance of the attention model slowly degrades to the performance of a vanilla LSTM model as the number of attractors increases, since it becomes more difficult for a random access memory mechanism to attend to the syntactic head in the presence of multiple attractors. For example, when there are five attractors, our attention model performs just as well as our vanilla LSTM model. ",
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+        "table_footnote": [
+            "Table 2: Accuracies on the Linzen number prediction dataset. 0, 1, 2, 3, 4, and 5 refer to the number of attractors between the subject and the predicted verb (see text for details). "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"6\">Number of attractors</td><td rowspan=\"2\">Acc.</td><td rowspan=\"2\">Ppx.</td></tr><tr><td>0</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>Best LSTM</td><td>99.3</td><td>97.2</td><td>95.0</td><td>92.2</td><td>90.0</td><td>84.2</td><td>99.11</td><td>23.8</td></tr><tr><td>Best attention</td><td>99.4</td><td>97.7</td><td>95.9</td><td>92.9</td><td>90.7</td><td>84.2</td><td>99.18</td><td>22.7</td></tr><tr><td>Best stack</td><td>99.4</td><td>97.9</td><td>96.5</td><td>93.5</td><td>91.6</td><td>88.0</td><td>99.23</td><td>22.2</td></tr></table>",
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+        "table_caption": [
+            "Table 3: Examples of mistakes made by competing models on the Linzen number prediction dataset. $\\pmb { \\chi }$ indicates an incorrect prediction, whereas $\\checkmark$ indicates a correct prediction. In general, we observe that the mistakes made by both the LSTM and attention models that are correctly predicted by the stack model (row 2) typically involve longer sentences regardless of the number of attractors. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Model</td><td rowspan=\"2\">Example</td></tr><tr><td>LSTM</td><td>attention</td><td>stack</td></tr><tr><td>X</td><td>X</td><td>X</td><td>the NN notes and front cover title {is,are}</td></tr><tr><td>X</td><td>X</td><td>√</td><td>other NNS that in the recent past were part of the JJ parish {is,are}</td></tr><tr><td>X</td><td>√</td><td>X</td><td>the class ofall VBN sets with JJ functions as NNS {form,forms}</td></tr><tr><td>√</td><td>X</td><td>X</td><td>various brands of JJ compound or NN NN {helps,helpl</td></tr><tr><td>X</td><td>√</td><td>√</td><td>score based on penalties forfallen bars,NNS，{falls,fall}</td></tr><tr><td>√</td><td>X</td><td></td><td>the loss of basic needs providers VBG from VBN countries {has,havel</td></tr><tr><td>√</td><td>√</td><td>X</td><td>the construction of the JJwalls,floors,and VBGwalls {is,are}</td></tr></table>",
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+        "type": "text",
+        "text": "The stack model performs best on this dataset, across all numbers of attractors (except zero, tie with attention), Notably, the advantage of the stack model becomes more pronounced as the number of attractors increases. In $\\ S 3 . 4$ , we analyze how the model uses its stack. We take this collection of results as evidence that a hierarchical bias introduced by a stack-like data structure helps the language model to learn better syntactic natural language dependencies. ",
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+        "text": "We also investigate whether mistakes are made on the same test examples. Figure 2 shows a Venn diagram of mistakes made by each of the models. Most mistakes (6115) are the same across all three models. It is clear that adding a stack or an attention mechanism improves a vanilla LSTM model, as shown by the significant decrease in the number of mistakes that are made only by LSTM (3942) to 1938 and 2534 respectively. Nonetheless, since there are still a large number of mistakes that are complementary, an interesting future direction is to combine all three kinds of memory models efficiently in a single language model. Table 3 shows examples of mistakes made by each of these models. ",
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+        "img_path": "images/46f0957461fd509a24ea960e566c7874cc11226280178219366d32a6ac07ee39.jpg",
+        "image_caption": [
+            "Figure 2: A Venn diagram of mistakes made on the Linzen dataset. "
+        ],
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+        "text": "3.4 ANALYSIS ",
+        "text_level": 1,
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+        "text": "In this section, we analyze how our stack model uses its memory to improve predictions. Recall that our adaptive continuous stack has 22 possible stack states at each timestep— $\\mathrm { { S T A Y _ { 0 } } }$ , $\\mathrm { P U S H _ { \\mathrm { 0 } } }$ , $\\mathrm { S T A Y _ { 1 } }$ , $\\mathrm { P U S H _ { 1 } , \\ldots , S T A Y _ { 1 0 } }$ , $\\mathrm { P U S H _ { 1 0 } }$ —where the subscript indicates the number of pops that are performed, limited to $K = 1 0$ in our experiments. In Figure 3, we show the number of times each stack state has the highest probability on the Wik-2 test set (left figure, red denotes STAY and blue denotes PUSH) and the maximum action probability (right figure). The LSTM hidden states are not pushed onto the stack most of the times. We inspect what is pushed and find that the model uses the stack to mostly store hidden states for difficult predictions (more details below). While the overall distribution of the maximum action probability is not very peaky since the stack is not used most of the times, we also observe that the action probability is relatively peaky when the stack is activated. ",
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+        "img_path": "images/4edfd539dc9daa067c4c014d9f96ffc9939418f47f416a04273bbe2cc565fabe.jpg",
+        "image_caption": [
+            "Figure 3: Statistics of the stack actions on Wik-2 test dataset. The left plot shows the number of times (in log space) each stack state has the highest probability. The $x$ -axis represents the number of pops, and the color represents the last action taken after POP (red represents STAY, blue represents PUSH). We can see that most words are not pushed onto the stack and that the model takes advantage of the flexibility of the stack to pop multiple times. The right plot shows the value of the maximum action probability for all timesteps. We observe that for most words the maximum action probabilities are not peaky (the distribution is roughly uniform across all stack states). They tend to be peaky only when the model wants to use the stack. ",
+            "Figure 4: An illustration of how the stack memory is written and read for a correct prediction. We follow Linzen et al. (2016) and convert some words (e.g., emigrating, impoverished) to their part of speech tags–given inside the brackets in the example above—to limit the vocabulary size. "
+        ],
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+        "text": "We next investigate how the stack improves accuracy on the Linzen dataset by looking into how it operates when making both a correct prediction and a wrong prediction. In Figure 4, we show a randomly selected test sentence the loss of basic needs providers VBG from VBN countries {has, have}. Similar to Figure 3, the red and blue bars represent STAY and PUSH, while the $x$ -axis denotes the number of POPs. The $y$ -axis, on the other hand, denotes the probability of each action. The green dots above each word shows the magnitude of the norm of the contribution of the memory vector $\\mathbf { W } _ { h , m } \\mathbf { m } _ { t }$ to the final hidden state $\\tilde { \\mathbf { h } } _ { t }$ (bigger dots represent higher magnitudes). Interestingly, the model seems to use and push onto the stack when the next word prediction has a high entropy. For example, after the word the, of, or from, the model decides to increase its capacity by pushing the current hidden state onto the stack and activates its memory component (as illustrated by bigger green dots). In $\\ S 2 . 1$ , we discuss connections of our stack model to adaptive computation time and variable computation RNN, which are designed to explicitly increase their capacity for difficult predictions. Our analysis shows that our stack model also exhibits this kind of behavior. For the Linzen dataset, we conjecture that the stack model is able to perform the best because it uses the stack as a controller to allow the sequential memory component from its base LSTM to carry longer term dependencies (e.g., tracking the subject of the sentence). ",
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+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For comparison, we also show how the stack operates when it makes an incorrect prediction in Figure 5. Here, the stack behaves similarly, being mostly active for higher entropy predictions, although the model was unable to predict the correct verb right after walls. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/7baddf218e2ada04c49ac6de9edfa4c4b977e95bd0c02141e8c7443e5d6b909d.jpg",
+        "image_caption": [
+            "Figure 5: An illustration of how the stack memory is written and read for an incorrect prediction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            161,
+            790,
+            502
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "4 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            560,
+            318,
+            575
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We proposed a generalization of the continuous stack model that allows a variable number of pop operations, and compared sequential, random access, and stack memory architectures for recurrent neural network language models. Our experiments on PTB and Wik-2 showed that adding the stack memory eliminates the need to tune window size in the random access attention model to achieve the best perplexity. We also evaluated these models on the Linzen syntactic dependencies dataset and demonstrated that the stack augmented model outperforms other methods in terms of both accuracy and perplexity, especially as the number of syntactic attractors increases. ",
+        "bbox": [
+            174,
+            592,
+            825,
+            689
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            708,
+            334,
+            720
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The authors thank Edward Grefenstette for valuable feedback on an earlier draft of this paper, Tal Linzen for his assistance with the Linzen dataset, and the DeepMind language group for helpful discussions. ",
+        "bbox": [
+            176,
+            733,
+            825,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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parse/train/V5V1vGrI2z/V5V1vGrI2z.md

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+# Exploring Cross-Video and Cross-Modality Signals for Weakly-Supervised Audio-Visual Video Parsing
+
+Yan-Bo Lin1,2 Hung-Yu Tseng3 Hsin-Ying Lee4 Yen-Yu Lin1 Ming-Hsuan Yang3,5,6
+
+1National Yang Ming Chiao Tung University 2UNC Chapel Hill 3UC Merc 4Snap Research 5Google Research 6Yonsei University yblin@unc.edu htseng6@ucmerced.edu hlee5@snap.com lin@cs.nctu.edu.tw mhyang@ucmerced.edu
+
+# Abstract
+
+The audio-visual video parsing task aims to temporally parse a video into audio or visual event categories. However, it is labor-intensive to temporally annotate audio and visual events and thus hampers the learning of a parsing model. To this end, we propose to explore additional cross-video and cross-modality supervisory signals to facilitate weakly-supervised audio-visual video parsing. The proposed method exploits both the common and diverse event semantics across videos to identify audio or visual events. In addition, our method explores event co-occurrence across audio, visual, and audio-visual streams. We leverage the explored cross-modality co-occurrence to localize segments of target events while excluding irrelevant ones. The discovered supervisory signals across different videos and modalities can greatly facilitate the training with only video-level annotations. Quantitative and qualitative results demonstrate that the proposed method performs favorably against existing methods on weakly-supervised audio-visual video parsing.
+
+# 1 Introduction
+
+Humans perceive multisensory signals via seeing, hearing, touching, etc., and obtain multimodal information while exploring the surrounding environments. Visual and audio signals, the most common modalities, motivate researchers to jointly comprehend audio-visual events (e.g., see people singing and hear their sounds) [1, 2, 3, 4, 5, 6, 7]. Events visible in images while hearable in audio are referred to as audio-visual events. However, learning-based models tend to recognize a particular audio-visual event by using the data from the dominant modality with richer information and overlook clues from either audio only or visual only events which still contribute to holistic video understanding. Therefore, the resultant models can generalize well on audio-visual events only instead of comprehensively understanding all kinds of video events. To address this issue, we target at audio-visual video parsing [4, 6] where predictions for audio, visual, and audio-visual events with temporal boundaries are all required but separately evaluated.
+
+The time-consuming and labor-intensive annotation process poses a major challenge for the audiovisual video parsing task. To address this issue, Tian et al. [4] handle this task in a weakly-supervised manner given only video-level labels, which indicate events of presence without temporal boundaries and detailed modalities. They develop an audio-visual co-attention mechanism to assemble discriminative multimodal representations and use multiple instance learning to aggregate frame-level predictions into video-level ones. However, video-level labels alone cannot identify which modality events are from. Wu et al. [6] then propose to perform label refinement by swapping the audio and visual tracks of different videos to estimate and remove irrelevant event categories for each modality. They further adopt temporal contrastive learning to align audio and visual representations from the same frame. However, the contrastive learning is based on the assumption that audio and visual signals are synchronized, which may not hold in practical scenarios with complex events. Furthermore, these methods [4, 6] only consider audio and visual tracks of a single video without exploiting the relationship across categories and videos, which also provide rich shared semantics regarding event categories.
+
+In this work, we propose to leverage audio and visual data across different videos to explore shared information of each category. For example, videos with singing events may have similar patterns whatever in an audio or a visual modality. By observing all videos in a training batch, we can not only explore the shared semantics among audio-visual data but also exclude unrelated events. In addition to the relationship across different videos, we exploit the dependency between event categories. For example, when people are singing, there is usually a music accompaniment. Therefore, we propose to treat audio, visual, and audio-visual streams separately and adopt an audio-visual class co-occurrence module that jointly explores the relationship of different categories among all streams. By measuring the similarity of event categories from audio, visual, and audio-visual events, the correlated events are more likely to be correctly determined as the presence or absence of event categories. Such a strategy can robustly learn the correlation of categories within/across modality and fully exploit video data. The proposed strategy can be applied to existing methods on video parsing.
+
+We evaluate the proposed method on the LLP [4] dataset. Videos are parsed into audio, visual, and audio-visual events under both segment and event levels, and evaluated with F-scores metrics. Both qualitative and quantitative results demonstrate the effectiveness of the proposed method on the audio-visual video parsing task. The main contributions of this work are summarized as follows:
+
+• We leverage audio and visual data across different videos and tracks, which can learn common semantics of the same events and discern unrelated clues. • We develop an audio-visual event co-occurrence module that jointly considers the relationship of categories in audio, visual, and audio-visual modalities, which can prevent models from differentiating the representations of the related events. • Qualitative and quantitative experimental results on the benchmark dataset demonstrate that the proposed method performs favorably against the state-of-the-arts in various settings.
+
+# 2 Related Work
+
+Audio-Visual Representation Learning. Implicit correlation between audio and visual data from videos provides rich information for audio-visual representation learning. First, the audio-visual pairs from the same video clip [8, 9, 10, 11, 12, 13, 14, 15, 16, 17] are strongly correlated based on the assumption that audio and visual data from a video are synchronized and highly correlated. Moreover, features extracted from unpaired video clips tend to be more diverse than those from the same clips. Second, by exploring audio-visual temporal synchronization [18, 19], temporal information can be served as a training guidance. Given a video sequence, existing methods [18, 19] distinguish audio and visual features from different frames while correlating features from the same frames. Such an idea enhances robust audio-visual representation learning that is essential to several tasks such as audio-visual event localization/parsing/recognition [1, 2, 3, 4, 5, 6, 7, 20], sound separation [21, 22, 23, 24, 25, 26, 27, 28, 29, 30], audio spatialization [31, 32, 33, 34, 35, 36, 37, 38], and sound localization [39, 40, 41, 42, 43, 44]. Instead of random sampling sound and images, our method selects both related and irreverent videos to explore common semantics and discern dissimilar events.
+
+Audio-Visual Video Event Localization and Parsing. Audio-visual video parsing aims to detect events in videos and identify audio, visual, and audio-visual events (e.g., seeing the event and hearing its sound) and activities. Videos can be parsed with event categories and boundaries in both audio and visual modalities. Early researches [5, 7, 2, 3] aim to jointly derive audiovisual information in each local segment of the input video for audio-visual event localization, which emphasizes to detect only audio-visual events. However, due to the inconsistent information observed from audio and visual signals, data from either modality with insufficient clues may degrade the performance of prediction. Therefore, the work [7] focuses on audio/visual data with relevant categorical events to tackle this issue. Although methods of this category present favorable results, they are applicable to audio-visual event localization, which considers only synchronous audio-visual events or not. Recently, multi-modal multiple instance learning (MMIL) based methods with hybrid attention [4] carry out weakly-supervised audio-visual video parsing. These methods aggregate segment-level predictions into video-level ones, with which optimizing a model by using video-level or weak labels is enabled. Since video-level labels are typically insufficient to identify either audio or visual events, Wu et al. [6] generate pseudo labels for each modality by exchanging audio and visual tracks between unrelated videos. However, we notice that videos with replaced sounds or images may share some common semantics. Our method can exploit videos in a training batch to extract their common semantics for a categorical event and discern unrelated clues. Furthermore, we can leverage the relationship between event classes to find out related events (e.g., singing may accompany music).
+
+![](images/270ce2aeb0adebd19f77c3fe3a37657e6dbf1b204e12e9a0401ee9485ffe46c2.jpg)  
+Figure 1: Algorithmic overview. Our framework consists of a visual feature extractor, an audio feature extractor, a feature aggregation module, MMIL pooling, shared cross-modality semantics, and an cross-modality co-occurrence module. Given $n$ videos of $T$ seconds, the visual and audio feature extractors compute their visual and audio features. The feature aggregation module [4] conducts self- and cross-modality attention to aggregate segment-wise audio $\bar { \hat { \mathbf { f } ^ { a } } }$ and visual $\hat { \mathbf { f } ^ { v } }$ representations. We map segment-wise aggregated features to class-specific features by exploring cross-modality co-occurrence. By performing self- and cross-modality attention for class features, we identify within and cross modalities relationship between classes for event predictions. Note that $\otimes$ denotes matrix multiplication with the softmax operation performing on each row, and the green block only shows the example for segment-wise visual prediction at time $t$ . We also leverage the aggregated features of all $n$ videos to figure out common semantics regrading events by maximizing the similarities between related videos while minimizing those between unrelated videos with Eq. 8. The MMIL Pooling [4] is an attention-based pooling function that aggregates segment-wise results to produce video-level ones, which are optimized by the binary cross entropy loss described in Eq. 3 and Eq. 6.
+
+# 3 Proposed Method
+
+In this paper, we propose a novel framework for weakly-supervised audio-visual video parsing. In order to explore common semantics across videos and dependency across event categories, the proposed model leverages all audio and visual signals across videos in a training batch and the correlation between classes for each training instance. In Section 3.1, we first define the notations and settings considered in this paper and revisit the common backbone [4, 6] for weakly-supervised audio-visual video parsing, which consists of feature aggregation and multi-modal multiple instance learning (MMIL) pooling. Then in Section 3.2 and Section 3.3, we detail the modules we propose to capture dependency across different events and information across different videos, respectively.
+
+# 3.1 Preliminaries
+
+Problem Formulation and Notations. Given a video sequence $S$ with $T$ seconds long, we obtain $T$ non-overlapping audio and visual segments where each segment is one-second long. Models are aiming to predict the event labels for each segment, which may contain several or no events. At time $t$ , there are three targets for audio, visual, and audio-visual events: $\mathbf { y } _ { t } ^ { a } \in \mathbb { R } ^ { 1 \times C } , \mathbf { y } _ { t } ^ { v } \in \mathbb { R } ^ { 1 \times C }$ and $\mathbf { y } _ { t } ^ { a v } \in \mathbb { R } ^ { 1 \times C }$ are multi-class event label with $C$ event categories. $\mathbf { y } _ { t } ^ { a } , \mathbf { y } _ { t } ^ { v }$ , and ${ \bf y } _ { t } ^ { a v }$ denote audio, visual, and audio-visual event labels, respectively. We note that detailed annotations (e.g., $\mathbf { y } _ { t } ^ { a }$ , $\mathbf { y } _ { t } ^ { a }$ , and ${ \bf y } _ { t } ^ { a v }$ ) are not accessible during training and only available during evaluation. As for training, only video-level annotations are available during training. Video-level annotations only contain action event categories without indicating specific times slots or modalities (e.g., audio and visual event).
+
+Revisit of Weakly-Supervised Audio-Visual Video Parsing. The previous method [4] presents promising results with feature aggregation based on transformers and multimodal multiple instance learning (MMIL) pooling. Given a video sequence $S$ of $T$ frames, we denote its audio and visual feature sets by ${ \bf F } ^ { a ^ { * } } = \{ { \bf f } _ { 1 } ^ { a ^ { * } } , . . . , { \bf f } _ { T } ^ { a } \} \in \mathbb { R } ^ { T \times d }$ and $\mathbf { F } ^ { v } = \{ \mathbf { f } _ { 1 } ^ { v } , . . . , \mathbf { f } _ { T } ^ { v } \} \in \mathbb { R } ^ { T \times d }$ , respectively, where $d$ is the feature dimension. The transformer encoder [45] is employed to aggregate both within-modality and cross-modality information using multi-head attention blocks:
+
+$$
+\begin{array} { l } { { \phi _ { s e l f } ( { \bf f } _ { t } ^ { a } , { \bf F } ^ { a } , { \bf F } ^ { a } ) = \mathrm { S o f t m a x } ( \frac { { \bf f } _ { t } ^ { a } { \bf F } ^ { a } ^ { \top } } { \sqrt { d } } ) { \bf F } ^ { a } , } } \\ { { \phi _ { c r o s s } ( { \bf f } _ { t } ^ { a } , { \bf F } ^ { v } , { \bf F } ^ { v } ) = \mathrm { S o f t m a x } ( \frac { { \bf f } _ { t } ^ { a } { \bf F } ^ { v } ^ { \top } } { \sqrt { d } } ) { \bf F } ^ { v } , } } \end{array}
+$$
+
+where $\phi _ { s e l f } ( \cdot )$ and $\phi _ { c r o s s } ( \cdot )$ are self-attention and cross-modality attention functions respectively. They perform dot-product on features across time stamps by using non-shared MLPs. Then the jointly aggregated representations are described as follows:
+
+$$
+\begin{array} { r } { \hat { \mathbf { f } } _ { t } ^ { a } = \mathbf { f } _ { t } ^ { a } + \phi _ { s e l f } ( \mathbf { f } _ { t } ^ { a } , \mathbf { F } ^ { a } , \mathbf { F } ^ { a } ) + \phi _ { c r o s s } ( \mathbf { f } _ { t } ^ { a } , \mathbf { F } ^ { v } , \mathbf { F } ^ { v } ) , } \\ { \hat { \mathbf { f } } _ { t } ^ { v } = \mathbf { f } _ { t } ^ { v } + \phi _ { s e l f } ( \mathbf { f } _ { t } ^ { v } , \mathbf { F } ^ { v } , \mathbf { F } ^ { v } ) + \phi _ { c r o s s } ( \mathbf { f } _ { t } ^ { v } , \mathbf { F } ^ { a } , \mathbf { F } ^ { a } ) , } \end{array}
+$$
+
+With the aggregated audio and visual features $\hat { \mathbf { f } } _ { t } ^ { a }$ and $\hat { \mathbf { f } } _ { t } ^ { v }$ , we can obtain the frame-wise event prediction $\hat { \mathbf { p } } _ { t } ^ { a } \in \mathbb { R } ^ { 1 \times \widetilde { C } }$ and $\hat { \mathbf { p } } _ { t } ^ { v } \in \mathbb { R } ^ { 1 \times C }$ , and the attention weights computed by MLPs and normalized by a softmax function for audio, visual, and audio-visual streams (i.e., $\mathbf { w } _ { t } ^ { a } \in \mathbb { R } ^ { 1 \times C }$ , $\mathbf { w } _ { t } ^ { v } \in \mathbb { R } ^ { 1 \times C }$ , and $\mathbf { w } _ { t } ^ { a v } \in \mathbb { R } ^ { 2 \times C } ,$ ). Then the video-level prediction is gathered with the MMIL pooling:
+
+$$
+\bar { \mathbf { p } } ^ { a } = \sum _ { t = 1 } ^ { T } \mathbf { w } _ { t } ^ { a } \hat { \mathbf { p } } _ { t } ^ { a } , \bar { \mathbf { p } } ^ { v } = \sum _ { t = 1 } ^ { T } \mathbf { w } _ { t } ^ { v } \hat { \mathbf { p } } _ { t } ^ { v } , \mathrm { a n d } \bar { \mathbf { p } } ^ { a v } = \sum _ { t = 1 } ^ { T } \mathbf { w } _ { t } ^ { a v } [ 0 ] \mathbf { w } _ { t } ^ { a } \hat { \mathbf { p } } _ { t } ^ { a } + \mathbf { w } _ { t } ^ { a v } [ 1 ] \mathbf { w } _ { t } ^ { v } \hat { \mathbf { p } } _ { t } ^ { v } .
+$$
+
+The model can then be optimized using the binary cross-entropy loss function between $\bar { \bf p }$ and a video-level weak label $\bar { \mathbf { y } } \in \mathbb { R } ^ { 1 \times C }$ , which does not indicate time boundaries and modalities for events.
+
+# 3.2 Cross-Modality Co-Occurrence
+
+Videos with multi-label events contain rich information among event categories because the related events are likely to present at the same time. The correlation is useful for models to robustly predict the presence or absence of events.
+
+Similar to [46], to explicitly model the relationship between event categories in different modalities, we first obtain the representations for each class and then measure the correlation. We note that the class relationships may be different in audio and visual modalities. That is why the work [46] cannot be directly applied to audio-visual video parsing since audio or visual events can be partially or jointly presented at a single frame. Thus, jointly understanding the class relationship within a modality and across two modalities can benefit the audio-visual video parsing task.
+
+In order to map the frame-wise audio and visual features into class-level ones, the nonlinear transformation with MLPs is formulated as follows:
+
+$$
+\begin{array} { r } { \mathbf { a } _ { t , c } = \operatorname { R e L U } ( \hat { \mathbf { f } } _ { t } ^ { a } \mathbf { M } _ { c } ^ { a } + \mathbf { b } _ { c } ^ { a } ) , } \\ { \mathbf { v } _ { t , c } = \operatorname { R e L U } ( \hat { \mathbf { f } } _ { t } ^ { v } \mathbf { M } _ { c } ^ { v } + \mathbf { b } _ { c } ^ { v } ) , } \end{array}
+$$
+
+where $\mathbf { a } _ { t , c }$ and $\mathbf { v } _ { t , c }$ are audio and visual class-level features for class $c$ at time $t$ with dimension $1 \times d _ { c }$ , respectively. The weights and biases for class $c$ for audio and visual features are denoted as $\mathbf { M } _ { c } ^ { a }$ $\mathbf { \Psi } _ { : } ^ { i } , \mathbf { M } _ { c } ^ { i } \in \mathbb { R } ^ { d \times } \mathbf { \tilde { { d } } } _ { c }$ and ${ \bf b } _ { c } ^ { a }$ $\mathbf { \bar { b } } _ { c } ^ { v } \in \mathbb { R } ^ { 1 \times d _ { c } }$ . With class-level representations, we can further model the relationship between event categories within and across modalities by self-attention and crossmodality co-attention mechanism:
+
+$$
+\begin{array} { r } { \hat { \mathbf { a } } _ { t , c } = \mathbf { a } _ { t , c } + \phi _ { s e l f } ( \mathbf { a } _ { t , c } , \mathbf { A } _ { t } , \mathbf { A } _ { t } ) + \phi _ { c r o s s } ( \mathbf { a } _ { t , c } , \mathbf { V } _ { t } , \mathbf { V } _ { t } ) , } \\ { \hat { \mathbf { v } } _ { t , c } = \mathbf { v } _ { t , c } + \phi _ { s e l f } ( \mathbf { v } _ { t , c } , \mathbf { V } _ { t } , \mathbf { V } _ { t } ) + \phi _ { c r o s s } ( \mathbf { v } _ { t , c } , \mathbf { A } _ { t } , \mathbf { A } _ { t } ) , } \end{array}
+$$
+
+where $\mathbf { A } _ { t } = \{ \mathbf { a } _ { t , 1 } , \dotsc , \mathbf { a } _ { t , C } \}$ and $\mathbf { V } _ { t } = \{ \mathbf { v } _ { t , 1 } , \dots , \mathbf { v } _ { t , C } \}$ are sets of audio and visual class features at time $t$ . $\hat { \mathbf { a } } _ { t , c }$ and $\hat { \mathbf { v } } _ { t , c }$ are now co-occcurence features that consider the relationships between categories within and across modalities. We can then predict the probability for each event at time $t$ by MLPs and aggregate every segment-wise predictions into video-level ones i.e.,
+
+$$
+\begin{array} { r l } & { \hat { \mathbf { p } } _ { t } ^ { a } = \sigma ( \mathrm { M L P } _ { a } ( \{ \hat { \mathbf { a } } _ { t , 1 } , \dots , \hat { \mathbf { a } } _ { t , C } \} ) ) , \quad \hat { \mathbf { p } } _ { t } ^ { v } = \sigma ( \mathrm { M L P } _ { v } ( \{ \hat { \mathbf { v } } _ { t , 1 } , \dots , \hat { \mathbf { v } } _ { t , C } \} ) ) , } \\ & { \bar { \mathbf { p } } ^ { a } , \bar { \mathbf { p } } ^ { v } , \bar { \mathbf { p } } ^ { a v } = \mathrm { M M I L } ( \{ \hat { \mathbf { p } } _ { 1 } ^ { a } , \dots , \hat { \mathbf { p } } _ { T } ^ { a } \} , \{ \hat { \mathbf { p } } _ { 1 } ^ { v } , \dots , \hat { \mathbf { p } } _ { T } ^ { v } \} ) } \end{array}
+$$
+
+where $\sigma$ is the sigmoid function, and $\mathrm { M M L } ( \cdot )$ is the multi-modal multiple instance learning pooling described in Eq. 3 taking all segment-wise predictions as inputs. The video-level prediction can be optimized by the binary cross-entropy loss function with a video-level weak label $\bar { \mathbf { y } }$ .
+
+# 3.3 Shared Cross-Modality Semantics across Videos
+
+The information across different videos provides rich supervisory signals that benefit the training of weakly-supervised audio-visual video parsing. By observing videos in a training batch, we can discover both the common and diverse event semantics. With video-level labels, we can initially associate related and irrelevant videos. In order to obtain a discriminative categorical representation, we would like to encourage audio and visual representations from related events to be similar and differentiate those from irrelevant videos. However, targeting at segment-wise representations with specific events is difficult due to the lack of temporal annotations. Therefore, we seek event-related frames through the weights from MMIL pooling in Eq. 3:
+
+$$
+\tilde { \mathbf { f } } ^ { a } = \sum _ { t = 1 } ^ { T } \Big [ \frac { \exp \bigl ( g \bigl ( \bar { \mathbf { y } } \odot \mathbf { w } _ { t } ^ { a } \bigr ) \bigr ) } { \sum _ { t ^ { \prime } = 1 } ^ { T } \exp \bigl ( g \bigl ( \bar { \mathbf { y } } \odot \mathbf { w } _ { t ^ { \prime } } ^ { a } \bigr ) \bigr ) } \hat { \mathbf { f } } _ { t } ^ { a } \Big ] , \quad \tilde { \mathbf { f } } ^ { v } = \sum _ { t = 1 } ^ { T } \Big [ \frac { \exp \bigl ( g \bigl ( \bar { \mathbf { y } } \odot \mathbf { w } _ { t } ^ { v } \bigr ) \bigr ) } { \sum _ { t ^ { \prime } = 1 } ^ { T } \exp \bigl ( g \bigl ( \bar { \mathbf { y } } \odot \mathbf { w } _ { t ^ { \prime } } ^ { v } \bigr ) \bigr ) } \hat { \mathbf { f } } _ { t } ^ { v } \Big ] ,
+$$
+
+where $\odot$ and $g ( . )$ are element-wise dot product and summation function over all elements respectively.
+
+With video-level labels and features ${ \tilde { \mathbf { f } } } ^ { a }$ and $\tilde { \mathbf { f } } ^ { v }$ ), we adopt contrastive learning [47, 48, 49] to encourage features across modalities with the same event category (at least one) to be close and those with different events to be far away from each other. We leverage all $n$ videos in a batch to explore diverse semantics, where the sets of audio and visual features are denoted as $\{ \widetilde { \bf f } _ { ( 0 ) } ^ { a } , . . . , \widetilde { \bf f } _ { ( n ) } ^ { a } \}$ and $\{ \tilde { \mathbf { f } } _ { ( 0 ) } ^ { v } , . . . , \tilde { \mathbf { f } } _ { ( n ) } ^ { v } \}$ respectively with video-level labels $\left\{ \bar { \mathbf { y } } _ { ( 0 ) } , . . . , \bar { \mathbf { y } } _ { ( n ) } \right\}$ . The relationship across videos can be optimized by the proposed training objective as follows:
+
+$$
+\mathcal { L } _ { \mathrm { c o n t r a s t } } = - \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \Big [ \log \frac { \sum _ { j = 1 } ^ { n } f ( \bar { \bf y } _ { i } \cdot \bar { \bf y } _ { j } ) \exp ( \tilde { \bf f } _ { ( i ) } ^ { a } \cdot \tilde { \bf f } _ { ( j ) } ^ { v } / \tau ) } { \sum _ { j = 1 } ^ { n } \exp ( \tilde { \bf f } _ { ( i ) } ^ { a } \cdot \tilde { \bf f } _ { ( j ) } ^ { v } / \tau ) } \Big ] ,
+$$
+
+where $f ( \cdot )$ is a clipping function that clips values over 1, and $\tau$ denotes a hyper-parameter controlling the temperature. Thus, the proposed method can be optimized by joint the binary cross-entropy loss mentioned in Section 3.1 and the contrastive learning loss in Eq. 8. Our training strategy can exploit cross-modality information across videos and event categories to understand common semantics while ignoring irrelevant ones.
+
+# 4 Experimental Results
+
+Datasets. We use the Look, Listen and Parse (LLP) Dataset [4] for all experiments. The LLP dataset consists of 11, 849 10-seconds video clips annotated with 25 event categories. It covers various real-life scenes such as speech, music performances, car, cheering, dog, etc. Particularly, there are 7202 video clips labeled with more than one event category. We use the 10000 video clips with only video-level event annotations for model training. The detailed annotations (e.g., individual audio and visual events per second) are available for the remaining 1849 validation and test videos. For all experiments, we use the official data splits from the LLP dataset.
+
+Evaluation Metrics. Following previous work [4, 6], we adopt F-scores as the evaluation metrics. Note that all types of events (audio, visual, and audio-visual) are measured under both segmentlevel and event-level metrics. The segment-level metrics can evaluate snippet-wise prediction results. As for the event-level metrics, the clips are extracted by concatenating positive consecutive segments in the same events. Then, we compute the event-level F-scores with $\mathrm { m I o U } = 0 . 5$ as the threshold. Furthermore, the overall Type $\ @ \mathbf { A V }$ performance on audio-visual scene is also considered by computing the averaged audio, visual, and audio-visual event evaluation results. Instead of directly averaging results from different event types, Event@AV considers all audio and visual event categories for each sample.
+
+Table 1: Quantitative results of weakly-supervised audio-visual video parsing. We evaluate all methods on the LLP dataset [4] with F-scores in five different event types and two kinds of segments. The first row indicates five different event types (audio, visual, audio-visual, Type@AV, and Event@AV). In the second row, two kinds of segments are shown: Seg. and Event are segmentlevel and event-level; and $^ *$ indicates only label refinement is utilized for fair comparisons.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Audio</td><td colspan="2">Visual</td><td colspan="2">Audio-visual</td><td colspan="2">Type@AV</td><td colspan="2">Event@AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>AVE [5]</td><td>47.2</td><td>40.4</td><td>37.1</td><td>34.7</td><td>35.4</td><td>31.6</td><td>39.9</td><td>35.5</td><td>41.6</td><td>36.5</td></tr><tr><td>AVSDN [2]</td><td>47.8</td><td>34.1</td><td>52.0</td><td>46.3</td><td>37.1</td><td>26.5</td><td>45.7</td><td>35.6</td><td>50.8</td><td>37.7</td></tr><tr><td>AVSDN + Ours</td><td>48.3</td><td>41.2</td><td>52.4</td><td>48.5</td><td>46.9</td><td>40.0</td><td>49.2</td><td>43.2</td><td>53.2</td><td>40.1</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + Ours</td><td>59.2</td><td>51.3</td><td>59.9</td><td>55.5</td><td>53.4</td><td>46.2</td><td>57.5</td><td>51.0</td><td>58.1</td><td>49.7</td></tr><tr><td>MA [6]</td><td>60.3</td><td>53.6</td><td>60.0</td><td>56.4</td><td>55.1</td><td>49.0</td><td>58.9</td><td>53.0</td><td>57.9</td><td>50.6</td></tr><tr><td>MA*</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA*+Ours</td><td>60.8</td><td>53.8</td><td>63.5</td><td>58.9</td><td>57.0</td><td>49.5</td><td>60.5</td><td>54.0</td><td>59.5</td><td>52.1</td></tr></table>
+
+Implementation Details. We implement the proposed method using PyTorch [50], and conduct the training and evaluation processes on a single NVIDIA GTX 1080 Ti GPU with 11 GB memory. Following [4, 6], we use the same visual and audio encoders for fair comparisons. We adopt both ResNet-152 [51] pre-trained on ImageNet [52] and 3D ResNet [53] pre-trained on Kinetics-400 [54] as visual feature extractors. Visual frames are sampled at 8 fps and their 2D and 3D visual features are extracted. The 2D and 3D visual features are concatenated and then processed by an MLP as the segment-wise representations. As for audio data, we utilize VGGish [55] pre-trained on AudioSet [56] to extract 128-dimensional audio features. The code and models are publicly available.
+
+Evaluated methods. We compare the proposed method based on several baselines to the following weakly-unsupervised approaches to the audio-visual video parsing task:
+
+• AVE [5] consists of an audio-guided co-attention mechanism to adaptively learn the sounding regions. We note that AVE [5] deals with the audio-visual event localization task. Thus, we follow [4] and add additional audio and visual parsing branches for the weakly-supervised audio-visual video parsing task as a baseline.   
+• AVSDN [2] is a sequence-to-sequence-based model to integrate global audio and visual features to local ones. Since AVSDN [2] also deals with the audio-visual event localization task, we make the same modifications to AVSDN as those to AVE.   
+• HAN [4] is a multi-modal multiple instance learning-based method with a hybrid attention network.   
+• MA [6] reports the state-of-the-art performance on the weakly-supervised audio-visual video parsing task. It is a method based on HAN with the label refinement and the audio-visual contrastive learning differentiating temporal segments.
+
+# 4.1 Quantitative Evaluation
+
+Table 1 shows the quantitative comparisons on the LLP dataset [4]. The proposed method performs favorably against the competing approaches on the weakly-supervised audio-visual video parsing task. Since our method can be easily extended to existing methods, we extend the proposed on three baselines. The third, fifth, and last rows in Table 1 indicate that the proposed method generally benefits three baselines on several metrics of the audio-visual video parsing task by a large margin. We note that $\mathbf { M A } ^ { * }$ [6] only utilizes label refinement to refine labels for each modality, and temporal difference audio-visual contrastive learning [6] is not implemented.
+
+Table 2: Ablation study. We investigate the effect of using different design components in the proposed method. We show how proposed cross-modality co-occurrence (CM-Co) in Section 3.3 and shared cross-modality semantics across videos (CM-S) module in Section 3.2 improve the baselines.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Audio</td><td colspan="2">Visual</td><td colspan="2">Audio-visual</td><td colspan="2">Type@AV</td><td colspan="2">Event@AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + CM-S</td><td>58.1</td><td>49.6</td><td>58.3</td><td>53.6</td><td>53.2</td><td>46.3</td><td>56.5</td><td>49.8</td><td>55.9</td><td>47.5</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>51.4</td><td>57.4</td><td>52.4</td><td>51.9</td><td>44.2</td><td>56.3</td><td>49.3</td><td>57.4</td><td>48.5</td></tr><tr><td>HAN + Ours</td><td>59.2</td><td>51.3</td><td>59.9</td><td>55.5</td><td>53.4</td><td>46.2</td><td>57.5</td><td>51.0</td><td>58.1</td><td>49.7</td></tr><tr><td>MA [28]</td><td>60.3</td><td>53.6</td><td>60.0</td><td>56.4</td><td>55.1</td><td>49.0</td><td>58.9</td><td>53.0</td><td>57.9</td><td>50.6</td></tr><tr><td>MA + CM-Co</td><td>61.1</td><td>53.3</td><td>61.7</td><td>57.3</td><td>56.3</td><td>49.0</td><td>59.7</td><td>53.0</td><td>58.9</td><td>51.2</td></tr><tr><td>MA*</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA* + CM-S</td><td>60.4</td><td>53.5</td><td>60.7</td><td>56.5</td><td>55.8</td><td>47.5</td><td>58.9</td><td>52.5</td><td>58.6</td><td>51.0</td></tr><tr><td>MA* + CM-Co</td><td>60.5</td><td>53.6</td><td>61.3</td><td>56.5</td><td>54.9</td><td>46.7</td><td>58.9</td><td>52.3</td><td>59.1</td><td>51.4</td></tr><tr><td>MA* +Ours</td><td>60.8</td><td>53.8</td><td>63.5</td><td>58.9</td><td>57.0</td><td>49.5</td><td>60.5</td><td>54.0</td><td>59.5</td><td>52.1</td></tr></table>
+
+We notice that our method significantly improves baselines in the metrics of visual, audio-visual, Type $@ \mathrm { A V } ,$ and Event $@$ AV. By observing the class distribution of training sets, we find that $3 1 \%$ , $7 \%$ , and $9 \%$ training videos contain speech, singing, and violin events. These events are more likely to present in the audio modality. Therefore, the video-level labels would limit the performance regarding visual events. The proposed method can leverage additional cross-video and cross-modality supervisory signals to explore common semantics, which can improve results in vision-related metrics.
+
+# 4.2 Ablation Study
+
+Cross-Modality Co-Occurrence and Semantics across Video. We conduct the ablation study to analyze the individual impact of each developed component in the proposed method. The results are presented in Table 2. CM-Co represents the usage of the cross-modality co-occurrence module described in Section 3.2, which leverages the relationship between categories within and cross modalities. CM-S indicates the shared cross-modality semantics across videos module described in Section 3.3, which considers all audio and visual information across videos in a batch.
+
+In Table 2, we note that both CM-S and CM-Co can improve baselines in several metrics. By exploring common semantics among training videos (CM-S), we improve the performance on visual and audio-visual evaluation by a large margin. Such a strategy can exploit additional information from videos to address the potential drawback of video-level labels described in Section 4.1. Furthermore, the proposed cross-modality co-occurrence module (CM-Co) also presents favorable results. We note that the significant improvement in Event@AV evaluation with the usage of CM-Co can verify the efficacy of considering the relationship between categories within and across modalities. Since Event $@$ AV considers all audio and visual events for the F-score (e.g., truth positive from both audio and visual events), the improvement of Event@AV indicates our cross-modality co-occurrence can perform well on video parsing when events present in an audio or a visual modality.
+
+In the second group of the evaluated methods in Table 2, we verify if the proposed CM-S works better than the contrastive learning method in MA. We perform our CM-S on the MA model. The CM-S exploits information across different videos to address the issue that audio and visual tracks may not be synchronized. Instead, the contrastive learning method in MA is developed based on the assumption of synchronization to associate the audio-visual representation in a single video. Since our CM-S learns diverse and common semantics, it is effective and complementary to the contrastive learning approach in MA performing on a single video. We note that our CM-S generally improves the performance over all segment-level metrics, which supports our claim.
+
+Self-attention and Cross-Modality Co-attention in Co-Occurrence. Since our cross-modality co-occurrence module exploits self-attention among class-level features in the same modality and cross-modality co-attention on cross-modality class-level representations to model the relationship between categories in the same and different modalities. Taking class-level audio features in Eq. 5 as an example, the class-level self-attention and cross-modality co-attention are $\mathrm { a t t n } ( \mathbf { a } _ { t , c } , \mathbf { A } _ { t } , \mathbf { A } _ { t } )$ and $\operatorname { a t t n } ( \mathbf { a } _ { t , c } , \mathbf { V } _ { t } , \mathbf { V } _ { t } ) .$ , respectively.
+
+Table 3: Ablation study. We investigate the effect of different developed mechanisms in the proposed cross-modality co-occurrence (CM-Co) module in Section 3.2. In Eq. 5, class-level features are processed by self-attention and cross-modality co-attention mechanisms. A Only and $\mathbf { V }$ Only indicate only self-attention performs for individual audio and visual events respectively. AV denotes performing self-attention for audio and visual events. CM-Co is the proposed method that considers relationship between categories within and cross modalities by both self-attention and cross-modality co-attention mechanisms.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Audio</td><td colspan="2">Visual</td><td colspan="2">Audio-visual</td><td colspan="2">Type@AV</td><td colspan="2">Event@ AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + A Only</td><td>60.5</td><td>52.3</td><td>49.8</td><td>43.9</td><td>45.6</td><td>38.3</td><td>52.0</td><td>44.8</td><td>55.7</td><td>45.9</td></tr><tr><td>HAN + V Only</td><td>56.1</td><td>44.5</td><td>56.8</td><td>53.2</td><td>49.7</td><td>40.7</td><td>54.2</td><td>46.1</td><td>54.1</td><td>44.6</td></tr><tr><td>HAN + AV</td><td>59.5</td><td>50.3</td><td>55.1</td><td>50.5</td><td>48.6</td><td>40.3</td><td>54.4</td><td>47.0</td><td>56.0</td><td>47.4</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>51.4</td><td>57.4</td><td>52.4</td><td>51.9</td><td>44.2</td><td>56.3</td><td>49.3</td><td>57.4</td><td>48.5</td></tr><tr><td>MA*[6]</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA*+ A Only</td><td>60.7</td><td>52.7</td><td>53.9</td><td>47.9</td><td>50.1</td><td>42.2</td><td>54.9</td><td>47.6</td><td>57.0</td><td>47.1</td></tr><tr><td>MA* + V Only</td><td>46.8</td><td>34.4</td><td>60.8</td><td>57.0</td><td>42.8</td><td>31.1</td><td>50.1</td><td>40.9</td><td>52.6</td><td>40.4</td></tr><tr><td>MA*+ AV</td><td>58.3</td><td>50.4</td><td>59.4</td><td>55.2</td><td>53.9</td><td>46.9</td><td>57.2</td><td>50.8</td><td>56.7</td><td>48.5</td></tr><tr><td>MA*+ CM-Co</td><td>60.5</td><td>53.6</td><td>61.3</td><td>56.5</td><td>54.9</td><td>46.7</td><td>58.9</td><td>52.3</td><td>59.1</td><td>51.4</td></tr></table>
+
+Table 4: Ablation study. We evaluate the proposed method in accuracy, efficiency, and model sizes. We show the numbers of parameters and FLOPs for the proposed cross-modality co-occurrence (CM-Co) and HAN [4] with a few layers.   
+Note that the results are all in the segment level.   
+
+<table><tr><td>Method</td><td>Audio</td><td>Visual</td><td>Audio-visual</td><td>Type@AV</td><td>Event@AV</td><td>GFLOPs</td><td>Params</td></tr><tr><td>HAN 1 Layer</td><td>60.1</td><td>52.9</td><td>48.9</td><td>54.0</td><td>55.4</td><td>6.63</td><td>2.4M</td></tr><tr><td>HAN 2 Layers</td><td>58.2</td><td>55.4</td><td>50.6</td><td>54.7</td><td>54.9</td><td>7.28</td><td>2.9M</td></tr><tr><td>HAN 3 Layers</td><td>58.1</td><td>55.2</td><td>50.3</td><td>54.5</td><td>54.6</td><td>7.97</td><td>3.5M</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>57.4</td><td>51.9</td><td>56.3</td><td>57.4</td><td>6.99</td><td>2.8M</td></tr></table>
+
+Table 3 presents the results in various modifications of the cross-modality co-occurrence module. We note that the design of co-occurrence in the same and cross modalities can generally improve the results in several metrics. We also evaluate the co-occurrence module in a single modality. The results are shown in the second, third, seventh, and eighth rows in Table 3, where A Only and $\mathbf { V }$ Only indicate the co-occurrence module only leverages the relationship between categories in audio or visual data respectively. As the results shown in the second and seventh rows, training with co-occurrence in audio events only (i.e., A Only) can slightly improve the performance on audio events. Similarly, considering visual event only (i.e., V Only) can benefit the results regarding visual events. Furthermore, the co-occurrence for both audio and visual categories (AV) in the fourth and ninth rows can contribute to the results in general metrics such as Type $@$ AV and Event $@ \mathrm { A V } .$ . We then further consider the correlation between events across modalities. That is the cross-modality co-occurrence module (CM-Co) in the fifth and tenth rows. The results can confirm the efficacy of the proposed cross-modality co-occurrence module in all metrics except segment-level audio events caused by similar reasons discussed in Section 4.1.
+
+Model Capacity. Since our cross-modality co-occurrence module leverages class-level representations, it would increase the capability of models on capturing information. For fair comparisons, we add extra parameters to HAN [4] to analyze whether more parameters can contribute to performance gain. Specifically, we increase the number of layers in its transformer-based feature aggregation to 2 and 3, respectively.
+
+In Table 4, we report the results in accuracy, computational costs, and model sizes. The first three rows show the performance of HAN with different numbers of layers. We note that HAN with one extra layer has more parameters than the proposed co-occurrence module. However, the results of HAN with extra layers indicate that using more parameters/layers for HAN does not improve the performance. The proposed cross-modality co-occurrence module enhances HAN more effectively.
+
+![](images/3a3e5bddfd4d0b8b3df955da6588c994004175b863c809999ccff152b083248e.jpg)  
+Figure 2: Qualitative comparisons. We compare the proposed method with the state-of-the-art weakly-supervised audio-visual video parsing method on the LLP dataset [4]. The frame-wise annotations are shown in gray and purple bars. The gray bar denotes visual events, and the purple bar represents audio events. GT_V and GT_A are the ground-truth visual and audio events respectively. Our results are shown in the green block, and the results by the competing method, MA [6], are present in the blue block.
+
+![](images/21cead53b19ede061abdc05068f5d07b4594cffe449abe292ef47d1a62306ab2.jpg)  
+Figure 3: Audio feature distribution by using t-SNE. The upper figure shows the distribution by our method. The lower figure presents that by $\mathbf { M A } ^ { * }$ . The legend lists all event combinations.
+
+# 4.3 Qualitative Evaluation
+
+Qualitative Results. We present the qualitative results of the evaluated methods in Figure 2. GT_V and GT_A show the ground-truth annotations for visual and audio events, respectively. Pred_V and Pred_A present the predictions made by our method and the state-of-the-art competing method, MA [6], respectively. Our results are shown in the green block, while the results of MA are present in the blue block. In general, our method presents more accurate predictions in both audio and visual events than MA. We note that the whole violin is shown after 7 seconds. That would hamper models for understanding visual events e.g., MA predicts wrong results on violin visual events before 6 seconds. Since our method leverages the relationship between categories, it can still predict correct temporal boundaries for guitar events by jointly considering cello events in the videos.
+
+![](images/ed423064930570c6abeaee1db5cadd585aab212a9182498adeec79bb58aa61f7.jpg)  
+Figure 4: Visual feature distribution by using t-SNE. The upper figure shows the distribution by our method. The lower figure presents that by MA∗. The legend lists all event combinations.
+
+Feature Distribution Visualized by t-SNE. We apply t-SNE to the aggregated audio and visual features from each segment described in Eq. 2. The visualization results are present in Figure 3 and Figure 4, respectively. The legends list all the combinations of multiple labels. For example, in Figure 3, audio events of singing are present as blue spots, and the mixed sounds of singing and violin are shown as purple spots. We note that the related events including multiple events are shown in similar colors. In Figure 4, the proposed method achieves better performance in the sense that similar color spots are closer than the spots in $\mathbf { M A } ^ { * }$ .
+
+# 5 Conclusions
+
+In this paper, we present a novel audio-visual video parsing framework in a weakly-supervised manner that can be applied to existing methods. We propose two modules to exploit the relationship across videos, modalities, and event categories, and explore additional supervisory signals that can benefit audio-visual video parsing. The shared cross-modality semantics module leverages common and diverse event semantics across videos to learn robust cross-modality representations that facilitate models to identify audio, visual, and audio-visual events. Furthermore, the cross-modality co-occurrence module aims to learn the relationship between event categories. It helps localize segments of target events and can exclude irrelevant ones by performing self-attention and crossmodality co-attention on class-wise features, Extensive experimental results show that our approach substantially improves several baselines and performs favorably against the state-of-the-art methods.
+
+Acknowledgments. This work was supported in part by the Ministry of Science and Technology under grants 109- 2221-E-009-113-MY3, 110-2628-E-A49-008, and 110-2634-F007-015. It was also funded in part by Qualcomm through a Taiwan University Research Collaboration Project, the Higher Education Sprout Project of the National Yang Ming Chiao Tung University, and Ministry of Education.
+
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parse/train/V5V1vGrI2z/V5V1vGrI2z_content_list.json

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+    {
+        "type": "text",
+        "text": "Exploring Cross-Video and Cross-Modality Signals for Weakly-Supervised Audio-Visual Video Parsing ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Yan-Bo Lin1,2 Hung-Yu Tseng3 Hsin-Ying Lee4 Yen-Yu Lin1 Ming-Hsuan Yang3,5,6 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1National Yang Ming Chiao Tung University 2UNC Chapel Hill 3UC Merc 4Snap Research 5Google Research 6Yonsei University yblin@unc.edu htseng6@ucmerced.edu hlee5@snap.com lin@cs.nctu.edu.tw mhyang@ucmerced.edu ",
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+        "type": "text",
+        "text": "Abstract ",
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+    },
+    {
+        "type": "text",
+        "text": "The audio-visual video parsing task aims to temporally parse a video into audio or visual event categories. However, it is labor-intensive to temporally annotate audio and visual events and thus hampers the learning of a parsing model. To this end, we propose to explore additional cross-video and cross-modality supervisory signals to facilitate weakly-supervised audio-visual video parsing. The proposed method exploits both the common and diverse event semantics across videos to identify audio or visual events. In addition, our method explores event co-occurrence across audio, visual, and audio-visual streams. We leverage the explored cross-modality co-occurrence to localize segments of target events while excluding irrelevant ones. The discovered supervisory signals across different videos and modalities can greatly facilitate the training with only video-level annotations. Quantitative and qualitative results demonstrate that the proposed method performs favorably against existing methods on weakly-supervised audio-visual video parsing. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Humans perceive multisensory signals via seeing, hearing, touching, etc., and obtain multimodal information while exploring the surrounding environments. Visual and audio signals, the most common modalities, motivate researchers to jointly comprehend audio-visual events (e.g., see people singing and hear their sounds) [1, 2, 3, 4, 5, 6, 7]. Events visible in images while hearable in audio are referred to as audio-visual events. However, learning-based models tend to recognize a particular audio-visual event by using the data from the dominant modality with richer information and overlook clues from either audio only or visual only events which still contribute to holistic video understanding. Therefore, the resultant models can generalize well on audio-visual events only instead of comprehensively understanding all kinds of video events. To address this issue, we target at audio-visual video parsing [4, 6] where predictions for audio, visual, and audio-visual events with temporal boundaries are all required but separately evaluated. ",
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+    {
+        "type": "text",
+        "text": "The time-consuming and labor-intensive annotation process poses a major challenge for the audiovisual video parsing task. To address this issue, Tian et al. [4] handle this task in a weakly-supervised manner given only video-level labels, which indicate events of presence without temporal boundaries and detailed modalities. They develop an audio-visual co-attention mechanism to assemble discriminative multimodal representations and use multiple instance learning to aggregate frame-level predictions into video-level ones. However, video-level labels alone cannot identify which modality events are from. Wu et al. [6] then propose to perform label refinement by swapping the audio and visual tracks of different videos to estimate and remove irrelevant event categories for each modality. They further adopt temporal contrastive learning to align audio and visual representations from the same frame. However, the contrastive learning is based on the assumption that audio and visual signals are synchronized, which may not hold in practical scenarios with complex events. Furthermore, these methods [4, 6] only consider audio and visual tracks of a single video without exploiting the relationship across categories and videos, which also provide rich shared semantics regarding event categories. ",
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+    {
+        "type": "text",
+        "text": "In this work, we propose to leverage audio and visual data across different videos to explore shared information of each category. For example, videos with singing events may have similar patterns whatever in an audio or a visual modality. By observing all videos in a training batch, we can not only explore the shared semantics among audio-visual data but also exclude unrelated events. In addition to the relationship across different videos, we exploit the dependency between event categories. For example, when people are singing, there is usually a music accompaniment. Therefore, we propose to treat audio, visual, and audio-visual streams separately and adopt an audio-visual class co-occurrence module that jointly explores the relationship of different categories among all streams. By measuring the similarity of event categories from audio, visual, and audio-visual events, the correlated events are more likely to be correctly determined as the presence or absence of event categories. Such a strategy can robustly learn the correlation of categories within/across modality and fully exploit video data. The proposed strategy can be applied to existing methods on video parsing. ",
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+    {
+        "type": "text",
+        "text": "We evaluate the proposed method on the LLP [4] dataset. Videos are parsed into audio, visual, and audio-visual events under both segment and event levels, and evaluated with F-scores metrics. Both qualitative and quantitative results demonstrate the effectiveness of the proposed method on the audio-visual video parsing task. The main contributions of this work are summarized as follows: ",
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+    {
+        "type": "text",
+        "text": "• We leverage audio and visual data across different videos and tracks, which can learn common semantics of the same events and discern unrelated clues. • We develop an audio-visual event co-occurrence module that jointly considers the relationship of categories in audio, visual, and audio-visual modalities, which can prevent models from differentiating the representations of the related events. • Qualitative and quantitative experimental results on the benchmark dataset demonstrate that the proposed method performs favorably against the state-of-the-arts in various settings. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    {
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+        "text": "Audio-Visual Representation Learning. Implicit correlation between audio and visual data from videos provides rich information for audio-visual representation learning. First, the audio-visual pairs from the same video clip [8, 9, 10, 11, 12, 13, 14, 15, 16, 17] are strongly correlated based on the assumption that audio and visual data from a video are synchronized and highly correlated. Moreover, features extracted from unpaired video clips tend to be more diverse than those from the same clips. Second, by exploring audio-visual temporal synchronization [18, 19], temporal information can be served as a training guidance. Given a video sequence, existing methods [18, 19] distinguish audio and visual features from different frames while correlating features from the same frames. Such an idea enhances robust audio-visual representation learning that is essential to several tasks such as audio-visual event localization/parsing/recognition [1, 2, 3, 4, 5, 6, 7, 20], sound separation [21, 22, 23, 24, 25, 26, 27, 28, 29, 30], audio spatialization [31, 32, 33, 34, 35, 36, 37, 38], and sound localization [39, 40, 41, 42, 43, 44]. Instead of random sampling sound and images, our method selects both related and irreverent videos to explore common semantics and discern dissimilar events. ",
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+    {
+        "type": "text",
+        "text": "Audio-Visual Video Event Localization and Parsing. Audio-visual video parsing aims to detect events in videos and identify audio, visual, and audio-visual events (e.g., seeing the event and hearing its sound) and activities. Videos can be parsed with event categories and boundaries in both audio and visual modalities. Early researches [5, 7, 2, 3] aim to jointly derive audiovisual information in each local segment of the input video for audio-visual event localization, which emphasizes to detect only audio-visual events. However, due to the inconsistent information observed from audio and visual signals, data from either modality with insufficient clues may degrade the performance of prediction. Therefore, the work [7] focuses on audio/visual data with relevant categorical events to tackle this issue. Although methods of this category present favorable results, they are applicable to audio-visual event localization, which considers only synchronous audio-visual events or not. Recently, multi-modal multiple instance learning (MMIL) based methods with hybrid attention [4] carry out weakly-supervised audio-visual video parsing. These methods aggregate segment-level predictions into video-level ones, with which optimizing a model by using video-level or weak labels is enabled. Since video-level labels are typically insufficient to identify either audio or visual events, Wu et al. [6] generate pseudo labels for each modality by exchanging audio and visual tracks between unrelated videos. However, we notice that videos with replaced sounds or images may share some common semantics. Our method can exploit videos in a training batch to extract their common semantics for a categorical event and discern unrelated clues. Furthermore, we can leverage the relationship between event classes to find out related events (e.g., singing may accompany music). ",
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+    {
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+        "img_path": "images/270ce2aeb0adebd19f77c3fe3a37657e6dbf1b204e12e9a0401ee9485ffe46c2.jpg",
+        "image_caption": [
+            "Figure 1: Algorithmic overview. Our framework consists of a visual feature extractor, an audio feature extractor, a feature aggregation module, MMIL pooling, shared cross-modality semantics, and an cross-modality co-occurrence module. Given $n$ videos of $T$ seconds, the visual and audio feature extractors compute their visual and audio features. The feature aggregation module [4] conducts self- and cross-modality attention to aggregate segment-wise audio $\\bar { \\hat { \\mathbf { f } ^ { a } } }$ and visual $\\hat { \\mathbf { f } ^ { v } }$ representations. We map segment-wise aggregated features to class-specific features by exploring cross-modality co-occurrence. By performing self- and cross-modality attention for class features, we identify within and cross modalities relationship between classes for event predictions. Note that $\\otimes$ denotes matrix multiplication with the softmax operation performing on each row, and the green block only shows the example for segment-wise visual prediction at time $t$ . We also leverage the aggregated features of all $n$ videos to figure out common semantics regrading events by maximizing the similarities between related videos while minimizing those between unrelated videos with Eq. 8. The MMIL Pooling [4] is an attention-based pooling function that aggregates segment-wise results to produce video-level ones, which are optimized by the binary cross entropy loss described in Eq. 3 and Eq. 6. "
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+    {
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+        "text": "3 Proposed Method ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we propose a novel framework for weakly-supervised audio-visual video parsing. In order to explore common semantics across videos and dependency across event categories, the proposed model leverages all audio and visual signals across videos in a training batch and the correlation between classes for each training instance. In Section 3.1, we first define the notations and settings considered in this paper and revisit the common backbone [4, 6] for weakly-supervised audio-visual video parsing, which consists of feature aggregation and multi-modal multiple instance learning (MMIL) pooling. Then in Section 3.2 and Section 3.3, we detail the modules we propose to capture dependency across different events and information across different videos, respectively. ",
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+    {
+        "type": "text",
+        "text": "3.1 Preliminaries ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Problem Formulation and Notations. Given a video sequence $S$ with $T$ seconds long, we obtain $T$ non-overlapping audio and visual segments where each segment is one-second long. Models are aiming to predict the event labels for each segment, which may contain several or no events. At time $t$ , there are three targets for audio, visual, and audio-visual events: $\\mathbf { y } _ { t } ^ { a } \\in \\mathbb { R } ^ { 1 \\times C } , \\mathbf { y } _ { t } ^ { v } \\in \\mathbb { R } ^ { 1 \\times C }$ and $\\mathbf { y } _ { t } ^ { a v } \\in \\mathbb { R } ^ { 1 \\times C }$ are multi-class event label with $C$ event categories. $\\mathbf { y } _ { t } ^ { a } , \\mathbf { y } _ { t } ^ { v }$ , and ${ \\bf y } _ { t } ^ { a v }$ denote audio, visual, and audio-visual event labels, respectively. We note that detailed annotations (e.g., $\\mathbf { y } _ { t } ^ { a }$ , $\\mathbf { y } _ { t } ^ { a }$ , and ${ \\bf y } _ { t } ^ { a v }$ ) are not accessible during training and only available during evaluation. As for training, only video-level annotations are available during training. Video-level annotations only contain action event categories without indicating specific times slots or modalities (e.g., audio and visual event). ",
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+    {
+        "type": "text",
+        "text": "Revisit of Weakly-Supervised Audio-Visual Video Parsing. The previous method [4] presents promising results with feature aggregation based on transformers and multimodal multiple instance learning (MMIL) pooling. Given a video sequence $S$ of $T$ frames, we denote its audio and visual feature sets by ${ \\bf F } ^ { a ^ { * } } = \\{ { \\bf f } _ { 1 } ^ { a ^ { * } } , . . . , { \\bf f } _ { T } ^ { a } \\} \\in \\mathbb { R } ^ { T \\times d }$ and $\\mathbf { F } ^ { v } = \\{ \\mathbf { f } _ { 1 } ^ { v } , . . . , \\mathbf { f } _ { T } ^ { v } \\} \\in \\mathbb { R } ^ { T \\times d }$ , respectively, where $d$ is the feature dimension. The transformer encoder [45] is employed to aggregate both within-modality and cross-modality information using multi-head attention blocks: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df014a1b954a2bcc0c32855513113f4fe555449e7a17c022439841acbd5fd03e.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\phi _ { s e l f } ( { \\bf f } _ { t } ^ { a } , { \\bf F } ^ { a } , { \\bf F } ^ { a } ) = \\mathrm { S o f t m a x } ( \\frac { { \\bf f } _ { t } ^ { a } { \\bf F } ^ { a } ^ { \\top } } { \\sqrt { d } } ) { \\bf F } ^ { a } , } } \\\\ { { \\phi _ { c r o s s } ( { \\bf f } _ { t } ^ { a } , { \\bf F } ^ { v } , { \\bf F } ^ { v } ) = \\mathrm { S o f t m a x } ( \\frac { { \\bf f } _ { t } ^ { a } { \\bf F } ^ { v } ^ { \\top } } { \\sqrt { d } } ) { \\bf F } ^ { v } , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            263,
+            645,
+            339
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\phi _ { s e l f } ( \\cdot )$ and $\\phi _ { c r o s s } ( \\cdot )$ are self-attention and cross-modality attention functions respectively. They perform dot-product on features across time stamps by using non-shared MLPs. Then the jointly aggregated representations are described as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6cff82b396fe130840dfd8f000dea5c20b6fe84a3191f893872089e8a3c5a7d.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\mathbf { f } } _ { t } ^ { a } = \\mathbf { f } _ { t } ^ { a } + \\phi _ { s e l f } ( \\mathbf { f } _ { t } ^ { a } , \\mathbf { F } ^ { a } , \\mathbf { F } ^ { a } ) + \\phi _ { c r o s s } ( \\mathbf { f } _ { t } ^ { a } , \\mathbf { F } ^ { v } , \\mathbf { F } ^ { v } ) , } \\\\ { \\hat { \\mathbf { f } } _ { t } ^ { v } = \\mathbf { f } _ { t } ^ { v } + \\phi _ { s e l f } ( \\mathbf { f } _ { t } ^ { v } , \\mathbf { F } ^ { v } , \\mathbf { F } ^ { v } ) + \\phi _ { c r o s s } ( \\mathbf { f } _ { t } ^ { v } , \\mathbf { F } ^ { a } , \\mathbf { F } ^ { a } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "With the aggregated audio and visual features $\\hat { \\mathbf { f } } _ { t } ^ { a }$ and $\\hat { \\mathbf { f } } _ { t } ^ { v }$ , we can obtain the frame-wise event prediction $\\hat { \\mathbf { p } } _ { t } ^ { a } \\in \\mathbb { R } ^ { 1 \\times \\widetilde { C } }$ and $\\hat { \\mathbf { p } } _ { t } ^ { v } \\in \\mathbb { R } ^ { 1 \\times C }$ , and the attention weights computed by MLPs and normalized by a softmax function for audio, visual, and audio-visual streams (i.e., $\\mathbf { w } _ { t } ^ { a } \\in \\mathbb { R } ^ { 1 \\times C }$ , $\\mathbf { w } _ { t } ^ { v } \\in \\mathbb { R } ^ { 1 \\times C }$ , and $\\mathbf { w } _ { t } ^ { a v } \\in \\mathbb { R } ^ { 2 \\times C } ,$ ). Then the video-level prediction is gathered with the MMIL pooling: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/55d25e95859dcdbc0e18dcaa2f5b054e82fd68aa4b4e701a080a2cb7163a0878.jpg",
+        "text": "$$\n\\bar { \\mathbf { p } } ^ { a } = \\sum _ { t = 1 } ^ { T } \\mathbf { w } _ { t } ^ { a } \\hat { \\mathbf { p } } _ { t } ^ { a } , \\bar { \\mathbf { p } } ^ { v } = \\sum _ { t = 1 } ^ { T } \\mathbf { w } _ { t } ^ { v } \\hat { \\mathbf { p } } _ { t } ^ { v } , \\mathrm { a n d } \\bar { \\mathbf { p } } ^ { a v } = \\sum _ { t = 1 } ^ { T } \\mathbf { w } _ { t } ^ { a v } [ 0 ] \\mathbf { w } _ { t } ^ { a } \\hat { \\mathbf { p } } _ { t } ^ { a } + \\mathbf { w } _ { t } ^ { a v } [ 1 ] \\mathbf { w } _ { t } ^ { v } \\hat { \\mathbf { p } } _ { t } ^ { v } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The model can then be optimized using the binary cross-entropy loss function between $\\bar { \\bf p }$ and a video-level weak label $\\bar { \\mathbf { y } } \\in \\mathbb { R } ^ { 1 \\times C }$ , which does not indicate time boundaries and modalities for events. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 Cross-Modality Co-Occurrence ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Videos with multi-label events contain rich information among event categories because the related events are likely to present at the same time. The correlation is useful for models to robustly predict the presence or absence of events. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similar to [46], to explicitly model the relationship between event categories in different modalities, we first obtain the representations for each class and then measure the correlation. We note that the class relationships may be different in audio and visual modalities. That is why the work [46] cannot be directly applied to audio-visual video parsing since audio or visual events can be partially or jointly presented at a single frame. Thus, jointly understanding the class relationship within a modality and across two modalities can benefit the audio-visual video parsing task. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In order to map the frame-wise audio and visual features into class-level ones, the nonlinear transformation with MLPs is formulated as follows: ",
+        "bbox": [
+            174,
+            763,
+            825,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d313b9172ca9b6250573624cb928ff08260ac3770f3f987b57f8e82d08c0916.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { a } _ { t , c } = \\operatorname { R e L U } ( \\hat { \\mathbf { f } } _ { t } ^ { a } \\mathbf { M } _ { c } ^ { a } + \\mathbf { b } _ { c } ^ { a } ) , } \\\\ { \\mathbf { v } _ { t , c } = \\operatorname { R e L U } ( \\hat { \\mathbf { f } } _ { t } ^ { v } \\mathbf { M } _ { c } ^ { v } + \\mathbf { b } _ { c } ^ { v } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            795,
+            593,
+            839
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { a } _ { t , c }$ and $\\mathbf { v } _ { t , c }$ are audio and visual class-level features for class $c$ at time $t$ with dimension $1 \\times d _ { c }$ , respectively. The weights and biases for class $c$ for audio and visual features are denoted as $\\mathbf { M } _ { c } ^ { a }$ $\\mathbf { \\Psi } _ { : } ^ { i } , \\mathbf { M } _ { c } ^ { i } \\in \\mathbb { R } ^ { d \\times } \\mathbf { \\tilde { { d } } } _ { c }$ and ${ \\bf b } _ { c } ^ { a }$ $\\mathbf { \\bar { b } } _ { c } ^ { v } \\in \\mathbb { R } ^ { 1 \\times d _ { c } }$ . With class-level representations, we can further model the relationship between event categories within and across modalities by self-attention and crossmodality co-attention mechanism: ",
+        "bbox": [
+            173,
+            842,
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/72881a6d61d9ee642027dbd9fe83a19d454f3be0d7ee0d0382843a0d8348f6b1.jpg",
+        "text": "$$\n\\begin{array} { r } { \\hat { \\mathbf { a } } _ { t , c } = \\mathbf { a } _ { t , c } + \\phi _ { s e l f } ( \\mathbf { a } _ { t , c } , \\mathbf { A } _ { t } , \\mathbf { A } _ { t } ) + \\phi _ { c r o s s } ( \\mathbf { a } _ { t , c } , \\mathbf { V } _ { t } , \\mathbf { V } _ { t } ) , } \\\\ { \\hat { \\mathbf { v } } _ { t , c } = \\mathbf { v } _ { t , c } + \\phi _ { s e l f } ( \\mathbf { v } _ { t , c } , \\mathbf { V } _ { t } , \\mathbf { V } _ { t } ) + \\phi _ { c r o s s } ( \\mathbf { v } _ { t , c } , \\mathbf { A } _ { t } , \\mathbf { A } _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            87,
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+            123
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { A } _ { t } = \\{ \\mathbf { a } _ { t , 1 } , \\dotsc , \\mathbf { a } _ { t , C } \\}$ and $\\mathbf { V } _ { t } = \\{ \\mathbf { v } _ { t , 1 } , \\dots , \\mathbf { v } _ { t , C } \\}$ are sets of audio and visual class features at time $t$ . $\\hat { \\mathbf { a } } _ { t , c }$ and $\\hat { \\mathbf { v } } _ { t , c }$ are now co-occcurence features that consider the relationships between categories within and across modalities. We can then predict the probability for each event at time $t$ by MLPs and aggregate every segment-wise predictions into video-level ones i.e., ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c29410692dc2863506c070ce822f56ae4879f19b55b5ec100c21bc53504c3d84.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { \\mathbf { p } } _ { t } ^ { a } = \\sigma ( \\mathrm { M L P } _ { a } ( \\{ \\hat { \\mathbf { a } } _ { t , 1 } , \\dots , \\hat { \\mathbf { a } } _ { t , C } \\} ) ) , \\quad \\hat { \\mathbf { p } } _ { t } ^ { v } = \\sigma ( \\mathrm { M L P } _ { v } ( \\{ \\hat { \\mathbf { v } } _ { t , 1 } , \\dots , \\hat { \\mathbf { v } } _ { t , C } \\} ) ) , } \\\\ & { \\bar { \\mathbf { p } } ^ { a } , \\bar { \\mathbf { p } } ^ { v } , \\bar { \\mathbf { p } } ^ { a v } = \\mathrm { M M I L } ( \\{ \\hat { \\mathbf { p } } _ { 1 } ^ { a } , \\dots , \\hat { \\mathbf { p } } _ { T } ^ { a } \\} , \\{ \\hat { \\mathbf { p } } _ { 1 } ^ { v } , \\dots , \\hat { \\mathbf { p } } _ { T } ^ { v } \\} ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            253,
+            185,
+            740,
+            223
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is the sigmoid function, and $\\mathrm { M M L } ( \\cdot )$ is the multi-modal multiple instance learning pooling described in Eq. 3 taking all segment-wise predictions as inputs. The video-level prediction can be optimized by the binary cross-entropy loss function with a video-level weak label $\\bar { \\mathbf { y } }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 Shared Cross-Modality Semantics across Videos ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            284,
+            547,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The information across different videos provides rich supervisory signals that benefit the training of weakly-supervised audio-visual video parsing. By observing videos in a training batch, we can discover both the common and diverse event semantics. With video-level labels, we can initially associate related and irrelevant videos. In order to obtain a discriminative categorical representation, we would like to encourage audio and visual representations from related events to be similar and differentiate those from irrelevant videos. However, targeting at segment-wise representations with specific events is difficult due to the lack of temporal annotations. Therefore, we seek event-related frames through the weights from MMIL pooling in Eq. 3: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b75402bbc144e55bc1f513cac29fe6ded89ed203a5d73acdcd8a4f394e79fba.jpg",
+        "text": "$$\n\\tilde { \\mathbf { f } } ^ { a } = \\sum _ { t = 1 } ^ { T } \\Big [ \\frac { \\exp \\bigl ( g \\bigl ( \\bar { \\mathbf { y } } \\odot \\mathbf { w } _ { t } ^ { a } \\bigr ) \\bigr ) } { \\sum _ { t ^ { \\prime } = 1 } ^ { T } \\exp \\bigl ( g \\bigl ( \\bar { \\mathbf { y } } \\odot \\mathbf { w } _ { t ^ { \\prime } } ^ { a } \\bigr ) \\bigr ) } \\hat { \\mathbf { f } } _ { t } ^ { a } \\Big ] , \\quad \\tilde { \\mathbf { f } } ^ { v } = \\sum _ { t = 1 } ^ { T } \\Big [ \\frac { \\exp \\bigl ( g \\bigl ( \\bar { \\mathbf { y } } \\odot \\mathbf { w } _ { t } ^ { v } \\bigr ) \\bigr ) } { \\sum _ { t ^ { \\prime } = 1 } ^ { T } \\exp \\bigl ( g \\bigl ( \\bar { \\mathbf { y } } \\odot \\mathbf { w } _ { t ^ { \\prime } } ^ { v } \\bigr ) \\bigr ) } \\hat { \\mathbf { f } } _ { t } ^ { v } \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            424,
+            769,
+            468
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\odot$ and $g ( . )$ are element-wise dot product and summation function over all elements respectively. ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "With video-level labels and features ${ \\tilde { \\mathbf { f } } } ^ { a }$ and $\\tilde { \\mathbf { f } } ^ { v }$ ), we adopt contrastive learning [47, 48, 49] to encourage features across modalities with the same event category (at least one) to be close and those with different events to be far away from each other. We leverage all $n$ videos in a batch to explore diverse semantics, where the sets of audio and visual features are denoted as $\\{ \\widetilde { \\bf f } _ { ( 0 ) } ^ { a } , . . . , \\widetilde { \\bf f } _ { ( n ) } ^ { a } \\}$ and $\\{ \\tilde { \\mathbf { f } } _ { ( 0 ) } ^ { v } , . . . , \\tilde { \\mathbf { f } } _ { ( n ) } ^ { v } \\}$ respectively with video-level labels $\\left\\{ \\bar { \\mathbf { y } } _ { ( 0 ) } , . . . , \\bar { \\mathbf { y } } _ { ( n ) } \\right\\}$ . The relationship across videos can be optimized by the proposed training objective as follows: ",
+        "bbox": [
+            173,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/575eda778d80d968a5746712c2c81b94feb4fdca31b45c4f7807df8f1f5230ba.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { c o n t r a s t } } = - \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\Big [ \\log \\frac { \\sum _ { j = 1 } ^ { n } f ( \\bar { \\bf y } _ { i } \\cdot \\bar { \\bf y } _ { j } ) \\exp ( \\tilde { \\bf f } _ { ( i ) } ^ { a } \\cdot \\tilde { \\bf f } _ { ( j ) } ^ { v } / \\tau ) } { \\sum _ { j = 1 } ^ { n } \\exp ( \\tilde { \\bf f } _ { ( i ) } ^ { a } \\cdot \\tilde { \\bf f } _ { ( j ) } ^ { v } / \\tau ) } \\Big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            292,
+            592,
+            704,
+            636
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $f ( \\cdot )$ is a clipping function that clips values over 1, and $\\tau$ denotes a hyper-parameter controlling the temperature. Thus, the proposed method can be optimized by joint the binary cross-entropy loss mentioned in Section 3.1 and the contrastive learning loss in Eq. 8. Our training strategy can exploit cross-modality information across videos and event categories to understand common semantics while ignoring irrelevant ones. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experimental Results ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets. We use the Look, Listen and Parse (LLP) Dataset [4] for all experiments. The LLP dataset consists of 11, 849 10-seconds video clips annotated with 25 event categories. It covers various real-life scenes such as speech, music performances, car, cheering, dog, etc. Particularly, there are 7202 video clips labeled with more than one event category. We use the 10000 video clips with only video-level event annotations for model training. The detailed annotations (e.g., individual audio and visual events per second) are available for the remaining 1849 validation and test videos. For all experiments, we use the official data splits from the LLP dataset. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Metrics. Following previous work [4, 6], we adopt F-scores as the evaluation metrics. Note that all types of events (audio, visual, and audio-visual) are measured under both segmentlevel and event-level metrics. The segment-level metrics can evaluate snippet-wise prediction results. As for the event-level metrics, the clips are extracted by concatenating positive consecutive segments in the same events. Then, we compute the event-level F-scores with $\\mathrm { m I o U } = 0 . 5$ as the threshold. Furthermore, the overall Type $\\ @ \\mathbf { A V }$ performance on audio-visual scene is also considered by computing the averaged audio, visual, and audio-visual event evaluation results. Instead of directly averaging results from different event types, Event@AV considers all audio and visual event categories for each sample. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/084b18971d504f27393e74bd41c53c9cc07fa98ff1b6441b9862e1db77699897.jpg",
+        "table_caption": [
+            "Table 1: Quantitative results of weakly-supervised audio-visual video parsing. We evaluate all methods on the LLP dataset [4] with F-scores in five different event types and two kinds of segments. The first row indicates five different event types (audio, visual, audio-visual, Type@AV, and Event@AV). In the second row, two kinds of segments are shown: Seg. and Event are segmentlevel and event-level; and $^ *$ indicates only label refinement is utilized for fair comparisons. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Audio</td><td colspan=\"2\">Visual</td><td colspan=\"2\">Audio-visual</td><td colspan=\"2\">Type@AV</td><td colspan=\"2\">Event@AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>AVE [5]</td><td>47.2</td><td>40.4</td><td>37.1</td><td>34.7</td><td>35.4</td><td>31.6</td><td>39.9</td><td>35.5</td><td>41.6</td><td>36.5</td></tr><tr><td>AVSDN [2]</td><td>47.8</td><td>34.1</td><td>52.0</td><td>46.3</td><td>37.1</td><td>26.5</td><td>45.7</td><td>35.6</td><td>50.8</td><td>37.7</td></tr><tr><td>AVSDN + Ours</td><td>48.3</td><td>41.2</td><td>52.4</td><td>48.5</td><td>46.9</td><td>40.0</td><td>49.2</td><td>43.2</td><td>53.2</td><td>40.1</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + Ours</td><td>59.2</td><td>51.3</td><td>59.9</td><td>55.5</td><td>53.4</td><td>46.2</td><td>57.5</td><td>51.0</td><td>58.1</td><td>49.7</td></tr><tr><td>MA [6]</td><td>60.3</td><td>53.6</td><td>60.0</td><td>56.4</td><td>55.1</td><td>49.0</td><td>58.9</td><td>53.0</td><td>57.9</td><td>50.6</td></tr><tr><td>MA*</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA*+Ours</td><td>60.8</td><td>53.8</td><td>63.5</td><td>58.9</td><td>57.0</td><td>49.5</td><td>60.5</td><td>54.0</td><td>59.5</td><td>52.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Implementation Details. We implement the proposed method using PyTorch [50], and conduct the training and evaluation processes on a single NVIDIA GTX 1080 Ti GPU with 11 GB memory. Following [4, 6], we use the same visual and audio encoders for fair comparisons. We adopt both ResNet-152 [51] pre-trained on ImageNet [52] and 3D ResNet [53] pre-trained on Kinetics-400 [54] as visual feature extractors. Visual frames are sampled at 8 fps and their 2D and 3D visual features are extracted. The 2D and 3D visual features are concatenated and then processed by an MLP as the segment-wise representations. As for audio data, we utilize VGGish [55] pre-trained on AudioSet [56] to extract 128-dimensional audio features. The code and models are publicly available. ",
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+        "type": "text",
+        "text": "Evaluated methods. We compare the proposed method based on several baselines to the following weakly-unsupervised approaches to the audio-visual video parsing task: ",
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+        "text": "• AVE [5] consists of an audio-guided co-attention mechanism to adaptively learn the sounding regions. We note that AVE [5] deals with the audio-visual event localization task. Thus, we follow [4] and add additional audio and visual parsing branches for the weakly-supervised audio-visual video parsing task as a baseline.   \n• AVSDN [2] is a sequence-to-sequence-based model to integrate global audio and visual features to local ones. Since AVSDN [2] also deals with the audio-visual event localization task, we make the same modifications to AVSDN as those to AVE.   \n• HAN [4] is a multi-modal multiple instance learning-based method with a hybrid attention network.   \n• MA [6] reports the state-of-the-art performance on the weakly-supervised audio-visual video parsing task. It is a method based on HAN with the label refinement and the audio-visual contrastive learning differentiating temporal segments. ",
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+    {
+        "type": "text",
+        "text": "4.1 Quantitative Evaluation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 1 shows the quantitative comparisons on the LLP dataset [4]. The proposed method performs favorably against the competing approaches on the weakly-supervised audio-visual video parsing task. Since our method can be easily extended to existing methods, we extend the proposed on three baselines. The third, fifth, and last rows in Table 1 indicate that the proposed method generally benefits three baselines on several metrics of the audio-visual video parsing task by a large margin. We note that $\\mathbf { M A } ^ { * }$ [6] only utilizes label refinement to refine labels for each modality, and temporal difference audio-visual contrastive learning [6] is not implemented. ",
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+    {
+        "type": "table",
+        "img_path": "images/3c22ac25c5061d220d82ddeb6f86b8e8cce2d27d122739cee7c6f25d9429a171.jpg",
+        "table_caption": [
+            "Table 2: Ablation study. We investigate the effect of using different design components in the proposed method. We show how proposed cross-modality co-occurrence (CM-Co) in Section 3.3 and shared cross-modality semantics across videos (CM-S) module in Section 3.2 improve the baselines. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Audio</td><td colspan=\"2\">Visual</td><td colspan=\"2\">Audio-visual</td><td colspan=\"2\">Type@AV</td><td colspan=\"2\">Event@AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + CM-S</td><td>58.1</td><td>49.6</td><td>58.3</td><td>53.6</td><td>53.2</td><td>46.3</td><td>56.5</td><td>49.8</td><td>55.9</td><td>47.5</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>51.4</td><td>57.4</td><td>52.4</td><td>51.9</td><td>44.2</td><td>56.3</td><td>49.3</td><td>57.4</td><td>48.5</td></tr><tr><td>HAN + Ours</td><td>59.2</td><td>51.3</td><td>59.9</td><td>55.5</td><td>53.4</td><td>46.2</td><td>57.5</td><td>51.0</td><td>58.1</td><td>49.7</td></tr><tr><td>MA [28]</td><td>60.3</td><td>53.6</td><td>60.0</td><td>56.4</td><td>55.1</td><td>49.0</td><td>58.9</td><td>53.0</td><td>57.9</td><td>50.6</td></tr><tr><td>MA + CM-Co</td><td>61.1</td><td>53.3</td><td>61.7</td><td>57.3</td><td>56.3</td><td>49.0</td><td>59.7</td><td>53.0</td><td>58.9</td><td>51.2</td></tr><tr><td>MA*</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA* + CM-S</td><td>60.4</td><td>53.5</td><td>60.7</td><td>56.5</td><td>55.8</td><td>47.5</td><td>58.9</td><td>52.5</td><td>58.6</td><td>51.0</td></tr><tr><td>MA* + CM-Co</td><td>60.5</td><td>53.6</td><td>61.3</td><td>56.5</td><td>54.9</td><td>46.7</td><td>58.9</td><td>52.3</td><td>59.1</td><td>51.4</td></tr><tr><td>MA* +Ours</td><td>60.8</td><td>53.8</td><td>63.5</td><td>58.9</td><td>57.0</td><td>49.5</td><td>60.5</td><td>54.0</td><td>59.5</td><td>52.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "We notice that our method significantly improves baselines in the metrics of visual, audio-visual, Type $@ \\mathrm { A V } ,$ and Event $@$ AV. By observing the class distribution of training sets, we find that $3 1 \\%$ , $7 \\%$ , and $9 \\%$ training videos contain speech, singing, and violin events. These events are more likely to present in the audio modality. Therefore, the video-level labels would limit the performance regarding visual events. The proposed method can leverage additional cross-video and cross-modality supervisory signals to explore common semantics, which can improve results in vision-related metrics. ",
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+        "type": "text",
+        "text": "4.2 Ablation Study ",
+        "text_level": 1,
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+        "text": "Cross-Modality Co-Occurrence and Semantics across Video. We conduct the ablation study to analyze the individual impact of each developed component in the proposed method. The results are presented in Table 2. CM-Co represents the usage of the cross-modality co-occurrence module described in Section 3.2, which leverages the relationship between categories within and cross modalities. CM-S indicates the shared cross-modality semantics across videos module described in Section 3.3, which considers all audio and visual information across videos in a batch. ",
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+        "type": "text",
+        "text": "In Table 2, we note that both CM-S and CM-Co can improve baselines in several metrics. By exploring common semantics among training videos (CM-S), we improve the performance on visual and audio-visual evaluation by a large margin. Such a strategy can exploit additional information from videos to address the potential drawback of video-level labels described in Section 4.1. Furthermore, the proposed cross-modality co-occurrence module (CM-Co) also presents favorable results. We note that the significant improvement in Event@AV evaluation with the usage of CM-Co can verify the efficacy of considering the relationship between categories within and across modalities. Since Event $@$ AV considers all audio and visual events for the F-score (e.g., truth positive from both audio and visual events), the improvement of Event@AV indicates our cross-modality co-occurrence can perform well on video parsing when events present in an audio or a visual modality. ",
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+    {
+        "type": "text",
+        "text": "In the second group of the evaluated methods in Table 2, we verify if the proposed CM-S works better than the contrastive learning method in MA. We perform our CM-S on the MA model. The CM-S exploits information across different videos to address the issue that audio and visual tracks may not be synchronized. Instead, the contrastive learning method in MA is developed based on the assumption of synchronization to associate the audio-visual representation in a single video. Since our CM-S learns diverse and common semantics, it is effective and complementary to the contrastive learning approach in MA performing on a single video. We note that our CM-S generally improves the performance over all segment-level metrics, which supports our claim. ",
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+        "type": "text",
+        "text": "Self-attention and Cross-Modality Co-attention in Co-Occurrence. Since our cross-modality co-occurrence module exploits self-attention among class-level features in the same modality and cross-modality co-attention on cross-modality class-level representations to model the relationship between categories in the same and different modalities. Taking class-level audio features in Eq. 5 as an example, the class-level self-attention and cross-modality co-attention are $\\mathrm { a t t n } ( \\mathbf { a } _ { t , c } , \\mathbf { A } _ { t } , \\mathbf { A } _ { t } )$ and $\\operatorname { a t t n } ( \\mathbf { a } _ { t , c } , \\mathbf { V } _ { t } , \\mathbf { V } _ { t } ) .$ , respectively. ",
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+    {
+        "type": "table",
+        "img_path": "images/6d2690df8819bd3c6c2c8effa4b228db879b7c72a522eedf405aa11e230fcf3e.jpg",
+        "table_caption": [
+            "Table 3: Ablation study. We investigate the effect of different developed mechanisms in the proposed cross-modality co-occurrence (CM-Co) module in Section 3.2. In Eq. 5, class-level features are processed by self-attention and cross-modality co-attention mechanisms. A Only and $\\mathbf { V }$ Only indicate only self-attention performs for individual audio and visual events respectively. AV denotes performing self-attention for audio and visual events. CM-Co is the proposed method that considers relationship between categories within and cross modalities by both self-attention and cross-modality co-attention mechanisms. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"2\">Audio</td><td colspan=\"2\">Visual</td><td colspan=\"2\">Audio-visual</td><td colspan=\"2\">Type@AV</td><td colspan=\"2\">Event@ AV</td></tr><tr><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td><td>Seg.</td><td>Event</td></tr><tr><td>HAN [4]</td><td>60.1</td><td>51.3</td><td>52.9</td><td>48.9</td><td>48.9</td><td>43.0</td><td>54.0</td><td>47.7</td><td>55.4</td><td>48.0</td></tr><tr><td>HAN + A Only</td><td>60.5</td><td>52.3</td><td>49.8</td><td>43.9</td><td>45.6</td><td>38.3</td><td>52.0</td><td>44.8</td><td>55.7</td><td>45.9</td></tr><tr><td>HAN + V Only</td><td>56.1</td><td>44.5</td><td>56.8</td><td>53.2</td><td>49.7</td><td>40.7</td><td>54.2</td><td>46.1</td><td>54.1</td><td>44.6</td></tr><tr><td>HAN + AV</td><td>59.5</td><td>50.3</td><td>55.1</td><td>50.5</td><td>48.6</td><td>40.3</td><td>54.4</td><td>47.0</td><td>56.0</td><td>47.4</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>51.4</td><td>57.4</td><td>52.4</td><td>51.9</td><td>44.2</td><td>56.3</td><td>49.3</td><td>57.4</td><td>48.5</td></tr><tr><td>MA*[6]</td><td>59.8</td><td>52.1</td><td>57.5</td><td>54.4</td><td>52.6</td><td>45.8</td><td>56.6</td><td>50.8</td><td>56.6</td><td>49.4</td></tr><tr><td>MA*+ A Only</td><td>60.7</td><td>52.7</td><td>53.9</td><td>47.9</td><td>50.1</td><td>42.2</td><td>54.9</td><td>47.6</td><td>57.0</td><td>47.1</td></tr><tr><td>MA* + V Only</td><td>46.8</td><td>34.4</td><td>60.8</td><td>57.0</td><td>42.8</td><td>31.1</td><td>50.1</td><td>40.9</td><td>52.6</td><td>40.4</td></tr><tr><td>MA*+ AV</td><td>58.3</td><td>50.4</td><td>59.4</td><td>55.2</td><td>53.9</td><td>46.9</td><td>57.2</td><td>50.8</td><td>56.7</td><td>48.5</td></tr><tr><td>MA*+ CM-Co</td><td>60.5</td><td>53.6</td><td>61.3</td><td>56.5</td><td>54.9</td><td>46.7</td><td>58.9</td><td>52.3</td><td>59.1</td><td>51.4</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/89f0e2785faa97f098bed9fc6e71d1f81760b2cf7c930cd9333bc2b3bd34ab9b.jpg",
+        "table_caption": [
+            "Table 4: Ablation study. We evaluate the proposed method in accuracy, efficiency, and model sizes. We show the numbers of parameters and FLOPs for the proposed cross-modality co-occurrence (CM-Co) and HAN [4] with a few layers. ",
+            "Note that the results are all in the segment level. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Audio</td><td>Visual</td><td>Audio-visual</td><td>Type@AV</td><td>Event@AV</td><td>GFLOPs</td><td>Params</td></tr><tr><td>HAN 1 Layer</td><td>60.1</td><td>52.9</td><td>48.9</td><td>54.0</td><td>55.4</td><td>6.63</td><td>2.4M</td></tr><tr><td>HAN 2 Layers</td><td>58.2</td><td>55.4</td><td>50.6</td><td>54.7</td><td>54.9</td><td>7.28</td><td>2.9M</td></tr><tr><td>HAN 3 Layers</td><td>58.1</td><td>55.2</td><td>50.3</td><td>54.5</td><td>54.6</td><td>7.97</td><td>3.5M</td></tr><tr><td>HAN + CM-Co</td><td>59.7</td><td>57.4</td><td>51.9</td><td>56.3</td><td>57.4</td><td>6.99</td><td>2.8M</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 3 presents the results in various modifications of the cross-modality co-occurrence module. We note that the design of co-occurrence in the same and cross modalities can generally improve the results in several metrics. We also evaluate the co-occurrence module in a single modality. The results are shown in the second, third, seventh, and eighth rows in Table 3, where A Only and $\\mathbf { V }$ Only indicate the co-occurrence module only leverages the relationship between categories in audio or visual data respectively. As the results shown in the second and seventh rows, training with co-occurrence in audio events only (i.e., A Only) can slightly improve the performance on audio events. Similarly, considering visual event only (i.e., V Only) can benefit the results regarding visual events. Furthermore, the co-occurrence for both audio and visual categories (AV) in the fourth and ninth rows can contribute to the results in general metrics such as Type $@$ AV and Event $@ \\mathrm { A V } .$ . We then further consider the correlation between events across modalities. That is the cross-modality co-occurrence module (CM-Co) in the fifth and tenth rows. The results can confirm the efficacy of the proposed cross-modality co-occurrence module in all metrics except segment-level audio events caused by similar reasons discussed in Section 4.1. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Model Capacity. Since our cross-modality co-occurrence module leverages class-level representations, it would increase the capability of models on capturing information. For fair comparisons, we add extra parameters to HAN [4] to analyze whether more parameters can contribute to performance gain. Specifically, we increase the number of layers in its transformer-based feature aggregation to 2 and 3, respectively. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Table 4, we report the results in accuracy, computational costs, and model sizes. The first three rows show the performance of HAN with different numbers of layers. We note that HAN with one extra layer has more parameters than the proposed co-occurrence module. However, the results of HAN with extra layers indicate that using more parameters/layers for HAN does not improve the performance. The proposed cross-modality co-occurrence module enhances HAN more effectively. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3a3e5bddfd4d0b8b3df955da6588c994004175b863c809999ccff152b083248e.jpg",
+        "image_caption": [
+            "Figure 2: Qualitative comparisons. We compare the proposed method with the state-of-the-art weakly-supervised audio-visual video parsing method on the LLP dataset [4]. The frame-wise annotations are shown in gray and purple bars. The gray bar denotes visual events, and the purple bar represents audio events. GT_V and GT_A are the ground-truth visual and audio events respectively. Our results are shown in the green block, and the results by the competing method, MA [6], are present in the blue block. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/21cead53b19ede061abdc05068f5d07b4594cffe449abe292ef47d1a62306ab2.jpg",
+        "image_caption": [
+            "Figure 3: Audio feature distribution by using t-SNE. The upper figure shows the distribution by our method. The lower figure presents that by $\\mathbf { M A } ^ { * }$ . The legend lists all event combinations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.3 Qualitative Evaluation ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Qualitative Results. We present the qualitative results of the evaluated methods in Figure 2. GT_V and GT_A show the ground-truth annotations for visual and audio events, respectively. Pred_V and Pred_A present the predictions made by our method and the state-of-the-art competing method, MA [6], respectively. Our results are shown in the green block, while the results of MA are present in the blue block. In general, our method presents more accurate predictions in both audio and visual events than MA. We note that the whole violin is shown after 7 seconds. That would hamper models for understanding visual events e.g., MA predicts wrong results on violin visual events before 6 seconds. Since our method leverages the relationship between categories, it can still predict correct temporal boundaries for guitar events by jointly considering cello events in the videos. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/ed423064930570c6abeaee1db5cadd585aab212a9182498adeec79bb58aa61f7.jpg",
+        "image_caption": [
+            "Figure 4: Visual feature distribution by using t-SNE. The upper figure shows the distribution by our method. The lower figure presents that by MA∗. The legend lists all event combinations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
+            90,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Feature Distribution Visualized by t-SNE. We apply t-SNE to the aggregated audio and visual features from each segment described in Eq. 2. The visualization results are present in Figure 3 and Figure 4, respectively. The legends list all the combinations of multiple labels. For example, in Figure 3, audio events of singing are present as blue spots, and the mixed sounds of singing and violin are shown as purple spots. We note that the related events including multiple events are shown in similar colors. In Figure 4, the proposed method achieves better performance in the sense that similar color spots are closer than the spots in $\\mathbf { M A } ^ { * }$ . ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "5 Conclusions ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we present a novel audio-visual video parsing framework in a weakly-supervised manner that can be applied to existing methods. We propose two modules to exploit the relationship across videos, modalities, and event categories, and explore additional supervisory signals that can benefit audio-visual video parsing. The shared cross-modality semantics module leverages common and diverse event semantics across videos to learn robust cross-modality representations that facilitate models to identify audio, visual, and audio-visual events. Furthermore, the cross-modality co-occurrence module aims to learn the relationship between event categories. It helps localize segments of target events and can exclude irrelevant ones by performing self-attention and crossmodality co-attention on class-wise features, Extensive experimental results show that our approach substantially improves several baselines and performs favorably against the state-of-the-art methods. ",
+        "bbox": [
+            174,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments. This work was supported in part by the Ministry of Science and Technology under grants 109- 2221-E-009-113-MY3, 110-2628-E-A49-008, and 110-2634-F007-015. It was also funded in part by Qualcomm through a Taiwan University Research Collaboration Project, the Higher Education Sprout Project of the National Yang Ming Chiao Tung University, and Ministry of Education. ",
+        "bbox": [
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+    {
+        "type": "text",
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Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In NeurIPS, 2017. 4   \n[46] Praveen Tirupattur, Kevin Duarte, Yogesh Rawat, and Mubarak Shah. Modeling multi-label action dependencies for temporal action localization. In CVPR, 2021. 4   \n[47] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv Preprint, 2018. 5   \n[48] Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance discrimination. In CVPR, 2018. 5   \n[49] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In CVPR, 2020. 5   \n[50] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-performance deep learning library. In NeurIPS, 2019. 6   \n[51] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016. 6   \n[52] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. ImageNet: A Large-Scale Hierarchical Image Database. In CVPR, 2009. 6   \n[53] Du Tran, Heng Wang, Lorenzo Torresani, Jamie Ray, Yann LeCun, and Manohar Paluri. A closer look at spatiotemporal convolutions for action recognition. In CVPR, 2018. 6   \n[54] João Carreira and Andrew Zisserman. Quo vadis, action recognition? A new model and the kinetics dataset. In CVPR, 2017. 6   \n[55] Shawn Hershey, Sourish Chaudhuri, Daniel P. W. Ellis, Jort F. Gemmeke, Aren Jansen, Channing Moore, Manoj Plakal, Devin Platt, Rif A. Saurous, Bryan Seybold, Malcolm Slaney, Ron Weiss, and Kevin Wilson. Cnn architectures for large-scale audio classification. In ICASSP, 2017. 6   \n[56] Jort F Gemmeke, Daniel PW Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R Channing Moore, Manoj Plakal, and Marvin Ritter. Audio set: An ontology and human-labeled dataset for audio events. In ICASSP, 2017. 6 ",
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parse/train/r1ejxnCctX/r1ejxnCctX.md

@@ -0,0 +1,252 @@
+# REPRESENTATION FLOW FOR ACTION RECOGNITION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+In this paper, we propose a convolutional layer inspired by optical flow algorithms to learn motion representations. Our representation flow layer is a fully-differentiable layer designed to capture the ‘flow’ of any representation channel within a convolutional neural network for action recognition. Its parameters for iterative flow optimization are learned in an end-to-end fashion together with the other model parameters, maximizing the action recognition performance. Furthermore, we newly introduce the concept of learning ‘flow of flow’ representations by stacking multiple representation flow layers. We conducted extensive experimental evaluations, confirming its advantages over previous recognition models using traditional optical flows in both computational speed and performance.
+
+# 1 INTRODUCTION
+
+Activity recognition is an important problem in computer vision with many societal applications including surveillance, robot perception, smart environment/city, and more. Use of video convolutional neural networks (CNNs) have become the standard method for this task, as they can learn more optimal representations for the problem. Two-stream networks (Simonyan & Zisserman, 2014), taking both RGB frames and optical flow as input, provide state-of-the-art results and have been extremely popular. 3-D spatio-temporal CNN models, e.g., (Carreira & Zisserman, 2017), with XYT convolutions also found that such two-stream design $\mathrm { R G B } +$ optical flow) increases their accuracy. Abstracting both appearance information and explicit motion flow benefits the recognition.
+
+However, optical flow is expensive to compute. It often requires hundreds of optimization iterations every frame, and causes learning of two separate CNN streams (i.e., RGB-stream and flow-stream). This requires significant computation cost and a great increase in the number of model parameters to learn. Further, this means that the model needs to compute optical flow every frame even during inference and run two parallel CNNs, limiting its real-time applications.
+
+There were previous works to learn representations capturing motion information without using optical flow as input, such as motion feature networks (Lee et al., 2018) and ActionFlowNet $( \mathrm { N g }$ et al., 2018). However, although they were more advantageous in terms of the number of model parameters and computation speed, they suffered from inferior performance compared to two-stream models on public datasets such as Kinetics (Kay et al., 2017) and HMDB (Kuehne et al., 2011). We hypothesize that the iterative optimization performed by optical flow methods produces an important feature that other methods fail to capture.
+
+In this paper, we propose a CNN layer inspired by optical flow algorithms to learn motion representations for action recognition without having to compute optical flow. Our representation flow layer is a fully-differentiable layer designed to capture ‘flow’ of any representation channels within the model. Its parameters for iterative flow optimization are learned together with other model parameters, maximizing the action recognition performance. This is also done without having/training multiple network streams, reducing the number of parameters in the model. Further, we newly introduce the concept of learning ‘flow of flow’ representations by stacking multiple representation flow layers. We conduct extensive action classification experimental evaluation of where to compute optical flow and various hyperparameters, learning parameters, and fusion techniques.
+
+# 2 RELATED WORKS
+
+Capturing motion and temporal information has been studied for activity recognition. Early, handcrafted approaches such as dense trajectories (Wang et al., 2011) captured motion information by tracking points through time. Many algorithms have been developed to compute optical flow as a way to capture motion in video (Fortun et al., 2015). Other works have explored learning the ordering of frames to summarize a video in a single ‘dynamic image’ used for activity recognition (Bilen et al., 2016).
+
+Convolutional neural networks (CNNs) have been applied to activity recognition. Initial approaches explored methods to combine temporal information based on pooling or temporal convolution $( \mathrm { N g }$ et al., 2015; Karpathy et al., 2014). Other works have explored using attention to capture sub-events of activities (Piergiovanni et al., 2017). Two-stream networks have been very popular: they take input of a single RGB frame (captures appearance information) and a stack of optical flow frames (captures motion information). Often, the two network streams of the model are separately trained and the final predictions are averaged together (Simonyan & Zisserman, 2014). There were other two-stream CNN works exploring different ways to ‘fuse’ or combine the motion CNN with the appearance CNN (Feichtenhofer et al., 2016b;a). There were also large 3D XYT CNNs learning spatio-temporal patterns (Carreira & Zisserman, 2017; Xie et al., 2017), enabled by large video datasets such as Kinetics (Kay et al., 2017). However, these approaches still rely on optical flow input to maximize their accuracies.
+
+Recently, there have been works on learning motion representations. Fan et al. (2018) implemented the TVL-1 method using deep learning libraries to increase its computational speed and allow for learning some parameters. The result was fed to a two-stream CNN for the recognition. Several works explored learning a CNN to predict optical flow, which also can be used for action recognition (Dosovitskiy et al., 2015; Hui et al., 2018; Gao et al., 2018; Sun et al., 2018; $\mathrm { N g }$ et al., 2018). Lee et al. (2018) shifted features from sequential frames to capture motion in a non-iterative fashion. Sun et al. (2018) proposed an optical flow guided feature (OFF) by computing the gradients of representations and temporal differences, but required RGB, optical flow and RGB differences to achieve state-of-the-art performance.
+
+Unlike prior works, our proposed model with the representation flow layers relies only on the RGB input, learning much fewer parameters while correctly representing motion with the iterative optimization. It is significantly faster than the video CNNs requiring optical flow input, while still performing as good as or even better than the two-stream models. It clearly outperforms existing motion representation methods including (Fan et al., 2018) in both speed and accuracy, which we experimentally confirm.
+
+# 3 APPROACH
+
+Our method is a fully-differentiable convolutional layer inspired by optical flow algorithms. Unlike traditional optical flow methods, all the parameters of our method can be learned end-to-end, maximizing action recognition performance. Furthermore, our layer is designed to compute the ‘flow’ of any representation channels, instead of limiting its input to be traditional RGB frames.
+
+# 3.1 REVIEW OF OPTICAL FLOW METHODS
+
+Before describing our layer, we briefly review how optical flow is computed. Optical flow methods are based on the brightness consistency assumption. That is, given sequential images $I _ { 1 } , I _ { 2 }$ , a point $x , y$ in $I _ { 1 }$ is located at $x + \Delta x , y + \Delta y$ in $I _ { 2 }$ , or $I _ { 1 } ( x , y ) = I _ { 2 } ( x + \Delta x , y + \Delta y )$ . These methods assume small movements between frames, so this can be approximated with a Taylor series: $\begin{array} { r } { I _ { 2 } = I _ { 1 } + \frac { \delta I } { \delta x } \Delta x + \frac { \delta I } { \delta y } \Delta y } \end{array}$ , where $\pmb { u } = [ \Delta x , \Delta y ]$ . These equations are solved for $\textbf { \em u }$ to obtain the flow, but can only be approximated due to the two unknowns.
+
+The standard, variational methods for approximating optical flow (e.g., Brox (Brox et al., 2004) and TVL-1 (Zach et al., 2007) methods) take sequential images $I _ { 1 } , I _ { 2 }$ as input. Variational optical flow methods estimate the flow field, $\textbf { \em u }$ , using an iterative optimization method. The tensor $\pmb { u } \in \dot { \mathcal { R } } ^ { 2 \times W \times H }$ is the $x$ and $y$ directional flow for every location in the image. Taking two sequential images as input, $I _ { 1 } , I _ { 2 }$ , the methods first compute the gradient in both $x$ and $y$ directions: $\nabla I _ { 2 }$ . The initial flow is set to 0, ${ \pmb u } = 0$ . Then $\rho$ , the residual, can be computed. For efficiency, the constant part of $\rho$ , $\rho _ { c }$ is pre-computed:
+
+$$
+\rho _ { c } = I _ { 2 } - \nabla _ { x } I _ { 2 } \cdot \pmb { u } _ { x } - \nabla _ { y } I _ { 2 } \cdot \pmb { u } _ { y } - I _ { 1 }
+$$
+
+The iterative optimization is then performed, each updating $\textbf { \em u }$ :
+
+$$
+\begin{array} { r l } & { \rho = \rho _ { c } + \nabla _ { x } I _ { 2 } \cdot \boldsymbol { u } _ { x } + \nabla _ { y } I _ { 2 } \cdot \boldsymbol { u } _ { y } } \\ & { \boldsymbol { v } = \left\{ \begin{array} { l l } { \boldsymbol { u } + \lambda \theta \nabla I _ { 2 } } & { \rho < - \lambda \theta | \nabla I _ { 2 } | ^ { 2 } } \\ { \boldsymbol { u } - \lambda \theta \nabla I _ { 2 } } & { \rho > \lambda \theta | \nabla I _ { 2 } | ^ { 2 } } \end{array} \right. } \\ & { \boldsymbol { u } - \rho \frac { \nabla I _ { 2 } } { | I _ { 2 } | ^ { 2 } } \quad \mathrm { o t h e r w i s e } } \\ & { \boldsymbol { u } = \boldsymbol { v } + \boldsymbol { \theta } \cdot \mathrm { d i v e r g e n c e } ( p ) } \\ & { p = \frac { p + \frac { \tau } { \theta } \nabla u } { 1 + \frac { \tau } { \theta } | \nabla u | } } \end{array}
+$$
+
+Here $\theta$ controls the weight of the TVL-1 regularization term, $\lambda$ controls the smoothness of the output and $\tau$ controls the time-step. These hyperparameters are manually set. $\pmb { p }$ is the dual vector fields, which are used to minimize the energy. The divergence of $\pmb { p }$ , or backward difference, is computed as:
+
+$$
+\mathrm { d i v e r g e n c e } ( p ) = p _ { x , i , j } - p _ { x , i - 1 , j } + p _ { y , i , j } - p _ { y , i , j - 1 }
+$$
+
+where ${ \pmb p } _ { x }$ is the $x$ direction and $\mathbf { \Delta } _ { p _ { y } }$ is the $y$ direction, and $\pmb { p }$ contains all the spatial locations in the image.
+
+The goal is to minimize the total variational energy:
+
+$$
+E = | \nabla \boldsymbol { u } | + \lambda | I _ { 1 } \ast \boldsymbol { u } + I _ { 1 } - I _ { 2 } |
+$$
+
+Approaches run this iterative optimization for multiple input scales, from small to large, and use the previous flow estimate $\textbf { \em u }$ to warp $I _ { 2 }$ at the larger scale, providing a coarse-to-fine optical flow estimation. These standard approaches require multiple scales and warpings to obtain a good flow estimate, taking thousands of iterations.
+
+# 3.2 REPRESENTATION FLOW LAYER
+
+Inspired by the optical flow algorithm, we design a fully-differentiable, learnable, convolutional representation flow layer by extending the general algorithm outlined above. The main differences are that (i) we allow the layer to capture flow of any CNN feature map, and that (ii) we learn its parameters including $\theta , \lambda .$ , and $\tau$ as well as the divergence weights. We also make several key changes to reduce computation time: (1) we only use a single scale, (2) we do not perform any warping, and (3) we compute the flow on a CNN tensor with a smaller spatial size. Multiple scale and warping are computationally expensive, each requiring many iterations. By learning the flow parameters, we can eliminate the need for these additional steps. Our method is applied on lower resolution CNN feature maps, instead of the RGB input, and is trained in an end-to-end fashion. This not only benefits its speed, but also allows the model to learn a motion representation optimized for activity recognition.
+
+We note that the brightness consistency assumption can similarly be applied to CNN feature maps. Instead of capturing pixel brightness, we capture feature value consistency. This same assumption holds because CNNs are designed to be spatially invariant; i.e., they produce roughly the same feature value for the same object as it moves.
+
+Given the input $F _ { 1 } , F _ { 2 }$ , a single channel from sequential CNN feature maps (or input image), we compute the gradient by convolving the input feature maps with the Sobel filter:
+
+$$
+\nabla F _ { 2 x } = \left[ \begin{array} { l l l } { 1 } & { 0 } & { - 1 } \\ { 2 } & { 0 } & { - 2 } \\ { 1 } & { 0 } & { - 1 } \end{array} \right] * F _ { 2 } , \nabla F _ { 2 y } = \left[ \begin{array} { l l l } { 1 } & { 2 } & { 1 } \\ { 0 } & { 0 } & { 0 } \\ { - 1 } & { - 2 } & { - 1 } \end{array} \right] * F _ { 2 }
+$$
+
+We set $\pmb { u } = 0 , \pmb { p } = 0$ initially, each having width and height matching the input, then we can compute $\rho _ { c } = F _ { 2 } - F _ { 1 }$ . Next, we run the iterative optimization for a fixed number of iterations,
+
+following Eqs. 2-5. To compute the divergence, we zero-pad $\pmb { p }$ on the first column ( $\mathbf { \dot { x } }$ -direction) or row (y-direction) then convolve it with weights, $w _ { x } , w _ { y }$ to compute Eq. 6:
+
+$$
+\mathrm { d i v e r g e n c e } ( p ) = p _ { x } * w _ { x } + p _ { y } * w _ { y }
+$$
+
+where initially $w _ { x } = [ - 1 \quad 1 ]$ and $w _ { y } = { \binom { - 1 } { 1 } }$ . Note that these parameters are differentiable and can be learned with backpropagation. We compute $\nabla \boldsymbol { u }$ as
+
+$$
+\nabla \pmb { u } _ { x } = \left[ \begin{array} { c c c } { 1 } & { 0 } & { - 1 } \\ { 2 } & { 0 } & { - 2 } \\ { 1 } & { 0 } & { - 1 } \end{array} \right] * \pmb { u } _ { x } , \nabla \pmb { u } _ { y } = \left[ \begin{array} { c c c } { 1 } & { 2 } & { 1 } \\ { 0 } & { 0 } & { 0 } \\ { - 1 } & { - 2 } & { - 1 } \end{array} \right] * \pmb { u } _ { y }
+$$
+
+Algorithm 1 shows the process of our representation flow layer. Our flow layer with multiple iterations could also be interpreted as having a sequence of convolutional layers with each layer behavior dependent on its previous layer. Note that our method is fully differentiable and allows for the learning of all parameters, including $( \tau , \lambda , \theta )$ and the divergence weights $( w _ { x } , w _ { y } )$ .
+
+<table><tr><td colspan="2">Algorithm1 Method for the representation flow layer</td></tr><tr><td colspan="2">function REPRESENTATIONFLOW(F1,F2)</td></tr><tr><td colspan="2">u=0,p=0</td></tr><tr><td colspan="2">Compute image/feature map gradients (Eq. 8)</td></tr><tr><td colspan="2">Pc=F2-F1</td></tr><tr><td colspan="2">for n iterations do</td></tr><tr><td rowspan="3">ρ= Pc+VxF2 :Ux +VyF2 :Uy</td><td></td></tr><tr><td>(u+λ0VF2 ρ&lt;-X0|VF2l²</td></tr><tr><td>v ={u- λ0VF2 ρ&gt; λ0|VF2|²</td></tr><tr><td rowspan="2">u-PF VF2 u = v + 0 · divergence(p)</td><td>otherwise</td></tr><tr><td></td></tr><tr><td>p+Vu p=</td><td></td></tr><tr><td colspan="2">1+Vu end for</td></tr><tr><td colspan="2">return u</td></tr><tr><td colspan="2">end function</td></tr></table>
+
+Computing Flow-of-Flow Standard optical flow algorithms compute the flow for two sequential images. An optical flow image contains information about the direction and magnitude of the motion. Applying the flow algorithm directly on two flow images means that we are tracking pixels/locations showing similar motion in two consecutive frames. In practice, this typically leads to a worse performance due to inconsistent optical flow results and non-rigid motion. On the other hand, our representation flow layer is trained for the data, and is able to suppress such inconsistency and better abstract/represent motion by having multiple regular convolutional layers between the flow layers. Fig. 4 illustrates such design, which we confirm its benefits in the experiment section. By stacking multiple representation flow layers, our model is able to capture longer temporal intervals and consider locations with motion consistency.
+
+Representation Flow within a CNN CNN feature maps may have hundreds or thousands of channels and our representation flow layer computes the flow for each channel, which can take significant time and memory. To address this, we apply a convolutional layer to reduce the number of channels from $C$ to $C ^ { \prime }$ before the flow layer (note that $C ^ { \prime }$ is still significantly more than traditional optical flow algorithms, which are only applied to a single channel, greyscale images). For numerical stability, we normalize this feature map to be in [0, 255], matching standard image values. We found that the CNN features were quite small on average $( < 0 . 5 )$ and the TVL-1 algorithm default hyperparameters are designed for standard images values in [0, 255], thus we found this normalization step important. Using the normalized feature, we compute the flow and stack the $x$ and $y$ flows, resulting in $2 C ^ { \prime }$ channels. Finally, we apply another convolutional layer to convert from $2 C ^ { \prime }$ channels to $C$ channels. This is passed to the remaining CNN layers for the prediction. We average predictions from many frames to classify each video, as shown in Fig. 1.
+
+![](images/4234d2e9ea73c6a3106678a0c4f7e1631e757b7c619ee17371cfbc905ce98c8e.jpg)  
+Figure 1: Illustration of a video-CNN with our representation flow layer. The CNN computes intermediate feature maps, and sequential feature maps are used as input to the flow layer. The outputs of the flow layer are used for prediction.
+
+# 3.3 ACTIVITY RECOGNITION MODEL
+
+We place the representation flow layer inside a standard activity recognition model taking a $T \times C \times$ $W \times H$ tensor as input to a CNN. Here, $C$ is 3 as our model uses direct RGB frames as an input. $T$ is the number of frames the model processes, and $W$ and $H$ are the spatial dims. The CNN outputs a prediction per-timestep and these are temporally averaged to produce a probability for each class. The model is trained to minimize cross-entropy:
+
+$$
+L ( v , c ) = - \sum _ { i } ^ { K } ( c = = i ) \log ( C N N ( v ) _ { i } )
+$$
+
+where $v$ is the video, $C N N$ is the classification CNN and $c$ represents which of the $K$ classes $v$ belongs. That is, the parameters in our flow layers are trained together with the other layers, so that it maximizes the final classification accuracy.
+
+# 4 EXPERIMENTS
+
+Implementation details We implemented our representation flow layer in PyTorch and will release our code and models upon publication. As training CNNs on videos is computationally expensive, we used a subset of the Kinetics dataset (Kay et al., 2017) with $1 0 0 \mathrm { k }$ videos from 150 classes: Tiny-Kinetics. This allowed testing many models more quickly, while still having sufficient data to train large CNNs. For most experiments, we used ResNet-34 (He et al., 2016) with input of size $1 6 \times 1 1 2 \times 1 1 2$ (i.e., 16 frames with spatial size of 112). As training video CNNs is computationally expensive, we used this smaller input, which reduces performance, but allowed us to use larger batch sizes and run many experiments more quickly. Our final models are trained on standard $2 2 4 \times 2 2 4$ images. See appendix A for specific training details.
+
+Where to compute flow? To determine where in the network to compute the flow, we compare applying our flow layer on the RGB input, after the first conv. layer, and after the each of the 5 residual blocks. The results are shown in Table 1. We find that computing the flow on the input provides poor performance, similar to the performance of the flow-only networks, but there is a significant jump after even 1 layer, suggesting that computing the flow of a feature is beneficial, capturing both the appearance and motion information. However, after 4 layers, the performance begins to decline as the spatial information is too abstracted/compressed (due to pooling and large spatial receptive field size), and sequential features become very similar, containing less motion information. Note that our HMDB performance in this table is quite low compared to state-of-the-art
+
+Table 1: Computing the optical flow representation after various number of CNN layers. Results are video classification accuracy on our Tiny-Kinetics and LowRes-HMDB51 datasets using 100 iterations to compute the flow representation.   
+Table 2: Comparison of learning different parameters. The flow was computed after Block 3 using 100 iterations.   
+
+<table><tr><td></td><td>Tiny-Kinetics</td><td>LowRes-HMDB</td></tr><tr><td>RGB CNN</td><td>55.2</td><td>35.5</td></tr><tr><td>Flow CNN</td><td>35.4</td><td>37.5</td></tr><tr><td>Two-Stream CNN</td><td>57.6</td><td>41.5</td></tr><tr><td>Flow Layer on RGB Input</td><td>37.4</td><td>40.5</td></tr><tr><td>After Block1</td><td>52.4</td><td>42.6</td></tr><tr><td>After Block 2</td><td>57.4</td><td>44.5</td></tr><tr><td>After Block 3</td><td>59.4</td><td>45.4</td></tr><tr><td>After Block 4</td><td>52.1</td><td>43.5</td></tr><tr><td>After Block 5</td><td>50.3</td><td>42.2</td></tr></table>
+
+<table><tr><td></td><td>Tiny-Kinetics</td><td>LowRes-HMDB</td></tr><tr><td>None (all fixed)</td><td>59.4</td><td>45.4</td></tr><tr><td>Sobel kernels</td><td>58.5</td><td>43.5</td></tr><tr><td>Divergence (wx, wy)</td><td>60.2</td><td>46.4</td></tr><tr><td>T,入,θ</td><td>59.9</td><td>46.2</td></tr><tr><td>All</td><td>59.2</td><td>46.2</td></tr><tr><td>Divergence + T, 入,0</td><td>60.7</td><td>46.8</td></tr></table>
+
+methods due to being trained from scratch using few frames and low spatial resolution $( 1 1 2 \times 1 1 2 )$ .   
+For the following experiments, unless otherwise noted, we apply the layer after the 3rd residual block.
+
+What to learn? As our method is fully differentiable, we can learn any of the parameters, such as the kernels used to compute image gradients, the kernels for the divergence computation and even $\tau , \lambda , \theta$ . In Table 2, we compare the effects of learning different parameters. We find that learning the Sobel kernel reduces performance, but learning the divergence and $\tau , \lambda , \theta$ is beneficial.
+
+How many iterations for flow? To confirm that the iterations are important and determine how many we need, we experiment with various numbers of iterations. We compare the number of iterations needed for both learning (divergence+ $\cdot \tau , \lambda , \theta )$ and not learning parameters. The flow is computed after 3 residual blocks. The results are shown in Table 3. We find that learning provides better performance with fewer iterations (similar to the finding in (Fan et al., 2018)), and that iteratively computing the feature is important. We use 10 or 20 iterations in the remaining experiments as they provide good perforamnce and are fast.
+
+Two-stream fusion? Two-stream CNNs fusing both RGB and optical flow features has been heavily studied (Simonyan & Zisserman, 2014; Feichtenhofer et al., 2016b). Based on these works, we compare various ways of fusing RGB and our flow representation, shown in Fig. 2. We compare no fusion, late fusion (i.e., separate RGB and flow CNNs) and addition/multiplication/concatenation fusion. In Table 4, we compare different fusion methods for different locations in the network. We find that fusing RGB information is very important “when computing flow directly from RGB input”. However, it is not as beneficial when computing the optical flow of representations as the CNN has already abstracted much appearance information away. We found that concatenation of the RGB and flow features perform poorly compared to the others. We do not use two-stream fusion in any other experiments, as we found that computing the representation flow after the 3rd residual block provides sufficient performance.
+
+Table 3: Effect of the number of iterations on our Tiny-Kinetics dataset for learning and not learning.   
+
+<table><tr><td></td><td>Not learned</td><td>Learned</td></tr><tr><td>1 iteration</td><td>46.7</td><td>49.5</td></tr><tr><td>5 iterations</td><td>51.3</td><td>55.4</td></tr><tr><td>10 iterations</td><td>52.4</td><td>59.4</td></tr><tr><td>20 iterations</td><td>53.6</td><td>60.7</td></tr><tr><td>50 iterations</td><td>59.2</td><td>60.9</td></tr><tr><td>100 iterations</td><td>59.4</td><td>60.7</td></tr></table>
+
+![](images/6e8c3e932994394863caa45ecee5ee3ae09be321f1fba131a0b803b452e812ce.jpg)
+
+![](images/987460efdd61be120c38a3d17eb21e208da49dac0a07d7276b3ff3cb814870f8.jpg)  
+Figure 2: Different approaches to fusing RGB and flow information. (a) No fusion (b) Late fusion (c) The circle represents elementwise addition/multiplication or concatenation.
+
+![](images/be45bcfbe4033ba98f4ae4abc655b39b6cd9903834517012943f2118ee1bf8b4.jpg)  
+Figure 3: (a) RGB (b) Flow (c) Flow-of-Flow.
+
+![](images/21a5d3739adf98f9c83a4a9ae0e407bdf9f3aed39958cfd107d75f9389dd8663.jpg)  
+Figure 4: Illustration of how our model computes the FoF.   
+Figure 5: Comparing the results of (b) TVL-1 and (c) our learned flow when applied to RGB images.
+
+Flow-of-flow We can stack our layer multiple times, computing the flow-of-flow (FoF). This has the advantage of combining more temporal information into a single feature. Our results are shown in Table 5. Applying the TVL-1 algorithm twice gives quite poor performance, as optical flow features do not really satisfy the brightness consistency assumption, as they capture magnitude and direction of motion (shown in Fig. 3). Applying our representation flow layer twice performs significantly better than TVL-1 twice, but still worse than our baseline of not doing so. However, we can add a convolutional layer between the first and second flow layer, flow-conv-flow (FcF), (Fig. 4), allowing the model to better learn longer-term flow representations. We find this performs best, as this intermediate layer is able to smooth the flow and produce a better input for the representation flow layer. However, we find adding a third flow layer reduces performance as the motion representation becomes unreliable, due to the large spatial receptive field size.
+
+Table 4: Different fusion methods for flow computed at different locations in the network on our Tiny-Kinetics dataset using 10 iterations and learning flow parameters.   
+
+<table><tr><td></td><td>RGB</td><td>1 Block</td><td>3 Blocks</td></tr><tr><td>None</td><td>37.4</td><td>52.4</td><td>59.4</td></tr><tr><td>Late</td><td>61.3</td><td>60.4</td><td>61.5</td></tr><tr><td>Add</td><td>59.7</td><td>57.2</td><td>56.5</td></tr><tr><td>Multiply</td><td>58.3</td><td>58.1</td><td>57.8</td></tr><tr><td>Layer + Multiply</td><td>60.1</td><td>61.7</td><td>61.7</td></tr><tr><td>Concat</td><td>42.4</td><td>48.5</td><td>47.6</td></tr></table>
+
+Table 5: Computing the FoF representation. TVL-1 twice provides poor performance, using two flow layers with a conv. in between provides the best performance. Experiments used 10 iterations and learning flow parameters.   
+
+<table><tr><td></td><td>Tiny-Kinetics</td></tr><tr><td>TVL-1 twice</td><td>12.2</td></tr><tr><td>Single Flow Layer</td><td>59.4</td></tr><tr><td>Flow-of-Flow</td><td>47.2</td></tr><tr><td>Flow-Conv-Flow (FcF)</td><td>62.3</td></tr><tr><td>Flow-Conv-Flow-Conv-Flow</td><td>56.5</td></tr></table>
+
+Table 6: Flow using 3D ResNet-18.   
+
+<table><tr><td>Tiny-Kinetics</td></tr><tr><td>RGB 3D ResNet-18 54.6</td></tr><tr><td>TVL-1 3D ResNet-18 37.6</td></tr><tr><td>Two-Stream 3DResNet 57.5</td></tr><tr><td>RGB-Only OFF (Sun et al., 2018) 54.8</td></tr><tr><td>Input (RGB) 38.5</td></tr><tr><td>After Block 1 58.4</td></tr><tr><td>After Block 3 59.7</td></tr></table>
+
+Table 7: Flow using $( 2 + 1 ) \mathrm { D }$ ResNet-18.   
+
+<table><tr><td></td><td>Tiny-Kinetics</td></tr><tr><td>RGB (2+1)D ResNet-18</td><td>53.4</td></tr><tr><td>TVL-1 (2+1)D ResNet-18</td><td>36.3</td></tr><tr><td>Two-Stream (2+1)D ResNet</td><td>55.6</td></tr><tr><td>RGB-Only OFF (Sun et al., 2018)</td><td>53.7</td></tr><tr><td>Input (RGB)</td><td>39.2</td></tr><tr><td>After Block1</td><td>57.3</td></tr><tr><td>After Block 3</td><td>60.7</td></tr></table>
+
+Flow of 3D CNN Feature Since 3D convolutions capture some temporal information, we test computing our flow representation on features from a 3D CNN. As 3D CNNs are expensive to train, we follow the method of Carreira & Zisserman (2017) to inflate a ResNet-18 pretrained on ImageNet to a 3D CNN for videos. We also compare to the $( 2 + 1 ) \mathrm { D }$ method of spatial conv. followed by temporal conv from (Xie et al., 2017), which produces a similar feature combining spatial and temporal information. We find our flow layer increases performance even with 3D and $( 2 + 1 ) \mathrm { D }$ CNNs already capturing some temporal information: Tables 6 and 7. These experiments used 10 iterations and learning the flow parameters. In this, FcF was not used.
+
+We also compared to the OFF (Sun et al., 2018) using $( 2 + 1 ) \mathrm { D }$ and 3D CNNs. We observe that this method does not result in meaningful performance increases using CNNs that capture temporal information, while our approach does.
+
+Comparison to other motion representations We compare to existing CNN-based motion representations methods to confirm that our iterative method to compute a representation is important. For these experiments, when available, we used code provided by the authors and otherwise implemented the methods ourselves. To better compare to existing works, we used $( 1 6 \times )$ $2 2 4 \times 2 2 4$ images. MFNet (Lee et al., 2018) captures motion by spatially shifting CNN feature maps, then summing the results, TVNet (Fan et al., 2018) applies a convolutional optical flow method to RGB inputs, and ActionFlowNet $\mathrm { N g }$ et al., 2018) trains a CNN to jointly predict optical flow and activity classes. We also compare to OFF (Sun et al., 2018) using only RGB inputs. Note that the HMDB performance in (Sun et al., 2018) is only reported using RGB, RGB-diff, and optical flow inputs, here we compare to RGB-only inputs. Our method, which applies the iterative optical flow method on CNN feature maps, performs the best.
+
+Table 8: Comparisons to other CNN-based motion representations, using 10 iterations and learning flow parameters. This is without FcF and two-stream fusion.   
+
+<table><tr><td></td><td>Tiny-Kinetics</td><td>HMDB</td></tr><tr><td>ActionFlownet (Ng et al., 2018)</td><td>51.8</td><td>56.2</td></tr><tr><td>MFNet (Lee et al., 2018)</td><td>52.5</td><td>56.8</td></tr><tr><td>TVNet (Fan et al., 2018)</td><td>39.4</td><td>57.5</td></tr><tr><td>RGB-OFF (Sun et al., 2018)</td><td>55.6</td><td>56.9</td></tr><tr><td>Ours</td><td>61.1</td><td>65.4</td></tr></table>
+
+Table 9: Comparison to the state-of-the-art action classifications. ‘HMDB $\mathrm { ( + K i n ) }$ ’ means that the model was pre-trained on Kinetics before training/testing with HMDB. Missing results are due to those papers not reporting that setting.   
+
+<table><tr><td></td><td>Kinetics</td><td>HMDB</td><td>HMDB(+Kin)</td><td>Run-time (ms)</td></tr><tr><td>2D CNNs</td><td></td><td></td><td></td><td></td></tr><tr><td>RGB</td><td>61.3</td><td>53.4</td><td></td><td>225±15</td></tr><tr><td>Flow</td><td>48.2</td><td>57.3</td><td></td><td>8039 ±140</td></tr><tr><td>Two-stream</td><td>64.5</td><td>62.4</td><td></td><td>8546 ±147</td></tr><tr><td>TVNet (+RGB) (Fan et al., 2018)</td><td>1</td><td>71.0</td><td></td><td>785 ±21</td></tr><tr><td>OFF (RGB Only) (Sun et al., 2018)</td><td>1</td><td>57.1</td><td></td><td>365±26</td></tr><tr><td>OFF (RGB + Flow + RGB Diff) (Sun et al., 2018)</td><td>-</td><td>74.2</td><td>-</td><td>9520 ±156</td></tr><tr><td>Ours (2D CNN + Rep.Flow)</td><td>68.5</td><td>73.5</td><td>76.4</td><td>524±24</td></tr><tr><td>Ours (2D CNN + FcF)</td><td>69.4</td><td>74.4</td><td>77.3</td><td>576±22</td></tr><tr><td>(2+1)D CNNs</td><td></td><td></td><td></td><td></td></tr><tr><td>RGB R(2+1)D (Tran et al., 2018)</td><td>74.3</td><td></td><td>74.5</td><td>471±18</td></tr><tr><td>Two-Stream R(2+1)D (Tran et al., 2018)</td><td>75.4</td><td></td><td>78.7</td><td>8623 ±152</td></tr><tr><td>Ours (2+1)D CNN + Rep. Flow)</td><td>75.5</td><td></td><td>77.1</td><td>622 ±23</td></tr><tr><td>Ours ((2+1)D CNN + FcF)</td><td>76.1</td><td></td><td>78.2</td><td>654±21</td></tr><tr><td>3D CNNs</td><td></td><td></td><td></td><td></td></tr><tr><td>RGB S3D (Xie et al., 2017)</td><td>74.7</td><td></td><td>75.9</td><td>525±22</td></tr><tr><td>Two-Stream S3D (Xie et al., 2017)</td><td>77.2</td><td>-</td><td>1</td><td>8886 ±162</td></tr><tr><td>I3D (RGB) (Carreira &amp; Zisserman,2017)</td><td>71.1</td><td>49.8</td><td>74.3</td><td>594±23</td></tr><tr><td>I3D (Flow)</td><td>63.4</td><td>61.9</td><td>77.3</td><td>8845 ±148</td></tr><tr><td>I3D (Two-Stream)</td><td>74.2</td><td>66.4</td><td>80.7</td><td>9354 ±154</td></tr></table>
+
+Computation time We compare our representation flow to state-of-the-art two-stream approaches in terms of run-time and number of parameters. All timings were measured using a single Pascal Titan X GPU, for a batch of videos with size $3 2 \times 2 2 4 \times 2 2 4$ . The flow/two-stream CNNs include the time to run the TVL-1 algorithm (OpenCV GPU version) to compute the optical flow. All CNNs were based on the ResNet-34 architecture. As also shown in Table 9, our method is significantly faster than two-stream models relying on TVL-1 or other optical flow methods, while performing similarly or better. The number of parameters our model has is half of its two-stream competitors (e.g., 21M vs. 42M in the case of our 2D CNN).
+
+Comparison to state-of-the-arts We also compared our action recognition accuracies with the state-of-the-arts on Kinetics and HMDB. For this, we train our models using $3 2 \times 2 2 4 \times 2 2 4$ inputs with the full kinetics dataset, using 8 V100s. We used the 2D ResNet-50 as the architecture. Based on our experiments, we applied our representation flow layer after the 3rd residual block, learned the hyperparameters and divergence kernels, and used 20 iterations. We also compare our flow-of-flow model. Following Szegedy et al. (2016), the evaluation is performed using a running average of the parameters over time. Our results, shown in Table 9, confirm that this approach outperforms existing models using only RGB as inputs and is competitive against expensive two-stream networks. Our model performs the best among those not using optical flow inputs (i.e., among the models only taking ${ \sim } 6 0 0 \mathrm { m s }$ per video). The models requiring optical flow were more than 10 times slower.
+
+# 5 CONCLUSION
+
+We introduced a learnable representation flow layer inspired by optical flow algorithms. We experimentally compared various forms of our layer to confirm that the iterative optimization and learnable parameters are important. Our model outperformed existing methods in both speed and accuracy on standard datasets. We also introduced the concept of ‘flow of flow’ to compute longer-term motion representations and showed it benefited performance.
+
+# REFERENCES
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+Will Kay, Joao Carreira, Karen Simonyan, Brian Zhang, Chloe Hillier, Sudheendra Vijayanarasimhan, Fabio Viola, Tim Green, Trevor Back, Paul Natsev, et al. The kinetics human action video dataset. arXiv preprint arXiv:1705.06950, 2017.
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+Myunggi Lee, Seung Eui Lee, Sung Joon Son, Gyutae Park, and Nojun Kwak. Motion feature network: Fixed motion filter for action recognition. In Proceedings of European Conference on Computer Vision (ECCV), 2018.
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+
+# A TRAINING AND IMPLEMENTATION DETAILS
+
+Implementation Details When applying the representation flow layer within a CNN, we first applied a 1x1 convolutional layer to reduce the number of channels from $C$ to 32. CNN feature maps often have hundreds of channels, but computing the representation flow for hundreds of channels is computationally expensive. We found 32 channels to be a good trade-off between performance and speed. The flow layer produces output with 64 channels, $x$ and $y$ flows for the 32 input channels, which are concatenated together. We apply a 3x3 convolutional layer to this representation to produce $C$ output channels. This allows us to apply the rest of the standard CNN to the representation flow feature.
+
+Two-stream networks stack 10 optical flow frames to capture temporal information (Simonyan & Zisserman, 2014). However, we found that stacking representation flows did not perform well. Instead, we computed the flow for sequential images and averaged the predictions from a sequence of 16 frames. We found this outperformed stacking flow representations.
+
+Training Details We trained the network using stochastic gradient descent with momentum set to 0.9. For Kinetics and Tiny-Kinetics, the initial learning rate was 0.1 and decayed by a factor of 10 every 50 epochs. The model was trained for 200 epochs. The 2D CNNs were trained using a batch size of 32 on 4 Titan X GPUs. The 3D and $( 2 + 1 ) \mathrm { D }$ CNNs were trained with a batch size of 24 using 8 V100 GPUs. When fine-tuning on HMDB, the learning rate started at 0.005 and decayed by a factor of 10 every 20 epochs. The network was fine-tuned for 50 epochs. When learning the optical flow parameters, the learning rate for the parameters (i.e., $\lambda , \tau , \theta$ , divergence kernels and Sobel filters) was set of $0 . 0 1 \cdot \mathrm { l r }$ , otherwise the model produced poor predictions. This is likely due to the accumulation of gradients from the many iterations of the algorithm. For Kinetics and Tiny-Kinetics, we used dropout at 0.5 and for HMDB it was set to 0.8.
+
+Testing Details For the results reported in Table 9, we classified actions by applying our model to 25 different random croppings of each video. As found in many previous works, this helps increasing the performance slightly. In all the other experiments (i.e., Tables 1-8), random cropping was not used. Also notice that only the results in Table 9 uses our full model with $3 2 \times 2 2 4 \times 2 2 4$ input resolution. The other experiments uses spatially and/or temporally smaller models.

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+    {
+        "type": "text",
+        "text": "REPRESENTATION FLOW FOR ACTION RECOGNITION ",
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+            400,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
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+            544,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a convolutional layer inspired by optical flow algorithms to learn motion representations. Our representation flow layer is a fully-differentiable layer designed to capture the ‘flow’ of any representation channel within a convolutional neural network for action recognition. Its parameters for iterative flow optimization are learned in an end-to-end fashion together with the other model parameters, maximizing the action recognition performance. Furthermore, we newly introduce the concept of learning ‘flow of flow’ representations by stacking multiple representation flow layers. We conducted extensive experimental evaluations, confirming its advantages over previous recognition models using traditional optical flows in both computational speed and performance. ",
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+            255,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            455,
+            336,
+            470
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Activity recognition is an important problem in computer vision with many societal applications including surveillance, robot perception, smart environment/city, and more. Use of video convolutional neural networks (CNNs) have become the standard method for this task, as they can learn more optimal representations for the problem. Two-stream networks (Simonyan & Zisserman, 2014), taking both RGB frames and optical flow as input, provide state-of-the-art results and have been extremely popular. 3-D spatio-temporal CNN models, e.g., (Carreira & Zisserman, 2017), with XYT convolutions also found that such two-stream design $\\mathrm { R G B } +$ optical flow) increases their accuracy. Abstracting both appearance information and explicit motion flow benefits the recognition. ",
+        "bbox": [
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+        "type": "text",
+        "text": "However, optical flow is expensive to compute. It often requires hundreds of optimization iterations every frame, and causes learning of two separate CNN streams (i.e., RGB-stream and flow-stream). This requires significant computation cost and a great increase in the number of model parameters to learn. Further, this means that the model needs to compute optical flow every frame even during inference and run two parallel CNNs, limiting its real-time applications. ",
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+        "type": "text",
+        "text": "There were previous works to learn representations capturing motion information without using optical flow as input, such as motion feature networks (Lee et al., 2018) and ActionFlowNet $( \\mathrm { N g }$ et al., 2018). However, although they were more advantageous in terms of the number of model parameters and computation speed, they suffered from inferior performance compared to two-stream models on public datasets such as Kinetics (Kay et al., 2017) and HMDB (Kuehne et al., 2011). We hypothesize that the iterative optimization performed by optical flow methods produces an important feature that other methods fail to capture. ",
+        "bbox": [
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+        "text": "In this paper, we propose a CNN layer inspired by optical flow algorithms to learn motion representations for action recognition without having to compute optical flow. Our representation flow layer is a fully-differentiable layer designed to capture ‘flow’ of any representation channels within the model. Its parameters for iterative flow optimization are learned together with other model parameters, maximizing the action recognition performance. This is also done without having/training multiple network streams, reducing the number of parameters in the model. Further, we newly introduce the concept of learning ‘flow of flow’ representations by stacking multiple representation flow layers. We conduct extensive action classification experimental evaluation of where to compute optical flow and various hyperparameters, learning parameters, and fusion techniques. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Capturing motion and temporal information has been studied for activity recognition. Early, handcrafted approaches such as dense trajectories (Wang et al., 2011) captured motion information by tracking points through time. Many algorithms have been developed to compute optical flow as a way to capture motion in video (Fortun et al., 2015). Other works have explored learning the ordering of frames to summarize a video in a single ‘dynamic image’ used for activity recognition (Bilen et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "Convolutional neural networks (CNNs) have been applied to activity recognition. Initial approaches explored methods to combine temporal information based on pooling or temporal convolution $( \\mathrm { N g }$ et al., 2015; Karpathy et al., 2014). Other works have explored using attention to capture sub-events of activities (Piergiovanni et al., 2017). Two-stream networks have been very popular: they take input of a single RGB frame (captures appearance information) and a stack of optical flow frames (captures motion information). Often, the two network streams of the model are separately trained and the final predictions are averaged together (Simonyan & Zisserman, 2014). There were other two-stream CNN works exploring different ways to ‘fuse’ or combine the motion CNN with the appearance CNN (Feichtenhofer et al., 2016b;a). There were also large 3D XYT CNNs learning spatio-temporal patterns (Carreira & Zisserman, 2017; Xie et al., 2017), enabled by large video datasets such as Kinetics (Kay et al., 2017). However, these approaches still rely on optical flow input to maximize their accuracies. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recently, there have been works on learning motion representations. Fan et al. (2018) implemented the TVL-1 method using deep learning libraries to increase its computational speed and allow for learning some parameters. The result was fed to a two-stream CNN for the recognition. Several works explored learning a CNN to predict optical flow, which also can be used for action recognition (Dosovitskiy et al., 2015; Hui et al., 2018; Gao et al., 2018; Sun et al., 2018; $\\mathrm { N g }$ et al., 2018). Lee et al. (2018) shifted features from sequential frames to capture motion in a non-iterative fashion. Sun et al. (2018) proposed an optical flow guided feature (OFF) by computing the gradients of representations and temporal differences, but required RGB, optical flow and RGB differences to achieve state-of-the-art performance. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Unlike prior works, our proposed model with the representation flow layers relies only on the RGB input, learning much fewer parameters while correctly representing motion with the iterative optimization. It is significantly faster than the video CNNs requiring optical flow input, while still performing as good as or even better than the two-stream models. It clearly outperforms existing motion representation methods including (Fan et al., 2018) in both speed and accuracy, which we experimentally confirm. ",
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+    {
+        "type": "text",
+        "text": "3 APPROACH",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our method is a fully-differentiable convolutional layer inspired by optical flow algorithms. Unlike traditional optical flow methods, all the parameters of our method can be learned end-to-end, maximizing action recognition performance. Furthermore, our layer is designed to compute the ‘flow’ of any representation channels, instead of limiting its input to be traditional RGB frames. ",
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+    {
+        "type": "text",
+        "text": "3.1 REVIEW OF OPTICAL FLOW METHODS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Before describing our layer, we briefly review how optical flow is computed. Optical flow methods are based on the brightness consistency assumption. That is, given sequential images $I _ { 1 } , I _ { 2 }$ , a point $x , y$ in $I _ { 1 }$ is located at $x + \\Delta x , y + \\Delta y$ in $I _ { 2 }$ , or $I _ { 1 } ( x , y ) = I _ { 2 } ( x + \\Delta x , y + \\Delta y )$ . These methods assume small movements between frames, so this can be approximated with a Taylor series: $\\begin{array} { r } { I _ { 2 } = I _ { 1 } + \\frac { \\delta I } { \\delta x } \\Delta x + \\frac { \\delta I } { \\delta y } \\Delta y } \\end{array}$ , where $\\pmb { u } = [ \\Delta x , \\Delta y ]$ . These equations are solved for $\\textbf { \\em u }$ to obtain the flow, but can only be approximated due to the two unknowns. ",
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+        "type": "text",
+        "text": "The standard, variational methods for approximating optical flow (e.g., Brox (Brox et al., 2004) and TVL-1 (Zach et al., 2007) methods) take sequential images $I _ { 1 } , I _ { 2 }$ as input. Variational optical flow methods estimate the flow field, $\\textbf { \\em u }$ , using an iterative optimization method. The tensor $\\pmb { u } \\in \\dot { \\mathcal { R } } ^ { 2 \\times W \\times H }$ is the $x$ and $y$ directional flow for every location in the image. Taking two sequential images as input, $I _ { 1 } , I _ { 2 }$ , the methods first compute the gradient in both $x$ and $y$ directions: $\\nabla I _ { 2 }$ . The initial flow is set to 0, ${ \\pmb u } = 0$ . Then $\\rho$ , the residual, can be computed. For efficiency, the constant part of $\\rho$ , $\\rho _ { c }$ is pre-computed: ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/99503061eb7e6f170045331aeed9be4765b79f02c3343540363f07c43a7dff0b.jpg",
+        "text": "$$\n\\rho _ { c } = I _ { 2 } - \\nabla _ { x } I _ { 2 } \\cdot \\pmb { u } _ { x } - \\nabla _ { y } I _ { 2 } \\cdot \\pmb { u } _ { y } - I _ { 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The iterative optimization is then performed, each updating $\\textbf { \\em u }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/5e7bc644d28f4c63ecaf68e036fd1da776d56815b105ece6b495c4b62d2ef17a.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\rho = \\rho _ { c } + \\nabla _ { x } I _ { 2 } \\cdot \\boldsymbol { u } _ { x } + \\nabla _ { y } I _ { 2 } \\cdot \\boldsymbol { u } _ { y } } \\\\ & { \\boldsymbol { v } = \\left\\{ \\begin{array} { l l } { \\boldsymbol { u } + \\lambda \\theta \\nabla I _ { 2 } } & { \\rho < - \\lambda \\theta | \\nabla I _ { 2 } | ^ { 2 } } \\\\ { \\boldsymbol { u } - \\lambda \\theta \\nabla I _ { 2 } } & { \\rho > \\lambda \\theta | \\nabla I _ { 2 } | ^ { 2 } } \\end{array} \\right. } \\\\ & { \\boldsymbol { u } - \\rho \\frac { \\nabla I _ { 2 } } { | I _ { 2 } | ^ { 2 } } \\quad \\mathrm { o t h e r w i s e } } \\\\ & { \\boldsymbol { u } = \\boldsymbol { v } + \\boldsymbol { \\theta } \\cdot \\mathrm { d i v e r g e n c e } ( p ) } \\\\ & { p = \\frac { p + \\frac { \\tau } { \\theta } \\nabla u } { 1 + \\frac { \\tau } { \\theta } | \\nabla u | } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here $\\theta$ controls the weight of the TVL-1 regularization term, $\\lambda$ controls the smoothness of the output and $\\tau$ controls the time-step. These hyperparameters are manually set. $\\pmb { p }$ is the dual vector fields, which are used to minimize the energy. The divergence of $\\pmb { p }$ , or backward difference, is computed as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/3bd236f687dc630ab37a0ce08aa72b632ad7e6d64a599ecb3956117db5e6588f.jpg",
+        "text": "$$\n\\mathrm { d i v e r g e n c e } ( p ) = p _ { x , i , j } - p _ { x , i - 1 , j } + p _ { y , i , j } - p _ { y , i , j - 1 }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where ${ \\pmb p } _ { x }$ is the $x$ direction and $\\mathbf { \\Delta } _ { p _ { y } }$ is the $y$ direction, and $\\pmb { p }$ contains all the spatial locations in the image. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The goal is to minimize the total variational energy: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/d94efab31c6b9eb79dc7e2b2f18d2264a16eaeb8cf1fd503f8e431e4ab352082.jpg",
+        "text": "$$\nE = | \\nabla \\boldsymbol { u } | + \\lambda | I _ { 1 } \\ast \\boldsymbol { u } + I _ { 1 } - I _ { 2 } |\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Approaches run this iterative optimization for multiple input scales, from small to large, and use the previous flow estimate $\\textbf { \\em u }$ to warp $I _ { 2 }$ at the larger scale, providing a coarse-to-fine optical flow estimation. These standard approaches require multiple scales and warpings to obtain a good flow estimate, taking thousands of iterations. ",
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+    {
+        "type": "text",
+        "text": "3.2 REPRESENTATION FLOW LAYER ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Inspired by the optical flow algorithm, we design a fully-differentiable, learnable, convolutional representation flow layer by extending the general algorithm outlined above. The main differences are that (i) we allow the layer to capture flow of any CNN feature map, and that (ii) we learn its parameters including $\\theta , \\lambda .$ , and $\\tau$ as well as the divergence weights. We also make several key changes to reduce computation time: (1) we only use a single scale, (2) we do not perform any warping, and (3) we compute the flow on a CNN tensor with a smaller spatial size. Multiple scale and warping are computationally expensive, each requiring many iterations. By learning the flow parameters, we can eliminate the need for these additional steps. Our method is applied on lower resolution CNN feature maps, instead of the RGB input, and is trained in an end-to-end fashion. This not only benefits its speed, but also allows the model to learn a motion representation optimized for activity recognition. ",
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+    {
+        "type": "text",
+        "text": "We note that the brightness consistency assumption can similarly be applied to CNN feature maps. Instead of capturing pixel brightness, we capture feature value consistency. This same assumption holds because CNNs are designed to be spatially invariant; i.e., they produce roughly the same feature value for the same object as it moves. ",
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+    {
+        "type": "text",
+        "text": "Given the input $F _ { 1 } , F _ { 2 }$ , a single channel from sequential CNN feature maps (or input image), we compute the gradient by convolving the input feature maps with the Sobel filter: ",
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+        "type": "equation",
+        "img_path": "images/de512bfec38d44a1143f22e20c639bdaaedc15184a210d59e3ac1014f6d08faa.jpg",
+        "text": "$$\n\\nabla F _ { 2 x } = \\left[ \\begin{array} { l l l } { 1 } & { 0 } & { - 1 } \\\\ { 2 } & { 0 } & { - 2 } \\\\ { 1 } & { 0 } & { - 1 } \\end{array} \\right] * F _ { 2 } , \\nabla F _ { 2 y } = \\left[ \\begin{array} { l l l } { 1 } & { 2 } & { 1 } \\\\ { 0 } & { 0 } & { 0 } \\\\ { - 1 } & { - 2 } & { - 1 } \\end{array} \\right] * F _ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We set $\\pmb { u } = 0 , \\pmb { p } = 0$ initially, each having width and height matching the input, then we can compute $\\rho _ { c } = F _ { 2 } - F _ { 1 }$ . Next, we run the iterative optimization for a fixed number of iterations, ",
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+    {
+        "type": "text",
+        "text": "following Eqs. 2-5. To compute the divergence, we zero-pad $\\pmb { p }$ on the first column ( $\\mathbf { \\dot { x } }$ -direction) or row (y-direction) then convolve it with weights, $w _ { x } , w _ { y }$ to compute Eq. 6: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/500398b9584ac20355119211fc2ed14f1932eb92f1b56af5bba94efc0bc5854a.jpg",
+        "text": "$$\n\\mathrm { d i v e r g e n c e } ( p ) = p _ { x } * w _ { x } + p _ { y } * w _ { y }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where initially $w _ { x } = [ - 1 \\quad 1 ]$ and $w _ { y } = { \\binom { - 1 } { 1 } }$ . Note that these parameters are differentiable and can be learned with backpropagation. We compute $\\nabla \\boldsymbol { u }$ as ",
+        "bbox": [
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+        "img_path": "images/7f804add1367d69be65504b84e0bcc8ec0e8ff64fba192e0b1cfdf19518412dc.jpg",
+        "text": "$$\n\\nabla \\pmb { u } _ { x } = \\left[ \\begin{array} { c c c } { 1 } & { 0 } & { - 1 } \\\\ { 2 } & { 0 } & { - 2 } \\\\ { 1 } & { 0 } & { - 1 } \\end{array} \\right] * \\pmb { u } _ { x } , \\nabla \\pmb { u } _ { y } = \\left[ \\begin{array} { c c c } { 1 } & { 2 } & { 1 } \\\\ { 0 } & { 0 } & { 0 } \\\\ { - 1 } & { - 2 } & { - 1 } \\end{array} \\right] * \\pmb { u } _ { y }\n$$",
+        "text_format": "latex",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Algorithm 1 shows the process of our representation flow layer. Our flow layer with multiple iterations could also be interpreted as having a sequence of convolutional layers with each layer behavior dependent on its previous layer. Note that our method is fully differentiable and allows for the learning of all parameters, including $( \\tau , \\lambda , \\theta )$ and the divergence weights $( w _ { x } , w _ { y } )$ . ",
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+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/25114b56fa2ed3f2d5eae792bec3315ca0386570c43d0f43d8d1ad1994bd332f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Algorithm1 Method for the representation flow layer</td></tr><tr><td colspan=\"2\">function REPRESENTATIONFLOW(F1,F2)</td></tr><tr><td colspan=\"2\">u=0,p=0</td></tr><tr><td colspan=\"2\">Compute image/feature map gradients (Eq. 8)</td></tr><tr><td colspan=\"2\">Pc=F2-F1</td></tr><tr><td colspan=\"2\">for n iterations do</td></tr><tr><td rowspan=\"3\">ρ= Pc+VxF2 :Ux +VyF2 :Uy</td><td></td></tr><tr><td>(u+λ0VF2 ρ&lt;-X0|VF2l²</td></tr><tr><td>v ={u- λ0VF2 ρ&gt; λ0|VF2|²</td></tr><tr><td rowspan=\"2\">u-PF VF2 u = v + 0 · divergence(p)</td><td>otherwise</td></tr><tr><td></td></tr><tr><td>p+Vu p=</td><td></td></tr><tr><td colspan=\"2\">1+Vu end for</td></tr><tr><td colspan=\"2\">return u</td></tr><tr><td colspan=\"2\">end function</td></tr></table>",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Computing Flow-of-Flow Standard optical flow algorithms compute the flow for two sequential images. An optical flow image contains information about the direction and magnitude of the motion. Applying the flow algorithm directly on two flow images means that we are tracking pixels/locations showing similar motion in two consecutive frames. In practice, this typically leads to a worse performance due to inconsistent optical flow results and non-rigid motion. On the other hand, our representation flow layer is trained for the data, and is able to suppress such inconsistency and better abstract/represent motion by having multiple regular convolutional layers between the flow layers. Fig. 4 illustrates such design, which we confirm its benefits in the experiment section. By stacking multiple representation flow layers, our model is able to capture longer temporal intervals and consider locations with motion consistency. ",
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+    {
+        "type": "text",
+        "text": "Representation Flow within a CNN CNN feature maps may have hundreds or thousands of channels and our representation flow layer computes the flow for each channel, which can take significant time and memory. To address this, we apply a convolutional layer to reduce the number of channels from $C$ to $C ^ { \\prime }$ before the flow layer (note that $C ^ { \\prime }$ is still significantly more than traditional optical flow algorithms, which are only applied to a single channel, greyscale images). For numerical stability, we normalize this feature map to be in [0, 255], matching standard image values. We found that the CNN features were quite small on average $( < 0 . 5 )$ and the TVL-1 algorithm default hyperparameters are designed for standard images values in [0, 255], thus we found this normalization step important. Using the normalized feature, we compute the flow and stack the $x$ and $y$ flows, resulting in $2 C ^ { \\prime }$ channels. Finally, we apply another convolutional layer to convert from $2 C ^ { \\prime }$ channels to $C$ channels. This is passed to the remaining CNN layers for the prediction. We average predictions from many frames to classify each video, as shown in Fig. 1. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/4234d2e9ea73c6a3106678a0c4f7e1631e757b7c619ee17371cfbc905ce98c8e.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of a video-CNN with our representation flow layer. The CNN computes intermediate feature maps, and sequential feature maps are used as input to the flow layer. The outputs of the flow layer are used for prediction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 ACTIVITY RECOGNITION MODEL ",
+        "text_level": 1,
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "We place the representation flow layer inside a standard activity recognition model taking a $T \\times C \\times$ $W \\times H$ tensor as input to a CNN. Here, $C$ is 3 as our model uses direct RGB frames as an input. $T$ is the number of frames the model processes, and $W$ and $H$ are the spatial dims. The CNN outputs a prediction per-timestep and these are temporally averaged to produce a probability for each class. The model is trained to minimize cross-entropy: ",
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+    {
+        "type": "equation",
+        "img_path": "images/f1874fe9e1b5faa5fd684373273cba1ab51390b98ecfda208f8dc55035a60fc6.jpg",
+        "text": "$$\nL ( v , c ) = - \\sum _ { i } ^ { K } ( c = = i ) \\log ( C N N ( v ) _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $v$ is the video, $C N N$ is the classification CNN and $c$ represents which of the $K$ classes $v$ belongs. That is, the parameters in our flow layers are trained together with the other layers, so that it maximizes the final classification accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Implementation details We implemented our representation flow layer in PyTorch and will release our code and models upon publication. As training CNNs on videos is computationally expensive, we used a subset of the Kinetics dataset (Kay et al., 2017) with $1 0 0 \\mathrm { k }$ videos from 150 classes: Tiny-Kinetics. This allowed testing many models more quickly, while still having sufficient data to train large CNNs. For most experiments, we used ResNet-34 (He et al., 2016) with input of size $1 6 \\times 1 1 2 \\times 1 1 2$ (i.e., 16 frames with spatial size of 112). As training video CNNs is computationally expensive, we used this smaller input, which reduces performance, but allowed us to use larger batch sizes and run many experiments more quickly. Our final models are trained on standard $2 2 4 \\times 2 2 4$ images. See appendix A for specific training details. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Where to compute flow? To determine where in the network to compute the flow, we compare applying our flow layer on the RGB input, after the first conv. layer, and after the each of the 5 residual blocks. The results are shown in Table 1. We find that computing the flow on the input provides poor performance, similar to the performance of the flow-only networks, but there is a significant jump after even 1 layer, suggesting that computing the flow of a feature is beneficial, capturing both the appearance and motion information. However, after 4 layers, the performance begins to decline as the spatial information is too abstracted/compressed (due to pooling and large spatial receptive field size), and sequential features become very similar, containing less motion information. Note that our HMDB performance in this table is quite low compared to state-of-the-art ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/3f93f7c00d0f67d8578606b237dc93948d9619f7e8ec803f68ad1ee551e8b676.jpg",
+        "table_caption": [
+            "Table 1: Computing the optical flow representation after various number of CNN layers. Results are video classification accuracy on our Tiny-Kinetics and LowRes-HMDB51 datasets using 100 iterations to compute the flow representation. ",
+            "Table 2: Comparison of learning different parameters. The flow was computed after Block 3 using 100 iterations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Tiny-Kinetics</td><td>LowRes-HMDB</td></tr><tr><td>RGB CNN</td><td>55.2</td><td>35.5</td></tr><tr><td>Flow CNN</td><td>35.4</td><td>37.5</td></tr><tr><td>Two-Stream CNN</td><td>57.6</td><td>41.5</td></tr><tr><td>Flow Layer on RGB Input</td><td>37.4</td><td>40.5</td></tr><tr><td>After Block1</td><td>52.4</td><td>42.6</td></tr><tr><td>After Block 2</td><td>57.4</td><td>44.5</td></tr><tr><td>After Block 3</td><td>59.4</td><td>45.4</td></tr><tr><td>After Block 4</td><td>52.1</td><td>43.5</td></tr><tr><td>After Block 5</td><td>50.3</td><td>42.2</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/696aba1df96fd98864a39130e1ff56ce6180ddc0d562a8db03098da9c4ddd466.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Tiny-Kinetics</td><td>LowRes-HMDB</td></tr><tr><td>None (all fixed)</td><td>59.4</td><td>45.4</td></tr><tr><td>Sobel kernels</td><td>58.5</td><td>43.5</td></tr><tr><td>Divergence (wx, wy)</td><td>60.2</td><td>46.4</td></tr><tr><td>T,入,θ</td><td>59.9</td><td>46.2</td></tr><tr><td>All</td><td>59.2</td><td>46.2</td></tr><tr><td>Divergence + T, 入,0</td><td>60.7</td><td>46.8</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "methods due to being trained from scratch using few frames and low spatial resolution $( 1 1 2 \\times 1 1 2 )$ .   \nFor the following experiments, unless otherwise noted, we apply the layer after the 3rd residual block. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "What to learn? As our method is fully differentiable, we can learn any of the parameters, such as the kernels used to compute image gradients, the kernels for the divergence computation and even $\\tau , \\lambda , \\theta$ . In Table 2, we compare the effects of learning different parameters. We find that learning the Sobel kernel reduces performance, but learning the divergence and $\\tau , \\lambda , \\theta$ is beneficial. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "How many iterations for flow? To confirm that the iterations are important and determine how many we need, we experiment with various numbers of iterations. We compare the number of iterations needed for both learning (divergence+ $\\cdot \\tau , \\lambda , \\theta )$ and not learning parameters. The flow is computed after 3 residual blocks. The results are shown in Table 3. We find that learning provides better performance with fewer iterations (similar to the finding in (Fan et al., 2018)), and that iteratively computing the feature is important. We use 10 or 20 iterations in the remaining experiments as they provide good perforamnce and are fast. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Two-stream fusion? Two-stream CNNs fusing both RGB and optical flow features has been heavily studied (Simonyan & Zisserman, 2014; Feichtenhofer et al., 2016b). Based on these works, we compare various ways of fusing RGB and our flow representation, shown in Fig. 2. We compare no fusion, late fusion (i.e., separate RGB and flow CNNs) and addition/multiplication/concatenation fusion. In Table 4, we compare different fusion methods for different locations in the network. We find that fusing RGB information is very important “when computing flow directly from RGB input”. However, it is not as beneficial when computing the optical flow of representations as the CNN has already abstracted much appearance information away. We found that concatenation of the RGB and flow features perform poorly compared to the others. We do not use two-stream fusion in any other experiments, as we found that computing the representation flow after the 3rd residual block provides sufficient performance. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/08fe9254154dff918d8f491f7f908f7f4e66dbd402bffb05a9a39c4393fe7886.jpg",
+        "table_caption": [
+            "Table 3: Effect of the number of iterations on our Tiny-Kinetics dataset for learning and not learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Not learned</td><td>Learned</td></tr><tr><td>1 iteration</td><td>46.7</td><td>49.5</td></tr><tr><td>5 iterations</td><td>51.3</td><td>55.4</td></tr><tr><td>10 iterations</td><td>52.4</td><td>59.4</td></tr><tr><td>20 iterations</td><td>53.6</td><td>60.7</td></tr><tr><td>50 iterations</td><td>59.2</td><td>60.9</td></tr><tr><td>100 iterations</td><td>59.4</td><td>60.7</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/6e8c3e932994394863caa45ecee5ee3ae09be321f1fba131a0b803b452e812ce.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    },
+    {
+        "type": "image",
+        "img_path": "images/987460efdd61be120c38a3d17eb21e208da49dac0a07d7276b3ff3cb814870f8.jpg",
+        "image_caption": [
+            "Figure 2: Different approaches to fusing RGB and flow information. (a) No fusion (b) Late fusion (c) The circle represents elementwise addition/multiplication or concatenation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/be45bcfbe4033ba98f4ae4abc655b39b6cd9903834517012943f2118ee1bf8b4.jpg",
+        "image_caption": [
+            "Figure 3: (a) RGB (b) Flow (c) Flow-of-Flow. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/21a5d3739adf98f9c83a4a9ae0e407bdf9f3aed39958cfd107d75f9389dd8663.jpg",
+        "image_caption": [
+            "Figure 4: Illustration of how our model computes the FoF. ",
+            "Figure 5: Comparing the results of (b) TVL-1 and (c) our learned flow when applied to RGB images. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Flow-of-flow We can stack our layer multiple times, computing the flow-of-flow (FoF). This has the advantage of combining more temporal information into a single feature. Our results are shown in Table 5. Applying the TVL-1 algorithm twice gives quite poor performance, as optical flow features do not really satisfy the brightness consistency assumption, as they capture magnitude and direction of motion (shown in Fig. 3). Applying our representation flow layer twice performs significantly better than TVL-1 twice, but still worse than our baseline of not doing so. However, we can add a convolutional layer between the first and second flow layer, flow-conv-flow (FcF), (Fig. 4), allowing the model to better learn longer-term flow representations. We find this performs best, as this intermediate layer is able to smooth the flow and produce a better input for the representation flow layer. However, we find adding a third flow layer reduces performance as the motion representation becomes unreliable, due to the large spatial receptive field size. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/4a14e764f39afc2e97dd13634072eaaad851bbfe0f7774779a2213030eca9306.jpg",
+        "table_caption": [
+            "Table 4: Different fusion methods for flow computed at different locations in the network on our Tiny-Kinetics dataset using 10 iterations and learning flow parameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>RGB</td><td>1 Block</td><td>3 Blocks</td></tr><tr><td>None</td><td>37.4</td><td>52.4</td><td>59.4</td></tr><tr><td>Late</td><td>61.3</td><td>60.4</td><td>61.5</td></tr><tr><td>Add</td><td>59.7</td><td>57.2</td><td>56.5</td></tr><tr><td>Multiply</td><td>58.3</td><td>58.1</td><td>57.8</td></tr><tr><td>Layer + Multiply</td><td>60.1</td><td>61.7</td><td>61.7</td></tr><tr><td>Concat</td><td>42.4</td><td>48.5</td><td>47.6</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/c4fa12c038c1337afb950b35815a4e83e1486b239b47eb4fb2bc59e81d63c0dc.jpg",
+        "table_caption": [
+            "Table 5: Computing the FoF representation. TVL-1 twice provides poor performance, using two flow layers with a conv. in between provides the best performance. Experiments used 10 iterations and learning flow parameters. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Tiny-Kinetics</td></tr><tr><td>TVL-1 twice</td><td>12.2</td></tr><tr><td>Single Flow Layer</td><td>59.4</td></tr><tr><td>Flow-of-Flow</td><td>47.2</td></tr><tr><td>Flow-Conv-Flow (FcF)</td><td>62.3</td></tr><tr><td>Flow-Conv-Flow-Conv-Flow</td><td>56.5</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/be5e317a02165c9b3cd461a71addffa86b0243ec9567531a90193d3fe0ad7733.jpg",
+        "table_caption": [
+            "Table 6: Flow using 3D ResNet-18. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Tiny-Kinetics</td></tr><tr><td>RGB 3D ResNet-18 54.6</td></tr><tr><td>TVL-1 3D ResNet-18 37.6</td></tr><tr><td>Two-Stream 3DResNet 57.5</td></tr><tr><td>RGB-Only OFF (Sun et al., 2018) 54.8</td></tr><tr><td>Input (RGB) 38.5</td></tr><tr><td>After Block 1 58.4</td></tr><tr><td>After Block 3 59.7</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/b33dc0f8e8ea66407b23b9496e1b13f84a04033dd887a2043e49a1cde3504723.jpg",
+        "table_caption": [
+            "Table 7: Flow using $( 2 + 1 ) \\mathrm { D }$ ResNet-18. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Tiny-Kinetics</td></tr><tr><td>RGB (2+1)D ResNet-18</td><td>53.4</td></tr><tr><td>TVL-1 (2+1)D ResNet-18</td><td>36.3</td></tr><tr><td>Two-Stream (2+1)D ResNet</td><td>55.6</td></tr><tr><td>RGB-Only OFF (Sun et al., 2018)</td><td>53.7</td></tr><tr><td>Input (RGB)</td><td>39.2</td></tr><tr><td>After Block1</td><td>57.3</td></tr><tr><td>After Block 3</td><td>60.7</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Flow of 3D CNN Feature Since 3D convolutions capture some temporal information, we test computing our flow representation on features from a 3D CNN. As 3D CNNs are expensive to train, we follow the method of Carreira & Zisserman (2017) to inflate a ResNet-18 pretrained on ImageNet to a 3D CNN for videos. We also compare to the $( 2 + 1 ) \\mathrm { D }$ method of spatial conv. followed by temporal conv from (Xie et al., 2017), which produces a similar feature combining spatial and temporal information. We find our flow layer increases performance even with 3D and $( 2 + 1 ) \\mathrm { D }$ CNNs already capturing some temporal information: Tables 6 and 7. These experiments used 10 iterations and learning the flow parameters. In this, FcF was not used. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We also compared to the OFF (Sun et al., 2018) using $( 2 + 1 ) \\mathrm { D }$ and 3D CNNs. We observe that this method does not result in meaningful performance increases using CNNs that capture temporal information, while our approach does. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparison to other motion representations We compare to existing CNN-based motion representations methods to confirm that our iterative method to compute a representation is important. For these experiments, when available, we used code provided by the authors and otherwise implemented the methods ourselves. To better compare to existing works, we used $( 1 6 \\times )$ $2 2 4 \\times 2 2 4$ images. MFNet (Lee et al., 2018) captures motion by spatially shifting CNN feature maps, then summing the results, TVNet (Fan et al., 2018) applies a convolutional optical flow method to RGB inputs, and ActionFlowNet $\\mathrm { N g }$ et al., 2018) trains a CNN to jointly predict optical flow and activity classes. We also compare to OFF (Sun et al., 2018) using only RGB inputs. Note that the HMDB performance in (Sun et al., 2018) is only reported using RGB, RGB-diff, and optical flow inputs, here we compare to RGB-only inputs. Our method, which applies the iterative optical flow method on CNN feature maps, performs the best. ",
+        "bbox": [
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+            611,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/f9dbd2e73d0e94cd90246a1c07fadb4efcd45a4a84c32c3f13b9e570961fcaea.jpg",
+        "table_caption": [
+            "Table 8: Comparisons to other CNN-based motion representations, using 10 iterations and learning flow parameters. This is without FcF and two-stream fusion. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Tiny-Kinetics</td><td>HMDB</td></tr><tr><td>ActionFlownet (Ng et al., 2018)</td><td>51.8</td><td>56.2</td></tr><tr><td>MFNet (Lee et al., 2018)</td><td>52.5</td><td>56.8</td></tr><tr><td>TVNet (Fan et al., 2018)</td><td>39.4</td><td>57.5</td></tr><tr><td>RGB-OFF (Sun et al., 2018)</td><td>55.6</td><td>56.9</td></tr><tr><td>Ours</td><td>61.1</td><td>65.4</td></tr></table>",
+        "bbox": [
+            294,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/6e86654a612d142b3d52268907d0b22b8e89c0fbfd09274d04c8cfbd90ac44a3.jpg",
+        "table_caption": [
+            "Table 9: Comparison to the state-of-the-art action classifications. ‘HMDB $\\mathrm { ( + K i n ) }$ ’ means that the model was pre-trained on Kinetics before training/testing with HMDB. Missing results are due to those papers not reporting that setting. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Kinetics</td><td>HMDB</td><td>HMDB(+Kin)</td><td>Run-time (ms)</td></tr><tr><td>2D CNNs</td><td></td><td></td><td></td><td></td></tr><tr><td>RGB</td><td>61.3</td><td>53.4</td><td></td><td>225±15</td></tr><tr><td>Flow</td><td>48.2</td><td>57.3</td><td></td><td>8039 ±140</td></tr><tr><td>Two-stream</td><td>64.5</td><td>62.4</td><td></td><td>8546 ±147</td></tr><tr><td>TVNet (+RGB) (Fan et al., 2018)</td><td>1</td><td>71.0</td><td></td><td>785 ±21</td></tr><tr><td>OFF (RGB Only) (Sun et al., 2018)</td><td>1</td><td>57.1</td><td></td><td>365±26</td></tr><tr><td>OFF (RGB + Flow + RGB Diff) (Sun et al., 2018)</td><td>-</td><td>74.2</td><td>-</td><td>9520 ±156</td></tr><tr><td>Ours (2D CNN + Rep.Flow)</td><td>68.5</td><td>73.5</td><td>76.4</td><td>524±24</td></tr><tr><td>Ours (2D CNN + FcF)</td><td>69.4</td><td>74.4</td><td>77.3</td><td>576±22</td></tr><tr><td>(2+1)D CNNs</td><td></td><td></td><td></td><td></td></tr><tr><td>RGB R(2+1)D (Tran et al., 2018)</td><td>74.3</td><td></td><td>74.5</td><td>471±18</td></tr><tr><td>Two-Stream R(2+1)D (Tran et al., 2018)</td><td>75.4</td><td></td><td>78.7</td><td>8623 ±152</td></tr><tr><td>Ours (2+1)D CNN + Rep. Flow)</td><td>75.5</td><td></td><td>77.1</td><td>622 ±23</td></tr><tr><td>Ours ((2+1)D CNN + FcF)</td><td>76.1</td><td></td><td>78.2</td><td>654±21</td></tr><tr><td>3D CNNs</td><td></td><td></td><td></td><td></td></tr><tr><td>RGB S3D (Xie et al., 2017)</td><td>74.7</td><td></td><td>75.9</td><td>525±22</td></tr><tr><td>Two-Stream S3D (Xie et al., 2017)</td><td>77.2</td><td>-</td><td>1</td><td>8886 ±162</td></tr><tr><td>I3D (RGB) (Carreira &amp; Zisserman,2017)</td><td>71.1</td><td>49.8</td><td>74.3</td><td>594±23</td></tr><tr><td>I3D (Flow)</td><td>63.4</td><td>61.9</td><td>77.3</td><td>8845 ±148</td></tr><tr><td>I3D (Two-Stream)</td><td>74.2</td><td>66.4</td><td>80.7</td><td>9354 ±154</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Computation time We compare our representation flow to state-of-the-art two-stream approaches in terms of run-time and number of parameters. All timings were measured using a single Pascal Titan X GPU, for a batch of videos with size $3 2 \\times 2 2 4 \\times 2 2 4$ . The flow/two-stream CNNs include the time to run the TVL-1 algorithm (OpenCV GPU version) to compute the optical flow. All CNNs were based on the ResNet-34 architecture. As also shown in Table 9, our method is significantly faster than two-stream models relying on TVL-1 or other optical flow methods, while performing similarly or better. The number of parameters our model has is half of its two-stream competitors (e.g., 21M vs. 42M in the case of our 2D CNN). ",
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+    {
+        "type": "text",
+        "text": "Comparison to state-of-the-arts We also compared our action recognition accuracies with the state-of-the-arts on Kinetics and HMDB. For this, we train our models using $3 2 \\times 2 2 4 \\times 2 2 4$ inputs with the full kinetics dataset, using 8 V100s. We used the 2D ResNet-50 as the architecture. Based on our experiments, we applied our representation flow layer after the 3rd residual block, learned the hyperparameters and divergence kernels, and used 20 iterations. We also compare our flow-of-flow model. Following Szegedy et al. (2016), the evaluation is performed using a running average of the parameters over time. Our results, shown in Table 9, confirm that this approach outperforms existing models using only RGB as inputs and is competitive against expensive two-stream networks. Our model performs the best among those not using optical flow inputs (i.e., among the models only taking ${ \\sim } 6 0 0 \\mathrm { m s }$ per video). The models requiring optical flow were more than 10 times slower. ",
+        "bbox": [
+            174,
+            592,
+            825,
+            731
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            768
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We introduced a learnable representation flow layer inspired by optical flow algorithms. We experimentally compared various forms of our layer to confirm that the iterative optimization and learnable parameters are important. Our model outperformed existing methods in both speed and accuracy on standard datasets. We also introduced the concept of ‘flow of flow’ to compute longer-term motion representations and showed it benefited performance. ",
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+        "page_idx": 8
+    },
+    {
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+    {
+        "type": "text",
+        "text": "A TRAINING AND IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Implementation Details When applying the representation flow layer within a CNN, we first applied a 1x1 convolutional layer to reduce the number of channels from $C$ to 32. CNN feature maps often have hundreds of channels, but computing the representation flow for hundreds of channels is computationally expensive. We found 32 channels to be a good trade-off between performance and speed. The flow layer produces output with 64 channels, $x$ and $y$ flows for the 32 input channels, which are concatenated together. We apply a 3x3 convolutional layer to this representation to produce $C$ output channels. This allows us to apply the rest of the standard CNN to the representation flow feature. ",
+        "bbox": [
+            173,
+            133,
+            825,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Two-stream networks stack 10 optical flow frames to capture temporal information (Simonyan & Zisserman, 2014). However, we found that stacking representation flows did not perform well. Instead, we computed the flow for sequential images and averaged the predictions from a sequence of 16 frames. We found this outperformed stacking flow representations. ",
+        "bbox": [
+            174,
+            252,
+            825,
+            308
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Training Details We trained the network using stochastic gradient descent with momentum set to 0.9. For Kinetics and Tiny-Kinetics, the initial learning rate was 0.1 and decayed by a factor of 10 every 50 epochs. The model was trained for 200 epochs. The 2D CNNs were trained using a batch size of 32 on 4 Titan X GPUs. The 3D and $( 2 + 1 ) \\mathrm { D }$ CNNs were trained with a batch size of 24 using 8 V100 GPUs. When fine-tuning on HMDB, the learning rate started at 0.005 and decayed by a factor of 10 every 20 epochs. The network was fine-tuned for 50 epochs. When learning the optical flow parameters, the learning rate for the parameters (i.e., $\\lambda , \\tau , \\theta$ , divergence kernels and Sobel filters) was set of $0 . 0 1 \\cdot \\mathrm { l r }$ , otherwise the model produced poor predictions. This is likely due to the accumulation of gradients from the many iterations of the algorithm. For Kinetics and Tiny-Kinetics, we used dropout at 0.5 and for HMDB it was set to 0.8. ",
+        "bbox": [
+            174,
+            323,
+            825,
+            462
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Testing Details For the results reported in Table 9, we classified actions by applying our model to 25 different random croppings of each video. As found in many previous works, this helps increasing the performance slightly. In all the other experiments (i.e., Tables 1-8), random cropping was not used. Also notice that only the results in Table 9 uses our full model with $3 2 \\times 2 2 4 \\times 2 2 4$ input resolution. The other experiments uses spatially and/or temporally smaller models. ",
+        "bbox": [
+            174,
+            478,
+            825,
+            547
+        ],
+        "page_idx": 11
+    }
+]

parse/train/rkg-mA4FDr/rkg-mA4FDr.md

@@ -0,0 +1,261 @@
+# PRE-TRAINING TASKS FOR EMBEDDING-BASED LARGE-SCALE RETRIEVAL
+
+Wei-Cheng Chang∗, Felix X. Yu, Yin-Wen Chang, Yiming Yang, Sanjiv Kumar Carnegie Mellon University & Google {wchang2,yiming}@cs.cmu.edu, {felixyu,yinwen,sanjivk}@google.com
+
+# ABSTRACT
+
+We consider the large-scale query-document retrieval problem: given a query (e.g., a question), return the set of relevant documents (e.g., paragraphs containing the answer) from a large document corpus. This problem is often solved in two steps. The retrieval phase first reduces the solution space, returning a subset of candidate documents. The scoring phase then re-ranks the documents. Critically, the retrieval algorithm not only desires high recall but also requires to be highly efficient, returning candidates in time sublinear to the number of documents. Unlike the scoring phase witnessing significant advances recently due to the BERT-style pre-training tasks on cross-attention models, the retrieval phase remains less well studied. Most previous works rely on classic Information Retrieval (IR) methods such as BM-25 (token matching $^ +$ TF-IDF weights). These models only accept sparse handcrafted features and can not be optimized for different downstream tasks of interest. In this paper, we conduct a comprehensive study on the embedding-based retrieval models. We show that the key ingredient of learning a strong embedding-based Transformer model is the set of pre-training tasks. With adequately designed paragraph-level pre-training tasks, the Transformer models can remarkably improve over the widely-used BM-25 as well as embedding models without Transformers. The paragraph-level pre-training tasks we studied are Inverse Cloze Task (ICT), Body First Selection (BFS), Wiki Link Prediction (WLP), and the combination of all three.
+
+# 1 INTRODUCTION
+
+We consider the large-scale retrieval problem: given a query, return the most relevant documents from a large corpus, where the size of the corpus can be hundreds of thousands or more. One can view this problem as learning a scoring function $f : \mathcal { X } \times \mathcal { Y }  \mathbb { R }$ , that maps a pair of a query and a document $( \mathbf { { q } } , \mathbf { { d } } ) \in \mathcal { X } \times \mathcal { Y }$ to a score $f ( \pmb { q } , \pmb { d } )$ . The function should be designed such that the relevant $( \mathbf { \boldsymbol { q } } , \mathbf { \boldsymbol { d } } )$ pairs have high scores, whereas the irrelevant ones have low scores. Many real-world applications besides query-document retrieval can be cast into this form. For example, in recommendation systems, $\pmb q$ represents a user query and $^ d$ represents a candidate item to recommend (Krichene et al., 2019). In extreme multi-label classification, $\pmb q$ represents a web-page document and $^ d$ represents the categories or hashtags of interests (Jain et al., 2019; Chang et al., 2019). In open-domain question answering, $\pmb q$ represents a question and $^ d$ represents an evidence passage containing the answer (Chen et al., 2017; Hu et al., 2019; Lee et al., 2019).
+
+Central to the above is designing the scoring function $f$ . Recently, BERT (Devlin et al., 2019), along with its many successors such as XLNet (Yang et al., 2019b) and RoBERTa (Liu et al., 2019), has led to significant improvements to many NLP tasks such as sentence pairs classification and question-answering. In BERT, the scoring function $f$ is a pre-trained deep bidirectional Transformer model. While BERT-style cross-attention models are very successful, it cannot be directly applied to large-scale retrieval problems because computing $f ( \boldsymbol { q } , \boldsymbol { d } )$ for every possible document can be prohibitively expensive. Thus, one typically first uses a less powerful but more efficient algorithm (another scoring function $f$ ) to reduce the solution space (the “retrieval phase”), and then use the BERT-style model to re-rank the retrieved documents (the “scoring phase”).
+
+The retrieval phase is critical. Ideally speaking, the algorithm should have a high recall; otherwise, many relevant documents won’t even be considered in the scoring phase. The algorithm also needs to be highly efficient: it should return a small subset of relevant documents in time sublinear to the number of all documents. Although significant developments are advancing the scoring algorithms, the retrieval algorithms remain less studied, and this is the focus of this paper.
+
+The retrieval algorithm can be put into two categories. The first type is classic information retrieval (IR) algorithms relying on token-based matching. One example is BM-25 (Robertson et al., 2009), which remains to be the most commonly-used (Nguyen et al., 2016; Yang et al., 2017; 2019a) and hard to beat (Chapelle & Chang, 2011; Lee et al., 2019) algorithm. Here the scoring function $f$ is based on token-matching between the two high-dimensional sparse vectors with TF-IDF token weights, and retrieval can be done in sublinear time using the inverted index. Despite the wide usage, these algorithms are handcrafted and therefore cannot be optimized for a specific task.
+
+The second option is an embedding-based model that jointly embeds queries and documents in the same embedding space and use an inner product or cosine distance to measure the similarity between queries and documents. Let the query embedding model be $\phi ( \cdot )$ and the document embedding model be $\psi ( \cdot )$ . The scoring function is
+
+$$
+f ( q , d ) = \langle \phi ( q ) , \psi ( d ) \rangle .
+$$
+
+In the inference stage, retrieving relevant documents then becomes finding the nearest neighbors of a query in the embedding space. Since the embeddings of all candidate documents can be precomputed and indexed, the inference can be done efficiently with approximate nearest neighbor search algorithms in the embedding space (Shrivastava & Li, 2014; Guo et al., 2016).
+
+In this paper, we refer to the above embedding-based model as the two-tower retrieval model, because the query and document embeddings are coming from two separate “towers” of neural networks. In the literature, it is also known as the Siamese network (Das et al., 2016; Triantafillou et al., 2017) or dual-encoder model (Cer et al., 2018; Mazare et al., 2018). Compared to the sparse ´ token-based models, the two-tower models can capture deeper semantic relationships within queries and documents, and the models can be optimized specifically for the task being considered.
+
+In the heart of two-tower models is the embedding functions $\phi ( \cdot )$ and $\psi ( \cdot )$ . A modern choice is using Transformers to model the attention within queries and within documents, rather than the cross-attention between them as in the BERT model. The token-level masked-LM (MLM) pretraining task is crucial to the success of BERT-style cross-attention models. Nevertheless, what pre-training tasks are useful for improving two-tower Transformer models in large-scale retrieval, remains a crucial yet unsolved research problem. In this paper, we aim to answer this question by studying different pre-training tasks for the two-tower Transformer models. We contribute the following insight:
+
+• The two-tower Transformer models with proper pre-training can significantly outperform the widely used BM-25 algorithm; • Paragraph-level pre-training tasks such as Inverse Cloze Task (ICT), Body First Selection (BFS), and Wiki Link Prediction (WLP) hugely improve the retrieval quality, whereas the most widely used pre-training task (the token-level masked-LM) gives only marginal gains. • The two-tower models with deep transformer encoders benefit more from paragraph-level pre-training compared to its shallow bag-of-word counterpart (BoW-MLP).
+
+To the best of our knowledge, this is the first comprehensive study on pre-training tasks for efficient large-scale retrieval algorithms. The rest of the paper is organized as follows. We start by introducing the two-tower retrieval model in Section 2. The pre-training tasks are presented in 3, and the experiments and analysis are presented in Section 4. Finally, we conclude this work in Section 5.
+
+# 2 THE TWO-TOWER RETRIEVAL MODEL
+
+Given a query $\pmb q \in \mathcal X$ and a document $\pmb { d } \in \mathcal { V }$ , we consider two-tower retrieval models that consist of two encoder functions, $\phi : \mathcal { X } \to \mathbb { R } ^ { k }$ and $\psi : \mathcal { V } \to \mathbb { R } ^ { k }$ which map a sequence of tokens in $\mathcal { X }$ and $\mathcal { V }$ to their associated embeddings $\phi ( { \pmb q } )$ and $\psi ( d )$ , respectively. The scoring function $f : \mathbb { R } ^ { k } \times \mathbb { R } ^ { k } \to \mathbb { R }$
+
+![](images/2a41632632b99bb29d6cff23bb7781060c349ab0f1044fb58293d5e02bf10dd5.jpg)  
+Figure 1: Difference between two-tower models and cross-attention models. Following previous works, we consider [CLS] embedding and average pooling as the aggregator’s output for the twotower Transformer model and the two-tower MLP model, respectively.
+
+is then defined to be the inner product1 of the embeddings
+
+$$
+f ( q , d ) = \langle \phi ( q ) , \psi ( d ) \rangle .
+$$
+
+In this paper, we are interested in parameterizing the encoders $\phi , \psi$ as deep Transformer models (Vaswani et al., 2017) due to its expressive power in modeling natural language.
+
+In the rest of this section, we illustrate the advantage of two-tower models in the inference phase; discuss the pros and cons of two-tower models in comparison with BERT-like cross-attention models; present the learning procedure of estimating model parameters under maximum likelihood principle; and review the related works.
+
+Inference The difference between two-tower models and cross-attention models is shown in Figure 1. The advantage of two-tower models is the efficiency in the inference time. First, all the document embeddings can be pre-computed. Then, given an unseen query $\pmb q$ , we only need to rank the document based on its inner product with the query embedding. This is way more efficient than running inference on a cross-attention BERT-style model (often used in the scoring stage). To see this, the scoring function of BERT-style model is with the form
+
+$$
+f _ { \boldsymbol { \theta } , { \boldsymbol { w } } } ( \mathbf { \boldsymbol { q } } , \mathbf { \boldsymbol { d } } ) = \psi _ { \boldsymbol { \theta } } ( \mathbf { \boldsymbol { q } } \oplus \mathbf { \boldsymbol { d } } ) ^ { T } \mathbf { \boldsymbol { w } } ,
+$$
+
+where $\oplus$ denotes the concatenate operation of the query and the document sequence and $\mathbf { \boldsymbol { w } } \in \mathbb { R } ^ { k }$ is an additional model parameters. In BERT, for each query, one has to make the above expensive inference on all documents. For example, with the 128-dimensional embedding space, inner product between 1000 query embeddings with 1 million document embeddings only takes hundreds of milliseconds on CPUs, while computing the same scores with cross-attention models takes hours if not more even on GPUs.
+
+Furthermore, retrieving the closest documents in the embedding space can be performed in sublinear time with the well-studied maximum inner product (MIPS) algorithms with almost no loss in recall (Shrivastava & Li, 2014; Guo et al., 2016).
+
+Learning One unique advantage of the two-tower retrieval model in comparison with classic IR algorithms is the ability to train it for specific tasks. In this paper, we assume that the training data is presented as relevant “positive” query-document pairs $\mathcal { T } ~ = ~ \{ ( q _ { i } , d _ { i } ) \} _ { i = 1 } ^ { | \mathcal { T } | }$ . Let $\theta$ be the model parameters. We estimate the model parameters by maximizing the log likelihood maxθ $\sum _ { ( \pmb { q } , \pmb { d } ) \in \mathcal { T } } \log p _ { \theta } ( \pmb { d } | \pmb { q } )$ where the conditional probability is defined by the Softmax:
+
+$$
+p _ { \theta } ( d | \mathbf { \epsilon } _ { q } ) = \frac { \exp \left( f _ { \theta } ( \mathbf { \epsilon } _ { q } , d ) \right) } { \sum _ { d ^ { \prime } \in \mathcal { D } } \exp \left( f _ { \theta } ( \mathbf { \epsilon } _ { q } , d ^ { \prime } ) \right) } ,
+$$
+
+and $\mathcal { D }$ is the set of all possible documents. The Softmax involves computing the expensive denominator of Equation (3), a.k.a, the partition function, that scales linearly to the number of documents. In practice, we use the Sampled Softmax, an approximation of the full-Softmax where we replace $\mathcal { D }$ by a small subset of documents in the current batch, with a proper correcting term to ensure the unbiasedness of the partition function (Bengio & Senecal, 2008). Sampled Softmax has been widely ´ used in language modeling (Chen et al., 2016; Grave et al., 2017), recommendation systems (Yu et al., 2017; Krichene et al., 2019) and extreme classification (Blanc & Rendle, 2018; Reddi et al., 2019).
+
+Since we often have a limited amount of supervised data from the downstream task, it is important to first train the retrieval model with positive pairs $\tau$ from a set of pre-training tasks. We then fine-tune it with positive pairs $\tau$ from the downstream task. We will present the set of pre-training tasks we study in Section 3.
+
+Related Works Cer et al. (2018) study the two-tower Transformer model as a universal sentence encoder. The model is learned with multiple tasks including the unsupervised Skip-Thought task (Kiros et al., 2015), the supervised conversation input-response task (Henderson et al., 2017), and the supervised sentence classification SNLI task (Bowman et al., 2015). Humeau et al. (2019) propose the Poly-encoders architecture to balance the computation/expressiveness tradeoff between two-tower models and cross-attention models. Reimers & Gurevych (2019) fine-tune the deep twotower models on two supervised datasets, SNLI and MNLI (Williams et al., 2018), then apply it in solving other downstream tasks. Unlike all the above works that consider training the two-tower Transformer models on a limited amount of supervised corpus for the sentence classification tasks, we study different pre-training tasks and their contributions in the large-scale retrieval settings.
+
+Another closely related topic is the open-domain question answering. Previous works consider using BM25 or other lexical matching methods to retrieve the top- $\mathbf { \nabla } _ { k }$ relevant passages efficiently and then deploy the more expensive cross-attention scoring function to find the answer (Chen et al., 2017; Yang et al., 2017; 2019a). Das et al. (2019) encode query and document separately with LSTM encoders. They employ a training procedure different from ours and do not consider pre-training. Very recently, Lee et al. (2019) propose to pre-train two-tower Transformer models with the Inverse Cloze Task (ICT) to replace BM25 in the passage retrieval phase. The advantage is that the retriever can be trained jointly with the reader/scorer. Nevertheless, their pre-trained two-tower models do not outperform BM25 on the SQuAD dataset, potentially because the fine-tuning is only performed on the query-tower.
+
+Model distillation (Hinton et al., 2015) can be used to compress expensive BERT-like cross-attention models into efficient two-tower Transformer models for large-scale retrieval problems. For example, Tang et al. (2019) demonstrate initial success in distilling the BERT model into a two-tower model with BiLSTM as encoders. The pre-training tasks we study in this paper can be used as additional supervision in the distillation process, and therefore complementary to model distillation.
+
+# 3 PRE-TRAINING TASKS OF DIFFERENT SEMANTIC GRANULARITIES
+
+As mentioned in Section 2, due to the limited amount of supervised data from downstream tasks, a crucial step of learning deep retrieval models is to pre-train the model with a set of pre-training tasks (we will verify this in Section 4). Sentence-level pre-training tasks have been studied before. One example is reconstructing the surface form of surrounding sentences given the encoded sentence (Le & Mikolov, 2014; Kiros et al., 2015), and another one is discriminating the next sentence from random candidates (Jernite et al., 2017; Logeswaran & Lee, 2018).
+
+In this paper, we assume that the pre-training data is defined as positive query-document $( q , d )$ pairs. A good pre-training task should have the following two properties. 1) It should be relevant to the downstream task. For example, when solving the question-answering retrieval problem, the model should capture different granularities of semantics between the query and document. The semantics can be the local context within a paragraph, global consistency within a document, and even semantic relation between two documents. 2) It should be cost-efficient to collect the pre-training data, ideally not requiring additional human supervision.
+
+![](images/cee9ffed37bc13879c0b3443175748368c1aaee1426b9b432db46c7e29502e1e.jpg)  
+Figure 2: An illustrative example of the three pre-training tasks where each query $\pmb q$ is highlighted in different colors. All queries are paired with the same text block $^ d$ . Concretely, $( q _ { 1 } , d )$ of ICT is defined locally within a paragraph; $( q _ { 2 } , d )$ of BFS is defined globally within an article; $( q _ { 3 } , d )$ o f WLP is defined distantly across two related articles hyper-linked by the Wikipedia entity.
+
+In light of the above requirements, we present three pre-training tasks that emphasize different aspects of semantics between queries and documents: Inverse Cloze Task (ICT), Body First Selection (BFS), and Wiki Link Prediction (WLP). In specific, BFS and WLP are newly proposed in this paper. The training data for all these tasks can be freely obtained based from Wikipedia without an additional manual labeling process. Figure 2 provides illustrative examples of these tasks.
+
+Inverse Cloze Task (ICT) Given a passage $\pmb { p }$ consisting of $n$ sentences, $\pmb { p } = \{ \pmb { s } _ { 1 } , \ldots , \pmb { s } _ { n } \}$ , the query $\pmb q$ is a sentence randomly drawn from the passage, $\mathbf { \Delta } q = s _ { i } , i \sim [ 1 , n ]$ , and the document $^ d$ is the rest of sentences, $d = \{ s _ { 1 } , \ldots , s _ { i - 1 } , s _ { i + 1 } , \ldots , s _ { n } \}$ . See $( q _ { 1 } , d )$ in Figure 2 as an example. This task captures the semantic context of a sentence and was originally proposed by Lee et al. (2019).
+
+Body First Selection (BFS) We propose BFS to capture semantic relationship outside of the local paragraph. Here, the query $\pmb { q } _ { 2 }$ is a random sentence in the first section of a Wikipedia page, and the document $^ d$ is a random passage from the same page (Figure 2). Since the first section of a Wikipedia article is often the description or summary of the whole page, we expect it to contain information central to the topic.
+
+Wiki Link Prediction (WLP) We propose WLP to capture inter-page semantic relation. The query $q _ { 3 }$ is a random sentence in the first section of a Wikipedia page, and the document $^ d$ is a passage from another page where there is a hyperlink link to the page of $\cdot$ (Figure 2). Intuitively, a hyperlink link indicates relationship between the two Wikipedia pages. Again, we take a sentence from the first section because it is often the description or summary of the topic.
+
+Masked LM (MLM) In addition to the above tasks, we also consider the classic masked language model (MLM) pre-training task as a baseline: predict the randomly masked tokens in a sentence. MLM is the primary pre-training task used in BERT (Devlin et al., 2019).
+
+<table><tr><td>Pre-training tasks ：</td><td>#tokens</td><td>#pairs</td><td>avg. #query tokens</td><td>#doc tokens</td></tr><tr><td>ICT</td><td>11.2B</td><td>50.2M</td><td>30.41</td><td>193.89</td></tr><tr><td>BFS</td><td>3.3B</td><td>17.5M</td><td>28.02</td><td>160.46</td></tr><tr><td>WLP</td><td>2.7B</td><td>24.9M</td><td>29.42</td><td>82.14</td></tr></table>
+
+Table 1: Data statistics of three pre-training tasks. #query tokens represent average number of tokens per query, and #doc tokens represent average number of tokens per passage.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETTING
+
+The two-tower retrieval model Each tower of the retrieval model follows the architecture and hyper-parameters of the 12 layers BERT-base model. For both towers, the final embedding is generated by applying a linear layer on the hidden state of the [CLS] token. The embedding dimension is 512. The sequence length for the query encoder and document encoder are set to be 64 and 288, respectively. We pre-train the model on 32 TPU v3 chips for 100K steps with an Adam optimizer and batch size of 8192. This process takes about 2.5 days. We use the Adam optimizer with an initial learning rate $1 \times 1 0 ^ { - 4 }$ with the warm-up ratio 0.1, followed by a linear learning rate decay. For fine-tuning, the learning rate of Adam is set to $5 \times 1 0 ^ { - 5 }$ with 2000 training steps and batch size 512.
+
+Pre-training tasks We compare the token-level pre-training task MLM with the three paragraphlevel pre-training tasks, ICT, BFS and WLP. The data of ICT, BFS and WLP are generated from the Wikipedia corpus. The data statistics are reported in Table 1. Note that #tokens represents the number of sub-words tokenized by WordPiece (Wu et al., 2016). The pre-training tasks define the positive $( q , d )$ pair for learning the two-tower Transformer models. For ICT, the $^ d$ is a pair of article title and passage separated by [SEP] symbol as input to the doc-tower.
+
+We propose to pre-train the two-tower Transformer models jointly with all three paragraph-level pretraining tasks, hence the name $1 mathsf { C T } + \mathsf { B F S } + \mathsf { W L P }$ . Here the model is pre-trained on one combined set of $( q , d )$ pairs, where each pair is uniformly sampled from the three pre-training tasks in Table 1. See Section 4.2 and 4.3 for its outstanding performance over other baselines.
+
+Downstream tasks We consider the Retrieval Question-Answering (ReQA) benchmark, proposed by Ahmad et al. (2019).2 The two QA datasets we consider are SQuAD and Natural Questions. Note that each entry of QA datasets is a tuple $( \pmb q , \pmb { a } , \pmb { p } )$ , where $\pmb q$ is the question, $\textbf { \em a }$ is the answer span, and $\pmb { p }$ is the evidence passage containing $\textbf { \em a }$ . Following Ahmad et al. (2019), we split a passage into sentences, $\pmb { p } = \pmb { s } _ { 1 } \pmb { s } _ { 2 } \dots \pmb { s } _ { n }$ and transform the original entry $( \pmb q , \pmb { a } , \pmb { p } )$ to a new tuple $( q , s _ { i } , p )$ where $\mathbf { \boldsymbol { s } } _ { i }$ is the sentence contains the answer span $^ { a }$ .
+
+The retrieval problem is that given a question $\pmb q$ , retrieve the correct sentence and evidence passage pair $( \boldsymbol { s } , \boldsymbol { p } )$ from all candidates. For each passage $\pmb { p }$ , we create a set of candidate pairs $( \boldsymbol { s } _ { i } , \boldsymbol { p } )$ where $i = 1 \dots n$ , and the retrieval candidate set is built by combining such pairs for all passages. This problem is more challenging than retrieving the evidence passage only since the larger number of candidates to be retrieved. The data statistics of the downstream ReQA benchmark are shown in Table 2. Note that, similar to Ahmad et al. (2019), the ReQA benchmark is not entirely opendomain QA retrieval as the candidates $( \boldsymbol { s } , \boldsymbol { p } )$ only cover the training set of QA dataset instead of entire Wikipedia articles. For the open-domain retrieval experiment, see details in Section 4.4.
+
+Evaluation For each dataset, we consider different training/test split of the data $( 1 \% / 9 9 \%$ , $5 \% / 9 5 \%$ and, $8 0 \% / 2 0 \% )$ in the fine-tuning stage and the $10 \%$ of training set is held out as the validation set for hyper-parameter tuning. The split is created assuming a cold-start retrieval scenario where the queries in the test (query, document) pairs are not seen in training.
+
+Table 2: Data statistics of ReQA benchmark. candidate represents all (sentence, passage) pairs.   
+
+<table><tr><td>ReQA Dataset</td><td>#query</td><td>#candidate</td><td>#tuples</td><td>#query tokens</td><td>#doc tokens</td></tr><tr><td>SQuAD</td><td>97,888</td><td>101,951</td><td>99,024</td><td>11.55</td><td>291.35</td></tr><tr><td>Natural Questions</td><td>74,097</td><td>239,008</td><td>74,097</td><td>9.29</td><td>352.67</td></tr></table>
+
+<table><tr><td rowspan=1 colspan=1>train/test ratio</td><td rowspan=1 colspan=1>Encoder</td><td rowspan=1 colspan=3>Pre-training task</td><td rowspan=1 colspan=1>R@1  R@5  R@10 R@50 R@100</td></tr><tr><td rowspan=5 colspan=1>1%/99%</td><td rowspan=5 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=2 colspan=3>No PretrainingNo Pretraining</td><td rowspan=1 colspan=1>41.86  58.00  63.64   74.15   77.91</td></tr><tr><td rowspan=1 colspan=2>NoPret</td><td rowspan=1 colspan=2>No Pretraining</td><td rowspan=1 colspan=1>0.14  0.35   0.49    1.13    1.72</td></tr><tr><td rowspan=3 colspan=3>ICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>+WLP</td><td rowspan=1 colspan=1>22.55  41.03  49.93   69.70   77.01</td></tr><tr><td rowspan=1 colspan=1>0.02   0.06   0.08    0.31    0.540.18  0.51   0.82    2.46    3.93</td></tr><tr><td rowspan=1 colspan=1>0.18  0.51   0.82    2.46    3.9337.43  61.48  70.18   85.37   89.85</td></tr><tr><td rowspan=6 colspan=1>5%/95%</td><td rowspan=6 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=6 colspan=3>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>41.87  57.98  63.63  74.17   77.91</td></tr><tr><td rowspan=1 colspan=1>1.13  2.68   3.62   7.16    9.55</td></tr><tr><td rowspan=1 colspan=1>26.23  46.49  55.68   75.28   81.89</td></tr><tr><td rowspan=1 colspan=1>0.17   0.36   0.54    1.43    2.17</td></tr><tr><td rowspan=1 colspan=1>1.19   3.59   5.40   12.52   17.41</td></tr><tr><td rowspan=1 colspan=1>45.90 70.89  78.47  90.49   93.64</td></tr><tr><td rowspan=6 colspan=1>80%/20%</td><td rowspan=6 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=6 colspan=3>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>41.77 57.95  63.55   73.94   77.49</td></tr><tr><td rowspan=1 colspan=1>19.65 36.31  44.19   62.40   69.19</td></tr><tr><td rowspan=1 colspan=1>32.24 55.26  65.49   83.37   88.50</td></tr><tr><td rowspan=1 colspan=1>12.32 26.88  34.46  53.74   61.53</td></tr><tr><td rowspan=1 colspan=1>27.34 49.59  58.17   74.89   80.33</td></tr><tr><td rowspan=1 colspan=1>58.35 82.76  88.44  95.87   97.49</td></tr></table>
+
+Table 3: Recall $@ \mathbf { k }$ on SQuAD. Numbers are in percentage $( \% )$
+
+For the evaluation metric, we focus on recall $\ @ \mathrm { k } ^ { 3 }$ because the goal of the retrieval phase is to capture the positives in the top-k results. The retrieval performance can be understood independently of the scoring model used by measuring recall at different k. In fact, in the extreme cases when the scoring model is either oracle or random, the final precision metric is proportional to recall $@ \mathbf { k }$ .
+
+# 4.2 MAIN RESULTS
+
+Table 3 and Table 4 compare the proposed combination of pre-training tasks, $1 C T + B F S + W L P$ , to various baselines on SQuAD and Natural Questions, respectively. In both benchmarks, $1 C T + B F S + W L P$ notably outperforms all other methods. This suggests that one should use a twotower Transformer model with properly designed pre-training tasks in the retrieval stage to replace the widely used BM-25 algorithm. We present some of the detailed findings below.
+
+The BM-25 baseline In retrieval, BM-25 is a simple but tough-to-beat unsupervised baseline using token-matching with TF-IDF weights as the scoring function. BM-25 performs especially well for the SQuAD benchmark, as the data collection process and human annotations of this dataset are biased towards question-answer pairs with overlapping tokens (Rajpurkar et al., 2016; Kwiatkowski et al., 2019). For instance, in the limited fine-tuning data scenario (e.g., $1 \%$ and $5 \%$ ), BM-25 outperforms the two-tower transformer models with no pre-training (No Pretraining) or with lesseffective pre-training tasks (MLM). This result verifies that BM-25 is a robust retrieval model and therefore widely used in recent works (Chen et al., 2017; Yang et al., 2017; Lee et al., 2019)4.
+
+Encoder architecture We justify the use of Transformer as encoders by comparing it with a shallow bag-of-word MLP model (BoW-MLP). Specifically, BoW-MLP looks up uni-grams from the embedding table5, aggregates the embeddings with average pooling, and passes them through a shallow two-layer MLP network with tanh activation to generate the final 512-dimensional query/document embeddings. For fair comparison, the BoW-MLP encoder has a comparable model size to the Transformer encoder (i.e., 128M v.s. 110M parameters, slightly favorable to BoW-MLP encoder).
+
+With a properly designed pre-training task (e.g., $\mathsf { I C T } + \mathsf { B F S } + \mathsf { W L P } )$ ), the Transformer encoder considerably outperforms its shallow counterpart (BoW-MLP), suggesting that the former benefits more from the unsupervised pre-training tasks. On the other hand, without any pre-training, the performance of the Transformer encoder is worse than BoW-MLP encoder, possibly because the former is over-fitting on the limited amount of labeled fine-tuning data.
+
+Pre-training tasks When pre-training the two-tower Transformer model, we compare the pretraining tasks to two baselines: No Pretraining and MLM. No Pretraining represents random initializing the model, and MLM is the token-level masked-LM task introduced in Section 3.
+
+On both datasets, the token-level pre-training task MLM only marginally improves over the nopretraining baseline (No Pretraining). In contrast, combining the paragraph-level pre-training tasks $1 C T + B F S + W L P$ provides a huge boost on the performance. This verifies our assumption that the design of task-related pre-training tasks is crucial. The performance of adding individual pre-training tasks is presented in the next section.
+
+<table><tr><td rowspan=1 colspan=1>train/test ratio</td><td rowspan=1 colspan=1>Encoder</td><td rowspan=1 colspan=1>Pre-training task</td><td rowspan=1 colspan=1>R@1  R@5  R@10 R@50 R@100</td></tr><tr><td rowspan=6 colspan=1>1%/99%</td><td rowspan=6 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=6 colspan=1>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>4.99  11.91  15.41   24.00   27.97</td></tr><tr><td rowspan=1 colspan=1>0.28   0.80   1.08    2.02    2.66</td></tr><tr><td rowspan=1 colspan=1>9.22  24.98  33.36   53.67   61.30</td></tr><tr><td rowspan=1 colspan=1>0.07   0.19   0.28    0.56     0.85</td></tr><tr><td rowspan=1 colspan=1>0.18   0.56   0.81    1.95    2.98</td></tr><tr><td rowspan=1 colspan=1>17.31  43.62  55.00   76.59   82.84</td></tr><tr><td rowspan=5 colspan=1>5%/95%</td><td rowspan=5 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=5 colspan=1>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>5.03  11.96  15.47   24.04   28.00</td></tr><tr><td rowspan=1 colspan=1>1.36   3.77   4.98   8.56    10.77</td></tr><tr><td rowspan=1 colspan=1>11.40  30.64  40.63   62.95   70.85</td></tr><tr><td rowspan=1 colspan=1>0.37   1.07   1.40   2.73    3.82</td></tr><tr><td rowspan=1 colspan=1>1.10   3.42   4.89   10.49   14.3721.46 51.03  62.99  83.04   88.05</td></tr><tr><td rowspan=6 colspan=1>80%/20%</td><td rowspan=6 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=6 colspan=1>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>4.93  11.52  14.96  23.64   27.77</td></tr><tr><td rowspan=1 colspan=1>9.78  26.76  34.16  50.34   56.44</td></tr><tr><td rowspan=1 colspan=1>13.58 37.78  50.40  76.11   82.98</td></tr><tr><td rowspan=1 colspan=1>7.49  20.11  25.40  38.26   43.75</td></tr><tr><td rowspan=1 colspan=1>16.74 40.48  49.53  67.91   73.91</td></tr><tr><td rowspan=1 colspan=1>30.27  63.97  75.85  91.84   94.60</td></tr></table>
+
+Table 4: Recall $@ \mathbf { k }$ on Natural Questions. Numbers are in percentage $( \% )$
+
+# 4.3 ABLATION STUDY
+
+We conduct a more thorough ablation study on Natural Questions involving (1) the number of layers in Transformer; (2) different pre-training tasks; and (3) dimension of the embedding space. The result is presented in Table 5.
+
+Index 1, 2, and 3 show the individual performance of three pre-training tasks. All of these tasks are much more effective than MLM. Among them, ICT has the best performance, followed by BFS, and then WLP. This suggests that the (query, document) pairs defined by local context within passage are suitable for the ReQA task.
+
+<table><tr><td rowspan="2">Index</td><td colspan="3">Ablation Configuration</td><td colspan="4">R @ 100 on different train/test ratio</td></tr><tr><td>#layer</td><td>Pre-training task</td><td>emb-dim|</td><td>1%</td><td>5%</td><td>10%</td><td>80%</td></tr><tr><td>1</td><td>4</td><td>ICT</td><td>128</td><td>77.13</td><td>82.03</td><td>84.22</td><td>91.88</td></tr><tr><td>2</td><td>4</td><td>BFS</td><td>128</td><td>72.99</td><td>78.34</td><td>80.47</td><td>89.82</td></tr><tr><td>3</td><td>4</td><td>WLP</td><td>128</td><td>56.94</td><td>68.08</td><td>72.51</td><td>86.15</td></tr><tr><td>4</td><td>12</td><td>No Pretraining</td><td>128</td><td>0.72</td><td>3.88</td><td>6.94</td><td>38.94</td></tr><tr><td>5</td><td>12</td><td>MLM</td><td>128</td><td>2.99</td><td>12.21</td><td>22.97</td><td>71.12</td></tr><tr><td>6</td><td>12</td><td>ICT</td><td>128</td><td>79.80</td><td>85.97</td><td>88.13</td><td>93.91</td></tr><tr><td>7</td><td>12</td><td>ICT+BFS+WLP</td><td>128</td><td>81.31</td><td>87.08</td><td>89.06</td><td>94.37</td></tr><tr><td>8</td><td>12</td><td>ICT+BFS+WLP</td><td>256</td><td>81.48</td><td>87.74</td><td>89.54</td><td>94.73</td></tr><tr><td>9</td><td>12</td><td>ICT+BFS+WLP</td><td>512</td><td>82.84</td><td>88.05</td><td>90.03</td><td>94.60</td></tr></table>
+
+Table 5: Ablation study on Natural Questions based on Recall $@$ 100. Index 9 represents the proposed method in Table 4.
+
+Also note from Index 6 and 7, $1 mathsf { C T } + \mathsf { B F S } + \mathsf { W L P }$ pre-training is better than ICT with $1 . 5 \%$ absolute improvement over ICT in the low-data regime. This reflects that, when there’s no sufficient downstream training data, more globally pre-training tasks is beneficial as it encodes multi-hop reasoning priors such as different passages within the same article (BFS) or even going beyond to different articles linked by the same entities (WLP).
+
+Finally, The advantage of increasing number of layers is manifest by comparing Index 1 and Index 6, while Index 7, 8 and 9 show the benefit of increasing the dimension of the embedding space.
+
+# 4.4 EVALUATION OF OPEN-DOMAIN RETRIEVAL
+
+We consider the open-domain retrieval setting by augmenting the candidate set of the ReQA benchmark with large-scale (sentence, evidence passage) pairs extracted from general Wikipedia articles. In particular, we preprocess/sub-sample the open-domain Wikipedia retrieval set of the DrQA paper (Chen et al., 2017) into one million (sentence, evidence passage) pairs, and add this external 1M candidate pairs into the existing retrieval candidate set of the ReQA benchmark.
+
+Table 6: Open-domain retrieval results of Natural Questions dataset, where existing candidates are augmented with additional 1M retrieval candidates (i.e., 1M of $( \boldsymbol { s } , \boldsymbol { p } )$ candidate pairs) extracted from open-domain Wikipedia articles.   
+
+<table><tr><td>train/test ratio</td><td>Pre-training task</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@50</td><td>R@100</td></tr><tr><td rowspan="3">1%/99%</td><td>BM-25</td><td>3.70</td><td>9.58</td><td>12.69</td><td>20.27</td><td>23.83</td></tr><tr><td>ICT</td><td>14.18</td><td>37.36</td><td>48.08</td><td>69.23</td><td>76.01</td></tr><tr><td>ICT+BFS+WLP</td><td>13.19</td><td>37.61</td><td>48.77</td><td>70.43</td><td>77.20</td></tr><tr><td rowspan="3">5%/95%</td><td>BM-25</td><td>3.21</td><td>8.62</td><td>11.50</td><td>18.59</td><td>21.78</td></tr><tr><td>ICT</td><td>17.94</td><td>45.65</td><td>57.11</td><td>76.87</td><td>82.60</td></tr><tr><td>ICT+BFS+WLP</td><td>17.62</td><td>45.92</td><td>57.75</td><td>78.14</td><td>83.78</td></tr><tr><td rowspan="3">80%/20%</td><td>BM-25</td><td>3.12</td><td>8.45</td><td>11.18</td><td>18.05</td><td>21.30</td></tr><tr><td>ICT</td><td>24.89</td><td>57.89</td><td>69.86</td><td>87.67</td><td>91.29</td></tr><tr><td>ICT+BFS+WLP</td><td>25.41</td><td>59.36</td><td>71.12</td><td>88.25</td><td>91.71</td></tr></table>
+
+The results of open-domain retrieval on Natural Questions are presented in Table 6. Firstly, we see that the two-tower Transformer models pretrained with $1 mathsf { C T } + \mathsf { B F S } + \mathsf { W L P }$ and ICT substantially outperform the BM-25 baseline. Secondly, $1 C T + B F S + W L P$ pre-training method consistently improves the ICT pre-training method in most cases. Interestingly, the improvements are more noticeable at $\mathrm { R @ 5 0 }$ and $\mathbf { R } @ \mathbf { l } 0 0$ , possibly due to that the distant multi-hop per-training supervision induces better retrieval quality at the latter part of the rank list. Finally, we conclude that the evaluation results of the 1M open-domain retrieval are consistent with our previous empirical evaluation on the ReQA benchmark with smaller retrieval candidate sets (Section 4.2).
+
+# 5 CONCLUSION
+
+We conducted a comprehensive study on how various pre-training tasks help in the large-scale retrieval problem such as evidence retrieval for question-answering. We showed that the two-tower Transformer models with random initialization (No Pretraining) or the unsuitable token-level pretraining task (MLM) are no better than the robust IR baseline BM-25 in most cases. With properly designed paragraph-level pre-training tasks inlcuding ICT, BFS and WLP, the two-tower Transformer models can considerably improve over the widely used BM-25 algorithm.
+
+For future works, we plan to study how the pre-training tasks apply to other types of encoders architectures, generating the pre-training data from corpora other than Wikipedia, and how pretraining compares with different types of regularizations.
+
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parse/train/rkg-mA4FDr/rkg-mA4FDr_content_list.json

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+    {
+        "type": "text",
+        "text": "PRE-TRAINING TASKS FOR EMBEDDING-BASED LARGE-SCALE RETRIEVAL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Wei-Cheng Chang∗, Felix X. Yu, Yin-Wen Chang, Yiming Yang, Sanjiv Kumar Carnegie Mellon University & Google {wchang2,yiming}@cs.cmu.edu, {felixyu,yinwen,sanjivk}@google.com ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            248,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider the large-scale query-document retrieval problem: given a query (e.g., a question), return the set of relevant documents (e.g., paragraphs containing the answer) from a large document corpus. This problem is often solved in two steps. The retrieval phase first reduces the solution space, returning a subset of candidate documents. The scoring phase then re-ranks the documents. Critically, the retrieval algorithm not only desires high recall but also requires to be highly efficient, returning candidates in time sublinear to the number of documents. Unlike the scoring phase witnessing significant advances recently due to the BERT-style pre-training tasks on cross-attention models, the retrieval phase remains less well studied. Most previous works rely on classic Information Retrieval (IR) methods such as BM-25 (token matching $^ +$ TF-IDF weights). These models only accept sparse handcrafted features and can not be optimized for different downstream tasks of interest. In this paper, we conduct a comprehensive study on the embedding-based retrieval models. We show that the key ingredient of learning a strong embedding-based Transformer model is the set of pre-training tasks. With adequately designed paragraph-level pre-training tasks, the Transformer models can remarkably improve over the widely-used BM-25 as well as embedding models without Transformers. The paragraph-level pre-training tasks we studied are Inverse Cloze Task (ICT), Body First Selection (BFS), Wiki Link Prediction (WLP), and the combination of all three. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider the large-scale retrieval problem: given a query, return the most relevant documents from a large corpus, where the size of the corpus can be hundreds of thousands or more. One can view this problem as learning a scoring function $f : \\mathcal { X } \\times \\mathcal { Y }  \\mathbb { R }$ , that maps a pair of a query and a document $( \\mathbf { { q } } , \\mathbf { { d } } ) \\in \\mathcal { X } \\times \\mathcal { Y }$ to a score $f ( \\pmb { q } , \\pmb { d } )$ . The function should be designed such that the relevant $( \\mathbf { \\boldsymbol { q } } , \\mathbf { \\boldsymbol { d } } )$ pairs have high scores, whereas the irrelevant ones have low scores. Many real-world applications besides query-document retrieval can be cast into this form. For example, in recommendation systems, $\\pmb q$ represents a user query and $^ d$ represents a candidate item to recommend (Krichene et al., 2019). In extreme multi-label classification, $\\pmb q$ represents a web-page document and $^ d$ represents the categories or hashtags of interests (Jain et al., 2019; Chang et al., 2019). In open-domain question answering, $\\pmb q$ represents a question and $^ d$ represents an evidence passage containing the answer (Chen et al., 2017; Hu et al., 2019; Lee et al., 2019). ",
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+        "type": "text",
+        "text": "Central to the above is designing the scoring function $f$ . Recently, BERT (Devlin et al., 2019), along with its many successors such as XLNet (Yang et al., 2019b) and RoBERTa (Liu et al., 2019), has led to significant improvements to many NLP tasks such as sentence pairs classification and question-answering. In BERT, the scoring function $f$ is a pre-trained deep bidirectional Transformer model. While BERT-style cross-attention models are very successful, it cannot be directly applied to large-scale retrieval problems because computing $f ( \\boldsymbol { q } , \\boldsymbol { d } )$ for every possible document can be prohibitively expensive. Thus, one typically first uses a less powerful but more efficient algorithm (another scoring function $f$ ) to reduce the solution space (the “retrieval phase”), and then use the BERT-style model to re-rank the retrieved documents (the “scoring phase”). ",
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+    {
+        "type": "text",
+        "text": "The retrieval phase is critical. Ideally speaking, the algorithm should have a high recall; otherwise, many relevant documents won’t even be considered in the scoring phase. The algorithm also needs to be highly efficient: it should return a small subset of relevant documents in time sublinear to the number of all documents. Although significant developments are advancing the scoring algorithms, the retrieval algorithms remain less studied, and this is the focus of this paper. ",
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+    {
+        "type": "text",
+        "text": "The retrieval algorithm can be put into two categories. The first type is classic information retrieval (IR) algorithms relying on token-based matching. One example is BM-25 (Robertson et al., 2009), which remains to be the most commonly-used (Nguyen et al., 2016; Yang et al., 2017; 2019a) and hard to beat (Chapelle & Chang, 2011; Lee et al., 2019) algorithm. Here the scoring function $f$ is based on token-matching between the two high-dimensional sparse vectors with TF-IDF token weights, and retrieval can be done in sublinear time using the inverted index. Despite the wide usage, these algorithms are handcrafted and therefore cannot be optimized for a specific task. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The second option is an embedding-based model that jointly embeds queries and documents in the same embedding space and use an inner product or cosine distance to measure the similarity between queries and documents. Let the query embedding model be $\\phi ( \\cdot )$ and the document embedding model be $\\psi ( \\cdot )$ . The scoring function is ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39173eac3daf06d077319bd3c6608084cf74f1fe97cf9df200ebaa6215eb8691.jpg",
+        "text": "$$\nf ( q , d ) = \\langle \\phi ( q ) , \\psi ( d ) \\rangle .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the inference stage, retrieving relevant documents then becomes finding the nearest neighbors of a query in the embedding space. Since the embeddings of all candidate documents can be precomputed and indexed, the inference can be done efficiently with approximate nearest neighbor search algorithms in the embedding space (Shrivastava & Li, 2014; Guo et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this paper, we refer to the above embedding-based model as the two-tower retrieval model, because the query and document embeddings are coming from two separate “towers” of neural networks. In the literature, it is also known as the Siamese network (Das et al., 2016; Triantafillou et al., 2017) or dual-encoder model (Cer et al., 2018; Mazare et al., 2018). Compared to the sparse ´ token-based models, the two-tower models can capture deeper semantic relationships within queries and documents, and the models can be optimized specifically for the task being considered. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the heart of two-tower models is the embedding functions $\\phi ( \\cdot )$ and $\\psi ( \\cdot )$ . A modern choice is using Transformers to model the attention within queries and within documents, rather than the cross-attention between them as in the BERT model. The token-level masked-LM (MLM) pretraining task is crucial to the success of BERT-style cross-attention models. Nevertheless, what pre-training tasks are useful for improving two-tower Transformer models in large-scale retrieval, remains a crucial yet unsolved research problem. In this paper, we aim to answer this question by studying different pre-training tasks for the two-tower Transformer models. We contribute the following insight: ",
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+    {
+        "type": "text",
+        "text": "• The two-tower Transformer models with proper pre-training can significantly outperform the widely used BM-25 algorithm; • Paragraph-level pre-training tasks such as Inverse Cloze Task (ICT), Body First Selection (BFS), and Wiki Link Prediction (WLP) hugely improve the retrieval quality, whereas the most widely used pre-training task (the token-level masked-LM) gives only marginal gains. • The two-tower models with deep transformer encoders benefit more from paragraph-level pre-training compared to its shallow bag-of-word counterpart (BoW-MLP). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To the best of our knowledge, this is the first comprehensive study on pre-training tasks for efficient large-scale retrieval algorithms. The rest of the paper is organized as follows. We start by introducing the two-tower retrieval model in Section 2. The pre-training tasks are presented in 3, and the experiments and analysis are presented in Section 4. Finally, we conclude this work in Section 5. ",
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+        "type": "text",
+        "text": "2 THE TWO-TOWER RETRIEVAL MODEL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given a query $\\pmb q \\in \\mathcal X$ and a document $\\pmb { d } \\in \\mathcal { V }$ , we consider two-tower retrieval models that consist of two encoder functions, $\\phi : \\mathcal { X } \\to \\mathbb { R } ^ { k }$ and $\\psi : \\mathcal { V } \\to \\mathbb { R } ^ { k }$ which map a sequence of tokens in $\\mathcal { X }$ and $\\mathcal { V }$ to their associated embeddings $\\phi ( { \\pmb q } )$ and $\\psi ( d )$ , respectively. The scoring function $f : \\mathbb { R } ^ { k } \\times \\mathbb { R } ^ { k } \\to \\mathbb { R }$ ",
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+        "img_path": "images/2a41632632b99bb29d6cff23bb7781060c349ab0f1044fb58293d5e02bf10dd5.jpg",
+        "image_caption": [
+            "Figure 1: Difference between two-tower models and cross-attention models. Following previous works, we consider [CLS] embedding and average pooling as the aggregator’s output for the twotower Transformer model and the two-tower MLP model, respectively. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "is then defined to be the inner product1 of the embeddings ",
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+    {
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+        "text": "$$\nf ( q , d ) = \\langle \\phi ( q ) , \\psi ( d ) \\rangle .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In this paper, we are interested in parameterizing the encoders $\\phi , \\psi$ as deep Transformer models (Vaswani et al., 2017) due to its expressive power in modeling natural language. ",
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+        "text": "In the rest of this section, we illustrate the advantage of two-tower models in the inference phase; discuss the pros and cons of two-tower models in comparison with BERT-like cross-attention models; present the learning procedure of estimating model parameters under maximum likelihood principle; and review the related works. ",
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+        "type": "text",
+        "text": "Inference The difference between two-tower models and cross-attention models is shown in Figure 1. The advantage of two-tower models is the efficiency in the inference time. First, all the document embeddings can be pre-computed. Then, given an unseen query $\\pmb q$ , we only need to rank the document based on its inner product with the query embedding. This is way more efficient than running inference on a cross-attention BERT-style model (often used in the scoring stage). To see this, the scoring function of BERT-style model is with the form ",
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+        "type": "equation",
+        "img_path": "images/e23eaa2d30e56cb19f0334afa08a6d6f8b6196a59edfa67af62761b4209b1089.jpg",
+        "text": "$$\nf _ { \\boldsymbol { \\theta } , { \\boldsymbol { w } } } ( \\mathbf { \\boldsymbol { q } } , \\mathbf { \\boldsymbol { d } } ) = \\psi _ { \\boldsymbol { \\theta } } ( \\mathbf { \\boldsymbol { q } } \\oplus \\mathbf { \\boldsymbol { d } } ) ^ { T } \\mathbf { \\boldsymbol { w } } ,\n$$",
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+        "type": "text",
+        "text": "where $\\oplus$ denotes the concatenate operation of the query and the document sequence and $\\mathbf { \\boldsymbol { w } } \\in \\mathbb { R } ^ { k }$ is an additional model parameters. In BERT, for each query, one has to make the above expensive inference on all documents. For example, with the 128-dimensional embedding space, inner product between 1000 query embeddings with 1 million document embeddings only takes hundreds of milliseconds on CPUs, while computing the same scores with cross-attention models takes hours if not more even on GPUs. ",
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+        "type": "text",
+        "text": "Furthermore, retrieving the closest documents in the embedding space can be performed in sublinear time with the well-studied maximum inner product (MIPS) algorithms with almost no loss in recall (Shrivastava & Li, 2014; Guo et al., 2016). ",
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+        "text": "Learning One unique advantage of the two-tower retrieval model in comparison with classic IR algorithms is the ability to train it for specific tasks. In this paper, we assume that the training data is presented as relevant “positive” query-document pairs $\\mathcal { T } ~ = ~ \\{ ( q _ { i } , d _ { i } ) \\} _ { i = 1 } ^ { | \\mathcal { T } | }$ . Let $\\theta$ be the model parameters. We estimate the model parameters by maximizing the log likelihood maxθ $\\sum _ { ( \\pmb { q } , \\pmb { d } ) \\in \\mathcal { T } } \\log p _ { \\theta } ( \\pmb { d } | \\pmb { q } )$ where the conditional probability is defined by the Softmax: ",
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+        "type": "equation",
+        "img_path": "images/79ad11c56a91bdf244ffe3d5f95196e390fd3948be8a23799e4a2e53fb670876.jpg",
+        "text": "$$\np _ { \\theta } ( d | \\mathbf { \\epsilon } _ { q } ) = \\frac { \\exp \\left( f _ { \\theta } ( \\mathbf { \\epsilon } _ { q } , d ) \\right) } { \\sum _ { d ^ { \\prime } \\in \\mathcal { D } } \\exp \\left( f _ { \\theta } ( \\mathbf { \\epsilon } _ { q } , d ^ { \\prime } ) \\right) } ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "and $\\mathcal { D }$ is the set of all possible documents. The Softmax involves computing the expensive denominator of Equation (3), a.k.a, the partition function, that scales linearly to the number of documents. In practice, we use the Sampled Softmax, an approximation of the full-Softmax where we replace $\\mathcal { D }$ by a small subset of documents in the current batch, with a proper correcting term to ensure the unbiasedness of the partition function (Bengio & Senecal, 2008). Sampled Softmax has been widely ´ used in language modeling (Chen et al., 2016; Grave et al., 2017), recommendation systems (Yu et al., 2017; Krichene et al., 2019) and extreme classification (Blanc & Rendle, 2018; Reddi et al., 2019). ",
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+        "type": "text",
+        "text": "Since we often have a limited amount of supervised data from the downstream task, it is important to first train the retrieval model with positive pairs $\\tau$ from a set of pre-training tasks. We then fine-tune it with positive pairs $\\tau$ from the downstream task. We will present the set of pre-training tasks we study in Section 3. ",
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+        "type": "text",
+        "text": "Related Works Cer et al. (2018) study the two-tower Transformer model as a universal sentence encoder. The model is learned with multiple tasks including the unsupervised Skip-Thought task (Kiros et al., 2015), the supervised conversation input-response task (Henderson et al., 2017), and the supervised sentence classification SNLI task (Bowman et al., 2015). Humeau et al. (2019) propose the Poly-encoders architecture to balance the computation/expressiveness tradeoff between two-tower models and cross-attention models. Reimers & Gurevych (2019) fine-tune the deep twotower models on two supervised datasets, SNLI and MNLI (Williams et al., 2018), then apply it in solving other downstream tasks. Unlike all the above works that consider training the two-tower Transformer models on a limited amount of supervised corpus for the sentence classification tasks, we study different pre-training tasks and their contributions in the large-scale retrieval settings. ",
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+        "type": "text",
+        "text": "Another closely related topic is the open-domain question answering. Previous works consider using BM25 or other lexical matching methods to retrieve the top- $\\mathbf { \\nabla } _ { k }$ relevant passages efficiently and then deploy the more expensive cross-attention scoring function to find the answer (Chen et al., 2017; Yang et al., 2017; 2019a). Das et al. (2019) encode query and document separately with LSTM encoders. They employ a training procedure different from ours and do not consider pre-training. Very recently, Lee et al. (2019) propose to pre-train two-tower Transformer models with the Inverse Cloze Task (ICT) to replace BM25 in the passage retrieval phase. The advantage is that the retriever can be trained jointly with the reader/scorer. Nevertheless, their pre-trained two-tower models do not outperform BM25 on the SQuAD dataset, potentially because the fine-tuning is only performed on the query-tower. ",
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+        "type": "text",
+        "text": "Model distillation (Hinton et al., 2015) can be used to compress expensive BERT-like cross-attention models into efficient two-tower Transformer models for large-scale retrieval problems. For example, Tang et al. (2019) demonstrate initial success in distilling the BERT model into a two-tower model with BiLSTM as encoders. The pre-training tasks we study in this paper can be used as additional supervision in the distillation process, and therefore complementary to model distillation. ",
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+        "type": "text",
+        "text": "3 PRE-TRAINING TASKS OF DIFFERENT SEMANTIC GRANULARITIES ",
+        "text_level": 1,
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+        "text": "As mentioned in Section 2, due to the limited amount of supervised data from downstream tasks, a crucial step of learning deep retrieval models is to pre-train the model with a set of pre-training tasks (we will verify this in Section 4). Sentence-level pre-training tasks have been studied before. One example is reconstructing the surface form of surrounding sentences given the encoded sentence (Le & Mikolov, 2014; Kiros et al., 2015), and another one is discriminating the next sentence from random candidates (Jernite et al., 2017; Logeswaran & Lee, 2018). ",
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+        "type": "text",
+        "text": "In this paper, we assume that the pre-training data is defined as positive query-document $( q , d )$ pairs. A good pre-training task should have the following two properties. 1) It should be relevant to the downstream task. For example, when solving the question-answering retrieval problem, the model should capture different granularities of semantics between the query and document. The semantics can be the local context within a paragraph, global consistency within a document, and even semantic relation between two documents. 2) It should be cost-efficient to collect the pre-training data, ideally not requiring additional human supervision. ",
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+        "image_caption": [
+            "Figure 2: An illustrative example of the three pre-training tasks where each query $\\pmb q$ is highlighted in different colors. All queries are paired with the same text block $^ d$ . Concretely, $( q _ { 1 } , d )$ of ICT is defined locally within a paragraph; $( q _ { 2 } , d )$ of BFS is defined globally within an article; $( q _ { 3 } , d )$ o f WLP is defined distantly across two related articles hyper-linked by the Wikipedia entity. "
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+        "text": "In light of the above requirements, we present three pre-training tasks that emphasize different aspects of semantics between queries and documents: Inverse Cloze Task (ICT), Body First Selection (BFS), and Wiki Link Prediction (WLP). In specific, BFS and WLP are newly proposed in this paper. The training data for all these tasks can be freely obtained based from Wikipedia without an additional manual labeling process. Figure 2 provides illustrative examples of these tasks. ",
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+        "text": "Inverse Cloze Task (ICT) Given a passage $\\pmb { p }$ consisting of $n$ sentences, $\\pmb { p } = \\{ \\pmb { s } _ { 1 } , \\ldots , \\pmb { s } _ { n } \\}$ , the query $\\pmb q$ is a sentence randomly drawn from the passage, $\\mathbf { \\Delta } q = s _ { i } , i \\sim [ 1 , n ]$ , and the document $^ d$ is the rest of sentences, $d = \\{ s _ { 1 } , \\ldots , s _ { i - 1 } , s _ { i + 1 } , \\ldots , s _ { n } \\}$ . See $( q _ { 1 } , d )$ in Figure 2 as an example. This task captures the semantic context of a sentence and was originally proposed by Lee et al. (2019). ",
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+        "text": "Body First Selection (BFS) We propose BFS to capture semantic relationship outside of the local paragraph. Here, the query $\\pmb { q } _ { 2 }$ is a random sentence in the first section of a Wikipedia page, and the document $^ d$ is a random passage from the same page (Figure 2). Since the first section of a Wikipedia article is often the description or summary of the whole page, we expect it to contain information central to the topic. ",
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+        "type": "text",
+        "text": "Wiki Link Prediction (WLP) We propose WLP to capture inter-page semantic relation. The query $q _ { 3 }$ is a random sentence in the first section of a Wikipedia page, and the document $^ d$ is a passage from another page where there is a hyperlink link to the page of $\\cdot$ (Figure 2). Intuitively, a hyperlink link indicates relationship between the two Wikipedia pages. Again, we take a sentence from the first section because it is often the description or summary of the topic. ",
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+        "type": "text",
+        "text": "Masked LM (MLM) In addition to the above tasks, we also consider the classic masked language model (MLM) pre-training task as a baseline: predict the randomly masked tokens in a sentence. MLM is the primary pre-training task used in BERT (Devlin et al., 2019). ",
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+    {
+        "type": "table",
+        "img_path": "images/32db23ecd4a9606f421cc6a66d3c034a566065e397e167bb47b274b51e2a15ba.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Data statistics of three pre-training tasks. #query tokens represent average number of tokens per query, and #doc tokens represent average number of tokens per passage. "
+        ],
+        "table_body": "<table><tr><td>Pre-training tasks ：</td><td>#tokens</td><td>#pairs</td><td>avg. #query tokens</td><td>#doc tokens</td></tr><tr><td>ICT</td><td>11.2B</td><td>50.2M</td><td>30.41</td><td>193.89</td></tr><tr><td>BFS</td><td>3.3B</td><td>17.5M</td><td>28.02</td><td>160.46</td></tr><tr><td>WLP</td><td>2.7B</td><td>24.9M</td><td>29.42</td><td>82.14</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 EXPERIMENTAL SETTING ",
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+        "text": "The two-tower retrieval model Each tower of the retrieval model follows the architecture and hyper-parameters of the 12 layers BERT-base model. For both towers, the final embedding is generated by applying a linear layer on the hidden state of the [CLS] token. The embedding dimension is 512. The sequence length for the query encoder and document encoder are set to be 64 and 288, respectively. We pre-train the model on 32 TPU v3 chips for 100K steps with an Adam optimizer and batch size of 8192. This process takes about 2.5 days. We use the Adam optimizer with an initial learning rate $1 \\times 1 0 ^ { - 4 }$ with the warm-up ratio 0.1, followed by a linear learning rate decay. For fine-tuning, the learning rate of Adam is set to $5 \\times 1 0 ^ { - 5 }$ with 2000 training steps and batch size 512. ",
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+        "text": "Pre-training tasks We compare the token-level pre-training task MLM with the three paragraphlevel pre-training tasks, ICT, BFS and WLP. The data of ICT, BFS and WLP are generated from the Wikipedia corpus. The data statistics are reported in Table 1. Note that #tokens represents the number of sub-words tokenized by WordPiece (Wu et al., 2016). The pre-training tasks define the positive $( q , d )$ pair for learning the two-tower Transformer models. For ICT, the $^ d$ is a pair of article title and passage separated by [SEP] symbol as input to the doc-tower. ",
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+        "type": "text",
+        "text": "We propose to pre-train the two-tower Transformer models jointly with all three paragraph-level pretraining tasks, hence the name $1 mathsf { C T } + \\mathsf { B F S } + \\mathsf { W L P }$ . Here the model is pre-trained on one combined set of $( q , d )$ pairs, where each pair is uniformly sampled from the three pre-training tasks in Table 1. See Section 4.2 and 4.3 for its outstanding performance over other baselines. ",
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+        "type": "text",
+        "text": "Downstream tasks We consider the Retrieval Question-Answering (ReQA) benchmark, proposed by Ahmad et al. (2019).2 The two QA datasets we consider are SQuAD and Natural Questions. Note that each entry of QA datasets is a tuple $( \\pmb q , \\pmb { a } , \\pmb { p } )$ , where $\\pmb q$ is the question, $\\textbf { \\em a }$ is the answer span, and $\\pmb { p }$ is the evidence passage containing $\\textbf { \\em a }$ . Following Ahmad et al. (2019), we split a passage into sentences, $\\pmb { p } = \\pmb { s } _ { 1 } \\pmb { s } _ { 2 } \\dots \\pmb { s } _ { n }$ and transform the original entry $( \\pmb q , \\pmb { a } , \\pmb { p } )$ to a new tuple $( q , s _ { i } , p )$ where $\\mathbf { \\boldsymbol { s } } _ { i }$ is the sentence contains the answer span $^ { a }$ . ",
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+        "type": "text",
+        "text": "The retrieval problem is that given a question $\\pmb q$ , retrieve the correct sentence and evidence passage pair $( \\boldsymbol { s } , \\boldsymbol { p } )$ from all candidates. For each passage $\\pmb { p }$ , we create a set of candidate pairs $( \\boldsymbol { s } _ { i } , \\boldsymbol { p } )$ where $i = 1 \\dots n$ , and the retrieval candidate set is built by combining such pairs for all passages. This problem is more challenging than retrieving the evidence passage only since the larger number of candidates to be retrieved. The data statistics of the downstream ReQA benchmark are shown in Table 2. Note that, similar to Ahmad et al. (2019), the ReQA benchmark is not entirely opendomain QA retrieval as the candidates $( \\boldsymbol { s } , \\boldsymbol { p } )$ only cover the training set of QA dataset instead of entire Wikipedia articles. For the open-domain retrieval experiment, see details in Section 4.4. ",
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+        "type": "text",
+        "text": "Evaluation For each dataset, we consider different training/test split of the data $( 1 \\% / 9 9 \\%$ , $5 \\% / 9 5 \\%$ and, $8 0 \\% / 2 0 \\% )$ in the fine-tuning stage and the $10 \\%$ of training set is held out as the validation set for hyper-parameter tuning. The split is created assuming a cold-start retrieval scenario where the queries in the test (query, document) pairs are not seen in training. ",
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+    {
+        "type": "table",
+        "img_path": "images/7d1579b7e1fcbb08cb6c8f52998673a77782c677a41acb67ffe103a59190c1d8.jpg",
+        "table_caption": [
+            "Table 2: Data statistics of ReQA benchmark. candidate represents all (sentence, passage) pairs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ReQA Dataset</td><td>#query</td><td>#candidate</td><td>#tuples</td><td>#query tokens</td><td>#doc tokens</td></tr><tr><td>SQuAD</td><td>97,888</td><td>101,951</td><td>99,024</td><td>11.55</td><td>291.35</td></tr><tr><td>Natural Questions</td><td>74,097</td><td>239,008</td><td>74,097</td><td>9.29</td><td>352.67</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/096c3c29e078e084ea64c4d8c595fc4dfabbc2db8f94d4cbc678cfe3332ef67c.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Recall $@ \\mathbf { k }$ on SQuAD. Numbers are in percentage $( \\% )$ "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>train/test ratio</td><td rowspan=1 colspan=1>Encoder</td><td rowspan=1 colspan=3>Pre-training task</td><td rowspan=1 colspan=1>R@1  R@5  R@10 R@50 R@100</td></tr><tr><td rowspan=5 colspan=1>1%/99%</td><td rowspan=5 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=2 colspan=3>No PretrainingNo Pretraining</td><td rowspan=1 colspan=1>41.86  58.00  63.64   74.15   77.91</td></tr><tr><td rowspan=1 colspan=2>NoPret</td><td rowspan=1 colspan=2>No Pretraining</td><td rowspan=1 colspan=1>0.14  0.35   0.49    1.13    1.72</td></tr><tr><td rowspan=3 colspan=3>ICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>+WLP</td><td rowspan=1 colspan=1>22.55  41.03  49.93   69.70   77.01</td></tr><tr><td rowspan=1 colspan=1>0.02   0.06   0.08    0.31    0.540.18  0.51   0.82    2.46    3.93</td></tr><tr><td rowspan=1 colspan=1>0.18  0.51   0.82    2.46    3.9337.43  61.48  70.18   85.37   89.85</td></tr><tr><td rowspan=6 colspan=1>5%/95%</td><td rowspan=6 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=6 colspan=3>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>41.87  57.98  63.63  74.17   77.91</td></tr><tr><td rowspan=1 colspan=1>1.13  2.68   3.62   7.16    9.55</td></tr><tr><td rowspan=1 colspan=1>26.23  46.49  55.68   75.28   81.89</td></tr><tr><td rowspan=1 colspan=1>0.17   0.36   0.54    1.43    2.17</td></tr><tr><td rowspan=1 colspan=1>1.19   3.59   5.40   12.52   17.41</td></tr><tr><td rowspan=1 colspan=1>45.90 70.89  78.47  90.49   93.64</td></tr><tr><td rowspan=6 colspan=1>80%/20%</td><td rowspan=6 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=6 colspan=3>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>41.77 57.95  63.55   73.94   77.49</td></tr><tr><td rowspan=1 colspan=1>19.65 36.31  44.19   62.40   69.19</td></tr><tr><td rowspan=1 colspan=1>32.24 55.26  65.49   83.37   88.50</td></tr><tr><td rowspan=1 colspan=1>12.32 26.88  34.46  53.74   61.53</td></tr><tr><td rowspan=1 colspan=1>27.34 49.59  58.17   74.89   80.33</td></tr><tr><td rowspan=1 colspan=1>58.35 82.76  88.44  95.87   97.49</td></tr></table>",
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+        "type": "text",
+        "text": "For the evaluation metric, we focus on recall $\\ @ \\mathrm { k } ^ { 3 }$ because the goal of the retrieval phase is to capture the positives in the top-k results. The retrieval performance can be understood independently of the scoring model used by measuring recall at different k. In fact, in the extreme cases when the scoring model is either oracle or random, the final precision metric is proportional to recall $@ \\mathbf { k }$ . ",
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+    {
+        "type": "text",
+        "text": "4.2 MAIN RESULTS ",
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+        "text": "Table 3 and Table 4 compare the proposed combination of pre-training tasks, $1 C T + B F S + W L P$ , to various baselines on SQuAD and Natural Questions, respectively. In both benchmarks, $1 C T + B F S + W L P$ notably outperforms all other methods. This suggests that one should use a twotower Transformer model with properly designed pre-training tasks in the retrieval stage to replace the widely used BM-25 algorithm. We present some of the detailed findings below. ",
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+        "type": "text",
+        "text": "The BM-25 baseline In retrieval, BM-25 is a simple but tough-to-beat unsupervised baseline using token-matching with TF-IDF weights as the scoring function. BM-25 performs especially well for the SQuAD benchmark, as the data collection process and human annotations of this dataset are biased towards question-answer pairs with overlapping tokens (Rajpurkar et al., 2016; Kwiatkowski et al., 2019). For instance, in the limited fine-tuning data scenario (e.g., $1 \\%$ and $5 \\%$ ), BM-25 outperforms the two-tower transformer models with no pre-training (No Pretraining) or with lesseffective pre-training tasks (MLM). This result verifies that BM-25 is a robust retrieval model and therefore widely used in recent works (Chen et al., 2017; Yang et al., 2017; Lee et al., 2019)4. ",
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+        "type": "text",
+        "text": "Encoder architecture We justify the use of Transformer as encoders by comparing it with a shallow bag-of-word MLP model (BoW-MLP). Specifically, BoW-MLP looks up uni-grams from the embedding table5, aggregates the embeddings with average pooling, and passes them through a shallow two-layer MLP network with tanh activation to generate the final 512-dimensional query/document embeddings. For fair comparison, the BoW-MLP encoder has a comparable model size to the Transformer encoder (i.e., 128M v.s. 110M parameters, slightly favorable to BoW-MLP encoder). ",
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+        "type": "text",
+        "text": "With a properly designed pre-training task (e.g., $\\mathsf { I C T } + \\mathsf { B F S } + \\mathsf { W L P } )$ ), the Transformer encoder considerably outperforms its shallow counterpart (BoW-MLP), suggesting that the former benefits more from the unsupervised pre-training tasks. On the other hand, without any pre-training, the performance of the Transformer encoder is worse than BoW-MLP encoder, possibly because the former is over-fitting on the limited amount of labeled fine-tuning data. ",
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+        "type": "text",
+        "text": "Pre-training tasks When pre-training the two-tower Transformer model, we compare the pretraining tasks to two baselines: No Pretraining and MLM. No Pretraining represents random initializing the model, and MLM is the token-level masked-LM task introduced in Section 3. ",
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+        "type": "text",
+        "text": "On both datasets, the token-level pre-training task MLM only marginally improves over the nopretraining baseline (No Pretraining). In contrast, combining the paragraph-level pre-training tasks $1 C T + B F S + W L P$ provides a huge boost on the performance. This verifies our assumption that the design of task-related pre-training tasks is crucial. The performance of adding individual pre-training tasks is presented in the next section. ",
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+        "type": "table",
+        "img_path": "images/db50032838c4f2d662ed4be29b73473c456eab7c6c0999d7679ff84bfde6bc8a.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 4: Recall $@ \\mathbf { k }$ on Natural Questions. Numbers are in percentage $( \\% )$ "
+        ],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>train/test ratio</td><td rowspan=1 colspan=1>Encoder</td><td rowspan=1 colspan=1>Pre-training task</td><td rowspan=1 colspan=1>R@1  R@5  R@10 R@50 R@100</td></tr><tr><td rowspan=6 colspan=1>1%/99%</td><td rowspan=6 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=6 colspan=1>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>4.99  11.91  15.41   24.00   27.97</td></tr><tr><td rowspan=1 colspan=1>0.28   0.80   1.08    2.02    2.66</td></tr><tr><td rowspan=1 colspan=1>9.22  24.98  33.36   53.67   61.30</td></tr><tr><td rowspan=1 colspan=1>0.07   0.19   0.28    0.56     0.85</td></tr><tr><td rowspan=1 colspan=1>0.18   0.56   0.81    1.95    2.98</td></tr><tr><td rowspan=1 colspan=1>17.31  43.62  55.00   76.59   82.84</td></tr><tr><td rowspan=5 colspan=1>5%/95%</td><td rowspan=5 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=5 colspan=1>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>5.03  11.96  15.47   24.04   28.00</td></tr><tr><td rowspan=1 colspan=1>1.36   3.77   4.98   8.56    10.77</td></tr><tr><td rowspan=1 colspan=1>11.40  30.64  40.63   62.95   70.85</td></tr><tr><td rowspan=1 colspan=1>0.37   1.07   1.40   2.73    3.82</td></tr><tr><td rowspan=1 colspan=1>1.10   3.42   4.89   10.49   14.3721.46 51.03  62.99  83.04   88.05</td></tr><tr><td rowspan=6 colspan=1>80%/20%</td><td rowspan=6 colspan=1>BM-25BoW-MLPBoW-MLPTransformerTransformerTransformer</td><td rowspan=6 colspan=1>No PretrainingNo PretrainingICT+BFS+WLPNo PretrainingMLMICT+BFS+WLP</td><td rowspan=1 colspan=1>4.93  11.52  14.96  23.64   27.77</td></tr><tr><td rowspan=1 colspan=1>9.78  26.76  34.16  50.34   56.44</td></tr><tr><td rowspan=1 colspan=1>13.58 37.78  50.40  76.11   82.98</td></tr><tr><td rowspan=1 colspan=1>7.49  20.11  25.40  38.26   43.75</td></tr><tr><td rowspan=1 colspan=1>16.74 40.48  49.53  67.91   73.91</td></tr><tr><td rowspan=1 colspan=1>30.27  63.97  75.85  91.84   94.60</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.3 ABLATION STUDY ",
+        "text_level": 1,
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+        "text": "We conduct a more thorough ablation study on Natural Questions involving (1) the number of layers in Transformer; (2) different pre-training tasks; and (3) dimension of the embedding space. The result is presented in Table 5. ",
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+        "type": "text",
+        "text": "Index 1, 2, and 3 show the individual performance of three pre-training tasks. All of these tasks are much more effective than MLM. Among them, ICT has the best performance, followed by BFS, and then WLP. This suggests that the (query, document) pairs defined by local context within passage are suitable for the ReQA task. ",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 5: Ablation study on Natural Questions based on Recall $@$ 100. Index 9 represents the proposed method in Table 4. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Index</td><td colspan=\"3\">Ablation Configuration</td><td colspan=\"4\">R @ 100 on different train/test ratio</td></tr><tr><td>#layer</td><td>Pre-training task</td><td>emb-dim|</td><td>1%</td><td>5%</td><td>10%</td><td>80%</td></tr><tr><td>1</td><td>4</td><td>ICT</td><td>128</td><td>77.13</td><td>82.03</td><td>84.22</td><td>91.88</td></tr><tr><td>2</td><td>4</td><td>BFS</td><td>128</td><td>72.99</td><td>78.34</td><td>80.47</td><td>89.82</td></tr><tr><td>3</td><td>4</td><td>WLP</td><td>128</td><td>56.94</td><td>68.08</td><td>72.51</td><td>86.15</td></tr><tr><td>4</td><td>12</td><td>No Pretraining</td><td>128</td><td>0.72</td><td>3.88</td><td>6.94</td><td>38.94</td></tr><tr><td>5</td><td>12</td><td>MLM</td><td>128</td><td>2.99</td><td>12.21</td><td>22.97</td><td>71.12</td></tr><tr><td>6</td><td>12</td><td>ICT</td><td>128</td><td>79.80</td><td>85.97</td><td>88.13</td><td>93.91</td></tr><tr><td>7</td><td>12</td><td>ICT+BFS+WLP</td><td>128</td><td>81.31</td><td>87.08</td><td>89.06</td><td>94.37</td></tr><tr><td>8</td><td>12</td><td>ICT+BFS+WLP</td><td>256</td><td>81.48</td><td>87.74</td><td>89.54</td><td>94.73</td></tr><tr><td>9</td><td>12</td><td>ICT+BFS+WLP</td><td>512</td><td>82.84</td><td>88.05</td><td>90.03</td><td>94.60</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Also note from Index 6 and 7, $1 mathsf { C T } + \\mathsf { B F S } + \\mathsf { W L P }$ pre-training is better than ICT with $1 . 5 \\%$ absolute improvement over ICT in the low-data regime. This reflects that, when there’s no sufficient downstream training data, more globally pre-training tasks is beneficial as it encodes multi-hop reasoning priors such as different passages within the same article (BFS) or even going beyond to different articles linked by the same entities (WLP). ",
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+    {
+        "type": "text",
+        "text": "Finally, The advantage of increasing number of layers is manifest by comparing Index 1 and Index 6, while Index 7, 8 and 9 show the benefit of increasing the dimension of the embedding space. ",
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+    {
+        "type": "text",
+        "text": "4.4 EVALUATION OF OPEN-DOMAIN RETRIEVAL ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider the open-domain retrieval setting by augmenting the candidate set of the ReQA benchmark with large-scale (sentence, evidence passage) pairs extracted from general Wikipedia articles. In particular, we preprocess/sub-sample the open-domain Wikipedia retrieval set of the DrQA paper (Chen et al., 2017) into one million (sentence, evidence passage) pairs, and add this external 1M candidate pairs into the existing retrieval candidate set of the ReQA benchmark. ",
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+    {
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+        "img_path": "images/11f57e7c5a243b6fa58b3333d9c1a4d043185d28605adb36dce6967d0f823454.jpg",
+        "table_caption": [
+            "Table 6: Open-domain retrieval results of Natural Questions dataset, where existing candidates are augmented with additional 1M retrieval candidates (i.e., 1M of $( \\boldsymbol { s } , \\boldsymbol { p } )$ candidate pairs) extracted from open-domain Wikipedia articles. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>train/test ratio</td><td>Pre-training task</td><td>R@1</td><td>R@5</td><td>R@10</td><td>R@50</td><td>R@100</td></tr><tr><td rowspan=\"3\">1%/99%</td><td>BM-25</td><td>3.70</td><td>9.58</td><td>12.69</td><td>20.27</td><td>23.83</td></tr><tr><td>ICT</td><td>14.18</td><td>37.36</td><td>48.08</td><td>69.23</td><td>76.01</td></tr><tr><td>ICT+BFS+WLP</td><td>13.19</td><td>37.61</td><td>48.77</td><td>70.43</td><td>77.20</td></tr><tr><td rowspan=\"3\">5%/95%</td><td>BM-25</td><td>3.21</td><td>8.62</td><td>11.50</td><td>18.59</td><td>21.78</td></tr><tr><td>ICT</td><td>17.94</td><td>45.65</td><td>57.11</td><td>76.87</td><td>82.60</td></tr><tr><td>ICT+BFS+WLP</td><td>17.62</td><td>45.92</td><td>57.75</td><td>78.14</td><td>83.78</td></tr><tr><td rowspan=\"3\">80%/20%</td><td>BM-25</td><td>3.12</td><td>8.45</td><td>11.18</td><td>18.05</td><td>21.30</td></tr><tr><td>ICT</td><td>24.89</td><td>57.89</td><td>69.86</td><td>87.67</td><td>91.29</td></tr><tr><td>ICT+BFS+WLP</td><td>25.41</td><td>59.36</td><td>71.12</td><td>88.25</td><td>91.71</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The results of open-domain retrieval on Natural Questions are presented in Table 6. Firstly, we see that the two-tower Transformer models pretrained with $1 mathsf { C T } + \\mathsf { B F S } + \\mathsf { W L P }$ and ICT substantially outperform the BM-25 baseline. Secondly, $1 C T + B F S + W L P$ pre-training method consistently improves the ICT pre-training method in most cases. Interestingly, the improvements are more noticeable at $\\mathrm { R @ 5 0 }$ and $\\mathbf { R } @ \\mathbf { l } 0 0$ , possibly due to that the distant multi-hop per-training supervision induces better retrieval quality at the latter part of the rank list. Finally, we conclude that the evaluation results of the 1M open-domain retrieval are consistent with our previous empirical evaluation on the ReQA benchmark with smaller retrieval candidate sets (Section 4.2). ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We conducted a comprehensive study on how various pre-training tasks help in the large-scale retrieval problem such as evidence retrieval for question-answering. We showed that the two-tower Transformer models with random initialization (No Pretraining) or the unsuitable token-level pretraining task (MLM) are no better than the robust IR baseline BM-25 in most cases. With properly designed paragraph-level pre-training tasks inlcuding ICT, BFS and WLP, the two-tower Transformer models can considerably improve over the widely used BM-25 algorithm. ",
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+    },
+    {
+        "type": "text",
+        "text": "For future works, we plan to study how the pre-training tasks apply to other types of encoders architectures, generating the pre-training data from corpora other than Wikipedia, and how pretraining compares with different types of regularizations. ",
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+    {
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