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+# CLIPORT: What and Where Pathways for Robotic Manipulation
+
+Mohit Shridhar 1,† Lucas Manuelli 2 Dieter Fox 1,2 1University of Washington 2NVIDIA mshr@cs.washington.edu lmanuelli@nvidia.com fox@cs.washington.edu
+
+cliport.github.io
+
+Abstract: How can we imbue robots with the ability to manipulate objects precisely but also to reason about them in terms of abstract concepts? Recent works in manipulation have shown that end-to-end networks can learn dexterous skills that require precise spatial reasoning, but these methods often fail to generalize to new goals or quickly learn transferable concepts across tasks. In parallel, there has been great progress in learning generalizable semantic representations for vision and language by training on large-scale internet data, however these representations lack the spatial understanding necessary for fine-grained manipulation. To this end, we propose a framework that combines the best of both worlds: a two-stream architecture with semantic and spatial pathways for vision-based manipulation. Specifically, we present CLIPORT, a language-conditioned imitationlearning agent that combines the broad semantic understanding (what) of CLIP [1] with the spatial precision (where) of Transporter [2]. Our end-to-end framework is capable of solving a variety of language-specified tabletop tasks from packing unseen objects to folding cloths, all without any explicit representations of object poses, instance segmentations, memory, symbolic states, or syntactic structures. Experiments in simulated and real-world settings show that our approach is data efficient in few-shot settings and generalizes effectively to seen and unseen semantic concepts. We even learn one multi-task policy for 10 simulated and 9 real-world tasks that is better or comparable to single-task policies.
+
+# 1 Introduction
+
+Ask a person to “get a scoop of coffee beans” or “fold the cloth in half” and they can naturally take concepts like scoop or fold and ground them in concrete physical actions within an accuracy of a few centimeters. We humans do this intuitively, without explicit geometric or kinematic models of coffee beans or cloths. Moreover, we can generalize to a broad range of tasks and concepts from a minimal set of examples on what needs to be achieved. How can we imbue robots with this ability to efficiently ground abstract semantic concepts in precise spatial reasoning?
+
+Recently, a number of end-to-end frameworks have been proposed for vision-based manipulation [2, 3, 4, 5]. While these methods do not use any explicit representations of object poses, instance segmentations, or symbolic states, they can only replicate demonstrations with a narrow range of variability and have no notion of the semantics underlying the tasks. Switching from packing red pens to blue pens involves collecting a new training set [2], or if using goal-conditioned policies, involves the user providing a goal-image from the scene [5, 6]. In realistic human-robot interaction settings, collecting additional demonstrations or providing goal-images is often infeasible and unscalable. A natural solution to both these problems is to condition policies with natural language. Language provides an intuitive interface for specifying goals and also for implicitly transferring concepts across tasks. While language-grounding for manipulation has been explored in the past [7, 8, 9, 10], these pipelines are limited by object-centric representations that cannot handle granular or deformable objects and often do not reason about perception and action in an integrated manner. In parallel, there has been great progress in learning models for visual representations [11, 12] and aligning representations of vision and language [13, 14, 15] by training on large-scale internet data. However, these models lack a fine-grained understanding on how to manipulate objects, i.e. physical affordances.
+
+![](images/3828e046605da1a58cfd55488e88f64400dd6b7ebfa4e8931b729aba5012e158.jpg)  
+Figure 1. Language-Conditioned Manipulation Tasks: CLIPORT is a broad framework applicable to a wide range of language-conditioned manipulation tasks in tabletop settings. We conduct large-scale experiments in Ravens [2] on 10 simulated tasks (a-j) with 1000s of unique instances per task. See Appendix A for challenges pertaining to each task. CLIPORT can even learn one multi-task model for all 10 tasks that achieves better or comparable performance to single-task models. Similarly, we demonstrate our approach on a Franka Panda manipulator with one multi-task model for 9 real-world tasks (k-o; only 5 shown) trained with just 179 image-action pairs.
+
+To this end, we propose the first framework that combines the best of both worlds: end-to-end learning for fine-grained manipulation with the multi-goal and multi-task generalization capabilities of vision-language grounding systems. We introduce a two-stream architecture for manipulation with semantic and spatial pathways broadly inspired by (or vaguely analogous to) the two-stream hypothesis in cognitive psychology [16, 17, 18]. Specifically, we present CLIPORT, a languageconditioned imitation-learning agent that integrates the semantic understanding (what) of CLIP [1] with the spatial precision (where) of Transporter [2]. Transporter has been applied to a wide range of rearragement tasks from industrial packing [2] to manipulating deformable objects [6]. The key insight of the approach is formulating tabletop manipulation as a series of pick-and-place affordance predictions, where the objective is to detect actions rather than detect objects and then learn a policy. This action-centric approach to perception [19] is data efficient and effective at circumventing the need for explicit “objectness” in learnt representations. However, Transporter is a tabula rasa system that learns all visual representations from scratch and so every new goal or task requires collecting a new set of demonstrations. To address this problem, we bake in a strong semantic prior while learning policies. We condition our semantic stream with visual and language-goal features from a pre-trained CLIP model [1]. Since CLIP is pre-trained to align image and language features from millions of image-caption pairs from the internet, it provides a powerful prior for grounding semantic concepts that are common across tasks like categories, parts, shapes, colors, texts, and other visual attributes, all without a top-down pipeline that requires bounding boxes or instance segmentations [13, 14, 15, 20]. This allows us to formulate tabletop rearrangement as a series of language-conditioned affordance predictions, a predominantly vision-based inference problem, and thus benefit from the strengths of data-driven paradigms like scale and generalization.
+
+To study these benefits, we conduct large-scale experiments in the Ravens [2] framework with a simulated suction-gripper robot. We propose 10 language-conditioned tasks with 1000s of unique instances per task that require both semantic and spatial reasoning (see Figure $1 \ \mathrm { a - j }$ ). CLIPORT is not only effective at solving these tasks, but surprisingly, it can even learn a multi-task model for all 10 tasks that achieves better or comparable performance to single-task models. Further, our evaluations indicate that our multi-task model can effectively transfer attributes like “pink block” across tasks, having never seen pink blocks or the word ‘pink’ in the context of the evaluation task. We also demonstrate our approach on a Franka Panda manipulator with one multi-task model for 9 real-world tasks trained with just 179 image-action pairs (see Figure $1 \ k { - } 0$ ).
+
+In summary, our contributions are as follows:
+
+• An extended benchmark of language-grounding tasks for manipulation in Ravens [2].   
+• Two-stream architecture for using internet pre-trained vision-language models for conditioning precise manipulation policies with language goals.   
+• Empirical results on a broad range of manipulation tasks, including multi-task models, validated with real-robot experiments.
+
+The benchmark, code, and pre-trained models are available at: cliport.github.io.
+
+# 2 Related Work
+
+Vision-based Manipulation. Traditionally, perception for manipulation has centered around object detectors, segmentors, and pose estimators [21, 22, 23, 24, 25, 26]. These methods cannot handle deformable objects, granular media, or generalize to unseen objects without object-specific training data. Alternatively, dense descriptors [27, 28, 29] and keypoint representations [30, 31, 32] forgo segmentation and pose representations, but do not reason about sequential actions and struggle to represent scenes with variable numbers of objects. On the other hand, end-to-end perception-toaction models can learn precise sequential policies [2, 4, 6, 33, 34, 35], but these methods have limited understanding of semantic concepts and rely on goal-images to condition policies. In contrast, Yen-Chen et. al [36] showed that pre-training on semantic tasks like classification and segmentation helps in improving efficiency and generalization of grasping predictions.
+
+Semantic Models. With the advent of large-scale models [37, 38, 39], a number of methods for learning joint vision and language representations have been proposed [13, 14, 15, 20, 40]. However, these methods are restricted to bounding boxes or instance segmentations, which make them inapplicable for detecting things like piles of coffee beans or squares on a chessboard. Alternatively, works in contrastive learning forgo top-down object-detection and learn continuous representations by pre-training on unlabeled data [11, 12]. Recently, CLIP [1] applied a similar approach to align vision and language representations by training on millions of image-caption pairs from the internet.
+
+Language Grounding for Robotics. Several works have proposed systems for instructing robots with natural language [7, 8, 9, 10, 41, 42, 43, 44, 45, 46, 47]. However, these methods use disentangled pipelines for perception and action with the language primarily being used to guide the perception. As such, these pipelines lack the spatial precision necessary for tasks like folding cloths. Recently, Lynch et. al [48] proposed an end-to-end system for grounding language in continuous control, but it requires several hours of human teleoperation data for a single simulated desk setting.
+
+Two-Stream Architectures are prevalent in action-recognition networks [49, 50, 51] and audiorecognition systems [52, 53]. In robotics, Zeng et. al [54] and Jang et. al [55] have proposed twostream pipelines for affordance predictions of novel objects. The former requires goal-images and the latter is restricted to one-step grasps with single-category goals. In contrast, our framework provides a rich and intuitive interface with composable language commands for sequential tasks.
+
+# 3 CLIPORT
+
+CLIPORT is an imitation-learning agent based on four key principles: (1) Manipulation through a two-step primitive where each action involves a start and final end-effector pose. (2) Visual representations of actions that are equivariant to translations and rotations [56, 57]. (3) Two separate pathways for semantic and spatial information. (4) Language-conditioned policies for specifying goals and also transferring concepts across tasks. Combining (1) and (2) from Transporter with (3) and (4) allows us to achieve generalizable policies that go beyond just imitating demonstrations.
+
+Section 3.1 describes the problem formulation, gives an overview of Transporter [2], and presents our language-conditioned model. Section 3.2 provides details on the training approach.
+
+# 3.1 Language-Conditioned Manipulation
+
+We consider the problem of learning a goal-conditioned policy $\pi$ that outputs actions $\mathbf { a } _ { t }$ given input $\gamma _ { t } = ( \mathbf { o } _ { t } , \mathbf { l } _ { t } )$ consisting of a visual observation $\mathbf { o } _ { t }$ and an English language instruction ${ \bf l } _ { t }$ :
+
+$$
+\pi ( \gamma _ { t } ) = \pi ( \mathbf { o } _ { t } , \mathbf { l } _ { t } )  \mathbf { a } _ { t } = ( { \mathcal { T } } _ { \mathrm { p i c k } } , { \mathcal { T } } _ { \mathrm { p l a c e } } ) \in \mathcal { A }
+$$
+
+![](images/7096ec9751b6431d9d588d6f507c70933b7da146cf1131293568d69619c96f9b.jpg)  
+Figure 2. CLIPORT Two-Stream Architecture. An overview of the semantic and spatial streams. The semantic stream uses a frozen CLIP ResNet50 [1] to encode RGB input, and its decoder layers are conditioned with tiled language features from the CLIP sentence encoder. The spatial stream encodes RGB-D input, and its decoder layers are laterally fused with the semantic stream. The final output is a map of dense pixelwise features that is used for pick or place affordance predictions. This same two-stream architecture is used in all 3 FullyConvolutional-Networks $f _ { \mathrm { p i c k } }$ , $\Phi _ { \mathrm { q u e r y } }$ , and $\Phi _ { \mathrm { k e y } }$ with $f _ { \mathrm { p i c k } }$ is used to predict pick actions, and $\Phi _ { \mathrm { q u e r y } }$ and $\Phi _ { \mathrm { k e y } }$ are used to predict place actions. See Appendix C for the exact architecture.
+
+The actions $\mathbf { a } = ( T _ { \mathrm { p i c k } } , T _ { \mathrm { p l a c e } } )$ specify the end-effector pose for picking and placing, respectively. We consider tabletop tasks where $\mathcal { T } _ { \mathrm { p i c k } } , \mathcal { T } _ { \mathrm { p l a c e } } \in \mathbf { S } \mathbf { E } ( 2 )$ . The visual observation $\mathbf { o } _ { t }$ is a top-down orthographic RGB-D reconstruction of the scene where each pixel corresponds to a point in 3D space. The language instruction ${ \bf l } _ { t }$ either specifies step-by-step instructions e.g. “pack the scissors” “pack the purple tape” etc., or a single goal description for the whole task e.g “pack all the blue and yellow boxes in the brown box”. See Figure 4 for specific examples.
+
+We assume access to a dataset $\mathcal { D } = \{ \zeta _ { 1 } , \zeta _ { 2 } , \ldots , \zeta _ { n } \}$ of $n$ expert demonstrations with associated discrete-time input-action pairs $\zeta _ { i } = \{ ( \mathbf { o } _ { 1 } , \mathbf { l } _ { 1 } , \mathbf { a } _ { 1 } ) , ( \mathbf { o } _ { 2 } , \mathbf { l } _ { 2 } , \mathbf { a } _ { 2 } ) , . . . \}$ where $\mathbf { a } _ { t } = ( T _ { \mathrm { p i c k } } , T _ { \mathrm { p l a c e } } )$ corresponds to expert pick-and-place coordinates at timestep $t$ . These expert demonstrations are used to supervise the policy $\pi$ .
+
+Transporter for Pick-and-Place. The policy $\pi$ is trained with Transporter [2] to perform spatial manipulation. The model first (i) attends to a local region to decide where to pick, then (ii) computes a placement location by finding the best match through cross-correlation of deep visual features.
+
+Following Transporter [2, 6], the policy $\pi$ is composed of two action-value modules (Q-functions): The pick module $\mathcal { Q } _ { \mathrm { p i c k } }$ decides where to pick, and conditioned on this pick action the place module $\mathcal { Q } _ { \mathrm { p l a c e } }$ decides where to place. These modules are implemented as Fully-Convolutional-Networks (FCNs) that are translationally equivariant by design. As we will describe in more detail below, we extend these networks to two-stream architectures that can handle language input. The pick FCN $f _ { \mathrm { p i c k } }$ takes input $\gamma _ { t } = \left( \mathbf { o } _ { t } , \mathbf { l } _ { t } \right)$ and outputs a dense pixelwise prediction $\mathbf { \bar { \mathcal { Q } } _ { \mathrm { p i c k } } } ^ { } \in \mathbb { R } ^ { H \times \dot { W } }$ of action-values, where are used to predict the pick action $\tau _ { \mathrm { p i c k } }$ :
+
+$$
+\mathcal { T } _ { \mathrm { p i c k } } = \underset { ( u , v ) } { \mathrm { a r g m a x } } \ : \mathcal { Q } _ { \mathrm { p i c k } } ( ( u , v ) | \gamma _ { t } )
+$$
+
+Since $\mathbf { o } _ { t }$ is an orthographic heightmap, each pixel location $( u , v )$ can be mapped to a 3D picking location using the known camera calibration. $f _ { \mathrm { p i c k } }$ is trained in a supervised manner to predict the pick action $\mathcal { T } _ { \mathrm { p i c k } }$ that imitates the expert demonstration with the specified language instruction at timestep $t$ .
+
+The second FCN $\Phi _ { \mathrm { q u e r y } }$ takes in $\gamma _ { t } [ \mathcal { T } _ { \mathrm { p i c k } } ]$ , which is a $c \times c$ crop of $\mathbf { o } _ { t }$ centered at $\tau _ { \mathrm { p i c k } }$ along with the language instruction ${ \bf l } _ { t }$ , and outputs a query feature embedding of shape $\mathbb { R } ^ { c \times c \times d }$ . The third FCN $\Phi _ { \mathrm { k e y } }$ consumes the full input $\gamma _ { t }$ and outputs a key feature embedding of shape $\mathbb { R } ^ { H \times W \times d }$ . The place action-values $\mathcal { Q } _ { \mathrm { p l a c e } }$ are then computed by cross-correlating the query and key features:
+
+$$
+\begin{array} { r } { \mathcal { Q } _ { \mathrm { p l a c e } } ( \Delta \tau | \gamma _ { t } , \mathcal { T } _ { \mathrm { p i c k } } ) = \left( \Phi _ { \mathrm { q u e r y } } ( \gamma _ { t } [ \mathcal { T } _ { \mathrm { p i c k } } ] ) * \Phi _ { \mathrm { k e y } } ( \gamma _ { t } ) \right) [ \Delta \tau ] } \end{array}
+$$
+
+where $ { \Delta \tau } \in S E ( 2 )$ represents a potential placement pose. Since $\mathbf { o } _ { t }$ is an orthographic heightmap, rotations in the placement pose $\Delta \tau$ can be captured by stacking $k$ discrete angle rotations of the crop before passing it through the query network $\Phi _ { \mathrm { q u e r y } }$ . Then $\begin{array} { r } { \mathcal { T } _ { \mathrm { p l a c e } } ^ { \mathrm { ~ \tiny ~ \bar { ~ } ~ } } = \arg \operatorname* { m a x } _ { \Delta \tau } \mathcal { Q } _ { \mathrm { p l a c e } } ( \Delta \tau | \gamma _ { t } , \mathcal { T } _ { \mathrm { p i c k } } ) , } \end{array}$ , where the place module is trained to imitate the placements in the expert demonstrations. For all models, we use $c = 6 4$ , $k = 3 6$ and $d = 3$ . As in Transporter [2, 6], our framework can be extended to handle any motion primitive like pushing, sliding, etc. that can be parameterized by two end-effector poses at each timestep. For more details, we refer the reader to the original paper [2].
+
+Two-Stream Architecture. In CLIPORT, we extend the network architecture of all three FCNs $f _ { \mathrm { p i c k } }$ , $\Phi _ { \mathrm { q u e r y } }$ and $\Phi _ { \mathrm { k e y } }$ from Transporter [2] to allow for language input and reasoning about high-level semantic concepts. We extend the FCNs to two-pathways: semantic (ventral) and spatial (dorsal). The semantic stream is conditioned with language features at the bottleneck and fused with intermediate features from the spatial stream. See Figure 2 for an overview of the architecture.
+
+The spatial stream is identical to the ResNet architecture in Transporter – a tabula rasa network that takes in RGB-D input $\mathbf { o } _ { t }$ and outputs dense features through an hourglass encoder-decoder model. The semantic stream uses a frozen pre-trained CLIP ResNet50 [1] to encode the RGB input2 $\tilde { \mathbf { o } } _ { t }$ up until the penultimate layer $\tilde { \mathbf { o } } _ { t }  \mathbf { v } _ { t } ^ { ( 0 ) } : \mathbb { R } ^ { 7 \times 7 \times 2 0 4 8 }$ and then introduces decoding layers that ! t upsample the feature tensors to mimic the spatial stream $\mathbf { v } _ { t } ^ { ( l - 1 ) } \to \mathbf { v } _ { t } ^ { ( l ) } : \mathbb { R } ^ { h \times w \times C }$ at each layer $l$ .
+
+The language instruction ${ \bf l } _ { t }$ is encoded with CLIP’s Transformer-based sentence encoder to produce a goal encoding $\mathbf { l } _ { t }  \mathbf { g } _ { t } : \mathbb { R } ^ { 1 0 2 4 }$ . This goal encoding $\mathbf { g } _ { t }$ is downsampled with fully-connected layers to match the channel dimension $C$ and tiled to match the spatial dimensions of the decoder features such that $\mathbf { g } _ { t } \ \to \ \mathbf { g } _ { t } ^ { ( l ) } \ : \ \mathbb { R } ^ { h \times w \times C }$ . The decoder features are then conditioned with the tiled goal features through an element-wise product $\mathbf { v } _ { t } ^ { ( l ) } \odot \mathbf { g } _ { t } ^ { ( l ) }$ (Hadamard product). Since CLIP was trained with contrastive loss on the dot-product alignment between pooled image features and language encodings, the element-wise product allows us to use this alignment while the tiling preserves the spatial dimensions of the visual features. This language conditioning is repeated for three subsequent layers after the bottleneck inspired by LingUNet [58]. We also add skip connections to these layers from the CLIP ResNet50 encoder to utilize different levels of semantic information from shapes to parts to object-level concepts [59]. Finally, following existing two-stream architectures in videoaction recognition [51], we add lateral connections from the spatial stream to the semantic stream. These connections invothe channel dimension ng , w $1 \times 1$ uceare $[ \mathbf { v } _ { t } ^ { ( l ) } \odot \mathbf { g } _ { t } ^ { ( l ) } ; \mathbf { d } _ { t } ^ { ( l ) } ] : \mathbb { R } ^ { h \times w \times C _ { \mathbf { v } } + C _ { \mathbf { d } } }  \mathbb { R } ^ { h \times w \times \check { C } _ { \mathbf { v } } }$ $\mathbf { v } _ { t } ^ { ( l ) }$ ) and d(l)t the semantic and spatial tensors at layer $l$ , respectively. For the final fusion of dense features, addition for $f _ { \mathrm { p i c k } }$ and $1 \times 1$ conv fusion for $\Phi _ { \mathrm { q u e r y } }$ and $\Phi _ { \mathrm { k e y } }$ worked the best empirically. See Appendix C for details on the exact architecture.
+
+# 3.2 Implementation Details
+
+Training from demonstrations. Similar to Transporter [2] we train CLIPORT through imitation learning from a set of expert demonstrations $\mathcal { D } = \{ \zeta _ { 1 } , \zeta _ { 2 } , . . . , \zeta _ { n } \}$ consisting of discrete-time inputaction pairs $\zeta _ { i } = \{ ( \mathbf { o } _ { 1 } , \mathbf { \bar { l } } _ { 1 } , \mathbf { a } _ { 1 } ) , ( \mathbf { o } _ { 2 } , \mathbf { l } _ { 2 } , \mathbf { a } _ { 2 } ) , \dots \}$ . During training, we randomly sample an inputaction pair from the dataset and supervise the model end-to-end with one-hot pixel encodings of demonstration actions $Y _ { \mathrm { p i c k } } : \mathbb { R } ^ { H \times \dot { W } \times k }$ and $Y _ { \mathrm { p l a c e } } : \mathbb { R } ^ { H \times W \times k }$ with $k$ discrete rotations. In simulated experiments with the suction-gripper, we use $k = 1$ for pick actions and $k = 3 6$ for place actions. The model is trained with cross-entropy loss: $\mathcal { L } = \dot { - } \mathbb { E } _ { Y _ { \mathrm { { p i c k } } } } [ \log \mathcal { V } _ { \mathrm { { p i c k } } } ] - \mathbb { E } _ { Y _ { \mathrm { { 0 i c e } } } } [ \log \mathcal { \bar { V } } _ { \mathrm { { p l a c e } } } ]$ where $\mathcal { V } _ { \mathrm { p i c k } } = \operatorname { s o f t m a x } \left( \mathcal { Q } _ { \mathrm { p i c k } } ( ( u , v ) | \gamma _ { t } ) \right)$ and $\mathcal { V } _ { \mathrm { p l a c e } } = \mathrm { s o f t m a x } ( \mathcal { Q } _ { \mathrm { p l a c e } } ( ( u ^ { \prime } , v ^ { \prime } , \omega ^ { \prime } ) | \gamma _ { t } , \dot { \mathcal { T } } _ { \mathrm { p i c k } } ) )$ . Compared to the original Transporter models that were trained for 40K iterations, we train our models for $2 0 0 \mathrm { K }$ iterations (with data augmentation; see Appendix E) to account for additional semantic variation in tasks – randomized colors, shapes, objects. All models are trained on a single commodity GPU for 2 days with a batch size of 1.
+
+Training multi-task models. Multi-task training is nearly identical to single-task training except for the sampling of training data. First, we randomly sample a task, and then select a random inputaction pair from that task in the dataset. Using this strategy, all tasks are equally likely to be sampled but longer horizon tasks are less likely to reach full coverage of input-action pairs available in the dataset. To compensate for this, we train all multi-task models $3 \times$ longer for 600K iterations or 6 GPU days.
+
+# 4 Results
+
+We perform experiments both in simulation and hardware aimed at answering the following questions: 1) How effective is the language-conditioned two-stream architecture for fine-grained manipulation compared to one-stream alternatives and other simpler baselines? 2) Is it possible to train a multi-task model for all tasks, and how well does it perform and generalize? 3) How well do these models generalize to seen and unseen semantic attributes like colors, shapes, and object categories?
+
+# 4.1 Simulation Setup
+
+Environment. All simulated experiments are based on a Universal Robot UR5e with a suction gripper. The setup provides a systematic and reproducible environment for evaluation, especially for benchmarking the ability to ground semantic concepts like colors and object categories. The input observation is a top-down RGB-D reconstruction from 3 cameras positioned around a rectangular table: one in the front, one on the left shoulder, and one on the right shoulder, all pointing towards the center. Each camera has a resolution of $6 4 0 \times 4 8 0$ and is noiseless.
+
+Language-Conditioned Manipulation Tasks. We extend the Ravens benchmark [2] set in PyBullet [60] with 10 language-conditioned manipulation tasks. See Figure 1 for examples and Table 3 for challenges associated with each task. Each task instance is constructed by sampling a set of objects and attributes: poses, colors, sizes, and object categories. 8 of the 10 tasks have two variants, denoted by seen and unseen, depending on whether the task has unseen attributes (e.g. color) at test time. For colors: $\mathbb { T } _ { \mathrm { s e e n c o l o r s } } = \{ \mathtt { y e l l o w }$ , brown, gray, cyan} and $\mathbb { T } _ { \mathrm { u n s e e n c o l o r s } } =$ {orange, purple, pink, white $\}$ with 3 overlapping colors $\mathbb { T } _ { \mathrm { a l l } } ~ = ~ \{ \mathrm { r e d }$ , green, blue} used in both the seen and unseen spilts. For packing objects, we use 56 tabletop objects from the Google Scanned Objects dataset [61] and split them into 37 seen and 19 unseen objects. The language instructions are constructed from templates for simulated experiments, and human-annotated for real-world experiments. For more details about individual tasks, see Appendix A.
+
+Evaluation Metric. We adopt the 0 (fail) to 100 (success) scores proposed in the Ravens benchmark [2]. The score assigns partial credit based on the task, e.g. $3 / 5 \Rightarrow 6 0 . 0$ for packing 3 out of 5 objects specified in the instructions, or $3 0 / 5 6 \Rightarrow 5 3 . 6$ for pushing 30 out of 56 particles into the correct zone. See Appendix A for the specific evaluation metric used in each task. During an evaluation episode, an agent keeps interacting with the scene until an oracle indicates task-completion. We report scores on 100 evaluation runs for agents trained with $n = 1 , 1 0$ , 100, 1000 demonstrations.
+
+# 4.2 Simulation Results
+
+Table 1 presents results from our large-scale experiments in Ravens [2] and Figure 3 summarizes these results with average scores across seen and unseen splits.
+
+![](images/2c8d085ed4e3b8280f7fea75e44ca5edb6cbae04f83dfd2885faf862f02ac008.jpg)  
+Figure 3. Average scores across seen and unseen splits for all tasks in Table 1.
+
+# Baseline Methods. To study
+
+the effectiveness of our two-stream architecture, we broadly compare against two baselines: Transporter-only and CLIP-only. Transporter-only is the original Transporter [2], or equivalently, the spatial stream of CLIPORT with RGB-D input. Although Transporter-only does not receive any language goals, it shows what can be achieved through chance by exploiting the most likely actions seen during training. On the other hand, CLIP-only is just the semantic stream of CLIPORT with RGB and language input. CLIP-only shows what can be achieved by fine-tuning a pre-trained semantic model for manipulation without spatial information, particularly depth.
+
+Two-Stream Performance. Figure 3 (seen) captures the essence of our main claims. The performance of Transporter-only saturates at $5 0 \%$ since it doesn’t use the language instruction to ground the desired goal. CLIP-only does have a goal, but lacks the spatial precision to go the last mile and thus saturates at $7 6 \%$ . Only CLIPORT (single) achieves more than $9 0 \%$ , which indicates that both the semantic and spatial streams are crucial for fine-grained manipulation. Further, CLIPORT (single) achieves $8 6 \%$ on most tasks with just 100 demonstrations, showcasing its efficiency.
+
+In addition to these baselines, we present various ablations and alternative one-stream and twostream models in Appendix F. To briefly summarize these results, CLIP is essential for few-shot
+
+<table><tr><td></td><td colspan="4">packing-box-pairs seen-colors</td><td colspan="4">packing-box-pairs unseen-colors</td><td colspan="4">packing-seen-google objects-seq</td><td colspan="4">packing-unseen-google objects-seq</td><td colspan="4">packing-seen-google objects-group</td><td colspan="4">packing-unseen-google objects-group</td></tr><tr><td>Method</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1 10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td></tr><tr><td>Transporter-only [2]</td><td>44.2 55.2 54.2</td><td></td><td></td><td></td><td></td><td></td><td>54.1</td><td>26.2 39.7</td><td></td><td>45.4</td><td></td><td></td><td>19.9 29.8</td><td>28.7</td><td>37.3</td><td></td><td></td><td></td><td>59.9</td><td></td><td></td><td>46.2 54.7 49.8</td><td></td></tr><tr><td>CLIP-only</td><td></td><td></td><td></td><td>52.4</td><td>34.648.747.2</td><td></td><td></td><td></td><td></td><td></td><td>46.3</td><td></td><td></td><td></td><td></td><td></td><td>60.0 54.3 61.5</td><td></td><td></td><td></td><td></td><td></td><td>：52.0</td></tr><tr><td>RN50-BERT</td><td></td><td></td><td></td><td></td><td>38.6 69.7 88.5 87.1 33.0 65.5 68.8</td><td></td><td>61.2</td><td></td><td>29.1 67.9</td><td>）89.3</td><td>95.8</td><td></td><td></td><td></td><td>37.1 49.4 60.4 57.8</td><td></td><td>52.5 62.0 89.6 92.7</td><td></td><td></td><td></td><td></td><td></td><td>43.465.9 73.1 70.0</td></tr><tr><td>CLIPORT (single)</td><td></td><td></td><td></td><td></td><td>36.2 64.0 94.7 90.3 31.4 52.7 65.6</td><td></td><td>72.1</td><td></td><td>32.948.4 87.9 94.0</td><td></td><td></td><td></td><td></td><td></td><td>）29.348.548.356.1 71.9</td><td>52.7</td><td></td><td></td><td></td><td></td><td></td><td></td><td>46.452.9 76.586.4 43.2 52.0 66.3 73.7</td></tr><tr><td>CLIPORT (multi)</td><td colspan="4">51.6 82.9 92.7 66.888.694.196.659.0 69.776.271.441.678.485.084.4 40.751.165.870.371.384.689.68.368.4 69.678.480.3</td><td colspan="4">98.2 45.6 65.3 68.6 71.5</td><td colspan="4"></td><td colspan="4">14.8 59.5 86.8 96.2 27.2 50.0 65.5</td><td colspan="4">67.0 84.1</td><td colspan="4">94.061.5 66.2 78.4 81.5</td></tr><tr><td rowspan="4"> CLIPORT (multi-attr)</td><td colspan="4"></td><td colspan="4">46.2 72.0 86.2 80.3</td><td colspan="4"></td><td colspan="4">35.4 45.1 78.9 87.4</td><td colspan="4"></td><td colspan="4">48.6 69.3 84.8 89.1</td></tr><tr><td></td><td>stack-block-pyramid seq-seen-colors</td><td></td><td></td><td>stack-block-pyramid</td><td>seq-unseen-colors</td><td></td><td></td><td>separating-piles seen-colors</td><td></td><td></td><td></td><td>separating-piles unseen-colors</td><td></td><td></td><td></td><td>towers-of-hanoi</td><td></td><td></td><td></td><td></td><td>towers-of-hanoi</td><td></td></tr><tr><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000 1</td><td>10</td><td>100</td><td>1000</td><td></td><td>10</td><td></td><td>1000</td><td>1</td><td>seq-seen-colors 10</td><td></td><td></td><td></td><td></td><td></td><td>seq-unseen-colors</td></tr><tr><td></td><td>4.5</td><td>2.3</td><td>5.2 4.5</td><td>3.0</td><td>4.0</td><td>2.3</td><td>5.8</td><td>42.7</td><td>52.3</td><td>42.0</td><td>48.4</td><td>1 41.2 49.2</td><td></td><td>100 44.7</td><td>52.3</td><td>67.9</td><td>98.0</td><td>100 1000 99.9</td><td></td><td>1</td><td>10 100 24.344.671.7 80.7</td><td>1000</td></tr><tr><td>Transporter-only [2] CLIP-only</td><td>6.328.7 55.7 54.8</td><td></td><td></td><td></td><td>2.012.2</td><td>18.3</td><td></td><td>19.543.5</td><td>55.0</td><td>84.9 90.25</td><td></td><td>59.949.6</td><td></td><td>73.0</td><td>71.0</td><td>25.4 9.4</td><td>52.6 88.645.3</td><td></td><td></td><td></td><td></td><td></td><td>24.747.067.0 58.0</td></tr><tr><td>RN50-BERT CLIPORT (single)</td><td>5.335.0 89.097.5 28.364.7 93.3 98.8</td><td></td><td></td><td></td><td>6.2 12.2 13.7 24.3</td><td>21.5 31.2</td><td>30.7 41.3</td><td>31.8 47.8 54.5 59.5</td><td></td><td>93.1</td><td>98.0</td><td>）47.2</td><td>46.546.533.444.4 51.0</td><td>41.3 76.6</td><td>44.9 75.2</td><td>59.4</td><td>28.0 66.1 92.9</td><td>91.392.1 97.4100</td><td></td><td>56.1</td><td></td><td></td><td>17.4 75.1 85.3 89.3 89.7 95.9 99.4</td></tr><tr><td>CLIPORT (multi)</td><td>33.5 75.3 96.8 96.5 23.3 26.831.7 22.2 48.9 72.4 90.3 89.0 56.6 62.6 64.9 62.8</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>49.9 51.8 48.2 59.8</td><td></td><td></td><td></td><td></td><td>61.6 96.3 98.7 98.1</td><td></td><td></td><td></td><td></td><td></td><td>60.1 65.6 76.7 68.7</td></tr><tr><td> CLIPORT (multi-attr)</td><td>一</td><td></td><td></td><td></td><td></td><td></td><td>15.5 51.5 59.3 79.8</td><td>assembling-kits-seq</td><td></td><td></td><td></td><td>assembling-kits-seq</td><td></td><td></td><td></td><td>put-blocks-in-bowls</td><td></td><td></td><td></td><td></td><td></td><td></td><td>56.7 78.0 88.3 96.9 put-blocks-in-bowls</td></tr><tr><td></td><td></td><td>align-rope</td><td></td><td></td><td> packing-unseen-shapes</td><td></td><td></td><td></td><td>seen-colors</td><td></td><td></td><td></td><td></td><td>unseen-colors</td><td></td><td></td><td>seen-colors</td><td></td><td></td><td></td><td></td><td>unseen-colors</td><td></td></tr><tr><td>Transporter-only [2]</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10 100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td></tr><tr><td>CLIP-only</td><td>6.930.633.1</td><td></td><td></td><td>51.5</td><td></td><td>16.0 20.0 22.0</td><td>22.0</td><td>5.8</td><td>11.6</td><td></td><td>28.6 29.6</td><td>7.8</td><td>17.6</td><td>25.6</td><td>28.4</td><td>16.8 333.3 62.7</td><td></td><td></td><td>64.7</td><td>11.7</td><td></td><td>17.2 14.8</td><td>18.7 11.2 34.2 33.2 44.5</td></tr><tr><td>RN50-BERT CLIPORT (single)</td><td></td><td>3.125.063.8</td><td>13.4 48.7 70.4</td><td>70.7 57.1</td><td>19.0 25.0</td><td>13.0 28.0 44.0 32.0</td><td>50.0 44.0</td><td>0.8 2.2</td><td>9.2 5.6</td><td>19.8 11.6</td><td>23.0 21.8</td><td>2.0 1.6</td><td>4.6 6.4</td><td>10.8 10.4</td><td>19.8 18.4</td><td>23.5 60.2 93.5 13.8 44.581.2</td><td></td><td>91.8</td><td>97.7</td><td></td><td></td><td></td><td>223.030.323.8</td></tr><tr></table>
+
+Table 1. Language-Conditioned Test Results. Task success scores (mean $\%$ ) from 100 evaluation instances vs. # of training demonstrations (1, 10, 100, or 1000). The challenges pertaining to each task are described in Appendix A. CLIPORT (single) models are trained on seen splits, and evaluated on both seen and unseen splits. CLIPORT (multi) models are trained on seen splits of all 10 tasks with 1T, 10T, 100T, and 1000T demonstrations where $\mathbb { T } = 1 0$ . CLIPORT (multi-attr) indicate CLIPORT (multi) models trained on seen-and-unseen splits from all tasks except for that one particular heldout task, for which it is trained only the seen split. See Figure 3 for an overview with average scores.
+
+learning (i.e. $n \geq 1 0 .$ ) in lieu of semantic stream alternatives like ImageNet-trained ResNet50 [62] with BERT [38]. Image-goal models outperform CLIPORT (single) in packing Google objects, but this is only because they do not have to solve the language-grounding problem.
+
+Multi-Task Performance. In realistic scenarios, we want the robot to be capable of any task, not just one task. We investigate this through CLIPORT (multi) in Table 1 with one multi-task model trained on all 10 tasks. CLIPORT (multi) models are trained only on seen-splits of tasks, so an unseen attribute like ‘pink’ is consistent throughout single and multi-task settings. Surprisingly, CLIPORT (multi) outperforms single-task CLIPORT (single) models in $4 1 / 7 2 = 5 7 \%$ of the evaluations in Table 1. This trend is also evident in Figure 3 (seen), especially in instances with 100 demonstrations or less. Although CLIPORT (multi) is trained on more diverse data from other tasks, both CLIPORT (multi) and CLIPORT (single) have access to the same amount of data per task. This supports our premise that language is a strong conditioning mechanism for reusing concepts from other tasks without learning them from scratch. It also validates a trait of data-driven approaches where training on lots of diverse data leads to more robust and generalizable representations [1, 63]. However, CLIPORT (multi) performs worse on longer-horizon tasks like align-rope. We hypothesize that this is because longer-horizon tasks get less coverage of input-action pairs in the dataset. Future works could use better sampling methods that balance tasks according to their average time horizon.
+
+Generalizing to Unseen Attributes. Tasks that require generalizing to novel colors, shapes, and objects are more difficult and all our agents achieve relatively lower performance on these tasks, as shown in Figure 3 (unseen). However, CLIPORT (single) models do substantially better than chance, i.e., Transporter-only. The lower performances are due to the difficulty of grounding unseen attributes such as ‘pink’ and ‘orange’ in the language instruction “put the pink block on the orange bowl”, when the agent has never encountered words ‘orange’, ‘pink’ or their corresponding visual characteristics in the context of the physical environment. Although pre-trained CLIP has been exposed to the attribute ‘pink’, it could correspond to different concepts in the physical setting depending on factors like lighting condition, and thus requires at least few examples to condition the trainable semantic decoder layers. Additionally, we notice that CLIPORT (single) is also less prone to overfitting compared to Transporter-only. As evidenced in towers-of-hanoi-seq-unseen-colors task in Table 1, Transporter-only suffers from a performance drop because of rings with unseen colors despite the fact that Tower of Hanoi can be solved without attending to the colors and simply focusing on the ring size. We hypothesize that since CLIP was trained on diverse internet data, it enables our agent to focus on task-relevant concepts while ignoring irrelevant aspects of the task.
+
+![](images/2d3a3c5d8f1664e75fb4cd0744b2b47254beccd56c4e1c6aecb38ce7542f4319.jpg)  
+Figure 4. Affordance predictions from CLIPORT (multi) models in sim (left two) and real settings (right three). More examples in Appendix H.
+
+Transferring Attributes across Tasks. One solution for dealing with unseen attributes is to explicitly learn these attributes from other tasks. We study this with CLIPORT (multi-attr) in Table 1 and Figure 3 (unseen). For these models, CLIPORT (multi) is trained on both seen-and-unseen splits from all tasks except for the task being evaluated on, for which it was only trained on the seen split. As such, this evaluation measures whether having seen pink blocks in put-blocks-in-bowl-unseen-colors helps solve “pack all the pink and cyan boxes” in packing-box-pairs-unseen-colors. Results indicate that such explicit transfers result in significant improvements. For instance, on the put-blocks-inbowls-unseen-colors task for $n = 1 0 0 0$ , CLIPORT (multi)’s performance increases from 45.8 to 75.7.
+
+# 4.3 Real-Robot Experiments
+
+We validated our results in hardware with a Franka Panda manipulator. See Appendix D for setup details. Table 2 reports success rates for a multi-task model trained and evaluated on 9 real-world tasks. Due to COVID restrictions, we could not conduct largescale user-studies, so we report on small train (5-10 demos) and test sets (5-10 runs) per task. Overall, CLIPORT (multi) is effective at few-shot learning with just 179 samples, and the performances roughly correspond to those in simulated experiments, with simple block manipulation tasks achieving $\sim 7 0 \%$ . We estimate that for more robust real-world performance at least 50 to 100 training demonstrations are necessary, as evident in Figure 3. Interestingly, we observed that the model sometimes exploits biases in the training data instead of learning to ground instructions. For instance, in Put Blocks in Bowl, the training set consisted of only one datapoint on “yellow blocks” being placed inside a “blue bowl”. This made it difficult to condition the model to place “yellow blocks” in non-blue bowls. But instances with just one or two examples where a colored block went to different colored bowls was sufficient to make the model pay attention to the language. In summary, unbiased datasets containing both a good coverage of expected skills and invariances, and a decent number of training demonstrations, are crucial for good real-world performance.
+
+<table><tr><td>Task # Train (Samples) #Test Succ.%</td></tr><tr><td>Stack Blocks</td></tr><tr><td>5 (13) 10 70.0 5(10) 10 65.0</td></tr><tr><td>Put Blocks in Bowl Pack Objects 10 (31) 10 60.0</td></tr><tr><td>Move Rook 4(29) 10 70.0</td></tr><tr><td>Fold Cloth 9(9) 10 57.0</td></tr><tr><td>Read Text 2(26) 10 55.0</td></tr><tr><td>Loop Rope 4(12) 10 60.0</td></tr><tr><td>Sweep Beans 5 (23) 5 60.6</td></tr><tr><td>Pick Cherries 4(26) 5 75.0</td></tr></table>
+
+Table 2. Success rates $( \% )$ of a multi-task model trained an evaluated 9 real-world tasks (see Figure 1). Samples indicate total image-action pairs, e.g 1 in Figure 9.
+
+# 5 Conclusion
+
+We introduced CLIPORT, an end-to-end framework for language-conditioned fine-grained manipulation. Our experiments, specifically with multi-task models, indicate that data-driven approaches to generalization have yet to be fully-exploited in robotics. Coupled with the right action abstraction and spatio-semantic priors, end-to-end methods can quickly learn new skills without requiring top-down pipelines that need task-specific engineering.
+
+While CLIPORT can solve a range of tabletop tasks, extending it to dexterous 6-DOF manipulation that goes beyond the two-step primitive remains a challenge. As such, it cannot handle complex partially-observable scenes, or output continuous control for multi-fingered hands, or predict task-completion (see Appendix I for an extended discussion). But overall, we are excited by the confluence of data and structural priors for building scalable and generalizable robotic systems.
+
+# Acknowledgments
+
+All simulated experiments were facilitated through the Hyak computing cluster funded by the STF at the University of Washington. We thank Mohak Bhardwaj for help with the Franka setup at UW. We are also grateful to our colleagues Chris Xie, Jesse Thomason, and Valts Blukis for providing feedback on the initial draft. This work was funded in part by ONR under award #1140209-405780.
+
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parse/train/9uFiX_HRsIL/9uFiX_HRsIL_content_list.json

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+        "type": "text",
+        "text": "CLIPORT: What and Where Pathways for Robotic Manipulation ",
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+        "text": "Mohit Shridhar 1,† Lucas Manuelli 2 Dieter Fox 1,2 1University of Washington 2NVIDIA mshr@cs.washington.edu lmanuelli@nvidia.com fox@cs.washington.edu ",
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+        "text": "cliport.github.io ",
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+        "text": "Abstract: How can we imbue robots with the ability to manipulate objects precisely but also to reason about them in terms of abstract concepts? Recent works in manipulation have shown that end-to-end networks can learn dexterous skills that require precise spatial reasoning, but these methods often fail to generalize to new goals or quickly learn transferable concepts across tasks. In parallel, there has been great progress in learning generalizable semantic representations for vision and language by training on large-scale internet data, however these representations lack the spatial understanding necessary for fine-grained manipulation. To this end, we propose a framework that combines the best of both worlds: a two-stream architecture with semantic and spatial pathways for vision-based manipulation. Specifically, we present CLIPORT, a language-conditioned imitationlearning agent that combines the broad semantic understanding (what) of CLIP [1] with the spatial precision (where) of Transporter [2]. Our end-to-end framework is capable of solving a variety of language-specified tabletop tasks from packing unseen objects to folding cloths, all without any explicit representations of object poses, instance segmentations, memory, symbolic states, or syntactic structures. Experiments in simulated and real-world settings show that our approach is data efficient in few-shot settings and generalizes effectively to seen and unseen semantic concepts. We even learn one multi-task policy for 10 simulated and 9 real-world tasks that is better or comparable to single-task policies. ",
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+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Ask a person to “get a scoop of coffee beans” or “fold the cloth in half” and they can naturally take concepts like scoop or fold and ground them in concrete physical actions within an accuracy of a few centimeters. We humans do this intuitively, without explicit geometric or kinematic models of coffee beans or cloths. Moreover, we can generalize to a broad range of tasks and concepts from a minimal set of examples on what needs to be achieved. How can we imbue robots with this ability to efficiently ground abstract semantic concepts in precise spatial reasoning? ",
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+        "text": "Recently, a number of end-to-end frameworks have been proposed for vision-based manipulation [2, 3, 4, 5]. While these methods do not use any explicit representations of object poses, instance segmentations, or symbolic states, they can only replicate demonstrations with a narrow range of variability and have no notion of the semantics underlying the tasks. Switching from packing red pens to blue pens involves collecting a new training set [2], or if using goal-conditioned policies, involves the user providing a goal-image from the scene [5, 6]. In realistic human-robot interaction settings, collecting additional demonstrations or providing goal-images is often infeasible and unscalable. A natural solution to both these problems is to condition policies with natural language. Language provides an intuitive interface for specifying goals and also for implicitly transferring concepts across tasks. While language-grounding for manipulation has been explored in the past [7, 8, 9, 10], these pipelines are limited by object-centric representations that cannot handle granular or deformable objects and often do not reason about perception and action in an integrated manner. In parallel, there has been great progress in learning models for visual representations [11, 12] and aligning representations of vision and language [13, 14, 15] by training on large-scale internet data. However, these models lack a fine-grained understanding on how to manipulate objects, i.e. physical affordances. ",
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+        "img_path": "images/3828e046605da1a58cfd55488e88f64400dd6b7ebfa4e8931b729aba5012e158.jpg",
+        "image_caption": [
+            "Figure 1. Language-Conditioned Manipulation Tasks: CLIPORT is a broad framework applicable to a wide range of language-conditioned manipulation tasks in tabletop settings. We conduct large-scale experiments in Ravens [2] on 10 simulated tasks (a-j) with 1000s of unique instances per task. See Appendix A for challenges pertaining to each task. CLIPORT can even learn one multi-task model for all 10 tasks that achieves better or comparable performance to single-task models. Similarly, we demonstrate our approach on a Franka Panda manipulator with one multi-task model for 9 real-world tasks (k-o; only 5 shown) trained with just 179 image-action pairs. "
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+        "text": "To this end, we propose the first framework that combines the best of both worlds: end-to-end learning for fine-grained manipulation with the multi-goal and multi-task generalization capabilities of vision-language grounding systems. We introduce a two-stream architecture for manipulation with semantic and spatial pathways broadly inspired by (or vaguely analogous to) the two-stream hypothesis in cognitive psychology [16, 17, 18]. Specifically, we present CLIPORT, a languageconditioned imitation-learning agent that integrates the semantic understanding (what) of CLIP [1] with the spatial precision (where) of Transporter [2]. Transporter has been applied to a wide range of rearragement tasks from industrial packing [2] to manipulating deformable objects [6]. The key insight of the approach is formulating tabletop manipulation as a series of pick-and-place affordance predictions, where the objective is to detect actions rather than detect objects and then learn a policy. This action-centric approach to perception [19] is data efficient and effective at circumventing the need for explicit “objectness” in learnt representations. However, Transporter is a tabula rasa system that learns all visual representations from scratch and so every new goal or task requires collecting a new set of demonstrations. To address this problem, we bake in a strong semantic prior while learning policies. We condition our semantic stream with visual and language-goal features from a pre-trained CLIP model [1]. Since CLIP is pre-trained to align image and language features from millions of image-caption pairs from the internet, it provides a powerful prior for grounding semantic concepts that are common across tasks like categories, parts, shapes, colors, texts, and other visual attributes, all without a top-down pipeline that requires bounding boxes or instance segmentations [13, 14, 15, 20]. This allows us to formulate tabletop rearrangement as a series of language-conditioned affordance predictions, a predominantly vision-based inference problem, and thus benefit from the strengths of data-driven paradigms like scale and generalization. ",
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+        "text": "To study these benefits, we conduct large-scale experiments in the Ravens [2] framework with a simulated suction-gripper robot. We propose 10 language-conditioned tasks with 1000s of unique instances per task that require both semantic and spatial reasoning (see Figure $1 \\ \\mathrm { a - j }$ ). CLIPORT is not only effective at solving these tasks, but surprisingly, it can even learn a multi-task model for all 10 tasks that achieves better or comparable performance to single-task models. Further, our evaluations indicate that our multi-task model can effectively transfer attributes like “pink block” across tasks, having never seen pink blocks or the word ‘pink’ in the context of the evaluation task. We also demonstrate our approach on a Franka Panda manipulator with one multi-task model for 9 real-world tasks trained with just 179 image-action pairs (see Figure $1 \\ k { - } 0$ ). ",
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+    {
+        "type": "text",
+        "text": "In summary, our contributions are as follows: ",
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+    {
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+        "text": "• An extended benchmark of language-grounding tasks for manipulation in Ravens [2].   \n• Two-stream architecture for using internet pre-trained vision-language models for conditioning precise manipulation policies with language goals.   \n• Empirical results on a broad range of manipulation tasks, including multi-task models, validated with real-robot experiments. ",
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+    {
+        "type": "text",
+        "text": "The benchmark, code, and pre-trained models are available at: cliport.github.io. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Vision-based Manipulation. Traditionally, perception for manipulation has centered around object detectors, segmentors, and pose estimators [21, 22, 23, 24, 25, 26]. These methods cannot handle deformable objects, granular media, or generalize to unseen objects without object-specific training data. Alternatively, dense descriptors [27, 28, 29] and keypoint representations [30, 31, 32] forgo segmentation and pose representations, but do not reason about sequential actions and struggle to represent scenes with variable numbers of objects. On the other hand, end-to-end perception-toaction models can learn precise sequential policies [2, 4, 6, 33, 34, 35], but these methods have limited understanding of semantic concepts and rely on goal-images to condition policies. In contrast, Yen-Chen et. al [36] showed that pre-training on semantic tasks like classification and segmentation helps in improving efficiency and generalization of grasping predictions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Semantic Models. With the advent of large-scale models [37, 38, 39], a number of methods for learning joint vision and language representations have been proposed [13, 14, 15, 20, 40]. However, these methods are restricted to bounding boxes or instance segmentations, which make them inapplicable for detecting things like piles of coffee beans or squares on a chessboard. Alternatively, works in contrastive learning forgo top-down object-detection and learn continuous representations by pre-training on unlabeled data [11, 12]. Recently, CLIP [1] applied a similar approach to align vision and language representations by training on millions of image-caption pairs from the internet. ",
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+    {
+        "type": "text",
+        "text": "Language Grounding for Robotics. Several works have proposed systems for instructing robots with natural language [7, 8, 9, 10, 41, 42, 43, 44, 45, 46, 47]. However, these methods use disentangled pipelines for perception and action with the language primarily being used to guide the perception. As such, these pipelines lack the spatial precision necessary for tasks like folding cloths. Recently, Lynch et. al [48] proposed an end-to-end system for grounding language in continuous control, but it requires several hours of human teleoperation data for a single simulated desk setting. ",
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+        "text": "Two-Stream Architectures are prevalent in action-recognition networks [49, 50, 51] and audiorecognition systems [52, 53]. In robotics, Zeng et. al [54] and Jang et. al [55] have proposed twostream pipelines for affordance predictions of novel objects. The former requires goal-images and the latter is restricted to one-step grasps with single-category goals. In contrast, our framework provides a rich and intuitive interface with composable language commands for sequential tasks. ",
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+    {
+        "type": "text",
+        "text": "3 CLIPORT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "CLIPORT is an imitation-learning agent based on four key principles: (1) Manipulation through a two-step primitive where each action involves a start and final end-effector pose. (2) Visual representations of actions that are equivariant to translations and rotations [56, 57]. (3) Two separate pathways for semantic and spatial information. (4) Language-conditioned policies for specifying goals and also transferring concepts across tasks. Combining (1) and (2) from Transporter with (3) and (4) allows us to achieve generalizable policies that go beyond just imitating demonstrations. ",
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+        "text": "Section 3.1 describes the problem formulation, gives an overview of Transporter [2], and presents our language-conditioned model. Section 3.2 provides details on the training approach. ",
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+        "text": "3.1 Language-Conditioned Manipulation ",
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+        "text": "We consider the problem of learning a goal-conditioned policy $\\pi$ that outputs actions $\\mathbf { a } _ { t }$ given input $\\gamma _ { t } = ( \\mathbf { o } _ { t } , \\mathbf { l } _ { t } )$ consisting of a visual observation $\\mathbf { o } _ { t }$ and an English language instruction ${ \\bf l } _ { t }$ : ",
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+        "img_path": "images/e011623dcb5c5c8146678648509e1d77e47b4b598f446a4cd2479ce7abaae3aa.jpg",
+        "text": "$$\n\\pi ( \\gamma _ { t } ) = \\pi ( \\mathbf { o } _ { t } , \\mathbf { l } _ { t } )  \\mathbf { a } _ { t } = ( { \\mathcal { T } } _ { \\mathrm { p i c k } } , { \\mathcal { T } } _ { \\mathrm { p l a c e } } ) \\in \\mathcal { A }\n$$",
+        "text_format": "latex",
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+        "img_path": "images/7096ec9751b6431d9d588d6f507c70933b7da146cf1131293568d69619c96f9b.jpg",
+        "image_caption": [
+            "Figure 2. CLIPORT Two-Stream Architecture. An overview of the semantic and spatial streams. The semantic stream uses a frozen CLIP ResNet50 [1] to encode RGB input, and its decoder layers are conditioned with tiled language features from the CLIP sentence encoder. The spatial stream encodes RGB-D input, and its decoder layers are laterally fused with the semantic stream. The final output is a map of dense pixelwise features that is used for pick or place affordance predictions. This same two-stream architecture is used in all 3 FullyConvolutional-Networks $f _ { \\mathrm { p i c k } }$ , $\\Phi _ { \\mathrm { q u e r y } }$ , and $\\Phi _ { \\mathrm { k e y } }$ with $f _ { \\mathrm { p i c k } }$ is used to predict pick actions, and $\\Phi _ { \\mathrm { q u e r y } }$ and $\\Phi _ { \\mathrm { k e y } }$ are used to predict place actions. See Appendix C for the exact architecture. "
+        ],
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+    {
+        "type": "text",
+        "text": "The actions $\\mathbf { a } = ( T _ { \\mathrm { p i c k } } , T _ { \\mathrm { p l a c e } } )$ specify the end-effector pose for picking and placing, respectively. We consider tabletop tasks where $\\mathcal { T } _ { \\mathrm { p i c k } } , \\mathcal { T } _ { \\mathrm { p l a c e } } \\in \\mathbf { S } \\mathbf { E } ( 2 )$ . The visual observation $\\mathbf { o } _ { t }$ is a top-down orthographic RGB-D reconstruction of the scene where each pixel corresponds to a point in 3D space. The language instruction ${ \\bf l } _ { t }$ either specifies step-by-step instructions e.g. “pack the scissors” “pack the purple tape” etc., or a single goal description for the whole task e.g “pack all the blue and yellow boxes in the brown box”. See Figure 4 for specific examples. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We assume access to a dataset $\\mathcal { D } = \\{ \\zeta _ { 1 } , \\zeta _ { 2 } , \\ldots , \\zeta _ { n } \\}$ of $n$ expert demonstrations with associated discrete-time input-action pairs $\\zeta _ { i } = \\{ ( \\mathbf { o } _ { 1 } , \\mathbf { l } _ { 1 } , \\mathbf { a } _ { 1 } ) , ( \\mathbf { o } _ { 2 } , \\mathbf { l } _ { 2 } , \\mathbf { a } _ { 2 } ) , . . . \\}$ where $\\mathbf { a } _ { t } = ( T _ { \\mathrm { p i c k } } , T _ { \\mathrm { p l a c e } } )$ corresponds to expert pick-and-place coordinates at timestep $t$ . These expert demonstrations are used to supervise the policy $\\pi$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Transporter for Pick-and-Place. The policy $\\pi$ is trained with Transporter [2] to perform spatial manipulation. The model first (i) attends to a local region to decide where to pick, then (ii) computes a placement location by finding the best match through cross-correlation of deep visual features. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Following Transporter [2, 6], the policy $\\pi$ is composed of two action-value modules (Q-functions): The pick module $\\mathcal { Q } _ { \\mathrm { p i c k } }$ decides where to pick, and conditioned on this pick action the place module $\\mathcal { Q } _ { \\mathrm { p l a c e } }$ decides where to place. These modules are implemented as Fully-Convolutional-Networks (FCNs) that are translationally equivariant by design. As we will describe in more detail below, we extend these networks to two-stream architectures that can handle language input. The pick FCN $f _ { \\mathrm { p i c k } }$ takes input $\\gamma _ { t } = \\left( \\mathbf { o } _ { t } , \\mathbf { l } _ { t } \\right)$ and outputs a dense pixelwise prediction $\\mathbf { \\bar { \\mathcal { Q } } _ { \\mathrm { p i c k } } } ^ { } \\in \\mathbb { R } ^ { H \\times \\dot { W } }$ of action-values, where are used to predict the pick action $\\tau _ { \\mathrm { p i c k } }$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/6b9544840e4983c86df667240465304222924a7efe8dd418f77513806d2c272d.jpg",
+        "text": "$$\n\\mathcal { T } _ { \\mathrm { p i c k } } = \\underset { ( u , v ) } { \\mathrm { a r g m a x } } \\ : \\mathcal { Q } _ { \\mathrm { p i c k } } ( ( u , v ) | \\gamma _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            739,
+            607,
+            767
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since $\\mathbf { o } _ { t }$ is an orthographic heightmap, each pixel location $( u , v )$ can be mapped to a 3D picking location using the known camera calibration. $f _ { \\mathrm { p i c k } }$ is trained in a supervised manner to predict the pick action $\\mathcal { T } _ { \\mathrm { p i c k } }$ that imitates the expert demonstration with the specified language instruction at timestep $t$ . ",
+        "bbox": [
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+            771,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The second FCN $\\Phi _ { \\mathrm { q u e r y } }$ takes in $\\gamma _ { t } [ \\mathcal { T } _ { \\mathrm { p i c k } } ]$ , which is a $c \\times c$ crop of $\\mathbf { o } _ { t }$ centered at $\\tau _ { \\mathrm { p i c k } }$ along with the language instruction ${ \\bf l } _ { t }$ , and outputs a query feature embedding of shape $\\mathbb { R } ^ { c \\times c \\times d }$ . The third FCN $\\Phi _ { \\mathrm { k e y } }$ consumes the full input $\\gamma _ { t }$ and outputs a key feature embedding of shape $\\mathbb { R } ^ { H \\times W \\times d }$ . The place action-values $\\mathcal { Q } _ { \\mathrm { p l a c e } }$ are then computed by cross-correlating the query and key features: ",
+        "bbox": [
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+            891
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d908c4b9f0b5dc88e3dc065fd7cf75a0b7e317db1f2cd2d5b6d4d8814e164bab.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { Q } _ { \\mathrm { p l a c e } } ( \\Delta \\tau | \\gamma _ { t } , \\mathcal { T } _ { \\mathrm { p i c k } } ) = \\left( \\Phi _ { \\mathrm { q u e r y } } ( \\gamma _ { t } [ \\mathcal { T } _ { \\mathrm { p i c k } } ] ) * \\Phi _ { \\mathrm { k e y } } ( \\gamma _ { t } ) \\right) [ \\Delta \\tau ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            895,
+            687,
+            914
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $ { \\Delta \\tau } \\in S E ( 2 )$ represents a potential placement pose. Since $\\mathbf { o } _ { t }$ is an orthographic heightmap, rotations in the placement pose $\\Delta \\tau$ can be captured by stacking $k$ discrete angle rotations of the crop before passing it through the query network $\\Phi _ { \\mathrm { q u e r y } }$ . Then $\\begin{array} { r } { \\mathcal { T } _ { \\mathrm { p l a c e } } ^ { \\mathrm { ~ \\tiny ~ \\bar { ~ } ~ } } = \\arg \\operatorname* { m a x } _ { \\Delta \\tau } \\mathcal { Q } _ { \\mathrm { p l a c e } } ( \\Delta \\tau | \\gamma _ { t } , \\mathcal { T } _ { \\mathrm { p i c k } } ) , } \\end{array}$ , where the place module is trained to imitate the placements in the expert demonstrations. For all models, we use $c = 6 4$ , $k = 3 6$ and $d = 3$ . As in Transporter [2, 6], our framework can be extended to handle any motion primitive like pushing, sliding, etc. that can be parameterized by two end-effector poses at each timestep. For more details, we refer the reader to the original paper [2]. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Two-Stream Architecture. In CLIPORT, we extend the network architecture of all three FCNs $f _ { \\mathrm { p i c k } }$ , $\\Phi _ { \\mathrm { q u e r y } }$ and $\\Phi _ { \\mathrm { k e y } }$ from Transporter [2] to allow for language input and reasoning about high-level semantic concepts. We extend the FCNs to two-pathways: semantic (ventral) and spatial (dorsal). The semantic stream is conditioned with language features at the bottleneck and fused with intermediate features from the spatial stream. See Figure 2 for an overview of the architecture. ",
+        "bbox": [
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+            194,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The spatial stream is identical to the ResNet architecture in Transporter – a tabula rasa network that takes in RGB-D input $\\mathbf { o } _ { t }$ and outputs dense features through an hourglass encoder-decoder model. The semantic stream uses a frozen pre-trained CLIP ResNet50 [1] to encode the RGB input2 $\\tilde { \\mathbf { o } } _ { t }$ up until the penultimate layer $\\tilde { \\mathbf { o } } _ { t }  \\mathbf { v } _ { t } ^ { ( 0 ) } : \\mathbb { R } ^ { 7 \\times 7 \\times 2 0 4 8 }$ and then introduces decoding layers that ! t upsample the feature tensors to mimic the spatial stream $\\mathbf { v } _ { t } ^ { ( l - 1 ) } \\to \\mathbf { v } _ { t } ^ { ( l ) } : \\mathbb { R } ^ { h \\times w \\times C }$ at each layer $l$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The language instruction ${ \\bf l } _ { t }$ is encoded with CLIP’s Transformer-based sentence encoder to produce a goal encoding $\\mathbf { l } _ { t }  \\mathbf { g } _ { t } : \\mathbb { R } ^ { 1 0 2 4 }$ . This goal encoding $\\mathbf { g } _ { t }$ is downsampled with fully-connected layers to match the channel dimension $C$ and tiled to match the spatial dimensions of the decoder features such that $\\mathbf { g } _ { t } \\ \\to \\ \\mathbf { g } _ { t } ^ { ( l ) } \\ : \\ \\mathbb { R } ^ { h \\times w \\times C }$ . The decoder features are then conditioned with the tiled goal features through an element-wise product $\\mathbf { v } _ { t } ^ { ( l ) } \\odot \\mathbf { g } _ { t } ^ { ( l ) }$ (Hadamard product). Since CLIP was trained with contrastive loss on the dot-product alignment between pooled image features and language encodings, the element-wise product allows us to use this alignment while the tiling preserves the spatial dimensions of the visual features. This language conditioning is repeated for three subsequent layers after the bottleneck inspired by LingUNet [58]. We also add skip connections to these layers from the CLIP ResNet50 encoder to utilize different levels of semantic information from shapes to parts to object-level concepts [59]. Finally, following existing two-stream architectures in videoaction recognition [51], we add lateral connections from the spatial stream to the semantic stream. These connections invothe channel dimension ng , w $1 \\times 1$ uceare $[ \\mathbf { v } _ { t } ^ { ( l ) } \\odot \\mathbf { g } _ { t } ^ { ( l ) } ; \\mathbf { d } _ { t } ^ { ( l ) } ] : \\mathbb { R } ^ { h \\times w \\times C _ { \\mathbf { v } } + C _ { \\mathbf { d } } }  \\mathbb { R } ^ { h \\times w \\times \\check { C } _ { \\mathbf { v } } }$ $\\mathbf { v } _ { t } ^ { ( l ) }$ ) and d(l)t the semantic and spatial tensors at layer $l$ , respectively. For the final fusion of dense features, addition for $f _ { \\mathrm { p i c k } }$ and $1 \\times 1$ conv fusion for $\\Phi _ { \\mathrm { q u e r y } }$ and $\\Phi _ { \\mathrm { k e y } }$ worked the best empirically. See Appendix C for details on the exact architecture. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 Implementation Details ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Training from demonstrations. Similar to Transporter [2] we train CLIPORT through imitation learning from a set of expert demonstrations $\\mathcal { D } = \\{ \\zeta _ { 1 } , \\zeta _ { 2 } , . . . , \\zeta _ { n } \\}$ consisting of discrete-time inputaction pairs $\\zeta _ { i } = \\{ ( \\mathbf { o } _ { 1 } , \\mathbf { \\bar { l } } _ { 1 } , \\mathbf { a } _ { 1 } ) , ( \\mathbf { o } _ { 2 } , \\mathbf { l } _ { 2 } , \\mathbf { a } _ { 2 } ) , \\dots \\}$ . During training, we randomly sample an inputaction pair from the dataset and supervise the model end-to-end with one-hot pixel encodings of demonstration actions $Y _ { \\mathrm { p i c k } } : \\mathbb { R } ^ { H \\times \\dot { W } \\times k }$ and $Y _ { \\mathrm { p l a c e } } : \\mathbb { R } ^ { H \\times W \\times k }$ with $k$ discrete rotations. In simulated experiments with the suction-gripper, we use $k = 1$ for pick actions and $k = 3 6$ for place actions. The model is trained with cross-entropy loss: $\\mathcal { L } = \\dot { - } \\mathbb { E } _ { Y _ { \\mathrm { { p i c k } } } } [ \\log \\mathcal { V } _ { \\mathrm { { p i c k } } } ] - \\mathbb { E } _ { Y _ { \\mathrm { { 0 i c e } } } } [ \\log \\mathcal { \\bar { V } } _ { \\mathrm { { p l a c e } } } ]$ where $\\mathcal { V } _ { \\mathrm { p i c k } } = \\operatorname { s o f t m a x } \\left( \\mathcal { Q } _ { \\mathrm { p i c k } } ( ( u , v ) | \\gamma _ { t } ) \\right)$ and $\\mathcal { V } _ { \\mathrm { p l a c e } } = \\mathrm { s o f t m a x } ( \\mathcal { Q } _ { \\mathrm { p l a c e } } ( ( u ^ { \\prime } , v ^ { \\prime } , \\omega ^ { \\prime } ) | \\gamma _ { t } , \\dot { \\mathcal { T } } _ { \\mathrm { p i c k } } ) )$ . Compared to the original Transporter models that were trained for 40K iterations, we train our models for $2 0 0 \\mathrm { K }$ iterations (with data augmentation; see Appendix E) to account for additional semantic variation in tasks – randomized colors, shapes, objects. All models are trained on a single commodity GPU for 2 days with a batch size of 1. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Training multi-task models. Multi-task training is nearly identical to single-task training except for the sampling of training data. First, we randomly sample a task, and then select a random inputaction pair from that task in the dataset. Using this strategy, all tasks are equally likely to be sampled but longer horizon tasks are less likely to reach full coverage of input-action pairs available in the dataset. To compensate for this, we train all multi-task models $3 \\times$ longer for 600K iterations or 6 GPU days. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Results ",
+        "text_level": 1,
+        "bbox": [
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+            266,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We perform experiments both in simulation and hardware aimed at answering the following questions: 1) How effective is the language-conditioned two-stream architecture for fine-grained manipulation compared to one-stream alternatives and other simpler baselines? 2) Is it possible to train a multi-task model for all tasks, and how well does it perform and generalize? 3) How well do these models generalize to seen and unseen semantic attributes like colors, shapes, and object categories? ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 Simulation Setup ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            202,
+            333,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Environment. All simulated experiments are based on a Universal Robot UR5e with a suction gripper. The setup provides a systematic and reproducible environment for evaluation, especially for benchmarking the ability to ground semantic concepts like colors and object categories. The input observation is a top-down RGB-D reconstruction from 3 cameras positioned around a rectangular table: one in the front, one on the left shoulder, and one on the right shoulder, all pointing towards the center. Each camera has a resolution of $6 4 0 \\times 4 8 0$ and is noiseless. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Language-Conditioned Manipulation Tasks. We extend the Ravens benchmark [2] set in PyBullet [60] with 10 language-conditioned manipulation tasks. See Figure 1 for examples and Table 3 for challenges associated with each task. Each task instance is constructed by sampling a set of objects and attributes: poses, colors, sizes, and object categories. 8 of the 10 tasks have two variants, denoted by seen and unseen, depending on whether the task has unseen attributes (e.g. color) at test time. For colors: $\\mathbb { T } _ { \\mathrm { s e e n c o l o r s } } = \\{ \\mathtt { y e l l o w }$ , brown, gray, cyan} and $\\mathbb { T } _ { \\mathrm { u n s e e n c o l o r s } } =$ {orange, purple, pink, white $\\}$ with 3 overlapping colors $\\mathbb { T } _ { \\mathrm { a l l } } ~ = ~ \\{ \\mathrm { r e d }$ , green, blue} used in both the seen and unseen spilts. For packing objects, we use 56 tabletop objects from the Google Scanned Objects dataset [61] and split them into 37 seen and 19 unseen objects. The language instructions are constructed from templates for simulated experiments, and human-annotated for real-world experiments. For more details about individual tasks, see Appendix A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evaluation Metric. We adopt the 0 (fail) to 100 (success) scores proposed in the Ravens benchmark [2]. The score assigns partial credit based on the task, e.g. $3 / 5 \\Rightarrow 6 0 . 0$ for packing 3 out of 5 objects specified in the instructions, or $3 0 / 5 6 \\Rightarrow 5 3 . 6$ for pushing 30 out of 56 particles into the correct zone. See Appendix A for the specific evaluation metric used in each task. During an evaluation episode, an agent keeps interacting with the scene until an oracle indicates task-completion. We report scores on 100 evaluation runs for agents trained with $n = 1 , 1 0$ , 100, 1000 demonstrations. ",
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+    {
+        "type": "text",
+        "text": "4.2 Simulation Results ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Table 1 presents results from our large-scale experiments in Ravens [2] and Figure 3 summarizes these results with average scores across seen and unseen splits. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/2c8d085ed4e3b8280f7fea75e44ca5edb6cbae04f83dfd2885faf862f02ac008.jpg",
+        "image_caption": [
+            "Figure 3. Average scores across seen and unseen splits for all tasks in Table 1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Baseline Methods. To study ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "the effectiveness of our two-stream architecture, we broadly compare against two baselines: Transporter-only and CLIP-only. Transporter-only is the original Transporter [2], or equivalently, the spatial stream of CLIPORT with RGB-D input. Although Transporter-only does not receive any language goals, it shows what can be achieved through chance by exploiting the most likely actions seen during training. On the other hand, CLIP-only is just the semantic stream of CLIPORT with RGB and language input. CLIP-only shows what can be achieved by fine-tuning a pre-trained semantic model for manipulation without spatial information, particularly depth. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Two-Stream Performance. Figure 3 (seen) captures the essence of our main claims. The performance of Transporter-only saturates at $5 0 \\%$ since it doesn’t use the language instruction to ground the desired goal. CLIP-only does have a goal, but lacks the spatial precision to go the last mile and thus saturates at $7 6 \\%$ . Only CLIPORT (single) achieves more than $9 0 \\%$ , which indicates that both the semantic and spatial streams are crucial for fine-grained manipulation. Further, CLIPORT (single) achieves $8 6 \\%$ on most tasks with just 100 demonstrations, showcasing its efficiency. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition to these baselines, we present various ablations and alternative one-stream and twostream models in Appendix F. To briefly summarize these results, CLIP is essential for few-shot ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e9189627928d3453b6e6a9b4c096c917ef1b617bf4768eccb992e241a5c360c8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">packing-box-pairs seen-colors</td><td colspan=\"4\">packing-box-pairs unseen-colors</td><td colspan=\"4\">packing-seen-google objects-seq</td><td colspan=\"4\">packing-unseen-google objects-seq</td><td colspan=\"4\">packing-seen-google objects-group</td><td colspan=\"4\">packing-unseen-google objects-group</td></tr><tr><td>Method</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1 10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td></tr><tr><td>Transporter-only [2]</td><td>44.2 55.2 54.2</td><td></td><td></td><td></td><td></td><td></td><td>54.1</td><td>26.2 39.7</td><td></td><td>45.4</td><td></td><td></td><td>19.9 29.8</td><td>28.7</td><td>37.3</td><td></td><td></td><td></td><td>59.9</td><td></td><td></td><td>46.2 54.7 49.8</td><td></td></tr><tr><td>CLIP-only</td><td></td><td></td><td></td><td>52.4</td><td>34.648.747.2</td><td></td><td></td><td></td><td></td><td></td><td>46.3</td><td></td><td></td><td></td><td></td><td></td><td>60.0 54.3 61.5</td><td></td><td></td><td></td><td></td><td></td><td>：52.0</td></tr><tr><td>RN50-BERT</td><td></td><td></td><td></td><td></td><td>38.6 69.7 88.5 87.1 33.0 65.5 68.8</td><td></td><td>61.2</td><td></td><td>29.1 67.9</td><td>）89.3</td><td>95.8</td><td></td><td></td><td></td><td>37.1 49.4 60.4 57.8</td><td></td><td>52.5 62.0 89.6 92.7</td><td></td><td></td><td></td><td></td><td></td><td>43.465.9 73.1 70.0</td></tr><tr><td>CLIPORT (single)</td><td></td><td></td><td></td><td></td><td>36.2 64.0 94.7 90.3 31.4 52.7 65.6</td><td></td><td>72.1</td><td></td><td>32.948.4 87.9 94.0</td><td></td><td></td><td></td><td></td><td></td><td>）29.348.548.356.1 71.9</td><td>52.7</td><td></td><td></td><td></td><td></td><td></td><td></td><td>46.452.9 76.586.4 43.2 52.0 66.3 73.7</td></tr><tr><td>CLIPORT (multi)</td><td colspan=\"4\">51.6 82.9 92.7 66.888.694.196.659.0 69.776.271.441.678.485.084.4 40.751.165.870.371.384.689.68.368.4 69.678.480.3</td><td colspan=\"4\">98.2 45.6 65.3 68.6 71.5</td><td colspan=\"4\"></td><td colspan=\"4\">14.8 59.5 86.8 96.2 27.2 50.0 65.5</td><td colspan=\"4\">67.0 84.1</td><td colspan=\"4\">94.061.5 66.2 78.4 81.5</td></tr><tr><td rowspan=\"4\"> CLIPORT (multi-attr)</td><td colspan=\"4\"></td><td colspan=\"4\">46.2 72.0 86.2 80.3</td><td colspan=\"4\"></td><td colspan=\"4\">35.4 45.1 78.9 87.4</td><td colspan=\"4\"></td><td colspan=\"4\">48.6 69.3 84.8 89.1</td></tr><tr><td></td><td>stack-block-pyramid seq-seen-colors</td><td></td><td></td><td>stack-block-pyramid</td><td>seq-unseen-colors</td><td></td><td></td><td>separating-piles seen-colors</td><td></td><td></td><td></td><td>separating-piles unseen-colors</td><td></td><td></td><td></td><td>towers-of-hanoi</td><td></td><td></td><td></td><td></td><td>towers-of-hanoi</td><td></td></tr><tr><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000 1</td><td>10</td><td>100</td><td>1000</td><td></td><td>10</td><td></td><td>1000</td><td>1</td><td>seq-seen-colors 10</td><td></td><td></td><td></td><td></td><td></td><td>seq-unseen-colors</td></tr><tr><td></td><td>4.5</td><td>2.3</td><td>5.2 4.5</td><td>3.0</td><td>4.0</td><td>2.3</td><td>5.8</td><td>42.7</td><td>52.3</td><td>42.0</td><td>48.4</td><td>1 41.2 49.2</td><td></td><td>100 44.7</td><td>52.3</td><td>67.9</td><td>98.0</td><td>100 1000 99.9</td><td></td><td>1</td><td>10 100 24.344.671.7 80.7</td><td>1000</td></tr><tr><td>Transporter-only [2] CLIP-only</td><td>6.328.7 55.7 54.8</td><td></td><td></td><td></td><td>2.012.2</td><td>18.3</td><td></td><td>19.543.5</td><td>55.0</td><td>84.9 90.25</td><td></td><td>59.949.6</td><td></td><td>73.0</td><td>71.0</td><td>25.4 9.4</td><td>52.6 88.645.3</td><td></td><td></td><td></td><td></td><td></td><td>24.747.067.0 58.0</td></tr><tr><td>RN50-BERT CLIPORT (single)</td><td>5.335.0 89.097.5 28.364.7 93.3 98.8</td><td></td><td></td><td></td><td>6.2 12.2 13.7 24.3</td><td>21.5 31.2</td><td>30.7 41.3</td><td>31.8 47.8 54.5 59.5</td><td></td><td>93.1</td><td>98.0</td><td>）47.2</td><td>46.546.533.444.4 51.0</td><td>41.3 76.6</td><td>44.9 75.2</td><td>59.4</td><td>28.0 66.1 92.9</td><td>91.392.1 97.4100</td><td></td><td>56.1</td><td></td><td></td><td>17.4 75.1 85.3 89.3 89.7 95.9 99.4</td></tr><tr><td>CLIPORT (multi)</td><td>33.5 75.3 96.8 96.5 23.3 26.831.7 22.2 48.9 72.4 90.3 89.0 56.6 62.6 64.9 62.8</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>49.9 51.8 48.2 59.8</td><td></td><td></td><td></td><td></td><td>61.6 96.3 98.7 98.1</td><td></td><td></td><td></td><td></td><td></td><td>60.1 65.6 76.7 68.7</td></tr><tr><td> CLIPORT (multi-attr)</td><td>一</td><td></td><td></td><td></td><td></td><td></td><td>15.5 51.5 59.3 79.8</td><td>assembling-kits-seq</td><td></td><td></td><td></td><td>assembling-kits-seq</td><td></td><td></td><td></td><td>put-blocks-in-bowls</td><td></td><td></td><td></td><td></td><td></td><td></td><td>56.7 78.0 88.3 96.9 put-blocks-in-bowls</td></tr><tr><td></td><td></td><td>align-rope</td><td></td><td></td><td> packing-unseen-shapes</td><td></td><td></td><td></td><td>seen-colors</td><td></td><td></td><td></td><td></td><td>unseen-colors</td><td></td><td></td><td>seen-colors</td><td></td><td></td><td></td><td></td><td>unseen-colors</td><td></td></tr><tr><td>Transporter-only [2]</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10 100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td><td>1</td><td>10</td><td>100</td><td>1000</td></tr><tr><td>CLIP-only</td><td>6.930.633.1</td><td></td><td></td><td>51.5</td><td></td><td>16.0 20.0 22.0</td><td>22.0</td><td>5.8</td><td>11.6</td><td></td><td>28.6 29.6</td><td>7.8</td><td>17.6</td><td>25.6</td><td>28.4</td><td>16.8 333.3 62.7</td><td></td><td></td><td>64.7</td><td>11.7</td><td></td><td>17.2 14.8</td><td>18.7 11.2 34.2 33.2 44.5</td></tr><tr><td>RN50-BERT CLIPORT (single)</td><td></td><td>3.125.063.8</td><td>13.4 48.7 70.4</td><td>70.7 57.1</td><td>19.0 25.0</td><td>13.0 28.0 44.0 32.0</td><td>50.0 44.0</td><td>0.8 2.2</td><td>9.2 5.6</td><td>19.8 11.6</td><td>23.0 21.8</td><td>2.0 1.6</td><td>4.6 6.4</td><td>10.8 10.4</td><td>19.8 18.4</td><td>23.5 60.2 93.5 13.8 44.581.2</td><td></td><td>91.8</td><td>97.7</td><td></td><td></td><td></td><td>223.030.323.8</td></tr><tr></table>",
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+        "text": "Table 1. Language-Conditioned Test Results. Task success scores (mean $\\%$ ) from 100 evaluation instances vs. # of training demonstrations (1, 10, 100, or 1000). The challenges pertaining to each task are described in Appendix A. CLIPORT (single) models are trained on seen splits, and evaluated on both seen and unseen splits. CLIPORT (multi) models are trained on seen splits of all 10 tasks with 1T, 10T, 100T, and 1000T demonstrations where $\\mathbb { T } = 1 0$ . CLIPORT (multi-attr) indicate CLIPORT (multi) models trained on seen-and-unseen splits from all tasks except for that one particular heldout task, for which it is trained only the seen split. See Figure 3 for an overview with average scores. ",
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+        "text": "learning (i.e. $n \\geq 1 0 .$ ) in lieu of semantic stream alternatives like ImageNet-trained ResNet50 [62] with BERT [38]. Image-goal models outperform CLIPORT (single) in packing Google objects, but this is only because they do not have to solve the language-grounding problem. ",
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+        "text": "Multi-Task Performance. In realistic scenarios, we want the robot to be capable of any task, not just one task. We investigate this through CLIPORT (multi) in Table 1 with one multi-task model trained on all 10 tasks. CLIPORT (multi) models are trained only on seen-splits of tasks, so an unseen attribute like ‘pink’ is consistent throughout single and multi-task settings. Surprisingly, CLIPORT (multi) outperforms single-task CLIPORT (single) models in $4 1 / 7 2 = 5 7 \\%$ of the evaluations in Table 1. This trend is also evident in Figure 3 (seen), especially in instances with 100 demonstrations or less. Although CLIPORT (multi) is trained on more diverse data from other tasks, both CLIPORT (multi) and CLIPORT (single) have access to the same amount of data per task. This supports our premise that language is a strong conditioning mechanism for reusing concepts from other tasks without learning them from scratch. It also validates a trait of data-driven approaches where training on lots of diverse data leads to more robust and generalizable representations [1, 63]. However, CLIPORT (multi) performs worse on longer-horizon tasks like align-rope. We hypothesize that this is because longer-horizon tasks get less coverage of input-action pairs in the dataset. Future works could use better sampling methods that balance tasks according to their average time horizon. ",
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+        "text": "Generalizing to Unseen Attributes. Tasks that require generalizing to novel colors, shapes, and objects are more difficult and all our agents achieve relatively lower performance on these tasks, as shown in Figure 3 (unseen). However, CLIPORT (single) models do substantially better than chance, i.e., Transporter-only. The lower performances are due to the difficulty of grounding unseen attributes such as ‘pink’ and ‘orange’ in the language instruction “put the pink block on the orange bowl”, when the agent has never encountered words ‘orange’, ‘pink’ or their corresponding visual characteristics in the context of the physical environment. Although pre-trained CLIP has been exposed to the attribute ‘pink’, it could correspond to different concepts in the physical setting depending on factors like lighting condition, and thus requires at least few examples to condition the trainable semantic decoder layers. Additionally, we notice that CLIPORT (single) is also less prone to overfitting compared to Transporter-only. As evidenced in towers-of-hanoi-seq-unseen-colors task in Table 1, Transporter-only suffers from a performance drop because of rings with unseen colors despite the fact that Tower of Hanoi can be solved without attending to the colors and simply focusing on the ring size. We hypothesize that since CLIP was trained on diverse internet data, it enables our agent to focus on task-relevant concepts while ignoring irrelevant aspects of the task. ",
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+        "image_caption": [
+            "Figure 4. Affordance predictions from CLIPORT (multi) models in sim (left two) and real settings (right three). More examples in Appendix H. "
+        ],
+        "image_footnote": [],
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+        "text": "Transferring Attributes across Tasks. One solution for dealing with unseen attributes is to explicitly learn these attributes from other tasks. We study this with CLIPORT (multi-attr) in Table 1 and Figure 3 (unseen). For these models, CLIPORT (multi) is trained on both seen-and-unseen splits from all tasks except for the task being evaluated on, for which it was only trained on the seen split. As such, this evaluation measures whether having seen pink blocks in put-blocks-in-bowl-unseen-colors helps solve “pack all the pink and cyan boxes” in packing-box-pairs-unseen-colors. Results indicate that such explicit transfers result in significant improvements. For instance, on the put-blocks-inbowls-unseen-colors task for $n = 1 0 0 0$ , CLIPORT (multi)’s performance increases from 45.8 to 75.7. ",
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+    {
+        "type": "text",
+        "text": "4.3 Real-Robot Experiments ",
+        "text_level": 1,
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+        "text": "We validated our results in hardware with a Franka Panda manipulator. See Appendix D for setup details. Table 2 reports success rates for a multi-task model trained and evaluated on 9 real-world tasks. Due to COVID restrictions, we could not conduct largescale user-studies, so we report on small train (5-10 demos) and test sets (5-10 runs) per task. Overall, CLIPORT (multi) is effective at few-shot learning with just 179 samples, and the performances roughly correspond to those in simulated experiments, with simple block manipulation tasks achieving $\\sim 7 0 \\%$ . We estimate that for more robust real-world performance at least 50 to 100 training demonstrations are necessary, as evident in Figure 3. Interestingly, we observed that the model sometimes exploits biases in the training data instead of learning to ground instructions. For instance, in Put Blocks in Bowl, the training set consisted of only one datapoint on “yellow blocks” being placed inside a “blue bowl”. This made it difficult to condition the model to place “yellow blocks” in non-blue bowls. But instances with just one or two examples where a colored block went to different colored bowls was sufficient to make the model pay attention to the language. In summary, unbiased datasets containing both a good coverage of expected skills and invariances, and a decent number of training demonstrations, are crucial for good real-world performance. ",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 2. Success rates $( \\% )$ of a multi-task model trained an evaluated 9 real-world tasks (see Figure 1). Samples indicate total image-action pairs, e.g 1 in Figure 9. "
+        ],
+        "table_body": "<table><tr><td>Task # Train (Samples) #Test Succ.%</td></tr><tr><td>Stack Blocks</td></tr><tr><td>5 (13) 10 70.0 5(10) 10 65.0</td></tr><tr><td>Put Blocks in Bowl Pack Objects 10 (31) 10 60.0</td></tr><tr><td>Move Rook 4(29) 10 70.0</td></tr><tr><td>Fold Cloth 9(9) 10 57.0</td></tr><tr><td>Read Text 2(26) 10 55.0</td></tr><tr><td>Loop Rope 4(12) 10 60.0</td></tr><tr><td>Sweep Beans 5 (23) 5 60.6</td></tr><tr><td>Pick Cherries 4(26) 5 75.0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We introduced CLIPORT, an end-to-end framework for language-conditioned fine-grained manipulation. Our experiments, specifically with multi-task models, indicate that data-driven approaches to generalization have yet to be fully-exploited in robotics. Coupled with the right action abstraction and spatio-semantic priors, end-to-end methods can quickly learn new skills without requiring top-down pipelines that need task-specific engineering. ",
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+        "type": "text",
+        "text": "While CLIPORT can solve a range of tabletop tasks, extending it to dexterous 6-DOF manipulation that goes beyond the two-step primitive remains a challenge. As such, it cannot handle complex partially-observable scenes, or output continuous control for multi-fingered hands, or predict task-completion (see Appendix I for an extended discussion). But overall, we are excited by the confluence of data and structural priors for building scalable and generalizable robotic systems. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "All simulated experiments were facilitated through the Hyak computing cluster funded by the STF at the University of Washington. We thank Mohak Bhardwaj for help with the Franka setup at UW. We are also grateful to our colleagues Chris Xie, Jesse Thomason, and Valts Blukis for providing feedback on the initial draft. This work was funded in part by ONR under award #1140209-405780. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing (EMNLP), 2018.   \n[59] G. Goh, N. C. †, C. V. †, S. Carter, M. Petrov, L. Schubert, A. Radford, and C. Olah. Multimodal neurons in artificial neural networks. Distill, 2021. doi:10.23915/distill.00030. https://distill.pub/2021/multimodal-neurons.   \n[60] E. Coumans and Y. Bai. Pybullet, a python module for physics simulation for games, robotics and machine learning. 2016.   \n[61] Google scanned objects dataset, 2020. URL https://app.ignitionrobotics.org/ GoogleResearch/fuel/collections/Google%20Scanned%20Objects.   \n[62] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In The IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 770–778, 2016.   \n[63] J. Lu, V. Goswami, M. Rohrbach, D. Parikh, and S. Lee. 12-in-1: Multi-task vision and language representation learning. In The IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.   \n[64] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al. Pytorch: An imperative style, high-performance deep learning library. arXiv preprint arXiv:1912.01703, 2019.   \n[65] V. Sanh, L. Debut, J. Chaumond, and T. Wolf. Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter. arXiv preprint arXiv:1910.01108, 2019.   \n[66] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.   \n[67] J. Mao, C. Gan, P. Kohli, J. B. Tenenbaum, and J. Wu. The neuro-symbolic concept learner: Interpreting scenes, words, and sentences from natural supervision. arXiv preprint arXiv:1904.12584, 2019.   \n[68] D. Ding, F. Hill, A. Santoro, and M. Botvinick. Object-based attention for spatio-temporal reasoning: Outperforming neuro-symbolic models with flexible distributed architectures. arXiv preprint arXiv:2012.08508, 2020.   \n[69] E. M. Bender, T. Gebru, A. McMillan-Major, and S. Shmitchell. On the dangers of stochastic parrots: Can language models be too big? In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, pages 610–623, 2021. ",
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parse/train/BJlgt2EYwr/BJlgt2EYwr.md

@@ -0,0 +1,266 @@
+# STABILIZING DARTS WITH AMENDED GRADIENT ESTIMATION ON ARCHITECTURAL PARAMETERS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Differentiable neural architecture search has been a popular methodology of exploring architectures for deep learning. Despite the great advantage of search efficiency, it often suffers weak stability, which hinders it from being applied to a large search space or being flexibly adjusted to different scenarios. This paper investigates DARTS, the currently most popular differentiable search algorithm, and points out an important factor of instability, which lies in its approximation on the gradients of architectural parameters. In the current status, the optimization algorithm can converge to another point which results in dramatic inaccuracy in the re-training process. Based on this analysis, we propose an amending term for computing architectural gradients by making use of a direct property of the optimality of network parameter optimization. Our approach mathematically guarantees that gradient estimation follows a roughly correct direction, which leads the search stage to converge on reasonable architectures. In practice, our algorithm is easily implemented and added to DARTS-based approaches efficiently. Experiments on CIFAR and ImageNet demonstrate that our approach enjoys accuracy gain and, more importantly, enables DARTS-based approaches to explore much larger search spaces that have not been studied before.
+
+# 1 INTRODUCTION
+
+Neural architecture search (NAS) has been an important topic in the research area of automated machine learning (AutoML). The idea is to replace the manual way of designing neural network architectures with an automatic algorithm, by which deep learning methods become more flexible in fitting complex data distributions, e.g., large-scale image datasets. Early efforts of NAS involved using heuristic search methods such as reinforcement learning (Zoph & Le, 2017; Zoph et al., 2018) and evolutionary algorithms (Real et al., 2017; Xie & Yuille, 2017) to sample networks from a large search space, and optimizing each sampled network individually to evaluate its quality. Despite notable successes obtained by this methodology, it often requires a vast amount of computation, which obstacles its applications in the scenarios of limited resources. Inspired by the idea of reusing and sharing parameters among trained networks, DARTS (Liu et al., 2019b) was designed as a ‘one-shot’ solution of NAS. The major difference from the aforementioned methods is a differentiable formulation of architecture search and an end-to-end mechanism which optimizes model weights (such as convolution) and architectural weights simultaneously. Recently, improvements upon DARTS were made in various aspects (Chen et al., 2019; Xu et al., 2019; Liang et al., 2019), making it a reasonable tradeoff between search cost and performance.
+
+Despite its broad applications, the current pipeline of DARTS (or, generally speaking, differentiable NAS approaches) suffers a critical weakness known as instability. Researchers reported (Liang et al., 2019) that DARTS-based algorithms can sometimes generate weird architectures that produce considerably worse accuracy than those generated in other individual runs, or even significantly worse than randomly generated architectures. There indeed exist tricks designed by human expertise (Chen et al., 2019; Nayman et al., 2019; Liang et al., 2019) to alleviate this issue, but we point out that these approaches violated the ideology of NAS, which is to maximally prevent human interventions. Moreover, even with such add-ons, a dramatic property of DARTS persists and has not been studied carefully in prior work. When DARTS, as well as its variants, gets trained for a longer time, e.g., from the default number of 50 epochs to 200 epochs, we surprisingly observe that all these approaches converge to very similar architectures, in which almost all edges are occupied by skip-connect (a.k.a., identity). These architectures, with fewer trainable parameters, are often far from producing high accuracy in particular on large datasets like ImageNet, but they somehow produce sufficiently high validation accuracy in the search stage. In other words, convergence in search often leads to bad performance in re-training. This is why some previous DARTS-based approaches advocated for early termination (Liang et al., 2019), a practical but non-essential solution. Also, we conjecture that early termination also contributes to the lack of stability and, more importantly, trustfulness, of DARTS-based approaches.
+
+This paper delves deep into the inconsistency between convergence and performance. We show that the devil lies in optimizing the loss function of the super-network, $\mathcal { L } _ { \mathrm { v a l } } ( \bar { \omega } ^ { \star } ( \alpha ) , \alpha )$ ( $\omega$ and $_ { \pmb { \alpha } }$ denote network and architectural parameters, respectively, and $\omega ^ { \star } ( \alpha )$ is the global optimum of $\omega$ given $_ \alpha$ ), in which ω and α get updated alternately. Following the chain rule, ∇αLval(ω?(α) , α)|α=αt equals to $\nabla _ { \alpha } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } + \left. \nabla _ { \alpha } \omega ^ { \star } ( \alpha ) \right| _ { \alpha = \alpha _ { t } }$ · $\nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } }$ , in which the first term is easy to compute while the second term is not, mainly because $\omega ^ { \star } ( \alpha )$ is difficult to estimate, and so is the term of ∇αω?(α)|α=αt . DARTS-based approaches performed inaccurate approximation for this purpose, in which the first-order version of DARTS directly discarded the second term – but this term is often numerically significant, and the second-order version of DARTS applied an approximation to this term which is not mathematically guaranteed (see Section 3.4). Consequently, the accuracy of $\nabla _ { \alpha } \mathcal { L } _ { \mathrm { v a l } } \big ( \omega ^ { \star } ( \alpha ) , \alpha \big ) | _ { \alpha = \alpha _ { t } }$ cannot be guaranteed, and hence the update of $_ { \pmb { \alpha } }$ can be problematic. To the best of our knowledge, this issue is not studied by existing DARTS-based approaches.
+
+To deal with this problem, we propose an alternative way of computing $\nabla _ { \alpha } \omega ^ { \star } ( \alpha ) | _ { \alpha = \alpha _ { t } }$ . We make use of an important property, i.e., $\nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha ^ { \dagger } ) , \alpha = \alpha ^ { \dagger } } \ \equiv \ \mathbf { 0 }$ holds for any $\alpha ^ { \dagger }$ , which directly comes from the optimality of $\omega ^ { \star } ( \alpha )$ . Differentiating both sides with respect to $\alpha ^ { \dagger }$ , we obtain a new equality which enables computing derives $\nabla _ { \alpha } \omega ^ { \star } ( \alpha ) | _ { \alpha = \alpha _ { t } }$ with the inverse of the Hesse matrix, $\nabla _ { \omega } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) \big | _ { \omega = \omega ^ { \star } ( \alpha ) , \alpha = \alpha _ { t } }$ . This idea enables us to achieve a more accurate approximation on $\nabla _ { \alpha } \mathcal { L } _ { \mathrm { v a l } } \big ( \omega ^ { \star } ( \alpha ) , \alpha \big ) | _ { \alpha = \alpha _ { t } }$ when $\omega ^ { \star } ( \alpha )$ is not available. Mathematically, we prove that when we have $\omega ^ { \mathrm { e s t } } \approx \omega ^ { \star } ( \alpha _ { t } )$ , the inner angle between the second term and our approximate term is smaller than 90 degrees. Note that this property does not hold in existing DARTS-based algorithms. Our final solution involves using the amended second term of ∇αLval(ω?(α) , α)|α=αt meanwhile keeping the first term unchanged, which goes one step further in optimizing the supernetwork, which reflects in a higher validation accuracy in the search stage.
+
+Our approach is very easily implemented. The overall computational overhead is comparable to the second-order version of DARTS. Experiments are performed on image classification, with popular datasets including CIFAR and ImageNet being used. In all experiments, we allow the search stage to come to a complete convergence and report competitive accuracy among current state-of-thearts. The stability of our approach also enables us to close the gap between hyper-parameters of search and evaluation, as well as explore more complex search spaces, which are believed to be correct directions of NAS but existing DARTS-based approaches would mostly fail. Therefore, we believe our algorithm can expand the application scenario of differentiable NAS methods in particular DARTS-based approaches.
+
+The remainder of this paper is organized as follows. We briefly review related work in Section 2, and illustrate our approach of amending architectural gradients in Section 3. After experiments are shown in Section 4, we conclude this work in Section 5.
+
+# 2 RELATED WORK
+
+With the era of big data and powerful computational resources, deep learning (LeCun et al., 2015), in particular, deep neural networks (Krizhevsky et al., 2012), have rapidly grown up to be the standard tool for learning representations in a complicated feature space. Recent years have witnessed the trend of using deeper (He et al., 2016) and denser (Huang et al., 2017) networks to boost recognition performance, while there is no justification that whether these manually designed architectures are best for each specific task, e.g., image classification. To advance, researchers started considering the possibility of learning network architectures automatically from data, which led to the appearance of neural architecture search (NAS) (Zoph & Le, 2017), which is now popular and known as a sub research field in automated machine learning (AutoML).
+
+The common pipeline of NAS starts with a pre-defined space of network operators. Since the search space is often large (e.g., containing $1 0 ^ { 1 0 }$ or even more possible architectures), it is unlikely that exhaustive search is tractable, and thus heuristic search methods are widely applied for speedup. Typical examples include reinforcement learning (Zoph & Le, 2017; Zoph et al., 2018; Liu et al., 2018a) and evolutionary algorithms (Real et al., 2017; Xie & Yuille, 2017; Real et al., 2019). These approaches followed a general pipeline that samples a set of architectures from a learnable distribution, evaluates them and learns from rewards by updating the distribution. In an early age, each sampled architecture underwent an individual training process from scratch and thus the overall computational overhead is large, e.g., hundreds of even thousands of GPU-days. To alleviate the burden, researchers started to share computation among training sampled architectures, with the key lying in reusing network weights trained previously (Cai et al., 2018) or starting from a well-trained super-network (Pham et al., 2018). These efforts shed light on the so-called one-shot architecture search methods, which required training the super-network only once and thus ran more efficiently, e.g., two or three orders of magnitude faster than conventional approaches.
+
+Within the scope of one-shot architecture search, an elegant solution lies in jointly formulating architecture search and approximation, so that it is possible to apply end-to-end optimization for training network and architectural parameters simultaneously. This methodology is known today as differentiable NAS, and a typical example is DARTS (Liu et al., 2019b), which constructed a super-network with all possible operators contained and decoupled, and the goal is to determine the weights of these architectural parameters, followed by pruning and re-training stages. This kind of approach allowed more flexible search space to be constructed, unlike conventional approaches with either reinforcement or evolutionary learning, which suffer from the computational burden and thus must constrain search within a relatively small search space (Tan & Le, 2019).
+
+Despite the inspirations brought by differentiable NAS, these approaches still suffer a few critical issues that narrow down their applications in practice. One significant drawback lies in the lack of stability, which reflects in the way that results of differentiable search can be impacted by very small perturbations, e.g., initialization of architectural weights, training hyper-parameters, and even randomness in the training process. Existing solutions include running search for several individual times and choosing the best one in validation (Liu et al., 2019b), or using other kinds of techniques such as decoupling modules (Cai et al., 2019; Guo et al., 2019), adjusting search space during optimization (Noy et al., 2019; Chen et al., 2019; Nayman et al., 2019), regularization $\mathrm { { X u } }$ et al., 2019), early termination (Liang et al., 2019), etc., however, these approaches seemed to develop heuristic remedies rather than analyze it from the mathematical fundamentals, e.g., how instability happens in mathematics.
+
+In this paper, we investigate the stability issue in mathematics and show that the results produced by the current approaches are much less reliable than people used to think. Then, we fix this issue by amending optimization of the architectural parameters, so that each step of the update gets closer to the correct direction. We show great improvement on stability in a fundamental task, image classification, while we believe our approach can be applied to a wide range of tasks including object detection (Ghiasi et al., 2019), semantic segmentation (Liu et al., 2019a), hyper-parameter learning (Cubuk et al., 2019), etc.
+
+# 3 STABILIZING DARTS WITH AMENDED GRADIENTS
+
+In this section, we first show that DARTS can fail dramatically when it gets trained till convergence, and then we mathematically analyze how this problem is related to inaccurate approximation in optimization, following which we present our solution to amend this error and thus stabilize DARTS.
+
+# 3.1 PRELIMINARIES: DIFFERENTIABLE NEURAL ARCHITECTURE SEARCH
+
+Differentiable NAS approaches start with defining a super-network, which is constrained in a search space with a pre-defined number of layers and a limited set of neural operators. The core idea is to introduce a ‘soft’ way operator selection (i.e., using a weighted sum over the outputs of a few operators instead of taking the output of only one), so that optimization can be done in an end-toend manner. Mathematically, the super-network is a function $\mathbf { f } \left( \mathbf { x } ; \omega , \alpha \right)$ , with $\mathbf { x }$ being input, and parameterized by network parameters $\omega$ (e.g., convolutional kernels) and architectural parameters $_ { \pmb { \alpha } }$ (e.g., indicating the importance of each operator between each pair of layers). $\mathbf { f } \left( \mathbf { x } ; \omega , \alpha \right)$ is differentiable to both $\omega$ and $_ { \pmb { \alpha } }$ , so that gradient-based approaches can be applied for optimization.
+
+![](images/90b275c4f31ff6a8aab6e67e8f3f69367a7a1877308a6bb9227e7ab562343780.jpg)  
+Figure 1: Left: a typical search process of the first-order DARTS, in which 200 search epochs are used. Red, green and blue lines indicate the average weight of none, the ratio of dominant skipconnect operators (over 14 normal edges) and the re-training accuracy, with respect to the number of search epochs, respectively. Right: the normal cell obtained after 200 search epochs, in which all preserved operators are skip-connect. We executed both first-order and second-order DARTS for several times, and such failure consistently happens in each individual run.
+
+In the example of DARTS, $\mathbf { f } \left( \mathbf { x } ; \omega , \alpha \right)$ is composed of a few cells, each of which contains $N$ nodes, and there is a pre-defined set, $\mathcal { E }$ , denoting which pairs of nodes are connected. For each connected node pair $( i , j ) , i < j$ , node $j$ takes the output of node $i$ , $\mathbf { x } _ { i }$ , as a part of its input, and propagate it through a pre-defined operator set, $\mathcal { O }$ , with all outputs summed up:
+
+$$
+\mathbf { y } ^ { ( i , j ) } \left( \mathbf { x } _ { i } \right) = \sum _ { o \in \mathcal { O } } \frac { \exp ( \alpha _ { o } ^ { ( i , j ) } ) } { \sum _ { o ^ { \prime } \in \mathcal { O } } \exp ( \alpha _ { o } ^ { ( i , j ) } ) } \cdot o ( \mathbf { x } _ { i } ) .
+$$
+
+Here, a softmax term is computed by architectural weights to normalize outputs. Within each unit of the search process, $\omega$ and $_ { \pmb { \alpha } }$ get optimized alternately. After search, the operator $o$ with the maximal value of $\alpha _ { o } ^ { ( i , j ) }$ is preserved for each edge $( i , j )$ . All network parameters $\omega$ are discarded and the obtained architecture is re-trained from scratch1.
+
+# 3.2 DARTS FAILS: THE CONTRADICTORY BETWEEN CONVERGENCE AND PERFORMANCE
+
+Our research is motivated by an observation that DARTS, at the end of a regular training process with, say, 50 epochs (Liu et al., 2019b), has not yet arrived at or even got close to convergence, yet the weight of the none (a.k.a., zero) operator is consistently the largest on each edge of the normal cell. To verify this, we increase the length of each training stage by 4 times, i.e., from 50 to 200 epochs. Two weird phenomena are observed, both of which are shown in the left part of Figure 1. First, the weight of the none operator monotonically goes up – at 200 epochs, the weight has achieved 0.97 on each edge of the normal cells, however, this operator is not considered in the final architecture. Second, almost all preserved operators are skip-connect (a.k.a., identity), a parameter-free operator that contributes little to feature learning – and surprisingly, it occupies $3 0 \%$ to $7 0 \%$ of the weight remained by none. Such a network has much fewer parameters than a well-designed one, and thus it usually reports unsatisfying performance at the re-training stage. This indicates that, in the context of DARTS, there exists a contradictory between search convergence and re-training accuracy.
+
+We point out that this is a critical issue, which suggests that when using DARTS-based approaches, one does not hope the search process to achieve convergence as it implies bad performance. In other words, each ‘successful’ architecture comes from an early-terminated search process. Consequently, the initialization of parameters $\scriptstyle { \mathbf { \alpha } } _ { \alpha }$ and $\omega$ ), the hyper-parameters of search (e.g., learning rate) and the time of terminating search become important and thus need to be determined by experience (Liang et al., 2019). This weakens the stability as well as explainability of search and, more importantly, violates the fundamental ideology of AutoML, i.e., maximally reducing human interference and determining the best architecture by training data.
+
+# 3.3 DELVING DEEP INTO MATHEMATICS: PROBLEM AND SOLUTION
+
+We point out that the reason lies in inaccurate estimation of the gradient with respect to $_ { \pmb { \alpha } }$ , namely, $\nabla _ { \alpha } \bar { \mathcal { L } } _ { \mathrm { v a l } } \big ( \omega ^ { \star } ( \alpha ) , \alpha \big ) | _ { \alpha = \alpha _ { t } }$ . Following the chain rule of gradients, this quantity equals to
+
+$$
+\nabla _ { \alpha } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } + \nabla _ { \alpha } \omega ^ { \star } ( \alpha ) | _ { \alpha = \alpha _ { t } } \cdot \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } ,
+$$
+
+in which the first term is relatively easy to compute (as is done by the first-order version of DARTS), while the second term, in particular $\mathrm { \nabla } \nabla _ { \alpha } \omega ^ { \star } ( \bar { \alpha } ) | _ { \alpha = \alpha _ { t } }$ , is not. The first-order version of DARTS directly discarded this term, but it often has significant numerical values which are not negligible. The second-order version of DARTS indeed proposed an approximation to this term, but, as we shall see in the next subsection, can incur a large approximation error (the inner-product between the correct and estimated directions can be smaller than 0). Consequently, there can be a significant gap between the estimated and true values of $\nabla _ { \alpha } \mathcal { L } _ { \mathrm { v a l } } \big ( \omega ^ { \star } ( \alpha ) , \alpha \big ) | _ { \alpha = \alpha _ { t } }$ . Such inaccuracy accumulates with every update on $_ { \pmb { \alpha } }$ , and gradually causes $_ { \pmb { \alpha } }$ to converge to weird solutions that are far from optimum, e.g., the entire super-network is dominated by none and skip-connect operators. We name this phenomenon as the gradient trap during optimization.
+
+To estimate $\nabla _ { \alpha } \omega ^ { \star } ( \alpha ) | _ { \alpha = \alpha _ { t } }$ , we first make a reasonable assumption that $\nabla _ { \alpha } \omega ^ { \star } ( \alpha ) | _ { \alpha = \alpha _ { t } }$ has finite values. Then, we make use of an important property, i.e., $\nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha ^ { \dagger } ) , \alpha = \alpha ^ { \dagger } } \ \equiv \ \mathbf { 0 }$ holds for any $\alpha ^ { \dagger }$ . This is property directly comes from the optimality of $\omega ^ { \star } ( \alpha )$ , but it has never been used by existing approaches. Applying differentiation with respect to any $\alpha ^ { \dagger }$ to both sides of this equality, we have $\bar { \nabla _ { \alpha ^ { \dagger } } } \left( \nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \bar { \alpha } ) | _ { \omega = \omega ^ { \star } ( \alpha ^ { \dagger } ) , \alpha = \alpha ^ { \dagger } } \right) \equiv \mathbf { 0 }$ . When $ { \alpha } ^ { \dagger } =  { \alpha } _ { t }$ , it becomes:
+
+$$
+\nabla _ { \alpha ^ { \dagger } } \left. \left( \nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) \vert _ { \omega = \omega ^ { \star } ( \alpha ^ { \dagger } ) , \alpha = \alpha ^ { \dagger } } \right) \right. _ { \alpha ^ { \dagger } = \alpha _ { t } } = 0 .
+$$
+
+Again, applying the chain rule to the left-hand side gives:
+
+$$
+\nabla _ { \alpha , \omega } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) \big | _ { \omega = \omega ^ { * } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } + \nabla _ { \alpha } \omega ^ { \star } ( \alpha ) \big | _ { \alpha = \alpha _ { t } } \cdot \left. \nabla _ { \omega } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) \right| _ { \omega = \omega ^ { * } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } = \mathbf { 0 } ,
+$$
+
+where we use the notation $\nabla _ { { \boldsymbol { \alpha } } , \omega } ^ { 2 } ( \cdot ) ~ \equiv ~ \nabla _ { { \boldsymbol { \alpha } } } ( \nabla _ { \omega } ( \cdot ) )$ throughout the remaining part of this paper. Here, ∇2ωLtrain(ω, α)ω=ω?(α ),α=α is the Hesse matrix corresponding to the optimum $\omega ^ { \star } ( \alpha _ { t } )$ , which is symmetric and positive-definite, and thus invertible. This gives us an estimation that $\nabla _ { \alpha } \omega ^ { \star } ( \alpha ) | _ { \alpha = \alpha _ { t } } = - \left. \nabla _ { \alpha , \omega } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) \right| _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } \cdot \mathbf { H } ^ { - 1 } .$ . Substituting it into Equation 2 gives:
+
+$$
+\begin{array} { r l r } & { } & { \nabla _ { \alpha } \mathcal { L } _ { \mathrm { v a l } } ( \omega ^ { \star } ( \alpha ) , \alpha ) | _ { \alpha = \alpha _ { t } } = \nabla _ { \alpha } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } - } \\ & { } & { \nabla _ { \alpha , \omega } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } \cdot \mathbf { H } ^ { - 1 } \cdot \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } . } \end{array}
+$$
+
+Note that Equation 5 does not involve any approximation. The only issue comes from the term of ${ \bf H } ^ { - 1 }$ , which is computationally intractable due to the large dimensionality of $\mathbf { H }$ (it is related to the number of network parameters, which often exceeds one million in a typical super-network).
+
+# 3.4 APPROXIMATIONS IN COMPUTING THE INVERSE HESSE MATRIX
+
+Let us denote Equation 5 in an abbreviated form of $\mathbf { g } = \mathbf { g } _ { 1 } + \mathbf { g } _ { 2 }$ , in which $\mathbf { g } _ { 1 }$ , the first-order term of DARTS, is easily computed, while $\mathbf { g } _ { 2 }$ is not due to the computation of ${ \bf H } ^ { - 1 }$ . Here, we propose an alternative solution which constructs an approximation term $\mathbf { g } _ { 2 } ^ { \prime }$ :
+
+$$
+\begin{array} { r } { \mathbf { g } _ { 2 } ^ { \prime } = - \eta \cdot \left. \nabla _ { \alpha , \omega } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) \right| _ { \omega = \omega ^ { * } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } \cdot \mathbf { H } \cdot \left. \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) \right| _ { \omega = \omega ^ { * } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } , } \end{array}
+$$
+
+where $\eta > 0$ is named the amending coefficient. In what follows, we show that $\mathbf { g } _ { 2 } ^ { \prime }$ is indeed a reasonable approximation of $\mathbf { g } _ { 2 }$ . Since
+
+$$
+\begin{array} { r l } & { \langle \mathbf { g } _ { 2 } ^ { \prime } , \mathbf { g } _ { 2 } \rangle = \boldsymbol { \eta } \cdot \left. \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) \right| _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } \boldsymbol { \mathsf { T } } \cdot \left. \mathbf { H } ^ { - 1 } \cdot \nabla _ { \omega , \alpha } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) \right| _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } \cdot } \\ & { \qquad \nabla _ { \alpha , \omega } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \alpha ) \big | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } \cdot \mathbf { H } \cdot \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) \big | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } , } \end{array}
+$$
+
+the product of the two terms between ${ \bf H } ^ { - 1 }$ and $\mathbf { H }$ is a semi-positive-definite matrix, and so is the matrix after similarity transformation, which directly gives $\left. \mathbf { g } _ { 2 } ^ { \prime } , \mathbf { g } _ { 2 } \right. \geqslant 0$ .
+
+In summary, we decompose the gradient of architectural parameters into two terms, $\mathbf { g } _ { 1 }$ and $\mathbf { g } _ { 2 }$ , compute $\mathbf { g } _ { 1 }$ directly and use an approximation to $\mathbf { g } _ { 2 }$ so that the angle between the accurate and approximated terms is smaller than 90 degrees. In comparison, existing DARTS-based approaches either discarded $\mathbf { g } _ { 2 }$ entirely or used a mathematically non-explainable approximation $\mathbf { \bar { g } } _ { 2 } ^ { \prime \prime } = \mathbf { \bar { \nu } } \eta \cdot \nabla _ { \alpha , \omega } ^ { 2 } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega , \mathbf { \bar { \alpha } } ) | _ { \omega = \omega ^ { * } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } \cdot \mathbf { I } \cdot \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \mathbf { \bar { \alpha } } ) | _ { \omega = \omega ^ { * } ( \alpha _ { t } ) , \alpha = \alpha _ { t } } .$ . None of them are reasonable because $\mathbf { g } _ { 2 }$ can be large, yet there is no guarantee that $\left. \mathbf { g } _ { 2 } ^ { \prime \prime } , \mathbf { g } _ { 2 } \right. \geqslant 0$ , i.e., the secondorder DARTS can lead the algorithm to a wrong direction.
+
+The rationality of our approach is also verified by the validation process, i.e., in updating architectural parameters. The first-order DARTS, by directly discarding $\mathbf { g } _ { 2 }$ , reported an average validation accuracy of $9 0 . 5 \%$ in search space $S _ { 1 }$ (see Section 4.1.3) on the CIFAR10 dataset. The second-order DARTS added $\mathbf { g } _ { 2 } ^ { \prime \prime }$ , which has no guarantee that $\left. { { \bf { g } } _ { 2 } ^ { \prime \prime } , { \bf { g } } _ { 2 } } \right. \mathrm { ~ \ \ \geqslant ~ 0 ~ }$ , and thus resulted in a reduced validation accuracy. Our approach, by adding $\mathbf { g } _ { 2 } ^ { \prime }$ , achieves a validation accuracy of $9 1 . 5 \%$ , implying that our optimization works better than DARTS. This eventually results in the advantage of searched architectures, which will be verified in Section 4.1.2.
+
+The remainder part of computing $\mathbf { g } _ { 2 } ^ { \prime }$ simply follows conventions, which we replace $\omega ^ { \star } ( \alpha )$ with the current $\omega ^ { \mathrm { e s t } }$ as the most accurate approximation we can get2. Computing ${ \bf g } _ { 2 } ^ { \bar { \prime } }$ with Equation 6 requires both ∇2α,ωLtrain(ω, α)ω=ω?(αt),α=αt and $\nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) | _ { \omega = \omega ^ { \star } ( \alpha _ { t } ) , \alpha = \alpha _ { t } }$ , while DARTS needs the former one with $\omega ^ { \star } ( \alpha _ { t } )$ estimated in two steps. Therefore, computing Equation 6 requires similar computational overhead compared to the second-order version of DARTS. In experiments, each search epoch requires around 0.02 GPU-days on the standard 8-cell space on CIFAR10.
+
+# 3.5 DISCUSSIONS AND RELATIONSHIP TO PRIOR WORK
+
+The core benefit brought by our approach is the consistency between search and evaluation. This is indeed a fundamental idea of NAS, but it was ignored by existing approaches since they have been perplexed by a more significant error caused by inaccurate optimization. After the error, we point out a few prior conventions that need to be adjusted accordingly, including using different depths (e.g., DARTS used 8 cells in search and 20 cells in evaluation) and widths (e.g., DARTS used a basic channel number of 16 in search and 36 in evaluation) during search and evaluation, as well as using different training strategies (e.g., during re-training, a few regularization techniques including Cutout (DeVries & Taylor, 2017), Dropout (Srivastava et al., 2014) and auxiliary loss were used, but none of them appeared in search). More importantly, the search process was followed by edge removal (8 out of 14 connections were preserved) which caused a significant difference between the network architectures of search and evaluation. Our approach provides the opportunity to bridge the gap between search and evaluation, which we will show in Section 4.1.2 that unifying these hyper-parameters leads to better performance.
+
+A few prior differentiable search approaches noticed the issue of instability, but they chose to solve it in different manners. For example, P-DARTS (Chen et al., 2019) fixed the number of preserved skip-connect operators, PC-DARTS (Xu et al., 2019) used edge normalization to eliminate the none operator, while XNAS (Nayman et al., 2019) and DARTS $^ { + }$ (Liang et al., 2019) introduced a few human expertise to stabilize search. However, we point out that (i) either P-DARTS or PC-DARTS, with carefully designed methods or tricks, can also fail in a sufficiently long search process (more than 200 epochs); and that (ii) XNAS and DART $\vdots +$ , by adding human expertise, somewhat violated the design nature of AutoML, in which one is expected to avoid introducing too many hand-designed rules.
+
+Another line of NAS, besides differentiable methods, is to use either reinforcement learning or an evolutionary algorithm as a controller of heuristic search and train each sampled network to get some kind of rewards, e.g., validation accuracy. In the viewpoint of optimization, this pipeline mainly differs from the differentiable one in that optimizing $_ { \pmb { \alpha } }$ is decoupled from optimizing $\omega$ , so that it does not require $\omega$ to arrive at $\omega ^ { \ast }$ , but only need a reasonable approximation of $\omega ^ { \ast }$ to predict model performance – this is an important reason that such algorithms often produce stable results. Our approach sheds light on introducing a similar property, i.e., robustness to approximated $\omega ^ { \star }$ , which helps in stabilizing differentiable search approaches.
+
+# 4 EXPERIMENTS
+
+# 4.1 RESULTS ON CIFAR10
+
+The CIFAR10 dataset (Krizhevsky & Hinton, 2009) has 50,000 training and 10,000 testing images, sized $3 2 \times 3 2$ , and equally distributed over 10 classes. We mainly use this dataset to evaluate the stability of our approach, as well as analyze the impacts of different search options and parameters.
+
+# 4.1.1 IMPACT OF THE AMENDING COEFFICIENT
+
+We first investigate how the amending coefficient, $\eta$ , defined in Equation 6, impacts architecture search. We search and re-train similarly as DARTS. During the search, all operators are assigned equal weights on each edge. We use a base channel number of 16, and a batch size of 96. An Adam optimizer is used to update architectural parameters, with a learning rate of 0.0003, a weight decay of 0.001 and a momentum of (0.5, 0.999). The number of epochs is to be discussed later. During re-training, the base channel number is increased to 36. An SGD optimizer is used with an initial learning rate of 0.025, decaying following the cosine annealing rule and arriving at 0 after 600 epochs. The weight decay is set to be 0.0003, and the momentum is 0.9.
+
+To arrive at convergence, we run the search stage for 500 epochs. We evaluate different $\eta$ values from 0 to 1, and the architectures corresponding to $\eta = 0$ (equivalent to DARTS), $\eta = 0 . 1$ and $\eta = 1$ are summarized in Figure 2. We can see that $\eta = 0 . 1$ converges, after 500, into a reasonable architecture that achieves an error rate of $3 . 0 8 \%$ on CIFAR10. We emphasize that, even with more search epochs, this architecture is not likely to change, as the preserved operator on each edge has a weight not smaller than 0.5, and most of these weights are still growing gradually.
+
+When $\eta$ is very small, e.g., $\eta = 0 . 0 0 1$ or $\eta = 0 . 0 1$ , the change brought by this amending term to architecture search is ignorable, and our approach shows almost the same behavior as the first-order version of DARTS, i.e., $\eta = 0$ . In addition, when $\eta$ grows up, e.g., from 0.001 to 0.01, although the search process eventually runs into an architecture with all skip-connect operators, the number of epochs needed for a complete failure is significantly postponed, which verifies that the amending term indeed pulls architecture search away from the gradient trap.
+
+![](images/06de76b891eab13d3496f86280dc133177d768ef6a253623c6e6097e0abcaa33.jpg)  
+Figure 2: Normal (left) and reduction (right) cells (the standard DARTS space) obtained by different amending coefficients, namely, $\eta = 0$ (top), $\eta = 0 . 1$ (middle) and $\eta = 1$ (bottom).
+
+On the other hand, if we use a sufficiently large $\eta$ value, e.g., $\eta = 1$ , the amending term, $\mathbf { g } _ { 2 } ^ { \prime }$ , can dominate optimization, so that the first term, i.e., the gradient of architectural parameters, has limited effects in updating $_ \alpha$ . Note that the amending term is closely related to network regularization, therefore, in the scenarios of a large $\eta$ , the network significantly prefers avg-pooling to other operators, as avg-pooling can smooth feature maps and avoid over-fitting. However, avg-pooling is also a parameter-free operator, so the performance of such architectures is also below satisfaction.
+
+Following these analyses, we simply use $\eta = 0 . 1$ for all later experiments. We do not tune $\eta$ very carefully, though it is possible to determine $\eta$ automatically using a held-out validation set. Besides, we find that the best architecture barely changes after 100 search epochs, which we fix the total length to be 100 epochs in all remaining experiments.
+
+# 4.1.2 CONSISTENCY BETWEEN SEARCH AND EVALUATION
+
+As discussed in Section 3.5, it is important to alleviate the difference between search and evaluation. We make the following modifications, listed from most to least important. First, to avoid edge removal, we fix the edges in each cell, so that each node $i$ is connected to node $i - 1$ and the least indexed node (denoted by $c _ { k - 2 }$ in most conventions), resulting in 8 edges in each cell. Note that our approach also works well with all 14 edges preserved, but we have used 8 edges to be computationally fair to DARTS. Second, we unify the width (the number of basic channels) as 36 for both search and evaluation. Third, we add normalization techniques, including Cutout (DeVries & Taylor, 2017), Dropout (Srivastava et al., 2014) and an auxiliary loss tower, into the search stage.
+
+We use the amending coefficient $\eta = 0 . 1$ learned from previous experiments, i.e., without modification, the searched architecture, denoted by $\mathbb { A } _ { \mathrm { o r i g } }$ , is shown in the middle column of Figure 2. After modification, the obtained architecture, denoted by $\mathbb { A } _ { \mathrm { n e w } }$ , is shown in Figure 3. We re-train both networks on CIFAR10, with or without the option that stacking duplicate cells to make the network deeper (with 20 cells). With a standard re-training process, $\mathbb { A } _ { \mathrm { o r i g } }$ reports a $3 . 6 7 \%$ error with 8 cells, and a $3 . 0 8 \%$ error with 20 cells; and the corresponding numbers are $3 . 2 0 \%$ and $2 . 8 1 \%$ for $\mathbb { A } _ { \mathrm { n e w } }$ . We find that $\mathbb { A } _ { \mathrm { n e w } }$ is consistently better than $\mathbb { A } _ { \mathrm { o r i g } }$ , which suggests that alleviating the gap indeed helps. This also reminds us of the significant depth gap (Chen et al., 2019) between search and re-training (the network has 8 cells in search, but 20 cells in re-training), and this gap also obstructs our approach from achieving better performance. We will investigate this issue in the following part.
+
+# 4.1.3 EXPLORING MORE COMPLEX SEARCH SPACES
+
+We note tremendous efforts made by existing approaches to alleviate the depth gap, while our solution is a direct one, thanks to the stability of our approach which enables us to directly explore larger search spaces. Here, we denote the original search space used in DARTS as $S _ { 1 }$ , which has six normal cells and two reduction cells, and all normal cells share the same set of architectural parameters and so do the reduction cells. Note that we have fixed the edges in this space, resulting in the total number of possible architectures to reduce from $1 . 1 \times 1 0 ^ { 1 8 }$ to $3 . 3 \times 1 0 ^ { 1 \bar { 3 } }$ . We also explore a more complex search space, denoted by $S _ { 2 }$ , in which we relax the constraint that either normal cells or reduction cells should be the same, and also the number of cells increases from 8 to 20, to be applied in re-training. Here, limited by GPU memory, we cannot support all seven operators to be searched, so we only choose two, namely skip-connect and sep-conv- $3 x 3$ , which have very different properties. This setting allows a total of $1 . 5 \times 1 0 ^ { 4 8 }$ architectures to appear, much larger than $S _ { 1 }$ .
+
+Results are listed in Table 1. In $S _ { 1 }$ , our approach achieves a moderate error rate of $2 . 8 1 \%$ , mainly because the assumption of consistency between search and re-training does not hold. In $S _ { 2 }$ , with directly searching in deep architectures, our result is significantly boosted (an error rate of $2 . 6 0 \%$ , the architecture is shown in the middle row of Figure 3). Again, we emphasize that we report retraining results based on an converged architecture, which stands out from existing DARTS-based approaches which required early termination.
+
+In $S _ { 2 }$ , we compare our approach against both DARTS (no amending term) and random search. We observe that DARTS produces weird architectures (the bottom row of Figure 3), that high-layer cells are mostly occupied by skip-connect, which is not likely to fully utilize the ability of the supernetwork. Regarding random search, we follow DARTS by randomly sampling 20 valid architectures from each of $S _ { 1 }$ and $S _ { 2 }$ , and using a small validation dataset to choose the best two architectures for re-training. Given a fixed number of probes, it becomes more and more difficult to sufficiently explore a large space. The deficits of DARTS and random search on CIFAR10 are $0 . 2 5 \%$ and $0 . { \bar { 2 } } 9 \%$ , respectively, which do not seem to be very large arguably because CIFAR10 is relatively easy and our edge-fixed search space guarantees sufficient depth. However, when we transfer these architectures to ImageNet, the deficits become much larger, i.e., with $1 . 7 \%$ and $0 . 8 \%$ top-1 accuracy drops, respectively (DARTS produces inferior performance to random search). These results remind us of prior work (Xie et al., 2019) which claimed that random search works sufficiently well in large search spaces. Here, we leave a comment on this debate, demonstrating that a large space indeed raises challenges to architecture search, but a stabilized search algorithm still has the ability of to find more powerful architectures.
+
+![](images/c1a067dcdf9289c598f7f77055e3e2886cc4b81851cc2828c839c5d62aae3a70.jpg)  
+Figure 3: Top: normal and reduction cells found in $ { \boldsymbol { S } } _ { 1 }$ . Middle & Bottom: the overall architecture found in $S _ { 2 }$ by DARTS, with and without the amended term, in which red and blue edges indicate skip-connect and sep-conv- $. 3 x 3$ operators, respectively.
+
+# 4.1.4 COMPARISON TO THE STATE-OF-THE-ARTS
+
+Finally, we compare our approach with recent approaches, in particular, differentiable ones. Result are shown in Table 1. Our approach produces competitive results among state-of-the-arts, although it does not seem to beat others. We note that existing approaches often used additional tricks, e.g., PDARTS assumed a fixed number of skip-connect operators, which shrinks the search space (so as to guarantee stability). More importantly, all these differentiable search approaches must be terminated in an early stage, which makes them less convincing as search has not arrived at convergence. These tricks somewhat violate the ideology of neural architecture search; in comparison, our research, though not producing the best performance, seems going along a correct and promising direction.
+
+<table><tr><td rowspan="2">Architecture</td><td>Test Err.</td><td>Params</td><td>Search Cost</td><td rowspan="2">Search Method</td></tr><tr><td>(%)</td><td>(M)</td><td>(GPU-days)</td></tr><tr><td>DenseNet-BC (Huang et al.,2017)</td><td>3.46</td><td>25.6</td><td>-</td><td>manual</td></tr><tr><td>ENAS (Pham et al.,2018) w/ Cutout</td><td>2.89</td><td>4.6</td><td>0.5</td><td>RL</td></tr><tr><td>NASNet-A (Zoph et al.,2018) w/Cutout</td><td>2.65</td><td>3.3</td><td>1800</td><td>RL</td></tr><tr><td>NAONet-WS (Luo et al.,2018)</td><td>3.53</td><td>3.1</td><td>0.4</td><td>NAO</td></tr><tr><td>Hireachical Evolution (Liu et al.,2018b)</td><td>3.75±0.12</td><td>15.7</td><td>300</td><td>evolution</td></tr><tr><td>AmoebaNet-B (Real et al.,2019) w/ Cutout</td><td>2.55±0.05</td><td>2.8</td><td>3150</td><td>evolution</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>3.41±0.09</td><td>3.2</td><td>225</td><td>SMBO</td></tr><tr><td>DARTS (first-order) (Liu et al.,2019b) w/ Cutout</td><td>3.00±0.14</td><td>3.3</td><td>0.4</td><td>gradient-based</td></tr><tr><td>DARTS (second-order) (Liu et al.,20i9b) w/Cutout</td><td>2.76±0.09</td><td>3.3</td><td>1.0</td><td>gradient-based</td></tr><tr><td>SNAS (moderate) (Xie etal.,2018) w/ Cutout</td><td>2.85±0.02</td><td>2.8</td><td>1.5</td><td>gradient-based</td></tr><tr><td>ProxylessNAS (Cai et al.,2019) w/Cutout</td><td>2.08</td><td>-</td><td>4.0</td><td>gradient-based</td></tr><tr><td>P-DARTS (Chen et al.,2019) w/ Cutout</td><td>2.50</td><td>3.4</td><td>0.3</td><td>gradient-based</td></tr><tr><td>BayesNAS (Zhou et al.,2019) w/ Cutout</td><td>2.81±0.04</td><td>3.4</td><td>0.2</td><td>gradient-based</td></tr><tr><td>PC-DARTS (Xu et al.,2019) w/ Cutout</td><td>2.57±0.07</td><td>3.6</td><td>0.1</td><td>gradient-based</td></tr><tr><td>Amended-DARTS,S1,w/Cutout</td><td>2.81±0.21</td><td>3.5</td><td>1.0</td><td>gradient-based</td></tr><tr><td>Amended-DARTS,S2,w/Cutout</td><td>2.60±0.15</td><td>3.6</td><td>1.1</td><td>gradient-based</td></tr></table>
+
+Table 1: Comparison with state-of-the-art network architectures on CIFAR10.
+
+<table><tr><td rowspan="2">Architecture</td><td colspan="2">Test Err. (%)</td><td rowspan="2">Params (M)</td><td rowspan="2">×+ (M)</td><td rowspan="2">Search Cost (GPU-days)</td><td rowspan="2">Search Method</td></tr><tr><td>top-1</td><td>top-5</td></tr><tr><td>Inception-v1 (Szegedy et al.,2015)</td><td>30.2</td><td>10.1</td><td>6.6</td><td>1448</td><td>-</td><td>manual</td></tr><tr><td>MobileNet (Howard etal.,2017)</td><td>29.4</td><td>10.5</td><td>4.2</td><td>569</td><td></td><td>manual</td></tr><tr><td>ShuffleNet 2× (v1) (Zhang et al.,2018)</td><td>26.4</td><td>10.2</td><td>~5</td><td>524</td><td></td><td>manual</td></tr><tr><td>ShuffleNet 2× (v2) (Ma et al.,2018)</td><td>25.1</td><td>-</td><td>~5</td><td>591</td><td>-</td><td>manual</td></tr><tr><td>NASNet-A (Zoph et al., 2018)</td><td>26.0</td><td>8.4</td><td>5.3</td><td>564</td><td>1800</td><td>RL</td></tr><tr><td>MnasNet-92 (Tan et al.,2019)</td><td>25.2</td><td>8.0</td><td>4.4</td><td>388</td><td>-</td><td>RL</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>25.8</td><td>8.1</td><td>5.1</td><td>588</td><td>225</td><td>SMBO</td></tr><tr><td>AmoebaNet-C (Real et al.,2019)</td><td>24.3</td><td>7.6</td><td>6.4</td><td>570</td><td>3150</td><td>evolution</td></tr><tr><td>DARTS (second-order) (Liu et al.,2019b)</td><td>26.7</td><td>8.7</td><td>4.7</td><td>574</td><td>4.0</td><td>gradient-based</td></tr><tr><td>SNAS (mild) (Xie et al., 2018)</td><td>27.3</td><td>9.2</td><td>4.3</td><td>522</td><td>1.5</td><td>gradient-based</td></tr><tr><td>BayesNAS (Zhou et al.,2019)</td><td>26.5</td><td>8.9</td><td>3.9</td><td>-</td><td>0.2</td><td>gradient-based</td></tr><tr><td>P-DARTS (CIFAR10) (Chen et al.,2019)</td><td>24.4</td><td>7.4</td><td>4.9</td><td>557</td><td>0.3</td><td>gradient-based</td></tr><tr><td>ProxylessNAS (GPU) (Cai et al., 2019)</td><td>24.9</td><td>7.5</td><td>7.1</td><td>465</td><td>8.3</td><td>gradient-based</td></tr><tr><td>PC-DARTS (Xu et al., 2019)‡</td><td>24.2</td><td>7.3</td><td>5.3</td><td>597</td><td>3.8</td><td>gradient-based</td></tr><tr><td>DARTS+ (Liang et al., 2019)‡</td><td>23.9</td><td>7.4</td><td>5.1</td><td>582</td><td>6.8</td><td>gradient-based</td></tr><tr><td>Amended-DARTS, S2</td><td>24.3</td><td>7.4</td><td>5.5</td><td>590</td><td>1.1</td><td>gradient-based</td></tr></table>
+
+Table 2: Comparison with state-of-the-art architectures on ILSVRC2012. All searched architectures are fit into the mobile setting. ‡ indicates architectures searched on ImageNet.
+
+# 4.2 RESULTS ON ILSVRC2012
+
+ILSVRC2012 (Russakovsky et al., 2015) is the most commonly used subset of ImageNet (Deng et al., 2009). It contains 1.3M training images and 50K testing images, which are almost evenly distributed over all 1,000 categories. We directly use the $S _ { 2 }$ architecture obtained from CIFAR10 experiments and enlarge it with a basic number of channels of 42, so that the FLOPs of our model is 590M, i.e., fitting the mobile setting. During re-training, there are a total of 250 epochs. We use an SGD optimizer with an initial learning rate of 0.5 (decaying linearly after each epoch), a momentum of 0.9 and a weight decay of $3 \times 1 0 ^ { - 5 }$ . On NVIDIA Tesla V100 GPUs, the entire re-training process takes around 24 GPU-days.
+
+The comparison of our approach to existing approaches is shown in Table 2. Our approach achieves a top-1 error rate of $2 4 . { \bar { 3 } } \bar { \% }$ without any common optimization tricks such as AutoAugment (Cubuk et al., 2019) and Squeeze-and-Excitation modules (Hu et al., 2018). This result is competitive among state-of-the-arts, and it is obtained after convergence is achieved in the search stage. On the other hand, without the amending term, DARTS converges to a solution that skip-connected and sep-conv$3 x 3$ are largely separated, on which the re-training performance is even inferior to random search.
+
+# 5 CONCLUSIONS
+
+In this paper, we present an effective approach for stabilizing differentiable neural architecture search. Our motivation comes from that DARTS-based approaches mostly generate all-skip-connect architectures when they are executed for a sufficient number of epochs. We analyze this weird phenomenon mathematically and find the reason to lie in the dramatic inaccuracy in estimating gradients of architectural parameters. With an alternative approximation based on the optimality of network parameters, we guarantee the update of architectural parameters to be in a correct direction. In DARTS-based search spaces on CIFAR10 and ImageNet, our approach shows improved stability, in particular in large search spaces, as well as improved performance in the re-training stage.
+
+Our research sheds light on future research on NAS in several aspects. First, we reveal that previous differentiable approaches were mostly built upon a dangerous pipeline, and mostly introduced heavy human expertise to avoid failure. By fixing the ‘system error’ of this pipeline, we provide a platform that NAS approaches can compete in the ability of NAS. Second, our approach enables researchers to explore even bigger search spaces that have not been studied before (due to search instability).
+
+REFERENCES   
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+
+# A APPENDIX
+
+# A.1 A TOY EXAMPLE TO SHOW THE IMPORTANCE OF $\mathbf { g } _ { 2 }$
+
+Let the loss function be $\begin{array} { l l l } { { \mathcal L } \left( \omega , \alpha ; x \right) } & { = } & { \left( \omega x - \alpha \right) ^ { 2 } } \end{array}$ . Then, the only difference between $\mathcal { L } _ { \mathrm { t r a i n } } \left( \omega , \alpha \right) = \mathcal { L } \left( \omega , \alpha ; x _ { \mathrm { t r a i n } } \right)$ and $\mathcal { L } _ { \mathrm { v a l i d } } \left( \omega , \alpha \right) = \mathcal { L } \left( \omega , \alpha ; x _ { \mathrm { v a l i d } } \right)$ lies in the input, $_ { \textbf { \em x } }$ . Assume the input of training data is ${ x } _ { \mathrm { t r a i n } } = 1$ and the input validation data is $\pmb { x } _ { \mathrm { v a l i d } } = 2$ . It is easy to derive that the local optimum of $\mathcal { L } _ { \mathrm { t r a i n } } \left( \omega , \alpha \right)$ is $\omega ^ { \ast } \left( \alpha \right) = \alpha$ . Substituting $\pmb { x } _ { \mathrm { v a l i d } } = 2$ into $\mathcal { L } _ { \mathrm { v a l i d } } \left( \omega , \alpha \right)$ yields ${ \mathcal { L } } _ { \mathrm { v a l i d } } \left( \omega , \alpha \right) = \left( 2 \omega - \alpha \right) ^ { 2 }$ . When $\mathbf { \alpha } _ { \alpha } = \mathbf { \alpha } _ { \mathbf { \alpha } _ { \mathbf { t } } }$ , $\omega$ arrives at $\omega ^ { \ast } \left( \alpha _ { \mathrm { t } } \right)$ , so $\mathbf { g } _ { 1 } = 2 \left( \alpha _ { \mathrm { t } } - 2 \alpha _ { \mathrm { t } } \right) = - 2 \alpha _ { \mathrm { t } }$ and $\mathbf { g } _ { 2 } = 4 \alpha _ { \mathrm { t } }$ . When $\omega$ arrives at ${ { \omega } ^ { * } } \left( \alpha \right) , { { \mathcal { L } } _ { v a l i d } } \left( { { \omega } ^ { * } } \left( \alpha \right) , \alpha \right) = { { \alpha } ^ { 2 } }$ , $\mathbf { g } \left( \alpha _ { \mathrm { t } } \right) = 2 \alpha _ { \mathrm { t } } = \mathbf { g } _ { 1 } + \mathbf { g } _ { 2 }$ . In summary, both $\mathbf { g } _ { 1 }$ and $\mathbf { g } _ { 2 }$ are important, but DARTS chose to ignore $\mathbf { g } _ { 2 }$ which can cause a dramatic error in approximation.
+
+# A.2 THE COMPLEXITY OF $\mathbf { g } _ { 2 } ^ { ' }$ CAN BE SUBSTANTIALLY REDUCED USING THE FINITEDIFFERENCE APPROXIMATION
+
+We use the finite difference approximation just like DARTS. Let $\epsilon$ be a small scalar, $\omega _ { 1 } \ = \ \omega + \epsilon \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha )$ , $\omega _ { 2 } ~ = ~ \omega - \epsilon \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha )$ . Then: $\begin{array} { r l } { \mathbf { H } \cdot \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega , \alpha ) } & { = } \end{array}$ $\frac { \nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega _ { 1 } , \alpha ) - \nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega _ { 2 } , \alpha ) } { \mathrm { . } }$ . $\begin{array} { r l r } { \omega _ { 3 } } & { { } = } & { \omega \ + \ \frac { \nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega _ { 1 } , \alpha ) - \nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega _ { 2 } , \alpha ) } { 2 } } \end{array}$ , $\omega _ { 4 } = \omega - \frac { } { }$ 2 . Then: $\frac { \nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega _ { 1 } , \alpha ) ^ { - \infty } \nabla _ { \omega } \mathcal { L } _ { \mathrm { t r a i n } } ( \omega _ { 2 } , \alpha ) } { 2 }$ $\begin{array} { r } { \mathbf { g } _ { 2 } ^ { \prime } = - \pmb { \eta } \times \frac { \nabla _ { \omega } \mathcal { L } _ { \mathrm { v a l } } ( \omega _ { 3 } , \alpha ) - \bar { \nabla _ { \omega } } \mathcal { L } _ { \mathrm { v a l } } ( \omega _ { 4 } , \alpha ) } { 2 \epsilon } } \end{array}$
+
+# A.3 A TOY EXAMPLE TO SHOW THE IMPORTANCE OF THE “GRADIENT TRAP”
+
+We have a small toy case to show the influence of the “gradient trap”. We searched for a small super-network in the DARTS’s search space, which only has two cells(we searched for 600 epochs).
+
+When we train the super-network in the “training sets in search phase” and validate in the “validation sets in search phase”, the test error using $\mathbf { g } _ { 2 } ^ { ' }$ is $1 0 . 5 \%$ while the test error without ${ \bf g } _ { 2 } ^ { ' }$ is $1 2 . 8 \%$ .
+
+When we train the super-network in the training sets and validate in the validation sets, the test error using ${ \bf g } _ { 2 } ^ { ' }$ is $5 . 4 \%$ while the test error without $\mathbf { g } _ { 2 } ^ { ' }$ is $7 . 4 \%$ .
+
+When we generalize and train the network in the training sets and validate in the validation sets, the test error using ${ \bf g } _ { 2 } ^ { ' }$ is $6 . 2 \%$ while the test error without $\mathbf { g } _ { 2 } ^ { \prime }$ is $7 . 4 \%$ .
+
+In this case the “gradient trap” will cause a dramatic accuracy drop of the super-network.
+
+# A.4 SEARCH WITH DIFFERENT SEEDS
+
+We ran our search algorithms with different seeds for 5 times in $S _ { 2 }$ and evaluated each discovered architecture for 3 times. The lowest test error is $2 . 5 7 { \pm } 0 . 1 1 \%$ and the highest is $2 . 6 3 { \pm } 0 . 1 3 \%$ . We did the same thing on $S _ { 1 }$ and the lowest and the highest test errors are $2 . 7 1 \pm 0 . 1 5 \%$ and $2 . 9 2 \pm 0 . 0 9 \%$ , respectively.
+
+As we expected, the results in $S _ { 1 }$ are less robust than those in $S _ { 2 }$ . The main reason is the difference between search and evaluation, including the different depths of search and evaluation networks and the the discretization stage of the standard DARTS method.
+
+More importantly, our approach survives after 500 (and even more) epochs, while DARTS degenerates to an all-skip-connect architecture in all $( 1 0 + )$ individual runs.
+
+A.5 THEORETICAL ANALYSIS OF SEMI-POSITIVE-DEFINITE MATRIX AFTER SIMILARITY TRANSFORMATION
+
+$\mathbf { A } = \mathbf { C } ^ { T } \cdot \mathbf { C } .$ , A is a semi-positive-definite matrix. In this case, the number of different eigenvalues is far less than the dimension of A(hundreds compared to millions). $\mathbf { H }$ is a real symmetric positivedefinite matrix.
+
+Let $\big \{ \alpha _ { \mathrm { i } } \big \}$ be a set of eigenvectors w.r.t $\mathbf { A } ( \alpha _ { \mathrm { i } } ^ { T } \cdot \alpha _ { \mathrm { i } } = 1 )$ ), then $\{ \mathbf { H } \cdot \mathbf { \boldsymbol { \alpha } } _ { \mathrm { i } } \}$ is a set of eigenvectors w.r.t $\mathbf { H } \cdot \dot { \mathbf { A } } \cdot \dot { \mathbf { H } } ^ { - 1 }$ , $\{ \mathbf { H } ^ { - 1 } \cdot \mathbf { \bar { \alpha } } _ { \mathrm { i } } \}$ is a set of eigenvectors w.r.t $\mathbf { H } ^ { - 1 } \cdot \mathbf { A } \cdot \mathbf { H }$ , and $\{ \lambda _ { \mathrm { i } } \}$ is a set of eigenvectors shared by them.
+
+Let $\beta$ be an eigenvector $\begin{array} { r } { \mathbf { \operatorname { p f } } \mathbf { H } { \cdot } \mathbf { A } { \cdot } \mathbf { H } ^ { - 1 } + \mathbf { H } ^ { - 1 } { \cdot } \mathbf { A } { \cdot } \mathbf { H } , \beta = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } a _ { \mathrm { i } } \times \mathbf { H } \cdot \alpha _ { \mathrm { i } } = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \times \mathbf { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } . } \end{array}$  $\begin{array} { r l } & { \qquad \mathrm { E } : = \dots \qquad \mathrm { A } : \mathrm { H } ^ { - 1 } + \mathrm { H } ^ { - 1 } \cdot \mathrm { A } \cdot \mathrm { H } \cdot \mathrm { \Lambda } \mathrm { H } \cdot \mathrm { \Lambda } \beta = \mathrm { \Lambda } \lambda \beta , \quad \mathrm { ~ a } : = 1 \times \mathrm { H } \cdot \mathrm { \Lambda } \alpha _ { \mathrm { i } } + \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \lambda _ { \mathrm { i } } \times \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } = } \\ & { \underset { \mathrm { - i } = 1 } { \sum } a _ { \mathrm { i } } \lambda \times \mathrm { H } \cdot \alpha _ { \mathrm { i } } = \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } b _ { \mathrm { i } } \lambda \times \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } . } \\ & { \underset { \mathrm { - i } = 1 } { \sum } \left( \lambda _ { \mathrm { i } } - \lambda \right) a _ { \mathrm { i } } b _ { \mathrm { i } } = - \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) b _ { \mathrm { i } } \alpha _ { \mathrm { i } } ^ { T } \cdot \mathrm { H } ^ { - 1 } \cdot \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) b _ { \mathrm { i } } \mathrm { H } ^ { - 1 } \cdot \alpha _ { \mathrm { i } } \le 0 } \end{array}$
+
+A is a real symmetric matrix, so we have many sets of $\big \{ \alpha _ { \mathrm { i } } \big \}$ that is orthogonal to each other. $\beta ^ { T } \cdot \beta =$ $\textstyle \sum _ { \mathrm { i } = 1 } ^ { \mathrm { n } } a _ { \mathrm { i } } b _ { \mathrm { i } } \geq 0$ , For every eigenvalue, the dimension of the subspace will be very high in the case of neural architecture search, so we assume that we can choose a set of $\big \{ \alpha _ { \mathrm { i } } \big \}$ orthogonal to each other from the subspace satisfying Pni+1j=ni $\sum _ { \mathrm { j = n _ { i } } } ^ { \mathrm { n _ { i + 1 } - 1 } } a _ { \mathrm { j } } b _ { \mathrm { j } } \geq 0$ in most cases $\mathrm { { \acute { n } _ { i } } }$ is the id of the first eigenvector w.r.t $\lambda _ { \mathrm { i } }$ ).
+
+$\begin{array} { r } { \sum _ { \mathrm { i = 1 } } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) a _ { \mathrm { i } } b _ { \mathrm { i } } = \sum _ { \mathrm { i = 1 } } ^ { \mathrm { n } } \left( \lambda _ { \mathrm { i } } - \lambda \right) \sum _ { \mathrm { j = n _ { i } } } ^ { \mathrm { n _ { i + 1 } - 1 } } a _ { \mathrm { j } } b _ { \mathrm { j } } \le 0 } \end{array}$ so $\boldsymbol { \lambda }$ cannot be smaller than zero.
+
+All of the eigenvalues w.r.t real symmetric matrix $\mathbf { H } \cdot \mathbf { A } \cdot \mathbf { H } ^ { - 1 } + \mathbf { H } ^ { - 1 } \cdot \mathbf { A } \cdot \mathbf { H }$ is not smaller than zero, so it is semi-positive-definite. Then we get that $\mathbf { H } ^ { - 1 } \cdot \mathbf { A } \cdot \mathbf { H }$ is semi-positive-definite.

parse/train/BJlgt2EYwr/BJlgt2EYwr_content_list.json

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+    {
+        "type": "text",
+        "text": "STABILIZING DARTS WITH AMENDED GRADIENT ESTIMATION ON ARCHITECTURAL PARAMETERS ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "Differentiable neural architecture search has been a popular methodology of exploring architectures for deep learning. Despite the great advantage of search efficiency, it often suffers weak stability, which hinders it from being applied to a large search space or being flexibly adjusted to different scenarios. This paper investigates DARTS, the currently most popular differentiable search algorithm, and points out an important factor of instability, which lies in its approximation on the gradients of architectural parameters. In the current status, the optimization algorithm can converge to another point which results in dramatic inaccuracy in the re-training process. Based on this analysis, we propose an amending term for computing architectural gradients by making use of a direct property of the optimality of network parameter optimization. Our approach mathematically guarantees that gradient estimation follows a roughly correct direction, which leads the search stage to converge on reasonable architectures. In practice, our algorithm is easily implemented and added to DARTS-based approaches efficiently. Experiments on CIFAR and ImageNet demonstrate that our approach enjoys accuracy gain and, more importantly, enables DARTS-based approaches to explore much larger search spaces that have not been studied before. ",
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+        "text": "1 INTRODUCTION ",
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+        "text": "Neural architecture search (NAS) has been an important topic in the research area of automated machine learning (AutoML). The idea is to replace the manual way of designing neural network architectures with an automatic algorithm, by which deep learning methods become more flexible in fitting complex data distributions, e.g., large-scale image datasets. Early efforts of NAS involved using heuristic search methods such as reinforcement learning (Zoph & Le, 2017; Zoph et al., 2018) and evolutionary algorithms (Real et al., 2017; Xie & Yuille, 2017) to sample networks from a large search space, and optimizing each sampled network individually to evaluate its quality. Despite notable successes obtained by this methodology, it often requires a vast amount of computation, which obstacles its applications in the scenarios of limited resources. Inspired by the idea of reusing and sharing parameters among trained networks, DARTS (Liu et al., 2019b) was designed as a ‘one-shot’ solution of NAS. The major difference from the aforementioned methods is a differentiable formulation of architecture search and an end-to-end mechanism which optimizes model weights (such as convolution) and architectural weights simultaneously. Recently, improvements upon DARTS were made in various aspects (Chen et al., 2019; Xu et al., 2019; Liang et al., 2019), making it a reasonable tradeoff between search cost and performance. ",
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+        "text": "Despite its broad applications, the current pipeline of DARTS (or, generally speaking, differentiable NAS approaches) suffers a critical weakness known as instability. Researchers reported (Liang et al., 2019) that DARTS-based algorithms can sometimes generate weird architectures that produce considerably worse accuracy than those generated in other individual runs, or even significantly worse than randomly generated architectures. There indeed exist tricks designed by human expertise (Chen et al., 2019; Nayman et al., 2019; Liang et al., 2019) to alleviate this issue, but we point out that these approaches violated the ideology of NAS, which is to maximally prevent human interventions. Moreover, even with such add-ons, a dramatic property of DARTS persists and has not been studied carefully in prior work. When DARTS, as well as its variants, gets trained for a longer time, e.g., from the default number of 50 epochs to 200 epochs, we surprisingly observe that all these approaches converge to very similar architectures, in which almost all edges are occupied by skip-connect (a.k.a., identity). These architectures, with fewer trainable parameters, are often far from producing high accuracy in particular on large datasets like ImageNet, but they somehow produce sufficiently high validation accuracy in the search stage. In other words, convergence in search often leads to bad performance in re-training. This is why some previous DARTS-based approaches advocated for early termination (Liang et al., 2019), a practical but non-essential solution. Also, we conjecture that early termination also contributes to the lack of stability and, more importantly, trustfulness, of DARTS-based approaches. ",
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+        "text": "This paper delves deep into the inconsistency between convergence and performance. We show that the devil lies in optimizing the loss function of the super-network, $\\mathcal { L } _ { \\mathrm { v a l } } ( \\bar { \\omega } ^ { \\star } ( \\alpha ) , \\alpha )$ ( $\\omega$ and $_ { \\pmb { \\alpha } }$ denote network and architectural parameters, respectively, and $\\omega ^ { \\star } ( \\alpha )$ is the global optimum of $\\omega$ given $_ \\alpha$ ), in which ω and α get updated alternately. Following the chain rule, ∇αLval(ω?(α) , α)|α=αt equals to $\\nabla _ { \\alpha } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } + \\left. \\nabla _ { \\alpha } \\omega ^ { \\star } ( \\alpha ) \\right| _ { \\alpha = \\alpha _ { t } }$ · $\\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } }$ , in which the first term is easy to compute while the second term is not, mainly because $\\omega ^ { \\star } ( \\alpha )$ is difficult to estimate, and so is the term of ∇αω?(α)|α=αt . DARTS-based approaches performed inaccurate approximation for this purpose, in which the first-order version of DARTS directly discarded the second term – but this term is often numerically significant, and the second-order version of DARTS applied an approximation to this term which is not mathematically guaranteed (see Section 3.4). Consequently, the accuracy of $\\nabla _ { \\alpha } \\mathcal { L } _ { \\mathrm { v a l } } \\big ( \\omega ^ { \\star } ( \\alpha ) , \\alpha \\big ) | _ { \\alpha = \\alpha _ { t } }$ cannot be guaranteed, and hence the update of $_ { \\pmb { \\alpha } }$ can be problematic. To the best of our knowledge, this issue is not studied by existing DARTS-based approaches. ",
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+        "text": "To deal with this problem, we propose an alternative way of computing $\\nabla _ { \\alpha } \\omega ^ { \\star } ( \\alpha ) | _ { \\alpha = \\alpha _ { t } }$ . We make use of an important property, i.e., $\\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha ^ { \\dagger } ) , \\alpha = \\alpha ^ { \\dagger } } \\ \\equiv \\ \\mathbf { 0 }$ holds for any $\\alpha ^ { \\dagger }$ , which directly comes from the optimality of $\\omega ^ { \\star } ( \\alpha )$ . Differentiating both sides with respect to $\\alpha ^ { \\dagger }$ , we obtain a new equality which enables computing derives $\\nabla _ { \\alpha } \\omega ^ { \\star } ( \\alpha ) | _ { \\alpha = \\alpha _ { t } }$ with the inverse of the Hesse matrix, $\\nabla _ { \\omega } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) \\big | _ { \\omega = \\omega ^ { \\star } ( \\alpha ) , \\alpha = \\alpha _ { t } }$ . This idea enables us to achieve a more accurate approximation on $\\nabla _ { \\alpha } \\mathcal { L } _ { \\mathrm { v a l } } \\big ( \\omega ^ { \\star } ( \\alpha ) , \\alpha \\big ) | _ { \\alpha = \\alpha _ { t } }$ when $\\omega ^ { \\star } ( \\alpha )$ is not available. Mathematically, we prove that when we have $\\omega ^ { \\mathrm { e s t } } \\approx \\omega ^ { \\star } ( \\alpha _ { t } )$ , the inner angle between the second term and our approximate term is smaller than 90 degrees. Note that this property does not hold in existing DARTS-based algorithms. Our final solution involves using the amended second term of ∇αLval(ω?(α) , α)|α=αt meanwhile keeping the first term unchanged, which goes one step further in optimizing the supernetwork, which reflects in a higher validation accuracy in the search stage. ",
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+        "text": "Our approach is very easily implemented. The overall computational overhead is comparable to the second-order version of DARTS. Experiments are performed on image classification, with popular datasets including CIFAR and ImageNet being used. In all experiments, we allow the search stage to come to a complete convergence and report competitive accuracy among current state-of-thearts. The stability of our approach also enables us to close the gap between hyper-parameters of search and evaluation, as well as explore more complex search spaces, which are believed to be correct directions of NAS but existing DARTS-based approaches would mostly fail. Therefore, we believe our algorithm can expand the application scenario of differentiable NAS methods in particular DARTS-based approaches. ",
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+        "text": "The remainder of this paper is organized as follows. We briefly review related work in Section 2, and illustrate our approach of amending architectural gradients in Section 3. After experiments are shown in Section 4, we conclude this work in Section 5. ",
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+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "With the era of big data and powerful computational resources, deep learning (LeCun et al., 2015), in particular, deep neural networks (Krizhevsky et al., 2012), have rapidly grown up to be the standard tool for learning representations in a complicated feature space. Recent years have witnessed the trend of using deeper (He et al., 2016) and denser (Huang et al., 2017) networks to boost recognition performance, while there is no justification that whether these manually designed architectures are best for each specific task, e.g., image classification. To advance, researchers started considering the possibility of learning network architectures automatically from data, which led to the appearance of neural architecture search (NAS) (Zoph & Le, 2017), which is now popular and known as a sub research field in automated machine learning (AutoML). ",
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+        "text": "The common pipeline of NAS starts with a pre-defined space of network operators. Since the search space is often large (e.g., containing $1 0 ^ { 1 0 }$ or even more possible architectures), it is unlikely that exhaustive search is tractable, and thus heuristic search methods are widely applied for speedup. Typical examples include reinforcement learning (Zoph & Le, 2017; Zoph et al., 2018; Liu et al., 2018a) and evolutionary algorithms (Real et al., 2017; Xie & Yuille, 2017; Real et al., 2019). These approaches followed a general pipeline that samples a set of architectures from a learnable distribution, evaluates them and learns from rewards by updating the distribution. In an early age, each sampled architecture underwent an individual training process from scratch and thus the overall computational overhead is large, e.g., hundreds of even thousands of GPU-days. To alleviate the burden, researchers started to share computation among training sampled architectures, with the key lying in reusing network weights trained previously (Cai et al., 2018) or starting from a well-trained super-network (Pham et al., 2018). These efforts shed light on the so-called one-shot architecture search methods, which required training the super-network only once and thus ran more efficiently, e.g., two or three orders of magnitude faster than conventional approaches. ",
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+        "type": "text",
+        "text": "Within the scope of one-shot architecture search, an elegant solution lies in jointly formulating architecture search and approximation, so that it is possible to apply end-to-end optimization for training network and architectural parameters simultaneously. This methodology is known today as differentiable NAS, and a typical example is DARTS (Liu et al., 2019b), which constructed a super-network with all possible operators contained and decoupled, and the goal is to determine the weights of these architectural parameters, followed by pruning and re-training stages. This kind of approach allowed more flexible search space to be constructed, unlike conventional approaches with either reinforcement or evolutionary learning, which suffer from the computational burden and thus must constrain search within a relatively small search space (Tan & Le, 2019). ",
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+        "type": "text",
+        "text": "Despite the inspirations brought by differentiable NAS, these approaches still suffer a few critical issues that narrow down their applications in practice. One significant drawback lies in the lack of stability, which reflects in the way that results of differentiable search can be impacted by very small perturbations, e.g., initialization of architectural weights, training hyper-parameters, and even randomness in the training process. Existing solutions include running search for several individual times and choosing the best one in validation (Liu et al., 2019b), or using other kinds of techniques such as decoupling modules (Cai et al., 2019; Guo et al., 2019), adjusting search space during optimization (Noy et al., 2019; Chen et al., 2019; Nayman et al., 2019), regularization $\\mathrm { { X u } }$ et al., 2019), early termination (Liang et al., 2019), etc., however, these approaches seemed to develop heuristic remedies rather than analyze it from the mathematical fundamentals, e.g., how instability happens in mathematics. ",
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+        "type": "text",
+        "text": "In this paper, we investigate the stability issue in mathematics and show that the results produced by the current approaches are much less reliable than people used to think. Then, we fix this issue by amending optimization of the architectural parameters, so that each step of the update gets closer to the correct direction. We show great improvement on stability in a fundamental task, image classification, while we believe our approach can be applied to a wide range of tasks including object detection (Ghiasi et al., 2019), semantic segmentation (Liu et al., 2019a), hyper-parameter learning (Cubuk et al., 2019), etc. ",
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+        "type": "text",
+        "text": "3 STABILIZING DARTS WITH AMENDED GRADIENTS ",
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+        "text": "In this section, we first show that DARTS can fail dramatically when it gets trained till convergence, and then we mathematically analyze how this problem is related to inaccurate approximation in optimization, following which we present our solution to amend this error and thus stabilize DARTS. ",
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+        "type": "text",
+        "text": "3.1 PRELIMINARIES: DIFFERENTIABLE NEURAL ARCHITECTURE SEARCH ",
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+        "text": "Differentiable NAS approaches start with defining a super-network, which is constrained in a search space with a pre-defined number of layers and a limited set of neural operators. The core idea is to introduce a ‘soft’ way operator selection (i.e., using a weighted sum over the outputs of a few operators instead of taking the output of only one), so that optimization can be done in an end-toend manner. Mathematically, the super-network is a function $\\mathbf { f } \\left( \\mathbf { x } ; \\omega , \\alpha \\right)$ , with $\\mathbf { x }$ being input, and parameterized by network parameters $\\omega$ (e.g., convolutional kernels) and architectural parameters $_ { \\pmb { \\alpha } }$ (e.g., indicating the importance of each operator between each pair of layers). $\\mathbf { f } \\left( \\mathbf { x } ; \\omega , \\alpha \\right)$ is differentiable to both $\\omega$ and $_ { \\pmb { \\alpha } }$ , so that gradient-based approaches can be applied for optimization. ",
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+        "img_path": "images/90b275c4f31ff6a8aab6e67e8f3f69367a7a1877308a6bb9227e7ab562343780.jpg",
+        "image_caption": [
+            "Figure 1: Left: a typical search process of the first-order DARTS, in which 200 search epochs are used. Red, green and blue lines indicate the average weight of none, the ratio of dominant skipconnect operators (over 14 normal edges) and the re-training accuracy, with respect to the number of search epochs, respectively. Right: the normal cell obtained after 200 search epochs, in which all preserved operators are skip-connect. We executed both first-order and second-order DARTS for several times, and such failure consistently happens in each individual run. "
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+        "text": "In the example of DARTS, $\\mathbf { f } \\left( \\mathbf { x } ; \\omega , \\alpha \\right)$ is composed of a few cells, each of which contains $N$ nodes, and there is a pre-defined set, $\\mathcal { E }$ , denoting which pairs of nodes are connected. For each connected node pair $( i , j ) , i < j$ , node $j$ takes the output of node $i$ , $\\mathbf { x } _ { i }$ , as a part of its input, and propagate it through a pre-defined operator set, $\\mathcal { O }$ , with all outputs summed up: ",
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+    {
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+        "img_path": "images/d9c9c9acf15499a89f5d6e581cc61c1399cba2a14516fda900fc3b41ec03ce5b.jpg",
+        "text": "$$\n\\mathbf { y } ^ { ( i , j ) } \\left( \\mathbf { x } _ { i } \\right) = \\sum _ { o \\in \\mathcal { O } } \\frac { \\exp ( \\alpha _ { o } ^ { ( i , j ) } ) } { \\sum _ { o ^ { \\prime } \\in \\mathcal { O } } \\exp ( \\alpha _ { o } ^ { ( i , j ) } ) } \\cdot o ( \\mathbf { x } _ { i } ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Here, a softmax term is computed by architectural weights to normalize outputs. Within each unit of the search process, $\\omega$ and $_ { \\pmb { \\alpha } }$ get optimized alternately. After search, the operator $o$ with the maximal value of $\\alpha _ { o } ^ { ( i , j ) }$ is preserved for each edge $( i , j )$ . All network parameters $\\omega$ are discarded and the obtained architecture is re-trained from scratch1. ",
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+    {
+        "type": "text",
+        "text": "3.2 DARTS FAILS: THE CONTRADICTORY BETWEEN CONVERGENCE AND PERFORMANCE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our research is motivated by an observation that DARTS, at the end of a regular training process with, say, 50 epochs (Liu et al., 2019b), has not yet arrived at or even got close to convergence, yet the weight of the none (a.k.a., zero) operator is consistently the largest on each edge of the normal cell. To verify this, we increase the length of each training stage by 4 times, i.e., from 50 to 200 epochs. Two weird phenomena are observed, both of which are shown in the left part of Figure 1. First, the weight of the none operator monotonically goes up – at 200 epochs, the weight has achieved 0.97 on each edge of the normal cells, however, this operator is not considered in the final architecture. Second, almost all preserved operators are skip-connect (a.k.a., identity), a parameter-free operator that contributes little to feature learning – and surprisingly, it occupies $3 0 \\%$ to $7 0 \\%$ of the weight remained by none. Such a network has much fewer parameters than a well-designed one, and thus it usually reports unsatisfying performance at the re-training stage. This indicates that, in the context of DARTS, there exists a contradictory between search convergence and re-training accuracy. ",
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+        "type": "text",
+        "text": "We point out that this is a critical issue, which suggests that when using DARTS-based approaches, one does not hope the search process to achieve convergence as it implies bad performance. In other words, each ‘successful’ architecture comes from an early-terminated search process. Consequently, the initialization of parameters $\\scriptstyle { \\mathbf { \\alpha } } _ { \\alpha }$ and $\\omega$ ), the hyper-parameters of search (e.g., learning rate) and the time of terminating search become important and thus need to be determined by experience (Liang et al., 2019). This weakens the stability as well as explainability of search and, more importantly, violates the fundamental ideology of AutoML, i.e., maximally reducing human interference and determining the best architecture by training data. ",
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+        "text": "3.3 DELVING DEEP INTO MATHEMATICS: PROBLEM AND SOLUTION ",
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+        "type": "text",
+        "text": "We point out that the reason lies in inaccurate estimation of the gradient with respect to $_ { \\pmb { \\alpha } }$ , namely, $\\nabla _ { \\alpha } \\bar { \\mathcal { L } } _ { \\mathrm { v a l } } \\big ( \\omega ^ { \\star } ( \\alpha ) , \\alpha \\big ) | _ { \\alpha = \\alpha _ { t } }$ . Following the chain rule of gradients, this quantity equals to ",
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+    {
+        "type": "equation",
+        "img_path": "images/b2199ddb4ecbe5393fc3630854dc9a53e6bbda85391389ab2f2072a508966b88.jpg",
+        "text": "$$\n\\nabla _ { \\alpha } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } + \\nabla _ { \\alpha } \\omega ^ { \\star } ( \\alpha ) | _ { \\alpha = \\alpha _ { t } } \\cdot \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } ,\n$$",
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+        "text": "in which the first term is relatively easy to compute (as is done by the first-order version of DARTS), while the second term, in particular $\\mathrm { \\nabla } \\nabla _ { \\alpha } \\omega ^ { \\star } ( \\bar { \\alpha } ) | _ { \\alpha = \\alpha _ { t } }$ , is not. The first-order version of DARTS directly discarded this term, but it often has significant numerical values which are not negligible. The second-order version of DARTS indeed proposed an approximation to this term, but, as we shall see in the next subsection, can incur a large approximation error (the inner-product between the correct and estimated directions can be smaller than 0). Consequently, there can be a significant gap between the estimated and true values of $\\nabla _ { \\alpha } \\mathcal { L } _ { \\mathrm { v a l } } \\big ( \\omega ^ { \\star } ( \\alpha ) , \\alpha \\big ) | _ { \\alpha = \\alpha _ { t } }$ . Such inaccuracy accumulates with every update on $_ { \\pmb { \\alpha } }$ , and gradually causes $_ { \\pmb { \\alpha } }$ to converge to weird solutions that are far from optimum, e.g., the entire super-network is dominated by none and skip-connect operators. We name this phenomenon as the gradient trap during optimization. ",
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+        "type": "text",
+        "text": "To estimate $\\nabla _ { \\alpha } \\omega ^ { \\star } ( \\alpha ) | _ { \\alpha = \\alpha _ { t } }$ , we first make a reasonable assumption that $\\nabla _ { \\alpha } \\omega ^ { \\star } ( \\alpha ) | _ { \\alpha = \\alpha _ { t } }$ has finite values. Then, we make use of an important property, i.e., $\\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha ^ { \\dagger } ) , \\alpha = \\alpha ^ { \\dagger } } \\ \\equiv \\ \\mathbf { 0 }$ holds for any $\\alpha ^ { \\dagger }$ . This is property directly comes from the optimality of $\\omega ^ { \\star } ( \\alpha )$ , but it has never been used by existing approaches. Applying differentiation with respect to any $\\alpha ^ { \\dagger }$ to both sides of this equality, we have $\\bar { \\nabla _ { \\alpha ^ { \\dagger } } } \\left( \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\bar { \\alpha } ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha ^ { \\dagger } ) , \\alpha = \\alpha ^ { \\dagger } } \\right) \\equiv \\mathbf { 0 }$ . When $ { \\alpha } ^ { \\dagger } =  { \\alpha } _ { t }$ , it becomes: ",
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+        "img_path": "images/0215eec06eb37d6b585d42af7531d53b383ac213e0f07b9c7e8d794e68720b2b.jpg",
+        "text": "$$\n\\nabla _ { \\alpha ^ { \\dagger } } \\left. \\left( \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) \\vert _ { \\omega = \\omega ^ { \\star } ( \\alpha ^ { \\dagger } ) , \\alpha = \\alpha ^ { \\dagger } } \\right) \\right. _ { \\alpha ^ { \\dagger } = \\alpha _ { t } } = 0 .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Again, applying the chain rule to the left-hand side gives: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f7e5f4bccb150aa52ff125b8f45e9e65e19e6c30fd7e309403a537d294d62ee.jpg",
+        "text": "$$\n\\nabla _ { \\alpha , \\omega } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) \\big | _ { \\omega = \\omega ^ { * } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } + \\nabla _ { \\alpha } \\omega ^ { \\star } ( \\alpha ) \\big | _ { \\alpha = \\alpha _ { t } } \\cdot \\left. \\nabla _ { \\omega } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) \\right| _ { \\omega = \\omega ^ { * } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } = \\mathbf { 0 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            516,
+            810,
+            540
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where we use the notation $\\nabla _ { { \\boldsymbol { \\alpha } } , \\omega } ^ { 2 } ( \\cdot ) ~ \\equiv ~ \\nabla _ { { \\boldsymbol { \\alpha } } } ( \\nabla _ { \\omega } ( \\cdot ) )$ throughout the remaining part of this paper. Here, ∇2ωLtrain(ω, α)\f\fω=ω?(α ),α=α is the Hesse matrix corresponding to the optimum $\\omega ^ { \\star } ( \\alpha _ { t } )$ , which is symmetric and positive-definite, and thus invertible. This gives us an estimation that $\\nabla _ { \\alpha } \\omega ^ { \\star } ( \\alpha ) | _ { \\alpha = \\alpha _ { t } } = - \\left. \\nabla _ { \\alpha , \\omega } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) \\right| _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } \\cdot \\mathbf { H } ^ { - 1 } .$ . Substituting it into Equation 2 gives: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/70969b5eebc52e4aea260b1e957676600949026ba0a59758b8bd1c8432db03fc.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\nabla _ { \\alpha } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega ^ { \\star } ( \\alpha ) , \\alpha ) | _ { \\alpha = \\alpha _ { t } } = \\nabla _ { \\alpha } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } - } \\\\ & { } & { \\nabla _ { \\alpha , \\omega } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } \\cdot \\mathbf { H } ^ { - 1 } \\cdot \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            633,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Note that Equation 5 does not involve any approximation. The only issue comes from the term of ${ \\bf H } ^ { - 1 }$ , which is computationally intractable due to the large dimensionality of $\\mathbf { H }$ (it is related to the number of network parameters, which often exceeds one million in a typical super-network). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.4 APPROXIMATIONS IN COMPUTING THE INVERSE HESSE MATRIX ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let us denote Equation 5 in an abbreviated form of $\\mathbf { g } = \\mathbf { g } _ { 1 } + \\mathbf { g } _ { 2 }$ , in which $\\mathbf { g } _ { 1 }$ , the first-order term of DARTS, is easily computed, while $\\mathbf { g } _ { 2 }$ is not due to the computation of ${ \\bf H } ^ { - 1 }$ . Here, we propose an alternative solution which constructs an approximation term $\\mathbf { g } _ { 2 } ^ { \\prime }$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b27b59c61b490e625079812721645a930a45d23a6eebaf0fbd4926af1777f3c0.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { g } _ { 2 } ^ { \\prime } = - \\eta \\cdot \\left. \\nabla _ { \\alpha , \\omega } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) \\right| _ { \\omega = \\omega ^ { * } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } \\cdot \\mathbf { H } \\cdot \\left. \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) \\right| _ { \\omega = \\omega ^ { * } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            821,
+            774,
+            844
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta > 0$ is named the amending coefficient. In what follows, we show that $\\mathbf { g } _ { 2 } ^ { \\prime }$ is indeed a reasonable approximation of $\\mathbf { g } _ { 2 }$ . Since ",
+        "bbox": [
+            176,
+            848,
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+            877
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3345fee3bccca194eda3d7d4bdb8a53d6def562804357f0f0a991d6ccc033a07.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\langle \\mathbf { g } _ { 2 } ^ { \\prime } , \\mathbf { g } _ { 2 } \\rangle = \\boldsymbol { \\eta } \\cdot \\left. \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) \\right| _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } \\boldsymbol { \\mathsf { T } } \\cdot \\left. \\mathbf { H } ^ { - 1 } \\cdot \\nabla _ { \\omega , \\alpha } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) \\right| _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } \\cdot } \\\\ & { \\qquad \\nabla _ { \\alpha , \\omega } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\alpha ) \\big | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } \\cdot \\mathbf { H } \\cdot \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) \\big | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "the product of the two terms between ${ \\bf H } ^ { - 1 }$ and $\\mathbf { H }$ is a semi-positive-definite matrix, and so is the matrix after similarity transformation, which directly gives $\\left. \\mathbf { g } _ { 2 } ^ { \\prime } , \\mathbf { g } _ { 2 } \\right. \\geqslant 0$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In summary, we decompose the gradient of architectural parameters into two terms, $\\mathbf { g } _ { 1 }$ and $\\mathbf { g } _ { 2 }$ , compute $\\mathbf { g } _ { 1 }$ directly and use an approximation to $\\mathbf { g } _ { 2 }$ so that the angle between the accurate and approximated terms is smaller than 90 degrees. In comparison, existing DARTS-based approaches either discarded $\\mathbf { g } _ { 2 }$ entirely or used a mathematically non-explainable approximation $\\mathbf { \\bar { g } } _ { 2 } ^ { \\prime \\prime } = \\mathbf { \\bar { \\nu } } \\eta \\cdot \\nabla _ { \\alpha , \\omega } ^ { 2 } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega , \\mathbf { \\bar { \\alpha } } ) | _ { \\omega = \\omega ^ { * } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } \\cdot \\mathbf { I } \\cdot \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\mathbf { \\bar { \\alpha } } ) | _ { \\omega = \\omega ^ { * } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } } .$ . None of them are reasonable because $\\mathbf { g } _ { 2 }$ can be large, yet there is no guarantee that $\\left. \\mathbf { g } _ { 2 } ^ { \\prime \\prime } , \\mathbf { g } _ { 2 } \\right. \\geqslant 0$ , i.e., the secondorder DARTS can lead the algorithm to a wrong direction. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The rationality of our approach is also verified by the validation process, i.e., in updating architectural parameters. The first-order DARTS, by directly discarding $\\mathbf { g } _ { 2 }$ , reported an average validation accuracy of $9 0 . 5 \\%$ in search space $S _ { 1 }$ (see Section 4.1.3) on the CIFAR10 dataset. The second-order DARTS added $\\mathbf { g } _ { 2 } ^ { \\prime \\prime }$ , which has no guarantee that $\\left. { { \\bf { g } } _ { 2 } ^ { \\prime \\prime } , { \\bf { g } } _ { 2 } } \\right. \\mathrm { ~ \\ \\ \\geqslant ~ 0 ~ }$ , and thus resulted in a reduced validation accuracy. Our approach, by adding $\\mathbf { g } _ { 2 } ^ { \\prime }$ , achieves a validation accuracy of $9 1 . 5 \\%$ , implying that our optimization works better than DARTS. This eventually results in the advantage of searched architectures, which will be verified in Section 4.1.2. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The remainder part of computing $\\mathbf { g } _ { 2 } ^ { \\prime }$ simply follows conventions, which we replace $\\omega ^ { \\star } ( \\alpha )$ with the current $\\omega ^ { \\mathrm { e s t } }$ as the most accurate approximation we can get2. Computing ${ \\bf g } _ { 2 } ^ { \\bar { \\prime } }$ with Equation 6 requires both ∇2α,ωLtrain(ω, α)\f\fω=ω?(αt),α=αt and $\\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) | _ { \\omega = \\omega ^ { \\star } ( \\alpha _ { t } ) , \\alpha = \\alpha _ { t } }$ , while DARTS needs the former one with $\\omega ^ { \\star } ( \\alpha _ { t } )$ estimated in two steps. Therefore, computing Equation 6 requires similar computational overhead compared to the second-order version of DARTS. In experiments, each search epoch requires around 0.02 GPU-days on the standard 8-cell space on CIFAR10. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.5 DISCUSSIONS AND RELATIONSHIP TO PRIOR WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The core benefit brought by our approach is the consistency between search and evaluation. This is indeed a fundamental idea of NAS, but it was ignored by existing approaches since they have been perplexed by a more significant error caused by inaccurate optimization. After the error, we point out a few prior conventions that need to be adjusted accordingly, including using different depths (e.g., DARTS used 8 cells in search and 20 cells in evaluation) and widths (e.g., DARTS used a basic channel number of 16 in search and 36 in evaluation) during search and evaluation, as well as using different training strategies (e.g., during re-training, a few regularization techniques including Cutout (DeVries & Taylor, 2017), Dropout (Srivastava et al., 2014) and auxiliary loss were used, but none of them appeared in search). More importantly, the search process was followed by edge removal (8 out of 14 connections were preserved) which caused a significant difference between the network architectures of search and evaluation. Our approach provides the opportunity to bridge the gap between search and evaluation, which we will show in Section 4.1.2 that unifying these hyper-parameters leads to better performance. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A few prior differentiable search approaches noticed the issue of instability, but they chose to solve it in different manners. For example, P-DARTS (Chen et al., 2019) fixed the number of preserved skip-connect operators, PC-DARTS (Xu et al., 2019) used edge normalization to eliminate the none operator, while XNAS (Nayman et al., 2019) and DARTS $^ { + }$ (Liang et al., 2019) introduced a few human expertise to stabilize search. However, we point out that (i) either P-DARTS or PC-DARTS, with carefully designed methods or tricks, can also fail in a sufficiently long search process (more than 200 epochs); and that (ii) XNAS and DART $\\vdots +$ , by adding human expertise, somewhat violated the design nature of AutoML, in which one is expected to avoid introducing too many hand-designed rules. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Another line of NAS, besides differentiable methods, is to use either reinforcement learning or an evolutionary algorithm as a controller of heuristic search and train each sampled network to get some kind of rewards, e.g., validation accuracy. In the viewpoint of optimization, this pipeline mainly differs from the differentiable one in that optimizing $_ { \\pmb { \\alpha } }$ is decoupled from optimizing $\\omega$ , so that it does not require $\\omega$ to arrive at $\\omega ^ { \\ast }$ , but only need a reasonable approximation of $\\omega ^ { \\ast }$ to predict model performance – this is an important reason that such algorithms often produce stable results. Our approach sheds light on introducing a similar property, i.e., robustness to approximated $\\omega ^ { \\star }$ , which helps in stabilizing differentiable search approaches. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 RESULTS ON CIFAR10 ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The CIFAR10 dataset (Krizhevsky & Hinton, 2009) has 50,000 training and 10,000 testing images, sized $3 2 \\times 3 2$ , and equally distributed over 10 classes. We mainly use this dataset to evaluate the stability of our approach, as well as analyze the impacts of different search options and parameters. ",
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+    {
+        "type": "text",
+        "text": "4.1.1 IMPACT OF THE AMENDING COEFFICIENT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first investigate how the amending coefficient, $\\eta$ , defined in Equation 6, impacts architecture search. We search and re-train similarly as DARTS. During the search, all operators are assigned equal weights on each edge. We use a base channel number of 16, and a batch size of 96. An Adam optimizer is used to update architectural parameters, with a learning rate of 0.0003, a weight decay of 0.001 and a momentum of (0.5, 0.999). The number of epochs is to be discussed later. During re-training, the base channel number is increased to 36. An SGD optimizer is used with an initial learning rate of 0.025, decaying following the cosine annealing rule and arriving at 0 after 600 epochs. The weight decay is set to be 0.0003, and the momentum is 0.9. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To arrive at convergence, we run the search stage for 500 epochs. We evaluate different $\\eta$ values from 0 to 1, and the architectures corresponding to $\\eta = 0$ (equivalent to DARTS), $\\eta = 0 . 1$ and $\\eta = 1$ are summarized in Figure 2. We can see that $\\eta = 0 . 1$ converges, after 500, into a reasonable architecture that achieves an error rate of $3 . 0 8 \\%$ on CIFAR10. We emphasize that, even with more search epochs, this architecture is not likely to change, as the preserved operator on each edge has a weight not smaller than 0.5, and most of these weights are still growing gradually. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When $\\eta$ is very small, e.g., $\\eta = 0 . 0 0 1$ or $\\eta = 0 . 0 1$ , the change brought by this amending term to architecture search is ignorable, and our approach shows almost the same behavior as the first-order version of DARTS, i.e., $\\eta = 0$ . In addition, when $\\eta$ grows up, e.g., from 0.001 to 0.01, although the search process eventually runs into an architecture with all skip-connect operators, the number of epochs needed for a complete failure is significantly postponed, which verifies that the amending term indeed pulls architecture search away from the gradient trap. ",
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+    {
+        "type": "image",
+        "img_path": "images/06de76b891eab13d3496f86280dc133177d768ef6a253623c6e6097e0abcaa33.jpg",
+        "image_caption": [
+            "Figure 2: Normal (left) and reduction (right) cells (the standard DARTS space) obtained by different amending coefficients, namely, $\\eta = 0$ (top), $\\eta = 0 . 1$ (middle) and $\\eta = 1$ (bottom). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "On the other hand, if we use a sufficiently large $\\eta$ value, e.g., $\\eta = 1$ , the amending term, $\\mathbf { g } _ { 2 } ^ { \\prime }$ , can dominate optimization, so that the first term, i.e., the gradient of architectural parameters, has limited effects in updating $_ \\alpha$ . Note that the amending term is closely related to network regularization, therefore, in the scenarios of a large $\\eta$ , the network significantly prefers avg-pooling to other operators, as avg-pooling can smooth feature maps and avoid over-fitting. However, avg-pooling is also a parameter-free operator, so the performance of such architectures is also below satisfaction. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Following these analyses, we simply use $\\eta = 0 . 1$ for all later experiments. We do not tune $\\eta$ very carefully, though it is possible to determine $\\eta$ automatically using a held-out validation set. Besides, we find that the best architecture barely changes after 100 search epochs, which we fix the total length to be 100 epochs in all remaining experiments. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1.2 CONSISTENCY BETWEEN SEARCH AND EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "As discussed in Section 3.5, it is important to alleviate the difference between search and evaluation. We make the following modifications, listed from most to least important. First, to avoid edge removal, we fix the edges in each cell, so that each node $i$ is connected to node $i - 1$ and the least indexed node (denoted by $c _ { k - 2 }$ in most conventions), resulting in 8 edges in each cell. Note that our approach also works well with all 14 edges preserved, but we have used 8 edges to be computationally fair to DARTS. Second, we unify the width (the number of basic channels) as 36 for both search and evaluation. Third, we add normalization techniques, including Cutout (DeVries & Taylor, 2017), Dropout (Srivastava et al., 2014) and an auxiliary loss tower, into the search stage. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We use the amending coefficient $\\eta = 0 . 1$ learned from previous experiments, i.e., without modification, the searched architecture, denoted by $\\mathbb { A } _ { \\mathrm { o r i g } }$ , is shown in the middle column of Figure 2. After modification, the obtained architecture, denoted by $\\mathbb { A } _ { \\mathrm { n e w } }$ , is shown in Figure 3. We re-train both networks on CIFAR10, with or without the option that stacking duplicate cells to make the network deeper (with 20 cells). With a standard re-training process, $\\mathbb { A } _ { \\mathrm { o r i g } }$ reports a $3 . 6 7 \\%$ error with 8 cells, and a $3 . 0 8 \\%$ error with 20 cells; and the corresponding numbers are $3 . 2 0 \\%$ and $2 . 8 1 \\%$ for $\\mathbb { A } _ { \\mathrm { n e w } }$ . We find that $\\mathbb { A } _ { \\mathrm { n e w } }$ is consistently better than $\\mathbb { A } _ { \\mathrm { o r i g } }$ , which suggests that alleviating the gap indeed helps. This also reminds us of the significant depth gap (Chen et al., 2019) between search and re-training (the network has 8 cells in search, but 20 cells in re-training), and this gap also obstructs our approach from achieving better performance. We will investigate this issue in the following part. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.1.3 EXPLORING MORE COMPLEX SEARCH SPACES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We note tremendous efforts made by existing approaches to alleviate the depth gap, while our solution is a direct one, thanks to the stability of our approach which enables us to directly explore larger search spaces. Here, we denote the original search space used in DARTS as $S _ { 1 }$ , which has six normal cells and two reduction cells, and all normal cells share the same set of architectural parameters and so do the reduction cells. Note that we have fixed the edges in this space, resulting in the total number of possible architectures to reduce from $1 . 1 \\times 1 0 ^ { 1 8 }$ to $3 . 3 \\times 1 0 ^ { 1 \\bar { 3 } }$ . We also explore a more complex search space, denoted by $S _ { 2 }$ , in which we relax the constraint that either normal cells or reduction cells should be the same, and also the number of cells increases from 8 to 20, to be applied in re-training. Here, limited by GPU memory, we cannot support all seven operators to be searched, so we only choose two, namely skip-connect and sep-conv- $3 x 3$ , which have very different properties. This setting allows a total of $1 . 5 \\times 1 0 ^ { 4 8 }$ architectures to appear, much larger than $S _ { 1 }$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results are listed in Table 1. In $S _ { 1 }$ , our approach achieves a moderate error rate of $2 . 8 1 \\%$ , mainly because the assumption of consistency between search and re-training does not hold. In $S _ { 2 }$ , with directly searching in deep architectures, our result is significantly boosted (an error rate of $2 . 6 0 \\%$ , the architecture is shown in the middle row of Figure 3). Again, we emphasize that we report retraining results based on an converged architecture, which stands out from existing DARTS-based approaches which required early termination. ",
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+    {
+        "type": "text",
+        "text": "In $S _ { 2 }$ , we compare our approach against both DARTS (no amending term) and random search. We observe that DARTS produces weird architectures (the bottom row of Figure 3), that high-layer cells are mostly occupied by skip-connect, which is not likely to fully utilize the ability of the supernetwork. Regarding random search, we follow DARTS by randomly sampling 20 valid architectures from each of $S _ { 1 }$ and $S _ { 2 }$ , and using a small validation dataset to choose the best two architectures for re-training. Given a fixed number of probes, it becomes more and more difficult to sufficiently explore a large space. The deficits of DARTS and random search on CIFAR10 are $0 . 2 5 \\%$ and $0 . { \\bar { 2 } } 9 \\%$ , respectively, which do not seem to be very large arguably because CIFAR10 is relatively easy and our edge-fixed search space guarantees sufficient depth. However, when we transfer these architectures to ImageNet, the deficits become much larger, i.e., with $1 . 7 \\%$ and $0 . 8 \\%$ top-1 accuracy drops, respectively (DARTS produces inferior performance to random search). These results remind us of prior work (Xie et al., 2019) which claimed that random search works sufficiently well in large search spaces. Here, we leave a comment on this debate, demonstrating that a large space indeed raises challenges to architecture search, but a stabilized search algorithm still has the ability of to find more powerful architectures. ",
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+            "Figure 3: Top: normal and reduction cells found in $ { \\boldsymbol { S } } _ { 1 }$ . Middle & Bottom: the overall architecture found in $S _ { 2 }$ by DARTS, with and without the amended term, in which red and blue edges indicate skip-connect and sep-conv- $. 3 x 3$ operators, respectively. "
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+        "text": "4.1.4 COMPARISON TO THE STATE-OF-THE-ARTS ",
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+        "text": "Finally, we compare our approach with recent approaches, in particular, differentiable ones. Result are shown in Table 1. Our approach produces competitive results among state-of-the-arts, although it does not seem to beat others. We note that existing approaches often used additional tricks, e.g., PDARTS assumed a fixed number of skip-connect operators, which shrinks the search space (so as to guarantee stability). More importantly, all these differentiable search approaches must be terminated in an early stage, which makes them less convincing as search has not arrived at convergence. These tricks somewhat violate the ideology of neural architecture search; in comparison, our research, though not producing the best performance, seems going along a correct and promising direction. ",
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+        "table_footnote": [
+            "Table 1: Comparison with state-of-the-art network architectures on CIFAR10. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Architecture</td><td>Test Err.</td><td>Params</td><td>Search Cost</td><td rowspan=\"2\">Search Method</td></tr><tr><td>(%)</td><td>(M)</td><td>(GPU-days)</td></tr><tr><td>DenseNet-BC (Huang et al.,2017)</td><td>3.46</td><td>25.6</td><td>-</td><td>manual</td></tr><tr><td>ENAS (Pham et al.,2018) w/ Cutout</td><td>2.89</td><td>4.6</td><td>0.5</td><td>RL</td></tr><tr><td>NASNet-A (Zoph et al.,2018) w/Cutout</td><td>2.65</td><td>3.3</td><td>1800</td><td>RL</td></tr><tr><td>NAONet-WS (Luo et al.,2018)</td><td>3.53</td><td>3.1</td><td>0.4</td><td>NAO</td></tr><tr><td>Hireachical Evolution (Liu et al.,2018b)</td><td>3.75±0.12</td><td>15.7</td><td>300</td><td>evolution</td></tr><tr><td>AmoebaNet-B (Real et al.,2019) w/ Cutout</td><td>2.55±0.05</td><td>2.8</td><td>3150</td><td>evolution</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>3.41±0.09</td><td>3.2</td><td>225</td><td>SMBO</td></tr><tr><td>DARTS (first-order) (Liu et al.,2019b) w/ Cutout</td><td>3.00±0.14</td><td>3.3</td><td>0.4</td><td>gradient-based</td></tr><tr><td>DARTS (second-order) (Liu et al.,20i9b) w/Cutout</td><td>2.76±0.09</td><td>3.3</td><td>1.0</td><td>gradient-based</td></tr><tr><td>SNAS (moderate) (Xie etal.,2018) w/ Cutout</td><td>2.85±0.02</td><td>2.8</td><td>1.5</td><td>gradient-based</td></tr><tr><td>ProxylessNAS (Cai et al.,2019) w/Cutout</td><td>2.08</td><td>-</td><td>4.0</td><td>gradient-based</td></tr><tr><td>P-DARTS (Chen et al.,2019) w/ Cutout</td><td>2.50</td><td>3.4</td><td>0.3</td><td>gradient-based</td></tr><tr><td>BayesNAS (Zhou et al.,2019) w/ Cutout</td><td>2.81±0.04</td><td>3.4</td><td>0.2</td><td>gradient-based</td></tr><tr><td>PC-DARTS (Xu et al.,2019) w/ Cutout</td><td>2.57±0.07</td><td>3.6</td><td>0.1</td><td>gradient-based</td></tr><tr><td>Amended-DARTS,S1,w/Cutout</td><td>2.81±0.21</td><td>3.5</td><td>1.0</td><td>gradient-based</td></tr><tr><td>Amended-DARTS,S2,w/Cutout</td><td>2.60±0.15</td><td>3.6</td><td>1.1</td><td>gradient-based</td></tr></table>",
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+        "table_footnote": [
+            "Table 2: Comparison with state-of-the-art architectures on ILSVRC2012. All searched architectures are fit into the mobile setting. ‡ indicates architectures searched on ImageNet. "
+        ],
+        "table_body": "<table><tr><td rowspan=\"2\">Architecture</td><td colspan=\"2\">Test Err. (%)</td><td rowspan=\"2\">Params (M)</td><td rowspan=\"2\">×+ (M)</td><td rowspan=\"2\">Search Cost (GPU-days)</td><td rowspan=\"2\">Search Method</td></tr><tr><td>top-1</td><td>top-5</td></tr><tr><td>Inception-v1 (Szegedy et al.,2015)</td><td>30.2</td><td>10.1</td><td>6.6</td><td>1448</td><td>-</td><td>manual</td></tr><tr><td>MobileNet (Howard etal.,2017)</td><td>29.4</td><td>10.5</td><td>4.2</td><td>569</td><td></td><td>manual</td></tr><tr><td>ShuffleNet 2× (v1) (Zhang et al.,2018)</td><td>26.4</td><td>10.2</td><td>~5</td><td>524</td><td></td><td>manual</td></tr><tr><td>ShuffleNet 2× (v2) (Ma et al.,2018)</td><td>25.1</td><td>-</td><td>~5</td><td>591</td><td>-</td><td>manual</td></tr><tr><td>NASNet-A (Zoph et al., 2018)</td><td>26.0</td><td>8.4</td><td>5.3</td><td>564</td><td>1800</td><td>RL</td></tr><tr><td>MnasNet-92 (Tan et al.,2019)</td><td>25.2</td><td>8.0</td><td>4.4</td><td>388</td><td>-</td><td>RL</td></tr><tr><td>PNAS (Liu et al., 2018a)</td><td>25.8</td><td>8.1</td><td>5.1</td><td>588</td><td>225</td><td>SMBO</td></tr><tr><td>AmoebaNet-C (Real et al.,2019)</td><td>24.3</td><td>7.6</td><td>6.4</td><td>570</td><td>3150</td><td>evolution</td></tr><tr><td>DARTS (second-order) (Liu et al.,2019b)</td><td>26.7</td><td>8.7</td><td>4.7</td><td>574</td><td>4.0</td><td>gradient-based</td></tr><tr><td>SNAS (mild) (Xie et al., 2018)</td><td>27.3</td><td>9.2</td><td>4.3</td><td>522</td><td>1.5</td><td>gradient-based</td></tr><tr><td>BayesNAS (Zhou et al.,2019)</td><td>26.5</td><td>8.9</td><td>3.9</td><td>-</td><td>0.2</td><td>gradient-based</td></tr><tr><td>P-DARTS (CIFAR10) (Chen et al.,2019)</td><td>24.4</td><td>7.4</td><td>4.9</td><td>557</td><td>0.3</td><td>gradient-based</td></tr><tr><td>ProxylessNAS (GPU) (Cai et al., 2019)</td><td>24.9</td><td>7.5</td><td>7.1</td><td>465</td><td>8.3</td><td>gradient-based</td></tr><tr><td>PC-DARTS (Xu et al., 2019)‡</td><td>24.2</td><td>7.3</td><td>5.3</td><td>597</td><td>3.8</td><td>gradient-based</td></tr><tr><td>DARTS+ (Liang et al., 2019)‡</td><td>23.9</td><td>7.4</td><td>5.1</td><td>582</td><td>6.8</td><td>gradient-based</td></tr><tr><td>Amended-DARTS, S2</td><td>24.3</td><td>7.4</td><td>5.5</td><td>590</td><td>1.1</td><td>gradient-based</td></tr></table>",
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+        "text": "4.2 RESULTS ON ILSVRC2012 ",
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+        "text": "ILSVRC2012 (Russakovsky et al., 2015) is the most commonly used subset of ImageNet (Deng et al., 2009). It contains 1.3M training images and 50K testing images, which are almost evenly distributed over all 1,000 categories. We directly use the $S _ { 2 }$ architecture obtained from CIFAR10 experiments and enlarge it with a basic number of channels of 42, so that the FLOPs of our model is 590M, i.e., fitting the mobile setting. During re-training, there are a total of 250 epochs. We use an SGD optimizer with an initial learning rate of 0.5 (decaying linearly after each epoch), a momentum of 0.9 and a weight decay of $3 \\times 1 0 ^ { - 5 }$ . On NVIDIA Tesla V100 GPUs, the entire re-training process takes around 24 GPU-days. ",
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+        "type": "text",
+        "text": "The comparison of our approach to existing approaches is shown in Table 2. Our approach achieves a top-1 error rate of $2 4 . { \\bar { 3 } } \\bar { \\% }$ without any common optimization tricks such as AutoAugment (Cubuk et al., 2019) and Squeeze-and-Excitation modules (Hu et al., 2018). This result is competitive among state-of-the-arts, and it is obtained after convergence is achieved in the search stage. On the other hand, without the amending term, DARTS converges to a solution that skip-connected and sep-conv$3 x 3$ are largely separated, on which the re-training performance is even inferior to random search. ",
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+        "type": "text",
+        "text": "5 CONCLUSIONS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we present an effective approach for stabilizing differentiable neural architecture search. Our motivation comes from that DARTS-based approaches mostly generate all-skip-connect architectures when they are executed for a sufficient number of epochs. We analyze this weird phenomenon mathematically and find the reason to lie in the dramatic inaccuracy in estimating gradients of architectural parameters. With an alternative approximation based on the optimality of network parameters, we guarantee the update of architectural parameters to be in a correct direction. In DARTS-based search spaces on CIFAR10 and ImageNet, our approach shows improved stability, in particular in large search spaces, as well as improved performance in the re-training stage. ",
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+        "text": "Our research sheds light on future research on NAS in several aspects. First, we reveal that previous differentiable approaches were mostly built upon a dangerous pipeline, and mostly introduced heavy human expertise to avoid failure. By fixing the ‘system error’ of this pipeline, we provide a platform that NAS approaches can compete in the ability of NAS. Second, our approach enables researchers to explore even bigger search spaces that have not been studied before (due to search instability). ",
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+        "text": "REFERENCES   \nH. Cai, T. Chen, W. Zhang, Y. Yu, and J. Wang. Efficient architecture search by network transformation. In AAAI Conference on Artificial Intelligence, 2018.   \nH. Cai, L. Zhu, and S. Han. Proxylessnas: Direct neural architecture search on target task and hardware. 2019.   \nX. Chen, L. Xie, J. Wu, and Q. Tian. Progressive differentiable architecture search: Bridging the depth gap between search and evaluation. In International Conference on Computer Vision, 2019.   \nE. D. Cubuk, B. Zoph, D. Mane, V. Vasudevan, and Q. V. Le. Autoaugment: Learning augmentation policies from data. In Computer Vision and Pattern Recognition, 2019.   \nJ. Deng, W. Dong, R. Socher, L. J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009.   \nT. DeVries and G. W. Taylor. Improved regularization of convolutional neural networks with cutout. arXiv preprint arXiv:1708.04552, 2017.   \nG. Ghiasi, T. Y. Lin, and Q. V. Le. Nas-fpn: Learning scalable feature pyramid architecture for object detection. In Computer Vision and Pattern Recognition, 2019.   \nZ. Guo, X. Zhang, H. Mu, W. Heng, Z. Liu, Y. Wei, and J. Sun. Single path one-shot neural architecture search with uniform sampling. arXiv preprint arXiv:1904.00420, 2019.   \nK. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Computer Vision and Pattern Recognition, 2016.   \nA. G. Howard, M. Zhu, B. Chen, D. Kalenichenko, W. Wang, T. Weyand, M. Andreetto, and H. Adam. Mobilenets: Efficient convolutional neural networks for mobile vision applications. arXiv preprint arXiv:1704.04861, 2017.   \nJ. Hu, L. Shen, and G. Sun. Squeeze-and-excitation networks. In Computer Vision and Pattern Recognition, 2018.   \nG. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In Computer Vision and Pattern Recognition, 2017.   \nA. Krizhevsky, I. Sutskever, and G. E. Hinton. Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, 2012.   \nAlex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, 2009.   \nY. LeCun, Y. Bengio, and G. E. Hinton. Deep learning. Nature, 521(7553):436, 2015.   \nH. Liang, S. Zhang, J. Sun, X. He, W. Huang, K. Zhuang, and Z. Li. Darts+: Improved differentiable architecture search with early stopping. arXiv preprint arXiv:1909.06035, 2019.   \nC. Liu, B. Zoph, M. Neumann, J. Shlens, W. Hua, L. J. Li, L. Fei-Fei, A. L. Yuille, J. Huang, and K. Murphy. Progressive neural architecture search. In European Conference on Computer Vision, 2018a.   \nC. Liu, L. C. Chen, F. Schroff, H. Adam, W. Hua, A. L. Yuille, and L. Fei-Fei. Auto-deeplab: Hierarchical neural architecture search for semantic image segmentation. In Computer Vision and Pattern Recognition, 2019a.   \nH. Liu, K. Simonyan, O. Vinyals, C. Fernando, and K. Kavukcuoglu. Hierarchical representations for efficient architecture search. In International Conference on Learning Representations, 2018b.   \nH. Liu, K. Simonyan, and Y. Yang. Darts: Differentiable architecture search. In International Conference on Learning Representations, 2019b.   \nR. Luo, F. Tian, T. Qin, E. Chen, and T. Y. Liu. Neural architecture optimization. In Advances in Neural Information Processing Systems, 2018.   \nN. Ma, X. Zhang, H. T. Zheng, and J. Sun. Shufflenet v2: Practical guidelines for efficient cnn architecture design. In European Conference on Computer Vision, 2018.   \nN. Nayman, A. Noy, T. Ridnik, I. Friedman, R. Jin, and L. Zelnik-Manor. Xnas: Neural architecture search with expert advice. arXiv preprint arXiv:1906.08031, 2019.   \nA. Noy, N. Nayman, T. Ridnik, N. Zamir, S. Doveh, I. Friedman, R. Giryes, and L. Zelnik-Manor. Asap: Architecture search, anneal and prune. arXiv preprint arXiv:1904.04123, 2019.   \nH. Pham, M. Guan, B. 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Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In Computer Vision and Pattern Recognition, 2015.   \nM. Tan and Q. V. Le. Efficientnet: Rethinking model scaling for convolutional neural networks. In International Conference on Machine Learning, 2019.   \nM. Tan, B. Chen, R. Pang, V. Vasudevan, M. Sandler, A. Howard, and Q. V. Le. Mnasnet: Platformaware neural architecture search for mobile. In Computer Vision and Pattern Recognition, 2019.   \nL. Xie and A. L. Yuille. Genetic CNN. In International Conference on Computer Vision, 2017.   \nS. Xie, H. Zheng, C. Liu, and L. Lin. SNAS: Stochastic neural architecture search. arXiv preprint arXiv:1812.09926, 2018.   \nS. Xie, A. Kirillov, R. Girshick, and K. He. Exploring randomly wired neural networks for image recognition. In International Conference on Computer Vision, 2019.   \nY. Xu, L. Xie, X. Zhang, X. Chen, G. J. Qi, Q. Tian, and H. Xiong. Pc-darts: Partial channel connections for memory-efficient differentiable architecture search. arXiv preprint arXiv:1907.05737, 2019.   \nX. Zhang, X. Zhou, M. Lin, and J. Sun. Shufflenet: An extremely efficient convolutional neural network for mobile devices. In Computer Vision and Pattern Recognition, 2018.   \nH. Zhou, M. Yang, J. Wang, and W. Pan. Bayesnas: A bayesian approach for neural architecture search. arXiv preprint arXiv:1905.04919, 2019.   \nB. Zoph and Q. V. Le. Neural architecture search with reinforcement learning. In International Conference on Learning Representations, 2017.   \nB. Zoph, V. Vasudevan, J. Shlens, and Q. V. Le. Learning transferable architectures for scalable image recognition. In Computer Vision and Pattern Recognition, 2018. ",
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+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A.1 A TOY EXAMPLE TO SHOW THE IMPORTANCE OF $\\mathbf { g } _ { 2 }$ ",
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+        "type": "text",
+        "text": "Let the loss function be $\\begin{array} { l l l } { { \\mathcal L } \\left( \\omega , \\alpha ; x \\right) } & { = } & { \\left( \\omega x - \\alpha \\right) ^ { 2 } } \\end{array}$ . Then, the only difference between $\\mathcal { L } _ { \\mathrm { t r a i n } } \\left( \\omega , \\alpha \\right) = \\mathcal { L } \\left( \\omega , \\alpha ; x _ { \\mathrm { t r a i n } } \\right)$ and $\\mathcal { L } _ { \\mathrm { v a l i d } } \\left( \\omega , \\alpha \\right) = \\mathcal { L } \\left( \\omega , \\alpha ; x _ { \\mathrm { v a l i d } } \\right)$ lies in the input, $_ { \\textbf { \\em x } }$ . Assume the input of training data is ${ x } _ { \\mathrm { t r a i n } } = 1$ and the input validation data is $\\pmb { x } _ { \\mathrm { v a l i d } } = 2$ . It is easy to derive that the local optimum of $\\mathcal { L } _ { \\mathrm { t r a i n } } \\left( \\omega , \\alpha \\right)$ is $\\omega ^ { \\ast } \\left( \\alpha \\right) = \\alpha$ . Substituting $\\pmb { x } _ { \\mathrm { v a l i d } } = 2$ into $\\mathcal { L } _ { \\mathrm { v a l i d } } \\left( \\omega , \\alpha \\right)$ yields ${ \\mathcal { L } } _ { \\mathrm { v a l i d } } \\left( \\omega , \\alpha \\right) = \\left( 2 \\omega - \\alpha \\right) ^ { 2 }$ . When $\\mathbf { \\alpha } _ { \\alpha } = \\mathbf { \\alpha } _ { \\mathbf { \\alpha } _ { \\mathbf { t } } }$ , $\\omega$ arrives at $\\omega ^ { \\ast } \\left( \\alpha _ { \\mathrm { t } } \\right)$ , so $\\mathbf { g } _ { 1 } = 2 \\left( \\alpha _ { \\mathrm { t } } - 2 \\alpha _ { \\mathrm { t } } \\right) = - 2 \\alpha _ { \\mathrm { t } }$ and $\\mathbf { g } _ { 2 } = 4 \\alpha _ { \\mathrm { t } }$ . When $\\omega$ arrives at ${ { \\omega } ^ { * } } \\left( \\alpha \\right) , { { \\mathcal { L } } _ { v a l i d } } \\left( { { \\omega } ^ { * } } \\left( \\alpha \\right) , \\alpha \\right) = { { \\alpha } ^ { 2 } }$ , $\\mathbf { g } \\left( \\alpha _ { \\mathrm { t } } \\right) = 2 \\alpha _ { \\mathrm { t } } = \\mathbf { g } _ { 1 } + \\mathbf { g } _ { 2 }$ . In summary, both $\\mathbf { g } _ { 1 }$ and $\\mathbf { g } _ { 2 }$ are important, but DARTS chose to ignore $\\mathbf { g } _ { 2 }$ which can cause a dramatic error in approximation. ",
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+        "text": "A.2 THE COMPLEXITY OF $\\mathbf { g } _ { 2 } ^ { ' }$ CAN BE SUBSTANTIALLY REDUCED USING THE FINITEDIFFERENCE APPROXIMATION",
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+    {
+        "type": "text",
+        "text": "We use the finite difference approximation just like DARTS. Let $\\epsilon$ be a small scalar, $\\omega _ { 1 } \\ = \\ \\omega + \\epsilon \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha )$ , $\\omega _ { 2 } ~ = ~ \\omega - \\epsilon \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha )$ . Then: $\\begin{array} { r l } { \\mathbf { H } \\cdot \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega , \\alpha ) } & { = } \\end{array}$ $\\frac { \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega _ { 1 } , \\alpha ) - \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega _ { 2 } , \\alpha ) } { \\mathrm { . } }$ . $\\begin{array} { r l r } { \\omega _ { 3 } } & { { } = } & { \\omega \\ + \\ \\frac { \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega _ { 1 } , \\alpha ) - \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega _ { 2 } , \\alpha ) } { 2 } } \\end{array}$ , $\\omega _ { 4 } = \\omega - \\frac { } { }$ 2\u000f . Then: $\\frac { \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega _ { 1 } , \\alpha ) ^ { - \\infty } \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { t r a i n } } ( \\omega _ { 2 } , \\alpha ) } { 2 }$ $\\begin{array} { r } { \\mathbf { g } _ { 2 } ^ { \\prime } = - \\pmb { \\eta } \\times \\frac { \\nabla _ { \\omega } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega _ { 3 } , \\alpha ) - \\bar { \\nabla _ { \\omega } } \\mathcal { L } _ { \\mathrm { v a l } } ( \\omega _ { 4 } , \\alpha ) } { 2 \\epsilon } } \\end{array}$ ",
+        "bbox": [
+            174,
+            347,
+            825,
+            415
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 A TOY EXAMPLE TO SHOW THE IMPORTANCE OF THE “GRADIENT TRAP” ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            435,
+            717,
+            449
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We have a small toy case to show the influence of the “gradient trap”. We searched for a small super-network in the DARTS’s search space, which only has two cells(we searched for 600 epochs). ",
+        "bbox": [
+            171,
+            463,
+            823,
+            492
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "When we train the super-network in the “training sets in search phase” and validate in the “validation sets in search phase”, the test error using $\\mathbf { g } _ { 2 } ^ { ' }$ is $1 0 . 5 \\%$ while the test error without ${ \\bf g } _ { 2 } ^ { ' }$ is $1 2 . 8 \\%$ . ",
+        "bbox": [
+            174,
+            498,
+            825,
+            530
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "When we train the super-network in the training sets and validate in the validation sets, the test error using ${ \\bf g } _ { 2 } ^ { ' }$ is $5 . 4 \\%$ while the test error without $\\mathbf { g } _ { 2 } ^ { ' }$ is $7 . 4 \\%$ . ",
+        "bbox": [
+            173,
+            535,
+            823,
+            568
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "When we generalize and train the network in the training sets and validate in the validation sets, the test error using ${ \\bf g } _ { 2 } ^ { ' }$ is $6 . 2 \\%$ while the test error without $\\mathbf { g } _ { 2 } ^ { \\prime }$ is $7 . 4 \\%$ . ",
+        "bbox": [
+            173,
+            573,
+            823,
+            604
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this case the “gradient trap” will cause a dramatic accuracy drop of the super-network. ",
+        "bbox": [
+            171,
+            609,
+            754,
+            626
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 SEARCH WITH DIFFERENT SEEDS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            647,
+            446,
+            661
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We ran our search algorithms with different seeds for 5 times in $S _ { 2 }$ and evaluated each discovered architecture for 3 times. The lowest test error is $2 . 5 7 { \\pm } 0 . 1 1 \\%$ and the highest is $2 . 6 3 { \\pm } 0 . 1 3 \\%$ . We did the same thing on $S _ { 1 }$ and the lowest and the highest test errors are $2 . 7 1 \\pm 0 . 1 5 \\%$ and $2 . 9 2 \\pm 0 . 0 9 \\%$ , respectively. ",
+        "bbox": [
+            173,
+            675,
+            825,
+            732
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As we expected, the results in $S _ { 1 }$ are less robust than those in $S _ { 2 }$ . The main reason is the difference between search and evaluation, including the different depths of search and evaluation networks and the the discretization stage of the standard DARTS method. ",
+        "bbox": [
+            174,
+            738,
+            823,
+            781
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "More importantly, our approach survives after 500 (and even more) epochs, while DARTS degenerates to an all-skip-connect architecture in all $( 1 0 + )$ individual runs. ",
+        "bbox": [
+            173,
+            787,
+            823,
+            816
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.5 THEORETICAL ANALYSIS OF SEMI-POSITIVE-DEFINITE MATRIX AFTER SIMILARITY TRANSFORMATION ",
+        "bbox": [
+            173,
+            840,
+            787,
+            867
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "$\\mathbf { A } = \\mathbf { C } ^ { T } \\cdot \\mathbf { C } .$ , A is a semi-positive-definite matrix. In this case, the number of different eigenvalues is far less than the dimension of A(hundreds compared to millions). $\\mathbf { H }$ is a real symmetric positivedefinite matrix. ",
+        "bbox": [
+            176,
+            881,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Let $\\big \\{ \\alpha _ { \\mathrm { i } } \\big \\}$ be a set of eigenvectors w.r.t $\\mathbf { A } ( \\alpha _ { \\mathrm { i } } ^ { T } \\cdot \\alpha _ { \\mathrm { i } } = 1 )$ ), then $\\{ \\mathbf { H } \\cdot \\mathbf { \\boldsymbol { \\alpha } } _ { \\mathrm { i } } \\}$ is a set of eigenvectors w.r.t $\\mathbf { H } \\cdot \\dot { \\mathbf { A } } \\cdot \\dot { \\mathbf { H } } ^ { - 1 }$ , $\\{ \\mathbf { H } ^ { - 1 } \\cdot \\mathbf { \\bar { \\alpha } } _ { \\mathrm { i } } \\}$ is a set of eigenvectors w.r.t $\\mathbf { H } ^ { - 1 } \\cdot \\mathbf { A } \\cdot \\mathbf { H }$ , and $\\{ \\lambda _ { \\mathrm { i } } \\}$ is a set of eigenvectors shared by them. ",
+        "bbox": [
+            173,
+            102,
+            825,
+            146
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Let $\\beta$ be an eigenvector $\\begin{array} { r } { \\mathbf { \\operatorname { p f } } \\mathbf { H } { \\cdot } \\mathbf { A } { \\cdot } \\mathbf { H } ^ { - 1 } + \\mathbf { H } ^ { - 1 } { \\cdot } \\mathbf { A } { \\cdot } \\mathbf { H } , \\beta = \\sum _ { \\mathrm { i } = 1 } ^ { \\mathrm { n } } a _ { \\mathrm { i } } \\times \\mathbf { H } \\cdot \\alpha _ { \\mathrm { i } } = \\sum _ { \\mathrm { i } = 1 } ^ { \\mathrm { n } } b _ { \\mathrm { i } } \\times \\mathbf { H } ^ { - 1 } \\cdot \\alpha _ { \\mathrm { i } } . } \\end{array}$ \u0000 $\\begin{array} { r l } & { \\qquad \\mathrm { E } : = \\dots \\qquad \\mathrm { A } : \\mathrm { H } ^ { - 1 } + \\mathrm { H } ^ { - 1 } \\cdot \\mathrm { A } \\cdot \\mathrm { H } \\cdot \\mathrm { \\Lambda } \\mathrm { H } \\cdot \\mathrm { \\Lambda } \\beta = \\mathrm { \\Lambda } \\lambda \\beta , \\quad \\mathrm { ~ a } : = 1 \\times \\mathrm { H } \\cdot \\mathrm { \\Lambda } \\alpha _ { \\mathrm { i } } + \\sum _ { \\mathrm { i } = 1 } ^ { \\mathrm { n } } b _ { \\mathrm { i } } \\lambda _ { \\mathrm { i } } \\times \\mathrm { H } ^ { - 1 } \\cdot \\alpha _ { \\mathrm { i } } = } \\\\ & { \\underset { \\mathrm { - i } = 1 } { \\sum } a _ { \\mathrm { i } } \\lambda \\times \\mathrm { H } \\cdot \\alpha _ { \\mathrm { i } } = \\sum _ { \\mathrm { i } = 1 } ^ { \\mathrm { n } } b _ { \\mathrm { i } } \\lambda \\times \\mathrm { H } ^ { - 1 } \\cdot \\alpha _ { \\mathrm { i } } . } \\\\ & { \\underset { \\mathrm { - i } = 1 } { \\sum } \\left( \\lambda _ { \\mathrm { i } } - \\lambda \\right) a _ { \\mathrm { i } } b _ { \\mathrm { i } } = - \\sum _ { \\mathrm { i } = 1 } ^ { \\mathrm { n } } \\left( \\lambda _ { \\mathrm { i } } - \\lambda \\right) b _ { \\mathrm { i } } \\alpha _ { \\mathrm { i } } ^ { T } \\cdot \\mathrm { H } ^ { - 1 } \\cdot \\sum _ { \\mathrm { i } = 1 } ^ { \\mathrm { n } } \\left( \\lambda _ { \\mathrm { i } } - \\lambda \\right) b _ { \\mathrm { i } } \\mathrm { H } ^ { - 1 } \\cdot \\alpha _ { \\mathrm { i } } \\le 0 } \\end{array}$ ",
+        "bbox": [
+            173,
+            150,
+            830,
+            208
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A is a real symmetric matrix, so we have many sets of $\\big \\{ \\alpha _ { \\mathrm { i } } \\big \\}$ that is orthogonal to each other. $\\beta ^ { T } \\cdot \\beta =$ $\\textstyle \\sum _ { \\mathrm { i } = 1 } ^ { \\mathrm { n } } a _ { \\mathrm { i } } b _ { \\mathrm { i } } \\geq 0$ , For every eigenvalue, the dimension of the subspace will be very high in the case of neural architecture search, so we assume that we can choose a set of $\\big \\{ \\alpha _ { \\mathrm { i } } \\big \\}$ orthogonal to each other from the subspace satisfying Pni+1j=ni $\\sum _ { \\mathrm { j = n _ { i } } } ^ { \\mathrm { n _ { i + 1 } - 1 } } a _ { \\mathrm { j } } b _ { \\mathrm { j } } \\geq 0$ in most cases $\\mathrm { { \\acute { n } _ { i } } }$ is the id of the first eigenvector w.r.t $\\lambda _ { \\mathrm { i } }$ ). ",
+        "bbox": [
+            173,
+            237,
+            826,
+            313
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { \\sum _ { \\mathrm { i = 1 } } ^ { \\mathrm { n } } \\left( \\lambda _ { \\mathrm { i } } - \\lambda \\right) a _ { \\mathrm { i } } b _ { \\mathrm { i } } = \\sum _ { \\mathrm { i = 1 } } ^ { \\mathrm { n } } \\left( \\lambda _ { \\mathrm { i } } - \\lambda \\right) \\sum _ { \\mathrm { j = n _ { i } } } ^ { \\mathrm { n _ { i + 1 } - 1 } } a _ { \\mathrm { j } } b _ { \\mathrm { j } } \\le 0 } \\end{array}$ so $\\boldsymbol { \\lambda }$ cannot be smaller than zero. ",
+        "bbox": [
+            169,
+            318,
+            776,
+            339
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "All of the eigenvalues w.r.t real symmetric matrix $\\mathbf { H } \\cdot \\mathbf { A } \\cdot \\mathbf { H } ^ { - 1 } + \\mathbf { H } ^ { - 1 } \\cdot \\mathbf { A } \\cdot \\mathbf { H }$ is not smaller than zero, so it is semi-positive-definite. Then we get that $\\mathbf { H } ^ { - 1 } \\cdot \\mathbf { A } \\cdot \\mathbf { H }$ is semi-positive-definite. ",
+        "bbox": [
+            173,
+            344,
+            823,
+            375
+        ],
+        "page_idx": 13
+    }
+]

parse/train/DktZb97_Fx/DktZb97_Fx.md

@@ -0,0 +1,634 @@
+# SENSEI: SENSITIVE SET INVARIANCE FOR ENFORCING INDIVIDUAL FAIRNESS
+
+# Yuekai Sun
+
+Mikhail Yurochkin   
+IBM Research   
+MIT-IBM Watson AI Lab   
+mikhail.yurochkin@ibm.com
+
+Department of Statistics University of Michigan yuekai@umich.edu
+
+# ABSTRACT
+
+In this paper, we cast fair machine learning as invariant machine learning. We first formulate a version of individual fairness that enforces invariance on certain sensitive sets. We then design a transport-based regularizer that enforces this version of individual fairness and develop an algorithm to minimize the regularizer efficiently. Our theoretical results guarantee the proposed approach trains certifiably fair ML models. Finally, in the experimental studies we demonstrate improved fairness metrics in comparison to several recent fair training procedures on three ML tasks that are susceptible to algorithmic bias.
+
+# 1 INTRODUCTION
+
+As machine learning (ML) models replace humans in high-stakes decision-making and decisionsupport roles, concern regarding the consequences of algorithmic bias is growing. For example, ML models are routinely used in criminal justice and welfare to supplement humans, but they may have racial, class, or geographic biases (Metz & Satariano, 2020). In response, researchers proposed many formal definitions of algorithmic fairness as a first step towards combating algorithmic bias.
+
+Broadly speaking, there are two kinds of definitions of algorithmic fairness: group fairness and individual fairness. In this paper, we focus on enforcing individual fairness. At a high-level, the idea of individual fairness is the requirement that a fair algorithm should treat similar individuals similarly. Individual fairness was dismissed as impractical because there is no consensus on which users are similar for many ML tasks. Fortunately, there is a flurry of recent work that addresses this issue (Ilvento, 2019; Wang et al., 2019; Yurochkin et al., 2020; Mukherjee et al., 2020). In this paper, we assume there is a similarity metric for the ML task at hand and consider the task of enforcing individual fairness. Our main contributions are:
+
+1. we define distributional individual fairness, a variant of Dwork et al.’s original definition of individual fairness that is (i) more amenable to statistical analysis and (ii) easier to enforce by regularization;   
+2. we develop a stochastic approximation algorithm to enforce distributional individual fairness when training smooth ML models;   
+3. we show that the stochastic approximation algorithm converges and the trained ML model generalizes under standard conditions;   
+4. we demonstrate the efficacy of the approach on three ML tasks that are susceptible to algorithmic bias: income-level classification, occupation prediction, and toxic comment detection.
+
+# 2 ENFORCING INDIVIDUAL FAIRNESS WITH SENSITIVE SET INVARIANCE (SENSEI)
+
+# 2.1 A TRANSPORT-BASED DEFINITION OF INDIVIDUAL FAIRNESS
+
+Let $\mathcal { X }$ and $\mathcal { V }$ be the space of inputs and outputs respectively for the supervised learning task at hand. For example, in classification tasks, $\mathcal { V }$ may be the probability simplex. An ML model is a function $h : \mathcal { X }  \mathcal { Y }$ in a space of functions $\mathcal { H }$ (e.g. the set of all neural nets with a certain architecture).
+
+Dwork et al. (2011) define individual fairness as $L$ -Lipschitz continuity of an ML model $h$ with respect to appropriate metrics on $\mathcal { X }$ and $\mathcal { V }$ :
+
+$$
+d _ { \mathcal { V } } ( h ( x ) , h ( x ^ { \prime } ) ) \leq L d _ { \mathcal { X } } ( x , x ^ { \prime } )
+$$
+
+for all $x , x ^ { \prime } \in { \mathcal { X } }$ . The choice of $d _ { \mathcal { Y } }$ is often determined by the form of the output. For example, if the ML model outputs a vector of the logits, then we may pick the Euclidean norm as $d _ { \mathcal { Y } }$ (Kannan et al., 2018; Garg et al., 2018). The metric $d _ { \mathcal { X } }$ is the crux of (2.1) because it encodes our intuition of which inputs are similar for the ML task at hand. For example, in natural language processing tasks, $d _ { \mathcal { X } }$ may be a metric on word/sentence embeddings that ignores variation in certain sensitive directions. In light of the importance of the similarity metric in (2.1) to enforcing individual fairness, there is also a line of work on learning the similarity metric from data (Ilvento, 2019; Wang et al., 2019; Mukherjee et al., 2020). In our experiments, we adapt the methods from Yurochkin et al. (2020) to learn similarity metrics.
+
+Although intuitive, individual fairness is statistically and computationally intractable. Statistically, it is generally impossible to detect violations of individual fairness on zero measure subset of the sample space. Computationally, individual fairness is a Lipschitz restriction, and such restrictions are hard to enforce. In this paper, we address both issues by lifting (2.1) to the space of probability distributions on $\mathcal { X }$ to obtain an “average case” version of individual fairness. This version (i) is more amenable to statistical analysis, (ii) is easy to enforce by minimizing a data-dependent regularizer, and (iii) preserves the intuition behind Dwork et al. (2011)’s original definition of individual fairness.
+
+Definition 2.1 (distributional individual fairness (DIF)). Let , $\delta > 0$ be tolerance parameters and $\Delta ( { \mathcal { X } } \times { \mathcal { X } } )$ be the set of probability measures on $\mathcal { X } \times \mathcal { X }$ . Define
+
+$$
+R ( h ) \triangleq \left\{ \begin{array} { l l } { \operatorname* { s u p } _ { \Pi \in \Delta ( { \mathcal { X } } \times { \mathcal { X } } ) } } & { { \mathbf { E } } _ { \Pi } \big [ d _ { { \mathcal { Y } } } ( h ( X ) , h ( X ^ { \prime } ) ) \big ] } \\ { s u b j e c t t o } & { { \mathbf { E } } _ { \Pi } \big [ d _ { { \mathcal { X } } } ( X , X ^ { \prime } ) \big ] \leq \epsilon } \\ & { \Pi ( \cdot , { \mathcal { X } } ) = P _ { X } } \end{array} \right\} .
+$$
+
+where $P _ { X }$ is the (marginal) distribution of the inputs in the ML task at hand. An ML model $h$ is $( \epsilon , \delta )$ -distributionally individually fair $( D I F )$ iff $R ( h ) \leq \delta$ .
+
+We remark that DIF only depends on $h$ and $P _ { X }$ . It does not depend on the (conditional) distribution of the labels $P _ { Y \mid X }$ , so it does not depend on the performance of the ML model. In other words, it is possible for a model to perform poorly and be perfectly DIF (e.g. the constant model $h ( x ) = 0$ ).
+
+The optimization problem in (2.2) formalizes correspondence studies in the empirical literature (Bertrand & Duflo, 2016). Here is a prominent example.
+
+Example 2.2. Bertrand & Mullainathan studied racial bias in the US labor market. The investigators responded to help-wanted ads in Boston and Chicago newspapers with fictitious resumes. To manipulate the perception of race, they randomly assigned African-American or white sounding names to the resumes. The investigators concluded there is discrimination against African-Americans because the resumes assigned white names received $50 \%$ more callbacks for interviews than the resumes.
+
+We view Bertrand & Mullainathan’s investigation as evaluating the objective in (2.3) at a special $T$ . Let $\mathcal { X }$ be the space of resumes, and $h : \mathcal { X }  \{ 0 , 1 \}$ be the decision rule that decides whether a resume receives a callback. Bertrand & Mullainathan implicitly pick the $T$ that reassigns the name on a resume from an African-American sounding name to a white one (or vice versa) and measures discrimination with the difference between callback rates before and after reassignment:
+
+$$
+\mathbf { E } _ { P } \left[ 1 \{ h ( X ) \neq h ( T ( X ) ) \} \right] = \mathbf { P } \{ h ( X ) \neq h ( T ( X ) ) \} .
+$$
+
+We consider distributional individual fairness a variant of Dwork et al.’s original definition. It is not a fundamentally new definition of algorithmic fairness because it encodes the same intuition as Dwork et al.’s original definition. Most importantly, it remains an individual notion of algorithmic fairness because it compares individuals to similar (close in $d _ { \mathcal { X } }$ ) individuals.
+
+# 2.2 DIF VERSUS INDIVIDUAL FAIRNESS
+
+It is hard to directly compare Dwork et al.’s original definition of individual fairness and DIF directly. First, they are parameterized differently: the original definition (2.1) is parameterized by a Lipschitz constant $L$ , while DIF is parameterized by an $( \epsilon , \delta )$ pair. Intuitively, (2.1) enforces (approximate) invariance at all scales (at any $\epsilon > 0$ ), while DIF only enforces invariance at one scale (determined by the input tolerance parameter). Second, (2.1) enforces invariance uniformly on $\mathcal { X }$ , while (2.2) enforces invariance on average. Although DIF seems a weaker notion of algorithmic fairness than (2.1) (average fairness vs uniform fairness), DIF is actually more stringent in some ways because the constraints in (2.2) are looser. This is evident in the Mongé version of the optimization problem in (2.2):
+
+$$
+\begin{array} { r l } { \operatorname* { s u p } _ { T : \mathcal { X } \to \mathcal { X } } } & { \mathbf { E } _ { P } \left[ d _ { \mathcal { Y } } ( h ( X ) , h ( T ( X ) ) ) \right] } \\ { \mathrm { s u b j e c t ~ t o ~ } } & { \mathbf { E } _ { P } \left[ d _ { \mathcal { X } } ( X , T ( X ) ) \right] \le \epsilon . } \end{array}
+$$
+
+The map corresponding to (2.1)
+
+$$
+\begin{array} { r } { T _ { \mathrm { I F } } ( x ) \triangleq \arg \operatorname* { m a x } _ { d _ { \mathcal { X } } ( x , x ^ { \prime } ) \leq \epsilon } d _ { \mathcal { Y } } ( h ( x ) , h ( x ^ { \prime } ) ) } \end{array}
+$$
+
+maps each $x$ to its worse-case counterpart $x ^ { \prime }$ in (2.1). It is not hard to see that $T _ { \mathrm { I F } }$ is a feasible map for (2.3), but it may not be optimal. This is because (2.3) only restricts $T$ to transport points by at most $\epsilon$ on average; the optimal $T$ may transport some points by more than $\epsilon$ .
+
+To make the two definitions more comparable, we consider an $\epsilon { - } \delta$ version of individual fairness. A model $h : \mathcal { X } \to \mathcal { Y }$ satisfies $( \epsilon , \delta )$ -individual fairness at $x \in \mathcal { X }$ if and only if
+
+$$
+\begin{array} { r } { d _ { \mathcal { Y } } ( h ( x ) , h ( T _ { \mathrm { I F } } ( x ) ) ) = \operatorname* { s u p } _ { d _ { \mathcal { X } } ( x , x ^ { \prime } ) \leq \epsilon } d _ { \mathcal { Y } } ( h ( x ) , h ( x ^ { \prime } ) ) \leq \delta . } \end{array}
+$$
+
+To arrive at (2.4), we start by observing that (2.1) is equivalent to
+
+$$
+\begin{array} { r } { \operatorname* { s u p } _ { x \in \mathcal { X } } d _ { \mathcal { Y } } ( h ( x ) , h ( T _ { \mathrm { I F } } ( x ) ) ) \le L \epsilon \mathrm { f o r } \operatorname * { a n y } x \in \mathcal { X } \mathrm { ~ a n d ~ } \epsilon > } \end{array}
+$$
+
+We fix $x$ and $\epsilon$ and re-parameterize the right side with $\delta$ to obtain (2.4). It is possible to show that if $h$ is $( \epsilon , \delta )$ -DIF, then there exists $\delta ^ { \prime }$ such that it satisfies $( \epsilon , \delta ^ { \prime } )$ -individual fairness for “most” $x$ ’s. We formally state this result in a proposition.
+
+Proposition 2.3. If $h : \mathcal { X } \to \mathcal { Y }$ is $( \epsilon , \delta )$ -DIF, then
+
+$$
+\begin{array} { r } { P _ { X } ( d _ { \mathcal { Y } } ( h ( X ) , h ( T _ { \mathrm { I F } } ( X ) ) ) \ge \tau ) \le \frac { \delta } { \tau } f o r a n y \tau > 0 . } \end{array}
+$$
+
+# 2.3 ENFORCING DIF
+
+There are two general approaches to enforcing invariance conditions such as (2.2). The first is distributionally robust optimization (DRO):
+
+$$
+\begin{array} { r } { \operatorname* { m i n } _ { h \in \mathcal { H } } L _ { \mathrm { a d v } } ( h ) \triangleq \operatorname* { s u p } _ { P ^ { \prime } : W _ { d } ( P , P ^ { \prime } ) \leq \epsilon } { \bf E } _ { P ^ { \prime } } \big [ \ell ( Y ^ { \prime } , h ( X ^ { \prime } ) ) \big ] , } \end{array}
+$$
+
+where $\ell$ is a loss function and $W _ { d } ( P , Q )$ is the Wasserstein distance between distributions on $\mathcal { X } \times \mathcal { V }$ induced by the transport cost function $c ( ( x , y ) , ( x ^ { \prime } y ^ { \prime } ) ) \triangleq d _ { \mathcal { X } } ( x , x ^ { \prime } ) + \infty \cdot 1 \{ y \neq y ^ { \prime } \}$ . This approach is very similar to adversarial training, and it was considered by Yurochkin et al. (2020) for enforcing (their modification of) individual fairness. In this paper, we consider a regularization approach to enforcing (2.2):
+
+$$
+\begin{array} { r } { \operatorname* { m i n } _ { h \in \mathcal { H } } L ( h ) + \rho R ( h ) , \quad L ( h ) \triangleq \mathbf { E } \big [ \ell ( Y , h ( X ) ) \big ] , } \end{array}
+$$
+
+where $\rho > 0$ is a regularization parameter and the regularizer $R$ is defined in (2.2). An obvious advantage of the regularization approach is it allows the user to fine-tune the trade-off between goodness-of-fit and fairness by adjusting $\rho$ (see Figure 1; in Figure 3 of Appendix D we show the lack of such flexibility in the method of Yurochkin et al. (2020)). As we shall see, although the two approaches share many theoretical properties, we show in Section 4 that the regularization approach has superior empirical performance. We defer a more in-depth comparison between the two approaches to subsection 2.4.
+
+At first blush, the regularized risk minimization problem (2.6) is not amenable to stochastic optimization because $R$ is not an expected value of a function of the training examples. Fortunately, by appealing to duality, it is possible to obtain a dual formulation of $R$ that is suitable for stochastic optimization.
+
+Theorem 2.4 (dual form of $R$ ). If $d y ( h ( x ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( x , x ^ { \prime } )$ is continuous (in $( x , x ^ { \prime } )$ ) for any $\lambda \geq 0$ , then
+
+$$
+\begin{array} { r } { R ( h ) = \operatorname* { i n f } _ { \lambda \geq 0 } \{ \lambda \epsilon + { \bf E } _ { P x } \left[ r _ { \lambda } ( h , X ) \right] \} , r _ { \lambda } ( h , X ) \triangleq \operatorname* { s u p } _ { x ^ { \prime } \in \mathcal { X } } \{ d y ( h ( X ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( X , x ^ { \prime } ) \} . } \end{array}
+$$
+
+![](images/eebc18d7d2044098b1c973923f671dada37fee4da7e8c75c7e311fd74b42c1a3.jpg)  
+Figure 1: The decision surface of a one hidden layer neural network trained with SenSeI as the fair regularization parameter $\rho$ varies. In this ML task, points on a horizontal line (points with identical $y$ -values) are similar, but the training data is biased because $P _ { Y \mid X }$ is not constant on horizontal lines. We see that fair regularization (eventually) corrects the bias in the data.
+
+We defer the proof of this result to Appendix A. In light of the dual form of the fair regularizer, the regularized risk minimization problem is equivalently
+
+$$
+\begin{array} { r } { \operatorname* { m i n } _ { h \in \mathcal { H } } \operatorname* { i n f } _ { \lambda \ge 0 } { \bf E } _ { P } \big [ \ell ( Y , h ( X ) ) + \rho ( \lambda \epsilon + r _ { \lambda } ( h , X ) ) \big ] , } \end{array}
+$$
+
+where $r _ { \lambda }$ is defined in Theorem 2.4, which has the form of minimizing an expected value of a function of the training examples. To optimize with respect to $h$ , we parameterize the function space $\mathcal { H }$ with a parameter $\mathbf { \bar { \theta } } \in \Theta \mathsf { \bar { \Lambda } } \subset \mathbf { R } ^ { d }$ and consider a stochastic approximation approach to finding the best parameter. Let $w \triangleq ( \theta , \lambda )$ and $Z \triangleq ( X , Y )$ . The stochastic optimization problem we wish to solve is
+
+$\begin{array} { r } { \operatorname* { m i n } _ { w \in \Theta \times { \bf R } _ { + } } F ( w ) \triangleq { \bf E } _ { P } \big [ f ( w , Z ) \big ] , \quad f ( w , Z ) \triangleq \ell ( Y , h _ { \theta } ( X ) ) + \rho ( \lambda \epsilon + r _ { \lambda } ( h _ { \theta } , X ) ) . } \end{array}$ (2.8) It is not hard to see that (2.8) is a stochastic optimization problem. We summarize Sensitive Set Invariance (SenSeI) for stochastic optimization of (2.8) in Algorithm 1.
+
+# Algorithm 1 SenSeI: Sensitive Set Invariance
+
+<table><tr><td>inputs: starting point (0o,Xo), step sizes (nt)</td></tr><tr><td>repeat</td></tr><tr><td>(Xt1,Yt1),...,(Xtb,YtB)~P &gt; sample mini-batch from P</td></tr><tr><td>xtb←argmax&#x27;∈x{dy(hθ(Xtb),h(x&#x27;))-Xtdx(Xtb,x&#x27;)},b∈[B] generate worst-case examples</td></tr><tr><td></td></tr><tr><td>λt+1←max{0,Xt-ntp(∈-B∑b=1dx(Xtb,xtb)},</td></tr><tr><td>0t+1←t-nt(∑-{e(Yth(t）)）+p{dy(h(Xt），(x））)） until converged</td></tr></table>
+
+# 2.4 ADVERSARIAL TRAINING VS FAIR REGULARIZATION
+
+Adversarial training is a popular approach to training invariant ML models. It was originally developed to defend ML models against adversarial examples (Goodfellow et al., 2014; Madry et al., 2017). There are many versions of adversarial training; the Wasserstein distributionally robust optimization (DRO) version by Sinha et al. (2017) is most closely related to (2.5). The direct goal of adversarial training is training ML models whose risk is small on adversarial examples, and the robust risk that adversarial training seeks to minimize (2.5) is exactly the risk on adversarial examples.
+
+An indirect consequence of adversarial training is invariance to imperceptible changes to the inputs. Recall an adversarial example is a training example with the same label as a non-adversarial training example whose inputs differ imperceptibly from those of a training example. Thus (successful) adversarial training leads to ML models that ignore such imperceptible changes and are thus invariant in “imperceptible neighborhoods” of the training examples.
+
+Unlike adversarial training, which leads to invariance as an indirect consequence of adversarial robustness, fair regularization enforces fairness by explicitly minimizing a fair regularizer. A key benefit of invariance regularization is it permits practitioners to fine-tune the trade-off between goodness-of-fit and invariance by adjusting the regularization parameter (see Figure 1).
+
+# 2.5 RELATED WORK
+
+There are three lines of work on enforcing individual fairness. There is a line of work that enforces group fairness with respect to many (possibly overlapping) groups to avoid disparate treatment of individuals (Hébert-Johnson et al., 2017; Kearns et al., 2017; Kim et al., 2018a;b). At a high-level, these methods repeatedly find groups in which group fairness is violated and updates the ML model to correct violations. Compared to these methods that approximate individual fairness with group fairness, we directly enforce individual fairness.
+
+There is another line of work on enforcing individual fairness without knowledge of the similarity metric. Gillen et al. (2018); Rothblum & Yona (2018); Jung et al. (2019) reduce the problem of enforcing individual fairness to a supervised learning problem by minimizing the number of violations. Instead of a similarity metric, these algorithms rely on an oracle that detects violations of individual fairness. Garg et al. (2018) enforce individual fairness by penalizing the expected difference in predictions between counterfactual inputs. Instead of a similarity metric, this algorithm relies a way of generating counterfactual inputs. Our approach complements these methods by relying on a similarity metric instead of such oracles. This allows us to take advantage of recent work on learning fair metrics (Ilvento, 2019; Wang et al., 2019; Yurochkin et al., 2020; Mukherjee et al., 2020).
+
+Most similar to our work is SenSR (Yurochkin et al., 2020) that also assumes access to a similarity metric. SenSR is based on adversarial training, i.e. it enforces a risk-based notion of individual fairness that only requires the risk of the ML model to be similar on similar inputs. In contrast, our method is based on fair regularization, i.e. it enforces a notion of individual fairness that requires the outputs of the ML model to be similar. The latter is stronger (it implies the former) and is much closer to Dwork et al.’s original definition. In addition, the risk-based notion of SenSR ties accuracy to fairness. Our approach separates these two (usually conflicting) goals and allows practitioners to more easily adjust the trade-off between accuracy and fairness as demonstrated in our experimental studies.
+
+Finally, there is a line of work that proposes pre-processing techniques to learn a fair representation of the individuals and training an ML model that accepts the fair representation as input (Zemel et al., 2013; Bower et al., 2018; Madras et al., 2018; Lahoti et al., 2019). These works are complimentary to ours as we propose an in-processing algorithm.
+
+# 3 THEORETICAL PROPERTIES OF SENSEI
+
+In this section, we describe some theoretical properties of SenSeI. We defer all proofs to Appendix A. First, Algorithm 1 is an instance of a stochastic gradient method, and its convergence properties are well-studied. Even if $f ( w , Z )$ is non-convex in $w$ , the algorithm converges (globally) to a stationary point (see Appendix A for a rigorous statement). Second, the fair regularizer is data-dependent, and it is unclear whether minimizing its empirical counterpart (3.1) enforces distributional fairness. We show that the fair regularizer generalizes under standard conditions. Consequently,
+
+1. it is possible for practitioners to certify that an ML model $h$ is DIF a posteriori by checking $\hat { R } ( h )$ (even if $h$ was trained by minimizing $\widehat { R } )$ ;   
+2. fair regularization enforces distributional individual fairness (as long as the hypothesis class includes DIF ML models);
+
+Notation Let $\{ ( X _ { i } , Y _ { i } ) \} _ { i = 1 } ^ { n }$ be the training set and ${ \widehat { P } } _ { X }$ be the empirical distribution of the inputs. Define $\widehat { L } : \mathcal H \to \mathbf { R }$ as the empirical risk and $\widehat { R } : \mathcal { H } \to \mathbf { R }$ as the empirical counterpart of the fair regularizer $R$ (2.2):
+
+$$
+\widehat { R } ( h ) \triangleq \left\{ \begin{array} { l l } { \operatorname* { m a x } _ { \Pi \in \Delta ( \mathcal { X } \times \mathcal { X } ) } } & { \mathbf { E } _ { \Pi } \big [ d y ( h ( X ) , h ( X ^ { \prime } ) ) \big ] } \\ { \mathrm { s u b j e c t ~ t o ~ } } & { \mathbf { E } _ { \Pi } \big [ d _ { \mathcal { X } } ( X , X ^ { \prime } ) \big ] \leq \epsilon } \\ & { \Pi ( \cdot , \mathcal { X } ) = \widehat { P } _ { X } , } \end{array} \right\} .
+$$
+
+Define the loss class $\mathcal { L }$ and its counterpart for the fair regularizer as
+
+$$
+\begin{array} { r l r } & { } & { \mathcal { L } \triangleq \{ \ell _ { h } : \mathcal { Z } \to { \bf R } \mid h \in \mathcal { H } \} , \quad \ell _ { h } ( z ) \triangleq \ell ( h ( x ) , y ) , } \\ & { } & { \mathcal { D } \triangleq \{ d _ { h } : \mathcal { X } \times \mathcal { X } \to { \bf R } _ { + } \mid h \in \mathcal { H } \} , \quad d _ { h } ( x , x ^ { \prime } ) \triangleq d _ { \mathcal { V } } ( h ( x ) , h ( x ^ { \prime } ) ) . } \end{array}
+$$
+
+We measure the complexity of $\mathcal { D }$ and $\mathcal { L }$ with their entropy integrals with respect to to the uniform metric: $\begin{array} { r } { J ( \mathcal { D } / \mathcal { L } ) \triangleq \int _ { 0 } ^ { \infty } \log N ( \mathcal { D } / \mathcal { L } , \| \cdot \| _ { \infty } , \epsilon ) ^ { \frac { 1 } { 2 } } d \epsilon } \end{array}$ , where $N ( \mathcal { D } , \| \cdot \| _ { \infty } , \epsilon )$ is the $\epsilon$ -covering number of $\mathcal { D }$ in the uniform metric. The main benefit of using entropy integrals instead of Rademacher or Gaussian complexities to measure the complexity of $\mathcal { D }$ and $\mathcal { L }$ is it does not depend on the distribution of the training examples. This allows us to obtain generalization error bounds for counterfactual training sets that are similar to the (observed) training set. Finally, define the diameter of $\mathcal { X }$ in the $d _ { \mathcal { X } }$ metric, that of $\mathcal { V }$ in the $d _ { \mathcal { Y } }$ metric as $\begin{array} { r } { D _ { \mathcal { X } } \triangleq \operatorname* { s u p } _ { x , x ^ { \prime } \in \mathcal { X } } d _ { \mathcal { X } } ( x , x ^ { \prime } ) } \end{array}$ , $\begin{array} { r } { D _ { \mathcal { Y } } \triangleq \operatorname* { s u p } _ { y , y ^ { \prime } \in \mathcal { Y } } d _ { \mathcal { Y } } ( y , y ^ { \prime } ) } \end{array}$ . The first result shows that the fair regularizer $R ( h )$ generalizes. We assume $D _ { \mathcal { X } }$ , $D _ { c Y } < \infty$ . This is a boundedness condition on $\mathcal { X } \times \mathcal { V }$ , and it is a common simplifying assumption in statistical learning theory. We also assume $J ( \mathcal { D } ) < \infty$ . This is a standard assumption that appears in many uniform convergence results.
+
+Theorem 3.1. As long as $D _ { \mathcal { X } } , D _ { \mathcal { Y } }$ , and $J ( { \mathcal { G } } )$ are all finite, with probability at least $1 - t$ :
+
+$$
+\begin{array} { r } { \operatorname* { s u p } _ { h \in \mathcal { H } } | \widehat { R } ( h ) - R ( h ) | \leq \frac { 4 8 ( J ( \mathcal { D } ) + \frac { 1 } { \epsilon } D _ { \mathcal { X } } D _ { \mathcal { Y } } ) } { \sqrt { n } } + D _ { \mathcal { Y } } ( \frac { \log \frac { 2 } { t } } { 2 n } ) ^ { \frac { 1 } { 2 } } . } \end{array}
+$$
+
+Theorem 3.1 implies it is possible to certify that an ML model $h$ satisfies distributional individual fairness (modulo error terms that vanish in the large-sample limit) by inspecting $\widehat { R } ( h )$ . This is important because a practitioner may inspect $\widehat { R } ( h )$ after training to verify whether the trained ML model $h$ is fair enough. Theorem 3.1 assures the user that $\widehat { R } ( h )$ is close to $R ( h )$ .
+
+# 4 COMPUTATIONAL EXPERIMENTS
+
+In this section we present empirical evidence that SenSeI trains individually fair ML models in practice and study the trade-off between accuracy and fairness parametrized by $\rho$ (defined in (2.6)).
+
+Baselines We compare SenSeI to empirical risk minimization (Baseline) and two recent approaches for training individually fair ML models: Sensitive Subspace Robustness (SenSR) (Yurochkin et al., 2020) that uses DRO to achieve robustness to perturbations in a fair metric, and Counterfactual Logit Pairing (CLP) (Garg et al., 2018) that penalizes the differences in the output of an ML model on training examples and hand-crafted counterfactuals. We provide implementation details of SenSeI and the baselines in Appendix B.
+
+# 4.1 TOXIC COMMENT DETECTION
+
+We consider the task of training a classifier to identify toxic comments, i.e. rude or disrespectful messages in online conversations. Identifying and moderating toxic comments is crucial for facilitating inclusive online conversations. Data is available through the “Toxic Comment Classification Challenge” Kaggle competition. We utilize the subset of the dataset that is labeled with a range of identity contexts (e.g. “muslim”, “white”, “black”, “homosexual gay or lesbian”). Many of the toxic comments in the train data also relate to these identities leading to a classifier with poor test performance on the sets of comments with some of the identity contexts (group fairness violation) and prediction rule utilizing words such as “gay” to flag a comment as toxic (individual fairness violation). To obtain good features we use last layer representation of BERT (base, uncased) (Devlin et al., 2018) fine tuned on a separate subset of $5 0 0 \mathrm { k }$ randomly selected comments without identity labels. We then train a 2000 hidden units neural network with these BERT features.
+
+Counterfactuals and fair metric. CLP (Garg et al., 2018) requires defining a set of counterfactual tokens. The training proceeds by taking an input comment and if it contains a counterfactual token replacing it with another random counterfactual token. For example, if “gay” and “straight” are among the counterfactual tokens, comment “Some people are gay” may be modified to “Some people are straight”. Then difference in logit outputs of the classifier on the original and modified comments is used as a regularizer. For toxicity classification Garg et al. (2018) adopted a set of 50 counterfactual tokens from (Dixon et al., 2018). Counterfactuals allow for a simple fair metric learning procedure via factor analysis: since any variation in representation of a data point and its counterfactuals is considered undesired, Yurochkin et al. (2020); Mukherjee et al. (2020) proposed to use a Mahalanobis metric with the major directions of variation among counterfactuals projected out. We utilize this approach here to obtain fair metric for training SenSR (Yurochkin et al., 2020) and SenSeI. See Appendix B.1 for additional details regarding the fair metric construction in the experiments.
+
+Table 1: Summary of Toxicity classification experiment over 10 restarts   
+
+<table><tr><td></td><td>BA</td><td>CTF score</td><td>PC</td><td>BA STD</td><td>ACC STD</td></tr><tr><td>Baseline</td><td>0.807±0.002</td><td>0.072±0.002</td><td>0.621±0.014</td><td>0.044±0.003</td><td>0.081±0.004</td></tr><tr><td>SenSeI</td><td>0.791±0.005</td><td>0.029±0.005</td><td>0.773±0.043</td><td>0.035±0.003</td><td>0.052±0.003</td></tr><tr><td>SenSR</td><td>0.794±0.003</td><td>0.043±0.005</td><td>0.729±0.044</td><td>0.036±0.002</td><td>0.059±0.004</td></tr><tr><td>CLP</td><td>0.795±0.006</td><td>0.032±0.006</td><td>0.763±0.048</td><td>0.038±0.003</td><td>0.056±0.004</td></tr></table>
+
+Table 2: Summary of Bios classification experiment over 10 restarts   
+
+<table><tr><td></td><td>BA</td><td>CTF score</td><td>PC</td><td>Gap RMS</td><td>Gap ABS</td></tr><tr><td>Baseline</td><td>0.842±0.002</td><td>0.028±0.001</td><td>0.942±0.001</td><td>0.120±0.004</td><td>0.080±0.003</td></tr><tr><td>SenSeI</td><td>0.843±0.003</td><td>0.003±0.000</td><td>0.977±0.001</td><td>0.086±0.005</td><td>0.054±0.003</td></tr><tr><td>SenSR</td><td>0.842±0.003</td><td>0.004±0.000</td><td>0.976±0.001</td><td>0.087±0.004</td><td>0.054±0.003</td></tr><tr><td>CLP</td><td>0.841±0.003</td><td>0.005±0.000</td><td>0.974±0.001</td><td>0.087±0.005</td><td>0.056±0.004</td></tr></table>
+
+Comparison metrics. To evaluate individual fairness of the classifiers we use test data and 50 counterfactuals to check if the classifier toxicity decision varies across counterfactuals. For example, is prediction for “Some people are gay” same as for “Some people are straight”? An intuitive fair metric should be 0 on a pair of such comments and prediction of the classifier should not change based on the Dwork et al.’s individual fairness definition. We report Counterfactual Token Fairness (CTF) score (Garg et al., 2018) that quantifies variance across counterfactuals of the predicted probability that a comment is toxic, and Prediction Consistency (PC) equal to the portion of test comments where prediction is the same across all 50 counterfactual variations.
+
+For goodness-of-fit we use balanced accuracy due to class imbalance. To quantify group fairness we follow accuracy parity notion (Zafar et al., 2017; Zhao et al., 2019). Here protected groups correspond to the identity context labels available in the data and accuracy parity quantifies whether a classifier is equally accurate on, e.g., comments labeled to have “white” context and those labeled with “black”. There are 9 protected groups and we report standard deviation of the accuracies and balanced accuracies across them. We present mathematical expressions for each of the metrics in Appendix C for completeness.
+
+Results. We repeat our experiment 10 times with random 70-30 train-test splits, every time utilizing a random subset of 25 counterfactuals during training. Results are summarized in Table 1. SenSeI on average outperforms other fair training methods on all individual and group fairness metrics at the cost of slightly lower balanced accuracy. SenSR has the lowest prediction consistency score suggesting that our fair regularization is more effective in enforcing individual fairness than adversarial training. This observation aligns with the empirical study by Yang et al. (2019) comparing various invariance enforcing techniques for spatial robustness in image recognition.
+
+SenSeI also outperforms CLP: our approach uses optimization to find worst case perturbations of the data according to a fair metric induced by counterfactuals, while CLP chooses a random perturbation among the counterfactuals. SenSeI is searching for a perturbation on every data point that potentially allows to generalize to unseen counterfactuals, while CLP can only perturb comments that explicitly contain a counterfactual known during training time. In Figure 2 we verify that both SenSeI and CLP allow to “trade” fairness and accuracy by varying the regularization strength $\rho$ (in the table we used $\rho = 5$ for both). We also notice the effect of worst case optimization opposed to random sampling: SenSeI has higher prediction consistency at the cost of balanced accuracy.
+
+# 4.2 OCCUPATION PREDICTION
+
+Online professional presence via dedicated services or personal websites is the common practice in many industries. ML systems can be trained using data from these sources to identify person’s occupation and used by recruiters to assist in finding suitable candidates for the job openings. Gender imbalances in occupations may trigger biases in such ML systems exacerbating societal inequality. De-Arteaga et al. (2019) proposed Bias in Bios dataset to study fairness in occupation prediction from a person’s bio. The dataset consists of $4 0 0 \mathrm { k }$ textual bio descriptions and the goal is to predict one of the 28 occupations. We again use BERT fine-tuned on the train data to obtain bio representations and then train a 2000 hidden neurons neural networks using each of the fair training methods.
+
+![](images/0d76a6ff8dfba0eac66b56b66be62af0a8f7c540fe6e1cdc1a812ceb68ed49b5.jpg)  
+Figure 2: Balanced accuracy (BA) and prediction consistency (PC) trade-off.
+
+Table 3: Adult experiment over 10 restarts. Prior methods are duplicated from Yurochkin et al. (2020)   
+
+<table><tr><td></td><td>BA,%</td><td>S-Con.</td><td>GR-Con.</td><td>GapG RMS</td><td>GapR RMS</td><td>GapG max</td><td>GapR max</td></tr><tr><td>SenSR</td><td>78.9</td><td>.934</td><td>.984</td><td>.068</td><td>.055</td><td>.087</td><td>.067</td></tr><tr><td>Baseline</td><td>82.9</td><td>.848</td><td>.865</td><td>.179</td><td>.089</td><td>.216</td><td>.105</td></tr><tr><td>Project</td><td>82.7</td><td>.868</td><td>1.00</td><td>.145</td><td>.064</td><td>.192</td><td>.086</td></tr><tr><td>Adv. debiasing</td><td>81.5</td><td>.807</td><td>.841</td><td>.082</td><td>.070</td><td>.110</td><td>.078</td></tr><tr><td>CoCL</td><td>79.0</td><td>1</td><td>1</td><td>.163</td><td>.080</td><td>.201</td><td>.109</td></tr><tr><td>SenSeI (p= 40)</td><td>76.8</td><td>.945</td><td>.963</td><td>.043</td><td>.054</td><td>.053</td><td>.064</td></tr></table>
+
+Counterfactuals and fair metric. Counterfactuals definition for this problem is the gender analog of the Bertrand & Mullainathan (2004) investigation of the racial bias in the labor market. For each bio we create a counterfactual bio by replacing male pronouns with the corresponding female ones and vice a versa. For the fair metric we use same approach as in the Toxicity study.
+
+Comparison metrics. We use the same individual fairness metrics. To compare group fairness we report root mean squared gap (Gap RMS) and mean absolute gap (Gap ABS) between male and female true positive rates for each of the occupations following prior studies of this dataset (Romanov et al., 2019; Prost et al., 2019). For performance we report balanced accuracy due to imbalance in occupation proportions in the data.
+
+Results. We repeat the experiment 10 times with 70-30 train-test splits and summarize results in Table 2 (for SenSeI and CLP we set $\rho = 5$ ). Comparing to the Toxicity experiment, we note that individual fairness metrics are much better. In particular, attaining prediction consistency is easier because there is only one type of counterfactuals. Overall, fairness metrics are comparable across fair training methods with a slight SenSeI advantage. It is interesting to note mild accuracy improvement: learning a classifier invariant to gender can help to avoid spurious correlations between occupations and gender present in the data. We present fairness accuracy “trade-off” in Figure 2 — increasing regularization strength has clear upward trend in terms of the prediction consistency without decreasing accuracy. This experiments is an example where fairness can be improved without ”trading” performance.
+
+We note two prior group fairness studies of the Bios dataset: Romanov et al. (2019) and Prost et al. (2019). Both reported worse classification results and fairness metrics. Better classification performance in our work is likely attributed to using BERT for obtaining bios feature vectors. For the group fairness, we note that relative to baseline improvement with SenSeI is more significant.
+
+# 4.3 INCOME PREDICTION
+
+The Adult dataset (Bache & Lichman, 2013) is a common benchmark in the group fairness literature. The task is to predict if a person earns more than $\$ 50\mathbf { k }$ per year using information about their education, gender, race, marital status, hours worked per week, etc. Yurochkin et al. (2020) studied individual fairness on Adult by considering prediction consistency with respect to demographic features: race and gender (GR-Con.) and marital status (S-Con., i.e. spouse consistency). To quantify group fairness they used RMS gaps and maximum gaps between true positive rates across genders $\bar { ( \mathrm { G a p } _ { G } ^ { \mathrm { R M S } } }$ and ${ \mathrm { G a p } } _ { G } ^ { \operatorname* { m a x } } ,$ ) and races ${ ( \mathrm { G a p } _ { R } ^ { \mathrm { R M S } } }$ and ${ \mathrm { G a p } } _ { R } ^ { \operatorname* { m a x } } .$ ). Due to class imbalance, performance is quantified with balanced accuracy (B-Acc). For the fair metric they used Mahalanobis distance with race, gender and a logistic regression vector predicting gender projected out. We note that CLP was proposed as a fair training method for text classification (Garg et al., 2018) and is not applicable on Adult because it is not clear how to define counterfactuals. We compare SenSeI to results reported in Yurochkin et al. (2020). In Table 3 we show that with sufficiently large regularization strength $\rho = 4 0$ SenSeI is able to further reduce all group fairness gaps and improve one of the individual fairness metrics, however trading off some accuracy.
+
+# 5 SUMMARY AND DISCUSSION
+
+In this paper, we studied a regularization approach to enforcing individual fairness. We defined distributional individual fairness, a variant of Dwork et al.’s original definition of individual fairness and a data-dependent regularizer that enforces this distributional fairness (see Definition 2.1). We also developed a stochastic approximation algorithm to solve regularized empirical risk minimization problems and showed that it trains ML models with distributional fairness guarantees. Finally, we showed that the algorithm mitigates algorithmic bias on three ML tasks that are susceptible to such biases: income-level classification, occupation prediction, and toxic comment detection.
+
+# ACKNOWLEDGEMENTS
+
+This paper is based upon work supported by the National Science Foundation (NSF) under grants no. 1830247 and 1916271. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF.
+
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+
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+
+# A THEORETICAL PROPERTIES
+
+We collect the proofs of all the theoretical results in the paper here. We restate the results before proving them for the reader’s convenience. We assume that $( \mathcal { X } , d _ { \mathcal { X } } )$ and $( \mathcal { V } , d _ { \mathcal { V } } )$ are complete and separable metric spaces (Polish spaces).
+
+# A.1 DIF AND INDIVIDUAL FAIRNESS
+
+Proposition A.1 (Proposition 2.3). If $h : \mathcal { X } \to \mathcal { Y }$ is $( \epsilon , \delta )$ -DIF, then
+
+$$
+\begin{array} { r } { P _ { X } ( \operatorname* { s u p } _ { d _ { \mathcal { X } } ( x , x ^ { \prime } ) \leq \epsilon } d y ( h ( x ) , h ( x ^ { \prime } ) ) \geq \tau ) \leq \frac { \delta } { \tau } . } \end{array}
+$$
+
+Proof. Recall
+
+$$
+\begin{array} { r } { T _ { \mathrm { I F } } \triangleq \arg \operatorname* { m a x } _ { d _ { \mathcal { X } } ( x , x ^ { \prime } ) \leq \epsilon } d _ { \mathcal { Y } } ( h ( x ) , h ( x ^ { \prime } ) ) . } \end{array}
+$$
+
+is feasible for (the Mongé version of) (2.2). Thus $R ( h ) \leq \delta$ implies ${ \bf E } _ { P x } \left[ d y ( X , T _ { \mathrm { I F } } ( X ) ) \right] \leq \delta$ Markov’s inequality implies $\begin{array} { r } { P _ { X } ( d _ { \mathcal { V } } ( X , T _ { \mathrm { I F } } ( X ) ) \ge \tau ) \le \frac { \delta } { \tau } } \end{array}$ for any $\tau > 0$ . □
+
+# A.2 CONVERGENCE PROPERTIES OF SENSEI
+
+Algorithm 1 is an instance of a stochastic gradient method, and its convergence properties are wellstudied. Even if $f ( w , Z )$ is non-convex in $w$ , the algorithm converges (globally) to a stationary point. This a well-known result in stochastic approximation, and we state it here for completeness.
+
+Theorem A.2 (Ghadimi & Lan (2013)). Let
+
+$$
+\begin{array} { r } { \sigma ^ { 2 } \geq { \bf E } \big [ \| \frac { 1 } { B } \sum _ { b = 1 } ^ { B } \partial _ { w } f ( w , Z _ { b } ) - \partial F ( w ) \| _ { 2 } ^ { 2 } \big ] } \end{array}
+$$
+
+e an upper bound of the variance of the stochastic gradient. As long as $F$ is $L$ -strongly smooth,
+
+$$
+\begin{array} { r } { F ( w ^ { \prime } ) \leq F ( w ) + \langle \partial F ( w ) , w ^ { \prime } - w \rangle + \frac { L } { 2 } \| w - w ^ { \prime } \| _ { 2 } ^ { 2 } } \end{array}
+$$
+
+for any $w , w ^ { \prime } \in \Theta \times \mathbf { R } _ { + }$ , then Algorithm $\cdot$ with constant step sizes $\begin{array} { r } { \eta _ { t } = \big ( \frac { 2 B \epsilon _ { 0 } } { L \sigma ^ { 2 } T } \big ) ^ { \frac { 1 } { 2 } } } \end{array}$ satisfies
+
+$$
+\begin{array} { r } { \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbf { E } \big [ \| \partial F ( w _ { t } ) \| _ { 2 } ^ { 2 } \big ] \leq \sigma ( \frac { 8 L \epsilon _ { 0 } } { B T } ) , } \end{array}
+$$
+
+where $\epsilon _ { \mathrm { 0 } }$ is any upper bound of the suboptimality of $w _ { 0 }$
+
+In other words, Algorithm 1 finds an $\epsilon$ -stationary point of (2.8) in at most $O ( \textstyle { \frac { 1 } { \epsilon ^ { 2 } } } )$ iterations. If $F$ has more structure (e.g. convexity), then Algorithm 1 may converge faster.
+
+# A.3 PROOF OF DUALITY RESULTS IN SECTION 2
+
+Theorem A.3 (Theorem 2.4). If $d y ( h ( x ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( x , x ^ { \prime } )$ is continuous (in $( x , x ^ { \prime } ) )$ for any $\lambda \geq 0$ , then
+
+$$
+\begin{array} { r l r } & { } & { R ( h ) = \operatorname* { i n f } _ { \lambda \ge 0 } \{ \lambda \epsilon + { \bf E } _ { P _ { X } } \left[ r _ { \lambda } ( h , X ) \right] \} , } \\ & { } & { r _ { \lambda } ( h , X ) \triangleq \operatorname* { s u p } _ { x ^ { \prime } \in \mathcal { X } } \{ d _ { \mathcal { V } } ( h ( X ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( X , x ^ { \prime } ) \} . } \end{array}
+$$
+
+Proof. We abuse notation and denote the function $d y ( h ( x ) , h ( x ^ { \prime } ) )$ as $d _ { \mathcal { Y } } \circ h$ . We recognize the optimization problem in (2.2) as an (infinite dimensional) linear optimization problem:
+
+$$
+R ( h ) \triangleq \left\{ \begin{array} { l l } { \operatorname* { s u p } _ { \Pi : \Delta ( \mathcal { X } \times \mathcal { X } ) } } & { \langle \Pi , d _ { \mathcal { Y } } \circ h \rangle = \mathbf { E } _ { \Pi } \big [ d _ { \mathcal { Y } } ( h ( X ) , h ( X ^ { \prime } ) ) \big ] } \\ { \mathrm { s u b j e c t ~ t o ~ } } & { \langle \Pi , d _ { \mathcal { X } } \rangle = \mathbf { E } _ { \Pi } \big [ d _ { \mathcal { X } } ( X , X ^ { \prime } ) \big ] \leq \epsilon } \\ & { \Pi ( \cdot , \mathcal { X } ) = P _ { \mathcal { X } } , } \end{array} \right.
+$$
+
+It is not hard check Slater’s condition: $d \Pi ( x , x ^ { \prime } ) = { \bf 1 } \{ x = x ^ { \prime } \} d P ( x )$ is strictly feasible. Thus we have strong duality (see Theorem 8.7.1 in (Luenberger, 1968)):
+
+$$
+\begin{array} { r l r } & { } & { R ( h ) = \operatorname* { s u p } _ { \Pi : \Pi ( \cdot , \mathcal { X } ) = P _ { X } } \operatorname* { i n f } _ { \lambda \geq 0 } \langle \Pi , d _ { \mathcal { Y } } \circ h \rangle + \lambda ( \epsilon - \langle \Pi , d _ { \mathcal { X } } \rangle ) } \\ & { } & { \quad = \operatorname* { s u p } _ { \Pi : \Pi ( \cdot , \mathcal { X } ) = P _ { X } } \operatorname* { i n f } _ { \lambda \geq 0 } \lambda \epsilon + \langle \Pi , d _ { \mathcal { Y } } \circ h - \lambda d _ { \mathcal { X } } \rangle \quad } \\ & { } & { \quad = \operatorname* { i n f } _ { \lambda \geq 0 } \{ \lambda \epsilon + \operatorname* { s u p } _ { \Pi : \Pi ( \cdot , \mathcal { X } ) = P _ { X } } \langle \Pi , d _ { \mathcal { Y } } \circ h - \lambda d _ { \mathcal { X } } \rangle \} , } \end{array}
+$$
+
+It remains to show
+
+$$
+\begin{array} { r } { \operatorname* { s u p } _ { \Pi : \Pi ( \cdot , x ) = P } \langle d _ { \mathcal { Y } } \circ h - \lambda d _ { \mathcal { X } } , \Pi \rangle = \mathbf { E } _ { P } \left[ \operatorname* { s u p } _ { x ^ { \prime } \in \mathcal { X } } \{ d _ { \mathcal { Y } } ( h ( X ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( X , x ^ { \prime } ) \} \right] . } \end{array}
+$$
+
+$\leq$ direction The integrands in (A.1) satisfy
+
+( $\begin{array} { r } { d _ { \mathcal { V } } \circ h - \lambda d _ { \mathcal { X } } ) ( x , x ^ { \prime } ) = d _ { \mathcal { V } } ( h ( x ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( x , x ^ { \prime } ) \leq \operatorname* { s u p } _ { x ^ { \prime } \in \mathcal { X } } \big \{ d _ { \mathcal { V } } ( h ( x ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( x , x ^ { \prime } ) \big \} , } \end{array}$ so the integrals satisfy the $\leq$ version of (A.1).
+
+$\geq$ direction Let $\mathcal { Q }$ be the set of all Markov kernels from $\mathcal { X }$ to $\mathcal { X }$ . We have
+
+$$
+\begin{array} { r l r } {  { \operatorname* { s u p } _ { \Pi : \Pi ( \cdot , \mathcal { X } ) = P } \langle d y \circ h - \lambda d _ { \mathcal { X } } , \Pi \rangle = \operatorname* { s u p } _ { Q \in \mathcal { Q } } \int _ { \mathcal { X } \times \mathcal { X } } d y ( h ( x ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( x , x ^ { \prime } ) d Q ( x ^ { \prime } \mid x ) d P ( x ) } } \\ & { } & { \ge \operatorname* { s u p } _ { T : \mathcal { X } \to \mathcal { X } } \int _ { \mathcal { X } } d y ( h ( x ) , h ( T ( x ) ) ) - \lambda d _ { \mathcal { X } } ( x , T ( x ) ) d P ( x ) , } \end{array}
+$$
+
+where we recalled $Q ( A \mid x ) = \mathbf { 1 } \{ T ( x ) \in A \}$ is a Markov kernel in the second step. (Technically, in the second step, we only sup over $T$ ’s that are decomposable (see Definition 14.59 in Rockafellar & Wets (2004)) with respect to $P$ , but we gloss over this detail here.) We appeal to the technology of integrands (Rockafellar & Wets, 2004) to interchange integration and maximization. We assumed $d _ { \mathscr { y } } \circ h - \lambda d _ { \mathscr { x } }$ is continuous, so it is a normal integrand (see Corollary 14.34 in Rockafellar & Wets (2004)). Thus it is OK to interchange integration and maximization (see Theorem 14.60 in Rockafellar & Wets (2004)):
+
+$$
+\operatorname* { s u p } _ { T : \mathcal { X } \to \mathcal { X } } \int _ { \mathcal { X } } d y ( h ( x ) , h ( T ( x ) ) ) - \lambda d _ { \mathcal { X } } ( x , T ( x ) ) d P ( x ) = \int _ { \mathcal { X } } \operatorname* { s u p } _ { x ^ { \prime } \in \mathcal { X } } \left\{ d y ( h ( x ) , h ( x ^ { \prime } ) ) - \lambda d _ { \mathcal { X } } ( x , x ^ { \prime } ) \right\} d P ( x ) .
+$$
+
+This shows the $\geq$ direction of (A.1).
+
+We remark that it is not necessary to rely on the technology of normal integrands to interchange expectation and maximization in the proof of Theorem 2.4. For example, Blanchet & Murthy (2016) prove a similar strong duality result without resorting to normal integrands. We do so here to simplify the proof.
+
+# A.4 PROOFS OF GENERALIZATION RESULTS IN SECTION 3
+
+Theorem A.4 (Theorem 3.1). As long as $D _ { \mathcal { X } } , D _ { \mathcal { Y } } , J ( \mathcal { G } )$ are all finite,
+
+$$
+\begin{array} { r } { \operatorname* { s u p } _ { f \in \mathcal { F } } | \mathbf { E } _ { \widehat { P } } \big [ f ( Z ) \big ] - \mathbf { E } _ { P } \big [ f ( Z ) \big ] | \leq \frac { 4 8 ( J ( \mathcal { D } ) + \frac { 1 } { \epsilon } D x D y ) } { \sqrt { n } } + D y ( \frac { \log \frac { 2 } { t } } { 2 n } ) ^ { \frac { 1 } { 2 } } } \end{array}
+$$
+
+with probability at least $1 - t$ .
+
+Proof. Let
+
+$$
+\widehat { R } ( h ) \triangleq \left\{ \begin{array} { l l } { \operatorname* { m a x } _ { \Pi \in \Delta ( \mathcal { X } \times \mathcal { X } ) } } & { \mathbf { E } _ { \Pi } \big [ d y ( h ( X ) , h ( X ^ { \prime } ) ) \big ] } \\ { \mathrm { s u b j e c t ~ t o ~ } } & { \mathbf { E } _ { \Pi } \big [ d _ { \mathcal { X } } ( X , X ^ { \prime } ) \big ] \leq \epsilon } \\ & { \Pi ( \cdot , \mathcal { X } ) = \widehat { P } _ { X } , } \end{array} \right\} .
+$$
+
+where ${ \widehat { P } } _ { X }$ is the empirical distribution of the inputs in the training set. By Theorem 2.4, we have
+
+$$
+\begin{array} { r l } & { \widehat { R } ( h ) - R ( h ) = \operatorname* { i n f } _ { \lambda \geq 0 } \{ \lambda \epsilon + \mathbf { E } _ { \widehat { P } _ { X } } \left[ r _ { \lambda } ( h , X ) \right] \} - \operatorname* { i n f } _ { \lambda \geq 0 } \{ \lambda \epsilon + \mathbf { E } _ { P _ { X } } \left[ r _ { \lambda } ( h , X ) \right] \} } \\ & { \qquad = \operatorname* { i n f } _ { \lambda \geq 0 } \{ \lambda \epsilon + \mathbf { E } _ { \widehat { P } _ { X } } \left[ r _ { \lambda } ( h , X ) \right] \} - \lambda _ { * } \epsilon - \mathbf { E } _ { \widehat { P } _ { X } } \left[ r _ { \lambda _ { * } } ( h , X ) \right] } \\ & { \qquad \leq \mathbf { E } _ { \widehat { P } _ { X } } \left[ r _ { \lambda _ { * } } ( h , X ) \right] \} - \mathbf { E } _ { P _ { X } } \left[ r _ { \lambda _ { * } } ( h , X ) \right] , } \end{array}
+$$
+
+where $\begin{array} { r } { \lambda _ { * } \in \arg \operatorname* { m i n } _ { \lambda \geq 0 } \lambda \epsilon + \mathbf { E } _ { P _ { X } } \left[ r _ { \lambda } ( h , X ) \right] } \end{array}$ . The infimum is attained because
+
+$$
+\begin{array} { r } { \operatorname* { i n f } _ { \lambda \ge 0 } \{ \lambda \epsilon + \mathbf { E } _ { P _ { X } } \left[ r _ { \lambda } ( h , X ) \right] \} } \end{array}
+$$
+
+is an strictly feasible (infinite-dimensional) linear optimization problem (see proof of Theorem 2.4). To bound $\lambda _ { * }$ , we observe that $r _ { \lambda } ( h , X ) \ge 0$ for any $h \in { \mathcal { H } } , \lambda \geq 0$ :
+
+$$
+\begin{array} { r l } & { r _ { \lambda } ( h , X ) = \operatorname* { s u p } _ { x ^ { \prime } \in { \mathcal { X } } } \{ d _ { { \mathcal { V } } } ( h ( X ) , h ( x ^ { \prime } ) ) - \lambda d _ { { \mathcal { X } } } ( X , x ^ { \prime } ) \} } \\ & { \qquad \geq d _ { { \mathcal { V } } } ( h ( X ) , h ( X ) ) - \lambda d _ { { \mathcal { X } } } ( X , X ) . } \end{array}
+$$
+
+This implies
+
+$$
+R ( h ) = \lambda _ { * } \epsilon + \mathbb { E } _ { P _ { X } } \left[ r _ { \lambda _ { * } } ( h , X ) \right] \geq \lambda _ { * } \epsilon .
+$$
+
+We rearrange to obtain a bound on $\lambda _ { * }$ :
+
+$$
+\lambda _ { * } \leq \frac { 1 } { \epsilon } R ( h ) \leq \frac { 1 } { \epsilon } D _ { \mathscr { y } } \triangleq \bar { \lambda } .
+$$
+
+This is admittedly a crude bound, but it is good enough here. Similarly,
+
+$$
+R ( h ) - \widehat { R } ( h ) \leq { \bf E } _ { P _ { X } } \left[ r _ { \widehat { \lambda } _ { * } } ( h , X ) \right] \} - { \bf E } _ { \widehat { P } _ { X } } \left[ r _ { \widehat { \lambda } _ { * } } ( h , X ) \right] ,
+$$
+
+where $\begin{array} { r } { \widehat { \lambda } _ { * } \in \arg \operatorname* { m i n } _ { \lambda \geq 0 } \lambda \epsilon + \mathbf { E } _ { \widehat { P } _ { X } } \left[ r _ { \lambda } ( h , X ) \right] } \end{array}$ , and $\widehat { \lambda } _ { * } \leq \bar { \lambda }$ . Combining (A.2) and (A.4), we obtain
+
+$$
+\begin{array} { r } { | \widehat { R } ( h ) - R ( h ) | \leq \operatorname* { s u p } _ { f \in \mathcal { F } } \big | \mathbf { E } _ { \widehat { P } } \big [ f ( Z ) \big ] - \mathbf { E } _ { P } \big [ f ( Z ) \big ] \big | , } \end{array}
+$$
+
+where $\mathcal { F } \triangleq \{ r _ { \lambda } ( h , \cdot ) \ | \ h \in \mathcal { H } , \lambda \in [ 0 , \bar { \lambda } ] \}$ . It is possible to bound $\begin{array} { r } { \operatorname* { s u p } _ { f \in \mathcal { F } } | \mathbf { E } _ { \widehat { P } } \big [ f ( Z ) \big ] - \mathbf { E } _ { P } \big [ f ( Z ) \big ] | } \end{array}$ bwith results from statistical learning theory. First, we observe that the functions in $\mathcal { F }$ are bounded:
+
+$$
+0 \leq r _ { \lambda } ( h , X ) \leq \frac { 1 } { \epsilon } \operatorname* { s u p } _ { y , y ^ { \prime } \in \mathcal { V } } d _ { \mathcal { V } } ( y , y ^ { \prime } ) .
+$$
+
+Thus $\begin{array} { r } { \operatorname* { s u p } _ { f \in \mathcal { F } } \left| \mathbf { E } _ { \widehat { P } } \big [ f ( Z ) \big ] - \mathbf { E } _ { P } \left[ f ( Z ) \right] \right| } \end{array}$ has bounded differences inequality, so it concentrates sharply baround its expectation. By the bounded-differences inequality and a standard symmetrization argument,
+
+$$
+\begin{array} { r } { \operatorname* { s u p } _ { f \in \mathcal { F } } \big | \mathbf { E } _ { \widehat { P } } \big [ f ( Z ) \big ] - \mathbf { E } _ { P } \big [ f ( Z ) \big ] \big | \leq 2 \Re _ { n } ( \mathcal { F } ) + D _ { \mathcal { V } } ( \frac { \log \frac { 2 } { t } } { 2 n } ) ^ { \frac { 1 } { 2 } } } \end{array}
+$$
+
+with probability at least $1 - t$ , where $\Re _ { n } ( \mathcal { F } )$ is the Rademacher complexity of $\mathcal { F }$
+
+$$
+\begin{array} { r } { \mathfrak { R } _ { n } ( \mathcal { F } ) = \mathbf { E } \big [ \operatorname* { s u p } _ { f \in \mathcal { F } } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \sigma _ { i } f ( Z _ { i } ) \big ] . } \end{array}
+$$
+
+It remains to study $\textstyle { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \sigma _ { i } f ( Z _ { i } )$ n is sub-Gaussian with respect to to the metric . First, we show that the -indexed Rademacher process $X _ { f }$ ,
+
+$$
+\begin{array} { r l } & { | \mathcal { F } \big ( ( h _ { 1 } , \lambda _ { 1 } ) , ( h _ { 2 } , \lambda _ { 2 } ) \big ) \triangleq \operatorname* { s u p } _ { x _ { 1 } , x _ { 2 } \in \mathcal { X } } \big | d y \big ( h _ { 1 } ( x _ { 1 } ) , h _ { 1 } ( x _ { 2 } ) \big ) - d y \big ( h _ { 2 } ( x _ { 1 } ) , h _ { 2 } ( x _ { 2 } ) \big ) \big | + D _ { \mathcal { X } } \big | \lambda _ { 1 } - \lambda _ { 1 } \big | : } \\ & { \mathrm { \Lambda } \mathrm { \Lambda } } \\ & { = \mathbf { E } \big [ \mathrm { e x p } ( t ( X _ { f _ { 1 } } - X _ { f _ { 2 } } ) \big ) \big ] } \\ & { \quad = \mathbf { E } \big [ \mathrm { e x p } \big ( \frac { t } { n } \sum _ { i = 1 } ^ { n } \sigma _ { i } ( r _ { \lambda _ { 1 } } ( h _ { 1 } , X _ { i } ) - r _ { \lambda _ { 2 } } ( h _ { 2 } , X _ { i } ) \big ) \big ) \big ] } \\ & { \quad = \mathbf { E } \big [ \mathrm { e x p } \big ( \frac { t } { n } \sigma ( r _ { \lambda _ { 1 } } ( h _ { 1 } , X _ { i } ) - r _ { \lambda _ { 2 } } ( h _ { 2 } , X _ { i } ) ) \big ) \big ) \big ] ^ { n } } \\ & { \quad = \mathbf { E } \big [ \mathrm { e x p } \Big ( \frac { t } { n } \sigma ( \operatorname* { s u p } _ { x _ { 1 } ^ { \prime } \in \mathcal { X } } \operatorname { i n f } _ { x _ { 2 } ^ { \prime } \in \mathcal { X } } d y \big ( h _ { 1 } ( X _ { i } ) , h _ { 1 } ( x _ { 1 } ^ { \prime } ) \big ) - \lambda _ { 1 } d _ { \mathcal { X } } ( x _ { 1 } , X ) - d _ { \mathcal { V } } ( h _ { 2 } ( X _ { i } ) , h _ { 2 } ( x _ { 2 } ^ { \prime } ) ) + \lambda _ { 2 } } \\ &  \quad \leq \mathbf { E } \big [ \mathrm { e x p } \big ( \frac { t } { n } \sigma ( \operatorname* { s u p } _ { x _ { 1 } \in \mathcal { X } } d y \big ( h _ { 1 } ( X _ { i } ) , h _ { 1 } ( x _ { 1 } ^ { \prime } ) \big ) - d y \big ( h _ { 2 } ( X _ { i } ) , h _ { 2 } ( x \end{array}
+$$
+
+Let $N ( \mathcal { F } , d _ { \mathcal { F } } , \epsilon )$ be the $\epsilon$ -covering number of $\mathcal { F }$ in the $d _ { \mathcal { F } }$ metric. We observe
+
+$$
+\begin{array} { r } { N ( \mathcal { F } , d _ { \mathcal { F } } , \epsilon ) \leq N ( \mathcal { D } , \| \cdot \| _ { \infty } , \frac { \epsilon } { 2 } ) \cdot N ( [ 0 , \bar { \lambda } ] , | \cdot | , \frac { \epsilon } { 2 D _ { \mathcal { X } } } ) . } \end{array}
+$$
+
+By Dudley’s entropy integral,
+
+$$
+\begin{array} { l } { \displaystyle \mathfrak { R } _ { n } ( \mathcal { F } ) \leq \frac { 1 2 } { \sqrt { n } } \int _ { 0 } ^ { \infty } \log N ( \mathcal { F } , d _ { \mathcal { F } } , \epsilon ) ^ { \frac { 1 } { 2 } } d \epsilon } \\ { \displaystyle \quad \leq \frac { 1 2 } { \sqrt { n } } \int _ { 0 } ^ { \infty } \left( \log N ( \mathcal { D } , \| \cdot \| _ { \infty } , \frac { \epsilon } { 2 } ) + N \big ( [ 0 , \bar { \lambda } ] , | \cdot | , \frac { \epsilon } { 2 D x } \big ) \right) ^ { \frac { 1 } { 2 } } d \epsilon } \\ { \displaystyle \quad \leq \frac { 1 2 } { \sqrt { n } } \left( \int _ { 0 } ^ { \infty } \log N ( \mathcal { D } , \| \cdot \| _ { \infty } , \frac { \epsilon } { 2 } ) ^ { \frac { 1 } { 2 } } d \epsilon + \int _ { 0 } ^ { \infty } N \big ( [ 0 , \bar { \lambda } ] , | \cdot | , \frac { \epsilon } { 2 D x } \big ) ^ { \frac { 1 } { 2 } } d \epsilon \right) } \\ { \displaystyle \quad \leq \frac { 2 4 J ( \mathcal { D } ) } { \sqrt { n } } + \frac { 2 4 D _ { x } \bar { \lambda } } { \sqrt { n } } \int _ { 0 } ^ { \frac { 1 } { 2 } } \log ( \frac { 1 } { \epsilon } ) d \epsilon . } \end{array}
+$$
+
+We check that $\begin{array} { r } { \int _ { 0 } ^ { \frac { 1 } { 2 } } \log ( \frac { 1 } { \epsilon } ) d \epsilon < 1 } \end{array}$ to arrive at Theorem 3.1.
+
+The chief technical novelty of this proof is the bound on $\lambda _ { * }$ in terms of the diameter of the output space. This bound allows us to restrict the relevant function class in a way that allows us to appeal to standard techniques from empirical process theory to obtain uniform convergence results. In prior work (e.g. Lee & Raginsky (2017)), this bound relies on smoothness properties of the loss, but this precludes non-smooth $d _ { \mathcal { Y } }$ in our problem setting.
+
+Corollary A.5. Assume there is $h _ { 0 } \in \mathcal { H }$ such that $L ( h _ { 0 } ) + \rho R ( h _ { 0 } ) < \delta _ { 0 }$ . As long as $D _ { \mathcal { X } } , D _ { \mathcal { Y } } , J ( \mathcal { L } )$ , and $J ( { \mathcal { G } } )$ are all finite, any global minimizer $\begin{array} { r } { \widehat { h } \in \arg \operatorname* { m i n } _ { h \in \mathcal { H } } \widehat { L } ( h ) + \rho \widehat { R } ( h ) } \end{array}$ satisfies
+
+$$
+L ( \hat { h } ) + \rho R ( \hat { h } ) \leq \delta _ { 0 } + 2 \left( \frac { 2 4 J ( \mathcal { L } ) + 4 8 \rho ( J ( \mathcal { D } ) + \frac { 1 } { \epsilon } D x D y ) } { \sqrt { n } } + ( \bar { L } + \rho D y ) ( \frac { \log \frac { 2 } { t } } { 2 n } ) ^ { \frac { 1 } { 2 } } \right)
+$$
+
+with probability at least $1 - 2 t$ .
+
+Proof. Let $F ( h ) \triangleq L ( h ) + \rho R ( h )$ and $\widehat F$ be its empirical counterpart. The optimality of $\hat { h }$ implies
+
+$$
+\begin{array} { r } { { \boldsymbol { \mathcal { F } } } ( \widehat { h } ) = F ( \widehat { h } ) - \widehat { F } ( \widehat { h } ) + \widehat { F } ( \widehat { h } ) - \widehat { F } ( h _ { 0 } ) + \widehat { F } ( h _ { 0 } ) - F ( h _ { 0 } ) + F ( h _ { 0 } ) \le \delta _ { 0 } + 2 \operatorname* { s u p } _ { h \in \mathcal { H } } \vert \widehat { F } ( h ) - F ( h ) \vert . } \end{array}
+$$
+
+We have
+
+$$
+\begin{array} { r } { \operatorname* { s u p } _ { h \in \mathcal { H } } | \widehat { F } ( h ) - F ( h ) | \leq \operatorname* { s u p } _ { h \in \mathcal { H } } | \widehat { L } ( h ) - L ( h ) | + \rho \operatorname* { s u p } _ { h \in \mathcal { H } } | \widehat { R } ( h ) - R ( h ) | . } \end{array}
+$$
+
+We assumed $\ell$ is bounded, so $\begin{array} { r } { \operatorname* { s u p } _ { h \in \mathcal { H } } | \widehat { L } ( h ) - L ( h ) | } \end{array}$ has bounded differences inequality, so it concentrates sharply around its expectation. By the bounded-differences inequality and a standard symmetrization argument,
+
+$$
+\operatorname* { s u p } _ { h \in \mathcal { H } } | \widehat { L } ( h ) - L ( h ) | \leq 2 \Re _ { n } ( \mathcal { L } ) + \bar { L } ( \frac { \log \frac { 2 } { t } } { 2 n } ) ^ { \frac { 1 } { 2 } }
+$$
+
+with probability at least $1 - t$ , where $\Re _ { n } ( \mathcal { L } )$ is the Rademacher complexity of $\mathcal { L }$ . By Dudley’s entropy integral,
+
+$$
+\Re _ { n } ( { \mathcal { L } } ) \leq { \frac { 1 2 } { \sqrt { n } } } \int _ { 0 } ^ { \infty } \log N ( { \mathcal { L } } , \| \cdot \| _ { \infty } , \epsilon ) ^ { \frac { 1 } { 2 } } d \epsilon ,
+$$
+
+so the first term on the right side of (A.6) is at most
+
+$$
+\operatorname* { s u p } _ { h \in \mathcal { H } } | \widehat { L } ( h ) - L ( h ) | \leq \frac { 2 4 J ( \mathcal { L } ) } { \sqrt { n } } + \bar { L } ( \frac { \log \frac { 2 } { t } } { 2 n } ) ^ { \frac { 1 } { 2 } }
+$$
+
+with probability at least $1 - t$ . Theorem 3.1 implies the second term on the right side of (A.6) is at most
+
+$$
+\operatorname* { s u p } _ { h \in \mathcal { H } } | \widehat { R } ( h ) - R ( h ) | \leq \frac { 4 8 ( J ( \mathcal { D } ) + \frac { 1 } { \epsilon } D _ { \mathcal { X } } D _ { \mathcal { Y } } ) } { \sqrt { n } } + D _ { \mathcal { Y } } ( \frac { \log \frac { 2 } { t } } { 2 n } ) ^ { \frac { 1 } { 2 } }
+$$
+
+with probability at least $1 - t$ . We combine the bounds to arrive at the stated result.
+
+# B SENSEI AND BASELINES IMPLEMENTATION DETAILS
+
+In this section we describe implementation details of all methods and hyperparameter selection to facilitate reproducibility of the experimental results reported in the main text.
+
+Improving balanced accuracy All three datasets we consider have noticeable class imbalances: over $80 \%$ comments in toxicity classification are non-toxic; several occupations in the Bias in Bios dataset are scarcely present (see Figure 1 in De-Arteaga et al. (2019) for details); about $7 5 \%$ of individuals in the Adult dataset make below $\$ 50\mathrm { k }$ a year. Because of this class imbalance we choose to report balanced accuracy to quantify classification performance of different methods. Balanced accuracy is simply an average of true positive rates of all classes. To improve balanced accuracy for all methods we use balanced mini-batches following Yurochkin et al. (2020), i.e. when sampling a mini-batch we enforce that every class is equally represented.
+
+Fair regularizer distance metric Recall that fair regularizer in Definition 2.1 of the main text requires selecting a distance metric on the classifier outputs $d y ( h ( x ) , h ( x ^ { \prime } ) )$ . This distance is also required for the implementation of Counterfactual Logit Pairing (CLP) (Garg et al., 2018). In a $K$ -class problem, let $h ( x ) \in \mathbf { R } ^ { K }$ denote a vector of $K$ logits of a classifier for an observation $x$ then for both SenSeI and CLP we define $\begin{array} { r } { d _ { \mathcal { V } } ( h ( x ) , h ( x ^ { \prime } ) ) \stackrel { = } { = } \frac { 1 } { K } \| h ( x ) - h ( x ^ { \prime } ) \| _ { 2 } ^ { 2 } } \end{array}$ , i.e. mean squared difference between logits of $x$ and $x ^ { \prime }$ . This is one of the choices empirically studied by Yang et al. (2019) for image classification. We defer exploring alternative fairness regularizer distance metrics for future work.
+
+Data processing and classifier architecture Data processing and classifier are shared across all methods in all experiments. In Toxicity experiment we utilized BERT (Devlin et al., 2018) finetuned on a random $33 \%$ subset of the data. We downloaded the fine-tuned model from one of the Kaggle kernels.1 In the Bios experiment for each train-test split we fine-tuned BERT-Base, Uncased2 for 3 epochs with mini-batch size 32, learning rate 2e−5 and 128 maximum sequence length. In both Toxicity and Bios experiments we obtained 768-dimensional sentence representations by average-pooling token embeddings of the corresponding fined-tuned BERTs. Then we trained a fully connected neural network with one hidden layer consisting of 2000 neurons with ReLU activations using BERT sentence representations as inputs.
+
+For Adult experiment we followed data processing and classifier choice (i.e. 100 hidden units neural network) as described in Yurochkin et al. (2020).
+
+Hyperparameters selection In Table 4 for each hyperparameter we summarize its meaning, abbreviation, name in the code provided with the submission3 and methods where it is used.
+
+To select hyperparameters for each experiment we performed a grid search on an independent train-test split. Then we fixed selected hyperparameters and ran 10 experiment repetitions with random train test splits (these results are reported in the main text). Hyperparameter choices for all experiments are summarized in Tables 5, 6, 7. For the Adult experiment we duplicated results for all prior methods from Yurochkin et al. (2020).
+
+Table 4: Hyperparameter names and notations   
+
+<table><tr><td></td><td>notation</td><td>name in code</td><td>relevant methods</td></tr><tr><td>Number of optimization steps</td><td>E</td><td>epoch</td><td>All</td></tr><tr><td>Mini-batch size</td><td>B</td><td>batch_size</td><td>All</td></tr><tr><td>Parameter learning rate n</td><td>n</td><td>lr</td><td>All</td></tr><tr><td>Subspace attack step size</td><td>S</td><td>adv_step</td><td>SenSeI, SenSR</td></tr><tr><td>Number of subspace attack steps</td><td>se</td><td>adv_epoch</td><td>SenSeI, SenSR</td></tr><tr><td>Full attack step size</td><td>f</td><td>l2_attack</td><td>SenSeI, SenSR</td></tr><tr><td>Number of full attack steps</td><td>fe</td><td>adv_epoch_full</td><td>SenSeI, SenSR</td></tr><tr><td>Attack budget E</td><td>E</td><td>ro</td><td>SenSeI, SenSR</td></tr><tr><td>Fair regularization strength p</td><td>p</td><td>fair_reg</td><td>SenSeI, CLP</td></tr></table>
+
+Table 5: Hyperparameter choices in Toxicity experiment   
+
+<table><tr><td></td><td>E</td><td>B</td><td>m</td><td>S</td><td>se</td><td>S</td><td>fe</td><td>E</td><td>p</td></tr><tr><td>Baseline</td><td>100k</td><td>256</td><td>1e-5</td><td>一</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SenSR</td><td>100k</td><td>256</td><td>1e-5</td><td>0.1</td><td>10</td><td>0</td><td>0</td><td>0</td><td></td></tr><tr><td>SenSeI</td><td>100k</td><td>256</td><td>1e-5</td><td>0.1</td><td>10</td><td>0</td><td>0</td><td>0</td><td>5</td></tr><tr><td>CLP</td><td>100k</td><td>256</td><td>1e-5</td><td>一</td><td></td><td></td><td></td><td>一</td><td>5</td></tr></table>
+
+Table 6: Hyperparameter choices in Bios experiment   
+
+<table><tr><td></td><td>E</td><td>B</td><td>n</td><td>S</td><td>se</td><td>S</td><td>fe</td><td>E</td><td>p</td></tr><tr><td>Baseline</td><td>100k</td><td>504</td><td>1e-6</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>SenSR</td><td>100k</td><td>504</td><td>1e-5</td><td>0.1</td><td>50</td><td>0.01</td><td>10</td><td>0.1</td><td></td></tr><tr><td>SenSeI</td><td>100k</td><td>504</td><td>1e-5</td><td>0.1</td><td>50</td><td>0.01</td><td>10</td><td>0.1</td><td>5</td></tr><tr><td>CLP</td><td>100k</td><td>504</td><td>1e-5</td><td>一</td><td></td><td>一</td><td></td><td>一</td><td>5</td></tr></table>
+
+Table 7: Hyperparameter choices in Adult experiment   
+
+<table><tr><td></td><td>E</td><td>B</td><td>n</td><td>S</td><td>se</td><td>f</td><td>fe</td><td>E</td><td>p</td></tr><tr><td>SenSeI</td><td>200k</td><td>1000</td><td>1e-5</td><td>5</td><td>50</td><td>0.001</td><td>50</td><td>0.01</td><td>40</td></tr></table>
+
+# B.1 FAIR METRIC LEARNING DETAILS
+
+Following Yurochkin et al. (2020); Mukherjee et al. (2020) we consider the fair metric of the form $d _ { \mathcal { X } } ( x , x ^ { \prime } ) \overset {  } { = } ( x - x ^ { \prime } ) ^ { T } \Sigma ( x - x ^ { \prime } )$ . We utilize their sensitive subspace idea writing $\Sigma = I - P _ { r a n ( A ) }$ i.e. an orthogonal complement projector of the subspace spanned by the columns of $A \in \mathbb { R } ^ { d \times k }$ . Here $A$ encodes the $k$ directions of sensitive variations that should be ignored by the fair metric $d$ is the data dimension), such as differences in sentence embeddings due to gender pronouns in the Bios experiment or due to identity (counterfactual) tokens in the Toxicity experiment.
+
+Synthetic experiment In the synthetic experiment in Figure 1 we consider a fair metric ignoring variation along the $x$ -axis coordinate, i.e. $\bar { A ^ { \mathbf { \alpha } } } = [ 1 \mathbf { \beta } 0 ] ^ { T }$ .
+
+Toxicity experiment To compute $A$ we utilize FACE algorithm of Mukherjee et al. (2020) (see section 2.1 and Algorithm 1 in their paper). Here groups of comparable samples are the BERT embeddings of sentences from the train data and their modifications obtained using 25 counterfactuals known at the training time. For example, suppose we have a sentence “Some people are gay” in the train data and the list of known counterfactuals is “gay”, “straight” and “muslim”. Then we can obtain two comparable sentences: “Some people are straight” and “Some people are muslim”. BERT embeddings of the original and two created sentences constitute a group of comparable sentences. Embeddings of groups of comparable sentences are the inputs to Algorithm 1 of Mukherjee et al. (2020), which consists of a per-group centering step followed by a singular value decomposition. Taking the top $k = 2 5$ singular vectors gives us matrix of sensitive directions $A$ defining the fair metric.
+
+Bios experiment We again utilize FACE algorithm of Mukherjee et al. (2020) to obtain the fair metric. Here the counterfactual modification is based on male and female gender pronouns. For example, sentence “He went to law school” is modified to “She went to law school”. As a result, each group of comparable samples consists of a pair of bios (original and modified). Let $\ b { X } \in \mathbb { R } ^ { n \times d }$ be the data matrix of BERT embeddings of the $n$ train bios, and let $X ^ { \prime } \in \mathbb { R } ^ { n \times d }$ be the corresponding modified bios. Here Algorithm 1 of Mukherjee et al. (2020) is equivalent to performing SVD on $X - X ^ { \prime }$ . We take the top $k = 2 5$ singular vectors to obtain sensitive directions $A$ and the corresponding fair metric.
+
+Adult experiment In this experiment $A$ consists of three vectors: a vector of zeros with 1 in the gender coordinate; a vector of zeros with 1 in the race coordinate; and a vector of logistic regression coefficients trained to predict gender using the remaining features (and 0 in the gender coordinate). This sensitive subspace construction replicates the approach Yurochkin et al. (2020) utilized in their Adult experiment for obtaining the fair metric. Please see Appendix B.1 and Appendix $\mathrm { D }$ in their paper for additional details.
+
+# C FAIRNESS EVALUATION METRICS DEFINITIONS
+
+Individual fairness To compare individual fairness we used two metrics: prediction consistency and Counterfactual Token Fairness (CTF) score of Garg et al. (2018). The idea behind these metrics is to quantify changes in prediction when modifying original data in ways that intuitively should not change behavior of an individually fair classifier.
+
+For Toxicity experiment an individually fair classifier should not change its prediction when a word “gay” in a comment is replaced with a word “straight”. For example, we expect toxicity predictions on “Some people are gay” and “Some people are straight” to be the same. Following prior work (Dixon et al., 2018; Garg et al., 2018) we considered a set of 50 tokens4 that should not affect the classifier when interchanged. For any comment that contains at least one of these 50 tokens we can create 49 versions of it via a simple word replacement and evaluate classifier prediction and probability of being toxic for each of the 50 variations (including the original). Prediction consistency is the proportion of comments (with at least one of the 50 tokens) where prediction is the same on all 50 variations. CTF score is the average (across all comments with at least one of the 50 tokens) standard deviation of the toxicity probability across 50 variations.
+
+We use similar individual fairness metrics for the Bios experiment. We create a single variation of each bio by interchanging “he” and “she”; “his” and “her”; “him” and “hers”; “himself” and “herself”; “mr” and “ms” or “mrs”; original name with a random name from a different gender sampled among those present in the data. Prediction consistency is computed as before using 2 variations (including the original one) of each bio. Note that although there are fewer variations, there are significantly more classes in the Bios dataset. CTF score in the average (across all bios) squared Euclidean distance between the vectors of class probabilities for the 2 bio variations.
+
+In the Adult experiment we compute same individual fairness metrics as in Yurochkin et al. (2020). S-Con. (spouse consistency) is the prediction consistency when creating data variations by altering marital status feature. GR-Con. (gender and race consistency) is the prediction consistency when creating data variations by altering race and gender features.
+
+Group fairness In our experiments we observed that enforcing individual fairness also has positive effect on group fairness metrics.
+
+In the Toxicity experiment we used accuracy parity (Zafar et al., 2017; Zhao et al., 2019) to quantify group fairness. There are multiple protected groups in the Toxicity dataset (e.g. “muslim”, “white”, “black”, “homosexual or lesbian”) that correspond to human annotated identity contexts (not necessarily mutually exclusive). To account for this when evaluating accuracy parity we computed accuracies for each of the protected groups and reported their standard deviation. Large standard deviation implies that classifier is significantly more accurate on some protected groups than on the others. Because of the class imbalance we also reported standard deviation of the corresponding balanced accuracies.
+
+For the Bios experiment we used same group fairness metrics as in the prior works studying this dataset (Romanov et al., 2019; Prost et al., 2019). Here protected attribute is binary: male or female genders. Let $\mathrm { T P R } _ { 0 , k }$ and $\mathrm { T P R } _ { 1 , k }$ denote true positive rates for class $k$ for protected attributes 0 and 1. Then TPR gap for class $k$ is $\mathrm { G a p } _ { k } = | \mathrm { T P R } _ { 0 , k } - \mathrm { T P R } _ { 1 , k } |$ . The summary statistics we report are Gap $\begin{array} { r } { \mathrm { R M S } = \sqrt { \frac { 1 } { K } \sum _ { k } \mathrm { G a p } _ { k } ^ { 2 } } } \end{array}$ and $\begin{array} { r } { \mathrm { G a p \ A B S } = \frac { 1 } { K } \sum _ { k } \mathrm { G a p } _ { k } . } \end{array}$ .
+
+For the Adult experiment we used same group fairness metrics as in Yurochkin et al. (2020), which correspond to Gap RMS described above and Gap $\mathbf { M A X } = \mathbf { m a x } _ { k } \mathbf { G a p } _ { k }$ , evaluated with respect to race and gender (both are binary protected attributes in the dataset).
+
+# D ADDITIONAL FAIRNESS-ACCURACY TRADE-OFF RESULTS
+
+Synthetic data experiment In Figure 1 we demonstrated how SenSeI allows to control fairnessaccuracy trade-off in simulations by plotting the decision boundary of the corresponding classifier for varying $\rho$ . In Figure 3 we show the lack of such flexibility in SenSR (Yurochkin et al., 2020): varying the radius of the DRO ball $\epsilon$ in their definition of individual fairness results in a horizontal decision boundary even for $\epsilon = 0$ . SenSR ties loss to fairness in its objective, and in this experiment loss can be increased significantly for anything but a horizontal decision boundary (fair metric allows free movement along the $\mathbf { X }$ -axis and a data-point can be perturbed in horizontal direction even for $\epsilon = 0$ ).
+
+![](images/9bf25b6bddd7b5fb4d54d10efe5717b46e7862a1e297f2fa3ad6a02acdd686fa.jpg)  
+Figure 3: The decision surface of a one hidden layer neural network trained with SenSR (Yurochkin et al., 2020) as the DRO radius $\epsilon$ varies. Problem setting is the same as in Figure 1. Even for $\epsilon = 0$ , SenSR prioritizes fairness over accuracy producing a horizontal decision surface. It is unable to achieve intermediate behaviors of SenSeI trading accuracy and fairness as in Figure 1 (a,b,c).
+
+Toxicity and Bios experiments In Figure 2 we presented trade-offs between prediction consistency and balanced accuracy for SenSeI and CLP (Garg et al., 2018) for the Toxicity and Bios experiments. For completeness we also present corresponding CTF score and balanced accuracy trade-offs in Figure 4. As with the prediction consistency, we see that increasing fair regularization strength $\rho$ allows to train classifiers with better individual fairness properties. SenSeI outperforms CLP as it trains classifiers with lower CTF score across all values of $\rho$ .
+
+![](images/30174e1d6471b61767eebd3e9d4dcc1ef9d9b23c8978959da564ed30bcce120c.jpg)  
+Figure 4: Balanced accuracy (BA) and CTF score trade-off on Toxicity and Bios experiments

parse/train/SJICXeWAb/SJICXeWAb.md

@@ -0,0 +1,311 @@
+# DEPTH SEPARATION AND WEIGHT-WIDTH TRADE-OFFSFOR SIGMOIDAL NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Some recent work has shown separation between the expressive power of depth-2 and depth-3 neural networks. These separation results are shown by constructing functions and input distributions, so that the function is well-approximable by a depth-3 neural network of polynomial size but it cannot be well-approximated under the chosen input distribution by any depth-2 neural network of polynomial size. These results are not robust and require carefully chosen functions as well as input distributions.
+
+We show a similar separation between the expressive power of depth-2 and depth3 sigmoidal neural networks over a large class of input distributions, as long as the weights are polynomially bounded. While doing so, we also show that depth2 sigmoidal neural networks with small width and small weights can be wellapproximated by low-degree multivariate polynomials.
+
+# 1 INTRODUCTION
+
+Understanding the remarkable success of deep neural networks in many domains is an important problem at present (e.g., LeCun et al. (2015)). This problem has many facets such as understanding generalization, expressive power, optimization algorithms in deep learning. In this paper, we focus on the question of understanding the expressive power of neural networks. In other words, we study what functions can and cannot be represented and approximated by neural networks of bounded size, depth, width and weights.
+
+The early results on the expressive power of neural networks showed that the depth-2 neural networks are universal approximators; that is to say, with only mild restrictions on the activation functions or neurons, the depth-2 neural networks are powerful enough to uniformly approximate arbitrary continuous functions on bounded domains in $\mathbb { R } ^ { d }$ , e.g., Cybenko (1989); Hornik et al. (1989); Barron (1994). However, the bounds that they provide on the size or width of these neural networks are quite general, and therefore, weak. Understanding what functions can be represented or wellapproximated by neural networks with bounded parameters is a general direction in the study of expressive power of neural networks. Here the parameters could mean the number of neurons, the width of hidden layers, the depth, and the magnitude of its weights etc.
+
+Natural signals (images, speech etc.) tend to be representable as compositional hierarchies LeCun et al. (2015), and deeper networks can be thought of as representing deeper hierarchies. The power of depth has been a subject of investigation in deep learning, e.g., He et al. (2016). We are interested in understanding the effect of depth on the expressive power. In particular, one may ask whether having more depth allows representation of more functions if the size bound remains the same.
+
+Eldan & Shamir (2016) show a separation between depth-2 and depth-3 neural networks. More precisely, they exhibit a function $\dot { g } : \mathbb { R } ^ { d }  \mathbb { R }$ and a probability distribution $\mu$ on $\mathbb { R } ^ { d }$ such that $g$ is bounded and supported on a ball of radius $O ( { \sqrt { d } } )$ and expressible by a depth-3 network of size polynomially bounded in $d$ . But any depth-2 network approximating $g$ in $L _ { 2 }$ -norm (or squared error) within a small constant under the distribution $\mu$ must be of size exponentially large in $d$ . Their separation works for all reasonable activation functions including ReLUs (Rectified Linear Units) and sigmoids. The function and the input distribution in Eldan & Shamir (2016) are carefully constructed and their proof techniques seem to crucially rely on the specifics of these constructions. Building upon this result, Safran $\&$ Shamir (2017) show that while the indicator function of the
+
+$L _ { 2 }$ -ball can be well-approximated by depth-3 networks of polynomial size, any good approximation to it by depth-2 networks must require exponential size. Here, the notion of approximation in the lower bound is the same as in Eldan & Shamir (2016) and a carefully constructed distribution that is arguably not quite natural.
+
+Daniely (2017) (see also Martens et al. (2013)) also gave a separation between depth-2 and depth-3 networks by exhibiting a function $g : \mathbb { S } ^ { d - 1 } \times \mathbb { S } ^ { d - \bar { 1 } } \to R$ which can be well-approximated by a depth-3 ReLU neural network of polynomially bounded size and weights but cannot be approximated by any depth-2 (sigmoid, ReLU or more general) neural network of polynomial size with (exponentially) bounded weights. This separation holds under uniform distribution on $\mathbb { S } ^ { d - 1 } \times \mathbb { S } ^ { d - 1 }$ , which is more natural than the previous distributions. However, the proof technique crucially uses harmonic analysis on the unit sphere, and does not seems robust or applicable to other distributions.
+
+Telgarsky (2016) shows a separation between depth- $\cdot 2 k ^ { 3 } + 8$ and depth- $k$ ReLU neural networks, for any positive integer $k$ , when the input is uniformly distributed over $[ - 1 , 1 ] ^ { d }$ . Liang & Srikant (2017) (see also Safran & Shamir (2017); Yarotsky (2016)) show that there are univariate functions on a bounded interval such that neural networks of constant depth require size at least $\Omega \left( \mathrm { p o l y } ( 1 / \epsilon ) \right)$ ) for a uniform $\epsilon$ -approximation over the interval, whereas deep networks (the depth can depend on $\epsilon$ ) can have size ${ \cal O } \left( \mathrm { p o l y l o g } ( 1 / \epsilon ) \right)$ .
+
+The above separation results all fit the following template: certain carefully constructed functions can be well approximated by deep networks, but are hard to approximate by shallow networks using a notion of error that uses a carefully defined distribution. (Only Liang & Srikant (2017) is distribution-independent as it deals with uniform approximation everywhere in the domain). Thus these results do not tell us the extent to which deeper networks are more expressive than the shallow ones. We would like to understand whether there are large classes of functions and distributions that witness the separation between deep and shallow networks. An answer to this question is also more likely to shed light on practical applications of neural networks. Shamir (2016); Shalev-Shwartz et al. (2017); Song et al. (2017) show that even functions computed by a depth-2 neural network of polynomial size can be hard to learn using gradient descent type of algorithms for a wide class of distributions. These results address questions about learnability rather than the expressive power of deep neural networks.
+
+Hanin (2017) shows that piecewise affine functions on $[ 0 , 1 ] ^ { d }$ with $N$ pieces can be exactly represented by a width $( d + 3 )$ network of depth at most $N$ . Lower bound of $\Omega ( ( N + d - 1 ) / ( d + 1 ) )$ on the depth is proven for functions of the above type when the network has width at most $( d + 1 )$ and very closely approximates the function.
+
+Our depth separation results apply to neural networks with bounds on the magnitudes of the weights. While we would prefer to prove our results without any weight restrictions, we now argue that small weights are natural. In training neural networks, often weights are not allowed to be too large to avoid overfitting. Weight decay is a commonly used regularization heuristic in deep learning to control the weights. Early stopping can also achieve this effect. Another motivation to keep the weights low is to keep the Lipschitz constant of the function computed by the network (w.r.t. changes in the input, while keeping the network parameters fixed) small. Goodfellow et al. (2016) contains many of these references. One of the surprising discoveries about neural networks has been the existence of adversarial examples (Szegedy et al. (2013)). These are examples obtained by adding a tiny perturbation to input from class so that the resulting input is misclassified by the network. The perturbations are imperceptible to humans. Existence of such examples for a network suggests that the Lipschitz constant of the network is high as noted in Szegedy et al. (2013). This lead them to suggest regularizing training of neural nets by penalizing high Lipschitz constant to improve the generalization error and, in particular, eliminate adversarial examples. This is carried out in Cisse et al. (2017), who find a way to control the Lipschitz constant by enforcing an orthonormality´ constraint on the weight matrices along with other tricks. They report better resilience to adversarial examples. On the other hand, Neyshabur et al. (2017) suggest that Lipschitz constant cannot tell the full story about generalization.
+
+# 2 OUR RESULTS
+
+We exhibit a simple function (derived from Daniely (2017)) over the unit ball $\mathbb { B } ^ { d }$ in $d$ -dimensions can be well-approximated by a depth-3 sigmoidal neural network with size and weights polynomially bounded in $d$ . However, its any reasonable approximation using a depth-2 sigmoidal neural network with polynomially bounded weights must have size exponentially large in $d$ .
+
+Our separation is robust and works for a general class of input distributions, as long as their density is at least $1 / \mathrm { p o l y } ( d )$ on some small ball of radius $1 / \mathrm { p o l y } ( \bar { d } )$ in $\mathbb { B } ^ { d }$ . The function we use can also be replaced by many other functions that are polynomially-Lipschitz but not close to any low-degree polynomial.
+
+As a by-product of our argument, we also show that constant-depth sigmoidal neural networks are well-approximated by low-degree multivariate polynomials (with a degree bound that allows the depth separation mentioned above).
+
+# 3 POLYNOMIAL APPROXIMATIONS TO SIGMOIDAL NEURAL NETWORKS
+
+In this section, we show that a sigmoid neuron can be well-approximated by a low-degree polynomial. As a corollary, we show that depth-2 (and in genenral, small-depth) sigmoidal neural networks can be well-approximated by low-degree multivariate polynomials. The main idea is to use Chebyshev polynomial approximation as in Shalev-Shwartz et al. (2011), which closely approximates the minimax polynomial (or the polynomial that has the smallest maximum deviation) to a given function. For the simplicity of presentation and arguments, we drop the bias term $b$ in the activation function $\sigma ( \langle { \bf w } , { \bf x } \rangle { } ^ { - } + b )$ . This is without loss of generality, as explained at the end of the last section.
+
+# 3.1 POLYNOMIAL APPROXIMATION TO A SIGMOID NEURON
+
+The activation function of a sigmoid neuron $\sigma : \mathbb { R }  \mathbb { R }$ is defined as
+
+$$
+\sigma ( t ) = \frac { 1 } { 1 + \exp ( - t ) } .
+$$
+
+Chebyshev polynomials of the first kind $\{ T _ { j } ( t ) \} _ { j \geq 0 }$ are defined recursively as $T _ { 0 } ( t ) = 1$ , $T _ { 1 } ( t ) = t$ , and $\dot { T } _ { j + 1 } ( t ) = 2 t \cdot T _ { j } ( t ) - T _ { j - 1 } ( t )$ . They form an orthonormal basis of polynomials over $[ - 1 , 1 ]$ with respect to the density $\textstyle \sum _ { j = 0 } ^ { \infty } c _ { j } T _ { j } ( t )$ over $[ - 1 , 1 ]$ is g iven b y $1 / \sqrt { 1 - t ^ { 2 } }$ . The coefficient $c _ { j }$ in the Chebyshev expansion of $\sigma ( w t ) =$
+
+$$
+c _ { j } = \frac { 1 + { \bf 1 } ( j > 0 ) } { \pi } \int _ { - 1 } ^ { 1 } \frac { \sigma ( w t ) T _ { j } ( t ) } { \sqrt { 1 - t ^ { 2 } } } d t .
+$$
+
+Proposition 1 (see Lemma B.1 in Shalev-Shwartz et al. (2011)) bounds the magnitude of coefficients $c _ { j }$ in the Chebyshev expansion of $\begin{array} { r } { \sigma ( w t ) = \sum _ { j = 0 } ^ { \infty } c _ { j } T _ { j } ( t ) } \end{array}$ .
+
+Proposition 1. For any $j > 1$ , the coefficient $c _ { j }$ in the Chebyshev expansion of a sigmoid neuron $\sigma ( w t )$ is bounded by
+
+$$
+| c _ { j } | \le \left( \frac { 4 } { | w | } + \frac { 2 } { \pi } \right) \left( 1 + \frac { \pi } { | w | } \right) ^ { - j } .
+$$
+
+Proposition 1 implies low-degree polynomial approximation to sigmoid neurons as follows. This observation appeared in Shalev-Shwartz et al. (2011) (see equation (B.7) in their paper). For completeness, we give the proof in Appendix A.
+
+Proposition 2. Given any $w ~ \in ~ \mathbb { R }$ with $| w | \ \leq \ B$ , there exists a polynomial $p$ of degree ${ \cal O } \left( B \log \left( B / \epsilon \right) \right)$ such that $| \sigma ( w t ) - p ( t ) | \leq \dot { \epsilon } ,$ for all $t \in [ - 1 , 1 ]$ .
+
+We use this $O \left( \log ( 1 / \epsilon ) \right)$ dependence in the above bound crucially in some of our results, e.g., a weaker version of Daniely’s separation result for depth-2 and depth-3 neural networks. Notice that this logarithmic dependence does not hold for a ReLU neuron; it is $O ( 1 / \epsilon )$ instead.
+
+A depth-2 sigmoidal neural network on input $t \in [ - 1 , 1 ]$ computes a linear combination of sigmoidal neurons $\sigma ( w _ { 1 } t ) , \sigma ( w _ { 2 } t ) , \ldots , \sigma ( w _ { n } \bar { t } )$ , for $w _ { 1 } , w _ { 2 } , \ldots , w _ { n } \in \mathbb { R }$ , and computes a function $f : [ - 1 , 1 ] \to \mathbb { R }$ given by
+
+$$
+f ( t ) = \sum _ { i = 1 } ^ { n } a _ { i } \sigma ( w _ { i } t )
+$$
+
+Here are a few propositions on polynomial approximations to small-depth neural networks. For completeness, their proofs are included in Appendix A.
+
+Proposition 3 shows that a depth-2 sigmoidal neural network of bounded weights and width is close to a low-degree polynomial.
+
+Proposition 3. Let $f : [ - 1 , 1 ] \to \mathbb { R }$ be a function computed by a depth-2 sigmoidal neural network of width n and weights bounded by $B$ . Then $f$ is $\delta$ -approximated (in $L _ { \infty }$ -norm) over $[ - 1 , 1 ]$ by $a$ polynomial of degree $O \left( B \log \left( n B ^ { 2 } / \delta \right) \right)$ .
+
+Now consider a depth-2 sigmoidal neural network on input $\textbf { x } \in \mathbb { B } ^ { d }$ , where $\mathbb { B } ^ { d } = \{ \mathbf { x } \in \mathbb { R } ^ { d } \quad :$ $\| \mathbf { x } \| \leq 1 \}$ . It is given by a linear combination of sigmoidal activations applied to linear functions $\left. \mathbf { w } _ { 1 } , \mathbf { x } \right. , \left. \mathbf { w } _ { 2 } , \mathbf { x } \right. , \ldots , \left. \mathbf { w } _ { n } , \mathbf { x } \right.$ (or affine functions when we have biases), for $\mathbf { w } _ { 1 } , \mathbf { w } _ { 2 } , \ldots , \mathbf { w } _ { n } \in \mathbb { R } ^ { d }$ and it computes a function $F : \mathbb { B } ^ { d }  \mathbb { R }$ given by
+
+$$
+F ( \mathbf { x } ) = \sum _ { i = 1 } ^ { n } a _ { i } \sigma ( \left. \mathbf { w } _ { i } , \mathbf { x } \right. )
+$$
+
+Proposition 4 below is a multivariate version of Proposition 3.
+
+Proposition 4. Let $F : \mathbb { B } ^ { d }  \mathbb { R }$ be a function computed by a depth-2 sigmoidal neural network with width $n$ and bounded weights, that is, $| a _ { i } | \le B$ and $\left\| \mathbf { w } _ { i } \right\| \leq B$ , for $1 \leq i \leq n$ . Then $F$ is $\delta$ - approximated (in $L _ { \infty }$ -norm) over $\mathbb { B } ^ { d }$ by a polynomial of degree $O \left( B \log \left( n B ^ { 2 } / \delta \right) \right)$ in $d$ variables given by the coordinates $\mathbf { x } = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { d } \right)$ .
+
+Note that its proof crucially uses the fact that Proposition 2 guarantees a low-degree polynomial that approximates a sigmoid neuron everywhere in $[ - 1 , 1 ]$ .
+
+A depth- $k$ sigmoidal neural network can be thought of as a composition – a depth-2 sigmoidal neural network on top, whose each input variable is a sigmoid applied to a depth- $\left( k - 2 \right)$ sigmoidal neural network. In other words, it computes a function $\bar { F } : \mathbb { B } ^ { d } \overset { \cdot \cdot } {  } \mathbb { R }$ given by
+
+$$
+F ( \mathbf { x } ) = \sum _ { i = 1 } ^ { n } a _ { i } \sigma \left( \left. \mathbf { w } _ { i } , \mathbf { y } \right. \right) ,
+$$
+
+where $\mathbf { y } = ( y _ { 1 } , y _ { 2 } , \dots y _ { m } )$ has each coordinate $y _ { j } = \sigma ( F _ { j } ( \mathbf { x } ) )$ , for $1 \leq j \leq m$ , such that each $F _ { i } : \mathbb { B } ^ { d }  \mathbb { R }$ is a function computed by a depth- $\left( k - 2 \right)$ sigmoidal neural network.
+
+Now we show an interesting consequence, namely, any constant-depth sigmoidal neural network with polynomial width and polynomially bounded weights can be well-approximated by a lowdegree multivariate polynomial. The bounds presented in Proposition 5 are not optimal but the qualitative statement is interesting in contrast with the depth separation result. The growth of the degree of polynomial approximation is dependent on the widths of hidden layers and it is also the subtle reason why a depth separation result is still possible (when the weights are bounded).
+
+Proposition 5. Let $F : \mathbb { B } ^ { d }  \mathbb { R }$ be a function computed by a depth- $k$ sigmoidal neural network of width at most $n$ in each layer and weights bounded by $B$ , then $F ( \mathbf { x } )$ can be $\delta$ -approximated (in $L _ { \infty }$ -norm) over $\mathbb { B } ^ { d }$ by a $d$ -variate polynomial of degree $O \left( ( n B ) ^ { k } \log ^ { k } { ( n B / \delta ) } \right)$ in each coordinate variable of $\mathbf { x } = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { d } \right)$ .
+
+Note that when $n$ and $B$ are polynomial in $d$ and the depth $k$ is constant, then this low-degree polynomial approximation also has degree polynomial in $d$ .
+
+# 4 $L _ { \infty }$ -SEPARATION OF DEPTH-2 VS. DEPTH-3 SEPARATION FOR GENERAL INPUT DISTRIBUTIONS
+
+Daniely shows that if $g : [ - 1 , 1 ] \to \mathbb { R }$ cannot be approximated by a polynomial of degree $O ( d ^ { 2 } )$ , then $\mathring { G ^ { \cdot } } \mathbb { S } ^ { d - 1 } \times \mathbb { S } ^ { d - 1 }  \mathring { \mathbb { R } }$ defined as $G ( \mathbf { x } , \mathbf { y } ) = \bar { g ( \mathbf { \langle x , y \rangle } ) }$ cannot be approximated by any depth2 neural network of polynomial size and (exponentially) bounded weights. Daniely shows this lower bound for a general neuron or activation function that includes sigmoids and ReLUs. Daniely then uses $G ( \mathbf { x } , \mathbf { y } ) { \overset { \cdot } { = } } \ g ( \langle \mathbf { x } , \mathbf { y } \rangle ) = \sin ( \pi d ^ { 3 } \left. \mathbf { x } , \mathbf { y } \right. )$ which, on the other hand, is approximable by a depth-3 ReLU neural network with polynomial size and polynomially bounded weights. This gives a separation between depth-2 and depth-3 ReLU neural networks w.r.t. uniform distribution over $\mathbb { S } ^ { d - 1 } \times \mathbb { S } ^ { d - 1 }$ . Daniely’s proof uses harmonic analysis on the unit sphere, and requires the uniform distribution on $\mathbb { S } ^ { d - 1 } \dot { \times } \mathbb { S } ^ { \dot { d } - 1 }$ in a crucial way.
+
+We show a simple proof of separation between depth-2 and depth-3 sigmoidal neural networks that compute functions $\mathbf { \dot { \boldsymbol { F } } } : \mathbb { B } ^ { d } \to \mathbf { \dot { \mathbb { R } } }$ . Our proof works for a large class of distributions on $\mathbb { B } ^ { d }$ but requires the weights to be polynomially bounded.
+
+The following lemma appears in Debao (1993). Assumption 1 in Eldan & Shamir (2016) and their version of this lemma for ReLU networks was used by Daniely (2017) in the proof of separation between the expressive power of depth-2 and depth-3 ReLU networks.
+
+Lemma 6. Let $f : [ - 1 , 1 ] \to \mathbb { R }$ be any $L$ -Lipschitz function. Then there exists a function $g :$ $[ - 1 , 1 ] \to \mathbb { R }$ computed by a depth-2 sigmoidal neural network such that
+
+$$
+g ( t ) = f ( 0 ) + \sum _ { i = 1 } ^ { n } a _ { i } \sigma ( w _ { i } t + b _ { i } ) ,
+$$
+
+the width $n$ as well as the weights are bounded by poly $( L , 1 / \epsilon )$ , and $| f ( t ) - g ( t ) | \leq \epsilon ,$ for all $t \in [ - 1 , 1 ]$ .
+
+Now we are ready to show the separation between depth-2 and depth-3 sigmoidal neural networks. The main idea, similar to Daniely (2017), is to exhibit a function that is Lipschitz but far from any low-degree polynomial. The Lipschitz property helps in showing that our function can be wellapproximated by a depth-3 neural network of small size and small weights. However, being far from any low-degree polynomial, it cannot be approximated by any depth-2 neural network.
+
+Theorem 7. Consider the function $G : \mathbb { B } ^ { d }  \mathbb { R }$ given by $G ( \mathbf { x } ) = \sin ( \pi d ^ { 5 } \left\| \mathbf { x } \right\| ^ { 2 } )$ . Then $G$ can be $\delta$ -approximated (in $L _ { \infty }$ -norm) by a depth-3 sigmoidal neural network of width and weights polynomially bounded in $d$ . However, any function $\bar { F } : \mathbb { B } ^ { d }  \mathbb { R }$ computed by a depth-2 sigmoidal neural network with weights $O ( d ^ { 2 } )$ cannot $\delta$ -approximate $G$ even when its width $n$ is $2 ^ { O ( d ) }$ .
+
+By modifying the function to $G ( \mathbf { x } ) = \sin ( \pi N \left\| \mathbf { x } \right\| ^ { 2 } )$ , this lower bound with $L _ { \infty }$ -norm holds for any distribution over $\mathbf { \mathbb { B } } ^ { d }$ whose support contains a radial line segment of length at least $1 / p o l y ( d )$ , by making $N = p o l y ( d )$ , for a large enough polynomial.
+
+Remark: Given any distribution $\mu$ over $\mathbb { B } ^ { d }$ whose probability density is at least $1 / p o l y ( d )$ on some small ball of radius $1 / p o l y ( d )$ , the lower bound or inapproximability by any depth-2 sigmoidal neural network can be made to work with $L _ { 2 }$ -norm (squared error), for a large enough $N = p o l y ( d )$ .
+
+Proof. First, we will show that $G ( \mathbf { x } )$ can be well-approximated by a depth-3 sigmoidal neural network of polynomial size and weights. The idea is similar to Daniely’s construction for ReLU networks in Daniely (2017). By Lemma 6, there exists a function $f \ \stackrel { \cdot } { : } \ [ - 1 , 1 ] \  \ \mathbb { R }$ computed by a depth-2 sigmoidal neural network of size and weights bounded by $\mathrm { p o l y } ( d , 1 / \epsilon )$ such that $\left| t ^ { 2 } - f ( t ) \right| \le \epsilon / 1 0 d ^ { 6 }$ , for all $t \in [ - 1 , 1 ]$ . Thus, we can compute $x _ { i } ^ { 2 }$ for each coordinate of $\mathbf { x }$ and add them up to get an $\epsilon$ -approximation to $\left\| \mathbf { x } \right\| ^ { 2 }$ over $\mathbb { B } ^ { d }$ . That is, there exists a function $S : \mathbb { B } ^ { d }  \mathbb { R }$ computed by a depth-2 sigmoidal neural network of size and weights bounded by poly $( d , 1 / \epsilon )$ such that $\left| S ( x ) - \left\| \mathbf { x } \right\| ^ { 2 } \right| \leq \epsilon / 1 0 d ^ { 5 }$ , for all $\mathbf { x } \in \mathbb { B } ^ { d }$ . Again, by Lemma 6, we can approximate $\sin ( \pi d ^ { 3 } t )$ over [0, 1] using $f : [ - 1 , 1 ] \to \mathbb { R }$ computed by another depth-2 sigmoidal neural network with size and weights bounded by $\mathrm { p o l y } ( d , 1 / \epsilon )$ such that $\left| \sin ( \pi d ^ { 3 } t ) - f ( t ) \right| \le \epsilon / 2$ , for all $t \in [ 0 , 1 ]$ . Note that the composition of these two depth-2 neural networks $f ( N ( \mathbf { x } ) )$ gives a depth-3 neural network as the output of the hidden layer of the bottom network can be fed into the top network as inputs.
+
+$$
+\begin{array} { r l } & { \left| G ( \mathbf { x } ) - f ( S ( \mathbf { x } ) ) \right| = \left| \sin ( \pi d ^ { 5 } \left\| \mathbf { x } \right\| ^ { 2 } ) - f ( S ( \mathbf { x } ) ) \right| } \\ & { \qquad \leq \left| \sin ( \pi d ^ { 5 } \left\| \mathbf { x } \right\| ^ { 2 } ) - f ( \left\| \mathbf { x } \right\| ^ { 2 } ) \right| + \left| f ( \left\| \mathbf { x } \right\| ^ { 2 } ) - f ( S ( \mathbf { x } ) ) \right| } \\ & { \qquad \leq \epsilon / 2 + 4 d ^ { 5 } \left| \left\| \mathbf { x } \right\| ^ { 2 } - S ( \mathbf { x } ) \right| } \end{array}
+$$
+
+$$
+\leq \epsilon / 2 + 4 d ^ { 5 } \cdot \epsilon / 1 0 d ^ { 5 } \leq \epsilon .
+$$
+
+Now we will show the lower bound. Consider any function $F : \mathbb { B } ^ { d }  \mathbb { R }$ computed by a depth-2 sigmoidal neural network whose weights are bounded by $B \ = \ O ( d ^ { 2 } )$ and width is $n$ . Proposition 4 shows that there exists a $d$ -variate polynomial $P ( \mathbf { x } )$ of degree $O \left( B \log ( n B ^ { 2 } / \delta ) \right) ~ =$ $O \left( d ^ { 2 } \log ( n / \delta ) + d ^ { 2 } \log d \right)$ in each variable such that $| F ( \mathbf { x } ) - P ( \mathbf { x } ) | \leq \delta$ , for all $\mathbf { x } \in \mathbb { B } ^ { d }$ . Let $\mu$ be any measure on $\mathbb { B } ^ { d }$ whose support contains some radial line segment of length at least $1 / \mathrm { p o l y } ( d )$ in $\mathbb { B } ^ { \bar { d } }$ . In other words, there exists a unit vector $\mathbf { u }$ such that the support of $\mu$ intersects the radial set $\{ \mathbf { x } ~ \in ~ \mathbb { B } ^ { d } \quad : \quad \mathbf { x } ~ = ~ t \mathbf { u } $ , for some $t \in [ - 1 , 1 ] \}$ in some line segment of length at least $1 / \mathrm { p o l y } ( d )$ . Then $P ( t \mathbf { u } )$ is a univariate polynomial of degree $O ( d ^ { 3 } \log ( \bar { n } / \delta ) + d ^ { 3 } \log \bar { d } )$ that $\delta$ - approximates $F ( t \mathbf { u } )$ , for all $t \in [ - 1 , 1 ]$ . By Lemma 8, using $\dot { D } = O \big ( d ^ { 3 } \log ( n / \delta ) + d ^ { 3 } \log d \big )$ , $l = 1 / \mathrm { p o l y } ( d )$ and $N = d ^ { 5 } / l$ , we get that if $n = 2 ^ { O ( d ) }$ , then there exists a $t _ { 0 } ~ \in ~ [ - 1 , 1 ]$ such that $\left| \sin ( \pi N t _ { 0 } ^ { 2 } ) - P ( t \mathbf { u } ) \right| \geq 1$ . Therefore, by triangle inequality, $\left| \sin ( \pi N \left\| t _ { 0 } \mathbf { u } \right\| ^ { 2 } ) - F ( t _ { 0 } \mathbf { u } ) \right| \geq$ $\left| \sin ( \pi N t _ { 0 } ^ { 2 } ) - P ( t _ { 0 } { \mathbf u } ) \right| - | P ( t _ { 0 } { \mathbf u } ) - F ( t _ { 0 } { \mathbf u } ) | \ge 1 - \delta > \delta _ { }$ , for $\delta < 1 / 2$ . This means that $G ( \mathbf { x } )$ cannot be well-approximated by any $F ( \mathbf { x } )$ computed by a depth-2 neural network with polynomially bounded weights even when it has width $2 ^ { O ( d ) }$ . □
+
+Now we show that the candidate function proposed by Daniely $g ( t ) = \sin ( \pi N t )$ , for large enough $N$ , is far from any low-degree polynomial w.r.t. any measure $\mu$ on $[ - 1 , 1 ]$ with a reasonable support.
+
+Lemma 8. Let $p$ be any polynomial of degree $D$ and $\mu$ be any measure on $[ - 1 , 1 ]$ whose support contains an interval of length at least $l$ . Then, for $N$ large enough to satisfy $N l > D + 3$ , there exists $t _ { 0 } \in [ - 1 , 1 ]$ such that $\mu ( t _ { 0 } ) > 0$ and $| \mathrm { s i n } ( \pi N t _ { 0 } ) - p ( t _ { 0 } ) | > 1 .$ . In other words, $\sin ( \pi N t )$ is 1-far (in $L _ { \infty }$ -norm) from any polynomial of degree $D$ over interval $[ - 1 , 1 ]$ with measure $\mu$ .
+
+Proof. Let $\mu ( t ) > 0$ for some interval $[ a , a + l ] \subseteq [ - 1 , 1 ]$ . Consider $S = \{ t \in [ a , a + l ] \colon t =$ $- 1 + ( i + 1 / 2 ) / N$ , for some integer $i \}$ . Then $S$ contains at least $N l - 2$ points where $\sin ( \pi N t )$ alternates as $\pm 1$ . Any polynomial $p$ of degree $D$ cannot match the sign of $\sin ( \pi N t )$ on all the points in $S$ . Otherwise, by intermediate value theorem, $p$ must have at least $N l - 3$ roots between the points of $S$ , which means $D \geq N l - 3$ , a contradiction. Thus, there exists $t _ { 0 } \in S$ such that $p ( t _ { 0 } )$ and $\sin ( \pi N t _ { 0 } )$ have opposite signs. Since $\sin ( \pi N t ) = \pm 1$ , for any $t \in S$ , the sign mismatch implies $| \mathrm { s i n } ( \pi N t _ { 0 } ) - p ( t _ { 0 } ) | > 1$ . □
+
+An important remark on biases: Even though we handled the case of sigmoid neurons without biases, the proof technique carries over to the sigmoid neurons with biases $\bar { \boldsymbol { \sigma } } ( \left. \mathbf { w } , \mathbf { x } \right. + b )$ . The idea is to consider a new $( d + 1 )$ -dimensional input $\mathbf { x } _ { \mathrm { n e w } } = \left( \mathbf { x } , x _ { d + 1 } \right) = \left( x _ { 1 } , x _ { 2 } , \ldots , x _ { d + 1 } \right)$ with $x _ { d + 1 } = 1$ , and consider the new weight vector $\mathbf { w } _ { \mathrm { n e w } } = ( \mathbf { w } , b )$ . Thus, $\left. \mathbf { w } _ { \mathrm { n e w } } , \mathbf { x } _ { \mathrm { n e w } } \right. = \left. \mathbf { w } , \mathbf { x } \right. + b$ . The new input lies on a $d$ -dimensional hyperplane slice of $\mathbb { B } ^ { d + 1 }$ , so we need to look at the restriction of the input distribution $\mu$ to this slice. Most of the ideas in our proofs generalize without any technical modifications. We defer the details to the full version.
+
+# 5 $L _ { 2 }$ -SEPARATION OF DEPTH-2 VS. DEPTH-3 SEPARATION FOR GENERAL INPUT DISTRIBUTIONS
+
+In this section we show lower bounds under the $L _ { 2 }$ -norm. The theorem below gives a technical condition on the class of densities $\mu$ on $\mathbb { B } ^ { d }$ for which our lower bound holds. Let’s give an example to illustrate that the condition on density is reasonable: Let $K \subset \mathbb { B } ^ { d }$ be a convex set such that every point in $K$ is at least $r$ away from the boundary of $\mathbb { B } ^ { d }$ (where $r = 1 / \mathrm { p o l y } ( d )$ is a parameter). Further assume that (1) the probability mass of $K$ is at least a constant and (2) for every point in $K$ the probability density is within a constant factor of the uniform density on $K$ . Then our lower bound applies to $\mu$ .
+
+Theorem 9. Consider the function $G : \mathbb { B } ^ { d }  \mathbb { R }$ given by $G ( \mathbf { x } ) = \sin ( \pi N \left\| \mathbf { x } \right\| ^ { 2 } )$ . Let $\mu$ be any probability density over $\mathbb { B } ^ { \dot { d } }$ such that there exists a subset $C \subseteq \mathbb { B } ^ { d }$ satisfying the following two conditions:
+
+• The $r$ -interior of $C$ defined as $C ^ { \prime } = \{ \mathbf { x } \in C ~ : ~ \mathbb { B } ( \mathbf { x } , r ) \subseteq C \}$ contains at least $\gamma$ fraction of the total probability mass for some $\gamma > 0$ , i.e., $\begin{array} { r } { \int _ { C ^ { \prime } } \mu ( \mathbf { x } ) d \mathbf { x } \geq \gamma } \end{array}$ . • For any affine line $\ell$ , the induced probability density on every segment of length at least $r$ in the intersection $\ell \cap C$ is $( \alpha , \beta )$ -uniform, i.e., it is at least $\alpha$ times and at most $\beta$ times the uniform density on that segment.
+
+Let $F : \mathbb { B } ^ { d }  \mathbb { R }$ be any function computed by a depth-2 sigmoidal neural network with weights bounded by $B$ and width $n$ . Then for any $0 < \delta \ll \alpha \gamma / 3 \beta$ and $N \gg ( B / r ^ { 2 } ) \log ( n B ^ { 2 } / \delta )$ , the function $F$ cannot $\delta$ -approximate $G$ on $\mathbb { B } ^ { d }$ under $L _ { 2 }$ -norm (squared error) under the probability density $\mu$ .
+
+In particular, if $\alpha , \beta , \gamma$ are constants, $B = p o l y ( d )$ , $n = 2 ^ { d }$ , and $r = 1 / p o l y ( d )$ , then it suffices to choose $N = p o l y ( d )$ for a sufficiently large degree polynomial.
+
+Proof. We show a lower bound on $L _ { 2 }$ -error of approximating $G ( \mathbf { x } )$ with any multivariate polynomial $P : \mathbb { B } ^ { d }  \mathbb { R }$ of degree $D$ under the distribution given by $\mu$ on $\mathbb { B } ^ { d }$ . For any fixed unit vector $\mathbf { v }$ , consider $\mathbf { u } \in \mathbb { B } ^ { d - 1 }$ orthogonal to $\mathbf { v }$ and let $\ell _ { \mathbf { u } }$ be the affine line going through $\mathbf { u }$ and parallel to the direction $\mathbf { v }$ given by $\ell _ { \mathbf { u } } = \{ \mathbf { x } = \mathbf { u } + t \mathbf { v } ~ : ~ t \in \mathbb { R } \}$ .
+
+$$
+\begin{array} { r l } {  { \int ( G ( \mathbf { x } ) - P ( \mathbf { x } ) ) ^ { 2 } \mu ( \mathbf { x } ) d \mathbf { x } } } \\ & { \ \stackrel { \mathrm { B } ^ { d } } { \geq } } \\ & { \ \geq \int ( G ( \mathbf { x } ) - P ( \mathbf { x } ) ) ^ { 2 } \mu ( \mathbf { x } ) d \mathbf { x } } \\ & { \ = \displaystyle \int \int \int ( G ( \mathbf { u } + t \mathbf { v } ) - P ( \mathbf { u } + t \mathbf { v } ) ) ^ { 2 } \mu ( \mathbf { u } + t \mathbf { v } ) \mathbb { I } _ { C } ( \mathbf { u } , t ) d t d \mathbf { u } } \end{array}
+$$
+
+$$
+\geq \int _ { \mathbb { R } ^ { d - 1 } } \int _ { \mathbb { R } } \left( G ( \mathbf { u } + t \mathbf { v } ) - P ( \mathbf { u } + t \mathbf { v } ) \right) ^ { 2 } \mu ( \mathbf { u } + t \mathbf { v } ) \mathbb { I } _ { \tilde { C } } ( \mathbf { u } , t ) d t d \mathbf { u }
+$$
+
+$$
+\begin{array} { r l } & { = \frac { \int _ { \mathbb { R } ^ { d - 1 } } \int _ { \mathbb { R } } \big ( G ( \mathbf { u } + t \mathbf { v } ) - P ( \mathbf { u } + t \mathbf { v } ) \big ) ^ { 2 } \mu ( \mathbf { u } + t \mathbf { v } ) \mathbb { I } _ { \boldsymbol { \bar { C } } } ( \mathbf { u } , t ) \ d t d \mathbf { u } } { \int _ { \mathbb { R } ^ { d - 1 } } \int _ { \mathbb { R } } \mu ( \mathbf { u } + t \mathbf { v } ) \mathbb { I } _ { \boldsymbol { \bar { C } } } ( \mathbf { u } , t ) \ d t d \mathbf { u } } \cdot \displaystyle \int _ { \mathbb { R } ^ { d - 1 } } \int _ { \mathbb { R } } \mu ( \mathbf { u } + t \mathbf { v } ) \mathbb { I } _ { \boldsymbol { \bar { C } } } ( \mathbf { u } , t ) \ d t d \mathbf { u } } \\ & { = \frac { \int _ { \mathbb { R } ^ { d - 1 } } \int _ { \mathbb { R } } \big ( G \big ( \mathbf { u } + t \mathbf { v } \big ) - P \big ( \mathbf { u } + t \mathbf { v } \big ) \big ) ^ { 2 } \mu ( \mathbf { u } + t \mathbf { v } ) \mathbb { I } _ { \boldsymbol { \bar { C } } } ( \mathbf { u } , t ) \ d t d \mathbf { u } } { \int _ { \mathbb { R } ^ { d - 1 } } \int _ { \mathbb { R } } \mu ( \mathbf { u } + t \mathbf { v } ) \int _ { \mathbb { \bar { C } } } ( \mathbf { u } , t ) \ d t d \mathbf { u } } \cdot \displaystyle \int _ { \boldsymbol { C } ^ { \prime } } \mu ( \mathbf { x } ) d \mathbf { x } } \end{array}
+$$
+
+$$
+\begin{array} { r l } & { \displaystyle \geq \operatorname* { m i n } _ { { \bf u } \in { \mathbb { R } } ^ { d - 1 } } \frac { \int _ { { \mathbb { R } } } \left( { \cal G } ( { \bf u } + t { \bf v } ) - P ( { \bf u } + t { \bf v } ) \right) ^ { 2 } \mu ( { \bf u } + t { \bf v } ) \mathbb { I } _ { { \widetilde { \cal C } } } ( { \bf u } , t ) d t } { \int _ { { \mathbb { R } } } \mu ( { \bf u } + t { \bf v } ) \mathbb { I } _ { { \widetilde { \cal C } } } ( { \bf u } , t ) d t } \cdot \displaystyle \int _ { { \cal C } ^ { \prime } } \mu ( { \bf x } ) d { \bf x } } \\ & { \displaystyle \geq \operatorname* { m i n } _ { { \bf u } \in { \mathbb { R } } ^ { d - 1 } } \frac { \alpha } { \beta } \cdot \frac { \int _ { { \mathbb { R } } } \left( { \cal G } ( { \bf u } + t { \bf v } ) - P ( { \bf u } + t { \bf v } ) \right) ^ { 2 } \mathbb { I } _ { { \widetilde { \cal C } } } ( { \bf u } , t ) d t } { \int _ { { \mathbb { R } } } \mathbb { I } _ { { \widetilde { \cal C } } } ( { \bf u } , t ) d t } \cdot \displaystyle \int _ { { \cal C } ^ { \prime } } \mu ( { \bf x } ) d { \bf x } } \end{array}
+$$
+
+because for any line $\ell$ , the distribution induced by $\mu ( \mathbf { x } )$ along any line segment of length at least $r$ in the intersection $\ell \cap C$ is $( \alpha , \beta )$ -uniform, for any line $\ell$ $\geq { \frac { \alpha } { \beta } } \cdot { \frac { \gamma } { 3 } } .$
+
+The last inequality is using the condition $\begin{array} { r } { \int _ { C ^ { \prime } } \mu ( \mathbf { x } ) d \mathbf { x } \geq \gamma } \end{array}$ given in Theorem 9 and an adaptation of the following idea from Lemma 5 of Daniely (2017). For any fixed $\mathbf { u }$ and $\mathbf { v }$ , $G ( \mathbf { u } + t \mathbf { v } ) =$ $\sin ( \pi N ( \left. \lvert \mathbf { u } \right. \rvert ^ { 2 } + t ^ { 2 } ) )$ and $P ( \mathbf { u } + t \mathbf { v } )$ is a polynomial of degree at most $D$ in $t$ . The function $\sin ( \pi N ( \left. \lvert \mathbf { u } \right. \rvert ^ { 2 } + t ^ { 2 } ) )$ alternates its sign as $\left\| \mathbf { u } \right\| ^ { 2 } + t ^ { 2 }$ takes values that are successive integer multiples of $1 / N$ . Consider $\acute { s } = t ^ { 2 } \in [ 0 , 1 ]$ and divide $[ 0 , 1 ]$ into $N$ disjoint segments using integer grid of step size $1 / N$ . For any polynomial $p ( s )$ of degree at most $D$ and any interval $I \subseteq [ 0 , 1 ]$ of length $r \gg D / N$ , there exists at least $N r - D - 2$ segments of length $1 / N$ each on which $\sin ( \pi N s )$ and $p ( s )$ do not change signs and have opposite signs. Now using $( \sin ( \pi N s ) - p ( s ) ) ^ { 2 } \ge \sin ^ { 2 } ( \pi N s ) \nonumber$ , integrating we get that $\begin{array} { r } { \int _ { I } ( \sin ( \pi N s ) - p ( s ) ) ^ { 2 } \bar { d s } \ge r / 2 } \end{array}$ . Extending this proof to $t$ instead of $s = t ^ { 2 }$ , using $\sin ^ { 2 } ( \pi N t ^ { 2 } ) t \leq \sin ^ { 2 } ( \pi N t )$ for all $t \in [ 0 , 1 ]$ , and incorporating the shift $\pi N \left. \mathbf { u } \right. ^ { 2 }$ , we can similarly show that $\begin{array} { r } { \int _ { I } \sin ^ { 2 } ( \pi N ( \left\| \mathbf { u } \right\| ^ { 2 } + t ^ { 2 } ) ) - P ( \mathbf { u } + t \mathbf { v } ) ) ^ { 2 } d t \ge r / 3 } \end{array}$ . Summing up over multiple such intervals gives the final inequality. □
+
+The $L _ { 2 }$ separation between depth-2 and depth-3 neural networks under probability density $\mu$ now follows by taking a small enough $\delta$ , and combining the following ingredients (i) Proposition 4 says that any depth-2 sigmoid neural networks of width $n = 2 ^ { d }$ and weights bounded by $B = { \mathfrak { p o l y } } ( d )$ can be $\delta$ -approximated in $L _ { \infty }$ (and hence, also $L _ { 2 }$ ) by a multivariate polynomials of degree $D = O ( B \mathbf { \bar { l o g } } ( n B ^ { 2 } / \delta ) ) = \mathrm { p o l y } ( d )$ , (ii) proof of Theorem 7 (initial part) says that $G ( \mathbf { x } )$ can be $\delta$ -approximated in $L _ { \infty }$ (and hence, also $L _ { 2 }$ ) by a depth-3 sigmoid neural network of width and size $\mathfrak { p o l y } ( d )$ , but (iii) Theorem 9 says that, for $N = { \mathrm { p o l y } } ( d )$ of large enough degree, $G ( \mathbf { x } )$ cannot be $3 \delta$ -approximated in $L _ { 2 }$ by any multivariate polynomial of degree $D$ , and (iv) triangle inequality.
+
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+
+Dmitry Yarotsky. Error bounds for approximations with deep relu networks. CoRR, abs/1610.01145, 2016. URL http://arxiv.org/abs/1610.01145.
+
+# A PROOFS OF POLYNOMIAL APPROXIMATIONS TO NEURAL NETWORKS
+
+# Proof of Proposition 2
+
+Proof. Consider the degree- $D$ approximation to $\sigma ( w t )$ given by the first $D$ terms in its Chebyshev expansion. The error of this approximation for any $t \in [ - 1 , 1 ]$ is bounded by
+
+$$
+\begin{array} { r l } {  { ( \sigma ( w t ) - p ( \ell ) | = | \sum _ { j \geq \ell } \varsigma _ { j } T _ { ( j ) } ( \ell ) | } } \\ & { \leq \sum _ { j \geq 0 } | \varsigma _ { j } | } \\ & { \leq ( \frac { 4 } { | w | } + \frac { 2 } { \pi } ) \sum _ { j > \ell } ( 1 + \frac { \pi } { | w | } ) ^ { - \beta } } \\ & { \leq ( \frac { 4 } { | w | } + \frac { 2 } { \pi } ) ( 1 + \frac { \pi } { | w | } ) ^ { - ( \beta + 1 ) } \sum _ { j = 0 } ^ { \infty } ( 1 + \frac { \pi } { | w | } ) ^ { - \beta } } \\ & { = ( \frac { 4 } { | w | } + \frac { 2 } { \pi } ) ( 1 + \frac { \pi } { | w | } ) ^ { - ( \beta + 1 ) } \cdot \frac { | w | } { \frac { \sqrt { w } } { 2 \pi } ( 1 + \frac { \pi } { | w | } ) ^ { - \beta } } } \\ & { = ( \frac { 4 } { | w | } + \frac { 2 } { \pi } ) ( 1 + \frac { \pi } { | w | } ) ^ { - ( \beta + 1 ) } \cdot \frac { | w | } { \pi } ( 1 + \frac { \pi } { | w | } ) } \\ & { \leq \epsilon _ { * } . } \end{array}
+$$
+
+using Proposition 1, $| w | \le B$ , and $D = O \left( B \log \left( B / \epsilon \right) \right)$ .
+
+# Proof of Proposition 3
+
+Proof. Let $f$ be computed by a depth-2 sigmoidal neural network given by $\begin{array} { r } { f ( t ) = \sum _ { i = 1 } ^ { n } a _ { i } \sigma ( w _ { i } t ) } \end{array}$ Define a parameter $\epsilon \ : = \ : \delta / n B$ . Proposition 2 guarantees polynomial $p _ { 1 } , p _ { 2 } , \ldots , p _ { n }$ of degree $O \left( B \log ( \mathbf { \bar { \boldsymbol { B } } } / \epsilon ) \right)$ such that $| \sigma ( w _ { i } t ) - \bar { p } _ { i } ( t ) | \ \leq \ \epsilon$ , for all $t ~ \in ~ [ - 1 , 1 ]$ . Thus, the polynomial $\begin{array} { r } { p ( t ) = \sum _ { i = 1 } ^ { n } a _ { i } p _ { i } ( t ) } \end{array}$ has degree $O \left( B \log ( B / \epsilon ) \right) = O \left( B \log ( n B ^ { 2 } / \delta ) \right)$ , and for any $t \in [ - 1 , 1 ]$ ,
+
+$$
+\begin{array} { r l } { \displaystyle \left. \sum _ { i = 1 } ^ { n } a _ { i } \sigma ( w _ { i } t ) - p ( t ) \right. = \displaystyle \left. \sum _ { i = 1 } ^ { n } a _ { i } \sigma ( w _ { i } t ) - \sum _ { i = 1 } ^ { n } a _ { i } p _ { i } ( t ) \right. } & { } \\ { \displaystyle \leq \sum _ { i = 1 } ^ { n } \lvert a _ { i } \rvert \lvert \sigma ( w _ { i } t ) - p _ { i } ( t ) \rvert } & { } \\ { \displaystyle \leq n B \epsilon } & { } \\ { \displaystyle = \delta } \end{array}
+$$
+
+# Proof of Proposition 4
+
+Proof. Let $F$ be computed by a depth-2 neural network given by $\begin{array} { r } { F ( \mathbf { x } ) \ = \ \sum _ { i = 1 } ^ { n } { a _ { i } \sigma } ( \left. \mathbf { w } _ { i } , \mathbf { x } \right. ) } \end{array}$ where $| a _ { i } | \le B$ and $\left\| \mathbf { w } _ { i } \right\| \leq B$ , for $1 \leq i \leq n$ . Thus, $\begin{array} { r } { F ( \mathbf { x } ) \ = \ \sum _ { i = 1 } ^ { n } { a _ { i } \sigma ( \left\| \mathbf { w } _ { i } \right\| t _ { i } ) } } \end{array}$ , where $t _ { i } = \left. \mathbf { w } _ { i } / \left\| \mathbf { w } _ { i } \right\| , \mathbf { x } \right. \in [ - 1 , 1 ]$ because $\mathbf { w } _ { i } / \left\| \mathbf { w } _ { i } \right\| \in \mathbb { B } ^ { d }$ , for $1 \leq i \leq n$ , and $\mathbf { x } \in \mathbb { B } ^ { d }$ .
+
+Define a parameter $\epsilon \ : = \ : \delta / n B$ . Proposition 2 guarantees polynomial $p _ { 1 } , p _ { 2 } , \ldots , p _ { n }$ of degree $O \left( B \log ( \mathbf { \bar { \boldsymbol { B } } } / \epsilon ) \right)$ such that $| \dot { \sigma } ( \| w _ { i } \| t ) - p _ { i } ( t ) | \ \leq \ \dot { \epsilon }$ , for all $t \in [ - 1 , 1 ]$ . Consider the following polynomial $\begin{array} { r } { P ( \mathbf { x } ) = P ( x _ { 1 } , x _ { 2 } , \ldots , x _ { d } ) = \sum _ { i = 1 } ^ { n } a _ { i } p _ { i } ( \langle \mathbf { w } _ { i } / \left\| \mathbf { w } _ { i } \right\| , \mathbf { x } \rangle ) } \end{array}$ . $P ( \mathbf { x } )$ is a $d$ -variate polynomial of degree $O \left( B \log ( B / \epsilon ) \right) = O \left( B \log ( n B ^ { 2 } / \delta ) \right)$ in each variable $x _ { 1 } , x _ { 2 } , \ldots , x _ { d }$ . For any $\mathbf { x } \in \mathbb { B } ^ { d }$ ,
+
+$$
+| F ( \mathbf { x } ) - P ( \mathbf { x } ) | = \left| \sum _ { i = 1 } ^ { n } a _ { i } \sigma ( \langle \mathbf { w } _ { i } , \mathbf { x } \rangle ) - \sum _ { i = 1 } ^ { n } a _ { i } p _ { i } ( \langle \mathbf { w } _ { i } / \| \mathbf { w } _ { i } \| , \mathbf { x } \rangle ) \right|
+$$
+
+$$
+\begin{array} { r l } & { \leq \displaystyle \sum _ { i = 1 } ^ { n }  a _ { i }   \sigma (  \mathbf { w } _ { i } , \mathbf { x }  ) - p _ { i } (  \mathbf { w } _ { i } /  \mathbf { w } _ { i }  , \mathbf { x } ) )  } \\ & { \leq B \displaystyle \sum _ { i = 1 } ^ { n }  \sigma (  \mathbf { w } _ { i }  t _ { i } ) - p _ { i } ( t _ { i } )  \qquad \mathrm { u s i n g ~ } t _ { i } =  \mathbf { w } _ { i } /  \mathbf { w } _ { i }  , \mathbf { x }  ; } \\ & { \leq \epsilon n B \qquad \mathrm { u s i n g ~ }  \sigma (  w _ { i }  t ) - p _ { i } ( t )  \leq \epsilon , \mathrm { f o r ~ a l l ~ } t \in [ - 1 , 1 ] } \\ & { = \delta . } \end{array}
+$$
+
+# Proof of Proposition 5
+
+Proof. We prove this by induction on the depth $k$ . By induction hypothesis each $F _ { j } ( \mathbf { x } )$ can be $\epsilon _ { 1 }$ -approximated (in $L _ { \infty }$ -norm) by a $d$ -variate polynomial $Q _ { j } ( \mathbf { x } )$ of degree $O \left( ( n B ) ^ { k - 2 } \log ^ { ( k - 2 ) } ( n B / \epsilon _ { 1 } ) \right)$ in each variable. Thus, $| F _ { j } ( \mathbf { x } ) - Q _ { j } ( \mathbf { x } ) | = \epsilon _ { 1 }$ , for any $\mathbf { x } \in \mathbb { B } ^ { d }$ and $1 \leq j \leq m$ . Because a sigmoid neuron is Lipschitz,
+
+$$
+| y _ { j } - \sigma ( Q _ { j } ( \mathbf x ) ) | = | \sigma ( F _ { j } ( \mathbf x ) ) - \sigma ( Q _ { j } ( \mathbf x ) ) | \le | F _ { j } ( \mathbf x ) - Q _ { j } ( \mathbf x ) | \le \epsilon _ { 1 } ,
+$$
+
+for any $\mathbf { x } \in \mathbb { B } ^ { d }$ and $1 \leq j \leq m$ .
+
+Since $F _ { j } ( \mathbf { x } )$ is the output of a depth- $\left( k - 2 \right)$ sigmoidal neural network of width at most $n$ and weights at most $B$ , we must have $| F _ { j } ( \mathbf { x } ) | \leq n B$ , for all $\mathbf { x } \in \mathbb { B } ^ { d }$ . Thus, $| Q _ { j } ( \mathbf { x } ) | \leq n B + \epsilon _ { 1 } \leq 2 n B$ . By Proposition 2, there exists a polynomial $q ( t )$ of degree at most $O \left( n B \log ( n B / \epsilon _ { 2 } ) \right)$ such that
+
+$$
+| \sigma ( Q _ { j } ( \mathbf x ) ) - q ( Q _ { j } ( \mathbf x ) ) | \le \epsilon _ { 2 } ,
+$$
+
+for all $\mathbf { x } \in \mathbb { B } ^ { d }$ and $1 \leq j \leq m$ .
+
+Consider $\mathbf { q } \in \mathbb { R } ^ { m }$ as $\mathbf { q } = ( q ( Q _ { 1 } ( \mathbf { x } ) ) , q ( Q _ { 2 } ( \mathbf { x } ) ) , \dots , q ( Q _ { m } ( \mathbf { x } ) ) )$ . Then, for any $\mathbf { x } \in \mathbb { B } ^ { d }$ , we have
+
+$$
+\begin{array} { r l } & { \left| \left. \mathbf { w } _ { i } , \mathbf { y } \right. - \left. \mathbf { w } _ { i } , \mathbf { q } \right. \right| = \left| \left. \mathbf { w } _ { i } , \mathbf { y } - \mathbf { q } \right. \right| } \\ & { \qquad \leq \left\| \mathbf { w } _ { i } \right\| \left\| \mathbf { y } - \mathbf { q } \right\| } \\ & { \qquad \leq B \left( \displaystyle \sum _ { j = 1 } ^ { m } ( y _ { j } - q ( Q _ { j } ( \mathbf { x } ) ) ) ^ { 2 } \right) ^ { 1 / 2 } } \\ & { \qquad \leq B \sqrt { m } \left( \epsilon _ { 1 } + \epsilon _ { 2 } \right) } \\ & { \qquad \leq B \sqrt { n } \left( \epsilon _ { 1 } + \epsilon _ { 2 } \right) . } \end{array}
+$$
+
+Again by Proposition 2, there is a polynomial $p$ of degree at most $O \left( n B \log ( n B / \epsilon ) \right)$ such that $| \sigma ( \langle \mathbf { w } _ { i } , \mathbf { q } \rangle ) - p ( \langle \mathbf { w } _ { i } , \mathbf { q } \rangle ) | \leq \epsilon$ , for all $\dot { \mathbf { x } } \in \mathbb { B } ^ { d }$ and $1 \leq i \leq n$ . This is because $| \langle \mathbf { w } _ { i } , \mathbf { q } \rangle | = O ( n B )$ .
+
+Let’s define $\begin{array} { r } { P ( \mathbf { x } ) = \sum _ { i = 1 } ^ { n } a _ { i } p ( \langle \mathbf { w } _ { i } , \mathbf { q } \rangle ) } \end{array}$ . Therefore, for any $\mathbf { x } \in \mathbb { B } ^ { d }$ ,
+
+$$
+\begin{array} { l } { \displaystyle | F ( \mathbf x ) - P ( \mathbf x ) | = \left| \displaystyle \sum _ { i = 1 } ^ { n } a _ { i } \sigma \big ( \langle \mathbf w _ { i } , \mathbf y \rangle \big ) - \displaystyle \sum _ { i = 1 } ^ { n } a _ { i } p \big ( \langle \mathbf w _ { i } , \mathbf q \rangle \big ) \right| } \\ { \displaystyle \leq \displaystyle \sum _ { i = 1 } ^ { n } \big | a _ { i } \big | | \sigma \big ( \langle \mathbf w _ { i } , \mathbf y \rangle \big ) - p \big ( \langle \mathbf w _ { i } , \mathbf q \rangle \big ) | } \\ { \displaystyle \leq \displaystyle \sum _ { i = 1 } ^ { n } \big | a _ { i } \big | \big ( | \sigma \big ( \langle \mathbf w _ { i } , \mathbf y \rangle \big ) - \sigma \big ( \langle \mathbf w _ { i } , \mathbf q \rangle \big ) \big | + | \sigma \big ( \langle \mathbf w _ { i } , \mathbf q \rangle \big ) - p \big ( \langle \mathbf w _ { i } , \mathbf q \rangle \big ) \big | \big ) } \\ { \displaystyle \leq \displaystyle \sum _ { i = 1 } ^ { n } a _ { i } \big | \big ( | \langle \mathbf w _ { i } , \mathbf y \rangle - \langle \mathbf w _ { i } , \mathbf q \rangle | + \epsilon \big ) } \\ { \displaystyle \leq n B \big ( B \sqrt { n } ( \epsilon _ { 1 } + \epsilon _ { 2 } ) + \epsilon \big ) } \\ { \displaystyle \leq \delta , } \end{array}
+$$
+
+if we use $\epsilon _ { 1 } = \epsilon _ { 2 } = \delta / 3 n ^ { 3 / 2 } B ^ { 2 }$ and $\epsilon = \delta / 3 n B$ .
+
+$P ( \mathbf { x } )$ is a $d$ -variate polynomial of degree
+
+$$
+\deg ( P ) \leq \deg ( p ) \deg ( q ) \cdot \deg ( Q _ { j } ) = O \left( ( n B ) ^ { k } \log ^ { k } \left( n B / \delta \right) \right) ,
+$$
+
+in each variable.

parse/train/SJICXeWAb/SJICXeWAb_content_list.json

@@ -0,0 +1,1556 @@
+[
+    {
+        "type": "text",
+        "text": "DEPTH SEPARATION AND WEIGHT-WIDTH TRADE-OFFSFOR SIGMOIDAL NEURAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            250
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Some recent work has shown separation between the expressive power of depth-2 and depth-3 neural networks. These separation results are shown by constructing functions and input distributions, so that the function is well-approximable by a depth-3 neural network of polynomial size but it cannot be well-approximated under the chosen input distribution by any depth-2 neural network of polynomial size. These results are not robust and require carefully chosen functions as well as input distributions. ",
+        "bbox": [
+            233,
+            267,
+            764,
+            364
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We show a similar separation between the expressive power of depth-2 and depth3 sigmoidal neural networks over a large class of input distributions, as long as the weights are polynomially bounded. While doing so, we also show that depth2 sigmoidal neural networks with small width and small weights can be wellapproximated by low-degree multivariate polynomials. ",
+        "bbox": [
+            233,
+            367,
+            764,
+            436
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            465,
+            336,
+            481
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Understanding the remarkable success of deep neural networks in many domains is an important problem at present (e.g., LeCun et al. (2015)). This problem has many facets such as understanding generalization, expressive power, optimization algorithms in deep learning. In this paper, we focus on the question of understanding the expressive power of neural networks. In other words, we study what functions can and cannot be represented and approximated by neural networks of bounded size, depth, width and weights. ",
+        "bbox": [
+            174,
+            497,
+            825,
+            580
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The early results on the expressive power of neural networks showed that the depth-2 neural networks are universal approximators; that is to say, with only mild restrictions on the activation functions or neurons, the depth-2 neural networks are powerful enough to uniformly approximate arbitrary continuous functions on bounded domains in $\\mathbb { R } ^ { d }$ , e.g., Cybenko (1989); Hornik et al. (1989); Barron (1994). However, the bounds that they provide on the size or width of these neural networks are quite general, and therefore, weak. Understanding what functions can be represented or wellapproximated by neural networks with bounded parameters is a general direction in the study of expressive power of neural networks. Here the parameters could mean the number of neurons, the width of hidden layers, the depth, and the magnitude of its weights etc. ",
+        "bbox": [
+            174,
+            588,
+            825,
+            713
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Natural signals (images, speech etc.) tend to be representable as compositional hierarchies LeCun et al. (2015), and deeper networks can be thought of as representing deeper hierarchies. The power of depth has been a subject of investigation in deep learning, e.g., He et al. (2016). We are interested in understanding the effect of depth on the expressive power. In particular, one may ask whether having more depth allows representation of more functions if the size bound remains the same. ",
+        "bbox": [
+            174,
+            719,
+            823,
+            790
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Eldan & Shamir (2016) show a separation between depth-2 and depth-3 neural networks. More precisely, they exhibit a function $\\dot { g } : \\mathbb { R } ^ { d }  \\mathbb { R }$ and a probability distribution $\\mu$ on $\\mathbb { R } ^ { d }$ such that $g$ is bounded and supported on a ball of radius $O ( { \\sqrt { d } } )$ and expressible by a depth-3 network of size polynomially bounded in $d$ . But any depth-2 network approximating $g$ in $L _ { 2 }$ -norm (or squared error) within a small constant under the distribution $\\mu$ must be of size exponentially large in $d$ . Their separation works for all reasonable activation functions including ReLUs (Rectified Linear Units) and sigmoids. The function and the input distribution in Eldan & Shamir (2016) are carefully constructed and their proof techniques seem to crucially rely on the specifics of these constructions. Building upon this result, Safran $\\&$ Shamir (2017) show that while the indicator function of the ",
+        "bbox": [
+            174,
+            796,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "$L _ { 2 }$ -ball can be well-approximated by depth-3 networks of polynomial size, any good approximation to it by depth-2 networks must require exponential size. Here, the notion of approximation in the lower bound is the same as in Eldan & Shamir (2016) and a carefully constructed distribution that is arguably not quite natural. ",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Daniely (2017) (see also Martens et al. (2013)) also gave a separation between depth-2 and depth-3 networks by exhibiting a function $g : \\mathbb { S } ^ { d - 1 } \\times \\mathbb { S } ^ { d - \\bar { 1 } } \\to R$ which can be well-approximated by a depth-3 ReLU neural network of polynomially bounded size and weights but cannot be approximated by any depth-2 (sigmoid, ReLU or more general) neural network of polynomial size with (exponentially) bounded weights. This separation holds under uniform distribution on $\\mathbb { S } ^ { d - 1 } \\times \\mathbb { S } ^ { d - 1 }$ , which is more natural than the previous distributions. However, the proof technique crucially uses harmonic analysis on the unit sphere, and does not seems robust or applicable to other distributions. ",
+        "bbox": [
+            174,
+            166,
+            825,
+            263
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Telgarsky (2016) shows a separation between depth- $\\cdot 2 k ^ { 3 } + 8$ and depth- $k$ ReLU neural networks, for any positive integer $k$ , when the input is uniformly distributed over $[ - 1 , 1 ] ^ { d }$ . Liang & Srikant (2017) (see also Safran & Shamir (2017); Yarotsky (2016)) show that there are univariate functions on a bounded interval such that neural networks of constant depth require size at least $\\Omega \\left( \\mathrm { p o l y } ( 1 / \\epsilon ) \\right)$ ) for a uniform $\\epsilon$ -approximation over the interval, whereas deep networks (the depth can depend on $\\epsilon$ ) can have size ${ \\cal O } \\left( \\mathrm { p o l y l o g } ( 1 / \\epsilon ) \\right)$ . ",
+        "bbox": [
+            174,
+            270,
+            825,
+            356
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The above separation results all fit the following template: certain carefully constructed functions can be well approximated by deep networks, but are hard to approximate by shallow networks using a notion of error that uses a carefully defined distribution. (Only Liang & Srikant (2017) is distribution-independent as it deals with uniform approximation everywhere in the domain). Thus these results do not tell us the extent to which deeper networks are more expressive than the shallow ones. We would like to understand whether there are large classes of functions and distributions that witness the separation between deep and shallow networks. An answer to this question is also more likely to shed light on practical applications of neural networks. Shamir (2016); Shalev-Shwartz et al. (2017); Song et al. (2017) show that even functions computed by a depth-2 neural network of polynomial size can be hard to learn using gradient descent type of algorithms for a wide class of distributions. These results address questions about learnability rather than the expressive power of deep neural networks. ",
+        "bbox": [
+            174,
+            361,
+            825,
+            527
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Hanin (2017) shows that piecewise affine functions on $[ 0 , 1 ] ^ { d }$ with $N$ pieces can be exactly represented by a width $( d + 3 )$ network of depth at most $N$ . Lower bound of $\\Omega ( ( N + d - 1 ) / ( d + 1 ) )$ on the depth is proven for functions of the above type when the network has width at most $( d + 1 )$ and very closely approximates the function. ",
+        "bbox": [
+            174,
+            535,
+            825,
+            590
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our depth separation results apply to neural networks with bounds on the magnitudes of the weights. While we would prefer to prove our results without any weight restrictions, we now argue that small weights are natural. In training neural networks, often weights are not allowed to be too large to avoid overfitting. Weight decay is a commonly used regularization heuristic in deep learning to control the weights. Early stopping can also achieve this effect. Another motivation to keep the weights low is to keep the Lipschitz constant of the function computed by the network (w.r.t. changes in the input, while keeping the network parameters fixed) small. Goodfellow et al. (2016) contains many of these references. One of the surprising discoveries about neural networks has been the existence of adversarial examples (Szegedy et al. (2013)). These are examples obtained by adding a tiny perturbation to input from class so that the resulting input is misclassified by the network. The perturbations are imperceptible to humans. Existence of such examples for a network suggests that the Lipschitz constant of the network is high as noted in Szegedy et al. (2013). This lead them to suggest regularizing training of neural nets by penalizing high Lipschitz constant to improve the generalization error and, in particular, eliminate adversarial examples. This is carried out in Cisse et al. (2017), who find a way to control the Lipschitz constant by enforcing an orthonormality´ constraint on the weight matrices along with other tricks. They report better resilience to adversarial examples. On the other hand, Neyshabur et al. (2017) suggest that Lipschitz constant cannot tell the full story about generalization. ",
+        "bbox": [
+            173,
+            598,
+            825,
+            848
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 OUR RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            323,
+            118
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We exhibit a simple function (derived from Daniely (2017)) over the unit ball $\\mathbb { B } ^ { d }$ in $d$ -dimensions can be well-approximated by a depth-3 sigmoidal neural network with size and weights polynomially bounded in $d$ . However, its any reasonable approximation using a depth-2 sigmoidal neural network with polynomially bounded weights must have size exponentially large in $d$ . ",
+        "bbox": [
+            174,
+            133,
+            823,
+            190
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our separation is robust and works for a general class of input distributions, as long as their density is at least $1 / \\mathrm { p o l y } ( d )$ on some small ball of radius $1 / \\mathrm { p o l y } ( \\bar { d } )$ in $\\mathbb { B } ^ { d }$ . The function we use can also be replaced by many other functions that are polynomially-Lipschitz but not close to any low-degree polynomial. ",
+        "bbox": [
+            173,
+            196,
+            825,
+            253
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As a by-product of our argument, we also show that constant-depth sigmoidal neural networks are well-approximated by low-degree multivariate polynomials (with a degree bound that allows the depth separation mentioned above). ",
+        "bbox": [
+            174,
+            258,
+            825,
+            301
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 POLYNOMIAL APPROXIMATIONS TO SIGMOIDAL NEURAL NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            171,
+            323,
+            774,
+            339
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we show that a sigmoid neuron can be well-approximated by a low-degree polynomial. As a corollary, we show that depth-2 (and in genenral, small-depth) sigmoidal neural networks can be well-approximated by low-degree multivariate polynomials. The main idea is to use Chebyshev polynomial approximation as in Shalev-Shwartz et al. (2011), which closely approximates the minimax polynomial (or the polynomial that has the smallest maximum deviation) to a given function. For the simplicity of presentation and arguments, we drop the bias term $b$ in the activation function $\\sigma ( \\langle { \\bf w } , { \\bf x } \\rangle { } ^ { - } + b )$ . This is without loss of generality, as explained at the end of the last section. ",
+        "bbox": [
+            173,
+            353,
+            825,
+            453
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 POLYNOMIAL APPROXIMATION TO A SIGMOID NEURON ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            470,
+            596,
+            484
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The activation function of a sigmoid neuron $\\sigma : \\mathbb { R }  \\mathbb { R }$ is defined as ",
+        "bbox": [
+            173,
+            494,
+            624,
+            511
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b98b6eab3a683e4c1574632514f4f429cfe5fd2693f071725e7463eae8a2272.jpg",
+        "text": "$$\n\\sigma ( t ) = \\frac { 1 } { 1 + \\exp ( - t ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            517,
+            571,
+            550
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Chebyshev polynomials of the first kind $\\{ T _ { j } ( t ) \\} _ { j \\geq 0 }$ are defined recursively as $T _ { 0 } ( t ) = 1$ , $T _ { 1 } ( t ) = t$ , and $\\dot { T } _ { j + 1 } ( t ) = 2 t \\cdot T _ { j } ( t ) - T _ { j - 1 } ( t )$ . They form an orthonormal basis of polynomials over $[ - 1 , 1 ]$ with respect to the density $\\textstyle \\sum _ { j = 0 } ^ { \\infty } c _ { j } T _ { j } ( t )$ over $[ - 1 , 1 ]$ is g iven b y $1 / \\sqrt { 1 - t ^ { 2 } }$ . The coefficient $c _ { j }$ in the Chebyshev expansion of $\\sigma ( w t ) =$ ",
+        "bbox": [
+            173,
+            564,
+            825,
+            625
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9852c269bae9c6f495b567adab16af9e80926d06eb49c0bf4bae2e886610f91b.jpg",
+        "text": "$$\nc _ { j } = \\frac { 1 + { \\bf 1 } ( j > 0 ) } { \\pi } \\int _ { - 1 } ^ { 1 } \\frac { \\sigma ( w t ) T _ { j } ( t ) } { \\sqrt { 1 - t ^ { 2 } } } d t .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            633,
+            632,
+            671
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 (see Lemma B.1 in Shalev-Shwartz et al. (2011)) bounds the magnitude of coefficients $c _ { j }$ in the Chebyshev expansion of $\\begin{array} { r } { \\sigma ( w t ) = \\sum _ { j = 0 } ^ { \\infty } c _ { j } T _ { j } ( t ) } \\end{array}$ . ",
+        "bbox": [
+            173,
+            684,
+            821,
+            715
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. For any $j > 1$ , the coefficient $c _ { j }$ in the Chebyshev expansion of a sigmoid neuron $\\sigma ( w t )$ is bounded by ",
+        "bbox": [
+            176,
+            718,
+            821,
+            747
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8451b801bcfb39f765b7d49a29a457a2e54e48561d4cd9436619a1b1220a8fcb.jpg",
+        "text": "$$\n| c _ { j } | \\le \\left( \\frac { 4 } { | w | } + \\frac { 2 } { \\pi } \\right) \\left( 1 + \\frac { \\pi } { | w | } \\right) ^ { - j } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            746,
+            617,
+            784
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 implies low-degree polynomial approximation to sigmoid neurons as follows. This observation appeared in Shalev-Shwartz et al. (2011) (see equation (B.7) in their paper). For completeness, we give the proof in Appendix A. ",
+        "bbox": [
+            174,
+            794,
+            825,
+            837
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. Given any $w ~ \\in ~ \\mathbb { R }$ with $| w | \\ \\leq \\ B$ , there exists a polynomial $p$ of degree ${ \\cal O } \\left( B \\log \\left( B / \\epsilon \\right) \\right)$ such that $| \\sigma ( w t ) - p ( t ) | \\leq \\dot { \\epsilon } ,$ for all $t \\in [ - 1 , 1 ]$ . ",
+        "bbox": [
+            173,
+            840,
+            823,
+            871
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We use this $O \\left( \\log ( 1 / \\epsilon ) \\right)$ dependence in the above bound crucially in some of our results, e.g., a weaker version of Daniely’s separation result for depth-2 and depth-3 neural networks. Notice that this logarithmic dependence does not hold for a ReLU neuron; it is $O ( 1 / \\epsilon )$ instead. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A depth-2 sigmoidal neural network on input $t \\in [ - 1 , 1 ]$ computes a linear combination of sigmoidal neurons $\\sigma ( w _ { 1 } t ) , \\sigma ( w _ { 2 } t ) , \\ldots , \\sigma ( w _ { n } \\bar { t } )$ , for $w _ { 1 } , w _ { 2 } , \\ldots , w _ { n } \\in \\mathbb { R }$ , and computes a function $f : [ - 1 , 1 ] \\to \\mathbb { R }$ given by ",
+        "bbox": [
+            173,
+            130,
+            825,
+            174
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25f764f0ac5e279bbe4c9de07f6ca48533f367400ead6d5497ce55b690c78c34.jpg",
+        "text": "$$\nf ( t ) = \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma ( w _ { i } t )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            175,
+            568,
+            217
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here are a few propositions on polynomial approximations to small-depth neural networks. For completeness, their proofs are included in Appendix A. ",
+        "bbox": [
+            174,
+            223,
+            823,
+            252
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3 shows that a depth-2 sigmoidal neural network of bounded weights and width is close to a low-degree polynomial. ",
+        "bbox": [
+            173,
+            257,
+            821,
+            286
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3. Let $f : [ - 1 , 1 ] \\to \\mathbb { R }$ be a function computed by a depth-2 sigmoidal neural network of width n and weights bounded by $B$ . Then $f$ is $\\delta$ -approximated (in $L _ { \\infty }$ -norm) over $[ - 1 , 1 ]$ by $a$ polynomial of degree $O \\left( B \\log \\left( n B ^ { 2 } / \\delta \\right) \\right)$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Now consider a depth-2 sigmoidal neural network on input $\\textbf { x } \\in \\mathbb { B } ^ { d }$ , where $\\mathbb { B } ^ { d } = \\{ \\mathbf { x } \\in \\mathbb { R } ^ { d } \\quad :$ $\\| \\mathbf { x } \\| \\leq 1 \\}$ . It is given by a linear combination of sigmoidal activations applied to linear functions $\\left. \\mathbf { w } _ { 1 } , \\mathbf { x } \\right. , \\left. \\mathbf { w } _ { 2 } , \\mathbf { x } \\right. , \\ldots , \\left. \\mathbf { w } _ { n } , \\mathbf { x } \\right.$ (or affine functions when we have biases), for $\\mathbf { w } _ { 1 } , \\mathbf { w } _ { 2 } , \\ldots , \\mathbf { w } _ { n } \\in \\mathbb { R } ^ { d }$ and it computes a function $F : \\mathbb { B } ^ { d }  \\mathbb { R }$ given by ",
+        "bbox": [
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+            351,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/95c652068e3430424bc5108bdc5a8ca4eff919ba3d7eddc724905039e9ffcbd6.jpg",
+        "text": "$$\nF ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma ( \\left. \\mathbf { w } _ { i } , \\mathbf { x } \\right. )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            416,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4 below is a multivariate version of Proposition 3. ",
+        "bbox": [
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+            465,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4. Let $F : \\mathbb { B } ^ { d }  \\mathbb { R }$ be a function computed by a depth-2 sigmoidal neural network with width $n$ and bounded weights, that is, $| a _ { i } | \\le B$ and $\\left\\| \\mathbf { w } _ { i } \\right\\| \\leq B$ , for $1 \\leq i \\leq n$ . Then $F$ is $\\delta$ - approximated (in $L _ { \\infty }$ -norm) over $\\mathbb { B } ^ { d }$ by a polynomial of degree $O \\left( B \\log \\left( n B ^ { 2 } / \\delta \\right) \\right)$ in $d$ variables given by the coordinates $\\mathbf { x } = \\left( x _ { 1 } , x _ { 2 } , \\ldots , x _ { d } \\right)$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that its proof crucially uses the fact that Proposition 2 guarantees a low-degree polynomial that approximates a sigmoid neuron everywhere in $[ - 1 , 1 ]$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A depth- $k$ sigmoidal neural network can be thought of as a composition – a depth-2 sigmoidal neural network on top, whose each input variable is a sigmoid applied to a depth- $\\left( k - 2 \\right)$ sigmoidal neural network. In other words, it computes a function $\\bar { F } : \\mathbb { B } ^ { d } \\overset { \\cdot \\cdot } {  } \\mathbb { R }$ given by ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/76512ff0331b38f0f2104227da71e9af179931eb7895a5ba7ed0f4dde320a760.jpg",
+        "text": "$$\nF ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma \\left( \\left. \\mathbf { w } _ { i } , \\mathbf { y } \\right. \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            645,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { y } = ( y _ { 1 } , y _ { 2 } , \\dots y _ { m } )$ has each coordinate $y _ { j } = \\sigma ( F _ { j } ( \\mathbf { x } ) )$ , for $1 \\leq j \\leq m$ , such that each $F _ { i } : \\mathbb { B } ^ { d }  \\mathbb { R }$ is a function computed by a depth- $\\left( k - 2 \\right)$ sigmoidal neural network. ",
+        "bbox": [
+            173,
+            695,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Now we show an interesting consequence, namely, any constant-depth sigmoidal neural network with polynomial width and polynomially bounded weights can be well-approximated by a lowdegree multivariate polynomial. The bounds presented in Proposition 5 are not optimal but the qualitative statement is interesting in contrast with the depth separation result. The growth of the degree of polynomial approximation is dependent on the widths of hidden layers and it is also the subtle reason why a depth separation result is still possible (when the weights are bounded). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 5. Let $F : \\mathbb { B } ^ { d }  \\mathbb { R }$ be a function computed by a depth- $k$ sigmoidal neural network of width at most $n$ in each layer and weights bounded by $B$ , then $F ( \\mathbf { x } )$ can be $\\delta$ -approximated (in $L _ { \\infty }$ -norm) over $\\mathbb { B } ^ { d }$ by a $d$ -variate polynomial of degree $O \\left( ( n B ) ^ { k } \\log ^ { k } { ( n B / \\delta ) } \\right)$ in each coordinate variable of $\\mathbf { x } = \\left( x _ { 1 } , x _ { 2 } , \\ldots , x _ { d } \\right)$ . ",
+        "bbox": [
+            173,
+            821,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that when $n$ and $B$ are polynomial in $d$ and the depth $k$ is constant, then this low-degree polynomial approximation also has degree polynomial in $d$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 $L _ { \\infty }$ -SEPARATION OF DEPTH-2 VS. DEPTH-3 SEPARATION FOR GENERAL INPUT DISTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            102,
+            790,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Daniely shows that if $g : [ - 1 , 1 ] \\to \\mathbb { R }$ cannot be approximated by a polynomial of degree $O ( d ^ { 2 } )$ , then $\\mathring { G ^ { \\cdot } } \\mathbb { S } ^ { d - 1 } \\times \\mathbb { S } ^ { d - 1 }  \\mathring { \\mathbb { R } }$ defined as $G ( \\mathbf { x } , \\mathbf { y } ) = \\bar { g ( \\mathbf { \\langle x , y \\rangle } ) }$ cannot be approximated by any depth2 neural network of polynomial size and (exponentially) bounded weights. Daniely shows this lower bound for a general neuron or activation function that includes sigmoids and ReLUs. Daniely then uses $G ( \\mathbf { x } , \\mathbf { y } ) { \\overset { \\cdot } { = } } \\ g ( \\langle \\mathbf { x } , \\mathbf { y } \\rangle ) = \\sin ( \\pi d ^ { 3 } \\left. \\mathbf { x } , \\mathbf { y } \\right. )$ which, on the other hand, is approximable by a depth-3 ReLU neural network with polynomial size and polynomially bounded weights. This gives a separation between depth-2 and depth-3 ReLU neural networks w.r.t. uniform distribution over $\\mathbb { S } ^ { d - 1 } \\times \\mathbb { S } ^ { d - 1 }$ . Daniely’s proof uses harmonic analysis on the unit sphere, and requires the uniform distribution on $\\mathbb { S } ^ { d - 1 } \\dot { \\times } \\mathbb { S } ^ { \\dot { d } - 1 }$ in a crucial way. ",
+        "bbox": [
+            173,
+            150,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We show a simple proof of separation between depth-2 and depth-3 sigmoidal neural networks that compute functions $\\mathbf { \\dot { \\boldsymbol { F } } } : \\mathbb { B } ^ { d } \\to \\mathbf { \\dot { \\mathbb { R } } }$ . Our proof works for a large class of distributions on $\\mathbb { B } ^ { d }$ but requires the weights to be polynomially bounded. ",
+        "bbox": [
+            174,
+            284,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The following lemma appears in Debao (1993). Assumption 1 in Eldan & Shamir (2016) and their version of this lemma for ReLU networks was used by Daniely (2017) in the proof of separation between the expressive power of depth-2 and depth-3 ReLU networks. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Lemma 6. Let $f : [ - 1 , 1 ] \\to \\mathbb { R }$ be any $L$ -Lipschitz function. Then there exists a function $g :$ $[ - 1 , 1 ] \\to \\mathbb { R }$ computed by a depth-2 sigmoidal neural network such that ",
+        "bbox": [
+            169,
+            378,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11337402465ca73498005d0513c9839ccd518058a1e07229c508b1be90d9c994.jpg",
+        "text": "$$\ng ( t ) = f ( 0 ) + \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma ( w _ { i } t + b _ { i } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            415,
+            609,
+            457
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "the width $n$ as well as the weights are bounded by poly $( L , 1 / \\epsilon )$ , and $| f ( t ) - g ( t ) | \\leq \\epsilon ,$ for all $t \\in [ - 1 , 1 ]$ . ",
+        "bbox": [
+            173,
+            462,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Now we are ready to show the separation between depth-2 and depth-3 sigmoidal neural networks. The main idea, similar to Daniely (2017), is to exhibit a function that is Lipschitz but far from any low-degree polynomial. The Lipschitz property helps in showing that our function can be wellapproximated by a depth-3 neural network of small size and small weights. However, being far from any low-degree polynomial, it cannot be approximated by any depth-2 neural network. ",
+        "bbox": [
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+            502,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Theorem 7. Consider the function $G : \\mathbb { B } ^ { d }  \\mathbb { R }$ given by $G ( \\mathbf { x } ) = \\sin ( \\pi d ^ { 5 } \\left\\| \\mathbf { x } \\right\\| ^ { 2 } )$ . Then $G$ can be $\\delta$ -approximated (in $L _ { \\infty }$ -norm) by a depth-3 sigmoidal neural network of width and weights polynomially bounded in $d$ . However, any function $\\bar { F } : \\mathbb { B } ^ { d }  \\mathbb { R }$ computed by a depth-2 sigmoidal neural network with weights $O ( d ^ { 2 } )$ cannot $\\delta$ -approximate $G$ even when its width $n$ is $2 ^ { O ( d ) }$ . ",
+        "bbox": [
+            174,
+            578,
+            823,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By modifying the function to $G ( \\mathbf { x } ) = \\sin ( \\pi N \\left\\| \\mathbf { x } \\right\\| ^ { 2 } )$ , this lower bound with $L _ { \\infty }$ -norm holds for any distribution over $\\mathbf { \\mathbb { B } } ^ { d }$ whose support contains a radial line segment of length at least $1 / p o l y ( d )$ , by making $N = p o l y ( d )$ , for a large enough polynomial. ",
+        "bbox": [
+            174,
+            643,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Remark: Given any distribution $\\mu$ over $\\mathbb { B } ^ { d }$ whose probability density is at least $1 / p o l y ( d )$ on some small ball of radius $1 / p o l y ( d )$ , the lower bound or inapproximability by any depth-2 sigmoidal neural network can be made to work with $L _ { 2 }$ -norm (squared error), for a large enough $N = p o l y ( d )$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proof. First, we will show that $G ( \\mathbf { x } )$ can be well-approximated by a depth-3 sigmoidal neural network of polynomial size and weights. The idea is similar to Daniely’s construction for ReLU networks in Daniely (2017). By Lemma 6, there exists a function $f \\ \\stackrel { \\cdot } { : } \\ [ - 1 , 1 ] \\  \\ \\mathbb { R }$ computed by a depth-2 sigmoidal neural network of size and weights bounded by $\\mathrm { p o l y } ( d , 1 / \\epsilon )$ such that $\\left| t ^ { 2 } - f ( t ) \\right| \\le \\epsilon / 1 0 d ^ { 6 }$ , for all $t \\in [ - 1 , 1 ]$ . Thus, we can compute $x _ { i } ^ { 2 }$ for each coordinate of $\\mathbf { x }$ and add them up to get an $\\epsilon$ -approximation to $\\left\\| \\mathbf { x } \\right\\| ^ { 2 }$ over $\\mathbb { B } ^ { d }$ . That is, there exists a function $S : \\mathbb { B } ^ { d }  \\mathbb { R }$ computed by a depth-2 sigmoidal neural network of size and weights bounded by poly $( d , 1 / \\epsilon )$ such that $\\left| S ( x ) - \\left\\| \\mathbf { x } \\right\\| ^ { 2 } \\right| \\leq \\epsilon / 1 0 d ^ { 5 }$ , for all $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ . Again, by Lemma 6, we can approximate $\\sin ( \\pi d ^ { 3 } t )$ over [0, 1] using $f : [ - 1 , 1 ] \\to \\mathbb { R }$ computed by another depth-2 sigmoidal neural network with size and weights bounded by $\\mathrm { p o l y } ( d , 1 / \\epsilon )$ such that $\\left| \\sin ( \\pi d ^ { 3 } t ) - f ( t ) \\right| \\le \\epsilon / 2$ , for all $t \\in [ 0 , 1 ]$ . Note that the composition of these two depth-2 neural networks $f ( N ( \\mathbf { x } ) )$ gives a depth-3 neural network as the output of the hidden layer of the bottom network can be fed into the top network as inputs. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            103,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb09e3ab7495de08e20ea81f27906b4e76daca3e9c3f06ab427e2e8931a4db19.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\left| G ( \\mathbf { x } ) - f ( S ( \\mathbf { x } ) ) \\right| = \\left| \\sin ( \\pi d ^ { 5 } \\left\\| \\mathbf { x } \\right\\| ^ { 2 } ) - f ( S ( \\mathbf { x } ) ) \\right| } \\\\ & { \\qquad \\leq \\left| \\sin ( \\pi d ^ { 5 } \\left\\| \\mathbf { x } \\right\\| ^ { 2 } ) - f ( \\left\\| \\mathbf { x } \\right\\| ^ { 2 } ) \\right| + \\left| f ( \\left\\| \\mathbf { x } \\right\\| ^ { 2 } ) - f ( S ( \\mathbf { x } ) ) \\right| } \\\\ & { \\qquad \\leq \\epsilon / 2 + 4 d ^ { 5 } \\left| \\left\\| \\mathbf { x } \\right\\| ^ { 2 } - S ( \\mathbf { x } ) \\right| } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            178,
+            125,
+            671,
+            208
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc06c404ad7154a9f9b679aa5a7fcfdb30bd689ca605acd6003d74fc8421e87d.jpg",
+        "text": "$$\n\\leq \\epsilon / 2 + 4 d ^ { 5 } \\cdot \\epsilon / 1 0 d ^ { 5 } \\leq \\epsilon .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            228,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Now we will show the lower bound. Consider any function $F : \\mathbb { B } ^ { d }  \\mathbb { R }$ computed by a depth-2 sigmoidal neural network whose weights are bounded by $B \\ = \\ O ( d ^ { 2 } )$ and width is $n$ . Proposition 4 shows that there exists a $d$ -variate polynomial $P ( \\mathbf { x } )$ of degree $O \\left( B \\log ( n B ^ { 2 } / \\delta ) \\right) ~ =$ $O \\left( d ^ { 2 } \\log ( n / \\delta ) + d ^ { 2 } \\log d \\right)$ in each variable such that $| F ( \\mathbf { x } ) - P ( \\mathbf { x } ) | \\leq \\delta$ , for all $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ . Let $\\mu$ be any measure on $\\mathbb { B } ^ { d }$ whose support contains some radial line segment of length at least $1 / \\mathrm { p o l y } ( d )$ in $\\mathbb { B } ^ { \\bar { d } }$ . In other words, there exists a unit vector $\\mathbf { u }$ such that the support of $\\mu$ intersects the radial set $\\{ \\mathbf { x } ~ \\in ~ \\mathbb { B } ^ { d } \\quad : \\quad \\mathbf { x } ~ = ~ t \\mathbf { u } $ , for some $t \\in [ - 1 , 1 ] \\}$ in some line segment of length at least $1 / \\mathrm { p o l y } ( d )$ . Then $P ( t \\mathbf { u } )$ is a univariate polynomial of degree $O ( d ^ { 3 } \\log ( \\bar { n } / \\delta ) + d ^ { 3 } \\log \\bar { d } )$ that $\\delta$ - approximates $F ( t \\mathbf { u } )$ , for all $t \\in [ - 1 , 1 ]$ . By Lemma 8, using $\\dot { D } = O \\big ( d ^ { 3 } \\log ( n / \\delta ) + d ^ { 3 } \\log d \\big )$ , $l = 1 / \\mathrm { p o l y } ( d )$ and $N = d ^ { 5 } / l$ , we get that if $n = 2 ^ { O ( d ) }$ , then there exists a $t _ { 0 } ~ \\in ~ [ - 1 , 1 ]$ such that $\\left| \\sin ( \\pi N t _ { 0 } ^ { 2 } ) - P ( t \\mathbf { u } ) \\right| \\geq 1$ . Therefore, by triangle inequality, $\\left| \\sin ( \\pi N \\left\\| t _ { 0 } \\mathbf { u } \\right\\| ^ { 2 } ) - F ( t _ { 0 } \\mathbf { u } ) \\right| \\geq$ $\\left| \\sin ( \\pi N t _ { 0 } ^ { 2 } ) - P ( t _ { 0 } { \\mathbf u } ) \\right| - | P ( t _ { 0 } { \\mathbf u } ) - F ( t _ { 0 } { \\mathbf u } ) | \\ge 1 - \\delta > \\delta _ { }$ , for $\\delta < 1 / 2$ . This means that $G ( \\mathbf { x } )$ cannot be well-approximated by any $F ( \\mathbf { x } )$ computed by a depth-2 neural network with polynomially bounded weights even when it has width $2 ^ { O ( d ) }$ . □ ",
+        "bbox": [
+            173,
+            262,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Now we show that the candidate function proposed by Daniely $g ( t ) = \\sin ( \\pi N t )$ , for large enough $N$ , is far from any low-degree polynomial w.r.t. any measure $\\mu$ on $[ - 1 , 1 ]$ with a reasonable support. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 8. Let $p$ be any polynomial of degree $D$ and $\\mu$ be any measure on $[ - 1 , 1 ]$ whose support contains an interval of length at least $l$ . Then, for $N$ large enough to satisfy $N l > D + 3$ , there exists $t _ { 0 } \\in [ - 1 , 1 ]$ such that $\\mu ( t _ { 0 } ) > 0$ and $| \\mathrm { s i n } ( \\pi N t _ { 0 } ) - p ( t _ { 0 } ) | > 1 .$ . In other words, $\\sin ( \\pi N t )$ is 1-far (in $L _ { \\infty }$ -norm) from any polynomial of degree $D$ over interval $[ - 1 , 1 ]$ with measure $\\mu$ . ",
+        "bbox": [
+            174,
+            529,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $\\mu ( t ) > 0$ for some interval $[ a , a + l ] \\subseteq [ - 1 , 1 ]$ . Consider $S = \\{ t \\in [ a , a + l ] \\colon t =$ $- 1 + ( i + 1 / 2 ) / N$ , for some integer $i \\}$ . Then $S$ contains at least $N l - 2$ points where $\\sin ( \\pi N t )$ alternates as $\\pm 1$ . Any polynomial $p$ of degree $D$ cannot match the sign of $\\sin ( \\pi N t )$ on all the points in $S$ . Otherwise, by intermediate value theorem, $p$ must have at least $N l - 3$ roots between the points of $S$ , which means $D \\geq N l - 3$ , a contradiction. Thus, there exists $t _ { 0 } \\in S$ such that $p ( t _ { 0 } )$ and $\\sin ( \\pi N t _ { 0 } )$ have opposite signs. Since $\\sin ( \\pi N t ) = \\pm 1$ , for any $t \\in S$ , the sign mismatch implies $| \\mathrm { s i n } ( \\pi N t _ { 0 } ) - p ( t _ { 0 } ) | > 1$ . □ ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "An important remark on biases: Even though we handled the case of sigmoid neurons without biases, the proof technique carries over to the sigmoid neurons with biases $\\bar { \\boldsymbol { \\sigma } } ( \\left. \\mathbf { w } , \\mathbf { x } \\right. + b )$ . The idea is to consider a new $( d + 1 )$ -dimensional input $\\mathbf { x } _ { \\mathrm { n e w } } = \\left( \\mathbf { x } , x _ { d + 1 } \\right) = \\left( x _ { 1 } , x _ { 2 } , \\ldots , x _ { d + 1 } \\right)$ with $x _ { d + 1 } = 1$ , and consider the new weight vector $\\mathbf { w } _ { \\mathrm { n e w } } = ( \\mathbf { w } , b )$ . Thus, $\\left. \\mathbf { w } _ { \\mathrm { n e w } } , \\mathbf { x } _ { \\mathrm { n e w } } \\right. = \\left. \\mathbf { w } , \\mathbf { x } \\right. + b$ . The new input lies on a $d$ -dimensional hyperplane slice of $\\mathbb { B } ^ { d + 1 }$ , so we need to look at the restriction of the input distribution $\\mu$ to this slice. Most of the ideas in our proofs generalize without any technical modifications. We defer the details to the full version. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 $L _ { 2 }$ -SEPARATION OF DEPTH-2 VS. DEPTH-3 SEPARATION FOR GENERAL INPUT DISTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            832,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section we show lower bounds under the $L _ { 2 }$ -norm. The theorem below gives a technical condition on the class of densities $\\mu$ on $\\mathbb { B } ^ { d }$ for which our lower bound holds. Let’s give an example to illustrate that the condition on density is reasonable: Let $K \\subset \\mathbb { B } ^ { d }$ be a convex set such that every point in $K$ is at least $r$ away from the boundary of $\\mathbb { B } ^ { d }$ (where $r = 1 / \\mathrm { p o l y } ( d )$ is a parameter). Further assume that (1) the probability mass of $K$ is at least a constant and (2) for every point in $K$ the probability density is within a constant factor of the uniform density on $K$ . Then our lower bound applies to $\\mu$ . ",
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+    {
+        "type": "text",
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+        "type": "text",
+        "text": "Theorem 9. Consider the function $G : \\mathbb { B } ^ { d }  \\mathbb { R }$ given by $G ( \\mathbf { x } ) = \\sin ( \\pi N \\left\\| \\mathbf { x } \\right\\| ^ { 2 } )$ . Let $\\mu$ be any probability density over $\\mathbb { B } ^ { \\dot { d } }$ such that there exists a subset $C \\subseteq \\mathbb { B } ^ { d }$ satisfying the following two conditions: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• The $r$ -interior of $C$ defined as $C ^ { \\prime } = \\{ \\mathbf { x } \\in C ~ : ~ \\mathbb { B } ( \\mathbf { x } , r ) \\subseteq C \\}$ contains at least $\\gamma$ fraction of the total probability mass for some $\\gamma > 0$ , i.e., $\\begin{array} { r } { \\int _ { C ^ { \\prime } } \\mu ( \\mathbf { x } ) d \\mathbf { x } \\geq \\gamma } \\end{array}$ . • For any affine line $\\ell$ , the induced probability density on every segment of length at least $r$ in the intersection $\\ell \\cap C$ is $( \\alpha , \\beta )$ -uniform, i.e., it is at least $\\alpha$ times and at most $\\beta$ times the uniform density on that segment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Let $F : \\mathbb { B } ^ { d }  \\mathbb { R }$ be any function computed by a depth-2 sigmoidal neural network with weights bounded by $B$ and width $n$ . Then for any $0 < \\delta \\ll \\alpha \\gamma / 3 \\beta$ and $N \\gg ( B / r ^ { 2 } ) \\log ( n B ^ { 2 } / \\delta )$ , the function $F$ cannot $\\delta$ -approximate $G$ on $\\mathbb { B } ^ { d }$ under $L _ { 2 }$ -norm (squared error) under the probability density $\\mu$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In particular, if $\\alpha , \\beta , \\gamma$ are constants, $B = p o l y ( d )$ , $n = 2 ^ { d }$ , and $r = 1 / p o l y ( d )$ , then it suffices to choose $N = p o l y ( d )$ for a sufficiently large degree polynomial. ",
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+        "type": "text",
+        "text": "Proof. We show a lower bound on $L _ { 2 }$ -error of approximating $G ( \\mathbf { x } )$ with any multivariate polynomial $P : \\mathbb { B } ^ { d }  \\mathbb { R }$ of degree $D$ under the distribution given by $\\mu$ on $\\mathbb { B } ^ { d }$ . For any fixed unit vector $\\mathbf { v }$ , consider $\\mathbf { u } \\in \\mathbb { B } ^ { d - 1 }$ orthogonal to $\\mathbf { v }$ and let $\\ell _ { \\mathbf { u } }$ be the affine line going through $\\mathbf { u }$ and parallel to the direction $\\mathbf { v }$ given by $\\ell _ { \\mathbf { u } } = \\{ \\mathbf { x } = \\mathbf { u } + t \\mathbf { v } ~ : ~ t \\in \\mathbb { R } \\}$ . ",
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+        "type": "equation",
+        "img_path": "images/ad3d8d5d69dbb8813fb87c3d8d1dea223ed8a76a19bb7395cc6c147b12e585c1.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { \\int ( G ( \\mathbf { x } ) - P ( \\mathbf { x } ) ) ^ { 2 } \\mu ( \\mathbf { x } ) d \\mathbf { x } } } \\\\ & { \\ \\stackrel { \\mathrm { B } ^ { d } } { \\geq } } \\\\ & { \\ \\geq \\int ( G ( \\mathbf { x } ) - P ( \\mathbf { x } ) ) ^ { 2 } \\mu ( \\mathbf { x } ) d \\mathbf { x } } \\\\ & { \\ = \\displaystyle \\int \\int \\int ( G ( \\mathbf { u } + t \\mathbf { v } ) - P ( \\mathbf { u } + t \\mathbf { v } ) ) ^ { 2 } \\mu ( \\mathbf { u } + t \\mathbf { v } ) \\mathbb { I } _ { C } ( \\mathbf { u } , t ) d t d \\mathbf { u } } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "equation",
+        "img_path": "images/e11033117d375e74a7653766bc08f28c42b1a44afff55d63b3ca04418e14aaaf.jpg",
+        "text": "$$\n\\geq \\int _ { \\mathbb { R } ^ { d - 1 } } \\int _ { \\mathbb { R } } \\left( G ( \\mathbf { u } + t \\mathbf { v } ) - P ( \\mathbf { u } + t \\mathbf { v } ) \\right) ^ { 2 } \\mu ( \\mathbf { u } + t \\mathbf { v } ) \\mathbb { I } _ { \\tilde { C } } ( \\mathbf { u } , t ) d t d \\mathbf { u }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "equation",
+        "img_path": "images/070758b48eb3af8b10768a56a811e14f394e9b39622955a7ced3f888ee230e27.jpg",
+        "text": "$$\n\\begin{array} { r l } & { = \\frac { \\int _ { \\mathbb { R } ^ { d - 1 } } \\int _ { \\mathbb { R } } \\big ( G ( \\mathbf { u } + t \\mathbf { v } ) - P ( \\mathbf { u } + t \\mathbf { v } ) \\big ) ^ { 2 } \\mu ( \\mathbf { u } + t \\mathbf { v } ) \\mathbb { I } _ { \\boldsymbol { \\bar { C } } } ( \\mathbf { u } , t ) \\ d t d \\mathbf { u } } { \\int _ { \\mathbb { R } ^ { d - 1 } } \\int _ { \\mathbb { R } } \\mu ( \\mathbf { u } + t \\mathbf { v } ) \\mathbb { I } _ { \\boldsymbol { \\bar { C } } } ( \\mathbf { u } , t ) \\ d t d \\mathbf { u } } \\cdot \\displaystyle \\int _ { \\mathbb { R } ^ { d - 1 } } \\int _ { \\mathbb { R } } \\mu ( \\mathbf { u } + t \\mathbf { v } ) \\mathbb { I } _ { \\boldsymbol { \\bar { C } } } ( \\mathbf { u } , t ) \\ d t d \\mathbf { u } } \\\\ & { = \\frac { \\int _ { \\mathbb { R } ^ { d - 1 } } \\int _ { \\mathbb { R } } \\big ( G \\big ( \\mathbf { u } + t \\mathbf { v } \\big ) - P \\big ( \\mathbf { u } + t \\mathbf { v } \\big ) \\big ) ^ { 2 } \\mu ( \\mathbf { u } + t \\mathbf { v } ) \\mathbb { I } _ { \\boldsymbol { \\bar { C } } } ( \\mathbf { u } , t ) \\ d t d \\mathbf { u } } { \\int _ { \\mathbb { R } ^ { d - 1 } } \\int _ { \\mathbb { R } } \\mu ( \\mathbf { u } + t \\mathbf { v } ) \\int _ { \\mathbb { \\bar { C } } } ( \\mathbf { u } , t ) \\ d t d \\mathbf { u } } \\cdot \\displaystyle \\int _ { \\boldsymbol { C } ^ { \\prime } } \\mu ( \\mathbf { x } ) d \\mathbf { x } } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "equation",
+        "img_path": "images/b5550bdfb5acac3fe1b13f7cb8ad300c36438893c75aaf9805e550791eb90d01.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\geq \\operatorname* { m i n } _ { { \\bf u } \\in { \\mathbb { R } } ^ { d - 1 } } \\frac { \\int _ { { \\mathbb { R } } } \\left( { \\cal G } ( { \\bf u } + t { \\bf v } ) - P ( { \\bf u } + t { \\bf v } ) \\right) ^ { 2 } \\mu ( { \\bf u } + t { \\bf v } ) \\mathbb { I } _ { { \\widetilde { \\cal C } } } ( { \\bf u } , t ) d t } { \\int _ { { \\mathbb { R } } } \\mu ( { \\bf u } + t { \\bf v } ) \\mathbb { I } _ { { \\widetilde { \\cal C } } } ( { \\bf u } , t ) d t } \\cdot \\displaystyle \\int _ { { \\cal C } ^ { \\prime } } \\mu ( { \\bf x } ) d { \\bf x } } \\\\ & { \\displaystyle \\geq \\operatorname* { m i n } _ { { \\bf u } \\in { \\mathbb { R } } ^ { d - 1 } } \\frac { \\alpha } { \\beta } \\cdot \\frac { \\int _ { { \\mathbb { R } } } \\left( { \\cal G } ( { \\bf u } + t { \\bf v } ) - P ( { \\bf u } + t { \\bf v } ) \\right) ^ { 2 } \\mathbb { I } _ { { \\widetilde { \\cal C } } } ( { \\bf u } , t ) d t } { \\int _ { { \\mathbb { R } } } \\mathbb { I } _ { { \\widetilde { \\cal C } } } ( { \\bf u } , t ) d t } \\cdot \\displaystyle \\int _ { { \\cal C } ^ { \\prime } } \\mu ( { \\bf x } ) d { \\bf x } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "because for any line $\\ell$ , the distribution induced by $\\mu ( \\mathbf { x } )$ along any line segment of length at least $r$ in the intersection $\\ell \\cap C$ is $( \\alpha , \\beta )$ -uniform, for any line $\\ell$ $\\geq { \\frac { \\alpha } { \\beta } } \\cdot { \\frac { \\gamma } { 3 } } .$ ",
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+    },
+    {
+        "type": "text",
+        "text": "The last inequality is using the condition $\\begin{array} { r } { \\int _ { C ^ { \\prime } } \\mu ( \\mathbf { x } ) d \\mathbf { x } \\geq \\gamma } \\end{array}$ given in Theorem 9 and an adaptation of the following idea from Lemma 5 of Daniely (2017). For any fixed $\\mathbf { u }$ and $\\mathbf { v }$ , $G ( \\mathbf { u } + t \\mathbf { v } ) =$ $\\sin ( \\pi N ( \\left. \\lvert \\mathbf { u } \\right. \\rvert ^ { 2 } + t ^ { 2 } ) )$ and $P ( \\mathbf { u } + t \\mathbf { v } )$ is a polynomial of degree at most $D$ in $t$ . The function $\\sin ( \\pi N ( \\left. \\lvert \\mathbf { u } \\right. \\rvert ^ { 2 } + t ^ { 2 } ) )$ alternates its sign as $\\left\\| \\mathbf { u } \\right\\| ^ { 2 } + t ^ { 2 }$ takes values that are successive integer multiples of $1 / N$ . Consider $\\acute { s } = t ^ { 2 } \\in [ 0 , 1 ]$ and divide $[ 0 , 1 ]$ into $N$ disjoint segments using integer grid of step size $1 / N$ . For any polynomial $p ( s )$ of degree at most $D$ and any interval $I \\subseteq [ 0 , 1 ]$ of length $r \\gg D / N$ , there exists at least $N r - D - 2$ segments of length $1 / N$ each on which $\\sin ( \\pi N s )$ and $p ( s )$ do not change signs and have opposite signs. Now using $( \\sin ( \\pi N s ) - p ( s ) ) ^ { 2 } \\ge \\sin ^ { 2 } ( \\pi N s ) \\nonumber$ , integrating we get that $\\begin{array} { r } { \\int _ { I } ( \\sin ( \\pi N s ) - p ( s ) ) ^ { 2 } \\bar { d s } \\ge r / 2 } \\end{array}$ . Extending this proof to $t$ instead of $s = t ^ { 2 }$ , using $\\sin ^ { 2 } ( \\pi N t ^ { 2 } ) t \\leq \\sin ^ { 2 } ( \\pi N t )$ for all $t \\in [ 0 , 1 ]$ , and incorporating the shift $\\pi N \\left. \\mathbf { u } \\right. ^ { 2 }$ , we can similarly show that $\\begin{array} { r } { \\int _ { I } \\sin ^ { 2 } ( \\pi N ( \\left\\| \\mathbf { u } \\right\\| ^ { 2 } + t ^ { 2 } ) ) - P ( \\mathbf { u } + t \\mathbf { v } ) ) ^ { 2 } d t \\ge r / 3 } \\end{array}$ . Summing up over multiple such intervals gives the final inequality. □ ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The $L _ { 2 }$ separation between depth-2 and depth-3 neural networks under probability density $\\mu$ now follows by taking a small enough $\\delta$ , and combining the following ingredients (i) Proposition 4 says that any depth-2 sigmoid neural networks of width $n = 2 ^ { d }$ and weights bounded by $B = { \\mathfrak { p o l y } } ( d )$ can be $\\delta$ -approximated in $L _ { \\infty }$ (and hence, also $L _ { 2 }$ ) by a multivariate polynomials of degree $D = O ( B \\mathbf { \\bar { l o g } } ( n B ^ { 2 } / \\delta ) ) = \\mathrm { p o l y } ( d )$ , (ii) proof of Theorem 7 (initial part) says that $G ( \\mathbf { x } )$ can be $\\delta$ -approximated in $L _ { \\infty }$ (and hence, also $L _ { 2 }$ ) by a depth-3 sigmoid neural network of width and size $\\mathfrak { p o l y } ( d )$ , but (iii) Theorem 9 says that, for $N = { \\mathrm { p o l y } } ( d )$ of large enough degree, $G ( \\mathbf { x } )$ cannot be $3 \\delta$ -approximated in $L _ { 2 }$ by any multivariate polynomial of degree $D$ , and (iv) triangle inequality. ",
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+    {
+        "type": "text",
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+        "text": "A PROOFS OF POLYNOMIAL APPROXIMATIONS TO NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof of Proposition 2 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            133,
+            330,
+            148
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof. Consider the degree- $D$ approximation to $\\sigma ( w t )$ given by the first $D$ terms in its Chebyshev expansion. The error of this approximation for any $t \\in [ - 1 , 1 ]$ is bounded by ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ed2d0844e1613dc0991c7fc3e34b73021bd295c098bd05d2e45580e75287c654.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { ( \\sigma ( w t ) - p ( \\ell ) | = | \\sum _ { j \\geq \\ell } \\varsigma _ { j } T _ { ( j ) } ( \\ell ) | } } \\\\ & { \\leq \\sum _ { j \\geq 0 } | \\varsigma _ { j } | } \\\\ & { \\leq ( \\frac { 4 } { | w | } + \\frac { 2 } { \\pi } ) \\sum _ { j > \\ell } ( 1 + \\frac { \\pi } { | w | } ) ^ { - \\beta } } \\\\ & { \\leq ( \\frac { 4 } { | w | } + \\frac { 2 } { \\pi } ) ( 1 + \\frac { \\pi } { | w | } ) ^ { - ( \\beta + 1 ) } \\sum _ { j = 0 } ^ { \\infty } ( 1 + \\frac { \\pi } { | w | } ) ^ { - \\beta } } \\\\ & { = ( \\frac { 4 } { | w | } + \\frac { 2 } { \\pi } ) ( 1 + \\frac { \\pi } { | w | } ) ^ { - ( \\beta + 1 ) } \\cdot \\frac { | w | } { \\frac { \\sqrt { w } } { 2 \\pi } ( 1 + \\frac { \\pi } { | w | } ) ^ { - \\beta } } } \\\\ & { = ( \\frac { 4 } { | w | } + \\frac { 2 } { \\pi } ) ( 1 + \\frac { \\pi } { | w | } ) ^ { - ( \\beta + 1 ) } \\cdot \\frac { | w | } { \\pi } ( 1 + \\frac { \\pi } { | w | } ) } \\\\ & { \\leq \\epsilon _ { * } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            200,
+            727,
+            433
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "using Proposition 1, $| w | \\le B$ , and $D = O \\left( B \\log \\left( B / \\epsilon \\right) \\right)$ . ",
+        "bbox": [
+            173,
+            436,
+            550,
+            454
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof of Proposition 3 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            468,
+            330,
+            483
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $f$ be computed by a depth-2 sigmoidal neural network given by $\\begin{array} { r } { f ( t ) = \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma ( w _ { i } t ) } \\end{array}$ Define a parameter $\\epsilon \\ : = \\ : \\delta / n B$ . Proposition 2 guarantees polynomial $p _ { 1 } , p _ { 2 } , \\ldots , p _ { n }$ of degree $O \\left( B \\log ( \\mathbf { \\bar { \\boldsymbol { B } } } / \\epsilon ) \\right)$ such that $| \\sigma ( w _ { i } t ) - \\bar { p } _ { i } ( t ) | \\ \\leq \\ \\epsilon$ , for all $t ~ \\in ~ [ - 1 , 1 ]$ . Thus, the polynomial $\\begin{array} { r } { p ( t ) = \\sum _ { i = 1 } ^ { n } a _ { i } p _ { i } ( t ) } \\end{array}$ has degree $O \\left( B \\log ( B / \\epsilon ) \\right) = O \\left( B \\log ( n B ^ { 2 } / \\delta ) \\right)$ , and for any $t \\in [ - 1 , 1 ]$ , ",
+        "bbox": [
+            173,
+            498,
+            825,
+            558
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/155f71629bf3d7473b4d41c935fd63ced36bca3e9071ab84652975893d7ae0c8.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\displaystyle \\left. \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma ( w _ { i } t ) - p ( t ) \\right. = \\displaystyle \\left. \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma ( w _ { i } t ) - \\sum _ { i = 1 } ^ { n } a _ { i } p _ { i } ( t ) \\right. } & { } \\\\ { \\displaystyle \\leq \\sum _ { i = 1 } ^ { n } \\lvert a _ { i } \\rvert \\lvert \\sigma ( w _ { i } t ) - p _ { i } ( t ) \\rvert } & { } \\\\ { \\displaystyle \\leq n B \\epsilon } & { } \\\\ { \\displaystyle = \\delta } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            232,
+            565,
+            593,
+            685
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof of Proposition 4 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            722,
+            330,
+            737
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $F$ be computed by a depth-2 neural network given by $\\begin{array} { r } { F ( \\mathbf { x } ) \\ = \\ \\sum _ { i = 1 } ^ { n } { a _ { i } \\sigma } ( \\left. \\mathbf { w } _ { i } , \\mathbf { x } \\right. ) } \\end{array}$ where $| a _ { i } | \\le B$ and $\\left\\| \\mathbf { w } _ { i } \\right\\| \\leq B$ , for $1 \\leq i \\leq n$ . Thus, $\\begin{array} { r } { F ( \\mathbf { x } ) \\ = \\ \\sum _ { i = 1 } ^ { n } { a _ { i } \\sigma ( \\left\\| \\mathbf { w } _ { i } \\right\\| t _ { i } ) } } \\end{array}$ , where $t _ { i } = \\left. \\mathbf { w } _ { i } / \\left\\| \\mathbf { w } _ { i } \\right\\| , \\mathbf { x } \\right. \\in [ - 1 , 1 ]$ because $\\mathbf { w } _ { i } / \\left\\| \\mathbf { w } _ { i } \\right\\| \\in \\mathbb { B } ^ { d }$ , for $1 \\leq i \\leq n$ , and $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ . ",
+        "bbox": [
+            173,
+            752,
+            825,
+            800
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Define a parameter $\\epsilon \\ : = \\ : \\delta / n B$ . Proposition 2 guarantees polynomial $p _ { 1 } , p _ { 2 } , \\ldots , p _ { n }$ of degree $O \\left( B \\log ( \\mathbf { \\bar { \\boldsymbol { B } } } / \\epsilon ) \\right)$ such that $| \\dot { \\sigma } ( \\| w _ { i } \\| t ) - p _ { i } ( t ) | \\ \\leq \\ \\dot { \\epsilon }$ , for all $t \\in [ - 1 , 1 ]$ . Consider the following polynomial $\\begin{array} { r } { P ( \\mathbf { x } ) = P ( x _ { 1 } , x _ { 2 } , \\ldots , x _ { d } ) = \\sum _ { i = 1 } ^ { n } a _ { i } p _ { i } ( \\langle \\mathbf { w } _ { i } / \\left\\| \\mathbf { w } _ { i } \\right\\| , \\mathbf { x } \\rangle ) } \\end{array}$ . $P ( \\mathbf { x } )$ is a $d$ -variate polynomial of degree $O \\left( B \\log ( B / \\epsilon ) \\right) = O \\left( B \\log ( n B ^ { 2 } / \\delta ) \\right)$ in each variable $x _ { 1 } , x _ { 2 } , \\ldots , x _ { d }$ . For any $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ , ",
+        "bbox": [
+            173,
+            804,
+            825,
+            880
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3079f0ed759454a3e11d2ec87f83be79632f44323b869efe554b29d17f2b5cba.jpg",
+        "text": "$$\n| F ( \\mathbf { x } ) - P ( \\mathbf { x } ) | = \\left| \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma ( \\langle \\mathbf { w } _ { i } , \\mathbf { x } \\rangle ) - \\sum _ { i = 1 } ^ { n } a _ { i } p _ { i } ( \\langle \\mathbf { w } _ { i } / \\| \\mathbf { w } _ { i } \\| , \\mathbf { x } \\rangle ) \\right|\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            886,
+            614,
+            929
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6da893eae7e2f4d1355d06c149f706808729f93e8be69a122e8d6fa3b90b596c.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\leq \\displaystyle \\sum _ { i = 1 } ^ { n }  a _ { i }   \\sigma (  \\mathbf { w } _ { i } , \\mathbf { x }  ) - p _ { i } (  \\mathbf { w } _ { i } /  \\mathbf { w } _ { i }  , \\mathbf { x } ) )  } \\\\ & { \\leq B \\displaystyle \\sum _ { i = 1 } ^ { n }  \\sigma (  \\mathbf { w } _ { i }  t _ { i } ) - p _ { i } ( t _ { i } )  \\qquad \\mathrm { u s i n g ~ } t _ { i } =  \\mathbf { w } _ { i } /  \\mathbf { w } _ { i }  , \\mathbf { x }  ; } \\\\ & { \\leq \\epsilon n B \\qquad \\mathrm { u s i n g ~ }  \\sigma (  w _ { i }  t ) - p _ { i } ( t )  \\leq \\epsilon , \\mathrm { f o r ~ a l l ~ } t \\in [ - 1 , 1 ] } \\\\ & { = \\delta . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            99,
+            730,
+            218
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof of Proposition 5 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            252,
+            330,
+            268
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. We prove this by induction on the depth $k$ . By induction hypothesis each $F _ { j } ( \\mathbf { x } )$ can be $\\epsilon _ { 1 }$ -approximated (in $L _ { \\infty }$ -norm) by a $d$ -variate polynomial $Q _ { j } ( \\mathbf { x } )$ of degree $O \\left( ( n B ) ^ { k - 2 } \\log ^ { ( k - 2 ) } ( n B / \\epsilon _ { 1 } ) \\right)$ in each variable. Thus, $| F _ { j } ( \\mathbf { x } ) - Q _ { j } ( \\mathbf { x } ) | = \\epsilon _ { 1 }$ , for any $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ and $1 \\leq j \\leq m$ . Because a sigmoid neuron is Lipschitz, ",
+        "bbox": [
+            173,
+            281,
+            825,
+            349
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e2567e1eb3564aed0171f0a25a07c0b1d93d9c4bba7476b54e36657761586a9.jpg",
+        "text": "$$\n| y _ { j } - \\sigma ( Q _ { j } ( \\mathbf x ) ) | = | \\sigma ( F _ { j } ( \\mathbf x ) ) - \\sigma ( Q _ { j } ( \\mathbf x ) ) | \\le | F _ { j } ( \\mathbf x ) - Q _ { j } ( \\mathbf x ) | \\le \\epsilon _ { 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            354,
+            730,
+            372
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "for any $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ and $1 \\leq j \\leq m$ . ",
+        "bbox": [
+            173,
+            377,
+            385,
+            395
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Since $F _ { j } ( \\mathbf { x } )$ is the output of a depth- $\\left( k - 2 \\right)$ sigmoidal neural network of width at most $n$ and weights at most $B$ , we must have $| F _ { j } ( \\mathbf { x } ) | \\leq n B$ , for all $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ . Thus, $| Q _ { j } ( \\mathbf { x } ) | \\leq n B + \\epsilon _ { 1 } \\leq 2 n B$ . By Proposition 2, there exists a polynomial $q ( t )$ of degree at most $O \\left( n B \\log ( n B / \\epsilon _ { 2 } ) \\right)$ such that ",
+        "bbox": [
+            173,
+            400,
+            825,
+            445
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56ee523f0ff544af8a162e17e1e23967e355a0c126d5927cc679a8efd5aebc09.jpg",
+        "text": "$$\n| \\sigma ( Q _ { j } ( \\mathbf x ) ) - q ( Q _ { j } ( \\mathbf x ) ) | \\le \\epsilon _ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            398,
+            449,
+            598,
+            468
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "for all $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ and $1 \\leq j \\leq m$ . ",
+        "bbox": [
+            173,
+            474,
+            379,
+            489
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Consider $\\mathbf { q } \\in \\mathbb { R } ^ { m }$ as $\\mathbf { q } = ( q ( Q _ { 1 } ( \\mathbf { x } ) ) , q ( Q _ { 2 } ( \\mathbf { x } ) ) , \\dots , q ( Q _ { m } ( \\mathbf { x } ) ) )$ . Then, for any $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ , we have ",
+        "bbox": [
+            171,
+            494,
+            808,
+            512
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8816108c847aa3ee5768f000b1e46169d15ee3e4ac71b42e8c7a1d902598f189.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\left| \\left. \\mathbf { w } _ { i } , \\mathbf { y } \\right. - \\left. \\mathbf { w } _ { i } , \\mathbf { q } \\right. \\right| = \\left| \\left. \\mathbf { w } _ { i } , \\mathbf { y } - \\mathbf { q } \\right. \\right| } \\\\ & { \\qquad \\leq \\left\\| \\mathbf { w } _ { i } \\right\\| \\left\\| \\mathbf { y } - \\mathbf { q } \\right\\| } \\\\ & { \\qquad \\leq B \\left( \\displaystyle \\sum _ { j = 1 } ^ { m } ( y _ { j } - q ( Q _ { j } ( \\mathbf { x } ) ) ) ^ { 2 } \\right) ^ { 1 / 2 } } \\\\ & { \\qquad \\leq B \\sqrt { m } \\left( \\epsilon _ { 1 } + \\epsilon _ { 2 } \\right) } \\\\ & { \\qquad \\leq B \\sqrt { n } \\left( \\epsilon _ { 1 } + \\epsilon _ { 2 } \\right) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            518,
+            679,
+            647
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Again by Proposition 2, there is a polynomial $p$ of degree at most $O \\left( n B \\log ( n B / \\epsilon ) \\right)$ such that $| \\sigma ( \\langle \\mathbf { w } _ { i } , \\mathbf { q } \\rangle ) - p ( \\langle \\mathbf { w } _ { i } , \\mathbf { q } \\rangle ) | \\leq \\epsilon$ , for all $\\dot { \\mathbf { x } } \\in \\mathbb { B } ^ { d }$ and $1 \\leq i \\leq n$ . This is because $| \\langle \\mathbf { w } _ { i } , \\mathbf { q } \\rangle | = O ( n B )$ . ",
+        "bbox": [
+            173,
+            657,
+            823,
+            688
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Let’s define $\\begin{array} { r } { P ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { n } a _ { i } p ( \\langle \\mathbf { w } _ { i } , \\mathbf { q } \\rangle ) } \\end{array}$ . Therefore, for any $\\mathbf { x } \\in \\mathbb { B } ^ { d }$ , ",
+        "bbox": [
+            173,
+            693,
+            625,
+            710
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ff52643846bca2c0e4edf96511ae3bb06e3e4cc456175440cba192250768cc0b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle | F ( \\mathbf x ) - P ( \\mathbf x ) | = \\left| \\displaystyle \\sum _ { i = 1 } ^ { n } a _ { i } \\sigma \\big ( \\langle \\mathbf w _ { i } , \\mathbf y \\rangle \\big ) - \\displaystyle \\sum _ { i = 1 } ^ { n } a _ { i } p \\big ( \\langle \\mathbf w _ { i } , \\mathbf q \\rangle \\big ) \\right| } \\\\ { \\displaystyle \\leq \\displaystyle \\sum _ { i = 1 } ^ { n } \\big | a _ { i } \\big | | \\sigma \\big ( \\langle \\mathbf w _ { i } , \\mathbf y \\rangle \\big ) - p \\big ( \\langle \\mathbf w _ { i } , \\mathbf q \\rangle \\big ) | } \\\\ { \\displaystyle \\leq \\displaystyle \\sum _ { i = 1 } ^ { n } \\big | a _ { i } \\big | \\big ( | \\sigma \\big ( \\langle \\mathbf w _ { i } , \\mathbf y \\rangle \\big ) - \\sigma \\big ( \\langle \\mathbf w _ { i } , \\mathbf q \\rangle \\big ) \\big | + | \\sigma \\big ( \\langle \\mathbf w _ { i } , \\mathbf q \\rangle \\big ) - p \\big ( \\langle \\mathbf w _ { i } , \\mathbf q \\rangle \\big ) \\big | \\big ) } \\\\ { \\displaystyle \\leq \\displaystyle \\sum _ { i = 1 } ^ { n } a _ { i } \\big | \\big ( | \\langle \\mathbf w _ { i } , \\mathbf y \\rangle - \\langle \\mathbf w _ { i } , \\mathbf q \\rangle | + \\epsilon \\big ) } \\\\ { \\displaystyle \\leq n B \\big ( B \\sqrt { n } ( \\epsilon _ { 1 } + \\epsilon _ { 2 } ) + \\epsilon \\big ) } \\\\ { \\displaystyle \\leq \\delta , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            222,
+            715,
+            776,
+            922
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "if we use $\\epsilon _ { 1 } = \\epsilon _ { 2 } = \\delta / 3 n ^ { 3 / 2 } B ^ { 2 }$ and $\\epsilon = \\delta / 3 n B$ . ",
+        "bbox": [
+            173,
+            102,
+            496,
+            119
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "$P ( \\mathbf { x } )$ is a $d$ -variate polynomial of degree ",
+        "bbox": [
+            176,
+            125,
+            442,
+            140
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2e7db8d8eb2745762419b01c091412a3b13a17023ce59275535add681bc0561f.jpg",
+        "text": "$$\n\\deg ( P ) \\leq \\deg ( p ) \\deg ( q ) \\cdot \\deg ( Q _ { j } ) = O \\left( ( n B ) ^ { k } \\log ^ { k } \\left( n B / \\delta \\right) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            284,
+            147,
+            709,
+            174
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "in each variable. ",
+        "bbox": [
+            173,
+            179,
+            282,
+            194
+        ],
+        "page_idx": 11
+    }
+]

parse/train/SkxbDsR9Ym/SkxbDsR9Ym.md

@@ -0,0 +1,531 @@
+# RELWALK – A LATENT VARIABLE MODEL APPROACH TO KNOWLEDGE GRAPH EMBEDDING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Knowledge Graph Embedding (KGE) is the task of jointly learning entity and relation embeddings for a given knowledge graph. Existing methods for learning KGEs can be seen as a two-stage process where (a) entities and relations in the knowledge graph are represented using some linear algebraic structures (embeddings), and (b) a scoring function is defined that evaluates the strength of a relation that holds between two entities using the corresponding relation and entity embeddings. Unfortunately, prior proposals for the scoring functions in the first step have been heuristically motivated, and it is unclear as to how the scoring functions in KGEs relate to the generation process of the underlying knowledge graph. To address this issue, we propose a generative account of the KGE learning task. Specifically, given a knowledge graph represented by a set of relational triples $( h , R , t )$ , where the semantic relation $R$ holds between the two entities $h$ (head) and $t$ (tail), we extend the random walk model (Arora et al., 2016a) of word embeddings to KGE. We derive a theoretical relationship between the joint probability $p ( h , R , t )$ and the embeddings of $h$ , $R$ and $t$ . Moreover, we show that marginal loss minimisation, a popular objective used by much prior work in KGE, follows naturally from the log-likelihood ratio maximisation under the probabilities estimated from the KGEs according to our theoretical relationship. We propose a learning objective motivated by the theoretical analysis to learn KGEs from a given knowledge graph. The KGEs learnt by our proposed method obtain state-of-the-art performance on FB15K237 and WN18RR benchmark datasets, providing empirical evidence in support of the theory.
+
+# 1 INTRODUCTION
+
+Knowledge graphs such as Freebase (Bollacker et al., 2008) organise information in the form of graphs, where entities are represented by vertices in the graph and the relation between two entities is represented by the edge that connects the corresponding two vertices. By embedding entities and relations that exist in a knowledge graph in some (possibly lower-dimensional and latent) space we can infer previously unseen relations between entities, thereby expanding a given knowledge graph (Nickel et al., 2016; Yang et al., 2015; Lin et al., 2015; Nickel et al., 2011; Trouillon et al., 2016; Wang et al., 2017; Bordes et al., 2011).
+
+Existing KGE methods can be seen as involving two main steps. First, given a knowledge graph represented by a set of relational triples $( h , R , t )$ , where a semantic relation $R$ holds between a head entity $h$ and a tail entity $t$ , entities and relations are represented using some mathematical structures such as vectors, matrices or tensors. Second, a scoring function is proposed that evaluates the relational strength of a triple $( h , R , t )$ and entity and relation embeddings that optimise the defined scoring function are learnt using some optimisation method. Table 1 shows some of the scoring functions proposed in prior work in KGE learning.
+
+Despite the wide applications of entity and relation embeddings created via KGE methods, the existing scoring functions are motivated heuristically to capture some geometric requirements of the embedding space. For example, TransE (Bordes et al., 2011) assumes that the entity and relation embeddings co-exist in the same (possibly lower dimensional) vector space and translating (shifting) the head entity embedding by the relation embedding must make it closer to the tail entity embedding, whereas ComplEx (Trouillon et al., 2016) models the asymmetry in relations using the component-wise multi-linear inner-product among entity and relation embeddings. Relational triples extracted from a given knowledge graph are used as positive training instances, whereas pseudo-negative (Bordes et al., 2011) instances are automatically generated by randomly corrupting positive instances. Finally, KGE are learnt such that the prediction loss computed over the positive and negative instances is minimised.
+
+Table 1: Score functions proposed in selected prior work on KGE. Entity embeddings $h , t \in \mathbb { R } ^ { d }$ are vectors in all models, except in ComplEx where $\pmb { h } , \pmb { t } \in \mathbb { C } ^ { d }$ . Here, $\pmb { x } _ { \ell _ { 1 / 2 } }$ denotes either $\ell _ { 1 }$ or $\ell _ { 2 }$ norm of the vector $_ { \textbf { \em x } }$ . In ComplEx, $\bar { \mathbf { x } }$ is the elementwise complex conjugate, and $\langle \cdot , \cdot , \cdot \rangle$ denotes the component-wise multi-linear inner-product.   
+
+<table><tr><td>Model</td><td>Score function f(h, R,t)</td><td>Relation parameters</td></tr><tr><td>Unstructured (Bordes et al., 2011)</td><td>/h-tlle1/2</td><td>none</td></tr><tr><td>Structured embeddings (Bordes et al., 2011)</td><td>|R1h-R2tlle1,2</td><td>R1,R2 ∈Rd×d</td></tr><tr><td>TransE (Bordes et al., 2011)</td><td>|/h +R-tlle1/2</td><td>ReRd</td></tr><tr><td>DistMult (Yang et al., 2015)</td><td>(h,R,t)</td><td>ReRd</td></tr><tr><td>RESCAL (Nickel et al., 2011)</td><td>hRt</td><td>Rdxd</td></tr><tr><td>ComplEx (Trouillon et al., 2016)</td><td>(h,R,t)</td><td>Re Cd</td></tr></table>
+
+Despite the good empirical performances of the existing KGE methods, theoretical understanding of KGE methods is comparatively under developed. For example, it is not clear how the heuristically defined KGE objectives relate to the generative process of a knowledge graph. In this paper, we attempt to fill this void by providing a theoretical analysis of KGE. Specifically, in section 2, we propose a generative process where we explain the formation of a relation $R$ between two entities $h$ and $t$ using the corresponding relation and entity embeddings. Following this generative story, we derive a relationship between the probability of $R$ holding between $h$ and $t$ , ${ \bar { p } } ( h , t \mid R )$ , and the embeddings of $R$ , $h$ and $t$ . Interestingly, the derived relationship is not covered by any of the previously proposed heuristically-motivated scoring functions, providing the first-ever KGE method with a provable generative explanation.
+
+Next, in section 3, we show that the margin loss, which has been popularly used as a training objective in prior work on KGE, naturally arises as the log-likelihood ratio computed from $p ( h , t \mid { \bar { R } } )$ . Based on this result, we derive a training objective that we subsequently optimise for learning KGEs that satisfy our theoretical relationship. Using standard benchmark datasets proposed in prior work on KGE learning, we evaluate the learnt KGEs on a link prediction task and a triple classification task. Experimental results show that the learnt KGEs obtain state-of-the-art performance on FB15K237 and WN18RR benchmarks, thereby providing empirical evidence to support the theoretical analysis.
+
+# 2 RELATIONAL WALK
+
+Let us consider a knowledge graph $\mathcal { D }$ where the knowledge is represented by relational triples $( h , R , t ) \in \mathcal { D }$ . Here, $R$ is a relational predicate of two arguments, where $h$ (head) and $t$ (tail) entities respectively filling the first and second arguments. We assume relations to be asymmetric in general. In other words, if $( h , R , t ) \in \mathcal { D }$ then it does not necessarily follow that $( t , \dot { R _ { \ l } } h ) \in \mathcal { D }$ . The goal of KGE is to learn embeddings (representations) for the relations and entities in the knowledge graph such that the entities that participate in similar relations are embedded closely to each other in the entity embedding space, while at the same time relations that hold between similar entities are embedded closely to each other in the relational embedding space. We call the learnt entity and relation embeddings collectively as KGEs. Following prior work on KGE (Bordes et al., 2011; Trouillon et al., 2016; Yang et al., 2015), we assume that entities and relations are embedded in the same vector space, allowing us to perform linear algebraic operations using the embeddings in the same vector space.
+
+Let us consider a random walk characterised by a time-dependent knowledge vector $c _ { k }$ , where $k$ is the current time step. The knowledge vector represents the knowledge we have about a particular group of entities and relations that express some facts about the world. For example, the knowledge that we have about people that are employed by companies can be expressed using entities of classes such as people and organisation, using relations such as CEO-of, employed-at, works-for, etc. We assume that entities $h$ and $t$ are represented by time-independent $d$ -dimensional vectors, respectively $h , t \in \mathbb { R } ^ { d }$ .
+
+We assume the task of generating a relational triple $( h , R , t )$ in a given knowledge graph to be a two-step process as described next. First, given the current knowledge vector at time $k$ , $\boldsymbol { c } = \boldsymbol { c } _ { k }$ and the relation $R$ , we assume that the probability of an entity $h$ satisfying the first argument of $R$ to be given by (1).
+
+$$
+p ( h \mid R , c ) = { \frac { 1 } { Z _ { c } } } \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) .
+$$
+
+Here, $\mathbf { R } _ { 1 } \in \mathbb { R } ^ { d \times d }$ is a relation-specific orthogonal matrix that evaluates the appropriateness of $h$ for the first argument of $R$ . For example, if $R$ is the CEO-of relation, we would require a person as the first argument and a company as the second argument of $R$ . However, note that the role of $\mathbf { R } _ { 1 }$ extends beyond simply checking the types of the entities that can fill the first argument of a relation. For our example above, not all people are CEOs and $\mathbf { R } _ { 1 }$ evaluates the likelihood of a person to be selected as the first argument of the CEO-of relation. $Z _ { c }$ is a normalisation coefficient such that $\begin{array} { r } { \sum _ { h \in \mathcal { V } } p ( h \mid R , \pmb { c } ) = \mathrm { i } } \end{array}$ , where the vocabulary $\nu$ is the set of all entities in the knowledge graph.1
+
+After generating $h$ , the state of our random walker changes to $\boldsymbol { c } ^ { \prime } = \boldsymbol { c } _ { k + 1 }$ , and we next generate the second argument of $R$ with the probability given by (2).
+
+$$
+p ( t \mid R , \pmb { c } ^ { \prime } ) = \frac { 1 } { Z _ { c ^ { \prime } } } \exp \left( \pmb { t } ^ { \top } \mathbf { R } _ { 2 } \pmb { c } ^ { \prime } \right) .
+$$
+
+Here, $\mathbf { R } _ { 2 } \in \mathbb { R } ^ { d \times d }$ is a relation-specific orthogonal matrix that evaluates the appropriateness of $t$ as the second argument of $R$ . $Z _ { c ^ { \prime } }$ is a normalisation coefficient such that $\begin{array} { r } { \sum _ { t \in \mathcal { V } } p ( \bar { t } \mid \bar { R } , { \pmb { c } } ) = 1 } \end{array}$ . Following our previous example of the CEO-of relation, $\mathbf { R } _ { 2 }$ evaluates the likelihood of an organisation to be a company with a CEO position. Importantly, $\mathbf { R } _ { 1 }$ and $\mathbf { R } _ { 2 }$ are representations of the relation $R$ and independent of the entities. Therefore, we consider $\mathbf { R } _ { 1 }$ and $\mathbf { R } _ { 2 }$ ) to collectively represent the embedding of $R$ . Orthogonality of ${ \bf R } _ { 1 } , { \bf R } _ { 2 }$ is a requirement for the mathematical proof and also act as a regularisation constraint to prevent overfitting by restricting the relational embedding space. We first perform our mathematical analysis for relational embeddings represented by orthogonal matrices and discuss later how this requirement can be relaxed.
+
+We assume a slow random walk where the knowledge vectors do not change significantly between consecutive time steps $( c _ { k } \approx c _ { k + 1 } )$ ). More specifically, we assume that $\| \pmb { c } _ { k } - \pmb { c } _ { k + 1 } \| \le \epsilon _ { 2 }$ for some small $\epsilon _ { 2 } > 0$ . This is a realistic assumption for generating the two entity arguments in the same relational triple because, if the knowledge vectors were significantly different in the two generation steps, then it is likely that the corresponding relations are also different, which would not be coherent with the above-described generative process. Moreover, we assume that the knowledge vectors are distributed uniformly in the unit sphere and denote the distribution of knowledge vectors by $\mathcal { C }$ .
+
+To learn KGEs, we must estimate the probability that $h$ and $t$ satisfy the relation $R$ , $p ( h , t \mid R )$ , which can be obtained by taking the expectation of $p ( h , t \mid R , c , c ^ { \prime } )$ w.r.t. $c , c ^ { \prime } \sim \mathcal { C }$ given by (3).
+
+$$
+\begin{array} { r l } & { p ( h , t \mid R ) = \mathbb { E } _ { c , c ^ { \prime } } \left[ p ( h , t \mid R , c , c ^ { \prime } ) \right] } \\ & { \quad \quad \quad = \mathbb { E } _ { c , c ^ { \prime } } \left[ p ( h \mid R , c ) p ( t \mid R , c ^ { \prime } ) \right] } \\ & { \quad \quad \quad = \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { \exp \big ( h ^ { \top } \mathbf { R } _ { 1 } c \big ) } { Z _ { c } } \frac { \exp \big ( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \big ) } { Z _ { c ^ { \prime } } } \right] . } \end{array}
+$$
+
+Here, partition functions are given by $\begin{array} { r l r } { Z _ { c } } & { { } = } & { \sum _ { h \in \mathcal { V } } \sum _ { c \in \mathcal { C } } \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) } \end{array}$ and $\begin{array} { r l } { Z _ { c ^ { \prime } } } & { { } = } \end{array}$ $\begin{array} { r l } { ~ } & { { } \sum _ { t \in \mathcal { V } } \sum _ { c ^ { \prime } \in \mathcal { C } } \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) } \end{array}$ . (4) follows from our two-step generative process where the generation of $h$ and $t$ in each step is independent given the relation and the corresponding knowledge vectors.
+
+Computing the expectation in (5) is generally difficult because of the two partition functions $Z _ { c }$ and $Z _ { c ^ { \prime } }$ . However, Lemma 1 shows that the partition functions are narrowly distributed around a constant value for all $c$ (or $c ^ { \prime }$ ) values with high probability.
+
+Lemma 1 (Concentration Lemma). If the entity embedding vectors satisfy the Bayesian prior $\mathbf { \nabla } \mathbf { \boldsymbol { v } } = s \hat { \mathbf { \nabla } } $ , where $\hat { v }$ is from the spherical Gaussian distribution, and s is a scalar random variable, which is always bounded by a constant $\kappa$ , then the entire ensemble of entity embeddings satisfies that
+
+$$
+\operatorname* { P r } _ { c \sim \mathcal { C } } [ ( 1 - \epsilon _ { z } ) Z \le Z _ { c } \le ( 1 + \epsilon _ { z } ) Z ] \ge 1 - \delta ,
+$$
+
+for $\epsilon _ { z } = O ( 1 / \sqrt { n } )$ , and $\delta = \exp ( - \Omega ( \log ^ { 2 } n ) )$ , where $n \geq d$ is the number of words and $Z _ { c }$ is the partition function for $c$ given by $\begin{array} { r } { \sum _ { c \in \mathcal { V } } \exp \left( { \pmb { h } } ^ { \top } { \bf R } _ { 1 } { \pmb { c } } \right) } \end{array}$ .
+
+proof: To prove the concentration lemma, we show that the mean $\mathbb { E } _ { h } [ Z _ { c } ]$ of $Z _ { c }$ is concentrated around a constant for all knowledge vectors $^ c$ and its variance is bounded. Recall that
+
+$$
+Z _ { c } = \sum _ { h \in \mathcal { V } } \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) .
+$$
+
+If $\mathbf { P }$ is an orthogonal matrix and $_ { \textbf { \em x } }$ is a vector, then $\| \mathbf { P } ^ { \top } \pmb { x } \| _ { 2 } ^ { 2 } = ( \mathbf { P } ^ { \top } \pmb { x } ) ^ { \top } ( \mathbf { P } ^ { \top } \pmb { x } ) = \pmb { x } ^ { \top } \mathbf { P } \mathbf { P } ^ { \top } \pmb { x } = \| \pmb { x } \| _ { 2 } ^ { 2 }$ , because $\mathbf { P } ^ { \top } \mathbf { P } \overset { = } \mathbf { I }$ . Therefore, from (7) and the orthogonality of the relational embeddings, we see that $\mathbf { R } _ { 1 } \boldsymbol { c }$ is a simple rotation of $^ c$ and does not alter the length of $^ c$ . We represent $\boldsymbol { h } = \boldsymbol { s } _ { h } \hat { \boldsymbol { h } }$ , where $s _ { h } = \left\| h \right\|$ and $\hat { h }$ is a unit vector (i.e. $\| \hat { h } \| _ { 2 } = 1 )$ distributed on the spherical Gaussian with zero mean and unit covariance matrix $\mathbf { I } _ { d } \in \mathbb { R } ^ { d \times d }$ . Let $s$ be a random variable that has the same distribution as $s _ { h }$ . Moreover, let us assume that $s$ is upper bounded by a constant $\kappa$ such that $s \leq \kappa$ . From the assumption of the knowledge vector $^ c$ , it is on the unit sphere as well, which is then rotated by $\mathbf { R } _ { 1 }$ .
+
+We can write the partition function using the inner-product between two vectors $^ { h }$ and $\mathbf { R } _ { 1 } \boldsymbol { c }$ , $Z _ { c } = $ $\begin{array} { r } { \sum _ { h \in \mathcal { V } } \exp \big ( { h } ^ { \top } ( \dot { \mathbf { R } } _ { 1 } { c } ) \big ) } \end{array}$ . Arora et al. (2016a) showed that (Lemma 2.1 in their paper) the expectation of a partition function of this form can be approximated as follows:
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { \pmb { c } } [ Z _ { c } ] = n \mathbb { E } _ { \pmb { c } } [ \exp \left( \pmb { h } ^ { \top } \mathbf { R } _ { 1 } \pmb { c } \right) ] } \\ & { \qquad \geq n \mathbb { E } _ { \pmb { c } } [ 1 + \pmb { h } ^ { \top } \mathbf { R } _ { 1 } \pmb { c } ] = n . } \end{array}
+$$
+
+where $n = | \mathcal { V } |$ is the number of entities in the vocabulary. (8) follows from the expectation of a sum and the independence of $^ { h }$ and $\mathbf { R } _ { 1 }$ from $^ c$ . The inequality of (9) is obtained by applying the Taylor expansion of the exponential series and the final equality is due to the symmetry of the spherical Gaussian. From the law of total expectation, we can write
+
+$$
+\operatorname { \mathbb { E } } _ { c } [ Z _ { c } ] = n \operatorname { \mathbb { E } } _ { c } [ \exp \left( { h ^ { \top } \mathbf { R } _ { 1 } c } \right) ] = n \operatorname { \mathbb { E } } _ { s _ { h } } \left[ \operatorname { \mathbb { E } } _ { x | s _ { h } } \left[ \exp \left( { h ^ { \top } \mathbf { R } _ { 1 } c } \right) \mid s _ { h } \right] \right] .
+$$
+
+where, $\underline { { x } } = \pmb { h } ^ { \top } \mathbf { R } _ { 1 } \pmb { c }$ . Note that conditioned on $s _ { h } , h$ is a Gaussian random variable with variance $\sigma ^ { 2 } = \acute { s } _ { h } ^ { 2 }$ . Therefore, conditioned on $s _ { h }$ , $x$ is a random variable with variance $\sigma ^ { 2 } = \sigma _ { h } ^ { 2 }$ . Using this distribution, we can evaluate $\mathbb { E } _ { \boldsymbol { x } | \boldsymbol { s } _ { h } } \left[ \exp \left( \pmb { h } ^ { \top } \mathbf { R } _ { 1 } \boldsymbol { c } \right) \right]$ as follows:
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } _ { x \mid s _ { h } } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \mid s _ { h } \right] = \int _ { x } \frac { 1 } { \sqrt { 2 \pi \sigma ^ { 2 } } } \exp \left( - \frac { x ^ { 2 } } { 2 \sigma ^ { 2 } } \right) \exp ( x ) d x } \\ { \displaystyle = \int _ { x } \frac { 1 } { \sqrt { 2 \pi \sigma ^ { 2 } } } \exp \left( - \frac { \left( x - \sigma ^ { 2 } \right) ^ { 2 } } { 2 \sigma ^ { 2 } } + \sigma ^ { 2 } / 2 \right) d x } \\ { \displaystyle = \exp ( \sigma ^ { 2 } / 2 ) . } \end{array}
+$$
+
+Therefore, it follows that
+
+$$
+\operatorname { \mathbb { E } } _ { c } [ Z _ { c } ] = n \operatorname { \mathbb { E } } _ { s _ { h } } [ \exp ( \sigma ^ { 2 } / 2 ) ] = n \operatorname { \mathbb { E } } _ { s _ { h } } [ \exp ( s _ { h } ^ { 2 } / 2 ) ] = n \exp ( s ^ { 2 } / 2 ) ,
+$$
+
+where $s$ is the variance of the $\ell _ { 2 }$ norms of the entity embeddings. Because the set of entities is given and fixed, both $n$ and $\sigma$ are constants, proving that $\mathbb { E } [ Z _ { c } ]$ does not depend on $c$ .
+
+Next, we calculate the variance $\mathbb { V } _ { c } [ Z _ { c } ]$ as follows:
+
+$$
+\begin{array} { r l } & { \mathbb { V } _ { c } [ Z _ { c } ] = \displaystyle \sum _ { h } \mathbb { V } _ { c } [ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) ] } \\ & { ~ \leq n \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \right] } \\ & { ~ = n \mathbb { E } _ { s _ { h } } \left[ \mathbb { E } _ { x \mid s _ { h } } \left[ \exp \left( 2 h ^ { \top } \mathbf { R } _ { 1 } t \right) \mid s _ { h } \right] \right] . } \end{array}
+$$
+
+Because $2 h ^ { \top } { \bf R } _ { 1 } t$ is a Gaussian random variable with variance $4 \sigma ^ { 2 } = 4 s _ { h } ^ { 2 }$ from a similar calculation as in (11) we obtain,
+
+$$
+\begin{array} { r } { \mathbb { E } _ { \boldsymbol { x } | s _ { h } } \left[ \exp \left( 2 \boldsymbol { h } ^ { \top } \mathbf { R } _ { 1 } t \right) \mid s _ { h } \right] = \exp ( 2 \sigma ^ { 2 } ) . } \end{array}
+$$
+
+By substituting (16) in (15) we have that
+
+$$
+\mathbb { V } _ { c } [ Z _ { c } ] \leq n \mathbb { E } _ { s _ { h } } \left[ \exp \left( 2 { \sigma } ^ { 2 } \right) \right] = n \mathbb { E } _ { s _ { h } } \left[ \exp ( 2 s ^ { 2 } ) \right] \leq \Lambda n
+$$
+
+for $\Lambda = \exp ( 8 \kappa ^ { 2 } )$ a constant bounding $s \leq \kappa$ as stated.
+
+From above, we have bounded both the mean and variance of the partition function by constants that are independent of the knowledge vector. Note that neither exp $\left( \hat { h ^ { \mathrm { { T } } } } \mathbf { R } _ { 1 } c \right)$ nor exp $( t ^ { \dagger } { \mathbf { R } } _ { 2 } c ^ { \prime } )$ are subGaussian nor sub-exponential. Therefore, standard concentration bounds derived for sub-Gaussian or sub-exponential random variables cannot be used in our analysis. However, the argument given in Appendix A.1 in Arora et al. (2016b) for a partition function with bounded mean and variance can be directly applied to $Z _ { c }$ in our case, which completes the proof of the concentration lemma. □
+
+From the symmetry between $h$ and $t$ , Lemma 1 also applies for the partition function $\begin{array} { r } { \sum _ { t \in \mathcal { V } } \left( { t ^ { { \top } } \mathbf { R } _ { 2 } c ^ { \prime } } \right) } \end{array}$ Under the conditions required to satisfy Lemma 1, the following main theorem of this paper holds:
+
+Theorem 1. Suppose that the entity embeddings satisfy (1). Then, we have
+
+$$
+\log p ( h , t \mid R ) = { \frac { \| \mathbf { R } _ { 1 } ^ { \mathsf { T } } h + \mathbf { R } _ { 2 } ^ { \mathsf { T } } t \| _ { 2 } ^ { 2 } } { 2 d } } - 2 \log Z \pm \epsilon .
+$$
+
+for $\epsilon = { O } ( 1 / { \sqrt { n } } ) + \widetilde { O } ( 1 / d ) $ , where
+
+$$
+Z = Z _ { c } = Z _ { c ^ { \prime } } .
+$$
+
+The complete proof of Theorem 1 is given in Appendix A. Below we briefly sketch the main steps.
+
+Proof sketch: Let $F$ be the event that both $c$ and $c ^ { \prime }$ are within $( 1 \pm \epsilon _ { z } ) Z$ . Then, from Lemma 1 and the union bound, event $F$ happens with probability at least $1 - 2 \exp ( - \Omega ( \log ^ { 2 } n ) )$ . The R.H.S. of (5) can be split into two parts $\dot { T } _ { 1 }$ and $T _ { 2 }$ according to whether $F$ happens or not.
+
+$$
+p ( h , t \mid R ) = \underbrace { \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) } { Z _ { c } } \frac { \exp \left( h ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) } { Z _ { c ^ { \prime } } } \mathbf { 1 } _ { F } \right] } _ { = T _ { 1 } } + \underbrace { \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) } { Z _ { c } } \frac { \exp \left( h ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) } { Z _ { c ^ { \prime } } } \mathbf { 1 } _ { F } \right] } _ { = T _ { 2 } } .
+$$
+
+$T _ { 1 }$ can be approximated as given by (21).
+
+$$
+T _ { 1 } = \frac { 1 \pm \mathcal { O } ( \epsilon _ { z } ) } { Z ^ { 2 } } \mathbb { E } _ { c , c ^ { \prime } } \left[ \exp \left( \pmb { h } ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( \pmb { t } ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \right]
+$$
+
+On the other hand, $T _ { 2 }$ can be shown to be a constant, independent of $d$ , given by (22).
+
+$$
+| T _ { 2 } | = \exp ( - \Omega ( \log ^ { 1 . 8 } n ) )
+$$
+
+The vocabulary size $n$ of real-world knowledge graphs is typically over $1 0 ^ { 5 }$ , for which $T _ { 2 }$ becomes negligibly small. Therefore, it suffices to consider only $T _ { 1 }$ . Because of the slowness of the random walk we have $c \approx c ^ { \prime }$
+
+Using the law of total expectation we can write $T _ { 1 }$ as follows:
+
+$$
+\begin{array} { l } { { \displaystyle { T _ { 1 } = \frac { 1 \pm \mathcal { O } ( \epsilon _ { z } ) } { Z ^ { 2 } } \mathbb { E } _ { c } \left[ \exp \left( { h ^ { \top } { \bf R } _ { 1 } c } \right) \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \exp \left( { t ^ { \top } { \bf R } _ { 2 } c ^ { \prime } } \right) \right] \right]}  } } \\ { { \displaystyle ~ = \frac { 1 \pm \mathcal { O } ( \epsilon _ { z } ) } { Z ^ { 2 } } \mathbb { E } _ { c } \left[ \exp \left( { h ^ { \top } { \bf R } _ { 1 } c } \right) A ( c ) \right] } } \end{array}
+$$
+
+where $A ( c ) : = \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \exp \left( { t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } } \right) \right]$ . Doing some further evaluations we show that
+
+$$
+A ( c ) = ( 1 \pm \epsilon _ { 2 } ) \exp \left( t ^ { \top } { \bf R } _ { 2 } c \right)
+$$
+
+Plugging (50) back in (23) provides the claim of the theorem.
+
+The relationship given by (18) indicates that head and tail entity embeddings are first transformed respectively by $\bar { \mathbf { R } _ { 1 } } ^ { \top }$ and $\mathbf { R } _ { 2 } ^ { \phantom { \dagger } }$ , and the squared $\ell _ { 2 }$ norm of the sum of the transformed vectors is proportional to the probability $p ( h , t \mid R )$ .
+
+# 3 LEARNING KNOWLEDGE GRAPH EMBEDDINGS
+
+In this section, we derive a training objective from Theorem 1 that we can then optimise to learn KGE. The goal is to empirically validate the theoretical result by evaluating the learnt KGEs. Knowledge graphs represent information about relations between two entities in the form of relational triples. The joint probability $p ( h , R , t )$ given by Theorem 1 is useful for determining whether a relation $R$ exists between two given entities $h$ and $t$ . For example, if we know that with a high probability that $R$ holds between $h$ and $t$ , then we can append $( h , R , t )$ to the knowledge graph. The task of expanding knowledge graphs by predicting missing links between entities or relations is known as the link prediction problem (Trouillon et al., 2016). In particular, if we can automatically append such previously unknown knowledge to the knowledge graph, we can expand the knowledge graph and address the knowledge acquisition bottleneck.
+
+To derive a criteria for determining whether a link must be predicted among entities and relations, let us consider a relational triple $( h , R , t ) \in \mathcal { D }$ that exists in a given knowledge graph $\mathcal { D }$ . We call such relational triples as positive triples because from the assumption it is known that $R$ holds between $h$ and $t$ . On the other hand, consider a negative relational triple $( h ^ { \prime } , R , t ^ { \prime } ) \in \mathcal { D }$ formed by, for example, randomly perturbing a positive triple. A popular technique for generating such (pseudo) negative triples is to replace $h$ or $t$ with a randomly selected different instance of the same entity type. As an alternative for random perturbation, Cai and Wang (2018) proposed a method for generating negative instances using adversarial learning. Here, we are not concerned about the actual method used for generating the negative triples but assume a set of negative triples, $\bar { \mathcal D }$ , generated using some method, to be given.
+
+Given a positive triple $( h , R , t ) \in \mathcal { D }$ and a negative triple $( h ^ { \prime } , R , t ^ { \prime } ) \in \bar { \mathcal { D } }$ , we would like to learn KGEs such that a higher probability is assigned to $( h , R , t )$ than that assigned to $( h ^ { \prime } , R , t ^ { \prime } )$ . We can formalise this requirement using the likelihood ratio given by (25).
+
+$$
+\frac { p ( h , R , t ) } { p ( h ^ { \prime } , R , t ^ { \prime } ) } \geq \eta
+$$
+
+Here, $\eta > 1$ is a threshold that determines how higher we would like to set the probabilities for the positive triples compares to that of the negative triples.
+
+By taking the logarithm of both sides in (25) we obtain
+
+$$
+\begin{array} { c } { \log p ( h , R , t ) - \log p ( h ^ { \prime } , R , t ^ { \prime } ) \geq \log { \eta } } \\ { \log \eta + \log p ( h ^ { \prime } , R , t ^ { \prime } ) - \log p ( h , R , t ) \geq 0 } \end{array}
+$$
+
+If a positive triple $( h , R , t )$ is correctly assigned a higher probability than a negative triple $p ( h ^ { \prime } , R , t ^ { \prime } )$ , then the left hand side of (26) will be negative, indicating that there is no loss incurred during this classification task. Therefore, we can re-write (26) to obtain the marginal loss Bordes et al. (2013; 2011), $L ( \mathcal { D } , \bar { \mathcal { D } } )$ , a popular choice as a learning objective in prior work in KGE, as shown in (27).
+
+$$
+\begin{array} { r l } & { L ( \mathcal { D } , \bar { \mathcal { D } } ) = \displaystyle \sum _ { ( h , R , t ) \in \mathcal { D } } \operatorname* { m a x } \left( 0 , \log \eta + \log p ( h ^ { \prime } , R , t ^ { \prime } ) - \log p ( h , R , t ) \right) } \\ & { \quad \quad \quad \quad ( h ^ { \prime } , R , t ^ { \prime } ) \in \bar { \mathcal { D } } } \\ & { \quad \quad \quad = \operatorname* { m a x } \left( 0 , 2 d \log \eta + \| \mathbf { R } _ { 1 } ^ { \top } h ^ { \prime } + \mathbf { R } _ { 2 } ^ { \top } t ^ { \prime } \| _ { 2 } ^ { 2 } - \| \mathbf { R } _ { 1 } ^ { \top } h + \mathbf { R } _ { 2 } ^ { \top } t \| _ { 2 } ^ { 2 } \right) } \end{array}
+$$
+
+We can assume $2 d \log \eta$ to be the margin for the constraint violation.
+
+Theorem 1 requires $\mathbf { R } _ { 1 }$ and ${ \bf R } _ { 2 }$ to be orthogonal. To reflect this requirement, we add two $\ell _ { 2 }$ regularisation terms $\lVert \mathbf { R } _ { 1 } ^ { \top } \mathbf { R } _ { 1 } - \mathbf { I } \rVert _ { 2 } ^ { 2 }$ and $| | \mathbf { R } _ { 2 } ^ { \top } \mathbf { \widetilde { R } } _ { 2 } - \mathbf { I } | | _ { 2 } ^ { 2 }$ respectively with regularisation coefficients $\lambda _ { 1 }$ and $\lambda _ { 2 }$ to the objective function given by (27). In our experiments, we compute the gradients (27) w.r.t. each of the parameters $\mathbf { \Sigma } _ { h , \ t , \ R _ { 1 } }$ and $R _ { 2 }$ and use stochastic gradient descent (SGD) for optimisation. This approach can be easily extended to learn from multiple negative triples as shown in Appendix B.
+
+# 4 RELATED WORK
+
+At a high-level of abstraction, KGE methods can be seen as differing in their design choices for the following two main problems: (a) how to represent entities and relations, and (b) how to model the interaction between two entities and a relation that holds between them. Next, we briefly discuss prior proposals to those two problems (refer (Wang et al., 2017; Nickel et al., 2015; Nguyen, 2017) for an extended survey on KGE).
+
+A popular choice for representing entities is to use vectors, whereas relations have been represented by vectors, matrices or tensors. For example, TransE (Bordes et al., 2011), TransH (Wang et al., 2014), TransD (Ji et al., 2015), TransG (Xiao et al., 2016), TransR (Lin et al., 2015), lppTransD (Yoon et al., 2016), DistMult (Yang et al., 2015), HolE (Nickel et al., 2016) and ComplEx (Trouillon et al., 2016) represent relations by vectors, whereas Structured Embeddings (Bordes et al., 2011), TranSparse (Ji et al., 2016), STransE (Nguyen et al., 2016), RESCAL (Nickel et al., 2011) use matrices and Neural Tensor Network (NTN) (Socher et al., 2013) uses 3D tensors. ComplEx (Trouillon et al., 2016) introduced complex vectors for KGEs to capture the asymmetry in semantic relations. (Ding et al., 2018) obtained state-of-the-art performance for KGE by imposing non-negativity and entailment constraints to ComplEx.
+
+Given entity and relation embeddings, a scoring function is defined that evaluates the strength of a relation $R$ between two entities $h$ and $t$ in a triple $( h , R , t )$ . The scoring functions that encode various intuitions have been proposed such as the $\ell _ { 1 }$ or $\ell _ { 2 }$ norms of the vector formed by a translation of the head entity embedding by the relation embedding over the target embedding, or by first performing a projection from the entity embedding space to the relation embedding space (Yoon et al., 2016) As an alternative to using vector norms as scoring functions, DistMult and ComplEx use the component-wise multi-linear dot product.
+
+Once a scoring function is defined, KGEs are learnt that assign better scores to relational triples in existing knowledge graphs (positive triples) over triples where the relation does not hold (negative triples) by minimising a loss function such as the logistic loss (RESCAL, DistMult, ComplEx) or marginal loss (TransE, TransH, TransD, TransD). Because knowledge graphs record only positive triples, a popular method to generate pseudo negative triples is to perturb a positive instance by replacing its head or tail entity by an entity selected uniformly at random from the vocabulary of the entities. However, uniformly sampled negative triples are likely to be obvious examples that do not provide much information to the learning process and can be detected by simply checking for the type of the entities in a triple. Cai and Wang (2018) proposed an adversarial learning approach where a generator assigns a probability to each relation triple and negative instances are sampled according to this probability distribution to train a discriminator that discriminates between positive and negative instances. (Xiao et al., 2016) proposed TransG, a generative model based on the Chinese restaurant process, to model multiple relations that exist between a pair of entities. However, their relation embeddings are designed to satisfy vector translation similar to TransE.
+
+As an alternative to directly learning embeddings from a graph, several methods (Grover and Leskovec, 2016; Perozzi et al., 2014; Ristoski et al., 2018) have considered the vertices visited during truncated random walks over the graph as pseudo sentences, and have applied popular word embedding learning algorithms such as skip-gram with negative sampling or continuous bag-of-words model (Mikolov et al., 2013) to learn vertex embeddings. However, pseudo sentences generated this way are syntactically very different from sentences in natural languages.
+
+On the other hand, our work extends the random walk analysis by Arora et al. (2016a) that derives a useful connection between the joint co-occurrence probability of two words and the $\ell _ { 2 }$ norm of the sum of the corresponding word embeddings. Specifically, they proposed a latent variable model where the words in a corpus are generated by a probabilistic model parametrised by a time-dependent discourse vector that performs a random walk. However, unlike in our work, they do not consider the relations between two co-occurring words in a corpus. Bollegala et al. (2018) extended the model proposed by Arora et al. (2016a) to capture co-occurrences involving more than two words. They defined the co-occurrence of $k$ unique words in a given context as a $k$ -way co-occurrence, where Arora et al. (2016a)’s result could be seen as a special case corersponding to $k = 2$ . Moreover, Bollegala et al. (2018) showed that it is possible to learn word embeddings that capture some types of semantic relations such as antonymy and collocation using 3-way co-occurrences more accurately than using 2-way co-occurrences. However, their model does not explicitly consider the relations between words/entities and uses only a corpus for learning the word embeddings.
+
+Table 2: Triple classification.   
+
+<table><tr><td></td><td colspan="2">Accuracy</td></tr><tr><td>Method</td><td>WN11</td><td>FB13</td></tr><tr><td>SE</td><td>53.0</td><td>75.2</td></tr><tr><td>TransE</td><td>75.9</td><td>81.5</td></tr><tr><td>TransR</td><td>85.9</td><td>82.5</td></tr><tr><td>TransG</td><td>87.4</td><td>87.3</td></tr><tr><td>NTN</td><td>70.4</td><td>87.1</td></tr><tr><td>RelWalk</td><td>75.48</td><td>87.5</td></tr></table>
+
+Table 3: Link prediction. Results marked with $[ \star ]$ are taken from Dettmers et al. (2017), $[ \bullet ]$ from Nguyen et al. (2017), [/] from and Cai and Wang (2018). All other results for the baselines are taken from their original papers.   
+
+<table><tr><td></td><td colspan="5">FB15K237</td><td colspan="5">WN18RR</td></tr><tr><td>Method</td><td>MRR</td><td>MR</td><td>H@1</td><td>H@3</td><td>H@10</td><td>MRR</td><td>MR</td><td>H@1</td><td>H@3</td><td>H@10</td></tr><tr><td>TransE</td><td>0.294</td><td>347</td><td>1</td><td>=</td><td>0.465</td><td>0.226</td><td>3384</td><td>=</td><td>=</td><td>0.50</td></tr><tr><td>TransD</td><td>0.28</td><td>-</td><td>1</td><td>1</td><td>0.453</td><td>1</td><td></td><td>1</td><td>1</td><td>0.43</td></tr><tr><td>DistMult*</td><td>0.241</td><td>254</td><td>0.155</td><td>0.263</td><td>0.419</td><td>0.43</td><td>5110</td><td>0.39</td><td>0.44</td><td>0.49</td></tr><tr><td>ComplEx*</td><td>0.247</td><td>339</td><td>0.158</td><td>0.275</td><td>0.428</td><td>0.44</td><td>5261</td><td>0.41</td><td>0.46</td><td>0.51</td></tr><tr><td>ConvE</td><td>0.316</td><td>246</td><td>0.239</td><td>0.35</td><td>0.491</td><td>0.46</td><td>5277</td><td>0.39</td><td>0.43</td><td>0.48</td></tr><tr><td>RelWalk</td><td>0.329</td><td>105</td><td>0.243</td><td>0.354</td><td>0.502</td><td>0.451</td><td>3232</td><td>0.42</td><td>0.47</td><td>0.51</td></tr></table>
+
+# 5 EMPIRICAL VALIDATION
+
+To empirically evaluate the theoretical result stated in Theorem 1, we learn KGEs (denoted by RelWalk) by minimising the marginal loss objective derived in section 3. We use the FB15k237, FB13 (subsets of Freebase) and WN11, WN18RR (subsets of WordNet) datasets, which are standard benchmarks for KGE. We use the standard training, validation and test splits as detailed in Table 4. We generate negative triples by replacing a head or a tail entity in a positive triple by a randomly selected different entity and learn KGEs. We train the model until convergence or at most 1000 epochs over the training data where each epoch is divided into $1 0 0 \mathrm { { m i n i } }$ -batches. The best model is selected by early stopping based on the performance of the learnt embeddings on the validation set (evaluated after each 20 epochs). The training details and hyperparameter settings are detailed in Appendix C. RelWalk is implemented in the open-source toolkit OpenKE (Han et al., 2018).2
+
+We conduct two evaluation tasks: link prediction (predict the missing head or tail entity in a given triple $( h , R , ? )$ or $( ? , R , t ) )$ ) (Bordes et al., 2011) and triple classification (predict whether a relation $R$ holds between $h$ and $t$ in a given triple $( h , R , t ) )$ (Socher et al., 2013). We evaluate the performance in the link prediction task using mean reciprocal rank (MRR), mean rank (MR (the average of the rank assigned to the original head or tail entity in a corrupted triple) and hits at ranks 1, 3 and 10 $( \mathbf { H } @ \mathbf { 1 } , \mathbf { 3 } , \mathbf { 1 0 } )$ , whereas in the triple classification task we use accuracy (percentage of the correctly classified test triples). We only report scores under the filtered setting Bordes et al. (2013), which removes all triples appeared in training, validating and testing sets from candidate triples before obtaining the rank of the ground truth triple. In link prediction, we consider all entities that appear in the corresponding argument in the entire knowledge graph as candidates.
+
+In Tables 2 and 3 we compare the KGEs learnt by RelWalk against prior work using the published results. For link prediction, RelWalk reports SoTA on both WN18RR and FB15K237 in all evaluation measures, except against ConvE in WN18RR measured by MRR. WN18RR excludes triples from WN18 that are simply inverted between train and test partitions (Toutanova and Chen, 2015; Dettmers et al., 2017). RelWalk’s consistently good performance on both versions of this dataset shows that it is considering the global structure in the knowledge graph when learning KGEs. For triple classification, RelWalk reports the best performance on FB13, whereas TransG reports the best performance on
+
+WN11. Considering that both TransG and RelWalk are generative models, it would be interesting to further investigate generative approaches for KGE in the future. Overall, the experimental results support our theoretical claim and emphasise the importance of theoretically motivating the scoring function design process.
+
+# 6 CONCLUSION
+
+We proposed RelWalk, a generative model of KGE and derived a theoretical relationship between the probability of a triple and entity, relation embeddings. We then proposed a learning objective based on the theoretical relationship we derived. Experimental results on a link prediction and a triple classification tasks show that RelWalk obtains strong performances in multiple benchmark datasets.
+
+# REFERENCES
+
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+
+Sanjeev Arora, Yuanzhi Li, Yingyu Liang, Tengyu Ma, and Andrej Risteski. Rand-walk: A latent variable model approach to word embeddings. arXiv, 2016b.
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+
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+
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+
+Bryan Perozzi, Rami Al-Rfou, and Steven Skiena. Deepwalk: Online learning of social representations. In Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’14, pages 701–710, New York, NY, USA, 2014. ACM. ISBN 978-1-4503-2956-9. doi: 10.1145/2623330.2623732. URL http://doi.acm.org/10.1145/ 2623330.2623732.
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+
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+
+# APPENDIX
+
+# A PROOF OF THEOREM 1
+
+Let us consider the probabilistic event that $( 1 - \epsilon _ { z } ) Z \le Z _ { c } \le ( 1 + \epsilon _ { z } ) Z$ to be $F _ { c }$ and $( 1 - \epsilon _ { z } ) Z \le$ $Z _ { c ^ { \prime } } \leq ( 1 + \epsilon _ { z } ) Z$ to be $F _ { c ^ { \prime } }$ . From Lemma 1 we have $\mathrm { P r } _ { c } [ F _ { c } ] \ge 1 - \delta$ . Then from the union bound we have,
+
+$$
+\begin{array} { r l } & { \mathrm { P r } [ \bar { F } _ { c } \vee \bar { F } _ { c ^ { \prime } } ] \leq \mathrm { P r } [ \bar { F } _ { c } ] + \mathrm { P r } [ \bar { F } _ { c ^ { \prime } } ] } \\ & { \qquad = 1 - \mathrm { P r } [ F _ { c } ] + 1 - \mathrm { P r } [ F _ { c ^ { \prime } } ] } \\ & { \qquad = 2 \delta . } \end{array}
+$$
+
+Moreover, let $F$ be the probabilistic event that both $F _ { c }$ and $F _ { c ^ { \prime } }$ being True. Then from ${ \mathrm { P r } } [ F ] =$ $1 - \operatorname* { P r } [ \bar { F } _ { c } \vee \bar { F } _ { c ^ { \prime } } ]$ we have, $\mathrm { P r } [ F ] \ge 1 - 2 \delta$ . We can decompose the expectation in the R.H.S. in (5) into two terms $T _ { 1 }$ and $T _ { 2 }$ depending on whether respectively $F$ is True or False as follows:
+
+$$
+\begin{array} { r } { \mathbf { \rho } ( h , t \mid r ) = \underbrace { \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { \exp \big ( h ^ { \top } \mathbf { R } _ { 1 } c \big ) } { Z _ { c } } \frac { \exp \big ( h ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \big ) } { Z _ { c ^ { \prime } } } \mathbf { 1 } _ { F } \right] } _ { = T _ { 1 } } + \underbrace { \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { \exp \big ( h ^ { \top } \mathbf { R } _ { 1 } c \big ) } { Z _ { c } } \frac { \exp \big ( h ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \big ) } { Z _ { c ^ { \prime } } } \mathbf { 1 } _ { \bar { F } } \right] } _ { = T _ { 2 } } . } \end{array}
+$$
+
+Here, ${ \bf 1 } _ { F }$ and ${ \mathbf { 1 } } _ { \bar { F } }$ are indicator functions given by:
+
+$$
+\mathbf { 1 } _ { F } = { \left\{ \begin{array} { l l } { 1 } & { { \mathrm { i f ~ } } F { \mathrm { ~ i s ~ T r u e , } } } \\ { 0 } & { { \mathrm { o t h e r w i s e , } } } \end{array} \right. }
+$$
+
+and
+
+$$
+\mathbf { 1 } _ { \bar { F } } = \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } F \mathrm { ~ i s ~ T r u e , } } \\ { 1 } & { \mathrm { o t h e r w i s e . } } \end{array} \right.
+$$
+
+Let us first show that $T _ { 2 }$ is negligibly small.
+
+For two real integrable functions $\psi _ { 1 } ( x )$ and $\psi _ { 2 } ( x )$ in $[ a , b ]$ , the Cauchy-Schwarz’s inequality states that
+
+$$
+\left[ \int _ { a } ^ { b } \psi _ { 1 } ( x ) \psi _ { 2 } ( x ) d x \right] ^ { 2 } \leq \int _ { a } ^ { b } \left[ \psi _ { 1 } ( x ) \right] ^ { 2 } d x \int _ { a } ^ { b } \left[ \psi _ { 2 } ( x ) \right] ^ { 2 } d x .
+$$
+
+Applying (32) to $T _ { 2 }$ in (29) we have:
+
+$$
+\begin{array} { r l } & { \left( \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { 1 } { Z _ { c } Z _ { c ^ { \prime } } } \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \mathbf { 1 } _ { \bar { F } } \right] \right) ^ { 2 } } \\ & { \leq \left( \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { 1 } { Z _ { c } ^ { 2 } } \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbf { 1 } _ { \bar { F } } \right] \right) \left( \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { 1 } { Z _ { c ^ { \prime } } ^ { 2 } } \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) ^ { 2 } \mathbf { 1 } _ { \bar { F } } \right] \right) } \\ & { = \left( \mathbb { E } _ { c } \left[ \frac { 1 } { Z _ { c } ^ { 2 } } \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] \right) \left( \mathbb { E } _ { c ^ { \prime } } \left[ \frac { 1 } { Z _ { c ^ { \prime } } ^ { 2 } } \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) ^ { 2 } \mathbb { E } _ { c \mid c ^ { \prime } } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] \right) } \end{array}
+$$
+
+Note that $Z _ { c } \geq 1$ because $Z _ { c }$ is the sum of positive numbers and if ${ \pmb h } ^ { \top } { \pmb R } _ { 1 } { \pmb c } \geq 0$ for at least one of the $h \in \mathcal V$ , then the total sum will be greater than 1. Therefore, by dropping $Z _ { c }$ term from the denominator we can further increase the first term in (33) as given by (34).
+
+$$
+\mathbb { E } _ { c } \left[ \frac { 1 } { Z _ { c } ^ { 2 } } \exp \left( \pmb { h } ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] \le \mathbb { E } _ { c } \left[ \exp \left( \pmb { h } ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right]
+$$
+
+Let us split the expectation on the R.H.S. of (34) into two cases depending on whether ${ \pmb h } ^ { \top } { \pmb R } _ { 1 } { \pmb c } > 0$ or otherwise, indicated respectively by 1(h>R1c>0) and 1(h>R1c≤0).
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { c } \left[ \exp \left( \boldsymbol { h } ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] } \\ & { = \mathbb { E } _ { c } \left[ \exp \left( \boldsymbol { h } ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbf { 1 } _ { \left( \boldsymbol { h } ^ { \top } \mathbf { R } _ { 1 } c > 0 \right) } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] + \mathbb { E } _ { c } \left[ \exp \left( \boldsymbol { h } ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbf { 1 } _ { \left( \boldsymbol { h } ^ { \top } \mathbf { R } _ { 1 } c \leq 0 \right) } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] } \end{array}
+$$
+
+The second term of (35) is upper bounded by
+
+$$
+\mathbb { E } _ { c , c ^ { \prime } } \left[ \mathbf { 1 } _ { \bar { F } } \right] \leq \exp \left( - \Omega ( \log ^ { 2 } n ) \right)
+$$
+
+The first term of (35) can be bounded as follows:
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbf { 1 } _ { ( h ^ { \top } \mathbf { R } _ { 1 } c > 0 ) } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] \leq \mathbb { E } _ { c } \left[ \exp \left( \alpha h ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbf { 1 } _ { ( h ^ { \top } \mathbf { R } _ { 1 } c > 0 ) } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] } \\ & { \qquad \leq \mathbb { E } _ { c } \left[ \exp \left( \alpha h ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] } \end{array}
+$$
+
+where $\alpha > 1$ . Therefore, it is sufficient to bound $\mathbb { E } _ { c } \left[ \exp ( \alpha h ^ { \top } \mathbf { R } _ { 1 } c ) ^ { 2 } \mathbb { E } _ { c ^ { \prime } | c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right]$ when $\left\| h \right\| =$ $\Omega ( { \sqrt { d } } )$ .
+
+Let us denote by $z$ the random variable $2 h ^ { \top } { \bf R } _ { 1 } c$ . Moreover, let $r ( z ) = \mathbb { E } _ { c ^ { \prime } | z } [ \mathbf { 1 } _ { \bar { F } } ]$ , which is a function of $z$ between $[ 0 , 1 ]$ . We wish to upper bound $\mathbb { E } _ { c } [ \exp ( z ) r ( z ) ]$ . The worst-case $r ( z )$ can be quantified using a continuous version of Abel’s inequality (proved as Lemma A.4 in Arora et al. (2016b)), we can upper bound $\mathbb { E } _ { c } \left[ \exp ( z ) r ( z ) \right]$ as follows:
+
+$$
+\mathbb { E } _ { c } \left[ \exp ( z ) r ( z ) \right] \leq \mathbb { E } \left[ \exp ( z ) \mathbf { 1 } _ { [ t , + \infty ] } ( z ) \right]
+$$
+
+where $t$ satisfies that $\begin{array} { r } { \mathbb { E } _ { c } [ \mathbf { 1 } _ { [ t , + \infty ] } ( z ) ] = \operatorname* { P r } [ z \geq t ] = \mathbb { E } _ { c } [ r ( z ) ] \leq \exp ( - \Omega ( \log ^ { 2 } n ) ) } \end{array}$ . Here, $\mathbf { 1 } _ { [ t , + \infty ] } ( z )$ is a function that takes the value 1 when $z \geq t$ and zero elsewhere. Then, we claim $\mathrm { P r } _ { c } [ z \ge t ] \le$ $\exp ( - \Omega ( \log ^ { 2 } n ) )$ implies that $t \geq \Omega ( \log ^ { . 9 } n )$ .
+
+If $c$ was distributed as $\mathcal { N } ( 0 , \frac { 1 } { d } \mathbf { I } )$ , this would be a simple tail bound. However, as $c$ is distributed uniformly on the sphere, this requires special care, and the claim follows by applying the tail bound for the spherical distribution given by Lemma A.1 in (Arora et al., 2016a) instead. Finally, applying Corollary A.3 in (Arora et al., 2016a), we have:
+
+$$
+\mathbb { E } [ \exp ( z ) r ( z ) ] \leq \mathbb { E } [ \exp ( z ) \mathbf { 1 } _ { [ t , + \infty ] } ( z ) ] = \exp ( - \Omega ( \log ^ { 1 . 8 } n ) )
+$$
+
+From a similar argument as above we can obtain the same bound for $c ^ { \prime }$ as well. Therefore, $T _ { 2 }$ in (29) can be upper bounded as follows:
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { c , c ^ { \prime } } \left[ \frac { 1 } { Z _ { c } Z _ { c ^ { \prime } } } \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \mathbf { 1 } _ { \bar { F } } \right] } \\ & { = \left( \mathbb { E } _ { c } \left[ \frac { 1 } { Z _ { c } ^ { 2 } } \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) ^ { 2 } \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right) \right) ^ { 1 / 2 } \left( \mathbb { E } _ { c ^ { \prime } } \left[ \frac { 1 } { Z _ { c ^ { \prime } } ^ { 2 } } \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) ^ { 2 } \mathbb { E } _ { c \mid c ^ { \prime } } \left[ \mathbf { 1 } _ { \bar { F } } \right] \right] \right) ^ { 1 / 2 } } \\ & { \leq \exp ( - \Omega ( \log ^ { 1 . 8 } n ) ) } \end{array}
+$$
+
+Because $n = | \mathcal { V } |$ , the size of the entity vocabulary, is large (ca. $n > 1 0 ^ { 5 }$ ) in most knowledge graphs, we can ignore the $T _ { 2 }$ term in (29). Combining this with (29) we obtain an upper bound for $p ( h , t \mid R )$ given by (41).
+
+$$
+\begin{array} { r l } & { p ( h , t \mid R ) \leq \left( 1 + \epsilon _ { z } \right) ^ { 2 } \frac { 1 } { Z ^ { 2 } } \mathbb { E } _ { c , c ^ { \prime } } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \mathbf { 1 } _ { F } \right] + | \mathcal { D } | \exp \left( - \Omega ( \log ^ { 1 . 8 } n ) \right) } \\ & { \phantom { \left( 1 + \epsilon _ { z } \right) ^ { 2 } } = \left( 1 + \epsilon _ { z } \right) ^ { 2 } \frac { 1 } { Z ^ { 2 } } \mathbb { E } _ { c , c ^ { \prime } } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \right] + \delta _ { 0 } \phantom { \left( 1 + \epsilon _ { z } \right) ^ { 2 } } ( \mathrm { a v e } ^ { 2 } + \mathrm { b } _ { 0 } ^ { 2 } ) , } \end{array}
+$$
+
+where $| \mathcal D |$ is the number of relational tuples $( h , R , t )$ in the KB $\mathcal { D }$ and $\delta _ { 0 } = | \mathcal { D } | \exp ( - \Omega ( \log ^ { 1 . 8 } n ) ) \leq$ $\exp ( - \Omega ( \log ^ { 1 . 8 } n ) )$ by the fact that $Z \le \exp ( 2 \kappa ) n = O ( n )$ , where $\kappa$ is the upper bound on $\mathbf { \Sigma } _ { h } \top \mathbf { R } _ { 1 } c$ and ${ \pmb t } ^ { \top } { \pmb R } _ { 2 } c ^ { \prime }$ , which is regarded as a constant.
+
+On the other hand, we can lower bound $p ( h , t \mid R )$ as given by (42).
+
+$$
+\begin{array} { r l } & { p ( h , t \mid R ) \ge ( 1 - \epsilon _ { z } ) ^ { 2 } \frac { 1 } { Z ^ { 2 } } \mathbb { E } _ { c , c ^ { \prime } } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \mathbf { 1 } _ { F } \right] } \\ & { \qquad \ge ( 1 - \epsilon _ { z } ) ^ { 2 } \frac { 1 } { Z ^ { 2 } } \mathbb { E } _ { c , c ^ { \prime } } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \right] - | \mathcal { D } | \exp ( - \Omega ( \log ^ { 1 . 8 } n ) ) } \\ & { \qquad \ge ( 1 - \epsilon _ { z } ) ^ { 2 } \frac { 1 } { Z ^ { 2 } } \mathbb { E } _ { c , c ^ { \prime } } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \right] - \delta _ { 0 } } \end{array}
+$$
+
+Taking the logarithm of both sides, from (41) and (42), the multiplicative error translates to an additive error given by (43).
+
+$$
+\begin{array} { r l r } { \log p ( h , t \mid R ) = \log \left( \mathbb { E } _ { c , c ^ { \prime } } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \right] \pm \delta _ { 0 } \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) } & \\ { = \log \left( \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \right] \right] \pm \delta _ { 0 } \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) } & \\ { = \log \left( \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) A ( c ) \pm \delta _ { 0 } \right] \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) } & { ( 4 3 \pi ) } & \end{array}
+$$
+
+where $A ( c ) : = \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c ^ { \prime } \right) \right]$ .
+
+We assumed that $^ c$ and $c ^ { \prime }$ are on the unit sphere and $\mathbf { R } _ { 1 }$ and $\mathbf { R } _ { 2 }$ to be orthogonal matrices. Therefore, $\mathbf { R } _ { 1 } \boldsymbol { c }$ and $\mathbf { R } _ { 2 } { \pmb { c } } ^ { \prime }$ are also on the unit sphere. Moreover, if we let the upper bound of the $\ell _ { 2 }$ norm of the entity embeddings to be $\kappa ^ { \prime } \sqrt { d }$ , then we have $\| h \| \leq \kappa ^ { \prime } \sqrt { d }$ and $\| \pmb { t } \| \leq \kappa ^ { \prime } \sqrt { d }$ . Therefore, we have
+
+$$
+\langle R _ { 1 } h , c ^ { \prime } - c \rangle \leq \| h \| \| c - c ^ { \prime } \| \leq \kappa ^ { \prime } \sqrt { d } \| c - c ^ { \prime } \|
+$$
+
+Then we can lower bound $A ( c )$ as follows:
+
+$$
+\begin{array} { r l } & { A ( c ) = \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c \right) \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \exp \left( t ^ { \top } \mathbf { R } _ { 2 } ( c ^ { \prime } - c ) \right) \right] } \\ & { \qquad \leq \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c \right) \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \exp \left( \kappa ^ { \prime } \sqrt { d } \| c ^ { \prime } - c \| \right) \right] } \\ & { \qquad \leq \left( 1 + \epsilon _ { 2 } \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c \right) } \end{array}
+$$
+
+For some $\epsilon _ { 2 } > 0$ . The last inequality holds because
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { c | c ^ { \prime } } \left[ \exp \left( \kappa ^ { \prime } \sqrt { d } \| c ^ { \prime } - c \| \right) \right] = \displaystyle \int \exp \left( \kappa ^ { \prime } \sqrt { d } \| c ^ { \prime } - c \| \right) p ( c ^ { \prime } | c ) d c ^ { \prime } } \\ & { \quad \quad \quad = \underbrace { \exp ( \kappa ^ { \prime } \sqrt { d } ) } _ { \geq 1 } \underbrace { \int \exp ( \| c - c ^ { \prime } \| ) p ( c ^ { \prime } | c ) d c ^ { \prime } } _ { \geq 1 } } \\ & { \quad \quad = 1 + \epsilon _ { 2 } } \end{array}
+$$
+
+To obtain a lower bound on $A ( c )$ from the first-order Taylor approximation of $\exp ( x ) \geq 1 + x$ we observe that
+
+$$
+\mathbb { E } _ { c | c ^ { \prime } } \left[ \exp \left( \kappa ^ { \prime } \sqrt { d } \| c ^ { \prime } - c \| \right) \right] + \mathbb { E } _ { c | c ^ { \prime } } \left[ \exp \left( - \kappa ^ { \prime } \sqrt { d } \| c ^ { \prime } - c \| \right) \right] \geq 2 .
+$$
+
+Therefore, from our model assumptions we have
+
+$$
+\begin{array} { r } { \mathbb { E } _ { c | c ^ { \prime } } \left[ \exp \left( - \kappa ^ { \prime } \sqrt { d } \| c ^ { \prime } - c \| \right) \right] \geq 1 - \epsilon _ { 2 } } \end{array}
+$$
+
+Hence,
+
+$$
+\begin{array} { r l } & { A ( c ) = \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c \right) \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \exp \left( t ^ { \top } \mathbf { R } _ { 2 } ( c ^ { \prime } - c ) \right) \right] } \\ & { \qquad \geq \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c \right) \mathbb { E } _ { c ^ { \prime } \mid c } \left[ \exp \left( - \kappa ^ { \prime } \sqrt { d } \| c ^ { \prime } - c \| \right) \right] } \\ & { \qquad \geq \left( 1 - \epsilon _ { 2 } \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c \right) } \end{array}
+$$
+
+Therefore, from (46) and (49) we have
+
+$$
+A ( c ) = ( 1 \pm \epsilon _ { 2 } ) \exp \left( t ^ { \top } { \bf R } _ { 2 } c \right)
+$$
+
+Plugging $A ( c )$ back in (43) we obtain
+
+$$
+\begin{array} { r l } & { \mathrm { o g } p ( h , t \mid R ) = \log \left( \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) A ( c ) \pm \delta _ { 0 } \right] \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) } \\ & { \qquad = \log \left( \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \left( 1 \pm \epsilon _ { 2 } \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c \right) \pm \delta _ { 0 } \right] \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) } \\ & { \qquad = \log \left( \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c \right) \exp \left( t ^ { \top } \mathbf { R } _ { 2 } c \right) \pm \delta _ { 0 } \right] \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) + \log ( 1 \pm \epsilon _ { 2 } ) } \\ & { \qquad = \log \left( \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c + t ^ { \top } \mathbf { R } _ { 2 } c \right) \pm \delta _ { 0 } \right] \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) + \log ( 1 \pm \epsilon _ { 2 } ) } \\ & { \qquad = \log \left( \mathbb { E } _ { c } \left[ \exp \left( h ^ { \top } \mathbf { R } _ { 1 } c + t ^ { \top } \mathbf { R } _ { 2 } c \right) \pm \delta _ { 0 } \right] \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) + \log ( 1 \pm \epsilon _ { 2 } ) } \\ & { \qquad = \log \left( \mathbb { E } _ { c } \left[ \exp \left( \mathbf { R } _ { 1 } ^ { \top } h + \mathbf { R } _ { 2 } ^ { \top } t \right) ^ { \top } c \pm \delta _ { 0 } \right] \right) - 2 \log Z + 2 \log ( 1 \pm \epsilon _ { z } ) + \log ( 1 \pm \epsilon _ { 2 } ) } \end{array}
+$$
+
+Note that $^ c$ has a uniform distribution over the unit sphere. In this case, from Lemma A.5 in (Arora et al., 2016b), (52) holds approximately.
+
+$$
+\mathbb { E } _ { c } \left[ \exp \left( { \mathbf { R } _ { 1 } } ^ { \top } { \pmb { h } } + { \mathbf { R } _ { 2 } } ^ { \top } { \pmb { t } } \right) ^ { \top } { \pmb { c } } \right] = \left( 1 \pm \epsilon _ { 3 } \right) \exp \left( \frac { \| { \mathbf { R } _ { 1 } } ^ { \top } { \pmb { h } } + { \mathbf { R } _ { 2 } } ^ { \top } { \pmb { t } } \| ^ { 2 } } { 2 d } \right)
+$$
+
+where $\epsilon _ { 3 } = \tilde { O } ( 1 / d )$ . Plugging (52) in (51) we have that
+
+$$
+\log p ( h , t \mid R ) = \frac { \| \mathbf R _ { 1 } ^ { \top } h + \mathbf R _ { 2 } ^ { \top } t \| _ { 2 } ^ { 2 } } { 2 d } + O ( \epsilon _ { z } ) + O ( \epsilon _ { 2 } ) + O ( \epsilon _ { 3 } ) + O ( \delta _ { 0 } ^ { \prime } ) - 2 \log Z
+$$
+
+where $\delta _ { 0 } ^ { \prime } = \delta _ { 0 } \cdot \left( { \mathbb E } _ { c } \left[ \exp \left( ( { \mathbf R _ { 1 } } ^ { \top } { \pmb h } + { \mathbf R _ { 2 } } ^ { \top } { \pmb t } ) ^ { \top } { \pmb c } \right) \right] \right) ^ { - 1 } = \exp ( - \Omega ( \log ^ { 1 . 8 } n ) )$ . Therefore, $\delta _ { 0 } ^ { \prime }$ can be ignored. Note that $\epsilon _ { 3 } = \tilde { O } ( 1 / d )$ and $\epsilon _ { z } = { \tilde { O } } ( 1 / { \sqrt { n } } )$ by assumption. Therefore, we obtain that
+
+$$
+\log p ( h , t \mid \boldsymbol { R } ) = \frac { \| \mathbf { R } _ { 1 } ^ { \top } \boldsymbol { h } + \mathbf { R } _ { 2 } ^ { \top } \boldsymbol { t } \| _ { 2 } ^ { 2 } } { 2 d } + O ( \epsilon _ { z } ) + O ( \epsilon _ { 2 } ) + \tilde { O } ( 1 / d ) - 2 \log Z
+$$
+
+# B LEARNING WITH MULTIPLE NEGATIVE TRIPLES
+
+In this section, we show how the margin loss-based learning objective derived in section 3 can be extended to learn from more than one negative triples per each positive triple. This formulation leads to rank-based loss objective used in prior work on KGE. Considering that negative triples are generated via random perturbation, it is important to consider multiple negative triples during training to better estimate the classification boundary.
+
+Let us consider that we are given a positive triple, $( h , R , t )$ and a set of $K$ negative triples $\{ ( h _ { k } ^ { \prime } , R , t _ { k } ^ { \prime } ) \} _ { k = 1 } ^ { K }$ . We would like our model to assign a probability, $p ( h , t \mid R )$ , to the positive triple that is higher than that assigned to any of the negative triples. This requirement can be written as (55).
+
+$$
+p ( h , t | R ) \geq \operatorname* { m a x } _ { k = 1 , \ldots , K } p ( h _ { k } ^ { \prime } , t _ { k } ^ { \prime } \mid R )
+$$
+
+We could further require the ratio between the probability of the positive triple and maximum probability over all negative triples to be greater than a threshold $\eta \geq 1$ to make the requirement of (55) to be tighter.
+
+$$
+\frac { p ( h , t \mid R ) } { \underset { k = 1 , \dots , K } { \operatorname* { m a x } } p ( h _ { k } ^ { \prime } , t _ { k } ^ { \prime } \mid R ) } \geq \eta
+$$
+
+By taking the logarithm of (56) we obtain
+
+$$
+\log p ( h , t \mid R ) - \log \left( \operatorname* { m a x } _ { k = 1 , \ldots , K } p ( h _ { k } ^ { \prime } , t _ { k } ^ { \prime } \mid R ) \right) \geq \log ( \eta )
+$$
+
+Therefore, we can define the margin loss for a misclassification as follows:
+
+$$
+L \left( ( h , R , t ) , \{ ( h _ { k } ^ { \prime } , R , t _ { k } ^ { \prime } ) \} _ { k = 1 } ^ { K } \right) = \operatorname* { m a x } \left( 0 , \log \left( \operatorname* { m a x } _ { k = 1 , \ldots , K } p ( h _ { k } ^ { \prime } , t _ { k } ^ { \prime } \mid R ) \right) + \log ( \eta ) - \log p ( h , t \mid R ) \right)
+$$
+
+However, from the monotonicity of the logarithm we have $\forall x _ { 1 } , x _ { 2 } \ > \ 0$ , if $\log ( x _ { 1 } ) \geq \log ( x _ { 2 } )$ then $x _ { 1 } \geq x _ { 2 }$ . Therefore, the logarithm of the maximum can be replaced by the maximum of the logarithms in (58) as shown in (59).
+
+$$
+L \left( ( h , R , t ) , \{ ( h _ { k } ^ { \prime } , R , t _ { k } ^ { \prime } ) \} _ { k = 1 } ^ { K } \right) = \operatorname* { m a x } \left( 0 , \operatorname* { m a x } _ { k = 1 , \ldots , K } \log \left( p ( h _ { k } ^ { \prime } , t _ { k } ^ { \prime } \mid R ) \right) + \log ( \eta ) - \log p ( h , t \mid R ) \right)
+$$
+
+By substituting (18) for the probabilities in (59) we obtain the rank-based loss given by (60).
+
+$$
+\overset { : } { \underset { ( ( h , R , t ) ,  } { ( ( h , R , t ) ,  } \{ ( h _ { k } ^ { \prime } , R , t _ { k } ^ { \prime } ) \} _ { k = 1 } ^ { K } ) } = \operatorname* { m a x } ( 0 , 2 d \log ( \eta ) + \underset { k = 1 , \ldots , K } { \operatorname* { m a x } } \| \mathbf { R } _ { 1 } ^ { \top } h _ { k } ^ { \prime } + \mathbf { R } _ { 2 } ^ { \top } t _ { k } ^ { \prime } \| _ { 2 } ^ { 2 } - \| \mathbf { R } _ { 1 } ^ { \top } h + \mathbf { R } _ { 2 } ^ { \top } t \| _ { 2 } ^ { 2 } )
+$$
+
+In practice, we can use $p ( h _ { k } ^ { \prime } , t _ { k } ^ { \prime } \mid { \cal R } )$ to select the negative triple with the highest probability for training with the positive triple.
+
+Table 4: Statistics of the datasets   
+
+<table><tr><td>Dataset</td><td>Relations</td><td>Entities</td><td>Train</td><td>Test</td><td>Validation</td></tr><tr><td>FB15K</td><td>1,345</td><td>14,951</td><td>483,142</td><td>59,071</td><td>50,000</td></tr><tr><td>FB15K237</td><td>237</td><td>14,541</td><td>272,115</td><td>17,535</td><td>20,466</td></tr><tr><td>WN18</td><td>18</td><td>40,943</td><td>141,442</td><td>5,000</td><td>5,000</td></tr><tr><td>WN18RR</td><td>11</td><td>40,943</td><td>86.835</td><td>3,134</td><td>3,034</td></tr><tr><td>WN11</td><td>11</td><td>38,588</td><td>112,581</td><td>10,544</td><td>2,609</td></tr><tr><td>FB13</td><td>13</td><td>75,043</td><td>316,232</td><td>23,733</td><td>5,908</td></tr></table>
+
+# C TRAINING DETAILS
+
+The statistics of the benchmark datasets are show in Table 4.
+
+We selected the initial learning rate $( \alpha )$ for SGD in $\{ 0 . 0 1 , 0 . 0 0 1 \}$ , the regularisation coefficients $( \lambda _ { 1 } , \lambda _ { 2 } )$ for the orthogonality constraints of relation matrices in $\{ 0 , 1 , 1 0 , 1 0 0 \}$ . The number of randomly generated negative triples $n _ { \mathrm { n { e g } } }$ for each positive example is varied in $\{ 1 , 1 0 , 2 0 , 5 0 , 1 0 0 \}$ and $d \in \{ 5 0 , 1 0 0 \}$ . Optimal hyperparameter settings were: $\lambda _ { 1 } = \lambda _ { 2 } = 1 0$ , $n _ { \mathrm { n { e g } } } = 1 0 0$ for all the datasets, $\alpha = 0 . 0 0 1$ for FB15K, FB15K237 and FB13, $\alpha = 0 . 0 1$ for WN18, WN18RR and WN11. For FB15K237 and WN18RR $d = 1 0 0$ was the best, whereas for all other datasets $d = 5 0$ performed best.

parse/train/ebS5NUfoMKL/ebS5NUfoMKL_content_list.json

@@ -0,0 +1,1922 @@
+[
+    {
+        "type": "text",
+        "text": "BOOST THEN CONVOLVE: GRADIENT BOOSTING MEETS GRAPH NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            759,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sergei Ivanov   \nCriteo AI Lab; Skoltech   \nParis, France   \ns.ivanov@criteo.com ",
+        "bbox": [
+            183,
+            170,
+            372,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Liudmila Prokhorenkova Yandex; HSE University; MIPT Moscow, Russia ostroumova-la@yandex-team.ru ",
+        "bbox": [
+            464,
+            170,
+            740,
+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph neural networks (GNNs) are powerful models that have been successful in various graph representation learning tasks. Whereas gradient boosted decision trees (GBDT) often outperform other machine learning methods when faced with heterogeneous tabular data. But what approach should be used for graphs with tabular node features? Previous GNN models have mostly focused on networks with homogeneous sparse features and, as we show, are suboptimal in the heterogeneous setting. In this work, we propose a novel architecture that trains GBDT and GNN jointly to get the best of both worlds: the GBDT model deals with heterogeneous features, while GNN accounts for the graph structure. Our model benefits from endto-end optimization by allowing new trees to fit the gradient updates of GNN. With an extensive experimental comparison to the leading GBDT and GNN models, we demonstrate a significant increase in performance on a variety of graphs with tabular features. The code is available: https://github.com/nd7141/bgnn. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "text": "Graph neural networks (GNNs) have shown great success in learning on graph-structured data with various applications in molecular design (Stokes et al., 2020), computer vision (Casas et al., 2019), combinatorial optimization (Mazyavkina et al., 2020), and recommender systems (Sun et al., 2020). The main driving force for progress is the existence of canonical GNN architecture that efficiently encodes the original input data into expressive representations, thereby achieving high-quality results on new datasets and tasks. ",
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+        "text": "Recent research has mostly focused on GNNs with sparse data representing either homogeneous node embeddings (e.g., one-hot encoded graph statistics) or bag-of-words representations. Yet tabular data with detailed information and rich semantics among nodes in the graph are more natural for many situations and abundant in real-world AI (Xiao et al., 2019). For example, in a social network, each person has socio-demographic characteristics (e.g., age, gender, date of graduation), which largely vary in data type, scale, and missing values. GNNs for graphs with tabular data remain unexplored, with gradient boosted decision trees (GBDTs) largely dominating in applications with such heterogeneous data (Bentejac et al., 2020). ´ ",
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+        "text": "GBDTs are so successful for tabular data because they possess certain properties: (i) they efficiently learn decision space with hyperplane-like boundaries that are common in tabular data; (ii) they are well-suited for working with variables of high cardinality, features with missing values, and of different scale; (iii) they provide qualitative interpretation for decision trees (e.g., by computing decrease in node impurity for every feature) or for ensembles via post-hoc analysis stage (Kaur et al., 2020); (iv) in practical applications, they mostly converge faster even for large amounts of data. ",
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+        "type": "text",
+        "text": "In contrast, a crucial feature of GNNs is that they take into account both the neighborhood information of the nodes and the node features to make a prediction, unlike GBDTs that require additional preprocessing analysis to provide the algorithm with graph summary (e.g., through unsupervised graph embeddings (Hu et al., 2020a)). Moreover, it has been shown theoretically that message-passing GNNs can compute any function on its graph input that is computable by a Turing machine, i.e., GNN is known to be the only learning architecture that possesses universality properties on graphs (approximation (Keriven & Peyre, 2019; Maron et al., 2019) and computability (Loukas, 2020)). ´ ",
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+        "type": "text",
+        "text": "Furthermore, gradient-based learning of neural networks can have numerous advantages over the treebased approach: (i) relational inductive bias imposed in GNNs alleviates the need to manually engineer features that capture the topology of the network (Battaglia et al., 2018); (ii) the end-to-end nature of training neural networks allows multi-stage (Fey et al., 2019) or multi-component (Wang et al., 2020) integration of GNNs in application-dependent solutions; (iii) pretraining representations with graph networks enriches transfer learning for many valuable tasks such as unsupervised domain adaptation (Wu et al., 2020), self-supervised learning (Hu et al., 2020b), and active learning regimes (Satorras & Estrach, 2018). ",
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+        "text": "Undoubtedly, there are major benefits in both GBDT and GNN methods. Is it possible to get advantages of both worlds? All previous approaches (Arik & Pfister, 2020; Popov et al., 2019; Badirli et al., 2020) that attempt to combine gradient boosting and neural networks are computationally heavy, do not consider graph-structured data, and suffer from the lack of relational bias imposed in GNN architectures, see Appendix A for a more detailed comparison with related literature. To the best of our knowledge, the current work is the first to explore using GBDT models for graph-structured data. ",
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+        "text": "In this paper, we propose a novel learning architecture for graphs with tabular data, BGNN, that combines GBDT’s learning on tabular node features with GNN that refines the predictions utilizing the graph’s topology. This allows BGNN to inherit the advantages of gradient boosting methods (heterogeneous learning and interpretability) and graph networks (representation learning and end-toend training). Overall, our contributions are the following: ",
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+        "text": "(1) We design a novel generic architecture that combines GBDT and GNN into a unique pipeline. To the best of our knowledge, this is the first work that systematically studies the application of GBDT to graph-structured data.   \n(2) We overcome the challenge of end-to-end training of GBDT by iteratively adding new trees that fit the gradient updates of GNN. This allows us to backpropagate the error signal from the topology of the network to GBDT.   \n(3) We perform an extensive evaluation of our approach against strong baselines in node prediction tasks. Our results consistently demonstrate significant performance improvements on heterogeneous node regression and node classification tasks over a variety of real-world graphs with tabular data.   \n(4) We show that our approach is also more efficient than the state-of-the-art GNN models due to much faster loss convergence during training. Furthermore, learned representations exhibit discernible structure in the latent space, which further demonstrates the expressivity of our approach. ",
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+        "text": "2 BACKGROUND ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Let $G = ( V , E )$ be a graph with nodes having features and target labels. In node prediction tasks (classification or regression), some target labels are known, and the goal is to predict the remaining ones. Throughout the text, by lowercase variables $\\mathbf { x } _ { v }$ $w \\in V ,$ ) or $\\mathbf { x }$ we denote features of individual nodes, and $\\mathbf { X }$ represents the matrix of all features for $v \\in V$ . Individual target labels are denoted by $y _ { v }$ , while $Y$ is the vector of known labels. ",
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+        "type": "text",
+        "text": "Graph Neural Networks (GNNs) use both the network’s connectivity and the node features to learn latent representations for all nodes $v \\in V$ . Many popular GNNs use a neighborhood aggregation approach, also called the message-passing mechanism, where the representation of a node $v$ is updated by applying a non-linear aggregation function of $v$ ’s neighbors representation (Fey & Lenssen, 2019). Formally, GNN is a differentiable, permutation-invariant function $g _ { \\boldsymbol { \\theta } } : ( G , \\mathbf { X } ) \\mapsto { \\widehat { Y } }$ , where $\\widehat { Y }$ is the vector of predicted labels. Similar to traditional neural networks, GNNs are composed of multiple layers, each representing a non-linear message-passing function: ",
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+        "type": "equation",
+        "img_path": "images/e95f7e0062b74a444395f5b318a15dde0d827b1c5c2c4516caf8d3923bbd79fe.jpg",
+        "text": "$$\n\\mathbf { x } _ { v } ^ { t } = \\mathbf { C O M B I N E } ^ { t } \\left( \\mathbf { x } _ { v } ^ { t - 1 } , \\mathbf { A G G R E G A T E } ^ { t } \\left( \\left\\{ \\left( \\mathbf { x } _ { w } ^ { t - 1 } , \\mathbf { x } _ { v } ^ { t - 1 } \\right) : \\left( w , v \\right) \\in E \\right\\} \\right) \\right) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\mathbf { x } _ { v } ^ { t }$ is the representation of node $v$ at layer $t$ , and COMBINEt and AGGREGATEt are (parametric) functions that aggregate representations from the local neighborhood of a node. Then, the GNN mapping $g _ { \\theta }$ includes multiple layers of aggregation (1). Parameters of GNN model $\\theta$ are optimized with gradient descent by minimizing an empirical loss function ${ \\cal L } _ { \\mathrm { G N N } } ( Y , g _ { \\theta } ( G , { \\bf X } ) )$ . ",
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+        "text": "Gradient Boosted Decision Trees (GBDT) is a well-known and widely used algorithm that is defined on non-graph tabular data (Friedman, 2001) and is particularly successful for tasks containing heterogeneous features and noisy data. ",
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+        "type": "text",
+        "text": "The core idea of gradient boosting is to construct a strong model by iteratively adding weak ones (usually decision trees). Formally, at each iteration $t$ of the gradient boosting algorithm, the model $f ( \\mathbf { x } )$ is updated in an additive manner: ",
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+    {
+        "type": "equation",
+        "img_path": "images/17610f16749cbcf09acb872f8b5d4107bf75dd55c40291419114a59025ed2d8a.jpg",
+        "text": "$$\nf ^ { t } ( { \\bf { x } } ) = f ^ { t - 1 } ( { \\bf { x } } ) + \\epsilon h ^ { t } ( { \\bf { x } } ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $f ^ { t - 1 }$ is a model constructed at the previous iteration, $h ^ { t }$ is a weak learner that is chosen from some family of functions $\\mathcal { H }$ , and $\\epsilon$ is a learning rate. The weak learner $h ^ { t } \\in \\mathcal { H }$ is chosen to approximate the negative gradient of a loss function $L$ w.r.t. the current model’s predictions: ",
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+        "type": "equation",
+        "img_path": "images/f6ad4c0f43afa849c7685f973cf39b3e473e75a57c7c775999324d8d683f0da8.jpg",
+        "text": "$$\nh ^ { t } = \\underset { h \\in \\mathcal { H } } { \\arg \\operatorname* { m i n } } \\sum _ { i } \\left( - \\frac { \\partial L ( f ^ { t - 1 } ( \\mathbf { x } _ { i } ) , y _ { i } ) } { \\partial f ^ { t - 1 } ( \\mathbf { x } _ { i } ) } - h ( \\mathbf { x } _ { i } ) \\right) ^ { 2 } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The gradient w.r.t. the current predictions indicates how one should change these predictions to improve the loss function. Informally, gradient boosting can be thought of as performing gradient descent in functional space. ",
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+        "type": "text",
+        "text": "The set of weak learners $\\mathcal { H }$ is usually formed by shallow decision trees. Decision trees are built by a recursive partition of the feature space into disjoint regions called leaves. This partition is usually constructed greedily to minimize the loss function (3). Each leaf $R _ { j }$ of the tree is assigned to a value $a _ { j }$ , which estimates the response $y$ in the corresponding region. In our case, $a _ { j }$ is equal to the average negative gradient value in the leaf $R _ { j }$ . To sum up, we can write $\\begin{array} { r } { h ( x ) = \\sum _ { j } \\overset { \\cdot } { a } _ { j } 1 _ { \\{ x \\in R _ { j } \\} } } \\end{array}$ . ",
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+        "type": "text",
+        "text": "3 GBDT MEETS GNN ",
+        "text_level": 1,
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+        "img_path": "images/1e72d21cce476df0095b8d7236f1fd696a8142e753ee8e141cecc0ce800b76d0.jpg",
+        "image_caption": [
+            "Figure 1: Training of BGNN, steps for one epoch are numbered. "
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+        "table_body": "<table><tr><td>Algorithm1TrainingofBGNN</td></tr><tr><td>Input: Graph G, node features X, targets Y Initialize GBDT targets V= Y for epoch i= 1 to N do # Train k trees of GBDT with eq. (2)-(3) fi← arg min LGBDT(fi(X),)) k fi f←f+fi # Train l steps of GNN on new node features</td></tr></table>",
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+        "type": "text",
+        "text": "Gradient boosting approach is successful for learning on tabular data; however, there are challenges of applying GBDT on graph-structured data: (i) how to propagate relational signal, in addition to node features, to otherwise inherently tabular model; and (ii) how to train it together with GNN in an end-to-end fashion. Indeed, optimizations of GBDT and GNN follow different approaches: the parameters of GNN are optimized via gradient descent, while GBDT is constructed iteratively, and the decision trees remain fixed after being built (decision trees are based on hard splits of the feature space, which makes them non-differentiable). ",
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+        "type": "text",
+        "text": "A straightforward approach would be to train the GBDT model only on the node features and then use the obtained predictions, jointly with the original input, as new node features for GNN. In this case, the graph-insensitive predictions of GBDT will further be refined by a graph neural network. ",
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+        "type": "text",
+        "text": "This approach (which we call Res-GNN) can already boost the performance of GNN for some tasks. However, in this case, the GBDT model completely ignores the graph structure and may miss descriptive features of the graph, providing inaccurate input data to GNN. ",
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+        "type": "text",
+        "text": "In contrast, we propose end-to-end training of GBDT and GNN called BGNN (for Boost-GNN). As before, we first apply GBDT and then GNN, but now we optimize both of them, taking into account the quality of final predictions. The training of BGNN is shown in Figure 1. Recall that one cannot tune already built decision trees due to their discrete structure, so we iteratively update the GBDT model by adding new trees that approximate the GNN loss function. ",
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+        "text": "In Algorithm 1, we present the training of BGNN that combines GBDT and GNN for any node-level prediction problem such as semi-supervised node regression or classification. In the first iteration, we build a GBDT model $f ^ { 1 } ( \\mathbf { x } )$ with $k$ decision trees by minimizing the loss function $L _ { \\mathrm { G B D T } } ( f ^ { 1 } ( \\mathbf { x } ) , y )$ (e.g., RMSE for regression or cross-entropy for classification) averaged over the train nodes, following the equations (2)-(3). Using all predictions $f ^ { 1 } ( { \\mathbf { X } } )$ , we update the node features to $\\mathbf { X } ^ { \\prime }$ that we pass to GNN. Possible update functions that we experiment with include concatenation with the original node features and their replacement by $f ^ { 1 } ( { \\mathbf { X } } )$ . Next, we train a graph neural network $g _ { \\boldsymbol { \\theta } }$ on a graph $G$ with node features $\\mathbf { X } ^ { \\prime }$ by minimizing $L _ { \\mathrm { G N N } } ( g _ { \\boldsymbol { \\theta } } ( G , \\mathbf { X } ^ { \\prime } ) , Y )$ with $l$ steps of gradient descent.1 Importantly, we optimize both the parameters $\\theta$ of GNN and the node features $\\mathbf { X } ^ { \\prime }$ . Then, we use the difference between the optimized node features $\\mathbf { X } _ { n e w } ^ { \\prime }$ and the input node features $\\mathbf { X } ^ { \\prime } = f ^ { 1 } ( \\mathbf { X } )$ as the target for the next decision trees built by GBDT. If $l = 1$ , the difference $\\mathbf { X } _ { n e w } ^ { \\prime } - \\mathbf { X } ^ { \\prime }$ exactly equals the negative gradient of the loss function w.r.t. the input features $\\mathbf { X } ^ { \\prime }$ multiplied by the learning rate $\\eta$ : ",
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+    {
+        "type": "equation",
+        "img_path": "images/973cc899045eefd0aa3ffbd1dfdfadeba6f6efae42286d192688d934635f7cf9.jpg",
+        "text": "$$\n{ \\bf { X } } _ { n e w } ^ { \\prime } = { \\bf { X } } ^ { \\prime } - \\eta \\frac { { \\partial { L _ { \\mathrm { { G N N } } } } { \\left( { { g \\theta \\left( { G , { \\bf { X } } ^ { \\prime } } \\right) } , Y } \\right) } } } { { \\partial { \\bf { X } } ^ { \\prime } } } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In the second iteration, we train a new GBDT model $f ^ { 2 }$ with the original input features $\\mathbf { X }$ but new target labels: $\\mathbf { X } _ { n e w } ^ { \\prime } - \\mathbf { X } ^ { \\prime }$ . Intuitively, $f ^ { 2 }$ fits the direction that would improve GNN prediction based on the first predictions $f ^ { 1 } ( { \\mathbf { X } } )$ . In other words, GBDT approximates the gradient steps made by GNN for the node features $\\mathbf { X } ^ { \\prime }$ . This is a regression problem, so here $L _ { \\mathrm { G B D T } }$ is the RMSE loss. ",
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+        "type": "text",
+        "text": "After $f ^ { 2 }$ is trained, we combine the predictions $f ( \\mathbf { X } ) = f ^ { 1 } ( \\mathbf { X } ) + f ^ { 2 } ( \\mathbf { X } )$ and pass the obtained values $\\mathbf { X } ^ { \\prime }$ to GNN as node features. GNN model $g _ { \\theta }$ again does $l$ steps of backpropagation and passes the new difference $\\mathbf { X } _ { n e w } ^ { \\prime } - \\mathbf { X } ^ { \\prime }$ as a target to the next iteration of GBDT. In total, the model is trained for $N$ epochs and outputs a GBDT model $f : \\mathbf { X } \\mapsto Y$ and GNN model $g _ { \\boldsymbol { \\theta } } : ( G , \\mathbf { X } ) \\mapsto Y$ , which can be used for downstream tasks. ",
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+        "type": "text",
+        "text": "Intuitively, BGNN model consists of two consecutive blocks, GBDT and GNN, which are trained end-to-end, and therefore can be interpreted from two angles: GBDT as an embedding layer for GNN or GNN as a parametric loss function for GBDT. In the former case, GBDT transforms the original input features $\\mathbf { X }$ to new node features $\\mathbf { X } ^ { \\prime }$ , which are then passed to GNN. In the latter case, one can see BGNN as a standard gradient boosted training where GNN acts as a complex loss function that depends on the graph topology. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "We have performed a comparative evaluation of BGNN and Res-GNN against a wide variety of strong baselines and previous approaches on heterogeneous node prediction problems, achieving significant improvement in performance across all of them. This section outlines our experimental setting, the results on node regression and classification problems, and extracted feature representations. ",
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+        "type": "text",
+        "text": "In our first experiments, we want to answer two questions: ",
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+        "type": "text",
+        "text": "Q1 Does combination of GBDT and GNN lead to better qualitative results in heterogeneous node regression and classification problems?   \nQ2 Is the end-to-end training proposed in Algorithm 1 better than a combination of pretrained GBDT with GNN? ",
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+        "text": "To answer these questions, we consider several strong baselines among GBDTs, GNNs, and pure neural networks. CatBoost is a recent GBDT implementation (Prokhorenkova et al., 2018) that uses oblivious trees as weak learners. LightGBM is another GBDT model (Ke et al., 2017) that is used extensively in ML competitions. Among GNNs, we tested four state-of-the-art recent models that showed superior performance in node prediction tasks: GAT (Velickovi ˇ c et al., 2018), ´ GCN (Kipf & Welling, 2017), AGNN (Thekumparampil et al., 2018), APPNP (Klicpera et al., 2019). Additionally, we test the performance of fully-connected neural network FCNN and its end-to-end combination with GNNs, FCNN-GNN. ",
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+    {
+        "type": "text",
+        "text": "We compare these baselines against two proposed approaches: the end-to-end BGNN model and not end-to-end Res-GNN. The BGNN model follows Algorithm 1 and builds each tree approximating the GNN error in the previous iteration. In contrast, Res-GNN first trains a GBDT model on the training set of nodes and then either appends its predictions for all nodes to the original node features or replaces the original features with the GBDT predictions, after which GNN is trained on the updated features, and GNN’s predictions are used to calculate metrics. Hence, Res-GNN is a twostage approach where the training of GBDT is independent of GNN. On the other hand, BGNN trains GBDT and GNN simultaneously in an end-to-end fashion. In most of our experiments, the GNN-component of FCNN-GNN, Res-GNN, and BGNN is based on GAT, while in Section 4.3 we analyze consistency of improvements across different GNN models. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We ensure that the comparison is done fairly by training each model until the convergence with a reasonable set of hyperparameters evaluated on the validation set. We run each hyperparameter setting three times and take the average of the results. Furthermore, we have five random splits of the data, and the final number represents the average performance of the model for all five random seeds. More details about hyperparameters can be found in Appendix B. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 NODE REGRESSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+            452,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We utilize five real-world node regression datasets with different properties outlined in Table 1. Four of these datasets are heterogeneous, i.e., the input features are of different types, scales, and meaning. For example, for the VK dataset, the node features are both numerical (e.g., last time seen on the platform) and categorical (e.g., country of living and university). On the other hand, Wiki dataset is homogeneous, i.e., the node features are interdependent and correspond to the bag-ofwords representations of Wikipedia articles. Additional details about the datasets can be found in Appendix C. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1f520086c7d9e559470cff5e551e4e8e4be20fde1de94c6f2da907fe639333c5.jpg",
+        "table_caption": [
+            "Table 1: Summary of regression datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>House</td><td>County</td><td>VK</td><td>Avazu</td><td>Wiki</td></tr><tr><td>Setting</td><td>Heterogeneous</td><td>Heterogeneous</td><td>Heterogeneous</td><td>Heterogeneous</td><td>Homogeneous</td></tr><tr><td>#Nodes</td><td>20640</td><td>3217</td><td>54028</td><td>1297</td><td>5201</td></tr><tr><td>#Edges</td><td>182146</td><td>12684</td><td>213644</td><td>54364</td><td>198493</td></tr><tr><td>#Features/Node</td><td>6</td><td>7</td><td>14</td><td>9</td><td>3148</td></tr><tr><td>Mean Target</td><td>2.06</td><td>5.44</td><td>35.47</td><td>0.08</td><td>27923.86</td></tr><tr><td>Min Target</td><td>0.14</td><td>1.7</td><td>13.48</td><td>0</td><td>16</td></tr><tr><td>Max Target</td><td>5.00</td><td>24.1</td><td>118.39</td><td>1</td><td>849131</td></tr><tr><td>Median Target</td><td>1.79</td><td>5</td><td>33.83</td><td>0</td><td>9225</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "4.1.2 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The results of our comparative evaluation for node regression are summarized in Table 2. We report the mean RMSE (with standard deviation) on the test set and the relative gap between RMSE of the GAT model (Velickovi ˇ c et al., 2018) and other methods, i.e., ´ ${ \\mathrm { g a p } } = ( r _ { m } - r _ { g n n } ) / r _ { g n n }$ , where $r _ { m }$ and $r _ { g n n }$ are RMSE of that model and of GAT, respectively. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Our results demonstrate significant improvement of BGNN over the baselines. In particular, in the heterogeneous case, BGNN achieves $8 \\%$ , $14 \\%$ , $4 \\%$ , and $4 \\%$ reduction of the error for House, County, VK, and Avazu datasets, respectively. Res-GNN model that uses a pretrained CatBoost model for the input of GNN also decreases RMSE, although not as much as the end-to-end model ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/c9c09465a3443da63a04eb156711c13ecf92ae1de68fe5b9b739b8938b05e206.jpg",
+        "table_caption": [
+            "Table 2: Summary of our results for node regression. Gap $\\%$ is relative difference w.r.t. GAT RMSE (the smaller the better). Top-2 results are highlighted in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"6\">Heterogeneous</td><td colspan=\"3\">Homogeneous</td></tr><tr><td colspan=\"2\"></td><td colspan=\"2\">House</td><td colspan=\"2\">County</td><td colspan=\"2\">VK Gap%</td><td colspan=\"2\">Avazu</td><td colspan=\"2\">Wiki Gap%</td></tr><tr><td>Method</td><td></td><td>RMSE</td><td>Gap%</td><td>RMSE</td><td>Gap %</td><td>RMSE</td><td></td><td>RMSE</td><td>Gap %</td><td>RMSE</td></tr><tr><td>GEPT</td><td>CatBoost LightGBM</td><td>0.63 ± 0.01</td><td>15.3</td><td>1.39 ± 0.07</td><td>-4.32</td><td>7.16 ± 0.20 7.2 ± 0.21</td><td>-0.82 -0.33</td><td>0.1172 ± 0.02</td><td>3.36 46359 ± 4508</td><td>0.97</td></tr><tr><td></td><td></td><td>0.63 ± 0.01</td><td>15.98</td><td>1.4 ± 0.07</td><td>-3.93</td><td></td><td>0.1171 ± 0.02</td><td>3.27</td><td>49915 ± 3643</td><td>8.71</td></tr><tr><td>GAT</td><td></td><td>0.54± 0.01</td><td>0</td><td>1.45 ± 0.06</td><td>0</td><td>7.22 ± 0.19</td><td>0 0.1134 ± 0.01</td><td>0</td><td>45916 ± 4527</td><td>0</td></tr><tr><td>NNO</td><td>GCN</td><td>0.63 ± 0.01</td><td>16.77</td><td>1.48 ± 0.08</td><td>2.06</td><td>7.25 ± 0.19</td><td>0.34 0.1141 ± 0.02</td><td>0.58</td><td>44936 ± 4083</td><td>-2.14</td></tr><tr><td></td><td>AGNN</td><td>0.59 ± 0.01</td><td>8.01</td><td>1.45 ± 0.08</td><td>-0.19 7.26 ± 0.20</td><td>0.54</td><td>0.1134 ± 0.02</td><td>-0.02</td><td>45982 ± 3058</td><td>0.14</td></tr><tr><td>APPNP</td><td></td><td>0.69 ± 0.01</td><td>27.11</td><td>1.5 ± 0.11</td><td>3.39 13.23 ± 0.12</td><td>83.19</td><td>0.1127 ± 0.01</td><td>-0.65</td><td>53426 ± 4159</td><td>16.36</td></tr><tr><td>M</td><td>FCNN</td><td>0.68 ±0.02</td><td>25.49</td><td>1.48 ± 0.07</td><td>1.56</td><td>7.29 ± 0.21</td><td>1.02 0.118 ± 0.02</td><td>4.07</td><td>51662± 2983</td><td>12.51</td></tr><tr><td></td><td>FCNN-GNN</td><td>0.53 ± 0.01</td><td>-2.48</td><td>1.39 ± 0.06</td><td>-4.68</td><td>7.22 ± 0.20</td><td>0.01 0.1114 ± 0.02</td><td>-1.82</td><td>48491± 7889</td><td>5.61</td></tr><tr><td></td><td>Res-GNN</td><td>0.51 ± 0.01</td><td>-6.39</td><td>1.33 ± 0.08</td><td>-8.35</td><td>7.07 ± 0.20 -2.04</td><td>0.1095 ± 0.01</td><td>-3.42</td><td>46747 ± 4639</td><td>1.81</td></tr><tr><td></td><td>BGNN</td><td>0.5 ± 0.01</td><td>-8.15</td><td>1.26 ± 0.08</td><td>-13.67</td><td>6.95 ± 0.21</td><td>-3.8 0.109 ± 0.01</td><td>-3.9</td><td>49222 ± 3743</td><td>7.2</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "BGNN. In the homogeneous dataset Wiki, CatBoost and, subsequently, Res-GNN and BGNN are outperformed by the GNN model. Intuitively, when the features are homogeneous, neural network approaches are sufficient to attain the best results. This shows that BGNN leads to better qualitative results and its end-to-end training outperforms other approaches in node prediction tasks for graphs with heterogeneous tabular data. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We can also observe that the end-to-end combination FCNN-GNN often leads to better performance than pure GNN. However, its improvement is smaller than for BGNN which uses the advantages of GBDT models. Moreover, CatBoost and LightGBM can be effective on their own, but their performance is not stable across all datasets. Overall, these experiments demonstrate the superiority of BGNN against other strong models. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.2 NODE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For node classification, we use five datasets with different properties. Due to the lack of publicly available datasets with heterogeneous node features, we adopt the datasets House class and VK class from the regression task by converting the target labels into several discrete classes. We additionally include two sparse node classification datasets SLAP and DBLP coming from heterogeneous information networks (HIN) with nodes having different types. We also include one homogeneous dataset OGB-ArXiv (Hu et al., 2020a). In this dataset, the node features correspond to a 128-dimensional feature vector obtained by averaging the embeddings of words in the title and abstract. Hence, the features are not heterogeneous, and therefore GBDT is not expected to outperform neural network approaches. More details about these datasets can be found in Appendix D. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/092c4a448a25cb39e5c87e996bc145bff52212a38fc5344054c7b0a129ce4383.jpg",
+        "table_caption": [
+            "Table 3: Summary of our results for node classification. Gap $\\%$ is the relative difference w.r.t. GAT accuracy (the higher the better). Top-2 results are highlighted in bold. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\"></td><td colspan=\"6\">Heterogeneous</td><td colspan=\"3\">Homogeneous</td></tr><tr><td colspan=\"2\">Method</td><td colspan=\"2\">House_class</td><td colspan=\"2\">VK_class</td><td colspan=\"2\">Slap Acc.</td><td colspan=\"2\">DBLP Acc.</td><td colspan=\"2\">OGB-ArXiv Acc. Gap%</td></tr><tr><td colspan=\"2\">CatBoost</td><td>Acc.</td><td>Gap %</td><td>Acc.</td><td>Gap % -1.26</td><td>0.922 ± 0.01</td><td>Gap % 15.12</td><td></td><td>Gap% -5.42</td><td></td><td>-36.35</td></tr><tr><td colspan=\"2\">GERI</td><td>0.52 ± 0.01 0.55 ± 0.00</td><td>-16.82 -11.98</td><td>0.57± 0.01 0.579 ± 0.01</td><td>0.26</td><td>0.963 ± 0.00</td><td>20.3</td><td>0.759 ± 0.03 0.913 ± 0.01</td><td>13.73</td><td>0.45 0.51</td><td>-26.97</td></tr><tr><td rowspan=\"4\">NNO</td><td>LightGBM</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.802 ± 0.01</td><td>0</td><td></td><td></td></tr><tr><td>GAT</td><td>0.625 ± 0.00</td><td>0</td><td>0.577 ± 0.00</td><td>0</td><td>0.801 ± 0.01</td><td>0</td><td></td><td></td><td>0.70</td><td>0</td></tr><tr><td>GCN</td><td>0.6±0.00</td><td>-3.98</td><td>0.574± 0.00</td><td>-0.6</td><td>0.878 ± 0.01</td><td>9.72</td><td>0.428 ± 0.04</td><td>-46.6</td><td>-</td><td>-</td></tr><tr><td>AGNN APPNP</td><td>0.614 ± 0.01 0.619 ± 0.00</td><td>-1.73</td><td>0.572 ± 0.00 0.573 ± 0.00</td><td>-0.79 -0.67</td><td>0.892 ± 0.01 0.895 ± 0.01</td><td>11.47 11.79</td><td>0.794 ± 0.01 0.83±0.02</td><td>-1.02 3.47</td><td>=</td><td>-</td></tr><tr><td rowspan=\"3\"></td><td></td><td></td><td>-0.89</td><td></td><td></td><td></td><td></td><td></td><td></td><td>-</td><td>-</td></tr><tr><td>FCNN</td><td>0.534 ± 0.01</td><td>-14.53</td><td>0.567 ± 0.01</td><td>-1.72</td><td>0.759 ±0.04</td><td>-5.24</td><td>0.623 ± 0.02</td><td>-22.3 0.94</td><td>0.50</td><td>-28.91</td></tr><tr><td>FCNN-GNN</td><td>0.64± 0.00</td><td>2.36</td><td>0.589 ± 0.00</td><td>2.13</td><td>0.89 ± 0.01</td><td>11.11</td><td>0.81 ± 0.01</td><td></td><td>0.71</td><td>0.54</td></tr><tr><td rowspan=\"2\"></td><td>Res-GNN</td><td>0.625 ± 0.01</td><td>-0.06</td><td>0.603 ± 0.00</td><td>4.45</td><td>0.905 ± 0.01</td><td>13.06</td><td>0.892 ± 0.01</td><td>11.11</td><td>0.70</td><td>-0.33</td></tr><tr><td>BGNN</td><td>0.682 ± 0.00</td><td>9.18</td><td>0.683 ± 0.00</td><td>18.3</td><td>0.95 ± 0.00</td><td>18.61</td><td>0.889 ± 0.01</td><td>10.77</td><td>0.67</td><td>-4.36</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As can be seen from Table 3, on the datasets with heterogeneous tabular features (House class and VK class), BGNN outperforms other approaches with a significant margin. For example, for the ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "VK class dataset BGNN achieves more than $18 \\%$ of relative increase in accuracy. This demonstrates that learned representations of GBDT together with GNN can be equally useful for node classification setting on data with heterogeneous features. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The other two datasets, Slap and DBLP, have sparse bag-of-words features that are particularly challenging for the GNN model. On these two datasets, GBDT is the strongest baseline. Moreover, since FCNN outperforms GNN, we conclude that graph structure does not help, hence BGNN is not supposed to beat GBDT. This is indeed the case: the final accuracy of BGNN is slightly worse than that of GBDT. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In the homogeneous OGB-ArXiv dataset, FCNN-GNN and GNN achieve the top performance followed by Res-GNN and BGNN models.2 In a nutshell, GBDT does not learn good predictions on the homogeneous input features and therefore reduces the discriminative power of GNN. Both cases, with sparse and with homogeneous features, show that the performance of BGNN is on par or higher than of GNN; however, lacking heterogeneous structure in the data may make the joint training of GBDT and GNN redundant. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 CONSISTENCY ACROSS DIFFERENT GNN MODELS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Seeing that our models perform significantly better than strong baselines on various datasets, we want to test whether the improvement is consistent if different GNN models are used. Thus, we ask: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Q3 Do different GNN models benefit from our approach of combination with GBDT? ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To answer this question, we compare GNN models that include GAT (Velickovi ˇ c et al., 2018), ´ GCN (Kipf & Welling, 2017), AGNN (Thekumparampil et al., 2018), and APPNP (Klicpera et al., 2019). We substitute each of these models to Res-GNN and BGNN and measure the relative change in performance with respect to the original GNN’s performance. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/5fc0f16cd9b806d268431dd1a4b3419c1289ff5d104c3699c28a0fcf7e7385a0.jpg",
+        "image_caption": [
+            "Figure 2: Relative difference for Res-GNN (yellow, diagonal) and BGNN (red, squared) for different GNN architectures w.r.t. GNN RMSE (the smaller the better). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In Figure 2 we report the relative RMSE gap between Res-GNN and BGNN for each of the GNN architectures, i.e., we compute $g a p = ( r _ { m } - r _ { g n n } ) / r _ { g n n }$ , where $r _ { m }$ and $r _ { g n n }$ are RMSE of that model and of GNN respectively. This experiment positively answers Q3 and shows that all tested GNN architectures significantly benefit from the proposed approach. For example, for House dataset the decrease in the mean squared error is $9 \\%$ , $18 \\%$ , $19 \\%$ , and $17 \\%$ for GAT, GCN, AGNN, and APPNP models respectively. Additionally, one can see that the end-to-end training of BGNN (red, squared) leads to larger improvements than a na¨ıve combination of CatBoost and GNN in Res-GNN (yellow, diagonal). Exact metrics and training time are in Appendix E. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 TRAINING TIME ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As the previous experiments demonstrated superior quality across various datasets and GNN models, it is important to understand if the additional GBDT part can become a bottleneck in terms of efficiency for training this model on real-world datasets. Hence, we ask: ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To answer this question, we measure the clock time to train each model until convergence, considering early stopping. Table 4 presents training time for each model. We can see that both BGNN and Res-GNN run faster than GNN in most cases. In other words, BGNN and Res-GNN models do not incur an increase in training time but actually are more efficient than GNN. For example, for VK dataset BGNN and Res-GNN run $3 \\mathbf { x }$ and $2 \\mathbf { x }$ faster than GNN, respectively. Moreover, BGNN is consistently faster than another end-to-end implementation FCNN-GNN that uses FCNN instead of CatBoost to preprocess the original input features. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/27d7be8c02679fc2d67c201a6fab862b0464164a28d6e15a3e87b154ca006e9d.jpg",
+        "table_caption": [
+            "Table 4: Training time (s) in node regression task. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td></td><td>House</td><td>County</td><td>VK</td><td>Wiki</td><td>Avazu</td></tr><tr><td rowspan=\"2\">GETT</td><td>CatBoost</td><td>4±1</td><td>2±1</td><td>24±4</td><td>10±1</td><td>2±2</td></tr><tr><td>LightGBM</td><td>3±0</td><td>1±0</td><td>5±3</td><td>3±2</td><td>0±0</td></tr><tr><td rowspan=\"4\"></td><td>GAT GCN</td><td>35±2</td><td>19±6</td><td>42±4</td><td>15±1</td><td>9±2</td></tr><tr><td></td><td>28±0</td><td>18±7</td><td>38±0</td><td>13±3</td><td>12±6</td></tr><tr><td>AGNN</td><td>38±5</td><td>28±3</td><td>48±3</td><td>19±5</td><td>14±8</td></tr><tr><td>APPNP</td><td>68±1</td><td>34±10</td><td>81±3</td><td>49± 26</td><td>24±15</td></tr><tr><td rowspan=\"2\">M</td><td>FCNN FCNN-GNN</td><td>16±5</td><td>2±1</td><td>109 ± 35</td><td>12±2</td><td>2±0</td></tr><tr><td></td><td>39±1</td><td>21±6</td><td>48±2</td><td>16±1</td><td>14±3</td></tr><tr><td rowspan=\"2\"></td><td>Res-GNN</td><td>36±7</td><td>7±3</td><td>41±7</td><td>31±9</td><td>7±2</td></tr><tr><td>BGNN</td><td>20±4</td><td>2±0</td><td>16±0</td><td>21±7</td><td>5±1</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "The reason for improved efficiency is that BGNN and Res-GNN converge with a much fewer number of iterations as demonstrated in Figure 3. We plot RMSE on the test set during training for all models (with winning hyperparameters). We can see that BGNN converges within the first ten iterations (for $k = 2 0$ ), leading to fast training. In contrast, Res-GNN is similar in terms of convergence to GNN for the first 100 epochs, but then it continues decreasing RMSE unlike GNN that requires much more epochs to converge. This behavior is similar for other datasets (see Appendix F). ",
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+    {
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+        "img_path": "images/7ef28e9ae2b9d9bd8faaefc3a8f2316640e7de3eaa2425d1d307982ff73e20be.jpg",
+        "image_caption": [
+            "Figure 3: RMSE on the test set during training for two node regression datasets. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.5 VISUALIZING PREDICTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To investigate the performance of BGNN, we plot the final predictions of trained models for observations in the training set. Our motivation is to scrutinize which points are correctly classified by different models. Figure 4 displays the predictions of GBDT, GNN, Res-GNN, and BGNN models as well as the true target value. To better understand the predictions of the BGNN model, in Figure 4(e) we show the values predicted by GBDT that was trained as a part of BGNN. This experiment is performed on House dataset, the plots for other datasets show similar trends and can be found in the supplementary materials. ",
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+    {
+        "type": "text",
+        "text": "Several observations can be drawn from these figures. First, the true target values change quite smoothly within local neighborhoods; however, there are a few outliers: single red points among many blue points and conversely. These points can mislead the model during the training, predicting the wrong target value for many observations in the outliers’ local neighborhoods. Hence, it is important for a model to make smoothed predictions in the local neighborhoods. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/980e8813be0fdcd802b8124062d45cb066afd363c94898e2e2a61abe4138b9ae.jpg",
+        "image_caption": [
+            "Figure 4: House dataset. True labels and predictions by trained GBDT, GNN, Res-GNN, and BGNN models (training points only). Point coordinates correspond to BGNN learned representations in the first hidden layer. Color represents the final predictions made by each model. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Second, comparing the prediction spaces of GBDT, GNN, Res-GNN, and BGNN models we can observe that predictions for GBDT are much more grainy with large variations in the neighborhoods of the vertices (high quality images can be found in the supplementary materials). Intuitively, because the GBDT model does not have access to the graph structure, it cannot propagate its predictions in the nodes’ vicinity. Alternatively, GNN, Res-GNN, and BGNN can extrapolate the outputs among local neighbors, smoothing out the final predictions as seen in Figures 4(c), 4(d), 4(f). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Third, focusing on the values of predictions (color bars on the right of each plot) of GBDT, GNN, and BGNN models we notice that the scale of final predictions for GBDT and BGNN models is closely aligned with the true predictions, while GNN’s predictions mismatch the true values by large margin. Our intuition is that the expressive power of GBDT to learn piecewise decision boundaries common in tabular datasets helps GBDT and BGNN to properly tune its final predictions with respect to the true range of values. In contrast, GNN relies solely on neural layers to learn complex decision rules. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Another observation comes from looking at the values predicted by GBDT trained as a part of BGNN (see Figure 4(e)). While this GBDT model is initialized using the true target labels, it was not forced to predict the target during the training. Interestingly, this model shows the same trend and clearly captures the regions on high/low target values. On the other hand, GBDT trained as a part of BGNN is much more conservative: on all datasets, the range of predicted values is significantly smaller than the true one. We hypothesize that GBDT is trained to scale its predictions to make them more suitable for further improvements by GNN. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have presented BGNN, a novel architecture for learning on graphs with heterogeneous tabular node features. BGNN takes advantages of the GBDT model to build hyperplane decision boundaries that are common for heterogeneous data, and then utilizes GNN to refine the predictions using relational information. Our approach is end-to-end and can be incorporated with any message-passing neural network and gradient boosting method. Extensive experiments demonstrate that the proposed architecture is superior to strong existing competitors in terms of accuracy of predictions and training time. A possible direction for future research is to analyze whether this approach is profitable for graph-level predictions such as graph classification or subgraph detection. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+    },
+    {
+        "type": "text",
+        "text": "The authors thank the Anonymous Reviewers for their reviews and Anton Tsitsulin for kindly sharing VK data. Liudmila Prokhorenkova also acknowledge the financial support from the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant 075-15-2019-1926 and from the Russian President grant supporting leading scientific schools of the Russian Federation NSh-2540.2020.1. ",
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Yongxin Yang, Irene Garcia Morillo, and Timothy M Hospedales. Deep neural decision trees. arXiv preprint arXiv:1806.06988, 2018. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Zhi-Hua Zhou and Ji Feng. Deep forest. National Science Review, 6(1):74–86, 2019. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A FURTHER RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+            102,
+            426,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, there are no approaches combining the benefits of GBDT and GNN models for representation learning on graphs with tabular data. However, there are many attempts to adapt non-graph neural networks for tabular data or to combine them with gradient boosting in different ways. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Several works (Popov et al., 2019; Yang et al., 2018; Zhou & Feng, 2019; Feng et al., 2018; Hazimeh et al., 2020) attempt to mitigate the non-differentiable nature of decision trees. For example, Popov et al. (2019) proposed to replace hard choices for tree splitting features and splitting thresholds with their continuous counterparts, using $\\alpha$ -entmax transformation (Peters et al., 2019). While such an approach becomes suitable for a union of decision trees with GNN, the computational burden of training both end-to-end becomes a bottleneck for large graphs. ",
+        "bbox": [
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+            280
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Another method (Badirli et al., 2020) uses neural networks as weak learners for the GBDT model. For graph representation problems such as node regression, one can replace standard neural networks with graph neural networks. However, training different GNN as weak classifiers at once would be exhaustive. Additionally, such a combination lacks some advantages of GBDT, like handling heterogeneous and categorical features and missing values. An approach called AdaGCN (Sun et al., 2019) incorporates AdaBoost ideas into the design of GNNs in order to construct deep models. Again, this method does not exploit the advantages of GBDT methods. ",
+        "bbox": [
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+            286,
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+            385
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Finally, Li et al. (2019) investigated different ways of combining decision-tree-based models and neural networks. While the motivation is similar to ours — get the benefits of both types of models — the paper focuses specifically on learning-to-rank problems. Additionally, while some of their methods are similar in spirit to Res-GNN, they do not update GBDT in an end-to-end manner, which is a substantial contribution of the current research. ",
+        "bbox": [
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+            391,
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+            462
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            481,
+            380,
+            497
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Parameters in brackets $\\{ \\}$ are selected by hyperparameter search on the validation set. ",
+        "bbox": [
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+            512,
+            735,
+            529
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "LightGBM: number of leaves is $\\{ 1 5 , 6 3 \\}$ , $| | \\lambda | | _ { 2 } = 0$ , boosting type is gbdt, number of epochs is 1000, early stopping rounds is 100. ",
+        "bbox": [
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+            534,
+            823,
+            563
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "CatBoost: depth is $\\{ 4 , 6 \\} , \\lvert \\lvert \\lambda \\rvert \\rvert _ { 2 } = 0$ , number of epochs is 1000, early stopping rounds is 100. ",
+        "bbox": [
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+            569,
+            792,
+            585
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "FCNN: number of layers is $\\{ 2 , 3 \\}$ , dropout is $\\{ 0 . , 0 . 5 \\}$ , hidden dimension is 64, number of epochs is 5000, early stopping rounds is 2000. ",
+        "bbox": [
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+            590,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "GNN: dropout rate is $\\{ 0 . , 0 . 5 \\}$ , hidden dimension is 64, number of epochs is 2000, early stopping rounds is 200. GAT, GCN, and AGNN models have two convolutional layers with dropout and ELU activation function (Clevert et al., 2016). APPNP has a two-layer fully-connected neural network with dropout and ELU activation followed by a convolutional layer with $k = 1 0$ and $\\alpha = 0 . 1$ . We use eight heads with eight hidden neurons for GAT model. ",
+        "bbox": [
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+            696
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Res-GNN: dropout rate is $\\{ 0 . , 0 . 5 \\}$ , hidden dimension is 64, number of epochs is 1000, early stopping rounds is 100. We also tune whether to use solely predictions of CatBoost model or append them to the input features. CatBoost model is trained for 1000 epochs. ",
+        "bbox": [
+            174,
+            703,
+            823,
+            746
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "BGNN: dropout rate is $\\{ 0 . , 0 . 5 \\}$ , hidden dimension is 64, number of epochs is 200, early stopping rounds is 10, number of trees and backward passes per epoch is $\\{ 1 0 , 2 0 \\}$ , depth of the tree is 6. We also tune whether to use solely predictions of CatBoost model or append them to the input features. ",
+        "bbox": [
+            174,
+            752,
+            823,
+            795
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For all models, we also perform a hyperparameter search on learning rate in $\\{ 0 . 1 , 0 . 0 1 \\}$ . Every hyperparameter setting is evaluated three times and an average is taken. We use five random splits for train/validation/test with $0 . 6 / 0 . 2 / 0 . 2$ ratio. The average across five seeds is reported in the tables. ",
+        "bbox": [
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+            801,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C REGRESSION DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            863,
+            406,
+            880
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In House dataset (Pace & Barry, 1997), nodes are the properties, edges connect the proximal nodes, and the target is the property’s price. We use the publicly available dataset (Pace & Barry, 1997) of all the block groups in California collected from the 1990 Census. We connect each block with at most five of its nearest neighbors if they lie within a ball of a certain radius, as measured by latitude and longitude. We keep the following node features: MedInc, HouseAge, AveRooms, AveBedrms, Population, AveOccup. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+            103,
+            825,
+            160
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "County dataset (Jia & Benson, 2020) is a county-level election map network. Each node is a county, and two nodes are connected if they share a border. We consider node features coming from the 2016 year. These features include DEM, GOP, MedianIncome, MigraRate, BirthRate, DeathRate, BachelorRate, UnemploymentRate. We follow the setup of the original paper and select UnemploymentRate as the target label. We filter out all nodes in the original data if they do not have features. ",
+        "bbox": [
+            173,
+            166,
+            825,
+            250
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "VK dataset (Tsitsulin et al., 2018) comes from a popular social network where people are mutually connected based on friendships, and the regression problem is to predict the age of a person. We use an open-access subsample of the VK social network of the first 1M users.3 Then, the dataset has been preprocessed to keep only the users who opt in to share their demographic information and preferences: country, city, has mobile, last seen platform, political, religion id, alcohol, smoking, relation, sex, university. ",
+        "bbox": [
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+            257,
+            825,
+            340
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Wiki dataset (Rozemberczki et al., 2019) represents a page-page network on a specific topic (squirrels) with the task of predicting average monthly traffic. The features are bag-of-words for informative nouns (3148 in total) that appeared in the main text of the Wikipedia article. The target is the average monthly traffic between October 2017 and November 2018 for each article. ",
+        "bbox": [
+            174,
+            348,
+            825,
+            404
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Avazu dataset (Song et al., 2019) represents a device-device network, with two devices being connected if they appear on the same site within the same application. For this dataset, the goal is to predict click-through-rate (CTR) for each device. We take the first 10M rows from the publicly available train log of user clicks.4 We compute CTR for each device id and filter those ids that do not have at least 10 ad displays. We connect two devices if they had ad displays on the same site id from the same application id. The node features are anonymized categories: C1, C14, C15, C16, C17, C18, C19, C20, C21. ",
+        "bbox": [
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+            410,
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+            507
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D CLASSIFICATION DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            529,
+            439,
+            544
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For node classification, we consider three types of node features: heterogeneous (VK and House), sparse (Slap and DBLP), and homogeneous (OGB-ArXiv). ",
+        "bbox": [
+            174,
+            559,
+            823,
+            588
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For House and VK, we transform the original numerical target value with respect to the bin it falls to. More specifically, for VK we consider the classes $< 2 0$ , $2 0 - 2 5 $ , $2 5 - 3 0 , \\dots , 4 5 - 5 0 , > 5 0$ for the age attribute. Similarly, for House dataset we replace the target value with the bin it falls to in the range [1, 1.5, 2, 2.5]. Hence, there are 7 and 5 classes for VK and House, respectively. ",
+        "bbox": [
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+            594,
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+            650
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/6cc1a4734307e4ef0fc51b1480f676e93a90b2d7368a472c2a5da9c7af52f6ae.jpg",
+        "table_caption": [
+            "Table 5: Summary of classification datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>SLAP</td><td>DBLP</td><td>OGB-ArXiv</td></tr><tr><td># Nodes</td><td>20419</td><td>14475</td><td>169343</td></tr><tr><td>#Edges</td><td>172248</td><td>40269</td><td>1166243</td></tr><tr><td>#Features</td><td>2701</td><td>5002</td><td>128</td></tr><tr><td>Classes</td><td>15</td><td>4</td><td>40</td></tr><tr><td>Min Class</td><td>103</td><td>745</td><td>29</td></tr><tr><td>Max Class</td><td>534</td><td>1197</td><td>27321</td></tr></table>",
+        "bbox": [
+            519,
+            695,
+            826,
+            803
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For datasets with sparse features, we consider two datasets coming from heterogeneous information networks (HIN), where nodes have a few different types. A common way to represent HIN is through meta-paths, i.e., a collection of all possible paths between nodes of a particular type. For example, for a citation network, one may specify paths of the type paper-author-paper (PAP) and the type papersubject-paper (PSP). Then the original graph is approximated as several adjacency matrices for different types. ",
+        "bbox": [
+            174,
+            657,
+            508,
+            810
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "DBLP dataset (Ren et al., 2019) is a network with three node types (authors, papers, conferences) and four target classes of the authors (database, data mining, information retrieval, and machine learning). To obtain a single graph, we use the adjacency matrix for the relation APA, which closely reflects the relationships between authors. Each author has a bag-of-words representation (300 words) of all the abstracts published by the author. Furthermore, for every node, we compute the degrees for all types of relations and append them as additional node features. Namely, we have two additional node features corresponding to degrees for paper nodes in APA and APCPA adjacency matrices. ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "SLAP dataset (Xiao et al., 2019) is a multiple-hub network in bioinformatics that contains node types such as chemical compound, gene, disease, pathway, etc. The goal is to predict one of 15 gene types. To obtain a single graph, we use the adjacency matrix for the relation GG between genes. Each gene has 3000 features that correspond to the extracted gene ontology terms (GO terms). As for DBLP, we compute the degrees for all types of relations and append them as additional node features. ",
+        "bbox": [
+            174,
+            138,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "As a dataset with homogeneous node features we consider OGB-ArXiv (Hu et al., 2020a). The node features correspond to a 128-dimensional feature vector obtained by averaging the embeddings of words in the title and abstract. Note that for this particular dataset we used the implementation of $\\mathrm { G A T } ^ { 5 }$ as a backbone architecture for GNN, Res-GNN, and BGNN models. This model scored the top place on the leaderboard.6 A summary of statistics for all datasets is outlined in Table 5. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E COMPARISON OF GNN MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            305,
+            472,
+            321
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In this section, we show the exact RMSE values and time for all tested GNN models on all regression datasets. We consider several state-of-the-art GNN models that include GAT (Velickovi ˇ c et al., 2018), ´ GCN (Kipf & Welling, 2017), AGNN (Thekumparampil et al., 2018), and APPNP (Klicpera et al., 2019). ",
+        "bbox": [
+            174,
+            337,
+            826,
+            392
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 6 demonstrates that for all considered models BGNN and Res-GNN achieve significant increase in performance compared to vanilla GNN. Additionally, end-to-end training of BGNN achieves typically better results than a straightforward implementation of Res-GNN. ",
+        "bbox": [
+            174,
+            400,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/7e9b4192111842530a605bbd517ccc308c5e3d25091046191974334d81b0a944.jpg",
+        "table_caption": [
+            "Table 6: Summary of our results for different GNN architectures for node regression. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Method</td><td colspan=\"2\">House</td><td colspan=\"2\">County</td><td colspan=\"2\">VK</td><td colspan=\"2\">Wiki</td><td colspan=\"2\">Avazu</td></tr><tr><td>RMSE</td><td>Time (s)</td><td>RMSE</td><td>Time (s)</td><td>RMSE</td><td>Time (s)</td><td>RMSE</td><td>Time (s)</td><td>RMSE</td><td>Time (s)</td></tr><tr><td rowspan=\"2\">5</td><td>GNN</td><td>0.54± 0.01</td><td>35±2</td><td>1.45 ± 0.06</td><td>19±6</td><td>7.22 ± 0.19</td><td>42±4</td><td>45916±4527</td><td>15±1</td><td>0.113 ± 0.01</td><td>9±2</td></tr><tr><td>Res-GNN</td><td>0.51 ± 0.01</td><td>36±7</td><td>1.33 ± 0.08</td><td>7±3</td><td>7.07 ± 0.20</td><td>41±7</td><td>46747 ± 4639</td><td>31±9</td><td>0.109 ± 0.01</td><td>7±2</td></tr><tr><td rowspan=\"3\"></td><td>BGNN</td><td>0.5±0.01</td><td>20±4</td><td>1.26 ± 0.08</td><td>2±0</td><td>6.95 ± 0.21</td><td>16±0</td><td>49222 ± 3743</td><td>21±7</td><td>0.109 ± 0.01</td><td>5±1</td></tr><tr><td>GNN</td><td>0.63 ± 0.01</td><td>28±0</td><td>1.48 ± 0.08</td><td>18±7</td><td>7.25 ± 0.19</td><td>38±0</td><td>44936 ± 4083</td><td>13±3</td><td>0.114 ± 0.02</td><td>12±6</td></tr><tr><td>Res-GNN</td><td>0.59 ± 0.01</td><td>25±2</td><td>1.35 ± 0.09</td><td>11 ±5</td><td>7.03±0.20</td><td>52±6</td><td>44876± 3777</td><td>21±5</td><td>0.111 ± 0.02</td><td>9±6</td></tr><tr><td rowspan=\"3\"></td><td>BGNN</td><td>0.54 ± 0.01</td><td>41 ±15</td><td>1.33 ± 0.13</td><td>12 ±8</td><td>7.12 ±0.21</td><td>76±6</td><td>47426 ± 4112</td><td>22±11</td><td>0.107 ± 0.01</td><td>4±1</td></tr><tr><td>GNN</td><td>0.59 ± 0.01</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Res-GNN</td><td>0.52 ±0.01</td><td>38±5 33±4</td><td>1.45 ± 0.08 1.3 ± 0.07</td><td>28±3 16±4</td><td>7.26±0.20 7.08±0.20</td><td>48±3 51±15</td><td>45982 ± 3058 46010 ± 2355</td><td>19±5 24±3</td><td>0.113 ± 0.02 0.111 ± 0.02</td><td>14±8 7±2</td></tr><tr><td rowspan=\"2\">ANNN</td><td>BGNN</td><td>0.49 ± 0.01</td><td>34±4</td><td>1.28 ± 0.08</td><td>3±1</td><td>6.89 ± 0.21</td><td>25±4</td><td>53080 ± 5117</td><td>47±37</td><td>0.108 ± 0.02</td><td>5±1</td></tr><tr><td>GNN</td><td>0.69 ± 0.01</td><td>68±1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=\"2\">JNPPA</td><td>Res-GNN</td><td>0.67 ± 0.01</td><td>58±12</td><td>1.5 ± 0.11 1.41 ± 0.12</td><td>34±10 19 ±10</td><td>13.23 ± 0.12</td><td>81±3 76±11</td><td>53426±4159</td><td>49±26</td><td>0.113 ± 0.01</td><td>24±15</td></tr><tr><td>BGNN</td><td>0.59 ± 0.01</td><td>21±7</td><td>1.33 ± 0.10</td><td>17±6</td><td>13.06 ± 0.17 12.36 ± 0.14</td><td>50±6</td><td>53206±4593 54359±4734</td><td>66±27 30±13</td><td>0.110 ± 0.01 0.108 ± 0.01</td><td>15±10 6±1</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "F LOSS CONVERGENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            383,
+            117
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "In Figure 5, we plot RMSE on the test set during training for the remaining datasets — County, Wiki, and Avazu. These results confirm that BGNN converges to its optimal value within the first ten iterations (for $k = 2 0$ ). Note that on the Wiki dataset, similarly to Figure 3, Res-GNN convergence is similar to GNN for the first 100 iterations and then the loss of Res-GNN decreases faster than of GNN. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/f8ae631dcb149f2fb9dfbce62e88ab9b50bef7c12b85b90c878ac590aa77b56a.jpg",
+        "image_caption": [
+            "Figure 5: Summary of RMSE of test set during training for node regression datasets. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            219,
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+        ],
+        "page_idx": 15
+    }
+]