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+# Neural Scene Flow Prior
+
+Xueqian Li∗1,2 Jhony Kaesemodel Pontes1 Simon Lucey2 1Argo AI 2The University of Adelaide
+
+# Abstract
+
+Before the deep learning revolution, many perception algorithms were based on runtime optimization in conjunction with a strong prior/regularization penalty. A prime example of this in computer vision is optical and scene flow. Supervised learning has largely displaced the need for explicit regularization. Instead, they rely on large amounts of labeled data to capture prior statistics, which are not always readily available for many problems. Although optimization is employed to learn the neural network, the weights of this network are frozen at runtime. As a result, these learning solutions are domain-specific and do not generalize well to other statistically different scenarios. This paper revisits the scene flow problem that relies predominantly on runtime optimization and strong regularization. A central innovation here is the inclusion of a neural scene flow prior, which uses the architecture of neural networks as a new type of implicit regularizer. Unlike learning-based scene flow methods, optimization occurs at runtime, and our approach needs no offline datasets—making it ideal for deployment in new environments such as autonomous driving. We show that an architecture based exclusively on multilayer perceptrons (MLPs) can be used as a scene flow prior. Our method attains competitive—if not better—results on scene flow benchmarks. Also, our neural prior’s implicit and continuous scene flow representation allows us to estimate dense long-term correspondences across a sequence of point clouds. The dense motion information is represented by scene flow fields where points can be propagated through time by integrating motion vectors. We demonstrate such a capability by accumulating a sequence of lidar point clouds.
+
+# 1 Introduction
+
+State-of-the-art results have recently been achieved by learning-based models [17, 30, 47, 60, 68] for the scene flow problem—the task of estimating 3D motion fields from dynamic scenes. However, such models heavily rely on large-scale data to capture prior knowledge, which is not always readily available. Scene flow annotations are expensive, and most methods train on synthetic and unrealistic scenarios to fine-tune on small real datasets.
+
+Poor generalization to unseen, out-of-the-distribution inputs is another problem. Prior information is generally limited to the statistics of the data used for training. Real-world applications such as autonomous driving require robust solutions to low-level vision tasks such as depth, optical, and scene flow estimation that work in statistically different scenarios. Inspired by recent innovations that make use of coordinate-based networks (i.e., pixels or 3D positions as inputs) [9, 36, 37, 39, 56] for 3D modeling and rendering, we investigate the use of such networks to regularize the scene flow problem without any learning directly from point clouds. Optimization happens at runtime, and instead of learning a prior from data, the network structure itself captures the prior information. It is not limited to the statistics of a specific dataset.
+
+Optimizing neural networks at execution time is not new. Ulyanov et al. [63] showed that a randomly initialized convolutional network could be used as a handcrafted prior for standard inverse problems such as image denoising, super-resolution, and inpainting. Ding and Feng [12] proposed a runtime optimization method (DeepMapping) for rigid pose estimation using deep neural networks. Although such deep image priors, deep mapping, and coordinate-based networks for neural scene representations have been successfully applied for inverse problems, rendering, and rigid registration, none has yet investigated (to the best of our knowledge) the use of network-based priors for regularizing scene flow directly from point clouds.
+
+Our proposed neural prior is based on a simple multilayer perceptron (MLP) architecture, and we show it is powerful enough to regularize scene flow given two point clouds implicitly. The input to the network is 3D points, and the output is a regularized scene flow.
+
+Our neural prior allows for a continuous scene flow representation instead of discrete such as in graph Laplacian-based priors, e.g., [45]. We show how the flow fields captured by our neural
+
+![](images/577ce96cb63b1a5d41e8fdcb8a10e98dc3b649626ddf6ea769bcd2e8f56dbedf.jpg)  
+Figure 1: Our neural scene flow prior method achieved higher accuracy while being ${ \sim } 1 0 \times$ faster than the recent runtime optimization method graph prior [45]. The evaluation was on the KITTI Scene Flow test set, where each point cloud size varies from $1 4 \mathrm { k }$ to $6 8 \mathrm { k }$ points. In our method, we fixed the number of hidden layers in the MLP to 4 and varied the number of hidden units. In the graph prior method, we varied the number of neighbors to create the graph. Accuracy uses the $A c c _ { 5 }$ metric as defined in the experiments section. Learningbased methods might still be $1 0 \times - 1 0 0 \times$ faster than the runtime optimization methods, but they still lack generalization and have memory issues when dealing with large point clouds—with tens of thousands of points.
+
+prior can be employed to estimate long-term correspondences across a sequence of point clouds. The continuous scene flow allows for better integration of motions across time.
+
+Our results are promising and competitive to supervised [30], self-supervised [38, 68], and nonlearning methods [1, 45] (see Table 1). Our method also scales to real-world point clouds with tens of thousands of points while achieving better accuracy and time complexity than recent runtime optimization methods (see Fig. 1).
+
+# 2 Related work
+
+Non-learning-based scene flow Scene flow is the uplift from the optical flow, which is proposed by Vedula et al. [64] as the non-rigid motion field in the 3D space. The authors proposed the optimization-based scene flow estimation using image sequences to infer reconstruction knowledge of the flow in 3D surfaces. Successive RGB/RGB-D image-based work [4, 18–21, 27, 43, 44, 50] used probability-based estimation, coarse-to-fine techniques, 6-DoF parameterization, or object segmentation, etc., to improve accuracy and computation time. Although image-based scene flow methods are widely used, direct estimation of the scene flow from the point cloud is still possible through non-rigid registration methods, such as [1, 10, 26, 42]. In this paper, we focus on point cloud-based scene flow estimation.
+
+Learning-based scene flow Image-based learning methods [6, 51, 54, 60, 70] use convolution and data supervision to solve scene flow from monocular or RGB-D images with the available depth information. Other image-based methods [22, 23, 32, 52, 53] take care of extra occlusion cues in the large-scale autonomous driving scenes. Point-based learning methods have become more prevalent with the rapid development of point cloud feature learning [28, 48, 49, 65, 67]. FlowNet3D [30] is a seminal work that estimates scene flow using PointNet+ $^ +$ [49]. Successive work [17, 31, 47, 66] extends point-based learning methods using different feature extraction techniques. One obvious drawback of these supervised learning methods is the demand for sufficient ground truth labels. Besides, supervised methods lack generalizability while eventually only fitting domain-specific data. Self-supervised methods [25, 38, 61, 68], on the other hand, replaced the loss between the prediction and the ground truth flow with a point distance loss to use the point cloud itself as supervision.
+
+Self-supervision can adapt to different datasets and maintain certain generalizability. Nonetheless, massive training data are still required for sufficient learning.
+
+The graph Laplacian method Graph Laplacian [2] is widely used to smooth the surface in mesh processing [5, 14, 57, 58], point cloud denoising [11, 71], etc. Here we talk about the recent scene flow estimation using Graph Laplacian [45]. The method explicitly constructed a graph of the point cloud to constrain the non-rigid scene flow as rigid within a specific range. While as a dataless runtime optimization, the method is heavily affected by the hyperparameters of the graph and loses scalability when the point cloud becomes larger or the neighbors in the graph grow.
+
+Deep neural prior, implicit functions, and neural rendering Although large-scale data helps in feature representation [59], end-to-end learning still requires high computation capacity and readily available training datasets. Instead, Ulyanov et al. [63] proposed a new style of optimization that uses a convolutional neural network to infer prior knowledge from the network architecture. One broader interest is to extend the idea of the network being an image function to the 3D shape modeling, and implicitly represent the continuous shape as level sets of neural networks. By directly mapping the 3D input to binary occupancy nets [9, 36], or signed distance functions [3, 39, 55], it is powerful to model the 3D geometries in a continuous space using the coordinate-based network. The following Scene Representation Networks [56] takes advantage of the coordinate-based network and renders view synthetic images. Mildenhall et al. proposed a seminal work NeRF [37], which is a novel way to do neural volume rendering using both point positions and viewing directions based on the coordinate-based network. Dynamic scene synthesis work [13, 16, 29, 40, 46, 62, 69] follows the NeRF framework, and integrates the motions to generate dynamic scenes. Some of the work [16, 29] use scene flow to further constrain or segment dynamic scenes. An interesting work that solves for the rigid alignment between point clouds using runtime optimization is DeepMapping [12]. However, this work only deals with rigid motion, and the network architecture is more complex than coordinate-based networks. In this work, we are interested in coordinate-based networks to address the large-scale, real-world scene flow problem.
+
+# 3 Approach
+
+Problem definition Let $S _ { 1 }$ and $S _ { 2 }$ be two 3D point clouds sampled from a dynamic scene at time $t$ -1 and $t$ . The number of points in each point cloud, $| S _ { 1 } |$ and $| S _ { 2 } |$ , are typically different and not in correspondence. A 3D point $\mathbf { p } \in S _ { 1 }$ moving from time $t \mathrm { - } 1$ to time $t$ can be modeled by a translational vector (or flow vector) $\bar { \mathbf { f } } \in \mathbb { R } ^ { 3 }$ , where $\mathbf { p ^ { \prime } } = \mathbf { p } + \mathbf { f }$ . The collection of flow vectors for all 3D points is the scene flow $\mathcal { F } = \{ \mathbf { f } _ { i } \} _ { i = 1 } ^ { | S _ { 1 } | }$ .
+
+Optimization We want to optimize for a scene flow $\mathcal { F }$ that minimizes the distance between the two point clouds, $S _ { 1 }$ and $S _ { 2 }$ . Given the non-rigidity assumption of the scene, the optimization is inherently unconstrained. Thus, a regularization term C is necessary to constrain the motion field. We therefore solve for scene flow as
+
+$$
+\mathcal { F } ^ { * } = \underset { \mathcal { F } } { \arg \operatorname* { m i n } } \sum _ { \mathbf { p } \in S _ { 1 } } \mathbf { D } \left( \mathbf { p } + \mathbf { f } , S _ { 2 } \right) + \lambda \mathbf { C } ,
+$$
+
+where $\mathrm { D }$ is a function to compute the distance from the perturbed point $\mathbf { p }$ by the flow vector f to its closest neighbor in $S _ { 2 }$ . C is a regularizer (e.g., Laplacian regularizer), and $\lambda$ is a weighting factor for the regularizer. In this paper, we want to investigate using a neural prior to regularize the scene flow.
+
+# 3.1 Neural scene flow prior
+
+Learning-based scene flow methods learn scene flow priors from a large number of examples. As in Deep Image Prior [63], we want to investigate if the structure of a neural network by itself is sufficient to capture a scene flow prior without any learning.
+
+Here we use a neural network as an implicit regularizer. The parameters are optimized as
+
+$$
+\Theta ^ { * } = \underset { \Theta } { \arg \operatorname* { m i n } } \sum _ { \mathbf { p } \in S _ { 1 } } \mathrm { D } \left( \mathbf { p } + g \left( \mathbf { p } ; \Theta \right) , S _ { 2 } \right) ,
+$$
+
+where $g$ is a neural network parameterized by $\Theta$ to regularize the scene flow $\mathcal { F }$ . The input to $g$ is $\mathbf { p }$ which is the point to be disturbed by the flow. The output of $g$ is $\mathbf { f }$ and thus $\mathbf { f } ^ { * } = g \left( \mathbf { p } ; \mathbf { \bar { \Theta } } \right)$ . For the distance function D, we define it as
+
+$$
+\mathbf { D } \left( \mathbf { p } , \pmb { S } \right) = \operatorname* { m i n } _ { \mathbf { x } \in \pmb { S } } \| \mathbf { p } - \mathbf { x } \| _ { 2 } ^ { 2 } .
+$$
+
+In practice, we use it bidirectionally for both point sets, which is equivalent to Chamfer distance [15].
+
+The objective in Eq. (2), is for the forward scene flow $\mathcal { F }$ , which is the one we are interested in. However, it has been shown in [30, 38], that a cycle consistency regularizer encourages better scene flow estimations. The extra regularizer simply enforces the backward flow to be similar to the forward flow, $\mathcal { F } _ { b w d } \approx \mathcal { F }$ . The optimal backward flow is defined as $\mathbf { f } _ { b w d } ^ { * } = g \left( \mathbf { p } ^ { \prime } ; \mathbf { \Theta } \Theta _ { b w d } ^ { * } \right)$ , where $\mathbf { p } ^ { \prime }$ is the shifted point by the forward flow as $\mathbf { p } + \mathbf { f }$ . Note that the network $g$ is the same but with different parameters, $\Theta _ { b w d }$ . Using the backward flow as an additional constraint, the optimal network weights are solved as
+
+$$
+\Theta ^ { * } , \Theta _ { b w d } ^ { * } = \underset { \Theta , \Theta _ { b w d } } { \mathrm { a r g } \mathrm { m i n } } \sum _ { \mathbf { p } \in S _ { 1 } } \mathbf { D } \left( \mathbf { p } + g \left( \mathbf { p } ; \Theta \right) , S _ { 2 } \right) + \sum _ { \mathbf { p } ^ { \prime } \in S _ { 1 } ^ { \prime } } \mathbf { D } \left( \mathbf { p } ^ { \prime } + g \left( \mathbf { p } ^ { \prime } ; \Theta _ { b w d } \right) , S _ { 1 } \right) ,
+$$
+
+where $ { \boldsymbol { S } } _ { 1 } ^ { \prime }$ is the shifted $S _ { 1 }$ by the forward flow, i.e., $S _ { 1 } ^ { \prime } = S _ { 1 } { + } \mathcal { F }$ . Please find more details in the supplementary material. For the network $g$ , we use MLPs with ReLU activations. The objective function in Eq. (4) can be optimized by gradient descent techniques using off-the-shelf frameworks with automatic differentiation. We show in the experiments section how the architecture of the neural prior affects performance by varying the number of hidden layers and units.
+
+Why use a neural scene flow prior? Deep learning relies on massive amounts of data and computational resources to capture prior statistics. Although learning methods have achieved impressive results in most tasks, they still struggle when deployed in environments where the statistics are different from those captured during learning. Our intuition is that a neural prior acts as a strong implicit regularizer that constrains dynamic motion fields to be as smooth as possible. A neural scene flow prior also scales to large scenes while achieving high-fidelity results at a low computational cost. Our proposed method with 8 hidden layers and 128 hidden units has about $1 1 6 \mathrm { k }$ parameters. FlowNet3D [30], for example, has about 1.2M parameters. Our method has ${ \sim } 1 0 \times$ fewer parameters than the state-of-the-art supervised methods while achieving competitive, if not better, results. Lastly, our deep scene flow prior captures a continuous flow field that allows us to perform better scene flow interpolation across a sequence of point clouds.
+
+# 4 Experiments
+
+We evaluated the performance (accuracy, generalizability, and computational cost) of our neural prior for scene flow on synthetic and real-world datasets. We performed experiments on different neural network settings and analyzed the performance of the neural prior to regularizing scene flow. Remarkably, we show that a simple MLP-based prior to regularize scene flow is enough to achieve competitive results to the state-of-the-art scene flow methods.
+
+Datasets We used four scene flow datasets: 1) FlyingThings3D [33] which is an extensive collection of randomly moving synthetic objects. We used the preprocessed data from [30]; 2) KITTI [34, 35] which has real-world self-driving scenes. We used the subset released by [30]; 3) Argoverse [8] and 4) nuScenes [7] are two large-scale autonomous driving datasets with challenging dynamic scenes. However, there are no official scene flow annotations. We followed the data processing method in [45] to collect pseudo-ground-truth scene flow. Ground points were removed from lidar point clouds as in [30] (please refer to the supplementary material for more details).
+
+Metrics We employed the widely used metrics as in [30, 38, 45, 68] to evaluate our method, which are: $\mathcal { E }$ to denote the end-point error (EPE), which is the mean absolute distance of two point clouds; $A c c _ { 5 }$ to denote the accuracy in percentage of estimated flows when $\mathcal { E } < 0 . 0 5 \mathrm { m }$ or $\mathcal { E } ^ { \prime } < 5 \%$ , where $\mathcal { E } ^ { \prime }$ is the relative error; $A c c _ { 1 0 }$ denotes the percentage of estimated flows where $\mathcal { E } < 0 . 1 \mathrm { m }$ or $\mathcal { E } ^ { \prime } < 1 0 \%$ ; and $\theta _ { \epsilon }$ which is the mean angle error between the estimated and ground-truth scene flows.
+
+Implementation details We defined our neural prior for scene flow as a simple coordinate-based MLP architecture with 8 hidden layers, a fixed length of 128 for the hidden units, Rectified Linear Unit (ReLU) activation and shared weights across points. The network input is the 3D point cloud $\mathbf { P } _ { t - 1 }$ , and the output is the scene flow F. We used PyTorch [41] for the implementation and optimized the objective function with Adam [24]. The weights were randomly initialized. We set a fixed learning rate of $8 \mathrm { { e } - 3 }$ and run the optimization for 5k iterations with early stopping on the loss. For our settings and datasets, we found the optimization to mostly converge in less than 1k iterations. All experiments were run on a machine with an NVIDIA Quadro P5000 GPU and a 16 Intel(R) Xeon(R) W-2145 CPU $\textcircled { a } 3 . 7 0 \mathrm { G H z }$ .
+
+Training setup for the learning-based methods We used the publicly available implementation of the learning-based methods to perform our experiments. The training settings for each method are: FlowNet3D and full-supervised PointPWC-Net: trained on FlyingThings3D with supervision; and self-supervised Just Go with the Flow and PointPWC-Net: trained on FlyingThings3D with supervision and fine-tuned on domain-matched datasets with self-supervision (i.e., fine-tuned and tested on statistically similar data, KITTI, nuScenes, Argoverse respectively). Note that FlowNet3D and full-supervised PointPWC-Net was only trained on the synthetic FlyingThings3D to demonstrate the poor generalizability of learning-based methods to other domains. Just Go with the Flow and self-supervised PointPWC-Net were trained using self-supervision, and although they do not require ground-truth annotations, they still require large-scale datasets for training to achieve competitive performance. Note that these self-supervised methods were both pretrained on fullylabeled FlyingThings3D to provide adequate full supervision.
+
+Optimization setup for the non-learning methods Non-rigid ICP [1] was originally proposed for mesh registration. We adapted it for point cloud registration. A graph prior was recently proposed in [45] to optimize scene flow from point clouds. We implemented the method using the hyperparameters defined by the authors. The weight for the graph prior term is set to 10, the number of neighbors $k$ to build the $k$ -NN graph, if not explicitly specified, is set to 50, the learning rate to 0.1, and the number of iterations to $1 . 5 \mathrm { k }$ .
+
+![](images/6ed524c4c1b3c3aa1576df39f5e971c34bb88f937c39f48cb63284eb463c5f50.jpg)  
+Figure 2: We analyzed the performance of our method on the KITTI test set when varying the number of hidden layers and hidden units of the MLP architecture. Accuracy is the $A c c _ { 5 }$ metric.
+
+# 4.1 Choosing the neural prior architecture
+
+Fig. 2 shows how the performance of our method
+
+is affected when varying the neural prior MLP architecture: number of hidden layers and hidden units. The experiments were performed on the KITTI test set and all points included (i.e., without point sampling). The average number of points for the KITTI dataset is about $3 0 \mathrm { k }$ . We ran our method five times with different random seeds to include the uncertainty levels in the plot.
+
+Overall, the performance of our method improved as we increased the number of hidden layers and hidden units. For small numbers of hidden layers (e.g., 1 and 2), the performance deteriorated when the number of hidden units is large, around 128 and 256 (or $2 ^ { 7 }$ and $2 ^ { 8 }$ ). We chose the MLP architecture with the best performance with relatively low computation time for our following experiments: 8 hidden layers and 128 $( 2 ^ { 7 } )$ hidden units.
+
+# 4.2 Comparing to other methods
+
+Table 1 shows how our method stands against other state-of-the-art methods on different datasets and metrics. We set the number of points to 2,048, which follows the experimental protocols as in FlowNet3D [30] and Graph prior [45]. We ran the experiments 5 times to report uncertainties for runtime optimization methods (i.e., our method and the graph prior method [45]) with uncertainties for each run. The learning-based methods and non-rigid ICP are deterministic during runtime.
+
+Our method achieved better performance in most datasets and metrics. We considered the well-known Non-rigid ICP as a baseline for the non-learning methods ( ). Our method outperformed the recent graph prior method by a large margin. The supervised FlowNet3D and PointPWC-Net (using fullsupervision loss) methods ( ) had better performance on FlyingThings3D because they were trained on it with supervision. If the dataset is out-of-the-distribution, these supervised methods produced unreliable results. The self-supervised methods ( ), Just Go with the Flow and PointPWC-Net (using self-supervision loss), despite not being exposed to ground-truth labels during training, still generated better results than supervised methods—showing that self-supervision is an important direction for scene flow estimation. Still, it is remarkable that with a simple MLP regularizer and an optimization framework, our method can robustly estimate scene flow from point clouds with great accuracy.
+
+Table 1: Performance of our method and others across different datasets and metrics. are supervised methods trained on the synthetic FlyingThings3D dataset. are self-supervised methods trained on FlyingThings3D with supervision and fine-tuned with self-supervision on matched datasets. $\bullet$ are non-learning methods that do not rely on training data. All experiments were run with 2,048 points. ↑ means larger values are better while $\downarrow$ means smaller values are better. We did no report standard deviations smaller than 1e−2.   
+
+<table><tr><td></td><td colspan="4">FlyingThings3D [33] Train:19,967 samples,Test: 2,000 samples</td><td colspan="4">nuScenes Scene Flow [7] Train:1,513 samples,Test:310 samples</td></tr><tr><td></td><td colspan="4"></td><td colspan="4"></td></tr><tr><td></td><td>ε(m)↓</td><td>Acc5(%) ↑</td><td>Acc10(%) ↑</td><td>0e(rad)↓</td><td>ε(m)↓</td><td>Acc5(%) ↑</td><td>Acc10(%) ↑</td><td>θe(rad)↓</td></tr><tr><td>·FlowNet3D [30]</td><td>0.134</td><td>22.64</td><td>54.17</td><td>0.305</td><td>0.505</td><td>2.12</td><td>10.81</td><td>0.620</td></tr><tr><td>·PointPWC-Net [68]</td><td>0.121</td><td>29.09</td><td>61.70</td><td>0.229</td><td>0.442</td><td>7.64</td><td>22.32</td><td>0.497</td></tr><tr><td>Just Go with the Flow [38]</td><td colspan="4"></td><td>0.625</td><td>6.09</td><td>0.139 22.42</td><td>0.432</td></tr><tr><td>·PointPWC-Net [68]</td><td></td><td></td><td></td><td></td><td>0.431</td><td>6.87</td><td></td><td>0.406</td></tr><tr><td>·Non-rigid ICP [1]</td><td>0.339 0.255</td><td>14.05</td><td>35.68</td><td>0.480</td><td>0.402</td><td>6.99</td><td>21.01</td><td>0.492 0.337</td></tr><tr><td>•Graph prior [45] ·Ours</td><td>0.234</td><td>16.56±0.02 19.16±0.23</td><td>42.05±0.02 46.74±0.46</td><td>0.362 0.341</td><td>0.289 0.175±0.01</td><td>20.12±0.01 35.18±1.32</td><td>43.54±0.02 63.45±0.46</td><td>0.279±0.04</td></tr><tr><td></td><td></td><td colspan="3">KITTI Scene Flow [34,35]</td><td colspan="4"></td></tr><tr><td></td><td colspan="4">Train:100 samples,Test: 50 samples</td><td colspan="4">Argoverse Scene Flow [8] Train:2,691 samples,Test: 212 samples</td></tr><tr><td></td><td>ε(m)↓</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>·FlowNet3D [30]</td><td></td><td>Acc5(%) ↑</td><td>Acc10(%) ↑</td><td>θ(rad)↓</td><td>ε(m)↓</td><td>Acc5(%) ↑</td><td>Acc10(%) ↑</td><td>θ(rad)↓</td></tr><tr><td>·PointPWC-Net [68]</td><td>0.199 0.142</td><td>10.44 29.91</td><td>38.89 59.83</td><td>0.386 0.239</td><td>0.455 0.405</td><td>1.34 8.25</td><td>6.12 25.47</td><td>0.736 0.674</td></tr><tr><td>• Just Go with the Flow [38]</td><td>0.218</td><td></td><td>34.38</td><td>0.254</td><td>0.542</td><td>8.80</td><td>20.28</td><td>0.715</td></tr><tr><td>·PointPWC-Net [68]</td><td>0.177</td><td>10.17 13.29</td><td>42.15</td><td>0.272</td><td>0.409</td><td>9.79</td><td>29.31</td><td>0.643</td></tr><tr><td>•Non-rigid ICP[1]</td><td>0.338</td><td>22.06</td><td>43.03</td><td>0.460</td><td>0.461</td><td>4.27</td><td>13.90</td><td>0.741</td></tr><tr><td>·Graph prior [45]</td><td>0.099</td><td>63.60±0.09</td><td>81.18±0.08</td><td>0.176</td><td>0.257</td><td>25.24±0.04</td><td>47.60±0.02</td><td>0.467</td></tr><tr><td>●Ours</td><td>0.050±0.01</td><td>81.68±2.00</td><td>93.19±1.30</td><td>0.133±0.01</td><td>0.159±0.01</td><td>38.43±0.48</td><td>63.08±0.59</td><td>0.374±0.01</td></tr></table>
+
+Our PointPWC-Net results were different from those reported in the PointPWC-Net paper. The reasons are: In the official PointPWC-Net implementation, there is a threshold to limit the lidar point cloud within 35 meters of distance from the center. In our experiments, we used all points available in all ranges (up to $8 5 \mathrm { m }$ ). Lidar point clouds get sparser as the distance increases, making it challenging to estimate scene flow in far ranges and sparse regions. Nevertheless, we did not shy away from this fact in our experiments. Also, in the original PointPWC-Net experiments, the authors used 8,192 points. In ours, we used 2,048 points. Naturally, there exists a performance gap between our reported results and theirs. We decided to use 2,048 points to follow the experiment protocols proposed in FlowNet3D [30] and graph prior [45] to facilitate comparisons across different datasets and models. Although simple and tested on sparse point clouds (2,048 points), our method achieved impressive results on different datasets. Please find further details and additional experiments in the supplementary material.
+
+# 4.3 Estimating scene flow from large point clouds with high density
+
+Real-world point clouds collected from depth sensors such as lidar typically have tens of thousands of points. We evaluated the performance of our method on large point clouds and compared it against the graph prior method. The KITTI and Argoverse Scene Flow datasets were used, given that both have large point clouds with high density.
+
+Fig. 3 shows the performance on the KITTI Scene Flow dataset, in terms of accuracy and computational time, of our method and the graph prior method when varying the number of points. Our method’s accuracy $( A c c _ { 5 } )$ increased as the number of points grew until around $2 0 \mathrm { k }$ points and then saturated, while the computational time slowly increased. In contrast, the graph prior achieved lower accuracy and dramatic growth in computation. The computational complexity of our MLPbased prior grows linearly in the number of points, ${ \mathcal { O } } ( n )$ , while the graph prior grows quadratically in the number of points, $O ( n ^ { 2 } )$ . The graph prior relies on the construction of a graph Laplacian matrix to use as a regularizer (i.e., $\mathbf { L } \in \mathbb { R } ^ { n \times n }$ , where $n$ is the number of points).
+
+![](images/7bfd34e47dd43d03bf9fd7c3f68ff60d0fcf519317338b9f561fcf07286cd321.jpg)  
+Figure 3: Performance of our neural prior and the graph prior [45] when varying the number of points. Our method achieved higher accuracy $( A c c _ { 5 } )$ and better time complexity. Results were averaged over the KITTI Scene Flow dataset.
+
+Table 2: Performance of our neural prior and the graph prior [45] when using all points available. Our method achieved better performance on all metrics by a margin while being ${ \sim } 5 \times$ faster if $k { = } 5 0$ and ${ \sim } 1 0 \times$ faster if $k { = } 2 0 0$ (for KITTI).   
+
+<table><tr><td></td><td colspan="5">KITTI Scene Flow Average number of points:~30k</td></tr><tr><td></td><td>m↓ (m)</td><td>Acc5↑ (%)</td><td>Acc10 ↑ (%）</td><td>↓ (rad)</td><td>Time↓ (s)</td></tr><tr><td>Graph prior (k=50)</td><td>0.225</td><td>65.50</td><td>70.32</td><td>0.277</td><td>162.97</td></tr><tr><td>Graph prior (k=200)</td><td>0.082</td><td>84.00</td><td>88.45</td><td>0.141</td><td>310.12</td></tr><tr><td>Ours</td><td>0.025</td><td>95.68</td><td>98.00</td><td>0.085</td><td>38.33</td></tr><tr><td></td><td colspan="5">Argoverse Scene Flow Average number of points:~50k</td></tr><tr><td></td><td>m↓ (m)</td><td>Acc5↑ (%</td><td>AcC10 ↑ (%）</td><td>0↓ (rad)</td><td>Time↓ (s)</td></tr><tr><td>Graph prior (k=50)</td><td>0.249</td><td>46.92</td><td>61.72</td><td>0.494</td><td>410.21</td></tr><tr><td>Ours</td><td>0.043</td><td>86.04</td><td>94.07</td><td>0.244</td><td>84.46</td></tr></table>
+
+The accuracy of the graph prior method degraded after $1 0 \mathrm { k }$ points. The graph regularizer needs more than $5 \mathrm { k }$ iterations for higher density point clouds to converge to a reasonable solution or carefully tuned schedulers to accelerate its convergence. Moreover, the $k$ -NN graph is built with 50 neighbors, and for higher density point clouds, larger graphs might be necessary for a better regularization.
+
+Table 2 shows a quantitative comparison between our method and the graph prior method on the KITTI and Argoverse Scene Flow datasets when using all points. Our method achieved better performance on all metrics. We also reported results for the graph prior method when setting the number of neighbors, $k$ , to 200. According to our results in Fig. 1, the scene flow accuracy saturated after $k { = } 2 0 0$ . Fig. 4 shows a qualitative example of a scene flow estimation using our method.
+
+These results show that our method scales to large point clouds with high density, and gain a great improvement in performance with much denser point clouds. In contrast, training supervised/selfsupervised models with high-density point clouds is not always practical due to high memory usage. Typically, such models are trained with up to $^ \mathrm { 8 k }$ points.
+
+Performance and inference time tradeoffs Estimating scene flow with runtime optimization is usually slower than using learning-based methods $1 0 \times - 1 0 0 \times$ slower). The trade-off, however, depends on the application. If robustness/generalizability is not an issue but rather the inference time, our proposed objective can train a self-supervised model and act as a surrogate of our non-learning method but inheriting the faster inference time from the trained model. We show an example of lidar point cloud densification (Section 4.6) to generate denser point clouds that can be used for robotics applications such as offline mapping, creating denser depth maps, etc., that would not require real-time inference.
+
+![](images/9906cdfb7e25e2253620a37a71751575201cd46bb2da6f98e13546830603e5da.jpg)  
+Figure 5: Example of a failure case. Partial scene from FlyingThings3D. Our nearest-neighbor-based loss might fail when handling large missing parts, occlusions, and bad correspondences. Green points are the target, and red points are the shifted blue points by the estimated scene flow (yellow arrows).
+
+# 4.4 Limitations
+
+Although our method achieved better computational complexity than other non-learning-based methods, the inference time is still limiting for some applications that demand real-time inferences.
+
+![](images/c3e17dbc70f1b47198675197465005670258e14285403e83ec79ed89d060791c.jpg)  
+Figure 4: Qualitative example of a scene flow estimation using our proposed method. The complex and highly dynamic driving scene is from the Argoverse Scene Flow dataset. The scene flow estimated by our method is close to the ground truth. We also show a prediction using the supervised FlowNet3D method trained on FlyingThings3D and fine-tuned on the KITTI Scene Flow dataset. Note how the scene flow deviated from the ground truth when the inference was performed on an out-of-the-distribution sample. The scene flow color encodes the magnitude (color intensity) and direction (angle) of the flow vectors. For example, the purplish vehicles are heading northeast.
+
+Another limitation is that the loss function we used relies on nearest neighbors, which might find bad correspondences due to partial point clouds and occlusions. Fig. 5 shows a failure case because of the nearest-neighbor-based distance loss. Few corresponding points due to missing parts in the scene might lead to incorrect flow estimations.
+
+# 4.5 A continuous scene flow field
+
+Our neural prior implicitly regularizes the scene flow through the coordinate-based MLP network. Thus, our method allows for a continuous scene flow representation instead of a discrete representation such as in graph-based priors.
+
+An advantage of a continuous scene flow representation is that we can reuse the optimal network weights to estimate dense long-term correspondences across a sequence of point clouds (see Section 4.6). Fig. 6 shows how the estimated scene flow and the continuous flow field change as the optimization converges to a solution.
+
+# 4.6 Application: scene flow integration
+
+Here we demonstrate how to use our method to perform scene flow integration. Given a temporal sequence of point sets, $\{ S _ { 0 } , S _ { 1 } , S _ { 2 } , \cdots , S _ { M } \}$ , we first optimize for the pairwise scene flows, $\{ \bar { \mathcal { F } } _ { 0  1 } , \mathcal { F } _ { 1  2 } , \cdot \cdot \cdot , \mathcal { F } _ { M - 1  M } \}$ , using our proposed method with the optimal neural prior parameters, $\{ \Theta _ { 0 \to 1 } ^ { * } , \Theta _ { 1 \to 2 } ^ { * } , \cdot \cdot \cdot , \Theta _ { M \cdot 1 \to M } ^ { * } \}$ , saved. Then, starting from $\mathbf { f } _ { 0 \to 1 }$ , we can integrate long-term flows using the classic Forward Euler method recursively for $m { = } 1 { : } M { - } 1$ iterations as
+
+$$
+\mathbf { f } _ { 0  m + 1 } = \mathbf { f } _ { 0  m } + g ( \mathbf { p } _ { 0 } + \mathbf { f } _ { 0  m } ; \mathbf { \Theta } _ { m  m + 1 } ^ { * } ) .
+$$
+
+![](images/7babee050ac204a5e621e3988b0b8bd4fe0ee47a084e7efd30aa5381f74778b7.jpg)  
+Figure 6: Example showing how the estimated scene flow and the continuous flow field (bottom) given by our neural prior change as the optimization converges to a solution. We show a top-view dynamic driving scene from Argoverse Scene Flow. The scene flow color encodes the magnitude (color intensity) and direction (angle) of the flow vectors. For example, the purplish vehicles are heading northeast. The red arrow shows the position and direction of travel of the autonomous vehicle, which is stopped, waiting for a pedestrian to cross the street. Note how the predicted scene flow is close to the ground truth at iteration 2k. At iteration 0, the scene flow is random, given the random initialization of the neural prior. Thus having very small magnitudes for the random directions. As the optimization went on, the flow fields became better constrained. A simple way to interpret the flow fields is to imagine sampling a point at any location in the continuous scene flow field to recover an estimated flow vector. For example, imagine sampling a point around the orange region in the flow field at iteration $2 \mathrm { k }$ (green arrow in the bottom right). The direction of the flow vector will be pointing southeast at a specific magnitude, similar to the vehicles in the orange region.
+
+This gives the long-term scene flow $\mathcal { F } _ { 0  M } = \{ ( \mathbf { f } _ { 0  M } ) _ { i } \} _ { i = 0 } ^ { | S _ { 0 } | }$ , from ${ \cal S } _ { 0 }  { \cal S } _ { M }$ . Note that we are not relying on discrete nearest-neighbor-based interpolations. Our neural scene flow prior is a continuous representation that naturally provides continuous scene flow estimations. Fig. 7 shows an example of an Argoverse scene where we applied such a technique to integrate 10 point clouds into a single frame to densify the point cloud.
+
+# 5 Conclusion
+
+We show that how hand-designed coordinate-based network architecture can serve as a new type of implicit regularizer in the runtime optimization for the scene flow problem. Our neural prior gets rid of the need for massive labeled/unlabeled training data while being scalable with dense point clouds. Additionally, since we infer prior knowledge from the network architectures instead of from data, our approach can generalize to out-of-the-distribution scenarios as compared to learning-based methods. The continuous flow representation also allows for flow integration across a long sequence that can be used in many robotics applications such as offline mapping. We believe this paper shows a promising direction for large-scale, real-world scene flow estimation without data supervision.
+
+# Broader impact
+
+Our proposed neural scene flow prior allows for estimating 3D motion fields from large-scale dynamic scenes without annotating massive data but preserving generalizability—making it useful for scenarios where motion prediction is required. Especially in computer vision and robotics communities, robust scene flow estimations is crucial. For example, autonomous vehicles need to predict future distribution of the surrounding objects to avoid catastrophe in dynamic environments; safe human-computer interaction is enabled with precise dynamic flow predictions. Our work also encourages further exploration in the combination of the innovative coordinate-based networks and classical runtime optimization algorithm. This dataless approach offers an affordable solution for many industrial problems without adequate supervision (e.g., autonomous driving).
+
+![](images/cdd654298f0a1f841aa96d169666b291d33e24d2ec537989e7069db918de6334.jpg)  
+Figure 7: Example of a scene flow integration to densify an Argoverse lidar point cloud. The left and middle columns are a top and front view of the point cloud, respectively. The rightmost column shows the accumulated point cloud projected onto the image. Note the smearing effect on the dynamic objects when rigidly accumulating the point clouds (middle row). Accumulation using our neural prior nicely produced a denser point cloud while taking care of all dynamic objects in the scene. Here, rigid means that the point cloud accumulation was performed using a rigid registration method (i.e., ICP) where rigid 6-DoF poses are used for the registrations.
+
+However, as AI-based research, it could be misused by malicious groups for nefarious purposes. For example, the collected data might contain sensitive information that potentially invades privacy, and can be used for illegal data trading. Moreover, such research has the potential to be used in autonomous weapons and military drones. The potential evil use of our method needs attention and needs to be prevented. We hope to motivate the community to take full advantage of the innovations of this work to benefit society.
+
+# Acknowledgments and Disclosure of Funding
+
+The authors would like to thank Chen-Hsuan Lin for useful discussions through the project, review and help with section 3. We thank Haosen Xing for careful review of the entire manuscript and assistance in several parts of the paper, Jianqiao Zheng for helpful discussions. We thank all anonymous reviewers for their valuable comments and suggestions to make our paper stronger.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] We described the limitations in the experiment section, and we also provided failure cases.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] We discussed in the Broader Impact section   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We cite all the data we used in the experiment section. And we will release code in personal GitHub repository.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] In the experiment section, we specify implementation details for all methods, and all data we used.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We report error bars for optimization-based methods which have uncertainties.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] We included in the implementation details of the experiments section.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] We have cited all the data we used.   
+(b) Did you mention the license of the assets? [N/A]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [N/A]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/4c1EiEvivpx/4c1EiEvivpx_content_list.json

@@ -0,0 +1,1190 @@
+[
+    {
+        "type": "text",
+        "text": "Neural Scene Flow Prior ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Xueqian Li∗1,2 Jhony Kaesemodel Pontes1 Simon Lucey2 1Argo AI 2The University of Adelaide ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+            535,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Before the deep learning revolution, many perception algorithms were based on runtime optimization in conjunction with a strong prior/regularization penalty. A prime example of this in computer vision is optical and scene flow. Supervised learning has largely displaced the need for explicit regularization. Instead, they rely on large amounts of labeled data to capture prior statistics, which are not always readily available for many problems. Although optimization is employed to learn the neural network, the weights of this network are frozen at runtime. As a result, these learning solutions are domain-specific and do not generalize well to other statistically different scenarios. This paper revisits the scene flow problem that relies predominantly on runtime optimization and strong regularization. A central innovation here is the inclusion of a neural scene flow prior, which uses the architecture of neural networks as a new type of implicit regularizer. Unlike learning-based scene flow methods, optimization occurs at runtime, and our approach needs no offline datasets—making it ideal for deployment in new environments such as autonomous driving. We show that an architecture based exclusively on multilayer perceptrons (MLPs) can be used as a scene flow prior. Our method attains competitive—if not better—results on scene flow benchmarks. Also, our neural prior’s implicit and continuous scene flow representation allows us to estimate dense long-term correspondences across a sequence of point clouds. The dense motion information is represented by scene flow fields where points can be propagated through time by integrating motion vectors. We demonstrate such a capability by accumulating a sequence of lidar point clouds. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "State-of-the-art results have recently been achieved by learning-based models [17, 30, 47, 60, 68] for the scene flow problem—the task of estimating 3D motion fields from dynamic scenes. However, such models heavily rely on large-scale data to capture prior knowledge, which is not always readily available. Scene flow annotations are expensive, and most methods train on synthetic and unrealistic scenarios to fine-tune on small real datasets. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Poor generalization to unseen, out-of-the-distribution inputs is another problem. Prior information is generally limited to the statistics of the data used for training. Real-world applications such as autonomous driving require robust solutions to low-level vision tasks such as depth, optical, and scene flow estimation that work in statistically different scenarios. Inspired by recent innovations that make use of coordinate-based networks (i.e., pixels or 3D positions as inputs) [9, 36, 37, 39, 56] for 3D modeling and rendering, we investigate the use of such networks to regularize the scene flow problem without any learning directly from point clouds. Optimization happens at runtime, and instead of learning a prior from data, the network structure itself captures the prior information. It is not limited to the statistics of a specific dataset. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Optimizing neural networks at execution time is not new. Ulyanov et al. [63] showed that a randomly initialized convolutional network could be used as a handcrafted prior for standard inverse problems such as image denoising, super-resolution, and inpainting. Ding and Feng [12] proposed a runtime optimization method (DeepMapping) for rigid pose estimation using deep neural networks. Although such deep image priors, deep mapping, and coordinate-based networks for neural scene representations have been successfully applied for inverse problems, rendering, and rigid registration, none has yet investigated (to the best of our knowledge) the use of network-based priors for regularizing scene flow directly from point clouds. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our proposed neural prior is based on a simple multilayer perceptron (MLP) architecture, and we show it is powerful enough to regularize scene flow given two point clouds implicitly. The input to the network is 3D points, and the output is a regularized scene flow. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our neural prior allows for a continuous scene flow representation instead of discrete such as in graph Laplacian-based priors, e.g., [45]. We show how the flow fields captured by our neural ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/577ce96cb63b1a5d41e8fdcb8a10e98dc3b649626ddf6ea769bcd2e8f56dbedf.jpg",
+        "image_caption": [
+            "Figure 1: Our neural scene flow prior method achieved higher accuracy while being ${ \\sim } 1 0 \\times$ faster than the recent runtime optimization method graph prior [45]. The evaluation was on the KITTI Scene Flow test set, where each point cloud size varies from $1 4 \\mathrm { k }$ to $6 8 \\mathrm { k }$ points. In our method, we fixed the number of hidden layers in the MLP to 4 and varied the number of hidden units. In the graph prior method, we varied the number of neighbors to create the graph. Accuracy uses the $A c c _ { 5 }$ metric as defined in the experiments section. Learningbased methods might still be $1 0 \\times - 1 0 0 \\times$ faster than the runtime optimization methods, but they still lack generalization and have memory issues when dealing with large point clouds—with tens of thousands of points. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "prior can be employed to estimate long-term correspondences across a sequence of point clouds. The continuous scene flow allows for better integration of motions across time. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our results are promising and competitive to supervised [30], self-supervised [38, 68], and nonlearning methods [1, 45] (see Table 1). Our method also scales to real-world point clouds with tens of thousands of points while achieving better accuracy and time complexity than recent runtime optimization methods (see Fig. 1). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related work ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Non-learning-based scene flow Scene flow is the uplift from the optical flow, which is proposed by Vedula et al. [64] as the non-rigid motion field in the 3D space. The authors proposed the optimization-based scene flow estimation using image sequences to infer reconstruction knowledge of the flow in 3D surfaces. Successive RGB/RGB-D image-based work [4, 18–21, 27, 43, 44, 50] used probability-based estimation, coarse-to-fine techniques, 6-DoF parameterization, or object segmentation, etc., to improve accuracy and computation time. Although image-based scene flow methods are widely used, direct estimation of the scene flow from the point cloud is still possible through non-rigid registration methods, such as [1, 10, 26, 42]. In this paper, we focus on point cloud-based scene flow estimation. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learning-based scene flow Image-based learning methods [6, 51, 54, 60, 70] use convolution and data supervision to solve scene flow from monocular or RGB-D images with the available depth information. Other image-based methods [22, 23, 32, 52, 53] take care of extra occlusion cues in the large-scale autonomous driving scenes. Point-based learning methods have become more prevalent with the rapid development of point cloud feature learning [28, 48, 49, 65, 67]. FlowNet3D [30] is a seminal work that estimates scene flow using PointNet+ $^ +$ [49]. Successive work [17, 31, 47, 66] extends point-based learning methods using different feature extraction techniques. One obvious drawback of these supervised learning methods is the demand for sufficient ground truth labels. Besides, supervised methods lack generalizability while eventually only fitting domain-specific data. Self-supervised methods [25, 38, 61, 68], on the other hand, replaced the loss between the prediction and the ground truth flow with a point distance loss to use the point cloud itself as supervision. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Self-supervision can adapt to different datasets and maintain certain generalizability. Nonetheless, massive training data are still required for sufficient learning. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The graph Laplacian method Graph Laplacian [2] is widely used to smooth the surface in mesh processing [5, 14, 57, 58], point cloud denoising [11, 71], etc. Here we talk about the recent scene flow estimation using Graph Laplacian [45]. The method explicitly constructed a graph of the point cloud to constrain the non-rigid scene flow as rigid within a specific range. While as a dataless runtime optimization, the method is heavily affected by the hyperparameters of the graph and loses scalability when the point cloud becomes larger or the neighbors in the graph grow. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Deep neural prior, implicit functions, and neural rendering Although large-scale data helps in feature representation [59], end-to-end learning still requires high computation capacity and readily available training datasets. Instead, Ulyanov et al. [63] proposed a new style of optimization that uses a convolutional neural network to infer prior knowledge from the network architecture. One broader interest is to extend the idea of the network being an image function to the 3D shape modeling, and implicitly represent the continuous shape as level sets of neural networks. By directly mapping the 3D input to binary occupancy nets [9, 36], or signed distance functions [3, 39, 55], it is powerful to model the 3D geometries in a continuous space using the coordinate-based network. The following Scene Representation Networks [56] takes advantage of the coordinate-based network and renders view synthetic images. Mildenhall et al. proposed a seminal work NeRF [37], which is a novel way to do neural volume rendering using both point positions and viewing directions based on the coordinate-based network. Dynamic scene synthesis work [13, 16, 29, 40, 46, 62, 69] follows the NeRF framework, and integrates the motions to generate dynamic scenes. Some of the work [16, 29] use scene flow to further constrain or segment dynamic scenes. An interesting work that solves for the rigid alignment between point clouds using runtime optimization is DeepMapping [12]. However, this work only deals with rigid motion, and the network architecture is more complex than coordinate-based networks. In this work, we are interested in coordinate-based networks to address the large-scale, real-world scene flow problem. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Approach ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Problem definition Let $S _ { 1 }$ and $S _ { 2 }$ be two 3D point clouds sampled from a dynamic scene at time $t$ -1 and $t$ . The number of points in each point cloud, $| S _ { 1 } |$ and $| S _ { 2 } |$ , are typically different and not in correspondence. A 3D point $\\mathbf { p } \\in S _ { 1 }$ moving from time $t \\mathrm { - } 1$ to time $t$ can be modeled by a translational vector (or flow vector) $\\bar { \\mathbf { f } } \\in \\mathbb { R } ^ { 3 }$ , where $\\mathbf { p ^ { \\prime } } = \\mathbf { p } + \\mathbf { f }$ . The collection of flow vectors for all 3D points is the scene flow $\\mathcal { F } = \\{ \\mathbf { f } _ { i } \\} _ { i = 1 } ^ { | S _ { 1 } | }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Optimization We want to optimize for a scene flow $\\mathcal { F }$ that minimizes the distance between the two point clouds, $S _ { 1 }$ and $S _ { 2 }$ . Given the non-rigidity assumption of the scene, the optimization is inherently unconstrained. Thus, a regularization term C is necessary to constrain the motion field. We therefore solve for scene flow as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/206835b510edd42b62d8275987d68999cc088c13bd3e6d629f5d9582df1f7a80.jpg",
+        "text": "$$\n\\mathcal { F } ^ { * } = \\underset { \\mathcal { F } } { \\arg \\operatorname* { m i n } } \\sum _ { \\mathbf { p } \\in S _ { 1 } } \\mathbf { D } \\left( \\mathbf { p } + \\mathbf { f } , S _ { 2 } \\right) + \\lambda \\mathbf { C } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { D }$ is a function to compute the distance from the perturbed point $\\mathbf { p }$ by the flow vector f to its closest neighbor in $S _ { 2 }$ . C is a regularizer (e.g., Laplacian regularizer), and $\\lambda$ is a weighting factor for the regularizer. In this paper, we want to investigate using a neural prior to regularize the scene flow. ",
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+    {
+        "type": "text",
+        "text": "3.1 Neural scene flow prior ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Learning-based scene flow methods learn scene flow priors from a large number of examples. As in Deep Image Prior [63], we want to investigate if the structure of a neural network by itself is sufficient to capture a scene flow prior without any learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here we use a neural network as an implicit regularizer. The parameters are optimized as ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/5a4520b308c33dab7b9391eb44dfc87a9a7c95af3442ccd98abf1a633d9cc974.jpg",
+        "text": "$$\n\\Theta ^ { * } = \\underset { \\Theta } { \\arg \\operatorname* { m i n } } \\sum _ { \\mathbf { p } \\in S _ { 1 } } \\mathrm { D } \\left( \\mathbf { p } + g \\left( \\mathbf { p } ; \\Theta \\right) , S _ { 2 } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $g$ is a neural network parameterized by $\\Theta$ to regularize the scene flow $\\mathcal { F }$ . The input to $g$ is $\\mathbf { p }$ which is the point to be disturbed by the flow. The output of $g$ is $\\mathbf { f }$ and thus $\\mathbf { f } ^ { * } = g \\left( \\mathbf { p } ; \\mathbf { \\bar { \\Theta } } \\right)$ . For the distance function D, we define it as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/72a5ddc38fee78f131ae4bde8331a3a84ef18dd8369ae3d03e127830010ffdef.jpg",
+        "text": "$$\n\\mathbf { D } \\left( \\mathbf { p } , \\pmb { S } \\right) = \\operatorname* { m i n } _ { \\mathbf { x } \\in \\pmb { S } } \\| \\mathbf { p } - \\mathbf { x } \\| _ { 2 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "In practice, we use it bidirectionally for both point sets, which is equivalent to Chamfer distance [15]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The objective in Eq. (2), is for the forward scene flow $\\mathcal { F }$ , which is the one we are interested in. However, it has been shown in [30, 38], that a cycle consistency regularizer encourages better scene flow estimations. The extra regularizer simply enforces the backward flow to be similar to the forward flow, $\\mathcal { F } _ { b w d } \\approx \\mathcal { F }$ . The optimal backward flow is defined as $\\mathbf { f } _ { b w d } ^ { * } = g \\left( \\mathbf { p } ^ { \\prime } ; \\mathbf { \\Theta } \\Theta _ { b w d } ^ { * } \\right)$ , where $\\mathbf { p } ^ { \\prime }$ is the shifted point by the forward flow as $\\mathbf { p } + \\mathbf { f }$ . Note that the network $g$ is the same but with different parameters, $\\Theta _ { b w d }$ . Using the backward flow as an additional constraint, the optimal network weights are solved as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/afd69c14313134c67b18cda9e78debe25311c406f3939cc9732958125d30b20d.jpg",
+        "text": "$$\n\\Theta ^ { * } , \\Theta _ { b w d } ^ { * } = \\underset { \\Theta , \\Theta _ { b w d } } { \\mathrm { a r g } \\mathrm { m i n } } \\sum _ { \\mathbf { p } \\in S _ { 1 } } \\mathbf { D } \\left( \\mathbf { p } + g \\left( \\mathbf { p } ; \\Theta \\right) , S _ { 2 } \\right) + \\sum _ { \\mathbf { p } ^ { \\prime } \\in S _ { 1 } ^ { \\prime } } \\mathbf { D } \\left( \\mathbf { p } ^ { \\prime } + g \\left( \\mathbf { p } ^ { \\prime } ; \\Theta _ { b w d } \\right) , S _ { 1 } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $ { \\boldsymbol { S } } _ { 1 } ^ { \\prime }$ is the shifted $S _ { 1 }$ by the forward flow, i.e., $S _ { 1 } ^ { \\prime } = S _ { 1 } { + } \\mathcal { F }$ . Please find more details in the supplementary material. For the network $g$ , we use MLPs with ReLU activations. The objective function in Eq. (4) can be optimized by gradient descent techniques using off-the-shelf frameworks with automatic differentiation. We show in the experiments section how the architecture of the neural prior affects performance by varying the number of hidden layers and units. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Why use a neural scene flow prior? Deep learning relies on massive amounts of data and computational resources to capture prior statistics. Although learning methods have achieved impressive results in most tasks, they still struggle when deployed in environments where the statistics are different from those captured during learning. Our intuition is that a neural prior acts as a strong implicit regularizer that constrains dynamic motion fields to be as smooth as possible. A neural scene flow prior also scales to large scenes while achieving high-fidelity results at a low computational cost. Our proposed method with 8 hidden layers and 128 hidden units has about $1 1 6 \\mathrm { k }$ parameters. FlowNet3D [30], for example, has about 1.2M parameters. Our method has ${ \\sim } 1 0 \\times$ fewer parameters than the state-of-the-art supervised methods while achieving competitive, if not better, results. Lastly, our deep scene flow prior captures a continuous flow field that allows us to perform better scene flow interpolation across a sequence of point clouds. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We evaluated the performance (accuracy, generalizability, and computational cost) of our neural prior for scene flow on synthetic and real-world datasets. We performed experiments on different neural network settings and analyzed the performance of the neural prior to regularizing scene flow. Remarkably, we show that a simple MLP-based prior to regularize scene flow is enough to achieve competitive results to the state-of-the-art scene flow methods. ",
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+    {
+        "type": "text",
+        "text": "Datasets We used four scene flow datasets: 1) FlyingThings3D [33] which is an extensive collection of randomly moving synthetic objects. We used the preprocessed data from [30]; 2) KITTI [34, 35] which has real-world self-driving scenes. We used the subset released by [30]; 3) Argoverse [8] and 4) nuScenes [7] are two large-scale autonomous driving datasets with challenging dynamic scenes. However, there are no official scene flow annotations. We followed the data processing method in [45] to collect pseudo-ground-truth scene flow. Ground points were removed from lidar point clouds as in [30] (please refer to the supplementary material for more details). ",
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+    {
+        "type": "text",
+        "text": "Metrics We employed the widely used metrics as in [30, 38, 45, 68] to evaluate our method, which are: $\\mathcal { E }$ to denote the end-point error (EPE), which is the mean absolute distance of two point clouds; $A c c _ { 5 }$ to denote the accuracy in percentage of estimated flows when $\\mathcal { E } < 0 . 0 5 \\mathrm { m }$ or $\\mathcal { E } ^ { \\prime } < 5 \\%$ , where $\\mathcal { E } ^ { \\prime }$ is the relative error; $A c c _ { 1 0 }$ denotes the percentage of estimated flows where $\\mathcal { E } < 0 . 1 \\mathrm { m }$ or $\\mathcal { E } ^ { \\prime } < 1 0 \\%$ ; and $\\theta _ { \\epsilon }$ which is the mean angle error between the estimated and ground-truth scene flows. ",
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+        "type": "text",
+        "text": "Implementation details We defined our neural prior for scene flow as a simple coordinate-based MLP architecture with 8 hidden layers, a fixed length of 128 for the hidden units, Rectified Linear Unit (ReLU) activation and shared weights across points. The network input is the 3D point cloud $\\mathbf { P } _ { t - 1 }$ , and the output is the scene flow F. We used PyTorch [41] for the implementation and optimized the objective function with Adam [24]. The weights were randomly initialized. We set a fixed learning rate of $8 \\mathrm { { e } - 3 }$ and run the optimization for 5k iterations with early stopping on the loss. For our settings and datasets, we found the optimization to mostly converge in less than 1k iterations. All experiments were run on a machine with an NVIDIA Quadro P5000 GPU and a 16 Intel(R) Xeon(R) W-2145 CPU $\\textcircled { a } 3 . 7 0 \\mathrm { G H z }$ . ",
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+    {
+        "type": "text",
+        "text": "Training setup for the learning-based methods We used the publicly available implementation of the learning-based methods to perform our experiments. The training settings for each method are: FlowNet3D and full-supervised PointPWC-Net: trained on FlyingThings3D with supervision; and self-supervised Just Go with the Flow and PointPWC-Net: trained on FlyingThings3D with supervision and fine-tuned on domain-matched datasets with self-supervision (i.e., fine-tuned and tested on statistically similar data, KITTI, nuScenes, Argoverse respectively). Note that FlowNet3D and full-supervised PointPWC-Net was only trained on the synthetic FlyingThings3D to demonstrate the poor generalizability of learning-based methods to other domains. Just Go with the Flow and self-supervised PointPWC-Net were trained using self-supervision, and although they do not require ground-truth annotations, they still require large-scale datasets for training to achieve competitive performance. Note that these self-supervised methods were both pretrained on fullylabeled FlyingThings3D to provide adequate full supervision. ",
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+    {
+        "type": "text",
+        "text": "Optimization setup for the non-learning methods Non-rigid ICP [1] was originally proposed for mesh registration. We adapted it for point cloud registration. A graph prior was recently proposed in [45] to optimize scene flow from point clouds. We implemented the method using the hyperparameters defined by the authors. The weight for the graph prior term is set to 10, the number of neighbors $k$ to build the $k$ -NN graph, if not explicitly specified, is set to 50, the learning rate to 0.1, and the number of iterations to $1 . 5 \\mathrm { k }$ . ",
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+    {
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+        "img_path": "images/6ed524c4c1b3c3aa1576df39f5e971c34bb88f937c39f48cb63284eb463c5f50.jpg",
+        "image_caption": [
+            "Figure 2: We analyzed the performance of our method on the KITTI test set when varying the number of hidden layers and hidden units of the MLP architecture. Accuracy is the $A c c _ { 5 }$ metric. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.1 Choosing the neural prior architecture ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Fig. 2 shows how the performance of our method ",
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+    {
+        "type": "text",
+        "text": "is affected when varying the neural prior MLP architecture: number of hidden layers and hidden units. The experiments were performed on the KITTI test set and all points included (i.e., without point sampling). The average number of points for the KITTI dataset is about $3 0 \\mathrm { k }$ . We ran our method five times with different random seeds to include the uncertainty levels in the plot. ",
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+        "type": "text",
+        "text": "Overall, the performance of our method improved as we increased the number of hidden layers and hidden units. For small numbers of hidden layers (e.g., 1 and 2), the performance deteriorated when the number of hidden units is large, around 128 and 256 (or $2 ^ { 7 }$ and $2 ^ { 8 }$ ). We chose the MLP architecture with the best performance with relatively low computation time for our following experiments: 8 hidden layers and 128 $( 2 ^ { 7 } )$ hidden units. ",
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+    {
+        "type": "text",
+        "text": "4.2 Comparing to other methods ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 1 shows how our method stands against other state-of-the-art methods on different datasets and metrics. We set the number of points to 2,048, which follows the experimental protocols as in FlowNet3D [30] and Graph prior [45]. We ran the experiments 5 times to report uncertainties for runtime optimization methods (i.e., our method and the graph prior method [45]) with uncertainties for each run. The learning-based methods and non-rigid ICP are deterministic during runtime. ",
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+    {
+        "type": "text",
+        "text": "Our method achieved better performance in most datasets and metrics. We considered the well-known Non-rigid ICP as a baseline for the non-learning methods ( ). Our method outperformed the recent graph prior method by a large margin. The supervised FlowNet3D and PointPWC-Net (using fullsupervision loss) methods ( ) had better performance on FlyingThings3D because they were trained on it with supervision. If the dataset is out-of-the-distribution, these supervised methods produced unreliable results. The self-supervised methods ( ), Just Go with the Flow and PointPWC-Net (using self-supervision loss), despite not being exposed to ground-truth labels during training, still generated better results than supervised methods—showing that self-supervision is an important direction for scene flow estimation. Still, it is remarkable that with a simple MLP regularizer and an optimization framework, our method can robustly estimate scene flow from point clouds with great accuracy. ",
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+    {
+        "type": "table",
+        "img_path": "images/6ab9aa581db998063030612dcf0d4e21ccba62bf34b0e1718a0f307f150bf1a7.jpg",
+        "table_caption": [
+            "Table 1: Performance of our method and others across different datasets and metrics. are supervised methods trained on the synthetic FlyingThings3D dataset. are self-supervised methods trained on FlyingThings3D with supervision and fine-tuned with self-supervision on matched datasets. $\\bullet$ are non-learning methods that do not rely on training data. All experiments were run with 2,048 points. ↑ means larger values are better while $\\downarrow$ means smaller values are better. We did no report standard deviations smaller than 1e−2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">FlyingThings3D [33] Train:19,967 samples,Test: 2,000 samples</td><td colspan=\"4\">nuScenes Scene Flow [7] Train:1,513 samples,Test:310 samples</td></tr><tr><td></td><td colspan=\"4\"></td><td colspan=\"4\"></td></tr><tr><td></td><td>ε(m)↓</td><td>Acc5(%) ↑</td><td>Acc10(%) ↑</td><td>0e(rad)↓</td><td>ε(m)↓</td><td>Acc5(%) ↑</td><td>Acc10(%) ↑</td><td>θe(rad)↓</td></tr><tr><td>·FlowNet3D [30]</td><td>0.134</td><td>22.64</td><td>54.17</td><td>0.305</td><td>0.505</td><td>2.12</td><td>10.81</td><td>0.620</td></tr><tr><td>·PointPWC-Net [68]</td><td>0.121</td><td>29.09</td><td>61.70</td><td>0.229</td><td>0.442</td><td>7.64</td><td>22.32</td><td>0.497</td></tr><tr><td>Just Go with the Flow [38]</td><td colspan=\"4\"></td><td>0.625</td><td>6.09</td><td>0.139 22.42</td><td>0.432</td></tr><tr><td>·PointPWC-Net [68]</td><td></td><td></td><td></td><td></td><td>0.431</td><td>6.87</td><td></td><td>0.406</td></tr><tr><td>·Non-rigid ICP [1]</td><td>0.339 0.255</td><td>14.05</td><td>35.68</td><td>0.480</td><td>0.402</td><td>6.99</td><td>21.01</td><td>0.492 0.337</td></tr><tr><td>•Graph prior [45] ·Ours</td><td>0.234</td><td>16.56±0.02 19.16±0.23</td><td>42.05±0.02 46.74±0.46</td><td>0.362 0.341</td><td>0.289 0.175±0.01</td><td>20.12±0.01 35.18±1.32</td><td>43.54±0.02 63.45±0.46</td><td>0.279±0.04</td></tr><tr><td></td><td></td><td colspan=\"3\">KITTI Scene Flow [34,35]</td><td colspan=\"4\"></td></tr><tr><td></td><td colspan=\"4\">Train:100 samples,Test: 50 samples</td><td colspan=\"4\">Argoverse Scene Flow [8] Train:2,691 samples,Test: 212 samples</td></tr><tr><td></td><td>ε(m)↓</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>·FlowNet3D [30]</td><td></td><td>Acc5(%) ↑</td><td>Acc10(%) ↑</td><td>θ(rad)↓</td><td>ε(m)↓</td><td>Acc5(%) ↑</td><td>Acc10(%) ↑</td><td>θ(rad)↓</td></tr><tr><td>·PointPWC-Net [68]</td><td>0.199 0.142</td><td>10.44 29.91</td><td>38.89 59.83</td><td>0.386 0.239</td><td>0.455 0.405</td><td>1.34 8.25</td><td>6.12 25.47</td><td>0.736 0.674</td></tr><tr><td>• Just Go with the Flow [38]</td><td>0.218</td><td></td><td>34.38</td><td>0.254</td><td>0.542</td><td>8.80</td><td>20.28</td><td>0.715</td></tr><tr><td>·PointPWC-Net [68]</td><td>0.177</td><td>10.17 13.29</td><td>42.15</td><td>0.272</td><td>0.409</td><td>9.79</td><td>29.31</td><td>0.643</td></tr><tr><td>•Non-rigid ICP[1]</td><td>0.338</td><td>22.06</td><td>43.03</td><td>0.460</td><td>0.461</td><td>4.27</td><td>13.90</td><td>0.741</td></tr><tr><td>·Graph prior [45]</td><td>0.099</td><td>63.60±0.09</td><td>81.18±0.08</td><td>0.176</td><td>0.257</td><td>25.24±0.04</td><td>47.60±0.02</td><td>0.467</td></tr><tr><td>●Ours</td><td>0.050±0.01</td><td>81.68±2.00</td><td>93.19±1.30</td><td>0.133±0.01</td><td>0.159±0.01</td><td>38.43±0.48</td><td>63.08±0.59</td><td>0.374±0.01</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Our PointPWC-Net results were different from those reported in the PointPWC-Net paper. The reasons are: In the official PointPWC-Net implementation, there is a threshold to limit the lidar point cloud within 35 meters of distance from the center. In our experiments, we used all points available in all ranges (up to $8 5 \\mathrm { m }$ ). Lidar point clouds get sparser as the distance increases, making it challenging to estimate scene flow in far ranges and sparse regions. Nevertheless, we did not shy away from this fact in our experiments. Also, in the original PointPWC-Net experiments, the authors used 8,192 points. In ours, we used 2,048 points. Naturally, there exists a performance gap between our reported results and theirs. We decided to use 2,048 points to follow the experiment protocols proposed in FlowNet3D [30] and graph prior [45] to facilitate comparisons across different datasets and models. Although simple and tested on sparse point clouds (2,048 points), our method achieved impressive results on different datasets. Please find further details and additional experiments in the supplementary material. ",
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+    {
+        "type": "text",
+        "text": "4.3 Estimating scene flow from large point clouds with high density ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Real-world point clouds collected from depth sensors such as lidar typically have tens of thousands of points. We evaluated the performance of our method on large point clouds and compared it against the graph prior method. The KITTI and Argoverse Scene Flow datasets were used, given that both have large point clouds with high density. ",
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+    {
+        "type": "text",
+        "text": "Fig. 3 shows the performance on the KITTI Scene Flow dataset, in terms of accuracy and computational time, of our method and the graph prior method when varying the number of points. Our method’s accuracy $( A c c _ { 5 } )$ increased as the number of points grew until around $2 0 \\mathrm { k }$ points and then saturated, while the computational time slowly increased. In contrast, the graph prior achieved lower accuracy and dramatic growth in computation. The computational complexity of our MLPbased prior grows linearly in the number of points, ${ \\mathcal { O } } ( n )$ , while the graph prior grows quadratically in the number of points, $O ( n ^ { 2 } )$ . The graph prior relies on the construction of a graph Laplacian matrix to use as a regularizer (i.e., $\\mathbf { L } \\in \\mathbb { R } ^ { n \\times n }$ , where $n$ is the number of points). ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/7bfd34e47dd43d03bf9fd7c3f68ff60d0fcf519317338b9f561fcf07286cd321.jpg",
+        "image_caption": [
+            "Figure 3: Performance of our neural prior and the graph prior [45] when varying the number of points. Our method achieved higher accuracy $( A c c _ { 5 } )$ and better time complexity. Results were averaged over the KITTI Scene Flow dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/dec1293dd77c7f46fe348db526b88149aab86e894544a646963d3fe64636fa34.jpg",
+        "table_caption": [
+            "Table 2: Performance of our neural prior and the graph prior [45] when using all points available. Our method achieved better performance on all metrics by a margin while being ${ \\sim } 5 \\times$ faster if $k { = } 5 0$ and ${ \\sim } 1 0 \\times$ faster if $k { = } 2 0 0$ (for KITTI). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"5\">KITTI Scene Flow Average number of points:~30k</td></tr><tr><td></td><td>m↓ (m)</td><td>Acc5↑ (%)</td><td>Acc10 ↑ (%）</td><td>↓ (rad)</td><td>Time↓ (s)</td></tr><tr><td>Graph prior (k=50)</td><td>0.225</td><td>65.50</td><td>70.32</td><td>0.277</td><td>162.97</td></tr><tr><td>Graph prior (k=200)</td><td>0.082</td><td>84.00</td><td>88.45</td><td>0.141</td><td>310.12</td></tr><tr><td>Ours</td><td>0.025</td><td>95.68</td><td>98.00</td><td>0.085</td><td>38.33</td></tr><tr><td></td><td colspan=\"5\">Argoverse Scene Flow Average number of points:~50k</td></tr><tr><td></td><td>m↓ (m)</td><td>Acc5↑ (%</td><td>AcC10 ↑ (%）</td><td>0↓ (rad)</td><td>Time↓ (s)</td></tr><tr><td>Graph prior (k=50)</td><td>0.249</td><td>46.92</td><td>61.72</td><td>0.494</td><td>410.21</td></tr><tr><td>Ours</td><td>0.043</td><td>86.04</td><td>94.07</td><td>0.244</td><td>84.46</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "The accuracy of the graph prior method degraded after $1 0 \\mathrm { k }$ points. The graph regularizer needs more than $5 \\mathrm { k }$ iterations for higher density point clouds to converge to a reasonable solution or carefully tuned schedulers to accelerate its convergence. Moreover, the $k$ -NN graph is built with 50 neighbors, and for higher density point clouds, larger graphs might be necessary for a better regularization. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows a quantitative comparison between our method and the graph prior method on the KITTI and Argoverse Scene Flow datasets when using all points. Our method achieved better performance on all metrics. We also reported results for the graph prior method when setting the number of neighbors, $k$ , to 200. According to our results in Fig. 1, the scene flow accuracy saturated after $k { = } 2 0 0$ . Fig. 4 shows a qualitative example of a scene flow estimation using our method. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "These results show that our method scales to large point clouds with high density, and gain a great improvement in performance with much denser point clouds. In contrast, training supervised/selfsupervised models with high-density point clouds is not always practical due to high memory usage. Typically, such models are trained with up to $^ \\mathrm { 8 k }$ points. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Performance and inference time tradeoffs Estimating scene flow with runtime optimization is usually slower than using learning-based methods $1 0 \\times - 1 0 0 \\times$ slower). The trade-off, however, depends on the application. If robustness/generalizability is not an issue but rather the inference time, our proposed objective can train a self-supervised model and act as a surrogate of our non-learning method but inheriting the faster inference time from the trained model. We show an example of lidar point cloud densification (Section 4.6) to generate denser point clouds that can be used for robotics applications such as offline mapping, creating denser depth maps, etc., that would not require real-time inference. ",
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+    {
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+        "img_path": "images/9906cdfb7e25e2253620a37a71751575201cd46bb2da6f98e13546830603e5da.jpg",
+        "image_caption": [
+            "Figure 5: Example of a failure case. Partial scene from FlyingThings3D. Our nearest-neighbor-based loss might fail when handling large missing parts, occlusions, and bad correspondences. Green points are the target, and red points are the shifted blue points by the estimated scene flow (yellow arrows). "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.4 Limitations ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Although our method achieved better computational complexity than other non-learning-based methods, the inference time is still limiting for some applications that demand real-time inferences. ",
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+    {
+        "type": "image",
+        "img_path": "images/c3e17dbc70f1b47198675197465005670258e14285403e83ec79ed89d060791c.jpg",
+        "image_caption": [
+            "Figure 4: Qualitative example of a scene flow estimation using our proposed method. The complex and highly dynamic driving scene is from the Argoverse Scene Flow dataset. The scene flow estimated by our method is close to the ground truth. We also show a prediction using the supervised FlowNet3D method trained on FlyingThings3D and fine-tuned on the KITTI Scene Flow dataset. Note how the scene flow deviated from the ground truth when the inference was performed on an out-of-the-distribution sample. The scene flow color encodes the magnitude (color intensity) and direction (angle) of the flow vectors. For example, the purplish vehicles are heading northeast. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Another limitation is that the loss function we used relies on nearest neighbors, which might find bad correspondences due to partial point clouds and occlusions. Fig. 5 shows a failure case because of the nearest-neighbor-based distance loss. Few corresponding points due to missing parts in the scene might lead to incorrect flow estimations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.5 A continuous scene flow field ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our neural prior implicitly regularizes the scene flow through the coordinate-based MLP network. Thus, our method allows for a continuous scene flow representation instead of a discrete representation such as in graph-based priors. ",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "An advantage of a continuous scene flow representation is that we can reuse the optimal network weights to estimate dense long-term correspondences across a sequence of point clouds (see Section 4.6). Fig. 6 shows how the estimated scene flow and the continuous flow field change as the optimization converges to a solution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.6 Application: scene flow integration ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Here we demonstrate how to use our method to perform scene flow integration. Given a temporal sequence of point sets, $\\{ S _ { 0 } , S _ { 1 } , S _ { 2 } , \\cdots , S _ { M } \\}$ , we first optimize for the pairwise scene flows, $\\{ \\bar { \\mathcal { F } } _ { 0  1 } , \\mathcal { F } _ { 1  2 } , \\cdot \\cdot \\cdot , \\mathcal { F } _ { M - 1  M } \\}$ , using our proposed method with the optimal neural prior parameters, $\\{ \\Theta _ { 0 \\to 1 } ^ { * } , \\Theta _ { 1 \\to 2 } ^ { * } , \\cdot \\cdot \\cdot , \\Theta _ { M \\cdot 1 \\to M } ^ { * } \\}$ , saved. Then, starting from $\\mathbf { f } _ { 0 \\to 1 }$ , we can integrate long-term flows using the classic Forward Euler method recursively for $m { = } 1 { : } M { - } 1$ iterations as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/290b2be3a3a378be7162151c508f4917dce76b82a8d9d672092754e70a449947.jpg",
+        "text": "$$\n\\mathbf { f } _ { 0  m + 1 } = \\mathbf { f } _ { 0  m } + g ( \\mathbf { p } _ { 0 } + \\mathbf { f } _ { 0  m } ; \\mathbf { \\Theta } _ { m  m + 1 } ^ { * } ) .\n$$",
+        "text_format": "latex",
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+        "page_idx": 7
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+            "Figure 6: Example showing how the estimated scene flow and the continuous flow field (bottom) given by our neural prior change as the optimization converges to a solution. We show a top-view dynamic driving scene from Argoverse Scene Flow. The scene flow color encodes the magnitude (color intensity) and direction (angle) of the flow vectors. For example, the purplish vehicles are heading northeast. The red arrow shows the position and direction of travel of the autonomous vehicle, which is stopped, waiting for a pedestrian to cross the street. Note how the predicted scene flow is close to the ground truth at iteration 2k. At iteration 0, the scene flow is random, given the random initialization of the neural prior. Thus having very small magnitudes for the random directions. As the optimization went on, the flow fields became better constrained. A simple way to interpret the flow fields is to imagine sampling a point at any location in the continuous scene flow field to recover an estimated flow vector. For example, imagine sampling a point around the orange region in the flow field at iteration $2 \\mathrm { k }$ (green arrow in the bottom right). The direction of the flow vector will be pointing southeast at a specific magnitude, similar to the vehicles in the orange region. "
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+        "text": "This gives the long-term scene flow $\\mathcal { F } _ { 0  M } = \\{ ( \\mathbf { f } _ { 0  M } ) _ { i } \\} _ { i = 0 } ^ { | S _ { 0 } | }$ , from ${ \\cal S } _ { 0 }  { \\cal S } _ { M }$ . Note that we are not relying on discrete nearest-neighbor-based interpolations. Our neural scene flow prior is a continuous representation that naturally provides continuous scene flow estimations. Fig. 7 shows an example of an Argoverse scene where we applied such a technique to integrate 10 point clouds into a single frame to densify the point cloud. ",
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+        "text": "5 Conclusion ",
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+        "text": "We show that how hand-designed coordinate-based network architecture can serve as a new type of implicit regularizer in the runtime optimization for the scene flow problem. Our neural prior gets rid of the need for massive labeled/unlabeled training data while being scalable with dense point clouds. Additionally, since we infer prior knowledge from the network architectures instead of from data, our approach can generalize to out-of-the-distribution scenarios as compared to learning-based methods. The continuous flow representation also allows for flow integration across a long sequence that can be used in many robotics applications such as offline mapping. We believe this paper shows a promising direction for large-scale, real-world scene flow estimation without data supervision. ",
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+        "text": "Our proposed neural scene flow prior allows for estimating 3D motion fields from large-scale dynamic scenes without annotating massive data but preserving generalizability—making it useful for scenarios where motion prediction is required. Especially in computer vision and robotics communities, robust scene flow estimations is crucial. For example, autonomous vehicles need to predict future distribution of the surrounding objects to avoid catastrophe in dynamic environments; safe human-computer interaction is enabled with precise dynamic flow predictions. Our work also encourages further exploration in the combination of the innovative coordinate-based networks and classical runtime optimization algorithm. This dataless approach offers an affordable solution for many industrial problems without adequate supervision (e.g., autonomous driving). ",
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+            "Figure 7: Example of a scene flow integration to densify an Argoverse lidar point cloud. The left and middle columns are a top and front view of the point cloud, respectively. The rightmost column shows the accumulated point cloud projected onto the image. Note the smearing effect on the dynamic objects when rigidly accumulating the point clouds (middle row). Accumulation using our neural prior nicely produced a denser point cloud while taking care of all dynamic objects in the scene. Here, rigid means that the point cloud accumulation was performed using a rigid registration method (i.e., ICP) where rigid 6-DoF poses are used for the registrations. "
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+        "text": "However, as AI-based research, it could be misused by malicious groups for nefarious purposes. For example, the collected data might contain sensitive information that potentially invades privacy, and can be used for illegal data trading. Moreover, such research has the potential to be used in autonomous weapons and military drones. The potential evil use of our method needs attention and needs to be prevented. We hope to motivate the community to take full advantage of the innovations of this work to benefit society. ",
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+        "text": "Acknowledgments and Disclosure of Funding ",
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+        "text": "The authors would like to thank Chen-Hsuan Lin for useful discussions through the project, review and help with section 3. We thank Haosen Xing for careful review of the entire manuscript and assistance in several parts of the paper, Jianqiao Zheng for helpful discussions. We thank all anonymous reviewers for their valuable comments and suggestions to make our paper stronger. ",
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+        "text": "References ",
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parse/train/Bke4KsA5FX/Bke4KsA5FX.md

@@ -0,0 +1,598 @@
+# GENERATIVE CODE MODELING WITH GRAPHS
+
+Marc Brockschmidt, Miltiadis Allamanis, Alexander Gaunt Microsoft Research   
+Cambridge, UK   
+{mabrocks,miallama,algaunt}@microsoft.com
+
+Oleksandr Polozov Microsoft Research Redmond, WA, USA polozov@microsoft.com
+
+# ABSTRACT
+
+Generative models for source code are an interesting structured prediction problem, requiring to reason about both hard syntactic and semantic constraints as well as about natural, likely programs. We present a novel model for this problem that uses a graph to represent the intermediate state of the generated output. Our model generates code by interleaving grammar-driven expansion steps with graph augmentation and neural message passing steps. An experimental evaluation shows that our new model can generate semantically meaningful expressions, outperforming a range of strong baselines.
+
+# 1 INTRODUCTION
+
+Learning to understand and generate programs is an important building block for procedural artificial intelligence and more intelligent software engineering tools. It is also an interesting task in the research of structured prediction methods: while imbued with formal semantics and strict syntactic rules, natural source code carries aspects of natural languages, since it acts as a means of communicating intent among developers. Early works in the area have shown that approaches from natural language processing can be applied successfully to source code (Hindle et al., 2012), whereas the programming languages community has had successes in focusing exclusively on formal semantics. More recently, methods handling both modalities (i.e., the formal and natural language aspects) have shown successes on important software engineering tasks (Raychev et al., 2015; Bichsel et al., 2016; Allamanis et al., 2018b) and semantic parsing (Yin & Neubig, 2017; Rabinovich et al., 2017).
+
+However, current generative models of source code mostly focus on only one of these modalities at a time. For example, program synthesis tools based on enumeration and deduction (Solar-Lezama, 2008; Polozov & Gulwani, 2015; Feser et al., 2015; Feng et al., 2018) are successful at generating programs that satisfy some (usually incomplete) formal specification but are often obviously wrong on manual inspection, as they cannot distinguish unlikely from likely, “natural” programs. On the other hand, learned code models have succeeded in generating realistic-looking programs (Maddison & Tarlow, 2014; Bielik et al., 2016; Parisotto et al., 2017; Rabinovich et al., 2017; Yin & Neubig, 2017). However, these programs often fail to be semantically relevant, for example because variables are not used consistently.
+
+In this work, we try to overcome these challenges for generative code models and present a general method for generative models that can incorporate structured information that is deterministically available at generation time. We focus our attention on generating source code and follow the ideas of program graphs (Allamanis et al., 2018b) that have been shown to learn semantically meaningful representations of (pre-existing) programs. To achieve this, we lift grammar-based tree decoder models into the graph setting, where the diverse relationships between various elements of the generated code can be modeled. For this, the syntax tree under generation is augmented with additional edges denoting known relationships (e.g., last use of variables). We then interleave the steps of the generative procedure with neural message passing (Gilmer et al., 2017) to compute more precise representations of the intermediate states of the program generation. This is fundamentally different from sequential generative models of graphs (Li et al., 2018; Samanta et al., 2018), which aim to generate all edges and nodes, whereas our graphs are deterministic augmentations of generated trees.
+
+To summarize, we present $a$ ) a general graph-based generative procedure for highly structured objects, incorporating rich structural information; $b$ ) ExprGen, a new code generation task focused on generating small, but semantically complex expressions conditioned on source code context; and $c$ ) a comprehensive experimental evaluation of our generative procedure and a range of baseline methods from the literature.
+
+<table><tr><td>Algorithm1 Pseudocode for Expand</td></tr><tr><td>Input: Context c,partial AST a,node v to expand 1:hy ← getRepresentation(c,a, v) 2: rhs ← pickProduction(u,hu) 3:for child node type l ∈ rhs do 4: (a,u)←insertChild(a,l) if l is nonterminal type then</td></tr></table>
+
+![](images/3eefeb5f4fcb3c569ae182457a795a26cc9afb96181aaaa89e9672922a8a7531.jpg)  
+Figure 1: Example for ExprGen, target expression to be generated is marked . Taken from BenchmarkDotNet, lightly edited for formatting.
+
+# 2 BACKGROUND & TASK
+
+The most general form of the code generation task is to produce a (partial) program in a programming language given some context information $c$ . This context information can be natural language (as in, e.g., semantic parsing), input-output examples (e.g., inductive program synthesis), partial program sketches, etc. Early methods generate source code as a sequence of tokens (Hindle et al., 2012; Hellendoorn & Devanbu, 2017) and sometimes fail to produce syntactically correct code. More recent models are sidestepping this issue by using the target language’s grammar to generate abstract syntax trees (ASTs) (Maddison & Tarlow, 2014; Bielik et al., 2016; Parisotto et al., 2017; Yin & Neubig, 2017; Rabinovich et al., 2017), which are syntactically correct by construction.
+
+In this work, we follow the AST generation approach. The key idea is to construct the AST $a$ sequentially, by expanding one node at a time using production rules from the underlying programming language grammar. This simplifies the code generation task to a sequence of classification problems, in which an appropriate production rule has to be chosen based on the context information and the partial AST generated so far. In this work, we simplify the problem further — similar to Maddison & Tarlow (2014); Bielik et al. (2016) — by fixing the order of the sequence to always expand the left-most, bottom-most nonterminal node. Alg. 1 illustrates the common structure of AST-generating models. Then, the probability of generating a given AST $a$ given some context $c$ is
+
+$$
+p ( a \mid c ) = \prod _ { t } p ( a _ { t } \mid c , a _ { < t } ) ,
+$$
+
+where $a _ { t }$ is the production choice at step $t$ and $a _ { < t }$ the partial syntax tree generated before step $t$ .
+
+Code Generation as Hole Completion We introduce the ExprGen task of filling in code within a hole of an otherwise existing program. This is similar, but not identical to the auto-completion function in a code editor, as we assume information about the following code as well and aim to generate whole expressions rather than single tokens. The ExprGen task also resembles program sketching (Solar-Lezama, 2008) but we give no other (formal) specification other than the surrounding code. Concretely, we restrict ourselves to expressions that have Boolean, arithmetic or string type, or arrays of such types, excluding expressions of other types or expressions that use project-specific APIs. An example is shown in Fig. 1. We picked this subset because it already has rich semantics that can require reasoning about the interplay of different variables, while it still only relies on few operators and does not require to solve the problem of open vocabularies of full programs, where an unbounded number of methods would need to be considered.
+
+In our setting, the context $c$ is the pre-existing code around a hole for which we want to generate an expression. This also includes the set of variables $v _ { 1 } , \ldots , v _ { \ell }$ that are in scope at this point, which can be used to guide the decoding procedure (Maddison & Tarlow, 2014). Note, however, that our method is not restricted to code generation and can be easily extended to all other tasks and domains that can be captured by variations of Alg. 1 (e.g. in NLP).
+
+# 3 GRAPH DECODING FOR SOURCE CODE
+
+To tackle the code generation task presented in the previous section, we have to make two design choices: (a) we need to find a way to encode the code context $c , v _ { 1 } , \ldots , v _ { \ell }$ and (b) we need to construct a model that can learn $p ( \dot { a } _ { t } \mid c , a _ { < t } )$ well. We do not investigate the question of encoding the context in this paper, and use two existing methods in our experiments in Sect. 5. Both these encoders yield a distributed vector representation for the overall context, representations $h _ { t _ { 1 } } , \ldots , h _ { t _ { T } }$ for all tokens in the context, and separate representations for each of the in-scope variables $v _ { 1 } , \ldots , v _ { \ell }$ summarizing how each variable is used in the context. This information can then be used in the generation process, which is the main contribution of our work and is described in this section.
+
+Overview Our decoder model follows the grammar-driven AST generation strategy of prior work as shown in Alg. 1. The core difference is in how we compute the representation of the node to expand. Maddison & Tarlow (2014) construct it entirely from the representation of its parent in the AST using a log-bilinear model. Rabinovich et al. (2017) construct the representation of a node using the parents of the AST node but also found it helpful to take the relationship to the parent node (e.g. “condition of a while”) into account. Yin & Neubig (2017) on the other hand propose to take the last expansion step into account, which may have finished a subtree “to the left”. In practice, these additional relationships are usually encoded by using gated recurrent units with varying input sizes.
+
+We propose to generalize and unify these ideas using a graph to structure the flow of information in the model. Concretely, we use a variation of attribute grammars (Knuth, 1967) from compiler theory to derive the structure of this graph. We associate each node in the AST with two fresh nodes representing inherited resp. synthesized information (or attributes). Inherited information is derived from the context and parts of the AST that are already generated, whereas synthesized information can be viewed as a “summary” of a subtree. In classical compiler theory, inherited attributes usually contain information such as declared variables and their types (to allow the compiler to check that only declared variables are used), whereas synthesized attributes carry information about a subtree “to the right” (e.g., which variables have been declared). Traditionally, to implement this, the language grammar has to be extended with explicit rules for deriving and synthesizing attributes.
+
+To transfer this idea to the deep learning domain, we represent attributes by distributed vector representations and train neural networks to learn how to compute attributes. Our method for getRepresentation from Alg. 1 thus factors into two parts: a deterministic procedure that turns a partial AST $a _ { < t }$ into a graph by adding additional edges that encode attribute relationships, and a graph neural network that learns from this graph.
+
+Notation Formally, we represent programs as graphs where nodes $u , v , \ldots$ are either the AST nodes or their associated attribute nodes, and typed directed edges $\langle u , \tau , v \rangle \in \mathcal { E }$ connect the nodes according to the flow of information in the model. The edge types $\tau$ represent different syntactic or semantic relations in the information flow, discussed in detail below. We write $\mathcal { E } _ { v }$ for the set of incoming edges into $v$ . We also use functions like parent $( a , v )$ and lastSibling $( a , v )$ that look up and return nodes from the AST $a$ (e.g. resp. the parent node of $v$ or the preceding AST sibling of $v$ ).
+
+Example Consider the AST of the expression $\mathrm { ~  ~ { ~ i ~ } ~ } - \mathrm { ~  ~ { ~ j ~ } ~ }$ shown in Fig. 2 (annotated with attribute relationships) constructed step by step by our model. The AST derivation using the programming language grammar is indicated by shaded backgrounds, nonterminal nodes are shown as rounded rectangles, and terminal nodes are shown as rectangles. We additionally show the variables given within the context as dashed rectangles at the bottom. First, the root node, Expr, was expanded using the production rule $( 1 ) : \mathtt { E x p r } \Longrightarrow \mathtt { E x p r } \ - \ \mathtt { E x p r }$ . Then, its two nonterminal children were in turn expanded to the set of known variables using the produc
+
+# Algorithm 2 Pseudocode for ComputeEdge
+
+Input: Partial AST $a$ , node $v$   
+1: Edge set $\mathcal { E }  \emptyset$   
+2: if $v$ is inherited then   
+3: $\mathcal { E }  \mathcal { E } \cup \{ \langle \mathsf { p a r e n t } ( a , v ) , C h i l d , v \rangle \}$   
+4: if $v$ is terminal node then   
+5: $\mathcal { E }  \mathcal { E } \cup \{ \langle | \mathsf { a s t T o k e n } ( a , v ) , N e x t T o k e n , v \rangle \}$   
+6: if $v$ is variable then   
+7: $\mathcal { E }  \mathcal { E } \cup \{ \langle | { \mathsf { a s t U s e } } ( a , v ) , N e x t U s e , v \rangle \}$   
+8: if $v$ is not first child then   
+9: $\mathcal { E } \gets \mathcal { E } \cup \{ \langle | { \mathsf { a s t } } \mathsf { S i b } | { \mathsf { i n g } } ( a , v ) , N e x t S i b , v \rangle \}$   
+10: else   
+11: $\mathcal { E }  \mathcal { E } \cup \{ \langle u , P a r e n t , v \rangle \mid u \in \mathsf { c h i l d r e n } ( a , v ) \}$   
+12: E ← E ∪ {hinheritedAttr(v), InhToSyn, vi}   
+13: return $\varepsilon$
+
+![](images/3fc653bdb98fab4920239277e9328578a582c0b977eade182f8983bf04276ad3.jpg)  
+Figure 2: Example AST with attribute dependencies, shown constructed step by step in the order of generation. Each AST node (labeled by a terminal or non-terminal) has either one or two associated attribute nodes, shown as its left/right parts. The node IDs are highlighted at the corresponding generation step. Edge color and label indicate edge type. Edges are computed using Alg. 2, but are only depicted after use in message passing. Best viewed in color.
+
+tion rule $( 2 ) : \mathtt { E x p r } \Longrightarrow \nu$ , choosing i for the first variable and $\dot { ] }$ for the second variable (cf. below for details on picking variables).
+
+Attribute nodes are shown overlaying their corresponding AST nodes. For example, the root node is associated with its inherited attributes node 0 and with node 10 for its synthesized attributes. For simplicity, we use the same representation for inherited and synthesized attributes of terminal nodes.
+
+Edges in $\mathbf { \delta } \mathbf { \delta } \mathbf { a } _ { < t }$ We discuss the edges used in our neural attribute grammars $( { \mathcal { N A } } { \mathcal { G } } )$ on our example below, and show them in Fig. 2 using different edge drawing styles for different edge types. Once a node is generated, the edges connecting this node can be deterministically added to $a _ { < t }$ (precisely defined in Alg. 2). The list of different edge types used in our model is as follows:
+
+Child (red) edges connect an inherited attribute node to the inherited attributes nodes of its children, as seen in the edges from node 0. These are the connections in standard syntaxdriven decoders (Maddison & Tarlow, 2014; Parisotto et al., 2017; Yin & Neubig, 2017; Rabinovich et al., 2017). • Parent (green) edges connect a synthesized attribute node to the synthesized attribute node of its AST parent, as seen in the edges leading to node 10. These are the additional connections used by the R3NN decoder introduced by Parisotto et al. (2017). • NextSib (black) edges connect the synthesized attribute node to the inherited attribute node of its next sibling (e.g. from node 5 to node 6). These allow information about the synthesized attribute nodes from a fully generated subtree to flow to the next subtree. NextUse (orange) edges connect the attribute nodes of a variable (since variables are always terminal nodes, we do not distinguish inherited from synthesized attributes) to their next use. Unlike Allamanis et al. (2018b), we do not perform a dataflow analysis, but instead just follow the lexical order. This can create edges from nodes of variables in the context $c$ (for example, from node 1 to 4 in Fig. 2), or can connect AST leaf nodes that represent multiple uses of the same variable within the generated expressions.
+
+• NextToken (blue) edges connect a terminal node (a token) to the next token in the program text, for example between nodes 4 and 6.
+
+• InhToSyn edges (not shown in Fig. 2) connect the inherited attributes nodes to its synthesized attribute nodes. This is not strictly adding any information, but we found it to help with training.
+
+The panels of Fig. 2 show the timesteps at which the representations of particular attribute nodes are computed and added to the graph. For example, in the second step, the attributes for the terminal token i (node 4) in Fig. 2 are computed from the inherited attributes of its AST parent Expr (node 3), the attributes of the last use of the variable i (node 1), and the node label i. In the third step, this computed attribute is used to compute the synthesized attributes of its AST parent Expr (node 5).
+
+Attribute Node Representations To compute the neural attribute representation $\mathbf { h } _ { v }$ of an attribute node $v$ whose corresponding AST node is labeled with $\ell _ { v }$ , we first obtain its incoming edges using Alg. 2 and then use the state update function from Gated Graph Neural Networks (GGNN) (Li et al., 2016). Thus, we take the attribute representations $\mathbf { h } _ { u _ { i } }$ at edge sources $u _ { i }$ , transform them according to the corresponding edge type $t _ { i }$ using a learned function $f _ { t _ { i } }$ , aggregate them (by elementwise summation) and combine them with the learned embedding $\mathsf { e m b } ( \ell _ { v } )$ of the node label $\ell _ { v }$ using a function $g$ :
+
+$$
+\mathbf { h } _ { v } = g ( \mathsf { e m b } ( \ell _ { v } ) , \sum _ { \langle u _ { i } , t _ { i } , v \rangle \in \mathcal { E } _ { v } } f _ { t _ { i } } ( \mathbf { h } _ { u _ { i } } ) )
+$$
+
+In practice, we use a single linear layer for $f _ { t _ { i } }$ and implement $g$ as a gated recurrent unit (Cho et al., 2014). We compute node representations in such an order that all $\mathbf { h } _ { u _ { i } }$ appearing on the right of (2) are already computed. This is possible as the graphs obtained by repeated application of Alg. 2 are directed acyclic graphs rooted in the inherited attribute node of the root node of the AST. We initialize the representation of the root inherited attribute to the representation returned by the encoder for the context information.
+
+Choosing Productions, Variables & Literals We can treat picking production rules as a simple classification problem over all valid production rules, masking out those choices that do not correspond to the currently considered nonterminal. For a nonterminal node $v$ with label $\ell _ { v }$ and inherited attributes $\mathbf { h } _ { v }$ , we thus define
+
+Here, $m \ell _ { v }$ is a mask vector whose value is 0 for valid productions $\ell _ { v } \Rightarrow . . .$ and $- \infty$ for all other productions. In practice, we implement $e$ using a linear layer.
+
+Similarly, we pick variables from the set of variables V in scope using their representations hvvar (initially the representation obtained from the context, and later the attribute representation of the last node in the graph in which they have been used) by using a pointer network (Vinyals et al., 2015). Concretely, to pick a variable at node $v$ , we use learnable linear function $k$ and define
+
+$$
+{ \mathsf { a } } ( { \mathcal { V } } , \mathbf { h } _ { v } ) = \underset { v a r \in \mathcal { V } } { \arg \operatorname* { m a x } } P ( v a r \mid \mathbf { h } _ { v } ) = \underset { v a r \in \mathcal { V } } { \arg \operatorname* { m a x } } k ( \mathbf { h } _ { v } , \mathbf { h } _ { v _ { v a r } } ) .
+$$
+
+Note that since the model always picks a variable from the set of in-scope variables $\nu$ , this generation model can never predict an unknown or out-of-scope variable.
+
+Finally, to generate literals, we combine a small vocabulary $\mathcal { L }$ of common literals observed in the training data and special UNK tokens for each type of literal with another pointer network that can copy one of the tokens $t _ { 1 } \ldots t _ { T }$ from the context. Thus, to pick a literal at node $v$ , we define
+
+$$
+\mathsf { t e r a l } ( \mathcal { V } , \mathbf { h } _ { v } ) = \operatorname * { a r g m a x } _ { l i t \in \mathcal { L } \cup \{ t _ { 1 } \ldots t _ { T } \} } P ( l i t \mid \mathbf { h } _ { v } ) .
+$$
+
+Note that this is the only operation that may produce an unknown token (i.e. an UNK literal). In practice, we implement this by learning two functions $s _ { \mathcal { L } }$ and $s _ { c }$ , such that $s _ { \mathcal { L } } ( \mathbf { h } _ { v } )$ produces a score for each token from the vocabulary and $s _ { c } ( \mathbf { h } _ { v } , h _ { t _ { i } } )$ computes a score for copying token $t _ { i }$ from the context. By computing a softmax over all resulting values and normalizing it by summing up entries corresponding to the same constant, we can learn to approximate the desired $\dot { P } ( l i t \mid \mathbf { h } _ { v } )$ .
+
+Training & Training Objective The different shapes and sizes of generated expressions complicate an efficient training regime. However, note that given a ground truth target tree, we can easily augment it with all additional edges according to Alg. 2. Given that full graph, we can compute a propagation schedule (intuitively, a topological ordering of the nodes in the graph, starting in the root node) that allows to repeatedly apply (2) to obtain representations for all nodes in the graph. By representing a batch of graphs as one large (sparse) graph with many disconnected components, similar to Allamanis et al. (2018b), we can train our graph neural network efficiently. We have released the code for this on https://github.com/Microsoft/graph-based-code-modelling.
+
+Our training procedure thus combines an encoder (cf. Sect. 5), whose output is used to initialize the representation of the root and context variable nodes in our augmented syntax graph, the sequential graph propagation procedure described above, and the decoder choice functions (3) and (4). We train the system end-to-end using a maximum likelihood objective without pre-trained components.
+
+Additional Improvements We extend (3) with an attention mechanism (Bahdanau et al., 2014; Luong et al., 2015) that uses the state $\mathbf { h } _ { v }$ of the currently expanded node $v$ as a key and the context token representations $h _ { t _ { 1 } } , \ldots , h _ { t _ { T } }$ as memories. Experimentally, we found that extending Eqs. 4, 5 similarly did not improve results, probably due to the fact that they already are highly dependent on the context information.
+
+Following Rabinovich et al. (2017), we provide additional information for Child edges. To allow this, we change our setup so that some edge types also require an additional label, which is used when computing the messages sent between different nodes in the graph. Concretely, we extend (2) by considering sets of unlabeled edges ${ \mathcal { E } } _ { v }$ and labeled edges $\mathcal { E } _ { v } ^ { \ell }$ :
+
+$$
+\mathbf { h } _ { v } = g ( \mathsf { e m b } ( \ell _ { v } ) , \sum _ { \substack { ( u _ { i } , t _ { i } , v ) \in \mathcal { E } _ { v } } } f _ { t _ { i } } ( \mathbf { h } _ { u _ { i } } ) + \sum _ { \substack { ( u _ { i } , t _ { i } , \ell _ { i } , v ) \in \mathcal { E } _ { v } ^ { \ell } } } f _ { t _ { i } } ( \mathbf { h } _ { u _ { i } } , \mathsf { e m b } _ { e } ( \ell _ { i } ) ) )
+$$
+
+Thus for labeled edge types, $f _ { t _ { i } }$ takes two inputs and we additionally introduce a learnable embedding for the edge labels. In our experiments, we found it useful to label Child with tuples consisting of the chosen production and the index of the child, i.e., in Fig. 2, we would label the edge from 0 to 3 with $( 2 , 0 )$ , the edge from 0 to 6 with $( 2 , 1 )$ , etc.
+
+Furthermore, we have extended pickProduction to also take the information about available variables into account. Intuitively, this is useful in cases of productions such as $\mathtt { E x p r } \Longrightarrow \mathtt { E x p r . L e n g t h } _ { \mathtt { i n d e m } }$ , which can only be used in a well-typed derivation if an array-typed variable is available. Thus, we extend $e ( \mathbf { h } _ { v } )$ from (3) to additionally take the representation of all variables in scope into account, i.e., $e ( \mathbf { h } _ { v } , r ( \{ \mathbf { h } _ { v _ { v a r } } \mid v a r \in \mathcal { V } \} )$ ), where we have implemented $r$ as a max pooling operation.
+
+# 4 RELATED WORK
+
+Source code generation has been studied in a wide range of different settings (Allamanis et al., 2018a). We focus on the most closely related works in language modeling here. Early works approach the task by generating code as sequences of tokens (Hindle et al., 2012; Hellendoorn & Devanbu, 2017), whereas newer methods have focused on leveraging the known target grammar and generate code as trees (Maddison & Tarlow, 2014; Bielik et al., 2016; Parisotto et al., 2017; Yin & Neubig, 2017; Rabinovich et al., 2017) (cf. Sect. 2 for an overview). While modern models succeed at generating “natural-looking” programs, they often fail to respect simple semantic rules. For example, variables are often used without initialization or written several times without being read inbetween.
+
+Existing tree-based generative models primarily differ in what information they use to decide which expansion rule to use next. Maddison & Tarlow (2014) consider the representation of the immediate parent node, and suggest to consider more information (e.g., nearby tokens). Parisotto et al. (2017) compute a fresh representation of the partial tree at each expansion step using R3NNs (which intuitively perform a leaf-to-root traversal followed by root-to-leaf traversal of the AST). The PHOG model (Bielik et al., 2016) conditions generation steps on the result of learned (decision tree-style) programs, which can do bounded AST traversals to consider nearby tokens and non-terminal nodes. The language also supports a jump to the last node with the same identifier, which can serve as syntactic approximation of data-flow analysis. Rabinovich et al. (2017) only use information about the parent node, but use neural networks specialized to different non-terminals to gain more fine-grained control about the flow of information to different successor nodes. Finally, Amodio et al. (2017) and Yin & Neubig (2017) follow a left-to-right, depth-first expansion strategy, but thread updates to single state (via a gated recurrent unit) through the overall generation procedure, thus giving the pickProduction procedure access to the full generation history as well as the representation of the parent node. Amodio et al. (2017) also suggest the use of attribute grammars, but use them to define a deterministic procedure that collects information throughout the generation process, which is provided as additional feature.
+
+As far as we are aware, previous work has not considered a task in which a generative model fills a hole in a program with an expression. Lanuage model-like methods take into account only the lexicographically previous context of code. The task of Raychev et al. (2014) is near to our ExprGen, but instead focuses on filling holes in sequences of API calls. There, the core problem is identifying the correct function to call from a potentially large set of functions, given a sequence context. In contrast, ExprGen requires to handle arbitrary code in the context, and then to build possibly complex expressions from a small set of operators. Allamanis et al. (2018b) consider similar context, but are only picking a single variable from a set of candidates, and thus require no generative modeling.
+
+# 5 EVALUATION
+
+Dataset We have collected a dataset for our ExprGen task from 593 highly-starred open-source C# projects on GitHub, removing any near-duplicate files, following the work of Lopes et al. (2017). We parsed all $C ^ { \# }$ files and identified all expressions of the fragment that we are considering (i.e., restricted to numeric, Boolean and string types, or arrays of such values; and not using any user-defined functions). We then remove the expression, perform a static analysis to determine the necessary context information and extract a sample. For each sample, we create an abstract syntax tree by coarsening the syntax tree generated by the $C ^ { \# }$ compiler Roslyn. This resulted in 343 974 samples overall with 4.3 $( \pm 3 . 8 ) $ tokens per expression to generate, or alternatively 3.7 $( \pm 3 . 1 )$ production steps. We split the data into four separate sets. A “test-only” dataset is made up from $\mathord { \sim } 1 0 0 \mathrm { k }$ samples generated from 114 projects. The remaining data we split into training-validation-test sets $( 3 : 1 : 1 )$ ), keeping all expressions collected from a single source file within a single fold. Samples from our dataset can be found in the supplementary material. Our decoder uses the grammar made up by 222 production rules observed in the ASTs of the training set, which includes rules such as $\mathtt { E x p r } \Longrightarrow \mathtt { E x p r } + \mathtt { E x p r }$ for binary operations, $\operatorname { E x p r } \Longrightarrow$ Expr.Equals(Expr) for built-in methods, etc.
+
+Encoders We consider two models to encode context information. Seq is a two-layer bi-directional recurrent neural network (using a GRU (Cho et al., 2014)) to encode the tokens before and after the “hole” in which we want to generate an expression. Additionally, it computes a representation for each variable var in scope in the context in a similar manner: For each variable var it identifies usages before/after the hole and encodes each of them independently using a second bi-directional two-layer GRU, which processes a window of tokens around each variable usage. It then computes a representation for var by average pooling of the final states of these GRU runs.
+
+The second encoder $\mathcal { G }$ is an implementation of the program graph approach introduced by Allamanis et al. (2018b). We follow the transformation used for the Varmisuse task presented in that paper, i.e., the program is transformed into a graph, and the target expression is replaced by a fresh dummy node. We then run a graph neural network for 8 steps to obtain representations for all nodes in the graph, allowing us to read out a representation for the “hole” (from the introduced dummy node) and for all variables in context. The used context information captured by the GNN is a superset of what existing methods (e.g. language models) consider.
+
+Baseline Decoders We compare our model to re-implementations of baselines from the literature. As our ExprGen task is new, re-using existing implementations is hard and problematic in comparison. Most recent baseline methods can be approximated by ablations of our model. We experimented with a simple sequence decoder with attention and copying over the input, but found it to be substantially weaker than other models in all regards. Next, we consider T ree, our model restricted to using only Child edges without edge labels. This can be viewed as an evolution of Maddison & Tarlow (2014), with the difference that instead of a log-bilinear network that does not maintain state during the generation, we use a GRU. $\mathcal { A } \mathcal { S } \mathcal { N }$ is similar to abstract syntax networks (Rabinovich et al., 2017) and arises as an extension of the T ree model by adding edge labels on Child that encode the chosen production and the index of the child (corresponding to the “field name” Rabinovich et al. (2017)). Finally, Syn follows the work of Yin & Neubig (2017), but uses a GRU instead of an LSTM. For this, we extend T ree by a new NextExp edge that connects nodes to each other in the expansion sequence of the tree, thus corresponding to the action flow (Yin & Neubig, 2017).
+
+Table 1: Evaluation of encoder and decoder combinations on predicting an expression from code context. $\dagger$ : PHOG (Bielik et al., 2016) is only conditioned on the tokens on the left of the expression.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Test (from seen projects)</td><td colspan="4">Test-only (from unseen projects)</td></tr><tr><td>Perplexity</td><td>Well-Typed</td><td>Acc@1</td><td>Acc@5</td><td>Perplexity</td><td>Well-Typed</td><td>Acc@1</td><td>Acc@5</td></tr><tr><td>PHOGt</td><td>1</td><td>1</td><td>34.8%</td><td>42.9%</td><td>1</td><td>1</td><td>28.0%</td><td>37.3%</td></tr><tr><td>Seq→Seq</td><td>87.48</td><td>32.4%</td><td>21.8%</td><td>28.1%</td><td>130.46</td><td>23.4%</td><td>10.8%</td><td>16.8%</td></tr><tr><td>Seq→NAg</td><td>6.81</td><td>53.2%</td><td>17.7%</td><td>33.7%</td><td>8.38</td><td>40.4%</td><td>8.4%</td><td>15.8%</td></tr><tr><td>g→Seq</td><td>93.31</td><td>40.9%</td><td>27.1%</td><td>34.8%</td><td>28.48</td><td>36.3%</td><td>17.2%</td><td>25.6%</td></tr><tr><td>g→Tree</td><td>4.37</td><td>49.3%</td><td>26.8%</td><td>48.9%</td><td>5.37</td><td>41.2%</td><td>19.9%</td><td>36.8%</td></tr><tr><td>g→ASN</td><td>2.62</td><td>78.7%</td><td>45.7%</td><td>62.0%</td><td>3.03</td><td>74.7%</td><td>32.4%</td><td>48.1%</td></tr><tr><td>g →Syn</td><td>2.71</td><td>84.9%</td><td>50.5%</td><td>66.8%</td><td>3.48</td><td>84.5%</td><td>36.0%</td><td>52.7%</td></tr><tr><td>9 →NAg</td><td>2.56</td><td>86.4%</td><td>52.3%</td><td>69.2%</td><td>3.07</td><td>84.5%</td><td>38.8%</td><td>57.0%</td></tr></table>
+
+In all cases, our re-implementations improve on prior work in our variable selection mechanism, which ensures that generated programs only use variables that are defined and in scope. Both Rabinovich et al. (2017) and Yin & Neubig (2017) instead use a copying mechanism from the context. On the other hand, they use RNN modules to generate function names and choose arguments from the context (Yin & Neubig, 2017) and to generate string literals (Rabinovich et al., 2017). Our ExprGen task limits the set of allowed functions and string literals substantially and thus no RNN decoder generating such things is required in our experiments.
+
+The authors of the PHOG (Bielik et al., 2016) language model kindly ran experiments on our data for the ExprGen task, to provide baseline results of a non-neural language model. Note, however, that PHOG does not consider the code context to the right of the expression to generate, and does no additional analyses to determine which variable choices are valid. Extending the model to take more context into account and do some analyses to restrict choices would certainly improve its results.
+
+# 5.1 QUANTITATIVE EVALUATION
+
+Metrics We are interested in the ability of a model to generate valid expressions based on the current code context. To evaluate this, we consider four metrics. As our ExprGen task requires a conditional language model of code, we first consider the per-token perplexity of the model; the lower the perplexity, the better the model fits the real data distribution. We then evaluate how often the generated expression is well-typed (i.e., can be typed in the original code context). We report these metrics for the most likely expression returned by beam search decoding with beam width 5. Finally, we compute how often the ground truth expression was generated (reported for the most likely expression, as well as for the top five expressions). This measure is stricter than semantic equivalence, as an expression $\mathrm { ~ \ j ~ } > \mathrm { ~ \ i ~ }$ will not match the equivalent $\dot { \mathrm { ~ \scriptsize ~ \perp ~ } } < \dot { \mathrm { ~ \scriptsize ~ \jmath ~ } }$ .
+
+Results We show the results of our evaluation in Tab. 1. Overall, the graph encoder architecture seems to be best-suited for this task. All models learn to generate syntactically valid code (which is relatively simple in our domain). However, the different encoder models perform very differently on semantic measures such as well-typedness and the retrieval of the ground truth expression. Most of the type errors are due to usage of an “UNK” literal (for example, the $\mathcal { G }  \mathcal { N A G }$ model only has $4 \%$ type error when filtering out such unknown literals). The results show a clear trend that correlates better semantic results with the amount of information about the partially generated programs employed by the generative models. Transferring a trained model to unseen projects with a new project-specific vocabulary substantially worsens results, as expected. Overall, our $\mathcal { N A G }$ model, combining and adding additional signal sources, seems to perform best on most measures, and seems to be leastimpacted by the transfer.
+
+![](images/7f8da9108d64d1ed4966bd020899b4a3c18291a176ccea5f3cba1c4bea0b40f3.jpg)
+
+Figure 3: Two lightly edited examples from our test set and expressions predicted by different models.   
+More examples can be found in the supplementary material.
+
+# 5.2 QUALITATIVE EVALUATION
+
+As the results in the previous section suggest, the proposed ExprGen task is hard even for the strongest models we evaluated, achieving no more than $50 \%$ accuracy on the top prediction. It is also unsolvable for classical logico-deductive program synthesis systems, as the provided code context does not form a precise specification. However, we do know that most instances of the task are (easily) solvable for professional software developers, and thus believe that machine learning systems can have considerable success on the task.
+
+Fig. 3 shows two (abbreviated) samples from our test set, together with the predictions made by the two strongest models we evaluated. In the first example, we can see that the $\mathcal { G }  \mathcal { N A G }$ model correctly identifies that the relationship between paramCount and methParamCount is important (as they appear together in the blocked guarded by the expression to generate), and thus generates comparison expressions between the two variables. The $\bar { \mathcal { G } }  \mathcal { A } \mathcal { S } \mathcal { N }$ model lacks the ability to recognize that paramCount (or any variable) was already used and thus fails to insert both relevant variables. We found this to be a common failure, often leading to suggestions using only one variable (possibly repeatedly). In the second example, both $\mathcal { G }  \mathcal { N A G }$ and $\mathcal { G }  S y n$ have learned the common if (var.StartsWith(...)) { var.Substring(num) ... } pattern, but of course fail to produce the correct string literal in the condition. We show results for all of our models for these examples, as well as for as additional examples, in the supplementary material B.
+
+# 6 DISCUSSION & CONCLUSIONS
+
+We presented a generative code model that leverages known semantics of partially generated programs to direct the generative procedure. The key idea is to augment partial programs to obtain a graph, and then use graph neural networks to compute a precise representation for the partial program. This representation then helps to better guide the remainder of the generative procedure. We have shown that this approach can be used to generate small but semantically interesting expressions from very imprecise context information. The presented model could be useful in program repair scenarios (where repair proposals need to be scored, based on their context) or in the code review setting (where it could highlight very unlikely expressions). We also believe that similar models could have applications in related domains, such as semantic parsing, neural program synthesis and text generation.
+
+REFERENCES   
+Miltiadis Allamanis, Earl T Barr, Premkumar Devanbu, and Charles Sutton. A survey of machine learning for big code and naturalness. ACM Computing Surveys, 2018a.   
+Miltiadis Allamanis, Marc Brockschmidt, and Mahmoud Khademi. Learning to represent programs with graphs. In International Conference on Learning Representations (ICLR), 2018b.   
+Matthew Amodio, Swarat Chaudhuri, and Thomas W. Reps. Neural attribute machines for program generation. arXiv preprint arXiv:1705.09231, 2017.   
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In International Conference on Learning Representations (ICLR), 2014.   
+Benjamin Bichsel, Veselin Raychev, Petar Tsankov, and Martin Vechev. Statistical deobfuscation of android applications. In Conference on Computer and Communications Security (CCS), 2016.   
+Pavol Bielik, Veselin Raychev, and Martin Vechev. PHOG: probabilistic model for code. In International Conference on Machine Learning (ICML), 2016.   
+Kyunghyun Cho, Bart van Merriënboer, Dzmitry Bahdanau, and Yoshua Bengio. On the properties of neural machine translation: Encoder–decoder approaches. Syntax, Semantics and Structure in Statistical Translation, 2014.   
+Yu Feng, Ruben Martins, Osbert Bastani, and Isil Dillig. Program synthesis using conflict-driven learning. In Programming Languages Design and Implementation (PLDI), 2018.   
+John K. Feser, Swarat Chaudhuri, and Isil Dillig. Synthesizing data structure transformations from input-output examples. In Programming Languages Design and Implementation (PLDI), 2015.   
+Justin Gilmer, Samuel S. Schoenholz, Patrick F. Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry. In International Conference on Machine Learning (ICML), 2017.   
+Vincent J. Hellendoorn and Premkumar Devanbu. Are deep neural networks the best choice for modeling source code? In Foundations of Software Engineering (FSE), 2017.   
+Abram Hindle, Earl T Barr, Zhendong Su, Mark Gabel, and Premkumar Devanbu. On the naturalness of software. In International Conference on Software Engineering (ICSE), 2012.   
+Donald E. Knuth. Semantics of context-free languages. Mathemtical Systems Theory, 2(2):127–145, 1967.   
+Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. In International Conference on Learning Representations (ICLR), 2016.   
+Yujia Li, Oriol Vinyals, Chris Dyer, Razvan Pascanu, and Peter Battaglia. Learning deep generative models of graphs. CoRR, abs/1803.03324, 2018.   
+Cristina V Lopes, Petr Maj, Pedro Martins, Vaibhav Saini, Di Yang, Jakub Zitny, Hitesh Sajnani, and Jan Vitek. DéjàVu: a map of code duplicates on GitHub. In Object-Oriented Programming, Systems, Languages, and Applications (OOPSLA), 2017.   
+Minh-Thang Luong, Hieu Pham, and Christopher D Manning. Effective approaches to attention-based neural machine translation. In Conference on Empirical Methods in Natural Language Processing (EMNLP), 2015.   
+Chris J Maddison and Daniel Tarlow. Structured generative models of natural source code. In International Conference on Machine Learning (ICML), 2014.   
+Emilio Parisotto, Abdel-rahman Mohamed, Rishabh Singh, Lihong Li, Dengyong Zhou, and Pushmeet Kohli. Neuro-symbolic program synthesis. In International Conference on Learning Representations (ICLR), 2017.   
+Oleksandr Polozov and Sumit Gulwani. FlashMeta: a framework for inductive program synthesis. In ObjectOriented Programming, Systems, Languages, and Applications (OOPSLA), 2015.   
+Maxim Rabinovich, Mitchell Stern, and Dan Klein. Abstract syntax networks for code generation and semantic parsing. In Annual Meeting of the Association for Computational Linguistics (ACL), 2017.   
+Veselin Raychev, Martin Vechev, and Eran Yahav. Code completion with statistical language models. In Programming Languages Design and Implementation (PLDI), 2014.
+
+Veselin Raychev, Martin Vechev, and Andreas Krause. Predicting program properties from Big Code. In Principles of Programming Languages (POPL), 2015.
+
+Bidisha Samanta, Abir De, Niloy Ganguly, and Manuel Gomez-Rodriguez. Designing random graph models using variational autoencoders with applications to chemical design. CoRR, abs/1802.05283, 2018.
+
+Armando Solar-Lezama. Program synthesis by sketching. University of California, Berkeley, 2008.
+
+Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems, 2015.
+
+Pengcheng Yin and Graham Neubig. A syntactic neural model for general-purpose code generation. In Annual Meeting of the Association for Computational Linguistics (ACL), 2017.
+
+# A DATASET SAMPLES
+
+Below we list some sample snippets from the training set for our ExprGen task. The highlighted expressions are to be generated.
+
+![](images/9bf5db9920bad5b86b50332389eb940df969b3495053d912b19f20a7d8e883a9.jpg)  
+Figure 4: Sample snippet from the Lean project. Formatting has been modified.
+
+![](images/980dced56cb62f9a552fa8e740530c65433016666277bb648b7bd58547c482d2.jpg)  
+Figure 5: Sample snippet from the BotBuilder project. Formatting has been modified.
+
+![](images/8bbb8ae37a302b987babedce16151a162516cda62d9ff75fd51d1255764f17bd.jpg)  
+Figure 6: Sample snippet from the Chocolatey project. Formatting has been modified.
+
+![](images/b2bfba5da90f7040e05a22ecd4ec9b9f0819e88507bfa8cf6104af735da43053.jpg)  
+Figure 7: Sample snippet from the Chocolatey project. Formatting has been modified and the snippet has been abbreviated.
+
+![](images/91f96abf7268b945a328273346f517540d9ad230a9d691f0e6f179c930bd4867.jpg)  
+Figure 8: Samples snippet in the CommonMark.NET project. Formatting has been modified.
+
+![](images/738703f100f2b0aff47045141a43ce3e3681f70700445f74d1f751a89469cb84.jpg)  
+Figure 9: Sample snippet from the Humanizer project. Formatting has been modified.
+
+![](images/2db9fd72aa16ded0bd99cfe288fafd5956b1e7fa8288ce79f1f68d12c6d55317.jpg)  
+Figure 10: Samples snippet from the Nancy project. Formatting has been modified.
+
+![](images/44ef1a3ea4a720565638384851fe1c6b80c3d878dce4cebb30224baa32ea02e3.jpg)  
+Figure 11: Sample snippet from the OpenLiveWriter project. Formatting has been modified.
+
+![](images/8bf540add536c256659c5f53525d2a7ab3802f93c71b728b218fbc04de6122a1.jpg)  
+Figure 12: Sample snippet from the OpenLiveWriter project. Formatting has been modified.
+
+![](images/e50448d0ec463dabace2f2f08910a5201e4faff679c19cd474f24562676596db.jpg)  
+Figure 13: Sample snippet from the OpenLiveWriter project. Formatting has been modified.
+
+# B SAMPLE GENERATIONS
+
+On the following pages, we list some sample snippets from the test set for our ExprGen task, together with suggestions produced by different models. The highlighted expressions are the ground truth expression that should be generated.
+
+# Sample 1
+
+if (context.Context $= =$ _MARKUP_CONTEXT_TYPE.CONTEXT_TYPE_Text && !String.IsNullOrEmpty(text)) { idx $=$ originalText.IndexOf(text) if (idx $\scriptstyle = = 0$ ) { // Drop this portion from the expected string originalText $=$ originalText.Substring(text.Length); // Update the current pointer beginDamagePointer.MoveToPointer(currentRange.End); } else if (idx > 0 && originalText.Substring(0, idx) .Replace("\r\n", string.Empty).Length $\scriptstyle = = 0$ ) { // Drop this portion from the expected string originalText $=$ originalText.Substring(text.Length $^ +$ idx); // Update the current pointer beginDamagePointer.MoveToPointer(currentRange.End); } else { return false; }   
+}
+
+Sample snippet from OpenLiveWriter. The following suggestions were made: $\underline { { S e q } }  S e q$ :
+
+UNK_TOKEN[i] $( 0 . 6 \% )$ input[inputOffset + 1] $( 0 . 3 \% )$ UNK_TOKEN & UNK_NUM_LITERAL $( 0 . 3 \% )$
+
+# $S e q  \mathcal { N A G } ;$
+
+MarshalUrlSupported.IndexOf(UNK_CHAR_LITERAL) $( 0 . 9 \%$ ) IsEditFieldSelected.IndexOf(UNK_CHAR_LITERAL) $( 0 . 8 \%$ ) marshalUrlSupported.IndexOf(UNK_CHAR_LITERAL) $( 0 . 7 \% )$ )
+
+# $\underline { { \mathcal { G } \to S e q } } .$
+
+UNK_TOKEN.IndexOf(UNK_CHAR_LITERAL) $( 2 1 . 6 \% )$ UNK_TOKEN.LastIndexOf(UNK_CHAR_LITERAL) $( 1 4 . 9 \% )$ ) UNK_TOKEN.GetHashCode() $( 8 . 1 \% )$
+
+# $\mathcal { G }  \mathcal { T } r e e .$
+
+UNK_CHAR_LITERAL.IndexOf(UNK_CHAR_LITERAL) $( 8 . 1 \% )$ UNK_CHAR_LITERAL.IndexOf(originalText) $( 8 . 1 \%$ ) originalText.IndexOf(UNK_CHAR_LITERAL) $( 8 . 1 \%$ )
+
+# $\mathcal { G }  \mathcal { A } S \mathcal { N }$
+
+originalText.GetHashCode() $( 3 7 . 8 \%$ ) originalText.IndexOf(UNK_CHAR_LITERAL) $( 1 4 . 8 \%$ ) originalText.LastIndexOf(UNK_CHAR_LITERAL) $( 6 . 2 \% )$ )
+
+# $\mathcal { G }  S y n$
+
+text.IndexOf(UNK_CHAR_LITERAL) $( 2 0 . 9 \%$ ) text.LastIndexOf(UNK_CHAR_LITERAL) $( 1 2 . 4 \% )$ ) originalText.IndexOf(UNK_CHAR_LITERAL) $( 1 1 . 6 \%$ )
+
+# $\mathcal { G }  \mathcal { N A G }$
+
+originalText.IndexOf(UNK_CHAR_LITERAL) $( 3 2 . 8 \%$ ) originalText.LastIndexOf(UNK_CHAR_LITERAL) $( 1 2 . 4 \% )$ originalText.IndexOf(text) $( 8 . 7 \% )$
+
+# Sample 2
+
+caretPos--;   
+if (caretPos $< ~ 0$ ) { caretPos $\qquad = \quad 0$ ;   
+}   
+int len $=$ inputString.Length;   
+if (caretPos $> =$ len) { caretPos $=$ len - 1 ;   
+}
+
+Sample snippet from acat. The following suggestions were made: $s e q \to S e q$ :
+
+UNK_TOKEN $^ { + 1 }$ $( 2 . 1 \% )$ )  
+UNK_TOKEN+UNK_TOKEN] $( 1 . 8 \% )$   
+UNK_TOKEN.IndexOf(UNK_CHAR_LITERAL) $( 1 . 3 \% )$
+
+# $S e q  \mathcal { N A G }$
+
+wordToReplace - 1 (3.2%) insertOrReplaceOffset - 1 $( 2 . 9 \%$ ) inputString - 1 $( 1 . 9 \% )$
+
+$\mathcal { G }  S e q$ :   
+$1 \mathsf { e n } ~ + ~ 1 \left( 3 5 . 6 \% \right)$   
+$\mathrm { l e n } \ - \ 1 \left( 1 1 . 3 \% \right)$   
+len $> >$ UNK_NUM_LITERAL $( 3 . 5 \% )$ )
+
+$$
+\begin{array} { r l } & { \underbrace { \mathcal { G } \to \mathcal { T } r e e } _ { \mathrm { l e n ~ + ~ \ l e n ~ ( 2 4 . 9 \% ) } } } \\ & { \mathrm { l e n ~ - ~ \ l e n ~ ( 1 0 . 7 \% ) } } \\ & { \mathrm { 1 ~ + ~ \ 1 e n ~ ( 3 . 7 \% ) } } \end{array}
+$$
+
+$$
+\begin{array} { l } { \underbrace { \mathcal { G }  \mathcal { A } S \mathcal { N } } _ { \mathrm { 1 e n ~ + ~ 1 ~ } ( 2 2 . 8 \% ) } \colon } \\ { \mathrm { ~ l e n ~ - ~ 1 ~ } ( 1 0 . 8 \% ) } \\ { \mathrm { ~ l e n ~ + ~ 1 ~ e n ~ } ( 1 0 . 3 \% ) } \end{array}
+$$
+
+$$
+\begin{array} { l } { { \underbrace { \mathcal { G } \to S y n } _ { \mathrm { 1 e n ~ + ~ 1 ~ } ( 1 3 . 7 \% ) } } } \\ { { \mathrm { 1 e n ~ - ~ 1 ~ } ( 1 1 . 5 \% ) } } \\ { { \mathrm { 1 e n ~ - ~ 1 } \mathrm { e n } ( 1 1 . 0 \% ) } } \end{array}
+$$
+
+# $\mathcal { G }  \mathcal { N A G } \mathrm { : }$
+
+$1 \mathrm { e n } + + \left( 3 3 . 6 \% \right)$ len-1 $( 2 1 . 9 \% )$ len $^ { + 1 }$ $( 1 4 . 6 \% )$ )
+
+# Sample 3
+
+public static String URItoPath(String uri) { if (System.Text.RegularExpressions .Regex.IsMatch(uri, "^file:\\\\[a-z,A-Z]:")) { return uri.Substring(6); } if uri.StartsWith(@"file:") { return uri.Substring(5); } return uri;   
+}
+
+Sample snippet from acat. The following suggestions were made: $s e q \to S e q$ :
+
+!UNK_TOKEN $( 1 1 . 1 \% )$ ) UNK_TOKEN $= = \mathrm { ~ 0 ~ } ( 3 . 6 \% )$ UNK_TOKEN $! = \mathrm { ~ 0 ~ } ( 3 . 4 \% )$
+
+# $S e q  { \mathcal { N A G } } \colon$
+
+!uri $( 7 . 6 \% )$ ， !MyVideos $( 4 . 7 \% )$ ) !MyDocuments $( 4 . 7 \% )$
+
+$\mathcal G  S e q \mathrm { : }$ action $= =$ UNK_STRING_LITERAL $( 2 2 . 6 \% )$ ) label $= =$ UNK_STRING_LITERAL $( 1 4 . 8 \%$ ) file.Contains(UNK_STRING_LITERAL) $( 4 . 6 \%$
+
+$\mathcal { G }  \mathcal { T } r e e .$ :   
+$\mathrm { u r i } \ = = \ \mathrm { u r i } \ ( 7 . 4 \% )$   
+uri.StartsWith(uri) $( 5 . 5 \% )$ uri.Contains(uri) $( 4 . 3 \% )$ 0
+
+# $\mathcal { G }  \mathcal { A } S \mathcal { N } \mathrm { : }$
+
+uri $= =$ UNK_STRING_LITERAL $( 1 1 . 7 \%$ ) uri.Contains(UNK_STRING_LITERAL) $( 1 1 . 7 \%$ ) uri.StartsWith(UNK_STRING_LITERAL) $( 8 . 3 \% )$
+
+# $\mathcal G \to S y n \colon$
+
+$\mathrm { { \ u r { \dot { 1 } } } ~ } = =$ UNK_STRING_LITERAL $( 2 6 . 4 \%$ ) $\mathsf { u r i } \ \mathsf { \Omega } = \mathsf { \Omega } ^ { \mathsf { m } \mathsf { n } } \left( 8 . 5 \% \right)$ uri.StartsWith(UNK_STRING_LITERAL) $( 6 . 7 \% )$
+
+# $\mathcal { G }  \underline { { \mathcal { N A G } } } ;$
+
+uri.Contains(UNK_STRING_LITERAL) $( 3 2 . 4 \%$ ) uri.StartsWith(UNK_STRING_LITERAL) $( 2 9 . 2 \% )$ uri.HasValue() $( 7 . 7 \% )$
+
+# Sample 4
+
+<table><tr><td>startPos = index + 1; int count = endPos - startPos + 1; word = (count &gt; 0）？ input.Substring(startPos， count) : String.Empty;</td></tr></table>
+
+# Sample snippet from acat. The following suggestions were made: $\underline { { S e q } }  S e q$ :
+
+UNK_TOKEN.Trim() $( 3 . 4 \% )$ UNK_TOKEN.Replace(UNK_STRING_LITERAL, UNK_STRING_LITERAL) $( 2 . 1 \% )$ UNK_TOKEN.Replace(‘UNK_CHAR‘, ‘UNK_CHAR‘) $( 3 . 4 \% )$
+
+# $S e q  \mathcal { N A G }$
+
+input[index] $( 1 . 4 \% )$ startPos[input] $( 0 . 9 \% )$ input[count] $( 0 . 8 \% )$
+
+# $\underline { { \mathcal { G } \to S e q } } :$
+
+val.Trim() $( 6 . 6 \% )$ )   
+input.Trim() $( 6 . 5 \% )$   
+input.Substring(UNK_NUM_LITERAL) $( 4 . 0 \% )$
+
+# $\mathcal { G }  \mathcal { T } r e e \mathrm { : }$
+
+UNK_STRING_LITERAL $^ +$ UNK_STRING_LITERAL $( 8 . 4 \% )$ ) UNK_STRING_LITERAL $^ +$ startPos $( 7 . 8 \%$ ) startPos $^ +$ UNK_STRING_LITERAL $( 7 . 8 \% )$
+
+# $\mathcal { G }  \mathcal { A } S \mathcal { N }$
+
+input.Trim() $( 1 5 . 6 \% )$ ,   
+input.Substring(0) $( 6 . 4 \% )$   
+input.Replace(UNK_STRING_LITERAL, UNK_STRING_LITERAL) $( 2 . 8 \% )$ )
+
+# $\mathcal { G }  S y n$
+
+input.Trim() $( 7 . 8 \% )$ input.ToLower() $( 6 . 4 \% )$ input $^ +$ UNK_STRING_LITERAL $( 5 . 6 \% )$ ,
+
+# $\mathcal { G }  \mathcal { N A G } ;$
+
+input+StartPos $( 1 1 . 8 \%$ 0   
+input+count $( 9 . 5 \% )$   
+input.Substring(startPos, endPos - count) $( 6 . 3 \% )$
+
+# Sample 5
+
+protected virtual void CrawlSite() { while ( !_crawlComplete { RunPreWorkChecks(); if (_scheduler.Count $> ~ 0$ ) { _threadManager.DoWork( () $= >$ ProcessPage(_scheduler.GetNext())); } else if (!_threadManager.HasRunningThreads()) { _crawlComplete $=$ true; else { _logger.DebugFormat("Waiting for links to be scheduled..."); Thread.Sleep(2500); } }   
+}
+
+Sample snippet from Abot. The following suggestions were made: $s e q \to S e q$ :
+
+!UNK_TOKEN $( 9 . 4 \% )$ UNK_TOKEN $> \ 0 \ ( 2 . 6 \% )$ UNK_TOKEN ! $=$ value $( 1 . 3 \% )$
+
+# $S e q  { \mathcal { N A G } } ;$
+
+!_maxPagesToCrawlLimitReachedOrScheduled $( 2 6 . 2 \%$ ) !_crawlCancellationReported $( 2 6 . 0 \% )$ !_crawlStopReported $( 2 1 . 8 \% )$
+
+# $\mathcal G \to S e q \mathrm { : }$
+
+!UNK_TOKEN $( 5 4 . 9 \% )$ ) !done $( 1 8 . 8 \%$ ) !throwOnError $( 3 . 3 \% )$ )
+
+# $\mathcal { G }  \mathcal { T } r e e \mathrm { : }$
+
+!_crawlCancellationReported $( 2 3 . 6 \%$ ) !_crawlStopReported $( 2 3 . 3 \% )$ !_maxPagesToCrawlLimitReachedOrScheduled $( 1 8 . 9 \% )$
+
+# $\mathcal { G }  \mathcal { A } S \mathcal { N } ;$
+
+!_crawlStopReported $( 2 6 . 6 \%$ ) !_crawlCancellationReported $( 2 6 . 5 \%$ ) !_maxPagesToCrawlLimitReachedOrScheduled $( 2 5 . 8 \%$ )
+
+# $\mathcal { G }  S y n$
+
+!_crawlStopReported $( 1 9 . 6 \%$ ) !_maxPagesToCrawlLimitReachedOrScheduled $( 1 9 . 0 \%$ ) !_crawlCancellationReported $( 1 5 . 7 \% )$ )
+
+# $\mathcal { G }  \mathcal { N A G }$
+
+!_crawlStopReported $( 3 8 . 4 \% )$ !_crawlCancellationReported $( 3 1 . 8 \%$ ) !_maxPagesToCrawlLimitReachedOrScheduled $( 2 7 . 0 \%$ )
+
+# Sample 6
+
+char character $=$ originalName[i];   
+if ( character == ’<’ { ++startTagCount; builder.Append(’‘���);   
+} else if (startTagCount $> ~ 0$ ) { if (character $\scriptstyle = = \prime > \prime$ ) { --startTagCount; }
+
+Sample snippet from StyleCop. The following suggestions were made: $\underline { { S e q } }  S e q \mathrm { . }$ :
+
+$\begin{array} { r l } { \mathrm { ~ x ~ } } & { { } = = } \end{array}$ UNK_CHAR_LITERAL $( 5 . 9 \% )$ ) UNK_TOKEN $= = \ 0 \ ( 3 . 3 \% )$ UNK_TOKEN $> \ 0 \ ( 2 . 7 \% )$
+
+$\displaystyle \frac { S e q \to \mathcal { N } A \mathcal { G } } { ! \mathrm { ~ i ~ \Gamma ~ \mathrm { ~ \Omega ~ \mathrm { ~ = ~ } ~ 0 ~ } ~ } ( 5 . 1 \% ) } \mathrm { ~ . ~ }$ character $< \mathrm { ~ 0 ~ } ( 2 . 7 \% )$ character $( 2 . 2 \% )$ )
+
+# $\underline { { \mathcal { G } \to S e q } } .$
+
+character $= =$ UNK_CHAR_LITERAL $( 7 0 . 8 \%$ )   
+character $= =$ UNK_CHAR_LITERAL || character $= =$ UNK_CHAR_LITERAL $( 5 . 8 \% )$ )   
+character ! $! =$ UNK_CHAR_LITERAL $( 3 . 1 \% )$ )
+
+# $\mathcal { G }  \mathcal { T } r e e \mathrm { : }$
+
+character $= =$ character $( 9 . 9 \%$ ) UNK_CHAR_LITERAL $= =$ character $( 8 . 2 \%$ ) character $= =$ UNK_CHAR_LITERAL $( 8 . 2 \% )$ )
+
+# $\mathcal { G }  \mathcal { A } S \mathcal { N } \mathrm { : }$
+
+character $= =$ UNK_CHAR_LITERAL $( 4 3 . 4 \%$ ) character || character $( 3 . 3 \% )$ character $= =$ UNK_CHAR_LITERAL $= =$ UNK_CHAR_LITERAL $( 3 . 0 \% )$
+
+# $\mathcal { G }  S y n$
+
+character $= =$ UNK_CHAR_LITERAL $( 3 9 . 6 \%$ ) character || character $= =$ UNK_STRING_LITERAL $( 5 . 2 \% )$ 0 character $= =$ UNK_STRING_LITERAL $( 2 . 8 \% )$
+
+# $\mathcal { G }  \mathcal { N A G } ;$
+
+character $= =$ UNK_CHAR_LITERAL $( 7 5 . 5 \%$ ) character $\mathrm { ~ \ -- ~ } \ ^ { \prime } \ ^ { \prime } \ ( 2 . 6 \% )$ character ! $=$ ’UNK_CHAR $( 2 . 5 \% )$
+
+# Sample 7
+
+public void AllowAccess(string path)   
+{ if (path $= =$ null) throw new ArgumentNullException("path"); if !path.StartsWith(" /") ) throw new ArgumentException( string.Format( "The path \"{0}\" is not application relative." + " It must start with \"\~/\".", path), "path"); paths.Add(path);   
+}
+
+Sample snippet from cassette. The following suggestions were made: $S e q  S e q$ :
+
+UNK_TOKEN $< \mathrm { ~ 0 ~ } ( 1 4 . 6 \% )$ !UNK_TOKEN $( 7 . 5 \% )$ ) UNK_TOKEN $= = \ 0 \ ( 3 . 3 \% )$
+
+$\underline { { S e q  \mathcal { N A G } } } ;$   
+path $= =$ UNK_STRING_LITERAL $( 1 8 . 1 \%$   
+path $< = \mathrm { ~ 0 ~ } ( 5 . 6 \% )$   
+path $\scriptstyle \mathbf { \mu = } \mathbf { \mu " } \mathbf { \mu " } \left( 4 . 8 \% \right)$ $\underline { { \mathcal { G } \to S e q } } .$   
+!UNK_TOKEN $( 4 8 . 0 \%$ )   
+!discardNulls $( 6 . 3 \% )$ ) !first $( 2 . 7 \% )$
+
+$\mathcal { G }  \mathcal { T } r e e \mathrm { : }$ !path $( 6 7 . 4 \% )$ ) path && path $( 8 . 4 \% )$ !!path $( 5 . 5 \% )$
+
+$\mathcal { G }  \mathcal { A } S \mathcal { N } \mathrm { : }$   
+!path $( 9 1 . 5 \%$ )   
+!path && !path $( 0 . 9 \%$ )   
+!path.Contains(UNK_STRING_LITERAL) $( 0 . 7 \% )$   
+$\mathcal { G }  S y n \mathrm { : }$ :   
+!path $( 8 9 . 6 \%$ )   
+!path && !path $( 1 . 5 \% )$ )   
+!path.Contains(UNK_STRING_LITERAL) $( 0 . 5 \% )$ )
+
+# $\mathcal { G }  \underline { { \mathcal { N A G } } } ;$
+
+!path $( 4 2 . 9 \% )$
+
+!path.StartsWith(UNK_STRING_LITERAL) $( 2 3 . 8 \% )$ !path.Contains(UNK_STRING_LITERAL) $( 5 . 9 \%$ )
+
+# Sample 8
+
+int methodParamCount $\qquad = \quad 0$ ;   
+IEnumerable<IParameterTypeInformation> moduleParameters $=$ Enumerable<IParameterTypeInformation>.Empty;   
+if (paramCount $> 0$ ) { IParameterTypeInformation[] moduleParameterArr $=$ this.GetModuleParameterTypeInformations(Dummy.Signature, paramCount); methodParamCount $=$ moduleParameterArr.Length; if (methodParamCount $> ~ 0$ ) moduleParameters $=$ IteratorHelper.GetReadonly(moduleParameterArr);   
+}   
+IEnumerabl $\beta < \mathrm { I P }$ arameterTypeInformation> moduleVarargsParameters $=$ Enumerable<IParameterTypeInformation>.Empty;   
+if ( paramCount > methodParamCount ) { IParameterTypeInformation[] moduleParameterArr $=$ this.GetModuleParameterTypeInformations( Dummy.Signature, paramCount - methodParamCount); if (moduleParameterArr.Length $> 0$ ) moduleVarargsParameters $=$ IteratorHelper.GetReadonly(moduleParameterArr);   
+}
+
+# Sample snippet from Afterthought. The following suggestions were made: $s e q \to S e q$ :
+
+!UNK_TOKEN $( 1 0 . 9 \%$ ) UNK_TOKEN $= =$ UNK_TOKEN $( 4 . 6 \% )$ UNK_TOKEN $= =$ UNK_STRING_LITERAL $( 3 . 3 \% )$
+
+$S e q  { \mathcal { N A G } } ;$ dummyPinned $! = \mathrm { ~ 0 ~ } ( 2 . 2 \% )$ paramCount $! = \mathrm { ~ 0 ~ } ( 2 . 1 \% )$ dummyPinned $= = \ 0 \ ( 1 . 5 \% )$
+
+# $\mathcal G  S e q \mathrm { : }$
+
+newValue > 0 (9.7%) zeroes $> \mathrm { ~ 0 ~ } ( 9 . 0 \% )$ paramCount $> \mathrm { ~ 0 ~ } ( 6 . 0 \% )$
+
+# $\mathcal { G }  \mathcal { T } r e e \mathrm { : }$
+
+methodParamCount $= =$ methodParamCount $( 3 . 4 \% )$ $0 \quad = =$ methodParamCount $( 2 . 8 \%$ ) methodParamCount $= =$ paramCount $( 2 . 8 \%$ )
+
+# $\mathcal { G }  \mathcal { A } S \mathcal { N } ;$
+
+paramCount $= = \ 0 \ ( 1 2 . 7 \% )$ paramCount $< \mathrm { ~ 0 ~ } ( 1 1 . 5 \% )$ paramCount $> \mathrm { ~ 0 ~ } ( 8 . 0 \% )$
+
+# $\mathcal { G }  S y n \mathrm { : }$
+
+methodParamCount $> \mathrm { ~ 0 ~ } ( 1 0 . 9 \% )$ paramCount $> \mathrm { ~ 0 ~ } ( 7 . 9 \% )$ methodParamCount $! = \mathrm { ~ 0 ~ } ( 5 . 6 \% )$
+
+# $\mathcal { G }  \underline { { \mathcal { N A G } } } ;$
+
+paramCount $>$ methodParamCount $( 3 4 . 4 \%$ ) paramCount $= =$ methodParamCount $( 1 1 . 4 \%$ ) paramCount $<$ methodParamCount $( 1 0 . 0 \%$ )
+
+# Sample 9
+
+public CodeLocation(int index, int endIndex, int indexOnLine, int endIndexOnLine, int lineNumber, int endLineNumber)   
+{ Param.RequireGreaterThanOrEqualToZero(index, "index"); Param.RequireGreaterThanOrEqualTo(endIndex, index, "endIndex"); Param.RequireGreaterThanOrEqualToZero(indexOnLine, "indexOnLine"); Param.RequireGreaterThanOrEqualToZero(endIndexOnLine, "endIndexOnLine") Param.RequireGreaterThanZero(lineNumber, "lineNumber"); Param.RequireGreaterThanOrEqualTo(endLineNumber, lineNumber, "endLineNumber"); // If the entire segment is on the same line, // make sure the end index is greater or equal to the start index. if $\mathrm { : | \ l i n e N u m b e r { \ell } = = \ e n d L i n e N u m b e r { \ell } | \ t a m b e r { \ell } | \ t a m b e r { \ell } { \ell } \ l }$ ) { Debug.Assert(endIndexOnLine $> =$ indexOnLine, "The end index must be greater than the start index," $^ +$ " since they are both on the same line."); } this.startPoint $=$ new CodePoint(index, indexOnLine, lineNumber); this.endPoint $=$ new CodePoint(endIndex, endIndexOnLine, endLineNumber);   
+}
+
+Sample snippet from StyleCop. The following suggestions were made: $\underline { { S e q } }  S e q$ :
+
+!UNK_TOKEN $( 1 4 . 0 \% )$ ) UNK_TOKEN $= = \ 0 \ ( 4 . 4 \% )$ UNK_TOKEN $> \mathrm { ~ 0 ~ } ( 3 . 5 \% )$
+
+# $S e q  { \mathcal { N A G } } ;$
+
+endIndex $< ~ 0$ $( 3 . 8 \% )$ endIndex $> 0$ $3 . 4 \% )$ endIndex $= = \ 0 \ ( 2 . 2 \% )$
+
+# $\mathcal G \to S e q \mathrm { : }$
+
+lineNumber $< \mathrm { ~ \ 0 ~ } ( 9 . 4 \% )$ lineNumber $= = \mathrm { ~ 0 ~ ( 7 . 4 \% ) ~ }$ lineNumber $< = \mathrm { ~ 0 ~ } ( 5 . 1 \% )$
+
+# $\mathcal { G }  \mathcal { T } r e e \mathrm { : }$
+
+lineNumber $= =$ lineNumber $( 3 . 4 \% )$ $0 \quad = =$ lineNumber $( 2 . 5 \% )$ , lineNumber $>$ lineNumber $( 2 . 5 \% )$
+
+# $\mathcal { G }  \mathcal { A } S \mathcal { N } ;$
+
+endLineNumber $\scriptstyle = = \ 0 \ ( 9 . 6 \% )$ endLineNumber $< \mathrm { ~ 0 ~ } ( 7 . 9 \% )$ endLineNumber $> \mathrm { ~ 0 ~ } ( 6 . 1 \% )$
+
+#
+
+$$
+\begin{array} { r l } & { \frac { \mathscr { G } \mathrm { ~  ~ } S y n \mathrm { : } } { \mathrm { 1 i n e N u m b e r ~ \gamma > ~ \gamma _ 0 ~ ( 1 1 . 3 \% ) } } } \\ & { \mathrm { ~ \bot ~ i n e N u m b e r ~ \gamma = = ~ \gamma _ 0 ~ ( 7 . 3 \% ) } } \\ & { \mathrm { ~ \bot ~ i n e N u m b e r ~ \gamma _ l = ~ \gamma _ 0 ~ ( 6 . 7 \% ) } } \end{array}
+$$
+
+# $\mathcal { G }  \underline { { \mathcal { N A G } } } ;$
+
+lineNumber $>$ endLineNumber $( 2 0 . 7 \%$ )lineNumber $<$ endLineNumber $( 1 6 . 5 \%$ )lineNumber $= =$ endLineNumber $( 1 6 . 2 \% )$
+
+# Sample 10
+
+public static Bitmap RotateImage(Image img, float angleDegrees, bool upsize, bool clip) { // Test for zero rotation and return a clone of the input image if (angleDegrees $\textstyle = = 0 \mathrm { { f } }$ ) return (Bitmap)img.Clone(); // Set up old and new image dimensions, assuming upsizing not wanted // and clipping OK int oldWidth $=$ img.Width; int oldHeight $=$ img.Height; int newWidth $=$ oldWidth; int newHeight $=$ oldHeight; float scaleFactor $\qquad = ~ \perp \pm$ ; // If upsizing wanted or clipping not OK calculate the size of the // resulting bitmap if upsize || !clip { double angleRadians $=$ angleDegrees $\star$ Math.PI / 180d; double cos $=$ Math.Abs(Math.Cos(angleRadians)); double sin $=$ Math.Abs(Math.Sin(angleRadians)); newWidth $=$ (int)Math.Round((oldWidth $\star$ cos) $^ +$ (oldHeight $\star$ sin)); newHeight $=$ (int)Math.Round((oldWidth $\star$ sin) $^ +$ (oldHeight $\star$ cos)); } // If upsizing not wanted and clipping not OK need a scaling factor if (!upsize && !clip) { scaleFactor $=$ Math.Min((float)oldWidth / newWidth, (float)oldHeight / newHeight); newWidth $=$ oldWidth; newHeight $=$ oldHeight; }
+
+Sample snippet from ShareX. The following suggestions were made: $\underline { { S e q } }  S e q \mathrm { . }$ :
+
+UNK_TOKEN $>$ 0 $( 8 . 3 \% )$ !UNK_TOKEN $( 4 . 4 \% )$ UNK_TOKEN $= = \ 0 \ ( 2 . 6 \% )$
+
+$S e q  { \mathcal { N A G } } ;$ newHeight $> \mathrm { ~ 0 ~ } ( 5 . 1 \% )$ $\beth \mathrm { { 1 i p } } ~ > ~ 0 ~ ( 3 . 2 \% )$ oldWidth $> 0$ $( 2 . 9 \% )$
+
+# $\underline { { \mathcal { G } \to S e q } } :$
+
+UNK_TOKEN && UNK_TOKEN $( 1 5 . 0 \%$ ) UNK_TOKEN || UNK_TOKEN $( 1 3 . 6 \% )$ ) trustedForDelegation && !appOnly $( 1 2 . 1 \% )$ )
+
+# $\mathcal { G }  \mathcal { T } r e e .$
+
+upsize && upsize $( 2 1 . 5 \% )$ ) upsize && clip $( 1 0 . 9 \%$ ) clip && upsize $( 1 0 . 9 \% )$
+
+# $\mathcal { G }  \mathcal { A } S \mathcal { N } ;$
+
+upsize && clip $( 1 3 . 9 \%$ ) upsize && !clip $( 9 . 8 \% )$ clip && clip $( 9 . 3 \% )$
+
+# $\mathcal { G }  S y n \mathrm { : }$
+
+upsize && !upsize $( 6 . 9 \%$ ) clip && !upsize $( 6 . 3 \% )$ ) upsize || upsize $( 5 . 7 \% )$
+
+# $\mathcal { G }  \mathcal { N A G } ;$
+
+upsize || clip $( 1 9 . 1 \%$ upsize && clip $( 1 8 . 8 \% )$ upsize && ! clip $( 1 2 . 2 \% )$ )

parse/train/G1jmxFOtY_/G1jmxFOtY_.md

@@ -0,0 +1,374 @@
+# Learning with User-Level Privacy
+
+Daniel Levy∗,1 Ziteng Sun∗,2 Kareem Amin3 Satyen Kale3   
+Alex Kulesza3 Mehryar Mohri3,4 Ananda Theertha Suresh3
+
+1Stanford University 2Cornell University 3Google Research 4Courant Institute danilevy@stanford.edu, zs335@cornell.edu, {kamin, satyenkale, kulesza, mohri, theertha}@google.com
+
+# Abstract
+
+We propose and analyze algorithms to solve a range of learning tasks under userlevel differential privacy constraints. Rather than guaranteeing only the privacy of individual samples, user-level DP protects a user’s entire contribution $m \geq 1$ samples), providing more stringent but more realistic protection against information leaks. We show that for high-dimensional mean estimation, empirical risk minimization with smooth losses, stochastic convex optimization, and learning hypothesis classes with finite metric entropy, the privacy cost decreases as $O ( 1 / \bar { \sqrt { m } } )$ as users provide more samples. In contrast, when increasing the number of users $n$ , the privacy cost decreases at a faster $O ( 1 / n )$ rate. We complement these results with lower bounds showing the minimax optimality of our algorithms for mean estimation and stochastic convex optimization. Our algorithms rely on novel techniques for private mean estimation in arbitrary dimension with error scaling as the concentration radius $\tau$ of the distribution rather than the entire range.
+
+# 1 Introduction
+
+Releasing seemingly innocuous functions of a data set can easily compromise the privacy of individuals, whether the functions are simple counts [35] or complex machine learning models like deep neural networks [52, 30]. To protect against such leaks, Dwork et al. proposed the notion of differential privacy (DP). Given some data from $n$ participants in a study, we say that a statistic of the data is differentially private if an attacker who already knows the data of $n - 1$ participants cannot reliably determine from the statistic whether the $n$ -th remaining participant is Alice or Bob. With the recent explosion of publicly available data, progress in machine learning, and widespread public release of machine learning models and other statistical inferences, differential privacy has become an important standard and is widely adopted by both industry and government [32, 5, 21, 55].
+
+The standard setting of DP described in [22] assumes that each participant contributes a single data point to the dataset, and preserves privacy by “noising” the output in a way that is commensurate with the maximum contribution of a single example. This is not the situation faced in many applications of machine learning models, where users often contribute multiple samples to the model—for example, when language and image recognition models are trained on the users’ own data, or in federated learning settings [37]. As a result, current techniques either provide privacy guarantees that degrade with a user’s increased participation or naively add a substantial amount of noise, relying on the group property of differential privacy, which significantly harms the performance of the deployed model.
+
+To remedy this issue, we consider user-level DP, which instead of guaranteeing privacy for individual samples, protects a user’s entire contribution $\textcircled { m } \geq 1$ samples). This is a more stringent but more realistic privacy desideratum. To hold, it requires that the output of our algorithm does not significantly change when changing user’s entire contribution—i.e. possibly swapping up to $m$ samples in total. We make this formal in Definition 1. Very recently, for the reasons outlined above, there has been increasing interest in user-level DP for applications such as estimating discrete distributions under user-level privacy constraints [46], PAC learning with user-level privacy [31], and bounding user contributions in ML models [4, 26]. Differentially private SQL with bounded user contributions was proposed in [59]. User-level privacy has been also studied in the context of learning models via federated learning [49, 48, 58, 6].
+
+In this paper, we tackle the problem of learning with user-level privacy in the central model of DP. In particular, we provide algorithms and analyses for the tasks of mean estimation, empirical risk minimization (ERM), stochastic convex optimization (SCO), and learning hypothesis classes with finite metric entropy. Our utility analyses assume that all users draw their samples i.i.d. from related distributions, a setting we refer to as limited heterogeneity. On these tasks, naively applying standard mechanisms, such as Laplace or Gaussian, or using the group property with item-level DP estimators, both yield a privacy error independent of $m$ . We first develop novel private mean estimators in high dimension with statistical and privacy error scaling with the (arbitrary) concentration radius rather than the range, and apply these to the statistical query setting [SQ; 41]. Our algorithms then rely on (privately) answering a sequence of adaptively chosen queries using users’ samples, e.g., gradient queries in stochastic gradient descent algorithms. We show that for these tasks, the additional error√ due to privacy constraints decreases as $\bar { O } ( 1 / \sqrt { m } )$ , contrasting with the naive rate—independent of $m$ . Interestingly, increasing $n$ , the number of users, decreases the privacy cost at a faster $O ( 1 / n )$ rate.
+
+Importantly, our results imply concrete practical recommendations on sample collection, regardless of the level of heterogeneity. Indeed, increasing $m$ will yield the most value in the i.i.d. setting and will yield no improvement when the users’ distributions are arbitrary. As the real-world will lie somewhere in between, our results exhibit a regime where, for any heterogeneity, it is strictly better to collect more users (increasing $n$ ) than more samples per user (increasing $m$ ).
+
+# 1.1 Our Contributions and Related Work
+
+We provide a theoretical tool to construct estimators for tasks with user-level privacy constraints and apply it to a range of learning problems.
+
+Optimal private mean estimation and uniformly concentrated queries (Section 3) We show that for a random variable in $[ - B , B ]$ concentrated in an unknown interval of radius $\tau$ (made precise in Definition 2), we can privately estimate its mean with error proportional to $\tau$ rather than $B$ , as we would obtain using standard private mean estimation techniques such as Laplace mechanism [24]. When data is concentrated in $\ell _ { \infty }$ -norm, several papers show that one can achieve an error scaling with $\tau$ rather than $B$ , either asymptotically [53], for Gaussian mean-estimation [40, 38], for sub-Gaussian symmetric distributions [18, 17] or for distributions with bounded $p$ -th moment [39]. We propose a private mean estimator (Algorithm 2) with error scaling with $\tau$ that works in arbitrary dimension when data is concentrated in $\ell _ { 2 }$ -norm (Theorem 2). In Corollary 1, we show it (optimally) solves mean estimation under user-level privacy constraints for random vectors bounded in $\ell _ { 2 }$ -norm. In Appendix D.6, we show that for uniformly concentrated queries (see Definition 3), sequentially applying Algorithm 2 privately answers $K$ adaptively chosen queries with privacy cost $\tilde { O } ( \tau \sqrt { K } / n \varepsilon )$ .
+
+Our conclusions relate to the growing literature in adaptive data analysis. While a sequence of work [25, 9, 27, 28] use techniques from differential privacy and their answers are $( \varepsilon , \delta )$ -DP with $\varepsilon = \Theta ( 1 )$ , our work guarantees privacy for arbitrary $\varepsilon$ with the additional assumption of uniform concentration.
+
+Empirical risk minimization (Section 4) An influential line of papers studies ERM under itemlevel privacy constraints [19, 42, 8]. Importantly, these papers assume arbitrary data, i.e., not necessarily samples from users’ distributions. The exact analog of ERM in the user-level setting is consequently less interesting as, for $n$ data points $\{ z _ { 1 } , \ldots , z _ { n } \}$ , in the worst case, each user $u \in [ n ]$ contributes $m$ copies of $z _ { u }$ and the problem reduces to the item-level setting. Instead, we consider the (related) problem of ERM when users contribute points sampled i.i.d. Assuming some regularity (A3 and A4), we develop and analyze algorithms for ERM under user-level DP constraints for convex, strongly-convex, and non-convex losses (Theorem 3).
+
+Optimal stochastic convex optimization (Section 5) Under item-level DP (or equivalently, userlevel DP with $m = 1$ ), a sequence of work [19, 8, 10, 11, 29] establishes the constrained minimax risk as $\tilde { \Theta } ( 1 / \sqrt { n } + \sqrt { d } / ( n \varepsilon ) )$ . In this paper, with the additional assumptions that the losses are individually smooth2 and the gradients are sub-Gaussian random vectors, we prove matching upper√ (Theorem 4) and lower bounds (Theorem 5) of order $\tilde { \Theta } ( 1 / \sqrt { n m } + \sqrt { d } / ( n \sqrt { m } \varepsilon ) )$ in a regime we make precise. We leave closing the gap outside of this regime to future work.
+
+Limit of learning with a fixed number of users (Appendix B) Finally, we resolve a conjecture of [4] and prove that with a fixed number of users, even in the limit $m  \infty$ (i.e., each user has an infinite number of samples), we cannot reach zero error. In particular, we prove that for all the learning tasks we consider, the risk under user-level privacy constraints is at least $\Omega ( e ^ { - \varepsilon n } )$ regardless of $m$ . Note that this does not contradict the results above since they require $n = \Omega ( ( \log m ) / \varepsilon )$ .
+
+Finally, we provide results in Appendix A for learning under pure user-level DP for function classes with finite metric entropy. We apply these to SCO with $\ell _ { \infty }$ constraints (Remark 1) and achieve (near)-optimal rates.
+
+# 2 Preliminaries
+
+Notation. Throughout this work, $d$ denotes the dimension, $n$ the number of users, and $m$ the number of samples per user. Generically, $\sigma$ will denote the sub-Gaussian parameter, $\tau$ the concentration radius, $\nu$ the variance of a random vector and $P$ a data distribution. We denote the optimization variable with $\theta \in \Theta \subset \mathbb { R } ^ { d }$ , use $z$ (or $Z$ when random) to denote the data sample supported on a space $\mathcal { Z }$ , and $\ell \colon \Theta \times \mathcal { Z }  \mathbb { R }$ for the loss function. Gradients (denoted $\nabla$ ) are always taken with respect to the optimization variable $\theta$ . For a convex set $\mathcal { C }$ , $\Pi _ { \mathcal { C } }$ denotes the euclidean projection on $\mathcal { C }$ , i.e. $\begin{array} { r } { \Pi _ { \mathcal { C } } ( y ) : = \operatorname * { a r g m i n } _ { z \in \mathcal { C } } \| y - z \| _ { 2 } } \end{array}$ . We use $\mathsf { A }$ to refer to (possibly random) private mechanisms and $X ^ { n }$ as a shorthand for the dataset $( X _ { 1 } , \ldots , X _ { n } )$ . For two distributions $P$ and $Q$ , we denote by $\| \boldsymbol { P } - \boldsymbol { Q } \| _ { \mathsf { T V } }$ their total variation distance and $D _ { \mathrm { k l } } \left( P \| Q \right)$ their Kullback-Leibler divergence. For a random vector $X \sim P$ supported on $\mathbb { R } ^ { d }$ , we use $\mathrm { V a r } ( P )$ or $\operatorname { V a r } ( X )$ to denote $\mathbb { E } \left[ \lVert X - \mathbb { E } [ X ] \rVert _ { 2 } ^ { 2 } \right]$ , which is equal to the trace of the covariance matrix of $X$ .
+
+Next, we consider differential privacy in the most general way, which only requires specifying a dataset space $\mathbb { S }$ and a distance $\mathrm { d }$ on $\mathbb { S }$ .
+
+Definition 1 (Differential Privacy). Let $\varepsilon , \delta \geq 0$ . Let $\ u : \mathbb { S } \to \Theta$ be a (potentially randomized) mechanism. We say that A is $( \varepsilon , \delta )$ - $D P$ with respect to d if for any measurable subset $O \subset \Theta$ and all $S , S ^ { \prime } \in \mathbb { S }$ satisfying $\mathrm { d } ( S , S ^ { \prime } ) \stackrel { \cdot } { = } 1$ ,
+
+$$
+\mathbb { P } ( \mathsf { A } ( S ) \in O ) \le e ^ { \varepsilon } \mathbb { P } ( \mathsf { A } ( S ^ { \prime } ) \in O ) + \delta .
+$$
+
+$I f \delta = 0$ , we refer to this guarantee as pure differential privacy.
+
+For a data space $\mathcal { Z }$ , choosing $\mathbb { S } = \mathcal { Z } ^ { n }$ and $\begin{array} { r } { \mathrm { d } ( S , S ^ { \prime } ) = \mathrm { d } _ { \mathsf { H a m } } ( S , S ^ { \prime } ) = \sum _ { i = 1 } ^ { n } 1 \{ z _ { i } \neq z _ { i } ^ { \prime } \} } \end{array}$ recovers the canonical setting considered in most of the literature—we refer to this as item-level differential privacy. When we wish to guarantee privacy for users rather than individual samples, we instead assume a structured dataset into which each of $n$ users contributes $m > 1$ samples. This corresponds to $\mathbb { S } = ( \mathcal { Z } ^ { m } ) ^ { n }$ such that for ${ \boldsymbol { s } } \in \mathbb { S }$ , we have
+
+$$
+{ \mathcal { S } } = ( S _ { 1 } , \ldots , S _ { n } ) , { \mathrm { ~ w h e r e ~ } } S _ { u } = \left\{ z _ { 1 } ^ { ( u ) } , \ldots , z _ { m } ^ { ( u ) } \right\} { \mathrm { ~ a n d ~ } } \mathrm { d } _ { { \mathfrak { u s e r } } } ( S , S ^ { \prime } ) : = \sum _ { u = 1 } ^ { n } 1 \{ S _ { u } \neq S _ { u } ^ { \prime } \} ,
+$$
+
+which means that, in this setting, two datasets are neighboring if at most one of the user’s contributions differ. We henceforth refer to this setting as user-level differential privacy.
+
+Distributional assumptions. In the case of user-level privacy with $n$ users each providing $m$ samples, we assume existence of a collection of distributions $\{ \bar { P _ { u } } \} _ { u \in [ n ] }$ over $\mathcal { Z }$ . One then observes the following user-level dataset3
+
+$$
+{ \mathcal { S } } = ( S _ { 1 } , \ldots , S _ { n } ) { \mathrm { ~ w h e r e ~ } } S _ { u } \stackrel { \mathrm { i i d } } { \sim } P _ { u } .
+$$
+
+In this paper, we consider the limited heterogeneity setting, i.e. when the users have related distributions. This setting is more reflective of practice, especially in light of growing interest towards federated learning applications [37, 60].
+
+Assumption A1 (Limited heterogeneity setting). There exists a distribution $P _ { 0 }$ over $\mathcal { Z }$ such that all the user distributions are close to $P _ { 0 }$ in total variation distance, i.e.
+
+$$
+\operatorname* { m a x } _ { u \in [ n ] } \lVert P _ { u } - P _ { 0 } \rVert \mathrm { { r v } } \leq \Delta ,
+$$
+
+where $\Delta \geq 0$ quantifies the level of heterogeneity. Note that $\Delta = 0$ corresponds to assumption $A 2$ .
+
+Note that our TV-based definition is natural in this setting as it is closely related to the notion of discrepancy (or $d _ { A }$ distance) which plays a key role in domain adaption scenarios [47, 12]. Lower bound results have been given in terms of the discrepancy measure (see [13]), which further justify the adoption of this definition in the presence of multiple distributions.
+
+In the case that $\Delta = 0$ , A1 reduces to the standard homogeneous setting. Many fundamental papers choose this setting when explicating minimax rates under constraints (e.g. in distributed optimization and federated learning [61] or under communication constraints [63, 15]).
+
+Assumption A2 (Homogeneous setting). The distributions of individual users are equal, meaning there exists $P _ { 0 }$ such that for all $u \in [ n ]$ , $P _ { u } = P _ { 0 }$ .
+
+In this paper, we develop techniques and provide matching upper and lower bounds for solving learning tasks in the homogeneous setting. In Appendix C, we prove that our techniques naturally apply to the heterogeneous setting in a black-box fashion, and for all considered problems provide meaningful guarantees under Assumption A1. Moreover, the algorithm achieves almost optimal rate whenever $\Delta$ is (polynomially) small. See the detailed statement in Theorem 9.
+
+# 2.1 ERM and stochastic convex optimization
+
+Assumptions on the loss. Throughout this work, we assume that the parameter space $\Theta$ is closed, convex, and satisfies $\lVert { \boldsymbol { \theta } } - { \boldsymbol { \vartheta } } \rVert _ { 2 } \leq R$ for all $\theta , \vartheta \in \Theta$ . We also assume that the loss $\ell \colon \Theta \times \mathcal { Z }  \mathbb { R }$ is $G$ -Lipschitz w.r.t. the $\ell _ { 2 } { \mathrm { - n o r m } } ^ { 4 }$ , meaning that for all $z \in { \mathcal { Z } }$ , for all $\theta \in \Theta$ , $\| \nabla \ell ( \theta ; z ) \| _ { 2 } \leq G$ . We further consider the following assumptions.
+
+Assumption A3. The function $\ell ( \cdot ; z )$ is $H$ -smooth. In other words, the gradient $\nabla \ell ( \theta ; z )$ is $H$ Lipschitz in the variable $\theta$ for all $z \in { \mathcal { Z } }$ .
+
+Assumption A4. The random vector $\nabla \ell ( \theta ; Z )$ is $\sigma ^ { 2 }$ -sub-Gaussian for all $\theta \in \Theta$ and $Z \sim P _ { 0 }$ Equivalently, for all $v \in \mathbb { R } ^ { d }$ , $\langle v , \nabla \ell ( \theta ; Z ) \rangle$ is a $\sigma ^ { 2 }$ -sub-Gaussian random variable, i.e.,
+
+$$
+\begin{array} { r } { \mathbb { E } \left[ \exp ( \langle v , \nabla \ell ( \theta ; Z ) - \mathbb { E } [ \nabla \ell ( \theta ; Z ) ] \rangle ) \right] \leq \exp \bigl ( \| v \| _ { 2 } ^ { 2 } \sigma ^ { 2 } / 2 \bigr ) . } \end{array}
+$$
+
+In this work, our rates often depend on the sub-Gaussianity and Lipschitz parameters $\sigma$ and $G$ , and thus we define the shorthands ${ \widetilde { G } } : = \sigma { \sqrt { d } }$ and $\underline { { G } } : = \operatorname* { m i n } \{ G , \widetilde { G } \}$ . Intuitively, the $G$ -Lipschitzness assumption bounds the gradient in a ball around 0 (independently of $\theta$ ), while sub-Gaussianity implies that, for each $\theta$ , $\nabla \ell ( \theta ; Z )$ likely lies in $\mathbb { B } _ { 2 } ^ { d } ( \nabla \mathcal { L } ( \theta ; \bar { P } _ { 0 } ) , \widetilde { G } )$ . Generically, there is no ordering between $G$ and $\widetilde { G }$ : for linear loss $\ell ( \theta ; z ) = \langle \theta , z \rangle$ , depending on $P _ { 0 }$ , it can hold that $G \ll \widetilde G$ (e.g., $P _ { 0 } = \mathsf { U n i f } \{ - v , v \}$ for $v \in \mathbb { R } ^ { d }$ ), $\widetilde G \ll G$ (e.g., $P _ { 0 }$ is $\mathsf { N } ( \mu , \sigma ^ { 2 } I _ { d } )$ truncated in a ball around $\mu$ , with $\| \mu \| _ { 2 } \gg \sigma { \sqrt { d } } )$ or $\boldsymbol { G } \approx \boldsymbol { \widetilde { G } }$ (e.g., $P _ { 0 } = \mathsf { U n i f } \{ - 1 , + 1 \} ^ { d } )$ .
+
+We introduce the tasks we consider in this work, namely empirical risk minimization (ERM) and stochastic convex optimization (SCO). For a collection of samples from $n$ users $\boldsymbol { S } = ( S _ { 1 } , \ldots , S _ { n } )$ , where each $S _ { u } = \{ z _ { 1 } ^ { ( u ) } , \dots , z _ { m } ^ { ( u ) } \} \in \mathcal { Z } ^ { m }$ , we define the empirical risk objectives
+
+$$
+\mathcal { L } ( \boldsymbol { \theta } ; S _ { u } ) : = \frac { 1 } { m } \sum _ { i = 1 } ^ { m } \ell ( \boldsymbol { \theta } ; z _ { i } ^ { ( u ) } ) \ \mathrm { ~ a n d ~ } \ \mathcal { L } ( \boldsymbol { \theta } ; S ) : = \frac { 1 } { n } \sum _ { u = 1 } ^ { n } \mathcal { L } ( \boldsymbol { \theta } ; S _ { u } ) = \frac { 1 } { m n } \sum _ { u = 1 } ^ { n } \sum _ { i = 1 } ^ { m } \ell ( \boldsymbol { \theta } ; z _ { i } ^ { ( u ) } ) .
+$$
+
+In the user-level setting we wish to minimize $\textstyle { \mathcal { L } } ( \theta ; S )$ under user-level privacy constraints. Going beyond the empirical risk, we also solve SCO [51], i.e. minimizing a convex population objective
+
+when provided with samples from each users’ distributions. In the user-level setting, for a convex loss $\ell$ and a convex constraint set $\Theta$ , we observe $\begin{array} { r } { \pmb { \mathcal { S } } = ( S _ { 1 } , \dots , S _ { n } ) \sim \otimes _ { \pmb { u } \in [ n ] } ( P _ { u } ) ^ { m } } \end{array}$ and wish to
+
+$$
+\operatorname* { m i n i m i z e } _ { \theta \in \Theta } \frac { 1 } { n } \sum _ { u \in [ n ] } \mathcal { L } ( \theta ; P _ { u } ) : = \frac { 1 } { n } \sum _ { u \in [ n ] } \mathbb { E } _ { P _ { u } } [ \ell ( \theta ; Z ) ] .
+$$
+
+In the homogeneous case (Assumption A2), this reduces to the classic SCO setting:
+
+$$
+\operatorname* { m i n i m i z e } _ { \theta \in \Theta } \mathcal { L } ( \theta ; P _ { 0 } ) : = \mathbb { E } _ { P _ { 0 } } [ \ell ( \theta ; Z ) ] .
+$$
+
+# 2.2 Uniform concentration of queries
+
+Let $\phi : \mathcal { Z } \to \mathbb { R } ^ { d }$ be a $d$ -dimensional query function. We define concentration of random variables and uniform concentration of multiple queries as follows.
+
+Definition 2. A (random) sample $X ^ { n }$ supported on $[ - B , B ] ^ { d }$ is $( \tau , \gamma )$ -concentrated (and we call $\tau$ the “concentration radius”) if there exists $x _ { 0 } \in [ - B , B ] ^ { d }$ such that with probability at least $1 - \gamma ,$
+
+$$
+\operatorname* { m a x } _ { i \in [ n ] } \lVert X _ { i } - x _ { 0 } \rVert _ { 2 } \leq \tau .
+$$
+
+Definition 3 (Uniform concentration of vector queries). Let $\mathcal { Q } _ { B } ^ { d } = \{ \phi \colon \mathcal { Z }  [ - B , B ] ^ { d } \}$ be $a$ family of queries with bounded range. For $Z ^ { n } = ( Z _ { 1 } , \ldots , Z _ { n } ) \stackrel { \mathrm { i i d } } { \sim } P$ , we say that $( Z ^ { n } , \mathcal { Q } _ { B } ^ { d } )$ is $( \tau , \gamma )$ -uniformly-concentrated if with probability at least $1 - \gamma$ , we have
+
+$$
+\operatorname* { m a x } _ { i \in [ n ] } \operatorname* { s u p } _ { \phi \in \mathcal { Q } _ { B } ^ { d } } \Big \| \phi ( Z _ { i } ) - \mathbb { E } _ { Z \sim P } [ \phi ( Z ) ] \Big \| _ { 2 } \leq \tau .
+$$
+
+In this work, we will often consider $\sigma ^ { 2 }$ -sub-Gaussian random variables (or vectors), which are concentrated according to Definition 2. For example, if $X ^ { n }$ is drawn i.i.d. from a $\sigma ^ { 2 }$ -sub-Gaussian random vector supported on $[ - B , B ] ^ { d }$ , then it is $( \sigma \sqrt { d \log ( 2 n / \gamma ) } , \gamma )$ -concentrated around its mean (see, e.g., [56]). Finally, we define a distance between random variables (and estimators).
+
+Definition 4 ( $\beta$ -close Random Variables). For any two random variables $X _ { 1 } \sim P _ { 1 }$ and $X _ { 2 } \sim P _ { 2 }$ , we say $X _ { 1 }$ and $X _ { 2 }$ are $\beta$ -close, if $\| P _ { 1 } - P _ { 2 } \| _ { \mathsf { T V } } \leq \bar { \beta }$ . We use the notation $X _ { 1 } \sim _ { \beta } X _ { 2 }$ if $X _ { 1 }$ and $X _ { 2 }$ are $\beta$ -close.
+
+$\beta$ -closeness is useful as, in many of our results, the private estimator we propose returns a simple unbiased estimate with high probability and is bounded otherwise. Thus, it suffices to do the analysis in the “nice” case and crudely bound the error otherwise.
+
+# 3 High Dimensional Mean Estimation and Uniformly Concentrated Queries
+
+In this section, we present a private mean estimator with privacy cost proportional to the concentration radius. Using these techniques, we show that, under uniform concentration, we answer adaptivelychosen queries with privacy cost proportional to the concentration radius instead of the whole range. Our theorems guarantee that the estimator is $\beta$ -close (with $\beta$ exponentially small in $n$ ) to a simple unbiased estimator with small noise. We further show how to directly translate these results into bounds on the estimator error, which we demonstrate by providing tight bounds on estimating the mean of $\ell _ { 2 }$ -bounded random vectors under user-level DP constraints (Corollary 1).
+
+Given i.i.d samples $X ^ { n }$ from a distribution $P$ supported on $\mathbb { R } ^ { d }$ with mean $\mu$ , the goal of mean estimation is to design a private estimator that minimizes the $\mathbb { E } \left[ \lVert \mathsf { A } ( X ^ { n } ) - \dot { \mu } \rVert _ { 2 } ^ { 2 } \right]$ . We focus on distributions with bounded supprot $[ - B , B ] ^ { d }$ . However, our algorithm also generalize to the case when the mean is guaranteed to be in $[ - B , B ] ^ { d }$ . In the user-level setting (in the homogeneous case), one observes a dataset $s$ sampled as in (2) and wishes to minimize $\mathbb { E } \bar { [ \| \mathsf { A } ( S ) - \mathbb { E } P _ { 0 } \| \bar { 2 } ] }$ under user-level privacy constraints. We first focus on the scalar case.
+
+Mean estimation in one dimension. The algorithm uses a two-stage procedure, similar in spirit to those of [53], [40], and [39]. In the first stage of this procedure, we use the approximate median estimation in [27], detailed in Algorithm 6 in Appendix D.1, to privately estimate a crude interval
+
+Require: $X ^ { n } : = ( X _ { 1 } , X _ { 2 } , . . . , X _ { n } ) \in [ - B , B ] ^ { n }$ , τ : concentration radius, privacy parameter $\varepsilon > 0$ . 1: $[ a , b ] = \mathbf { P r i v a t e R a n g e } ( X ^ { n } , \varepsilon / 2 , \tau , B )$ with $| b - a | = 4 \tau$ . {Algorithm 6 in Appendix D.1. $\}$ 2: Sample $\textstyle \xi \sim \mathrm { L a p } { \bigl ( } 0 , { \frac { 8 \tau } { \varepsilon n } } { \bigr ) }$ and return
+
+$$
+\bar { \mu } = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \Pi _ { [ a , b ] } ( X _ { i } ) + \xi ,
+$$
+
+where $\Pi _ { [ a , b ] } ( x ) = \operatorname* { m a x } \{ a , \operatorname* { m i n } \{ x , b \} \}$ .
+
+in which the means lie, with accuracy $\Theta ( \tau )$ . The second stage clips the mean around this interval, reducing the sensitivity from $O ( B )$ to $O ( \tau )$ , and adds the appropriate Laplace noise. With high probability, we can recover the guarantee of the Laplace mechanism with smaller sensitivity since the samples are concentrated in a radius $\tau$ . We present the formal guarantees of Algorithm 1 in Theorem 1 and defer its proof to Appendix D.2.
+
+Theorem 1. Let $X ^ { n }$ be a dataset supported on $[ - B , B ]$ . The output of Algorithm 1, denoted by ${ \mathsf { A } } ( X ^ { n } )$ , is ${ \varepsilon } { - } D P .$ Furthermore, if $X ^ { n }$ is $( \tau , \gamma )$ -concentrated, it holds that
+
+$$
+\mathsf { A } ( X ^ { n } ) \sim _ { \beta } \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i } + L a p \bigg ( \frac { 8 \tau } { n \varepsilon } \bigg ) ,
+$$
+
+where $\begin{array} { r } { \beta = \operatorname* { m i n } \left\{ 1 , \gamma + \frac { B } { \tau } \exp \left( - \frac { n \varepsilon } { 8 } \right) \right\} } \end{array}$ . Moreover, Algorithm $I$ runs in time ${ \tilde { O } } ( n + \log ( B / \tau ) )$
+
+Compared to [40, 38, 39], our algorithm runs in time ${ \tilde { O } } ( n + \log ( B / \tau ) )$ instead of ${ \tilde { O } } ( n + B / \tau )$ owing to the approximate median estimation algorithm in [27], which is faster when $\tau \ll B$ .
+
+Mean estimation in arbitrary dimension. In the general $d$ -dimensional case, if $X ^ { n }$ is concentrated in $\ell _ { \infty }$ -norm, one simply applies Algorithm 1 to each dimension. However, when $X ^ { n }$ is concentrated in $\ell _ { 2 }$ -norm, naively upper bounding $\ell _ { \infty }$ -norm by the $\ell _ { 2 }$ -norm will incur a superfluous $\sqrt { d }$ factor: if $\| v \| _ { 2 } \leq \rho$ , each $| v _ { j } |$ is possibly as large as $\rho$ . To remedy this issue, we use the random rotation trick in [3, 54]. This guarantees that all coordinates have roughly the same range: for √ $v \in \mathbb { R } ^ { d }$ , with high probability, $\| R \dot { v } \| _ { \infty } \leq \tilde { O } ( \| v \| _ { 2 } / \sqrt { d } )$ , where $R$ is the random rotation. We present this procedure in Algorithm 2 and its performance in Theorem 2.
+
+Require: $X ^ { n } : = ( X _ { 1 } , X _ { 2 } , . . . , X _ { n } ) , X _ { i } \ \in \ [ - B , B ] ^ { d } , \tau , \gamma$ : concentration radius and probability, privacy parameter $\varepsilon , \delta > 0$ .   
+1: Let $D = { \mathsf { D i a g } } ( \omega )$ where $\omega$ is sampled uniformly from $\{ \pm 1 \} ^ { d }$ .   
+2: Set $U = d ^ { - 1 / 2 } \mathbf { H } D$ , where $\mathbf { H }$ is a $d$ -dimensional Hadamard matrix. For all $i \in [ n ]$ , compute $Y _ { i } = U X _ { i }$ .   
+3: Let $\begin{array} { r } { \varepsilon ^ { \prime } = \frac { \varepsilon } { \sqrt { 8 d \log ( 1 / \delta ) } } , \tau ^ { \prime } = 1 0 \tau \sqrt { \frac { \log ( d n / \gamma ) } { d } } } \end{array}$ log(dn/γ)d . For j ∈ [d], compute $\overset { \sim } { \underset {  } { \bar { Y } } } ( j ) = \mathbf { W i n s o r i z e d M e a n 1 D } \Big ( \{ Y _ { i } ( j ) \} _ { i \in [ n ] } , \varepsilon ^ { \prime } , \tau ^ { \prime } , \sqrt { d } B \Big ) .$   
+4: return ${ \bar { X } } = U ^ { - 1 } { \bar { Y } }$ .
+
+Theorem 2. Let $\mathsf { A } ( X ^ { n } ) = { \mathsf { W i n s o r i z e d M e a n H i g h D } } ( X ^ { n } , \varepsilon , \delta , \tau , B , \gamma )$ be the output of Algorithm 2. $\mathsf { A } ( X ^ { n } )$ is $( \varepsilon , \delta )$ -DP. Furthermore, if $X ^ { n }$ is $( \tau , \gamma )$ -concentrated in $\ell _ { 2 }$ -norm, there exists an estimator $\mathsf { A } ^ { \prime } ( X ^ { n } )$ such that ${ \mathsf { A } } ( X ^ { n } ) \sim _ { \beta } { \mathsf { A } } ^ { \prime } ( { \bar { X } } ^ { n } )$ and
+
+$$
+\mathbb { E } [ \mathsf { A } ^ { \prime } ( X ^ { n } ) | X ^ { n } ] = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } X _ { i } a n d \mathrm { ~ } \mathrm { V a r } ( \mathsf { A } ^ { \prime } ( X ^ { n } ) | X ^ { n } ) \leq c _ { 0 } \frac { d \tau ^ { 2 } \log ( d n / \alpha ) \log ( 1 / \delta ) } { n ^ { 2 } \varepsilon ^ { 2 } } ,
+$$
+
+We present the proof of Theorem 2 in Appendix D.3. We are able to transfer both Theorem 1 and Theorem 2 into finite-sample estimation error bounds for various types of concentrated distributions and obtain near optimal guarantees (see Appendix D.5 for an example in mean estimation of subGaussian distributions). The next corollary characterizes the risk of mean estimation for distributions supported on an $\ell _ { 2 }$ -bounded domain with user-level DP guarantees (see Appendix D.4 for the proof).
+
+Corollary 1. Assume $A 2$ holds with $P _ { 0 }$ supported on $\mathbb { B } _ { 2 } ^ { d } ( 0 , B )$ with mean $\mu$ . Given ${ \boldsymbol { s } } \ =$ $( S _ { 1 } , S _ { 2 } , . . . , S _ { n } )$ , $| S _ { u } | = m$ , consisting of m i.i.d. samples from $P _ { u }$ . There exists an $( \varepsilon , \delta )$ -userlevel $D P$ algorithm $\mathsf { A } ( \boldsymbol { S } )$ such that, if $\dot { n } \geq ( c _ { 1 } \sqrt { d \log ( 1 / \delta ) } / \varepsilon ) \log ( m ( d n + n ^ { 2 } \varepsilon ^ { 2 } ) )$ for a numerical constant c1, we have5
+
+$$
+\mathbb { E } \left[ \| \mathsf { A } ( \pmb { \mathscr { S } } ) - \mu \| _ { 2 } ^ { 2 } \right] = \frac { \mathrm { V a r } ( P _ { 0 } ) } { m n } + \tilde { O } \bigg ( \frac { d B ^ { 2 } } { m n ^ { 2 } \varepsilon ^ { 2 } } \bigg ) .
+$$
+
+Note that $\mathrm { V a r } ( P _ { 0 } ) \le B ^ { 2 }$ for any $P _ { 0 }$ supported on $\mathbb { B } _ { 2 } ^ { d } ( 0 , B )$ . Replacing ${ \mathrm { V a r } } ( P _ { 0 } )$ by $B ^ { 2 }$ , the bound 2is minimax optimal up to logarithmic factors. When onsame error bounds holds (up to constant) for estimating $A l$ with for a $\Delta \leq \mathsf { p o l y } ( d , \frac { 1 } { n } , \frac { 1 } { m } , \frac { 1 } { \varepsilon } )$ , the $\mathbb { E } _ { Z \sim P _ { u } } [ Z ]$ $u \in [ n ]$
+
+Note that algorithms in [38, 39], which focus on estimating the mean of $d$ -dimensional subGaussian distributions, can also be used to estimate the mean of $\ell _ { 2 }$ -bounded distributions since bounded random variables are also subGaussian. However, applying these algorithms directly will incur a superfluous $d$ factor in the mean square error. We void this using the random rotation trick in Algorithm 2.
+
+Answering multiple queries. We end this section by noting that, when a family of queries $\mathcal { Q }$ is uniformly concentrated (as made precise in Definition 3), we answer sequences of √ $K d$ -dimensional, adaptively chosen queries with error scaling as ${ \tilde { O } } ( { \sqrt { d K } } \tau / ( n \varepsilon ) )$ by applying Algorithm 2 to $\{ \phi _ { k } ( Z _ { i } ) \} _ { i \in [ n ] }$ with the right $( \varepsilon _ { 0 } , \delta _ { 0 } )$ . We make this formal in Theorem 10 in Appendix D.6.
+
+# 4 Empirical Risk Minimization with User-Level Differential Privacy
+
+In this section, we present an algorithm to solve the ERM objective of (3) under user-level DP constraints. We apply the results of Section 3 by noting that the SQ framework encompasses stochastic gradient methods. Informally, one can sequentially choose queries $\phi _ { k } ( z ) = \nabla \ell ( \theta _ { k } ; z )$ and, for a stepsize $\eta$ , update $\theta _ { k + 1 } = \Pi _ { \Theta } \big ( \dot { \theta } _ { k } - \eta v _ { k } \big )$ , where $v _ { k }$ is the answer to the $k$ -th query. For the results to hold, we require a uniform concentration result over the appropriate class of queries.
+
+Uniform concentration of stochastic gradients The class of queries for stochastic gradient methods is $\mathcal { Q } _ { \sf e r m } ~ : = ~ \{ \nabla \ell ( \theta ; \cdot ) ~ : ~ \theta ~ \in ~ \bar { \Theta } \}$ . We prove that when assumptions A3 and A4 hold, $( \{ \nabla \ell ( \cdot ; S _ { u } ) \} _ { u \in [ n ] } , \mathcal { Q } _ { \mathrm { e r m } } )$ is $( \tilde { O } ( \sigma \sqrt { d / m } ) , \alpha )$ -uniformly concentrated. The next proposition is a simplification of the result of [50] under the (stronger) assumption A3 that $\ell$ is uniformly $H$ -smooth. The proof, which we defer to Appendix E.1, hinges on a covering number argument.
+
+Proposition 1 (Concentration of random gradients). Let $S _ { u } \overset { \mathrm { i i d } } { \sim } P _ { u }$ $| S _ { u } | = m$ for $u \in [ n ]$ and $\alpha \geq 0$ $1 - \alpha$
+
+$$
+\operatorname* { m a x } _ { u \in [ n ] } \operatorname* { s u p } _ { \theta \in \Theta } \| \nabla \mathcal { L } ( \theta ; S _ { u } ) - \nabla \mathcal { L } ( \theta ; P _ { u } ) \| _ { 2 } = O \left( \sigma \sqrt { \frac { d \log \left( \frac { R H m } { d \sigma } \right) } { m } + \frac { \log \left( \frac { n } { \alpha } \right) } { m } } \right) .
+$$
+
+Stochastic gradient methods We state classical convergence results for stochastic gradient methods for both convex and non-convex losses under smoothness. For a function $F : \Theta \to \mathbb { R }$ , we assume access to a first-order stochastic oracle $\mathsf { O } _ { F , \nu ^ { 2 } }$ , i.e., a random mapping such that for all $\theta \in \Theta$ ,
+
+$$
+\mathsf { O } _ { F , \nu ^ { 2 } } ( \theta ) = \nabla \widehat { F } ( \theta ) \mathrm { ~ w i t h ~ } \mathbb { E } \Big [ \nabla \widehat { F } ( \theta ) \Big ] = \nabla F ( \theta )
+$$
+
+$$
+\operatorname { V a r } \left( \nabla { \widehat { F } } ( \theta ) \right) \leq \nu ^ { 2 } .
+$$
+
+We abstract optimization algorithms in the following way: an algorithm consists of an output set $\mathcal { O }$ , a sub-routine $\mathsf { Q u e r y : } \mathcal { O } \to \Theta$ that takes the last output and indicates the next point to query and a sub-routine Update : ${ \mathcal { O } } \times \mathbb { R } ^ { d } \to { \mathcal { O } }$ that takes the previous output and a stochastic gradient and returns the next output. After $T$ steps, we call Aggregate : $O ^ { * } \to \Theta$ , which takes all the previous outputs and returns the final point. (See Algorithm 7 in Appendix E.2 for how to instantiate generic first-order optimization in this framework.) We detail in Proposition 4 in Appendix E.2 standard convergence results for variations of (projected) stochastic gradient descent (SGD). We introduce this abstraction to forego the details of each specific algorithm and instead focus on the privacy and utility guarantees.
+
+Algorithm We recall the ERM setting with user-level DP. We observe $\boldsymbol { S } = ( S _ { 1 } , \ldots , S _ { n } )$ with $S _ { u } \in \mathcal { Z } ^ { m }$ for $u \in [ n ]$ and wish to solve the constrained optimization problem with objective in (3). We present our method in Algorithm 3 and provide utility and privacy guarantees in Theorem 3.
+
+# Algorithm 3 Winsorized First-Order Optimization
+
+1: Input: Number of iterations $T$ , optimization algorithm $\{ \mathcal { O } , \mathsf { Q u e r y }$ , Update, Aggregate}, privacy   
+parameters $( \varepsilon , \delta )$ , data $\boldsymbol { S } = ( S _ { 1 } , \ldots , S _ { n } )$ , initial output $o _ { 0 }$ , parameter set $\Theta$ , concentration radius   
+2: Set $\tau$ , probability $\begin{array} { r } { \varepsilon ^ { \prime } = \frac { \varepsilon } { 2 \sqrt { 2 T \log ( 2 / \delta ) } } } \end{array}$ $\gamma$ . and $\begin{array} { r } { \delta ^ { \prime } = \frac { \delta } { 2 T } } \end{array}$   
+3: for $t = 0 , \ldots , T - 1$ do   
+4: $\theta _ { t } \gets \mathsf { Q u e r y } ( o _ { t } )$ .   
+5: For each user $u \in [ n ]$ , compute   
+6: Compute $\bar { g } _ { t } =$ Winso $\begin{array} { r l r } & { } & { g _ { t } ^ { ( u ) } = \nabla \mathcal { L } ( \theta _ { t } ; S _ { u } ) = \displaystyle \frac { 1 } { m } \sum _ { j \in [ m ] } \nabla \ell ( \theta _ { t } ; z _ { j } ^ { ( u ) } ) . } \\ & { } & { \mathrm { ~ } \mathrm { ~ r i z e d M e a n H i g h } \mathbf { D } ( \{ g _ { t } ^ { ( u ) } \} _ { u \in [ n ] } , \varepsilon ^ { \prime } , \delta ^ { \prime } , \tau , G , \gamma ) . } \end{array}$   
+7: $o _ { t + 1 } \gets \mathsf { U p d a t e } ( o _ { t } , \bar { g } _ { t } )$ .   
+8: end for   
+9: return $\bar { \theta } \gets \mathsf { A g g r e g a t e } ( o _ { 0 } , \ldots , o _ { T } )$ .
+
+Theorem 3 (Privacy and utility guarantees for ERM). Assume $A 2$ holds and recall that ${ \widetilde { G } } = \sigma { \sqrt { d } } ,$ , assume6 $n = \tilde { \Omega } ( \sqrt { d T } / \varepsilon )$ and let $\widehat { \theta }$ be the output of Algorithm 3. There exists variants of projected $S G D$ (e.g. the ones we present in Proposition 4) such that, with probability greater than $1 - \gamma$ :
+
+(i) If for all $z \in \mathcal { Z } , \ell ( \cdot ; z )$ is convex, then
+
+$$
+\mathbb { E } \bigg [ \mathcal { L } ( \widehat { \theta } ; \mathcal { S } ) - \operatorname* { i n f } _ { \theta ^ { \prime } \in \Theta } \mathcal { L } ( \theta ^ { \prime } ; \mathcal { S } ) \bigg | \mathcal { S } \bigg ] = \tilde { O } \Bigg ( \frac { R ^ { 2 } H } { T } + R \widetilde { G } \frac { \sqrt { d } } { n \sqrt { m } \varepsilon } \Bigg ) .
+$$
+
+(ii) If for all $z \in \mathcal { Z } , \ell ( \cdot ; z )$ is $\mu$ -strongly-convex, then
+
+$$
+\mathbb { E } \bigg [ \mathcal { L } ( \widehat { \theta } ; \mathcal { S } ) - \operatorname* { i n f } _ { \theta ^ { \prime } \in \Theta } \mathcal { L } ( \theta ^ { \prime } ; \mathcal { S } ) \bigg | \mathcal { S } \bigg ] = \widetilde { \mathcal { O } } \bigg ( G R \exp \big ( - \frac { \mu } { H } T \big ) + \widetilde { G } ^ { 2 } \frac { d } { \mu n ^ { 2 } m \varepsilon ^ { 2 } } \bigg ) .
+$$
+
+(iii) Otherwise, defining the gradient mapping7 $\begin{array} { r } { { \sf G } _ { F , \gamma } ( \theta ) : = \frac { 1 } { \gamma } [ \theta - \Pi _ { \Theta } ( \theta - \gamma \nabla F ( \theta ) ) ] . } \end{array}$ , we have
+
+$$
+\mathbb { E } \bigg [ \| \mathsf { G } _ { \mathcal { L } ( \cdot ; S ) , 1 / H } ( \widehat { \theta } ) \| _ { 2 } ^ { 2 } | S \bigg ] = \tilde { O } \bigg ( \frac { H ^ { 2 } R } { T } + H R \widetilde G \frac { \sqrt { d } } { n \sqrt { m } \varepsilon } \bigg ) .
+$$
+
+For $\varepsilon \le 1 , \delta > 0$ , Algorithm $^ 3$ instantiated with any first-order gradient algorithm is $( \varepsilon , \delta )$ -user-level $D P .$ In the case that only $A l$ holds, the same guarantees hold whenever $\Delta \leq \mathsf { p o l y } ( d , \frac { 1 } { n } , \frac { 1 } { m } , \frac { 1 } { \varepsilon } )$ .
+
+We present the proof in Appendix E.3. For the utility guarantees, the crux of the proof resides in Theorem 10: as well as ensuring small excess loss in expectation, the SQ algorithm produces with high probability a sample from the stochastic gradient oracle $\operatorname { O } _ { \mathcal { L } ( \cdot ; S ) , \nu ^ { 2 } }$ where $\begin{array} { r } { \nu ^ { 2 } = { \tilde { O } } ( T { \widetilde G } ^ { 2 } \frac { d } { n ^ { 2 } m \varepsilon ^ { 2 } } ) } \end{array}$ When this happens for all $T$ steps, the analysis of stochastic gradient methods provide the desired regret. The privacy guarantees follow from the strong composition theorem of [23].
+
+Importantly, when the function exhibits (some) strong-convexity (which will be the case for any regularized objective), we are able to localize the optimal parameter—up to the privacy cost—in ${ \cal \tilde { O } } ( H / \mu )$ steps. This will be particularly important in Section 5.
+
+Corollary 2 (Localization). Let $\widehat { \theta }$ be the output of Algorithm $^ 3$ on the ERM problem of (3). Assume that $\ell ( \cdot ; z )$ is $\mu$ -strongly-convex for all $z \in { \mathcal { Z } }$ , that $n = \tilde { \Omega } ( \sqrt { d H / \mu } )$ and set $T =$ $\begin{array} { r } { \frac { H } { \mu } \log \left( n ^ { 2 } m ( \underline { { G } } / \widetilde G ^ { 2 } ) \frac { \mu R \varepsilon ^ { 2 } } { d } \right) } \end{array}$ and $\begin{array} { r } { \gamma = { \frac { \sigma ^ { 2 } d ^ { 2 } } { \mu ^ { 2 } n ^ { 2 } m \varepsilon ^ { 2 } R ^ { 2 } } } } \end{array}$ . For $\theta _ { S } ^ { * } \in \mathrm { a r g m i n } _ { \theta ^ { \prime } \in \Theta } \mathcal { L } ( \theta ^ { \prime } ; S )$ , it holds8
+
+$$
+\mathbb { E } [ \| \widehat { \theta } - \theta _ { S } ^ { * } \| _ { 2 } ^ { 2 } ] = \tilde { O } \bigg ( \frac { \sigma ^ { 2 } d ^ { 2 } } { \mu ^ { 2 } n ^ { 2 } m \varepsilon ^ { 2 } } \bigg ) .
+$$
+
+# 5 Stochastic Convex Optimization with User-level Privacy
+
+In this section we address the SCO task of (5) under user-level DP constraints. Our approach (which we show in Algorithm 4) solves a sequence of carefully regularized ERM problems, drawing on√ the guarantees of the previous section. Recall that $\widetilde { G } = \sigma \sqrt { d }$ and $\underline { { G } } = \operatorname* { m i n } \{ G , \widetilde { G } \}$ , and that $\ell$ is $H$ -smooth under assumption A3. In this section, we assume that $\ell$ is convex. We first present our results and state an upper and lower bound for SCO with user-level privacy constraints.
+
+Theorem 4 (Phased ERM for SCO). Algorithm $^ { 4 }$ is user-level $( \varepsilon , \delta )$ -DP. When $A 2$ holds and $n =$ $\tilde { \Omega } ( \operatorname* { m i n } \{ \sqrt [ 3 ] { d ^ { 2 } m H ^ { 2 } R ^ { 2 } / ( G { \underline { { G } } } \varepsilon ^ { 4 } ) } , H R \sqrt { m } / ( \sigma \varepsilon ) \} )$ , or, equivalently, $\begin{array} { r } { H = \tilde { O } ( \sqrt { \frac { n ^ { 2 } \varepsilon ^ { 2 } \sigma ^ { 2 } } { R ^ { 2 } m } + \frac { G \bar { a } ^ { 3 } \varepsilon ^ { 4 } } { d ^ { 2 } R ^ { 2 } m } } ) f o r } \end{array}$ r all $P$ and $\ell$ satisfying Assumptions $A 3$ and A4, we have
+
+$$
+\mathbb { E } \left[ \mathcal { L } \big ( \mathsf { A } _ { \mathsf { P h a s e d E R M } } ( S ) ; P _ { 0 } \big ) \right] - \operatorname* { m i n } _ { \theta ^ { \prime } \in \Theta } \mathcal { L } ( \theta ^ { \prime } ; P _ { 0 } ) = \tilde { O } \left( \frac { R \sqrt { G G } } { \sqrt { m n } } + R \widetilde G \frac { \sqrt { d } } { n \sqrt { m } \varepsilon } \right) .
+$$
+
+Furthermore, our results still hold in the heterogeneous setting (Assumption $A I$ ) whenever $\Delta \le$ $\mathtt { p o l y } ( d , \frac { 1 } { n } , \frac { 1 } { m } , \frac { 1 } { \varepsilon } )$ ; the risk guarantee being with respect to any user distribution $P _ { u }$ .
+
+Theorem 5 (Lower bound for SCO). There exists a distribution $P$ and a loss $\ell$ satisfying Assumptions $A 3$ and A4 such that for any algorithm A satisfying $( \varepsilon , \delta )$ -DP at user-level, we have
+
+$$
+\mathbb { E } \left[ \mathcal { L } ( \mathsf { A } ( \mathcal { S } ) ; P ) \right] - \operatorname* { m i n } _ { \theta ^ { \prime } \in \Theta } \mathcal { L } ( \theta ^ { \prime } ; P ) = \Omega \Bigg ( \frac { R G } { \sqrt { m n } } + R \underline { { G } } \frac { \sqrt { d } } { n \sqrt { m } \varepsilon } \Bigg ) .
+$$
+
+When $G = \Theta ( \sigma { \sqrt { d } } )$ , the upper bound matches the lower bound up to logarithmic factors. We present the algorithm and proof for Theorem 4 in Section 5.1. Theorem 5 is proved in Section 5.2.
+
+# 5.1 Upper bound: minimizing a sequence of regularized ERM problems
+
+We now present Algorithm 4, which achieves the upper bound of Theorem 4. It is similar in spirit to Phased ERM [29] and EpochGD [34], in that at each round we minimize a regularized ERM problem with fresh samples and increased regularization, initializing each round from the final iterate of the previous round. This allows us to localize the optimum with exponentially increasing accuracy without blowing up our privacy budget. We solve each round using Algorithm 3 to guarantee privacy and obtain an approximate minimizer. We show the guarantee in Corollary 2 is enough to achieve optimal rates. We provide the proof of Theorem 4 in Appendix $\mathrm { F }$ and present a sketch here.
+
+Algorithm 4 APhasedERM: Phased ERM   
+
+<table><tr><td>parameterσ.</td><td>Require: Private dataset: S = (S1,...,Sn) ∈ (Zm)n : n × m i.i.d samples from P,H-smooth, convex loss function l,convex set Θ C Rd, privacy parameters ε ≤ 1,δ ≤1/n²,sub-Gaussian</td><td></td><td></td><td></td></tr><tr><td>1: SetT=[log2(</td><td>(Gn√mε)1,入=√ GG gd nm</td><td>/R</td><td></td><td></td></tr><tr><td>2:</td><td>fort=1toTdo</td><td>n²mε²</td><td></td><td></td></tr><tr><td>3: 4:</td><td>Setnt =，,xt =4t入</td><td></td><td></td><td></td></tr><tr><td></td><td> Sample St, nt users that have not participated in previous rounds. Using Algorithm 3,compute</td><td></td><td></td><td></td></tr><tr><td></td><td>an approximate minimizer 0t, to the accuracy of Corollary 2, for the objective</td><td>m</td><td></td><td></td></tr><tr><td></td><td>Lλt，t_(0;St)=</td><td>1 MM e（0,z</td><td>t 11-0t-12.</td><td></td></tr><tr><td></td><td></td><td>mnt</td><td>(u) 十 2</td><td></td></tr><tr><td>5: end for</td><td></td><td>uESt j=1</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>6:return</td><td>T</td><td></td><td></td><td></td></tr></table>
+
+Proof sketch of Theorem 4. The privacy guarantee comes directly from the privacy guarantee of Algorithm 3 and the fact that $S _ { t }$ are non-overlapping. The proof for utility is similar to the proof of Theorem 4.8 in [29]. In round $t$ of Algorithm 4, we consider the true minimizer $\theta _ { t } ^ { * }$ and the approximate minimizer $\widehat { \theta } _ { t }$ . By stability [14], we can bound the generalization error of $\theta _ { t } ^ { * }$ (see Proposition 5 in Appendix F) and, by Corollary 2, we can bound $\widehat { \mathbb { E } } \| \widehat { \theta } _ { t } - \theta _ { t } ^ { * } \| _ { 2 } ^ { 2 }$ . We finally choose $\{ ( \lambda _ { t } , n _ { t } ) \} _ { t \le T }$ such that the assumptions of Corollary 2 hold and to minimize the final error. □
+
+# 5.2 Lower bound: SCO is harder than Gaussian mean estimation
+
+First of all, note that it suffices to prove the lower bounds in the homogeneous setting as any level of heterogeneity only makes the problem harder. Theorem 5 holds for $( \varepsilon , \delta )$ -user-level DP—importantly, this is a setting for which lower bounds are generally more challenging (we provide a related lower bound for $\varepsilon$ -user-level DP in Appendix A.2). We present the proof in Appendix F.2 and a sketch here.
+
+Proof sketch of Theorem 5. The (constrained) minimax lower bound decomposes into a statistical rate and a privacy rate. The statistical rate is optimal (see, e.g., [44, 2]), thus we focus on the privacy rate. We consider linear losses of the form $\ell ( \theta ; z ) = - \langle \theta , z \rangle$ . We show that optimizing $\mathsf { \bar { L } } ( \theta ; \dot { P } ) = \mathbb { E } _ { P } [ \ell ( \theta ; Z ) ]$ over $\theta \in \Theta$ is harder than the mean estimation task for $P$ . Intuitively, $C ( \theta ; P ) = - \langle \theta , \mathbb { E } Z \rangle$ attains its minimum at $\theta ^ { * } = R \mathbb { E } [ Z ] / \| \mathbb { E } [ Z ] \| _ { 2 }$ and finding $\theta ^ { * }$ provides a good estimate of (the direction of) $\mathbb { E } [ Z ]$ . We make this formal in Proposition 6. Next, for Gaussian mean estimation, we reduce, in Proposition 3, user-level DP to item-level DP with lower variance by having each user contribute their sample average (which is a sufficient statistic). We conclude with the results of [38] (see Proposition 7) by proving in Corollary 6 that estimating the direction of the mean with item-level privacy is hard. □
+
+# Acknowledgments
+
+The authors would like to thank Hilal Asi and Karan Chadha for comments on an earlier draft as well as Yair Carmon, Peter Kairouz, Gautam Kamath, Sai Praneeth Karimireddy, Thomas Steinke and Sebastian Stich, for useful discussions and pointers to very relevant references.
+
+# References
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+        "text": "1Stanford University 2Cornell University 3Google Research 4Courant Institute danilevy@stanford.edu, zs335@cornell.edu, {kamin, satyenkale, kulesza, mohri, theertha}@google.com ",
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+        "type": "text",
+        "text": "Abstract ",
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+        "text": "We propose and analyze algorithms to solve a range of learning tasks under userlevel differential privacy constraints. Rather than guaranteeing only the privacy of individual samples, user-level DP protects a user’s entire contribution $m \\geq 1$ samples), providing more stringent but more realistic protection against information leaks. We show that for high-dimensional mean estimation, empirical risk minimization with smooth losses, stochastic convex optimization, and learning hypothesis classes with finite metric entropy, the privacy cost decreases as $O ( 1 / \\bar { \\sqrt { m } } )$ as users provide more samples. In contrast, when increasing the number of users $n$ , the privacy cost decreases at a faster $O ( 1 / n )$ rate. We complement these results with lower bounds showing the minimax optimality of our algorithms for mean estimation and stochastic convex optimization. Our algorithms rely on novel techniques for private mean estimation in arbitrary dimension with error scaling as the concentration radius $\\tau$ of the distribution rather than the entire range. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Releasing seemingly innocuous functions of a data set can easily compromise the privacy of individuals, whether the functions are simple counts [35] or complex machine learning models like deep neural networks [52, 30]. To protect against such leaks, Dwork et al. proposed the notion of differential privacy (DP). Given some data from $n$ participants in a study, we say that a statistic of the data is differentially private if an attacker who already knows the data of $n - 1$ participants cannot reliably determine from the statistic whether the $n$ -th remaining participant is Alice or Bob. With the recent explosion of publicly available data, progress in machine learning, and widespread public release of machine learning models and other statistical inferences, differential privacy has become an important standard and is widely adopted by both industry and government [32, 5, 21, 55]. ",
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+        "text": "The standard setting of DP described in [22] assumes that each participant contributes a single data point to the dataset, and preserves privacy by “noising” the output in a way that is commensurate with the maximum contribution of a single example. This is not the situation faced in many applications of machine learning models, where users often contribute multiple samples to the model—for example, when language and image recognition models are trained on the users’ own data, or in federated learning settings [37]. As a result, current techniques either provide privacy guarantees that degrade with a user’s increased participation or naively add a substantial amount of noise, relying on the group property of differential privacy, which significantly harms the performance of the deployed model. ",
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+        "type": "text",
+        "text": "To remedy this issue, we consider user-level DP, which instead of guaranteeing privacy for individual samples, protects a user’s entire contribution $\\textcircled { m } \\geq 1$ samples). This is a more stringent but more realistic privacy desideratum. To hold, it requires that the output of our algorithm does not significantly change when changing user’s entire contribution—i.e. possibly swapping up to $m$ samples in total. We make this formal in Definition 1. Very recently, for the reasons outlined above, there has been increasing interest in user-level DP for applications such as estimating discrete distributions under user-level privacy constraints [46], PAC learning with user-level privacy [31], and bounding user contributions in ML models [4, 26]. Differentially private SQL with bounded user contributions was proposed in [59]. User-level privacy has been also studied in the context of learning models via federated learning [49, 48, 58, 6]. ",
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+        "text": "In this paper, we tackle the problem of learning with user-level privacy in the central model of DP. In particular, we provide algorithms and analyses for the tasks of mean estimation, empirical risk minimization (ERM), stochastic convex optimization (SCO), and learning hypothesis classes with finite metric entropy. Our utility analyses assume that all users draw their samples i.i.d. from related distributions, a setting we refer to as limited heterogeneity. On these tasks, naively applying standard mechanisms, such as Laplace or Gaussian, or using the group property with item-level DP estimators, both yield a privacy error independent of $m$ . We first develop novel private mean estimators in high dimension with statistical and privacy error scaling with the (arbitrary) concentration radius rather than the range, and apply these to the statistical query setting [SQ; 41]. Our algorithms then rely on (privately) answering a sequence of adaptively chosen queries using users’ samples, e.g., gradient queries in stochastic gradient descent algorithms. We show that for these tasks, the additional error√ due to privacy constraints decreases as $\\bar { O } ( 1 / \\sqrt { m } )$ , contrasting with the naive rate—independent of $m$ . Interestingly, increasing $n$ , the number of users, decreases the privacy cost at a faster $O ( 1 / n )$ rate. ",
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+        "text": "Importantly, our results imply concrete practical recommendations on sample collection, regardless of the level of heterogeneity. Indeed, increasing $m$ will yield the most value in the i.i.d. setting and will yield no improvement when the users’ distributions are arbitrary. As the real-world will lie somewhere in between, our results exhibit a regime where, for any heterogeneity, it is strictly better to collect more users (increasing $n$ ) than more samples per user (increasing $m$ ). ",
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+        "type": "text",
+        "text": "1.1 Our Contributions and Related Work ",
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+        "type": "text",
+        "text": "We provide a theoretical tool to construct estimators for tasks with user-level privacy constraints and apply it to a range of learning problems. ",
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+        "text": "Optimal private mean estimation and uniformly concentrated queries (Section 3) We show that for a random variable in $[ - B , B ]$ concentrated in an unknown interval of radius $\\tau$ (made precise in Definition 2), we can privately estimate its mean with error proportional to $\\tau$ rather than $B$ , as we would obtain using standard private mean estimation techniques such as Laplace mechanism [24]. When data is concentrated in $\\ell _ { \\infty }$ -norm, several papers show that one can achieve an error scaling with $\\tau$ rather than $B$ , either asymptotically [53], for Gaussian mean-estimation [40, 38], for sub-Gaussian symmetric distributions [18, 17] or for distributions with bounded $p$ -th moment [39]. We propose a private mean estimator (Algorithm 2) with error scaling with $\\tau$ that works in arbitrary dimension when data is concentrated in $\\ell _ { 2 }$ -norm (Theorem 2). In Corollary 1, we show it (optimally) solves mean estimation under user-level privacy constraints for random vectors bounded in $\\ell _ { 2 }$ -norm. In Appendix D.6, we show that for uniformly concentrated queries (see Definition 3), sequentially applying Algorithm 2 privately answers $K$ adaptively chosen queries with privacy cost $\\tilde { O } ( \\tau \\sqrt { K } / n \\varepsilon )$ . ",
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+        "type": "text",
+        "text": "Our conclusions relate to the growing literature in adaptive data analysis. While a sequence of work [25, 9, 27, 28] use techniques from differential privacy and their answers are $( \\varepsilon , \\delta )$ -DP with $\\varepsilon = \\Theta ( 1 )$ , our work guarantees privacy for arbitrary $\\varepsilon$ with the additional assumption of uniform concentration. ",
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+        "type": "text",
+        "text": "Empirical risk minimization (Section 4) An influential line of papers studies ERM under itemlevel privacy constraints [19, 42, 8]. Importantly, these papers assume arbitrary data, i.e., not necessarily samples from users’ distributions. The exact analog of ERM in the user-level setting is consequently less interesting as, for $n$ data points $\\{ z _ { 1 } , \\ldots , z _ { n } \\}$ , in the worst case, each user $u \\in [ n ]$ contributes $m$ copies of $z _ { u }$ and the problem reduces to the item-level setting. Instead, we consider the (related) problem of ERM when users contribute points sampled i.i.d. Assuming some regularity (A3 and A4), we develop and analyze algorithms for ERM under user-level DP constraints for convex, strongly-convex, and non-convex losses (Theorem 3). ",
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+        "type": "text",
+        "text": "Optimal stochastic convex optimization (Section 5) Under item-level DP (or equivalently, userlevel DP with $m = 1$ ), a sequence of work [19, 8, 10, 11, 29] establishes the constrained minimax risk as $\\tilde { \\Theta } ( 1 / \\sqrt { n } + \\sqrt { d } / ( n \\varepsilon ) )$ . In this paper, with the additional assumptions that the losses are individually smooth2 and the gradients are sub-Gaussian random vectors, we prove matching upper√ (Theorem 4) and lower bounds (Theorem 5) of order $\\tilde { \\Theta } ( 1 / \\sqrt { n m } + \\sqrt { d } / ( n \\sqrt { m } \\varepsilon ) )$ in a regime we make precise. We leave closing the gap outside of this regime to future work. ",
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+        "text": "Limit of learning with a fixed number of users (Appendix B) Finally, we resolve a conjecture of [4] and prove that with a fixed number of users, even in the limit $m  \\infty$ (i.e., each user has an infinite number of samples), we cannot reach zero error. In particular, we prove that for all the learning tasks we consider, the risk under user-level privacy constraints is at least $\\Omega ( e ^ { - \\varepsilon n } )$ regardless of $m$ . Note that this does not contradict the results above since they require $n = \\Omega ( ( \\log m ) / \\varepsilon )$ . ",
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+        "text": "Finally, we provide results in Appendix A for learning under pure user-level DP for function classes with finite metric entropy. We apply these to SCO with $\\ell _ { \\infty }$ constraints (Remark 1) and achieve (near)-optimal rates. ",
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+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
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+        "type": "text",
+        "text": "Notation. Throughout this work, $d$ denotes the dimension, $n$ the number of users, and $m$ the number of samples per user. Generically, $\\sigma$ will denote the sub-Gaussian parameter, $\\tau$ the concentration radius, $\\nu$ the variance of a random vector and $P$ a data distribution. We denote the optimization variable with $\\theta \\in \\Theta \\subset \\mathbb { R } ^ { d }$ , use $z$ (or $Z$ when random) to denote the data sample supported on a space $\\mathcal { Z }$ , and $\\ell \\colon \\Theta \\times \\mathcal { Z }  \\mathbb { R }$ for the loss function. Gradients (denoted $\\nabla$ ) are always taken with respect to the optimization variable $\\theta$ . For a convex set $\\mathcal { C }$ , $\\Pi _ { \\mathcal { C } }$ denotes the euclidean projection on $\\mathcal { C }$ , i.e. $\\begin{array} { r } { \\Pi _ { \\mathcal { C } } ( y ) : = \\operatorname * { a r g m i n } _ { z \\in \\mathcal { C } } \\| y - z \\| _ { 2 } } \\end{array}$ . We use $\\mathsf { A }$ to refer to (possibly random) private mechanisms and $X ^ { n }$ as a shorthand for the dataset $( X _ { 1 } , \\ldots , X _ { n } )$ . For two distributions $P$ and $Q$ , we denote by $\\| \\boldsymbol { P } - \\boldsymbol { Q } \\| _ { \\mathsf { T V } }$ their total variation distance and $D _ { \\mathrm { k l } } \\left( P \\| Q \\right)$ their Kullback-Leibler divergence. For a random vector $X \\sim P$ supported on $\\mathbb { R } ^ { d }$ , we use $\\mathrm { V a r } ( P )$ or $\\operatorname { V a r } ( X )$ to denote $\\mathbb { E } \\left[ \\lVert X - \\mathbb { E } [ X ] \\rVert _ { 2 } ^ { 2 } \\right]$ , which is equal to the trace of the covariance matrix of $X$ . ",
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+        "type": "text",
+        "text": "Next, we consider differential privacy in the most general way, which only requires specifying a dataset space $\\mathbb { S }$ and a distance $\\mathrm { d }$ on $\\mathbb { S }$ . ",
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+        "text": "Definition 1 (Differential Privacy). Let $\\varepsilon , \\delta \\geq 0$ . Let $\\ u : \\mathbb { S } \\to \\Theta$ be a (potentially randomized) mechanism. We say that A is $( \\varepsilon , \\delta )$ - $D P$ with respect to d if for any measurable subset $O \\subset \\Theta$ and all $S , S ^ { \\prime } \\in \\mathbb { S }$ satisfying $\\mathrm { d } ( S , S ^ { \\prime } ) \\stackrel { \\cdot } { = } 1$ , ",
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+        "text": "$$\n\\mathbb { P } ( \\mathsf { A } ( S ) \\in O ) \\le e ^ { \\varepsilon } \\mathbb { P } ( \\mathsf { A } ( S ^ { \\prime } ) \\in O ) + \\delta .\n$$",
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+        "type": "text",
+        "text": "$I f \\delta = 0$ , we refer to this guarantee as pure differential privacy. ",
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+        "type": "text",
+        "text": "For a data space $\\mathcal { Z }$ , choosing $\\mathbb { S } = \\mathcal { Z } ^ { n }$ and $\\begin{array} { r } { \\mathrm { d } ( S , S ^ { \\prime } ) = \\mathrm { d } _ { \\mathsf { H a m } } ( S , S ^ { \\prime } ) = \\sum _ { i = 1 } ^ { n } 1 \\{ z _ { i } \\neq z _ { i } ^ { \\prime } \\} } \\end{array}$ recovers the canonical setting considered in most of the literature—we refer to this as item-level differential privacy. When we wish to guarantee privacy for users rather than individual samples, we instead assume a structured dataset into which each of $n$ users contributes $m > 1$ samples. This corresponds to $\\mathbb { S } = ( \\mathcal { Z } ^ { m } ) ^ { n }$ such that for ${ \\boldsymbol { s } } \\in \\mathbb { S }$ , we have ",
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+        "img_path": "images/ea458b724dd12507bcea011301af5e9b02a64134093d1d1441c8caf86309d486.jpg",
+        "text": "$$\n{ \\mathcal { S } } = ( S _ { 1 } , \\ldots , S _ { n } ) , { \\mathrm { ~ w h e r e ~ } } S _ { u } = \\left\\{ z _ { 1 } ^ { ( u ) } , \\ldots , z _ { m } ^ { ( u ) } \\right\\} { \\mathrm { ~ a n d ~ } } \\mathrm { d } _ { { \\mathfrak { u s e r } } } ( S , S ^ { \\prime } ) : = \\sum _ { u = 1 } ^ { n } 1 \\{ S _ { u } \\neq S _ { u } ^ { \\prime } \\} ,\n$$",
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+        "type": "text",
+        "text": "which means that, in this setting, two datasets are neighboring if at most one of the user’s contributions differ. We henceforth refer to this setting as user-level differential privacy. ",
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+        "type": "text",
+        "text": "Distributional assumptions. In the case of user-level privacy with $n$ users each providing $m$ samples, we assume existence of a collection of distributions $\\{ \\bar { P _ { u } } \\} _ { u \\in [ n ] }$ over $\\mathcal { Z }$ . One then observes the following user-level dataset3 ",
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+        "text": "$$\n{ \\mathcal { S } } = ( S _ { 1 } , \\ldots , S _ { n } ) { \\mathrm { ~ w h e r e ~ } } S _ { u } \\stackrel { \\mathrm { i i d } } { \\sim } P _ { u } .\n$$",
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+        "type": "text",
+        "text": "In this paper, we consider the limited heterogeneity setting, i.e. when the users have related distributions. This setting is more reflective of practice, especially in light of growing interest towards federated learning applications [37, 60]. ",
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+        "type": "text",
+        "text": "Assumption A1 (Limited heterogeneity setting). There exists a distribution $P _ { 0 }$ over $\\mathcal { Z }$ such that all the user distributions are close to $P _ { 0 }$ in total variation distance, i.e. ",
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+        "type": "equation",
+        "img_path": "images/81716f6b4f8e37a8193044adaf5d49987d29a37d8e67bbca13b67bf5e14b9156.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { u \\in [ n ] } \\lVert P _ { u } - P _ { 0 } \\rVert \\mathrm { { r v } } \\leq \\Delta ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\Delta \\geq 0$ quantifies the level of heterogeneity. Note that $\\Delta = 0$ corresponds to assumption $A 2$ . ",
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+        "type": "text",
+        "text": "Note that our TV-based definition is natural in this setting as it is closely related to the notion of discrepancy (or $d _ { A }$ distance) which plays a key role in domain adaption scenarios [47, 12]. Lower bound results have been given in terms of the discrepancy measure (see [13]), which further justify the adoption of this definition in the presence of multiple distributions. ",
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+        "type": "text",
+        "text": "In the case that $\\Delta = 0$ , A1 reduces to the standard homogeneous setting. Many fundamental papers choose this setting when explicating minimax rates under constraints (e.g. in distributed optimization and federated learning [61] or under communication constraints [63, 15]). ",
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+        "type": "text",
+        "text": "Assumption A2 (Homogeneous setting). The distributions of individual users are equal, meaning there exists $P _ { 0 }$ such that for all $u \\in [ n ]$ , $P _ { u } = P _ { 0 }$ . ",
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+        "text": "In this paper, we develop techniques and provide matching upper and lower bounds for solving learning tasks in the homogeneous setting. In Appendix C, we prove that our techniques naturally apply to the heterogeneous setting in a black-box fashion, and for all considered problems provide meaningful guarantees under Assumption A1. Moreover, the algorithm achieves almost optimal rate whenever $\\Delta$ is (polynomially) small. See the detailed statement in Theorem 9. ",
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+        "type": "text",
+        "text": "2.1 ERM and stochastic convex optimization ",
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+        "text": "Assumptions on the loss. Throughout this work, we assume that the parameter space $\\Theta$ is closed, convex, and satisfies $\\lVert { \\boldsymbol { \\theta } } - { \\boldsymbol { \\vartheta } } \\rVert _ { 2 } \\leq R$ for all $\\theta , \\vartheta \\in \\Theta$ . We also assume that the loss $\\ell \\colon \\Theta \\times \\mathcal { Z }  \\mathbb { R }$ is $G$ -Lipschitz w.r.t. the $\\ell _ { 2 } { \\mathrm { - n o r m } } ^ { 4 }$ , meaning that for all $z \\in { \\mathcal { Z } }$ , for all $\\theta \\in \\Theta$ , $\\| \\nabla \\ell ( \\theta ; z ) \\| _ { 2 } \\leq G$ . We further consider the following assumptions. ",
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+        "type": "text",
+        "text": "Assumption A3. The function $\\ell ( \\cdot ; z )$ is $H$ -smooth. In other words, the gradient $\\nabla \\ell ( \\theta ; z )$ is $H$ Lipschitz in the variable $\\theta$ for all $z \\in { \\mathcal { Z } }$ . ",
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+        "type": "text",
+        "text": "Assumption A4. The random vector $\\nabla \\ell ( \\theta ; Z )$ is $\\sigma ^ { 2 }$ -sub-Gaussian for all $\\theta \\in \\Theta$ and $Z \\sim P _ { 0 }$ Equivalently, for all $v \\in \\mathbb { R } ^ { d }$ , $\\langle v , \\nabla \\ell ( \\theta ; Z ) \\rangle$ is a $\\sigma ^ { 2 }$ -sub-Gaussian random variable, i.e., ",
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+        "img_path": "images/829fdaaec031b2d195d467511e45447e9aa210777c1caaeec97e5c2cd22df903.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } \\left[ \\exp ( \\langle v , \\nabla \\ell ( \\theta ; Z ) - \\mathbb { E } [ \\nabla \\ell ( \\theta ; Z ) ] \\rangle ) \\right] \\leq \\exp \\bigl ( \\| v \\| _ { 2 } ^ { 2 } \\sigma ^ { 2 } / 2 \\bigr ) . } \\end{array}\n$$",
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+        "text": "In this work, our rates often depend on the sub-Gaussianity and Lipschitz parameters $\\sigma$ and $G$ , and thus we define the shorthands ${ \\widetilde { G } } : = \\sigma { \\sqrt { d } }$ and $\\underline { { G } } : = \\operatorname* { m i n } \\{ G , \\widetilde { G } \\}$ . Intuitively, the $G$ -Lipschitzness assumption bounds the gradient in a ball around 0 (independently of $\\theta$ ), while sub-Gaussianity implies that, for each $\\theta$ , $\\nabla \\ell ( \\theta ; Z )$ likely lies in $\\mathbb { B } _ { 2 } ^ { d } ( \\nabla \\mathcal { L } ( \\theta ; \\bar { P } _ { 0 } ) , \\widetilde { G } )$ . Generically, there is no ordering between $G$ and $\\widetilde { G }$ : for linear loss $\\ell ( \\theta ; z ) = \\langle \\theta , z \\rangle$ , depending on $P _ { 0 }$ , it can hold that $G \\ll \\widetilde G$ (e.g., $P _ { 0 } = \\mathsf { U n i f } \\{ - v , v \\}$ for $v \\in \\mathbb { R } ^ { d }$ ), $\\widetilde G \\ll G$ (e.g., $P _ { 0 }$ is $\\mathsf { N } ( \\mu , \\sigma ^ { 2 } I _ { d } )$ truncated in a ball around $\\mu$ , with $\\| \\mu \\| _ { 2 } \\gg \\sigma { \\sqrt { d } } )$ or $\\boldsymbol { G } \\approx \\boldsymbol { \\widetilde { G } }$ (e.g., $P _ { 0 } = \\mathsf { U n i f } \\{ - 1 , + 1 \\} ^ { d } )$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We introduce the tasks we consider in this work, namely empirical risk minimization (ERM) and stochastic convex optimization (SCO). For a collection of samples from $n$ users $\\boldsymbol { S } = ( S _ { 1 } , \\ldots , S _ { n } )$ , where each $S _ { u } = \\{ z _ { 1 } ^ { ( u ) } , \\dots , z _ { m } ^ { ( u ) } \\} \\in \\mathcal { Z } ^ { m }$ , we define the empirical risk objectives ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/8915926112c209511902ba228dc665b6b6472fcf0e2d70c8ea77afcdf139ed15.jpg",
+        "text": "$$\n\\mathcal { L } ( \\boldsymbol { \\theta } ; S _ { u } ) : = \\frac { 1 } { m } \\sum _ { i = 1 } ^ { m } \\ell ( \\boldsymbol { \\theta } ; z _ { i } ^ { ( u ) } ) \\ \\mathrm { ~ a n d ~ } \\ \\mathcal { L } ( \\boldsymbol { \\theta } ; S ) : = \\frac { 1 } { n } \\sum _ { u = 1 } ^ { n } \\mathcal { L } ( \\boldsymbol { \\theta } ; S _ { u } ) = \\frac { 1 } { m n } \\sum _ { u = 1 } ^ { n } \\sum _ { i = 1 } ^ { m } \\ell ( \\boldsymbol { \\theta } ; z _ { i } ^ { ( u ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the user-level setting we wish to minimize $\\textstyle { \\mathcal { L } } ( \\theta ; S )$ under user-level privacy constraints. Going beyond the empirical risk, we also solve SCO [51], i.e. minimizing a convex population objective ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "when provided with samples from each users’ distributions. In the user-level setting, for a convex loss $\\ell$ and a convex constraint set $\\Theta$ , we observe $\\begin{array} { r } { \\pmb { \\mathcal { S } } = ( S _ { 1 } , \\dots , S _ { n } ) \\sim \\otimes _ { \\pmb { u } \\in [ n ] } ( P _ { u } ) ^ { m } } \\end{array}$ and wish to ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7126de2c481b3b937594c58ecddc1345129316b7c9f1aa8ee01f469f485e04cc.jpg",
+        "text": "$$\n\\operatorname* { m i n i m i z e } _ { \\theta \\in \\Theta } \\frac { 1 } { n } \\sum _ { u \\in [ n ] } \\mathcal { L } ( \\theta ; P _ { u } ) : = \\frac { 1 } { n } \\sum _ { u \\in [ n ] } \\mathbb { E } _ { P _ { u } } [ \\ell ( \\theta ; Z ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the homogeneous case (Assumption A2), this reduces to the classic SCO setting: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f01cb820ca012a2f779f5e1d0e336a5f0e8a32f8b46b909508e45f2e662c9419.jpg",
+        "text": "$$\n\\operatorname* { m i n i m i z e } _ { \\theta \\in \\Theta } \\mathcal { L } ( \\theta ; P _ { 0 } ) : = \\mathbb { E } _ { P _ { 0 } } [ \\ell ( \\theta ; Z ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.2 Uniform concentration of queries ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let $\\phi : \\mathcal { Z } \\to \\mathbb { R } ^ { d }$ be a $d$ -dimensional query function. We define concentration of random variables and uniform concentration of multiple queries as follows. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 2. A (random) sample $X ^ { n }$ supported on $[ - B , B ] ^ { d }$ is $( \\tau , \\gamma )$ -concentrated (and we call $\\tau$ the “concentration radius”) if there exists $x _ { 0 } \\in [ - B , B ] ^ { d }$ such that with probability at least $1 - \\gamma ,$ ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/142f78ac580eb155ce4b9b92be310815fed87b21ff47b2e931077900dff14885.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { i \\in [ n ] } \\lVert X _ { i } - x _ { 0 } \\rVert _ { 2 } \\leq \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 3 (Uniform concentration of vector queries). Let $\\mathcal { Q } _ { B } ^ { d } = \\{ \\phi \\colon \\mathcal { Z }  [ - B , B ] ^ { d } \\}$ be $a$ family of queries with bounded range. For $Z ^ { n } = ( Z _ { 1 } , \\ldots , Z _ { n } ) \\stackrel { \\mathrm { i i d } } { \\sim } P$ , we say that $( Z ^ { n } , \\mathcal { Q } _ { B } ^ { d } )$ is $( \\tau , \\gamma )$ -uniformly-concentrated if with probability at least $1 - \\gamma$ , we have ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/11673f14f1c2e10bf698ada845a587011732c8c9e0007de053757e2c21869761.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { i \\in [ n ] } \\operatorname* { s u p } _ { \\phi \\in \\mathcal { Q } _ { B } ^ { d } } \\Big \\| \\phi ( Z _ { i } ) - \\mathbb { E } _ { Z \\sim P } [ \\phi ( Z ) ] \\Big \\| _ { 2 } \\leq \\tau .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            421,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this work, we will often consider $\\sigma ^ { 2 }$ -sub-Gaussian random variables (or vectors), which are concentrated according to Definition 2. For example, if $X ^ { n }$ is drawn i.i.d. from a $\\sigma ^ { 2 }$ -sub-Gaussian random vector supported on $[ - B , B ] ^ { d }$ , then it is $( \\sigma \\sqrt { d \\log ( 2 n / \\gamma ) } , \\gamma )$ -concentrated around its mean (see, e.g., [56]). Finally, we define a distance between random variables (and estimators). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 4 ( $\\beta$ -close Random Variables). For any two random variables $X _ { 1 } \\sim P _ { 1 }$ and $X _ { 2 } \\sim P _ { 2 }$ , we say $X _ { 1 }$ and $X _ { 2 }$ are $\\beta$ -close, if $\\| P _ { 1 } - P _ { 2 } \\| _ { \\mathsf { T V } } \\leq \\bar { \\beta }$ . We use the notation $X _ { 1 } \\sim _ { \\beta } X _ { 2 }$ if $X _ { 1 }$ and $X _ { 2 }$ are $\\beta$ -close. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "$\\beta$ -closeness is useful as, in many of our results, the private estimator we propose returns a simple unbiased estimate with high probability and is bounded otherwise. Thus, it suffices to do the analysis in the “nice” case and crudely bound the error otherwise. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 High Dimensional Mean Estimation and Uniformly Concentrated Queries ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present a private mean estimator with privacy cost proportional to the concentration radius. Using these techniques, we show that, under uniform concentration, we answer adaptivelychosen queries with privacy cost proportional to the concentration radius instead of the whole range. Our theorems guarantee that the estimator is $\\beta$ -close (with $\\beta$ exponentially small in $n$ ) to a simple unbiased estimator with small noise. We further show how to directly translate these results into bounds on the estimator error, which we demonstrate by providing tight bounds on estimating the mean of $\\ell _ { 2 }$ -bounded random vectors under user-level DP constraints (Corollary 1). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Given i.i.d samples $X ^ { n }$ from a distribution $P$ supported on $\\mathbb { R } ^ { d }$ with mean $\\mu$ , the goal of mean estimation is to design a private estimator that minimizes the $\\mathbb { E } \\left[ \\lVert \\mathsf { A } ( X ^ { n } ) - \\dot { \\mu } \\rVert _ { 2 } ^ { 2 } \\right]$ . We focus on distributions with bounded supprot $[ - B , B ] ^ { d }$ . However, our algorithm also generalize to the case when the mean is guaranteed to be in $[ - B , B ] ^ { d }$ . In the user-level setting (in the homogeneous case), one observes a dataset $s$ sampled as in (2) and wishes to minimize $\\mathbb { E } \\bar { [ \\| \\mathsf { A } ( S ) - \\mathbb { E } P _ { 0 } \\| \\bar { 2 } ] }$ under user-level privacy constraints. We first focus on the scalar case. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Mean estimation in one dimension. The algorithm uses a two-stage procedure, similar in spirit to those of [53], [40], and [39]. In the first stage of this procedure, we use the approximate median estimation in [27], detailed in Algorithm 6 in Appendix D.1, to privately estimate a crude interval ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Require: $X ^ { n } : = ( X _ { 1 } , X _ { 2 } , . . . , X _ { n } ) \\in [ - B , B ] ^ { n }$ , τ : concentration radius, privacy parameter $\\varepsilon > 0$ . 1: $[ a , b ] = \\mathbf { P r i v a t e R a n g e } ( X ^ { n } , \\varepsilon / 2 , \\tau , B )$ with $| b - a | = 4 \\tau$ . {Algorithm 6 in Appendix D.1. $\\}$ 2: Sample $\\textstyle \\xi \\sim \\mathrm { L a p } { \\bigl ( } 0 , { \\frac { 8 \\tau } { \\varepsilon n } } { \\bigr ) }$ and return ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fdc861ce3f9148784eff58878efe514ea6ce441fa25732d71c41fc84ce4dfc9e.jpg",
+        "text": "$$\n\\bar { \\mu } = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } \\Pi _ { [ a , b ] } ( X _ { i } ) + \\xi ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            160,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\Pi _ { [ a , b ] } ( x ) = \\operatorname* { m a x } \\{ a , \\operatorname* { m i n } \\{ x , b \\} \\}$ . ",
+        "bbox": [
+            200,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "in which the means lie, with accuracy $\\Theta ( \\tau )$ . The second stage clips the mean around this interval, reducing the sensitivity from $O ( B )$ to $O ( \\tau )$ , and adds the appropriate Laplace noise. With high probability, we can recover the guarantee of the Laplace mechanism with smaller sensitivity since the samples are concentrated in a radius $\\tau$ . We present the formal guarantees of Algorithm 1 in Theorem 1 and defer its proof to Appendix D.2. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $X ^ { n }$ be a dataset supported on $[ - B , B ]$ . The output of Algorithm 1, denoted by ${ \\mathsf { A } } ( X ^ { n } )$ , is ${ \\varepsilon } { - } D P .$ Furthermore, if $X ^ { n }$ is $( \\tau , \\gamma )$ -concentrated, it holds that ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/185fa986ef5105d249e3ce7070641aeb25285f662d80cfae5b6bf2db89269282.jpg",
+        "text": "$$\n\\mathsf { A } ( X ^ { n } ) \\sim _ { \\beta } \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } X _ { i } + L a p \\bigg ( \\frac { 8 \\tau } { n \\varepsilon } \\bigg ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            362,
+            616,
+            404
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\beta = \\operatorname* { m i n } \\left\\{ 1 , \\gamma + \\frac { B } { \\tau } \\exp \\left( - \\frac { n \\varepsilon } { 8 } \\right) \\right\\} } \\end{array}$ . Moreover, Algorithm $I$ runs in time ${ \\tilde { O } } ( n + \\log ( B / \\tau ) )$ ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Compared to [40, 38, 39], our algorithm runs in time ${ \\tilde { O } } ( n + \\log ( B / \\tau ) )$ instead of ${ \\tilde { O } } ( n + B / \\tau )$ owing to the approximate median estimation algorithm in [27], which is faster when $\\tau \\ll B$ . ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Mean estimation in arbitrary dimension. In the general $d$ -dimensional case, if $X ^ { n }$ is concentrated in $\\ell _ { \\infty }$ -norm, one simply applies Algorithm 1 to each dimension. However, when $X ^ { n }$ is concentrated in $\\ell _ { 2 }$ -norm, naively upper bounding $\\ell _ { \\infty }$ -norm by the $\\ell _ { 2 }$ -norm will incur a superfluous $\\sqrt { d }$ factor: if $\\| v \\| _ { 2 } \\leq \\rho$ , each $| v _ { j } |$ is possibly as large as $\\rho$ . To remedy this issue, we use the random rotation trick in [3, 54]. This guarantees that all coordinates have roughly the same range: for √ $v \\in \\mathbb { R } ^ { d }$ , with high probability, $\\| R \\dot { v } \\| _ { \\infty } \\leq \\tilde { O } ( \\| v \\| _ { 2 } / \\sqrt { d } )$ , where $R$ is the random rotation. We present this procedure in Algorithm 2 and its performance in Theorem 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Require: $X ^ { n } : = ( X _ { 1 } , X _ { 2 } , . . . , X _ { n } ) , X _ { i } \\ \\in \\ [ - B , B ] ^ { d } , \\tau , \\gamma$ : concentration radius and probability, privacy parameter $\\varepsilon , \\delta > 0$ .   \n1: Let $D = { \\mathsf { D i a g } } ( \\omega )$ where $\\omega$ is sampled uniformly from $\\{ \\pm 1 \\} ^ { d }$ .   \n2: Set $U = d ^ { - 1 / 2 } \\mathbf { H } D$ , where $\\mathbf { H }$ is a $d$ -dimensional Hadamard matrix. For all $i \\in [ n ]$ , compute $Y _ { i } = U X _ { i }$ .   \n3: Let $\\begin{array} { r } { \\varepsilon ^ { \\prime } = \\frac { \\varepsilon } { \\sqrt { 8 d \\log ( 1 / \\delta ) } } , \\tau ^ { \\prime } = 1 0 \\tau \\sqrt { \\frac { \\log ( d n / \\gamma ) } { d } } } \\end{array}$ log(dn/γ)d . For j ∈ [d], compute $\\overset { \\sim } { \\underset {  } { \\bar { Y } } } ( j ) = \\mathbf { W i n s o r i z e d M e a n 1 D } \\Big ( \\{ Y _ { i } ( j ) \\} _ { i \\in [ n ] } , \\varepsilon ^ { \\prime } , \\tau ^ { \\prime } , \\sqrt { d } B \\Big ) .$   \n4: return ${ \\bar { X } } = U ^ { - 1 } { \\bar { Y } }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Theorem 2. Let $\\mathsf { A } ( X ^ { n } ) = { \\mathsf { W i n s o r i z e d M e a n H i g h D } } ( X ^ { n } , \\varepsilon , \\delta , \\tau , B , \\gamma )$ be the output of Algorithm 2. $\\mathsf { A } ( X ^ { n } )$ is $( \\varepsilon , \\delta )$ -DP. Furthermore, if $X ^ { n }$ is $( \\tau , \\gamma )$ -concentrated in $\\ell _ { 2 }$ -norm, there exists an estimator $\\mathsf { A } ^ { \\prime } ( X ^ { n } )$ such that ${ \\mathsf { A } } ( X ^ { n } ) \\sim _ { \\beta } { \\mathsf { A } } ^ { \\prime } ( { \\bar { X } } ^ { n } )$ and ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/228209eabd4cce5fc828341c9384e87453c2f4a17ee6f63daa5a251cd16a3d26.jpg",
+        "text": "$$\n\\mathbb { E } [ \\mathsf { A } ^ { \\prime } ( X ^ { n } ) | X ^ { n } ] = \\frac { 1 } { n } \\sum _ { i = 1 } ^ { n } X _ { i } a n d \\mathrm { ~ } \\mathrm { V a r } ( \\mathsf { A } ^ { \\prime } ( X ^ { n } ) | X ^ { n } ) \\leq c _ { 0 } \\frac { d \\tau ^ { 2 } \\log ( d n / \\alpha ) \\log ( 1 / \\delta ) } { n ^ { 2 } \\varepsilon ^ { 2 } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We present the proof of Theorem 2 in Appendix D.3. We are able to transfer both Theorem 1 and Theorem 2 into finite-sample estimation error bounds for various types of concentrated distributions and obtain near optimal guarantees (see Appendix D.5 for an example in mean estimation of subGaussian distributions). The next corollary characterizes the risk of mean estimation for distributions supported on an $\\ell _ { 2 }$ -bounded domain with user-level DP guarantees (see Appendix D.4 for the proof). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Corollary 1. Assume $A 2$ holds with $P _ { 0 }$ supported on $\\mathbb { B } _ { 2 } ^ { d } ( 0 , B )$ with mean $\\mu$ . Given ${ \\boldsymbol { s } } \\ =$ $( S _ { 1 } , S _ { 2 } , . . . , S _ { n } )$ , $| S _ { u } | = m$ , consisting of m i.i.d. samples from $P _ { u }$ . There exists an $( \\varepsilon , \\delta )$ -userlevel $D P$ algorithm $\\mathsf { A } ( \\boldsymbol { S } )$ such that, if $\\dot { n } \\geq ( c _ { 1 } \\sqrt { d \\log ( 1 / \\delta ) } / \\varepsilon ) \\log ( m ( d n + n ^ { 2 } \\varepsilon ^ { 2 } ) )$ for a numerical constant c1, we have5 ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a78bff70a427649200f89eafc00b02a2f1bae3d5e3b556f95040e385616bb3df.jpg",
+        "text": "$$\n\\mathbb { E } \\left[ \\| \\mathsf { A } ( \\pmb { \\mathscr { S } } ) - \\mu \\| _ { 2 } ^ { 2 } \\right] = \\frac { \\mathrm { V a r } ( P _ { 0 } ) } { m n } + \\tilde { O } \\bigg ( \\frac { d B ^ { 2 } } { m n ^ { 2 } \\varepsilon ^ { 2 } } \\bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that $\\mathrm { V a r } ( P _ { 0 } ) \\le B ^ { 2 }$ for any $P _ { 0 }$ supported on $\\mathbb { B } _ { 2 } ^ { d } ( 0 , B )$ . Replacing ${ \\mathrm { V a r } } ( P _ { 0 } )$ by $B ^ { 2 }$ , the bound 2is minimax optimal up to logarithmic factors. When onsame error bounds holds (up to constant) for estimating $A l$ with for a $\\Delta \\leq \\mathsf { p o l y } ( d , \\frac { 1 } { n } , \\frac { 1 } { m } , \\frac { 1 } { \\varepsilon } )$ , the $\\mathbb { E } _ { Z \\sim P _ { u } } [ Z ]$ $u \\in [ n ]$ ",
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+    {
+        "type": "text",
+        "text": "Note that algorithms in [38, 39], which focus on estimating the mean of $d$ -dimensional subGaussian distributions, can also be used to estimate the mean of $\\ell _ { 2 }$ -bounded distributions since bounded random variables are also subGaussian. However, applying these algorithms directly will incur a superfluous $d$ factor in the mean square error. We void this using the random rotation trick in Algorithm 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Answering multiple queries. We end this section by noting that, when a family of queries $\\mathcal { Q }$ is uniformly concentrated (as made precise in Definition 3), we answer sequences of √ $K d$ -dimensional, adaptively chosen queries with error scaling as ${ \\tilde { O } } ( { \\sqrt { d K } } \\tau / ( n \\varepsilon ) )$ by applying Algorithm 2 to $\\{ \\phi _ { k } ( Z _ { i } ) \\} _ { i \\in [ n ] }$ with the right $( \\varepsilon _ { 0 } , \\delta _ { 0 } )$ . We make this formal in Theorem 10 in Appendix D.6. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4 Empirical Risk Minimization with User-Level Differential Privacy ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            419,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this section, we present an algorithm to solve the ERM objective of (3) under user-level DP constraints. We apply the results of Section 3 by noting that the SQ framework encompasses stochastic gradient methods. Informally, one can sequentially choose queries $\\phi _ { k } ( z ) = \\nabla \\ell ( \\theta _ { k } ; z )$ and, for a stepsize $\\eta$ , update $\\theta _ { k + 1 } = \\Pi _ { \\Theta } \\big ( \\dot { \\theta } _ { k } - \\eta v _ { k } \\big )$ , where $v _ { k }$ is the answer to the $k$ -th query. For the results to hold, we require a uniform concentration result over the appropriate class of queries. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Uniform concentration of stochastic gradients The class of queries for stochastic gradient methods is $\\mathcal { Q } _ { \\sf e r m } ~ : = ~ \\{ \\nabla \\ell ( \\theta ; \\cdot ) ~ : ~ \\theta ~ \\in ~ \\bar { \\Theta } \\}$ . We prove that when assumptions A3 and A4 hold, $( \\{ \\nabla \\ell ( \\cdot ; S _ { u } ) \\} _ { u \\in [ n ] } , \\mathcal { Q } _ { \\mathrm { e r m } } )$ is $( \\tilde { O } ( \\sigma \\sqrt { d / m } ) , \\alpha )$ -uniformly concentrated. The next proposition is a simplification of the result of [50] under the (stronger) assumption A3 that $\\ell$ is uniformly $H$ -smooth. The proof, which we defer to Appendix E.1, hinges on a covering number argument. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 (Concentration of random gradients). Let $S _ { u } \\overset { \\mathrm { i i d } } { \\sim } P _ { u }$ $| S _ { u } | = m$ for $u \\in [ n ]$ and $\\alpha \\geq 0$ $1 - \\alpha$ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bedde13ab22a214a320297697f43f3c40635e91e980994f5539a0f904d120837.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { u \\in [ n ] } \\operatorname* { s u p } _ { \\theta \\in \\Theta } \\| \\nabla \\mathcal { L } ( \\theta ; S _ { u } ) - \\nabla \\mathcal { L } ( \\theta ; P _ { u } ) \\| _ { 2 } = O \\left( \\sigma \\sqrt { \\frac { d \\log \\left( \\frac { R H m } { d \\sigma } \\right) } { m } + \\frac { \\log \\left( \\frac { n } { \\alpha } \\right) } { m } } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Stochastic gradient methods We state classical convergence results for stochastic gradient methods for both convex and non-convex losses under smoothness. For a function $F : \\Theta \\to \\mathbb { R }$ , we assume access to a first-order stochastic oracle $\\mathsf { O } _ { F , \\nu ^ { 2 } }$ , i.e., a random mapping such that for all $\\theta \\in \\Theta$ , ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/693d2c93eef7449477fd9d66cddf499edb28faaff6ae1f6fec349122b4619c32.jpg",
+        "text": "$$\n\\mathsf { O } _ { F , \\nu ^ { 2 } } ( \\theta ) = \\nabla \\widehat { F } ( \\theta ) \\mathrm { ~ w i t h ~ } \\mathbb { E } \\Big [ \\nabla \\widehat { F } ( \\theta ) \\Big ] = \\nabla F ( \\theta )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/200d07018719d4cabb5fe72366bf6a8b203e173d16552c0c18524c07a8ad2cfc.jpg",
+        "text": "$$\n\\operatorname { V a r } \\left( \\nabla { \\widehat { F } } ( \\theta ) \\right) \\leq \\nu ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We abstract optimization algorithms in the following way: an algorithm consists of an output set $\\mathcal { O }$ , a sub-routine $\\mathsf { Q u e r y : } \\mathcal { O } \\to \\Theta$ that takes the last output and indicates the next point to query and a sub-routine Update : ${ \\mathcal { O } } \\times \\mathbb { R } ^ { d } \\to { \\mathcal { O } }$ that takes the previous output and a stochastic gradient and returns the next output. After $T$ steps, we call Aggregate : $O ^ { * } \\to \\Theta$ , which takes all the previous outputs and returns the final point. (See Algorithm 7 in Appendix E.2 for how to instantiate generic first-order optimization in this framework.) We detail in Proposition 4 in Appendix E.2 standard convergence results for variations of (projected) stochastic gradient descent (SGD). We introduce this abstraction to forego the details of each specific algorithm and instead focus on the privacy and utility guarantees. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Algorithm We recall the ERM setting with user-level DP. We observe $\\boldsymbol { S } = ( S _ { 1 } , \\ldots , S _ { n } )$ with $S _ { u } \\in \\mathcal { Z } ^ { m }$ for $u \\in [ n ]$ and wish to solve the constrained optimization problem with objective in (3). We present our method in Algorithm 3 and provide utility and privacy guarantees in Theorem 3. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 3 Winsorized First-Order Optimization ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "1: Input: Number of iterations $T$ , optimization algorithm $\\{ \\mathcal { O } , \\mathsf { Q u e r y }$ , Update, Aggregate}, privacy   \nparameters $( \\varepsilon , \\delta )$ , data $\\boldsymbol { S } = ( S _ { 1 } , \\ldots , S _ { n } )$ , initial output $o _ { 0 }$ , parameter set $\\Theta$ , concentration radius   \n2: Set $\\tau$ , probability $\\begin{array} { r } { \\varepsilon ^ { \\prime } = \\frac { \\varepsilon } { 2 \\sqrt { 2 T \\log ( 2 / \\delta ) } } } \\end{array}$ $\\gamma$ . and $\\begin{array} { r } { \\delta ^ { \\prime } = \\frac { \\delta } { 2 T } } \\end{array}$   \n3: for $t = 0 , \\ldots , T - 1$ do   \n4: $\\theta _ { t } \\gets \\mathsf { Q u e r y } ( o _ { t } )$ .   \n5: For each user $u \\in [ n ]$ , compute   \n6: Compute $\\bar { g } _ { t } =$ Winso $\\begin{array} { r l r } & { } & { g _ { t } ^ { ( u ) } = \\nabla \\mathcal { L } ( \\theta _ { t } ; S _ { u } ) = \\displaystyle \\frac { 1 } { m } \\sum _ { j \\in [ m ] } \\nabla \\ell ( \\theta _ { t } ; z _ { j } ^ { ( u ) } ) . } \\\\ & { } & { \\mathrm { ~ } \\mathrm { ~ r i z e d M e a n H i g h } \\mathbf { D } ( \\{ g _ { t } ^ { ( u ) } \\} _ { u \\in [ n ] } , \\varepsilon ^ { \\prime } , \\delta ^ { \\prime } , \\tau , G , \\gamma ) . } \\end{array}$   \n7: $o _ { t + 1 } \\gets \\mathsf { U p d a t e } ( o _ { t } , \\bar { g } _ { t } )$ .   \n8: end for   \n9: return $\\bar { \\theta } \\gets \\mathsf { A g g r e g a t e } ( o _ { 0 } , \\ldots , o _ { T } )$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3 (Privacy and utility guarantees for ERM). Assume $A 2$ holds and recall that ${ \\widetilde { G } } = \\sigma { \\sqrt { d } } ,$ , assume6 $n = \\tilde { \\Omega } ( \\sqrt { d T } / \\varepsilon )$ and let $\\widehat { \\theta }$ be the output of Algorithm 3. There exists variants of projected $S G D$ (e.g. the ones we present in Proposition 4) such that, with probability greater than $1 - \\gamma$ : ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(i) If for all $z \\in \\mathcal { Z } , \\ell ( \\cdot ; z )$ is convex, then ",
+        "bbox": [
+            210,
+            428,
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+            443
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cfec3b2602379c1216e278c2cb51e196dddbccdbf479917f66456d6b8a7e0ec2.jpg",
+        "text": "$$\n\\mathbb { E } \\bigg [ \\mathcal { L } ( \\widehat { \\theta } ; \\mathcal { S } ) - \\operatorname* { i n f } _ { \\theta ^ { \\prime } \\in \\Theta } \\mathcal { L } ( \\theta ^ { \\prime } ; \\mathcal { S } ) \\bigg | \\mathcal { S } \\bigg ] = \\tilde { O } \\Bigg ( \\frac { R ^ { 2 } H } { T } + R \\widetilde { G } \\frac { \\sqrt { d } } { n \\sqrt { m } \\varepsilon } \\Bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            446,
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+            489
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(ii) If for all $z \\in \\mathcal { Z } , \\ell ( \\cdot ; z )$ is $\\mu$ -strongly-convex, then ",
+        "bbox": [
+            204,
+            494,
+            555,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad3b4f8bd5a65cabc470a74d6771f8345a7b7b5105a4a52ccafe8b18dac15d1e.jpg",
+        "text": "$$\n\\mathbb { E } \\bigg [ \\mathcal { L } ( \\widehat { \\theta } ; \\mathcal { S } ) - \\operatorname* { i n f } _ { \\theta ^ { \\prime } \\in \\Theta } \\mathcal { L } ( \\theta ^ { \\prime } ; \\mathcal { S } ) \\bigg | \\mathcal { S } \\bigg ] = \\widetilde { \\mathcal { O } } \\bigg ( G R \\exp \\big ( - \\frac { \\mu } { H } T \\big ) + \\widetilde { G } ^ { 2 } \\frac { d } { \\mu n ^ { 2 } m \\varepsilon ^ { 2 } } \\bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            513,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(iii) Otherwise, defining the gradient mapping7 $\\begin{array} { r } { { \\sf G } _ { F , \\gamma } ( \\theta ) : = \\frac { 1 } { \\gamma } [ \\theta - \\Pi _ { \\Theta } ( \\theta - \\gamma \\nabla F ( \\theta ) ) ] . } \\end{array}$ , we have ",
+        "bbox": [
+            199,
+            553,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d0252ad4a0034f04fb9c9906841d082b72bf22ee01890704c0f3b4a32694f098.jpg",
+        "text": "$$\n\\mathbb { E } \\bigg [ \\| \\mathsf { G } _ { \\mathcal { L } ( \\cdot ; S ) , 1 / H } ( \\widehat { \\theta } ) \\| _ { 2 } ^ { 2 } | S \\bigg ] = \\tilde { O } \\bigg ( \\frac { H ^ { 2 } R } { T } + H R \\widetilde G \\frac { \\sqrt { d } } { n \\sqrt { m } \\varepsilon } \\bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            578,
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+            621
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For $\\varepsilon \\le 1 , \\delta > 0$ , Algorithm $^ 3$ instantiated with any first-order gradient algorithm is $( \\varepsilon , \\delta )$ -user-level $D P .$ In the case that only $A l$ holds, the same guarantees hold whenever $\\Delta \\leq \\mathsf { p o l y } ( d , \\frac { 1 } { n } , \\frac { 1 } { m } , \\frac { 1 } { \\varepsilon } )$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We present the proof in Appendix E.3. For the utility guarantees, the crux of the proof resides in Theorem 10: as well as ensuring small excess loss in expectation, the SQ algorithm produces with high probability a sample from the stochastic gradient oracle $\\operatorname { O } _ { \\mathcal { L } ( \\cdot ; S ) , \\nu ^ { 2 } }$ where $\\begin{array} { r } { \\nu ^ { 2 } = { \\tilde { O } } ( T { \\widetilde G } ^ { 2 } \\frac { d } { n ^ { 2 } m \\varepsilon ^ { 2 } } ) } \\end{array}$ When this happens for all $T$ steps, the analysis of stochastic gradient methods provide the desired regret. The privacy guarantees follow from the strong composition theorem of [23]. ",
+        "bbox": [
+            173,
+            662,
+            826,
+            737
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Importantly, when the function exhibits (some) strong-convexity (which will be the case for any regularized objective), we are able to localize the optimal parameter—up to the privacy cost—in ${ \\cal \\tilde { O } } ( H / \\mu )$ steps. This will be particularly important in Section 5. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Corollary 2 (Localization). Let $\\widehat { \\theta }$ be the output of Algorithm $^ 3$ on the ERM problem of (3). Assume that $\\ell ( \\cdot ; z )$ is $\\mu$ -strongly-convex for all $z \\in { \\mathcal { Z } }$ , that $n = \\tilde { \\Omega } ( \\sqrt { d H / \\mu } )$ and set $T =$ $\\begin{array} { r } { \\frac { H } { \\mu } \\log \\left( n ^ { 2 } m ( \\underline { { G } } / \\widetilde G ^ { 2 } ) \\frac { \\mu R \\varepsilon ^ { 2 } } { d } \\right) } \\end{array}$ and $\\begin{array} { r } { \\gamma = { \\frac { \\sigma ^ { 2 } d ^ { 2 } } { \\mu ^ { 2 } n ^ { 2 } m \\varepsilon ^ { 2 } R ^ { 2 } } } } \\end{array}$ . For $\\theta _ { S } ^ { * } \\in \\mathrm { a r g m i n } _ { \\theta ^ { \\prime } \\in \\Theta } \\mathcal { L } ( \\theta ^ { \\prime } ; S )$ , it holds8 ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8597fa46b29a44cfe93b88214a012fd93ff233a17dad664b911458024e507c80.jpg",
+        "text": "$$\n\\mathbb { E } [ \\| \\widehat { \\theta } - \\theta _ { S } ^ { * } \\| _ { 2 } ^ { 2 } ] = \\tilde { O } \\bigg ( \\frac { \\sigma ^ { 2 } d ^ { 2 } } { \\mu ^ { 2 } n ^ { 2 } m \\varepsilon ^ { 2 } } \\bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            136
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5 Stochastic Convex Optimization with User-level Privacy ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            148,
+            669,
+            167
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section we address the SCO task of (5) under user-level DP constraints. Our approach (which we show in Algorithm 4) solves a sequence of carefully regularized ERM problems, drawing on√ the guarantees of the previous section. Recall that $\\widetilde { G } = \\sigma \\sqrt { d }$ and $\\underline { { G } } = \\operatorname* { m i n } \\{ G , \\widetilde { G } \\}$ , and that $\\ell$ is $H$ -smooth under assumption A3. In this section, we assume that $\\ell$ is convex. We first present our results and state an upper and lower bound for SCO with user-level privacy constraints. ",
+        "bbox": [
+            173,
+            176,
+            826,
+            250
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4 (Phased ERM for SCO). Algorithm $^ { 4 }$ is user-level $( \\varepsilon , \\delta )$ -DP. When $A 2$ holds and $n =$ $\\tilde { \\Omega } ( \\operatorname* { m i n } \\{ \\sqrt [ 3 ] { d ^ { 2 } m H ^ { 2 } R ^ { 2 } / ( G { \\underline { { G } } } \\varepsilon ^ { 4 } ) } , H R \\sqrt { m } / ( \\sigma \\varepsilon ) \\} )$ , or, equivalently, $\\begin{array} { r } { H = \\tilde { O } ( \\sqrt { \\frac { n ^ { 2 } \\varepsilon ^ { 2 } \\sigma ^ { 2 } } { R ^ { 2 } m } + \\frac { G \\bar { a } ^ { 3 } \\varepsilon ^ { 4 } } { d ^ { 2 } R ^ { 2 } m } } ) f o r } \\end{array}$ r all $P$ and $\\ell$ satisfying Assumptions $A 3$ and A4, we have ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/859e5cdaac639a2a3be5e9536cfee245a369520343e8708311efd1ecb0a32c7c.jpg",
+        "text": "$$\n\\mathbb { E } \\left[ \\mathcal { L } \\big ( \\mathsf { A } _ { \\mathsf { P h a s e d E R M } } ( S ) ; P _ { 0 } \\big ) \\right] - \\operatorname* { m i n } _ { \\theta ^ { \\prime } \\in \\Theta } \\mathcal { L } ( \\theta ^ { \\prime } ; P _ { 0 } ) = \\tilde { O } \\left( \\frac { R \\sqrt { G G } } { \\sqrt { m n } } + R \\widetilde G \\frac { \\sqrt { d } } { n \\sqrt { m } \\varepsilon } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            246,
+            305,
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+            347
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, our results still hold in the heterogeneous setting (Assumption $A I$ ) whenever $\\Delta \\le$ $\\mathtt { p o l y } ( d , \\frac { 1 } { n } , \\frac { 1 } { m } , \\frac { 1 } { \\varepsilon } )$ ; the risk guarantee being with respect to any user distribution $P _ { u }$ . ",
+        "bbox": [
+            169,
+            353,
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+            382
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Theorem 5 (Lower bound for SCO). There exists a distribution $P$ and a loss $\\ell$ satisfying Assumptions $A 3$ and A4 such that for any algorithm A satisfying $( \\varepsilon , \\delta )$ -DP at user-level, we have ",
+        "bbox": [
+            171,
+            390,
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+            420
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7e2c58edf7fcc1e842b57082b13acb7382469d8b7f8d0f39a8298654ec62e660.jpg",
+        "text": "$$\n\\mathbb { E } \\left[ \\mathcal { L } ( \\mathsf { A } ( \\mathcal { S } ) ; P ) \\right] - \\operatorname* { m i n } _ { \\theta ^ { \\prime } \\in \\Theta } \\mathcal { L } ( \\theta ^ { \\prime } ; P ) = \\Omega \\Bigg ( \\frac { R G } { \\sqrt { m n } } + R \\underline { { G } } \\frac { \\sqrt { d } } { n \\sqrt { m } \\varepsilon } \\Bigg ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            290,
+            426,
+            705,
+            468
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "When $G = \\Theta ( \\sigma { \\sqrt { d } } )$ , the upper bound matches the lower bound up to logarithmic factors. We present the algorithm and proof for Theorem 4 in Section 5.1. Theorem 5 is proved in Section 5.2. ",
+        "bbox": [
+            173,
+            483,
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+            513
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.1 Upper bound: minimizing a sequence of regularized ERM problems ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            521,
+            689,
+            537
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We now present Algorithm 4, which achieves the upper bound of Theorem 4. It is similar in spirit to Phased ERM [29] and EpochGD [34], in that at each round we minimize a regularized ERM problem with fresh samples and increased regularization, initializing each round from the final iterate of the previous round. This allows us to localize the optimum with exponentially increasing accuracy without blowing up our privacy budget. We solve each round using Algorithm 3 to guarantee privacy and obtain an approximate minimizer. We show the guarantee in Corollary 2 is enough to achieve optimal rates. We provide the proof of Theorem 4 in Appendix $\\mathrm { F }$ and present a sketch here. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/dcb435fff179ac940622bfcce4a73a45db67bebc8692c04b18d082f14c6a9bcc.jpg",
+        "table_caption": [
+            "Algorithm 4 APhasedERM: Phased ERM "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>parameterσ.</td><td>Require: Private dataset: S = (S1,...,Sn) ∈ (Zm)n : n × m i.i.d samples from P,H-smooth, convex loss function l,convex set Θ C Rd, privacy parameters ε ≤ 1,δ ≤1/n²,sub-Gaussian</td><td></td><td></td><td></td></tr><tr><td>1: SetT=[log2(</td><td>(Gn√mε)1,入=√ GG gd nm</td><td>/R</td><td></td><td></td></tr><tr><td>2:</td><td>fort=1toTdo</td><td>n²mε²</td><td></td><td></td></tr><tr><td>3: 4:</td><td>Setnt =，,xt =4t入</td><td></td><td></td><td></td></tr><tr><td></td><td> Sample St, nt users that have not participated in previous rounds. Using Algorithm 3,compute</td><td></td><td></td><td></td></tr><tr><td></td><td>an approximate minimizer 0t, to the accuracy of Corollary 2, for the objective</td><td>m</td><td></td><td></td></tr><tr><td></td><td>Lλt，t_(0;St)=</td><td>1 MM e（0,z</td><td>t 11-0t-12.</td><td></td></tr><tr><td></td><td></td><td>mnt</td><td>(u) 十 2</td><td></td></tr><tr><td>5: end for</td><td></td><td>uESt j=1</td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>6:return</td><td>T</td><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Proof sketch of Theorem 4. The privacy guarantee comes directly from the privacy guarantee of Algorithm 3 and the fact that $S _ { t }$ are non-overlapping. The proof for utility is similar to the proof of Theorem 4.8 in [29]. In round $t$ of Algorithm 4, we consider the true minimizer $\\theta _ { t } ^ { * }$ and the approximate minimizer $\\widehat { \\theta } _ { t }$ . By stability [14], we can bound the generalization error of $\\theta _ { t } ^ { * }$ (see Proposition 5 in Appendix F) and, by Corollary 2, we can bound $\\widehat { \\mathbb { E } } \\| \\widehat { \\theta } _ { t } - \\theta _ { t } ^ { * } \\| _ { 2 } ^ { 2 }$ . We finally choose $\\{ ( \\lambda _ { t } , n _ { t } ) \\} _ { t \\le T }$ such that the assumptions of Corollary 2 hold and to minimize the final error. □ ",
+        "bbox": [
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "5.2 Lower bound: SCO is harder than Gaussian mean estimation ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First of all, note that it suffices to prove the lower bounds in the homogeneous setting as any level of heterogeneity only makes the problem harder. Theorem 5 holds for $( \\varepsilon , \\delta )$ -user-level DP—importantly, this is a setting for which lower bounds are generally more challenging (we provide a related lower bound for $\\varepsilon$ -user-level DP in Appendix A.2). We present the proof in Appendix F.2 and a sketch here. ",
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+    {
+        "type": "text",
+        "text": "Proof sketch of Theorem 5. The (constrained) minimax lower bound decomposes into a statistical rate and a privacy rate. The statistical rate is optimal (see, e.g., [44, 2]), thus we focus on the privacy rate. We consider linear losses of the form $\\ell ( \\theta ; z ) = - \\langle \\theta , z \\rangle$ . We show that optimizing $\\mathsf { \\bar { L } } ( \\theta ; \\dot { P } ) = \\mathbb { E } _ { P } [ \\ell ( \\theta ; Z ) ]$ over $\\theta \\in \\Theta$ is harder than the mean estimation task for $P$ . Intuitively, $C ( \\theta ; P ) = - \\langle \\theta , \\mathbb { E } Z \\rangle$ attains its minimum at $\\theta ^ { * } = R \\mathbb { E } [ Z ] / \\| \\mathbb { E } [ Z ] \\| _ { 2 }$ and finding $\\theta ^ { * }$ provides a good estimate of (the direction of) $\\mathbb { E } [ Z ]$ . We make this formal in Proposition 6. Next, for Gaussian mean estimation, we reduce, in Proposition 3, user-level DP to item-level DP with lower variance by having each user contribute their sample average (which is a sufficient statistic). We conclude with the results of [38] (see Proposition 7) by proving in Corollary 6 that estimating the direction of the mean with item-level privacy is hard. □ ",
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Hilal Asi and Karan Chadha for comments on an earlier draft as well as Yair Carmon, Peter Kairouz, Gautam Kamath, Sai Praneeth Karimireddy, Thomas Steinke and Sebastian Stich, for useful discussions and pointers to very relevant references. ",
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            484,
+            266,
+            501
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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parse/train/Hye9lnCct7/Hye9lnCct7.md

@@ -0,0 +1,389 @@
+# LEARNING ACTIONABLE REPRESENTATIONS WITH GOAL-CONDITIONED POLICIES
+
+Dibya Ghosh, Abhishek Gupta, & Sergey Levine ∗   
+Department of Electrical Engineering and Computer Science   
+University of California, Berkeley   
+Berkeley, CA 94703, USA
+
+# ABSTRACT
+
+Representation learning is a central challenge across a range of machine learning areas. In reinforcement learning, effective and functional representations have the potential to tremendously accelerate learning progress and solve more challenging problems. Most prior work on representation learning has focused on generative approaches, learning representations that capture all the underlying factors of variation in the observation space in a more disentangled or well-ordered manner. In this paper, we instead aim to learn functionally salient representations: representations that are not necessarily complete in terms of capturing all factors of variation in the observation space, but rather aim to capture those factors of variation that are important for decision making – that are “actionable.” These representations are aware of the dynamics of the environment, and capture only the elements of the observation that are necessary for decision making rather than all factors of variation, eliminating the need for explicit reconstruction. We show how these learned representations can be useful to improve exploration for sparse reward problems, to enable long horizon hierarchical reinforcement learning, and as a state representation for learning policies for downstream tasks. We evaluate our method on a number of simulated environments, and compare it to prior methods for representation learning, exploration, and hierarchical reinforcement learning.
+
+# 1 INTRODUCTION
+
+Representation learning refers to a transformation of an observation, such as a camera image or state observation, into a form that is easier to manipulate to deduce a desired output or perform a downstream task, such as prediction or control. In reinforcement learning (RL) in particular, effective representations are ones that enable generalizable controllers to be learned quickly for challenging and temporally extended tasks. While end-to-end representation learning with full supervision has proven effective in many scenarios, from supervised image recognition (Krizhevsky et al., 2012) to vision-based robotic control (Levine et al., 2015), devising representation learning methods that can use unlabeled data or experience effectively remains an open problem.
+
+Much of the prior work on representation learning in RL has focused on generative approaches. Learning these models is often challenging because of the need to model the interactions of all elements of the state. We instead aim to learn functionally salient representations: representations that are not necessarily complete in capturing all factors of variation in the observation space, but rather aim to capture factors of variation that are relevant for decision making – that are actionable.
+
+How can we learn a representation that is aware of the dynamical structure of the environment? We propose that a basic understanding of the world can be obtained from a goal-conditioned policy, a policy that can knows how to reach arbitrary goal states from a given state. Learning how to execute shortest paths between all pairs of states suggests a deep understanding of the environment dynamics, and we hypothesize that a representation incorporating the knowledge of a goal-conditioned policy can be readily used to accomplish more complex tasks. However, such a policy does not provide a readily usable state representation, and it remains to choose how an effective state representation should be extracted. We want to extract those factors of the state observation that are critical for deciding which action to take. We can do this by comparing which actions a goal-conditioned policy takes for two different goal states. Intuitively, if two goal states require different actions, then they are functionally different and vice-versa. This principle is illustrated in the diagram in Figure 1. Based on this principle, we propose actionable representations for control (ARC), representations in which Euclidean distances between states correspond to expected differences between actions taken to reach them. Such representations emphasize factors in the state that induce significant differences in the corresponding actions, and de-emphasize those features that are irrelevant for control.
+
+While learning a goal-conditioned policy to extract such a representation might itself represent a daunting task, it is worth noting that such a policy can be learned without any knowledge of downstream tasks, simply through unsupervised exploration of the environment. It is reasonable to postulate that, without active exploration, no representation learning method can possibly acquire a dynamics-aware representation, since understanding the dynamics requires experiencing transitions and interactions, rather than just observations of valid states. As we demonstrate in our experiments, representations extracted from goal-conditioned policies can be used to better learn more challenging tasks than simple goal reaching, which cannot be easily contextualized by goal states. The process of learning goal-conditioned policies can also be made recursive, so that the actionable representations learned from one goalconditioned policy can be used to quickly learn a better one.
+
+Actionable representations for control are useful for a number of downstream tasks: as representations for task-specific policies, as representations for hierarchical RL, and to construct well-shaped reward functions. We show that ARCs enable these applications better than representations that are learned using unsupervised generative models, predictive models, and other prior representation learning methods. We analyze structure of the learned representation, and compare the performance of ARC with a number of prior methods on downstream tasks in simulated robotic domains such as wheeled locomotion, legged locomotion, and robotic manipulation.
+
+![](images/326d12a64b3d1f209ffc25b6e8780386ab792473772e6a06b3e616bfe8b69c51.jpg)  
+Figure 1: Actionable representations: 3 houses A, B, C can only be reached by indicated roads. The actions taken to reach A, B, C are shown by arrows. Although A, B are very close in space, they are functionally different. The car has to take a completely different road to reach A, compared to $\mathbf { B }$ and C. Representations $z _ { A }$ , $z _ { B }$ , $z _ { C }$ learn these functional differences to differentiate A from B and C, while keeping B and C close.
+
+# 2 PRELIMINARIES
+
+Goal-conditioned reinforcement learning. In RL, the goal is to learn a policy $\pi _ { \boldsymbol { \theta } } \big ( a _ { t } | \boldsymbol { s } _ { t } \big )$ that maximizes the expected return $R _ { t } = \mathbb { E } _ { \pi _ { \theta } } [ \bar { \sum _ { t } r _ { t } } ]$ . Typically, RL learns a single task that optimizes for a particular reward function. If we instead would like to train a policy that can accomplish a variety of tasks, we might instead train a policy that is conditioned on another input – a goal. When the different tasks directly correspond to different states, this amounts to conditioning the policy $\pi$ on both the current and goal state. The policy $\pi _ { \boldsymbol { \theta } } ( a _ { t } | \boldsymbol { s } _ { t } , \boldsymbol { g } )$ is trained to reach goals from the state space $g \sim { \mathcal { S } }$ , by optimizing $\mathbb { E } _ { g \sim S } [ \mathbb { E } _ { \pi _ { \theta } ( a | s , g ) } ( R _ { g } ) ) ]$ , where $R _ { g }$ is a reward for reaching the goal $g$ .
+
+Maximum entropy RL. Maximum entropy RL algorithms modify the RL objective, and instead learns a policy to maximize the reward as well as the entropy of the policy (Haarnoja et al., 2017; Todorov, 2006), according to $\begin{array} { r } { \pi ^ { \star } = \arg \operatorname* { m a x } _ { \pi } E _ { \pi } [ r ( s , a ) ] \stackrel { } { + } \mathcal { H } ( \pi ) } \end{array}$ . In contrast to standard RL, where optimal policies in fully observed environments are deterministic, the solution in maximum entropy RL is a stochastic policy, where the entropy reflects the sensitivity of the rewards to the action: when the choice of action has minimal effect on future rewards, actions are more random, and when the choice of action is critical, the actions are more deterministic. In this way, the action distributions for a maximum entropy policy carry more information about the dynamics of the task.
+
+![](images/b3508108c7eb40b8fd735bc580ff522c024d2ea2813efb96c22dab519b1cf675.jpg)  
+Figure 2: An illustration of actionable representations. For a pair of states $s _ { 1 } , s _ { 2 }$ , the divergence between the goal-conditioned action distributions they induce defines the actionable distance $D _ { \mathrm { { A c t } } }$ which in turn is used to learn representation $\phi$ .
+
+# 3 LEARNING ACTIONABLE REPRESENTATIONS
+
+In this work, we extract a representation that can distinguish states based on actions required to reach them, which we term an actionable representation for control (ARC). In order to learn state representations $\phi$ that can capture the elements of the state which are important for decision making, we first consider defining actionable distances $D _ { \mathrm { A c t } } ( s _ { 1 } , s _ { 2 } )$ between states. Actionable distances are distances between states that capture the differences between the actions required to reach the different states, thereby implicitly capturing dynamics. If actions required for reaching state $s _ { 1 }$ are very different from the actions needed for reaching state $s _ { 2 }$ , then these states are functionally different, and should have large actionable distances. This subsequently allows us to extract a feature representation $( \phi ( s ) )$ of state, which captures elements that are important for decision making.
+
+To formally define actionable distances, we build on the framework of goal-conditioned RL. We assume that we have already trained a maximum entropy goal-conditioned policy $\pi _ { \boldsymbol { \theta } } ( a | \boldsymbol { s } , \boldsymbol { g } )$ that can start at an arbitrary state $s _ { 0 } \in S$ in the environment, and reach a goal state $s _ { g } \in \mathcal S$ . Although this is a significant assumption, we will discuss later how this is in fact reasonable in many settings. We can extract actionable distances by examining how varying the goal state affects action distributions for goal-conditioned policies. Formally, consider two different goal states $s _ { 1 }$ and $s _ { 2 }$ . At an intermediate state $s$ , the goal-conditioned policy induces different action distributions $\pi _ { \boldsymbol { \theta } } ( a | \boldsymbol { s } , \boldsymbol { s } _ { 1 } )$ and $\pi _ { \boldsymbol { \theta } } ( a | \boldsymbol { s } , \boldsymbol { s } _ { 2 } )$ to reach $s _ { 1 }$ and $s _ { 2 }$ respectively. If these distributions are similar over many intermediate states $s$ , this suggests that these states are functionally similar, while if these distributions are different, then the states must be functionally different. This motivates a definition for actionable distances $D _ { \mathrm { { A c t } } }$ as
+
+$$
+D _ { \mathrm { A c t } } ( s _ { 1 } , s _ { 2 } ) = \mathbb { E } _ { s } \left[ D _ { K L } ( \pi ( a | s , s _ { 1 } ) | | \pi ( a | s , s _ { 2 } ) ) + D _ { K L } ( \pi ( a | s , s _ { 2 } ) | | \pi ( a | s , s _ { 1 } ) ) \right] .
+$$
+
+The distance consists of the expected divergence over all initial states $s$ (refer to Section $\mathbf { B }$ for how we do this practically). If we focus on a subset of states, the distance may not capture action differences induced elsewhere, and can miss functional differences between states. Since maximum entropy policies learn unique optimal stochastic policies, the actionable distance is well-defined and unambiguous. Furthermore, because max-ent policies capture sensitivity of the value function, goals are similar under ARC if they require the same action and they are equally “easy” to reach.
+
+We can use $D _ { \mathrm { { A c t } } }$ to extract an actionable representation of state. To learn this representation $\phi ( s )$ , we optimize $\phi$ such that Euclidean distance between states in representation space corresponds to actionable distances $D _ { \mathrm { { A c t } } }$ between them. This optimization yields good representations of state because it emphasizes the functionally relevant elements of state, which significantly affect the actionable distance, while suppressing less functionally relevant elements of state. The problem is:
+
+$$
+\displaystyle \operatorname* { m i n } _ { \phi } \mathbb { E } _ { s _ { 1 } , s _ { 2 } } \bigg [ \| \phi ( s _ { 1 } ) - \phi ( s _ { 2 } ) \| _ { 2 } - D _ { \mathrm { A c t } } ( s _ { 1 } , s _ { 2 } ) \bigg ] ^ { 2 }
+$$
+
+This objective yields representations where Euclidean distances are meaningful. This is not necessarily true in the state space or in generative representations (Section 6.4). These representations are meaningful for several reasons. First, since we are leveraging a goal-conditioned policy, they are aware of dynamics and are able to capture local connectivity of the environment. Secondly, the representation is optimized so that it captures only the functionally relevant elements of state.
+
+Requirement for Goal Conditioned Policy: A natural question to ask is whether needing a goalconditioned policy is too strong of a prerequisite. However, it is worth noting that the GCP can be trained with existing RL methods (TRPO) using a sparse task-agnostic reward (Section 6.2, Appendix A.1) – obtaining such a policy is not especially difficult, and existing methods are quite capable of doing so ( Nair et al. (2018)). Furthermore, it is likely not possible to acquire a functionalityaware state representation without some sort of active environment interaction, since dynamics can only be understood by observing outcomes of actions, rather than individual states. Importantly, we discuss in the following section how ARCs help us solve tasks beyond what a simple goalconditioned policy can achieve.
+
+# 4 USING ACTIONABLE REPRESENTATIONS FOR DOWNSTREAM TASKS
+
+A natural question that emerges when learning representations from a goal-conditioned policy pertains to what such a representation enables over the goal-conditioned policy itself. Although goalconditioned policies enable reaching between arbitrary states, they suffer from fundamental limitations: they do not generalize very well to new states, and they are limited to solving only goalreaching tasks. We show in our empirical evaluation that the ARC representation expands meaningfully over these limitations of a goal-conditioned policy - to new tasks and to new regions of the environment. In this section, we detail how ARCs can be used to generalize beyond a goalconditioned policy to help solve tasks that cannot be expressed as goal reaching (Section 4.1), tasks involving larger regions of state space (Section 4.2), and temporally extended tasks which involve sequences of goals (Section 4.3).
+
+# 4.1 FEATURES FOR LEARNING POLICIES
+
+Goal-conditioned policies are trained with only a goal-reaching reward, and so are unaware of reward structures used for other tasks in the environment which do not involve simple goal-reaching. Tasks which cannot be expressed as simply reaching a goal are abundant in real life scenarios such as navigation under non-uniform preferences or manipulation with costs on quality of motion, and for such tasks, using the ARC representation as input for a policy or value function can make the learning problem easier. We can learn a policy for a downstream task, of the form $\pi _ { \theta } ( a | \phi ( s ) )$ , using the representation $\phi ( s )$ instead of state $s$ . The implicit understanding of the environment dynamics in the learned representation prioritizes the parts of the state that are most important for learning, and enables quicker learning for these tasks as we see in Section 6.6.
+
+# 4.2 REWARD SHAPING
+
+We can use ARC to construct better-shaped reward functions. It is common in continuous control to define rewards in terms of some distance to a desired state, oftentimes using Euclidean distance (Schulman et al., 2015; Lillicrap et al., 2015). However, Euclidean distance in state space is not necessarily a meaningful metric of functional proximity. ARC provides a better metric, since it directly accounts for reachability. We can use the actionable representation to define better-shaped reward functions for downstream tasks. We define a shaping of this form to be the negative Euclidean distance between two states in ARC space: $- | | \phi ( s _ { 1 } ) - \bar { \phi ( s _ { 2 } ) } | | _ { 2 }$ : for example, on a goal-reaching task $r ( s ) = r _ { \mathrm { s p a r s e } } ( s , s _ { g } ) - | | \phi ( s ) - \phi ( s _ { g } ) | | _ { 2 }$ . This allows us to explore and learn policies even in the presence of sparse reward functions.
+
+One may wonder whether, instead of using ARCs for reward shaping, we might directly use the goalconditioned policy to reach a particular goal. As we will illustrate in Section 6.5, the representation typically generalizes better than the goal-conditioned policy. Goal-conditioned policies typically can be trained on small regions of the state space, but don’t extrapolate well to new parts of the state space. We observe that ARC exhibits better generalization, and can provide effective reward shaping for goals that are very difficult to reach with the goal-conditioned policy.
+
+# 4.3 HIERARCHICAL REINFORCEMENT LEARNING
+
+Goal-conditioned policies can serve as low-level controllers for hierarchical tasks which require synthesizing a particular sequence of behaviours, and thus not expressible as a single goal-reaching objective. One approach to solving such tasks learns a high-level controller $\pi _ { \mathrm { m e t a } } ( g | s )$ via RL that produces desired goal states for a goal-conditioned policy to reach sequentially (Nachum et al., 2018). The high-level controller suggests a goal, which the goal conditioned policy attempts to reach for several time-steps, following which the high-level controller picks a new goal. For many tasks, naively training such a high-level controller which outputs goals directly in state space is unlikely to perform well, since such a controller must disentangle the relevant attributes in the goal for long horizon reasoning. We consider two schemes to use ARCs for hierarchical RL - learning a high level policy which commands directly in ARC space or commands in a clustered latent space.
+
+HRL directly in ARC space: ARC representations provide a better goal space for high-level controllers, since they de-emphasize components of the goal space irrelevant for determining the optimal action. In this scheme, the high-level controller $\pi _ { \mathrm { m e t a } } ( z | s )$ observes the current state and generates a distribution over points in the latent space. At every meta-step, a sample $z _ { h }$ is taken from $\pi _ { \mathrm { m e t a } } ( z | s )$ which represents the high level command. $z _ { h }$ is then translated into a goal $g _ { h }$ via a decoder which is trained to reconstruct states from their corresponding goals. This goal $g _ { h }$ can then be used to command the goal conditioned policy for several time steps, before resampling again from $\pi _ { \mathrm { m e t a } }$ . A high-level controller producing outputs in ARC space does not need to rediscover saliency in the goal space, which makes the search problem less noisy and more accurate. We show in Section 4.3 that using ARC as a hierarchical goal space enables significant improvement for waypoint navigation tasks.
+
+![](images/31fd69e95cab3603688c39d33ca06e8174f9d93d6e41e08862778e7d6356730b.jpg)  
+Figure 3: Hierarchical RL with ARC. Left: Directly commanding in ARC space Right: Commanding a cluster in ARC space
+
+Clustering in ARC space: Since ARC captures the topology of the environment, clusters in ARC space often correspond to semantically meaningful state abstractions. We utilize these clusters, with the intuition that a meta-controller searching in “cluster space” should learn faster than directly outputting states. In this scheme, we first build a discrete number of clusters by clustering the points that the goal conditioned policy is trained on using the $\mathbf { k }$ -means algorithm within the ARC representation space. We then train a high-level controller $\pi _ { \mathrm { m e t a } } ( c | s )$ which observes a state $s$ and generates a distribution over discrete clusters $c$ . At every meta-step, a cluster sample $c _ { h }$ is taken from $\pi _ { \mathrm { m e t a } } ( c | s )$ . A goal in state space $g _ { h }$ is then chosen uniformly at random from points within the cluster $c _ { h }$ and used to command the GCP for several time steps before the next cluster is sampled from $\pi _ { \mathrm { m e t a } }$ . We train a meta-policy to output clusters, instead of states: $\pi _ { m e t a } ( { \mathrm { c l u s t e r } } | s )$ . We see that for hierarchical tasks with less granular reward functions such as room navigation, performing RL in “cluster space” induced by ARC outperform cluster spaces induced by other representations, since the distance metric is much more meaningful in ARC space.
+
+# 5 RELATED WORK
+
+The capability to learn effective representations is a major advantage of deep neural network models. These representations can be acquired implicitly, through end-to-end training (Goodfellow et al., 2016), or explicitly, by formulating and optimizing a representation learning objective. A classic approach to representation learning is generative modeling, where a latent variable model is trained to model the data distribution, and the latent variables are then used as a representation (Rasmus et al., 2015; Dumoulin et al., 2016; Kingma & Welling, 2013; Finn et al., 2015; Ghadirzadeh et al., 2017; Curran et al., 2015; Goroshin et al., 2015; Higgins et al., 2017). In the context of control and sequence models, generative models have also been proposed to model transitions (Watter et al., 2015; Assael et al., 2015; Zhang et al., 2018b; Kurutach et al., 2018). While generative models are general and principled, they must not only explain the entirety of the input observation, but must also generate it. Several methods perform representation learning without generation, often based on contrastive losses (Sermanet et al., 2018; van den Oord et al., 2018; Belghazi et al., 2018; Chopra et al., 2005; Weinberger & Saul, 2009). While these methods avoid generation, they either still require modeling of the entire input, or utilize heuristics that encode user-defined information.
+
+In contrast, ARCs are directly trained to focus on decision-relevant features of input, providing a broadly applicable objective that is still selective about which aspects of input to represent.
+
+In the context of RL and control, representation learning methods have been used for many downstream applications (Lesort et al., 2018), including representing value functions (Barreto et al., 2016) and building models (Watter et al., 2015; Assael et al., 2015; Zhang et al., 2018b). Our approach is complementary: it can also be applied to these applications. Several works have sought to learn representations that are specifically suited for physical dynamical systems (Jonschkowski & Brock, 2015) and that use interaction to build up dynamics-aware features (Bengio et al., 2017; LaversanneFinot et al., 2018). In contrast to Jonschkowski & Brock (2015), our method does not attempt to encode all physically-relevant features of state, only those relevant for choosing actions. In contrast to Bengio et al. (2017); Laversanne-Finot et al. (2018), our approach does not try to determine which features of the state can be independently controlled, but rather which features are relevant for choosing controls. Srinivas et al. (2018) also consider learning representations through goal-directed behaviour, but receives supervision through demonstrations instead of active observation. Related methods learn features that are predictive of actions based on pairs of sequential states (so-called inverse models) (Agrawal et al., 2016; Pathak et al., 2017; Zhang et al., 2018a). More recent work such as (Burda et al., 2018b) perform a large scale study of these types of methods in the context of exploration. Unlike ARC, which is learned from a policy performing long-horizon control, inverse models are not obliged to represent all relevant features for multi-step control, and suffer from greedy reasoning.
+
+# 6 EXPERIMENTS
+
+The aim of our experimental evaluation is to study the following research questions:
+
+1. Can we learn ARCs for multiple continuous control environments? What are the properties   
+of these learned representations?   
+2. Can ARCs be used as feature representations for learning policies quickly on new tasks?   
+3. Can reward shaping with ARCs enable faster learning?   
+4. Do ARCs provide an effective mechanism for hierarchical RL?
+
+Full experimental and hyperparemeter tuning details are presented in the appendix.
+
+# 6.1 DOMAINS
+
+We study six simulated environments as illustrated in Figure 4: 2D navigation tasks in two settings, wheeled locomotion tasks in two settings, legged locomotion, and object pushing with a robotic gripper. The 2D navigation domains consist of either a room with a central divider wall or four rooms. Wheeled locomotion involves a two-wheeled differential drive robot, either in free space or with four rooms. For legged locomotion, we use a quadrupedal ant robot, where the state space consists of all joint angles, along with the Cartesian position of the center of mass (CoM). The manipulation task uses a simulated Sawyer arm to push an object, where the state consists of endeffector and object positions. Further details are presented in Appendix C.
+
+These environments present interesting representation learning challenges. In 2D navigation, the walls impose structure similar to those in Figure 1: geometrically proximate locations on either side of a wall are far apart in terms of reachability. The locomotion environments present an additional challenge: an effective representation must account for the fact that the internal joints of each robot (legs or wheel orientation) are less salient for long-horizon tasks than CoM. The original state representation does not reflect this structure: joint angles expressed in radians carry as much weight as CoM positions in meters. In the object manipulation task, a key representational challenge is to distinguish between pushing the block and simply moving the arm in free space.
+
+# 6.2 LEARNING THE GOAL-CONDITIONED POLICY AND ARC REPRESENTATION
+
+We first learn a stochastic goal-conditioned policy parametrized by a neural network which outputs actions given the current state and the desired goal. This goal-conditioned policy is trained using a sparse reward using entropy-regularized Trust Region Policy Optimization (TRPO) (Schulman et al., 2015). For a discussion of the assumption about the existence of a goal-conditioned policy, please refer to Section 3. Exact details about the training procedure, the reward function, and hyperparameters are presented in Appendix A.1. To train the ARC representation, we collect a dataset of 500 trajectories with horizon 100 from the goal-conditioned policy, where each trajectory has an arbitrary start state and intended goal state. We optimize Eqn 2 as a supervised learning problem using this dataset to train the representation, computing the relevant expectations by uniform sampling from states in the dataset. A detailed outline of the training procedure, along with hyperparameter and architecture choices, is presented in Appendix A.2.
+
+![](images/c93a6875cc668d79a256b1f2129fe2f4c1f5d35ce0637ecf25e350b3307113cb.jpg)  
+Figure 4: The tasks in our evaluation. The 2D navigation tasks allow for easy visualization and analysis, while the more complex tasks allow us to investigate how well ARC and prior methods can discern the most functionally-relevant features of the state.
+
+# 6.3 COMPARISONS WITH PRIOR WORK
+
+We compare ARC to other representation learning methods used in previous works for control: variational autoencoders (Kingma & Welling, 2013) (VAE), variational autoencoders trained for feature slowness (Jonschkowski & Brock, 2015) (slowness), features extracted from a predictive model (Oh et al., 2015) (predictive model), features extracted from inverse models (Agrawal et al., 2016; Burda et al., 2018a), and a na¨ıve baseline that uses the full state space as the representation (state). Details of the exact objectives used to train these methods is provided in Appendix B.
+
+For each downstream task in Section 4, we also compare with alternative approaches for solving the task not involving representation learning. For reward shaping, we compare with VIME (Houthooft et al., 2016), an exploration method based on novelty bonuses. For hierarchical RL, we compare with option critic (Klissarov et al., 2017) and an on-policy adaptation of HIRO (Nachum et al., 2018). We also compare to model-based reinforcement learning with MPC (Nagabandi et al., 2017), a method which explicitly learns and uses environment dynamics, as compared to the implicit dynamics learnt by ARC. Because sample complexity of model-based and model-free methods differ, all results with model-based reinforcement learning indicate final performance.
+
+To ensure a fair comparison between the methods, we provide the same information and trajectory data that ARC receives to all of the representation learning methods. Each representation is trained on the same dataset of trajectories collected from the goal-conditioned policy, ensuring that each comparisons receives data from the full state distribution and meaningful transitions.
+
+![](images/8e74cf78dae36a5c3eedf9a5639cef6964ce25dea0613de67f8b928918a38172.jpg)  
+6.4 ANALYSIS OF LEARNED ACTIONABLE REPRESENTATIONS   
+Figure 5: Visualization of ARC for 2D navigation. The states in the environment are colored to help visualize their position in representation space. For the wall task, points on opposite sides of the wall are clearly separated in ARC space (c). For four rooms, we see that ARCs provide a clear decomposition into room clusters (f), while VAEs do not (e).
+
+We analyze the structure of ARC space for the tasks described in Section 6.1, to identify which factors of state ARC chooses to emphasize, and how system dynamics affect the representation.
+
+In the 2D navigation tasks, we visualize the original state and learned representations in Figure 5. In both environments, ARC reflects the dynamics: points close by in Euclidean distance in the original state space are distant in representation space when they are functionally distinct. For instance, there is a clear separation in the latent space where the wall should be, and points on opposite sides of the wall are much further apart in ARC space (Figure 5) than in the original environment and in the VAE representation. In the room navigation task, the passages between rooms are clear bottlenecks, and the ARC representation separates the rooms according to these bottlenecks.
+
+![](images/05c44312c9391cf5d285b74e436946a1c6439a10fcfe5d0a4398ce5a231da8fa.jpg)  
+Figure 6: Perturbation analysis (Section 6.4): Effective representations vary significantly with perturbations to functionally relevant elements of state (shown in orange), and less for secondary elements (shown in purple). ARC exhibits this property, with a spread orange region - robot CoM or object position, and a suppressed purple region - joint angles and other secondary elements. The VAE and naive state representations do not capture this saliency, containing spread purple regions.
+
+The representations learned in more complex domains, such as wheeled or legged locomotion and block manipulation, also show meaningful patterns. We aim to understand which elements of state are being emphasized by the representation, by analyzing how distances in the latent space change as we perturb various elements of state. (Fig 6). For each environment, we determine two factors in the state: one which we consider salient for decision making (in orange), and one which is secondary (in purple). We expect a good representation to have a larger variation in distance as we perturb the important factor than when we perturb the secondary factor. In the legged locomotion environment, the CoM is the important factor and the joint angles are secondary. As we perturb the CoM, the representation should vary significantly, while the effect should be muted as we perturb the joints. For the wheeled environment, position of the car should cause large variations while the orientation should be secondary. For the object pushing, we expect block position to be salient and end-effector position to be secondary. Since distances in the high-dimensional representation space are hard to visualize, we project [ARC, VAE, State] representations of perturbed states into 2 dimensions (Fig 6) using multi-dimensional scaling (MDS) (Borg & Groenen, 2005), which projects points while preserving Euclidean distances. From Fig 6, we see that ARC captures the factors of interest; as the important factor is perturbed the representation changes significantly (spread out orange points), while when the secondary factor is perturbed the representation changes minimally (close together purple points). This implies that for Ant, ARC captures CoM while suppressing joint angles; for wheeled, ARC captures position while suppressing orientation; for block pushing, ARC captures block position, suppressing arm movement. Both VAE representations and original state space are unable to capture this.
+
+# 6.5 LEVERAGING ACTIONABLE REPRESENTATIONS FOR REWARD SHAPING
+
+As desribed in Section 4.2, distances in ARC space can be used for reward shaping to solve tasks that present a large exploration challenge with sparse reward functions. We investigate this on two challenging exploration tasks for wheeled locomotion and legged locomotion (seen in Fig 7). We acquire an ARC from a goal-conditioned policy in the region $s$ where the $\mathbf { \mathrm { C o M } }$ is within a $2 \mathrm { m }$ square. The learned representation is then used to guide learning via reward shaping for learning a goal-conditioned policy on a larger region $S ^ { \prime }$ , where the CoM is within a square of $8 \mathrm { m }$ . The task is to reach arbitrary goals in $S ^ { \prime }$ , but with only a sparse goal completion reward, so exploration is challenging.We shape the reward with a term corresponding to distance between the representation of the current and desired state: $r ( s , g ) = r _ { \mathrm { s p a r s e } } - \| \phi ( s ) - \phi ( g ) \| _ { 2 }$ . To ensure fairness, all comparisons initialize from the goal-conditioned policy on small region $s$ and train on the same data. Further details on the experimental setup for this domain can be found in Appendix A.3.
+
+![](images/f72bbaadb6b38942ee729b9bff6071a193c793281ff4204901973aaedd233da0.jpg)  
+Figure 7: Learning new tasks with reward shaping in representation space. ARC representations are more effective than other methods, and match the performance of a hand-specified shaping.
+
+As shown in Fig 7, ARC demonstrates faster learning speed and better asymptotic performance over all compared methods, when all are initialized from the goal conditioned policy trained on the small region. This can be attributed to the fact that, unlike the other representation learning algorithms, the ARC representation explicitly optimizes for functional distances in latent space, which generalizes well to a larger domain since the functionality in the new space is preserved. The performance of ARC is similar to a hand-designed reward shaping corresponding to distance in COM space, corroborating Figure 6 that ARC considers CoM to be the most salient feature. We notice that representations which are dynamics-aware (ARC, predictive models, inverse models) outperform VIME, which uses a novelty-based exploration strategy without considering environment dynamics, indicating that effectively incorporating dynamics information into representations can help tackle exploration challenges in large environments.
+
+# 6.6 LEVERAGING ACTIONABLE REPRESENTATIONS AS FEATURES FOR LEARNING POLICIES
+
+We consider using the ARC representation as a feature space for learning policies for tasks that cannot be expressed with a goal-reaching objective. We consider a quadruped ant robot task which requires the agent to reach a target (shown in green in Fig 8) while avoiding a dangerous region (shown in red in Fig 8). Instead of learning a policy from state $\pi ( a | s )$ , we learn a policy using a representation $\phi$ as features $\pi ( a | \phi ( s ) )$ . It is important to note that this task cannot be solved directly by a goal-conditioned policy (GCP), and a GCP attempting to reach the specified goal will walk through the dangerous region and receive a reward of -760. The reward function for this task and other experimental details are noted in Appendix A.4.
+
+![](images/89da5429fd0885db7970cc29458531bb45cf1836d94420c462b52b7cfb7c6b24.jpg)  
+Figure 8: ARCs as policy features. Top: Reach-while-avoiding task Bottom: Task learning curves
+
+Although all the methods ultimately learn to solve the task, policies using ARC features learn at a significantly faster rate (Figure 8). Policies using ARC features solve the task by Iteration 100, by which point all other methods can only solve with
+
+$5 \%$ success. We attribute the rapid learning progress to the ability of ARC to emphasize elements of the state that are important for multi-timestep control, rather than greedy features discovered by reconstruction or one-step prediction. Features which emphasize elements important for control make learning easier because they reduce redundancy and noise in the input, and allows the RL algorithm to effectively assign credit. We further note that other representation learning methods learn only as fast as the original state representation, and model-based MPC controllers (Nagabandi et al., 2017) also perform suboptimally. It is important to note that the same representation can be used to quickly train many different tasks, amortizing the cost of training a GCP.
+
+We consider using ARC representations to control high-level controllers for learning temporally extended navigation tasks in room and waypoint navigation settings, as described in Section 4. In the multi-room environments, the agent must navigate through a sequence of 50 rooms in order, receiving a sparse reward when it enters the correct room. In waypoint navigation, the ant must reach a sequence of waypoints in order with a similar sparse reward. These tasks are illustrated in $\operatorname { F i g } 9$ , and are described in detail in Appendix A.5.
+
+We evaluate the two schemes for hierarchical reasoning with ARCs detailed in Section 4.3: commanding directly in representation space or through a $k$ -means clustering of the representation space. We train a high-level controller $\pi _ { h }$ with TRPO which outputs as actions either a direct point in the latent space $z _ { h }$ or a cluster index $c _ { h }$ , from which a goal $g _ { h }$ is decoded and passed to the goal-conditioned policy to follow for 50 timesteps. Exact specifications and details are in Appendix A.5 and A.6.
+
+![](images/ce3cfabfdb9533d2b0cb83dce81a7c0b4f4aa706713721da26307a6de363c319.jpg)  
+Figure 9: Waypoint and multi-room HRL tasks
+
+![](images/b0af848bd13a6908ff5c83117016df3bb9b8b6e366f467141b70b8d4d44fd9a4.jpg)  
+Figure 10: Comparison on hierarchical tasks. ARCs perform significantly better than other representation methods, option-critic, and commanding goals in state space
+
+Using a hierarchical meta-policy with ARCs performs significantly better than those using alternative representations which do not properly capture abstraction and environment dynamics (Fig 10). For multi-rooms, ARC clusters very clearly capture different rooms (Fig 5), so commanding in cluster space reduces redundancy in action space, allowing for effective exploration. ARC likely works better than commanding goals in spaces learned by other representation learning algorithms, because the learned ARC space is more structured for high-level control, which makes search and clustering simpler. Semantically similar states like two points in the same room end up in the same ARC cluster, thus simplifying the high-level planning process for the meta-controller. As compared to learning from scratch via TRPO and standard HRL methods such as option critic (Klissarov et al., 2017) and an on-policy adaptation of HIRO (Nachum et al., 2018), commanding in representation space enables more effective search and high-level control. The failure of TRPO and option-critic, algorithms not using a goal-conditioned policy, emphasizes the task difficulty and indicates that a goal-conditioned policy trained on simple reaching tasks can be re-used to solve long-horizon problems. Commanding in ARC space is better than in state space using HIRO because state space has redundancies which makes search challenging.
+
+# 7 DISCUSSION
+
+In this work, we introduce actionable representations for control (ARC), which capture representations of state important for decision making. We build on the framework of goal-conditioned RL to extract state representations that emphasize features of state that are functionally relevant. The learned state representations are implicitly aware of the dynamics, and capture meaningful distances in representation space. ARCs are useful for tasks such as learning policies, HRL and exploration. While ARC are learned by first training a goal-conditioned policy, learning this policy using offpolicy data is a promising direction for future work. Interleaving the process of representation learning and learning of the goal-conditioned policy promises to scale ARC to more general tasks.
+
+Acknowledgements This research was supported by Berkeley DeepDrive, Honda, an ONR Young Investigator Program Award, Google, and computational resources from Amazon. Abhishek Gupta was supported by an NSF Graduate Research Fellowship. We thank Pim de Haan, Aviv Tamar, Vitchyr Pong, and Ignasi Clavera for helpful insights and discussions.
+
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+
+# A EXPERIMENTAL DETAILS
+
+# A.1 TRAINING THE GOAL-CONDITIONED POLICY
+
+We train a stochastic goal-conditioned policy $\pi ( \cdot | s , g )$ using TRPO with an entropy regularization term, where the goal space $\mathcal { G }$ coincides with the state space $s$ . In every episode, a starting state and a goal state $s , g \in S$ are sampled from a uniform distribution on states, with a sparse reward given of the form below, where $\epsilon$ is task-specific, and listed in the table below.
+
+$$
+r ( s , g ) = { \left\{ \begin{array} { l l } { 0 } & { \| s - g \| _ { \infty } > \epsilon } \\ { \epsilon - \| s - g \| _ { \infty } } & { \| s - g \| _ { \infty } < \epsilon } \end{array} \right. }
+$$
+
+For the Sawyer environment, although this sparse reward formulation can learn a goal-conditioned policy, it is highly sample inefficient, so in practice we use a shaped reward as detailed in Appendix C. For all of the other environments, in the free space and rooms environments, the goal-conditioned policy is trained using a sparse reward.
+
+The goal-conditioned policy is parameterized as $\pi _ { \boldsymbol { \theta } } ( a | s , g ) \sim \mathcal { N } ( \mu _ { \boldsymbol { \theta } } ( s , g ) , \Sigma _ { \boldsymbol { \theta } } )$ . The mean, $\mu _ { \boldsymbol { \theta } } ( \cdot , \cdot )$ is a fully-connected neural network which takes in the state and the desired goal state as a concatenated vector, and has three hidden layers containing 150, 100, and 50 units respectively. $\Sigma$ is a learned diagonal covariance matrix, and is initially set to $\Sigma = I$ .
+
+<table><tr><td></td><td>Navigation</td><td>Wheeled Navigation</td><td>Ant Navigation</td><td>Sawyer Pushing</td></tr><tr><td>State/Goal Space Dimension</td><td>2</td><td>6</td><td>15</td><td>6</td></tr><tr><td>Action Space Dimension</td><td>2</td><td>2</td><td>7</td><td>3</td></tr><tr><td>Sparse Reward Threshold (ε)</td><td>0.1</td><td>0.5</td><td>1</td><td>0.3</td></tr><tr><td># Trajectories per Iteration</td><td>100</td><td>200</td><td>250</td><td>500</td></tr><tr><td># Steps in Trajecotry</td><td>50</td><td>100</td><td>200</td><td>100</td></tr><tr><td># Iterations</td><td>200</td><td>1000</td><td>2000</td><td>2000</td></tr><tr><td>Entropy Penalty</td><td>1</td><td>1</td><td>0.1</td><td>0.1</td></tr><tr><td>Learning Rate</td><td>.01</td><td>.02</td><td>.02</td><td>.01</td></tr></table>
+
+# A.2 TRAINING THE REPRESENTATION
+
+After training a goal-conditioned policy $\pi$ on the specified region of interest, we collect 500 trajectories each of length 100 timesteps, where each trajectory starts at an arbitrary start state, going towards an arbitrary goal state, selected exactly as the goal-conditioned policy was trained in Appendix A.1. This dataset was chosen to be large enough so that the collected dataset has full coverage of the entire state space. Each of the representation learning methods evaluated is trained on this dataset, which means that each learning algorithm receives data from the full state space, and witnesses meaningful transitions between states $( s _ { t } , s _ { t + 1 } )$ .
+
+We evaluate ARCs against representations minimizing reconstruction error (VAE, slowness) and representations performing one-step prediction (predictive model, inverse dynamics). For each representation, each component is parametrized by a neural network with ReLU activations and linear outputs, and the objective function is optimized using Adam with a learning rate of $1 0 ^ { - 3 }$ , holding out $20 \%$ of the trajectories as a validation set. We perform coarse hyperparameter sweeps over various hyperparameters for all of the methods, including the dimensionality of the latent state, the size of the neural networks, and the parameters which weigh the various terms in the objectives. The exact objective functions for each representation are detailed further in Appendix B.
+
+# A.3 REWARD SHAPING
+
+We test the reward shaping capabilities of the learned representations with a set of navigation tasks on the Wheeled and Ant tasks. A goal-conditioned policy $\pi$ is trained on a $n \times n$ meter square of free space, and representations are learned (as specified above) on trajectories collected in this small region. We then attempt to generalize to an $m \times m$ meter square (where $m > > n$ ), and consider the set of tasks of reaching an arbitrary goal in the larger region: a start state and goal state are chosen uniformly at random every episode. The environment setup is identical to that in Appendix A.1, although with a larger region, and policy training is done the same with two distinctions. Instead of training with the sparse reward $r _ { s p a r s e } ( s , g )$ , we train on a ”shaped” surrogate reward
+
+$$
+r _ { s h a p e d , \phi } ( s , g ) = r _ { s p a r s e } ( s , g ) - \alpha \| \phi ( s ) - \phi ( g ) \| _ { 2 }
+$$
+
+where $\alpha$ weights between the euclidean distance and the sparse reward terms. Second, the policy is initialized to the parameters of the original goal-conditioned policy $\pi$ which was previously trained on the small region to help exploration. As a heuristic for the best possible shaping term, we compare with a ”hand-specified” In addition to reward shaping with the various representations, we also compare to a dedicated exploration algorithm, VIME (Houthooft et al., 2016), which also uses TRPO as a base algorithm. Understanding that different representation learning methods may learn representations with varying scales, we performed a hyperparameter sweep on $\alpha$ for all the representation methods. For VIME, we performed a hyperparameter sweep on $\eta$ . The parameters used for TRPO are exactly those in Appendix A.1, albeit for 3000 iterations.
+
+# A.4 FEATURES FOR POLICIES
+
+We test the ability of the representation to be used as features for a policy learning some downstream task within the Ant environment. The downstream task is a ”reach-while-avoid” task, in which the Ant requires the quadruped robot to start at the point $( - 1 . 5 , - 1 . 5 )$ and reach the point (1.5, 1.5) while avoiding a circular region centered at the origin with radius 1 (all units in meters). Letting $d _ { g o a l } ( s )$ be the distance of the agent to (1.5, 1.5) and $d _ { o r i g i n } ( s )$ to be the distance of the agent to the origin, the reward function for the task is
+
+$$
+r ( s ) = - d _ { g o a l } ( s ) - 4 * \mathbf { 1 } \{ d _ { o r i g i n } ( s ) < 1 \}
+$$
+
+For any given representation $\phi ( s )$ , we train a policy which uses the feature representation as input as follows. We use TRPO to train a stochastic policy $\pi ( a | \phi ( s ) )$ , which is of the form $\mathcal { N } ( \mu _ { \boldsymbol { \theta } } ( \phi ( s ) ) , \Sigma _ { \boldsymbol { \theta } } )$ . The mean is a fully connected neural network which takes in the representation, and has two layers of size 50 each, and $\Sigma$ is a learned diagonal covariance matrix initially set to $\Sigma = I$ . Note that gradients do not flow through the representation, so only the policy is adapted and the representation is fixed for the entirety of the experiment.
+
+# A.5 HIERARCHICAL REINFORCEMENT LEARNING IN LATENT SPACE
+
+We provide comparisons on using the learned representation to direct a goal-conditioned policy for long-horizon sequential tasks. In particular, we consider a waypoint reaching task for the Ant, in which the agent must navigate to a sequence of 50 target locations in order: $\left\{ ( x _ { 1 } , y _ { 1 } ) , ( x _ { 2 } , y _ { 2 } ) , \dots ( x _ { 5 0 } , y _ { 5 0 } ) \right\}$ . The agent receives as input the state of the ant and the checkpoint number that it is currently trying to reach (encoded as a one-hot vector). When the agent gets within $0 . 5 \mathrm { m }$ of the checkpoint, it receives $+ 1$ reward, and the checkpoint is moved to the next point, making this a highly sparse reward. Target locations are sampled uniformly at random from a $8 \times 8$ meter region, but are fixed for the entirety of the experiment.
+
+We consider learning a high-level policy $\pi _ { h } ( z _ { h } | s )$ which outputs goals in latent space, which are then executed by a goal-conditioned policy as described in Appendix A.1. Specifically, when the high-level policy outputs a goal in latent space $z _ { h }$ , we use a reconstruction network $\psi$ , which is described below, to receive a goal state $g _ { h } = \psi ( z _ { h } )$ . The goal-conditioned policy executes for 50 timesteps according to $\pi ( a | s , \bar { g } _ { h } )$ . The high-level policy is trained with TRPO with the reward being equal to the sum of the rewards obtained by running the goal-conditioned policy for every meta-step. We parametrize the high-level policy $\pi _ { h } ( z _ { h } | s )$ as having a Gaussian distribution in the latent space, with the mean being specified as a MLP with two layers of 50 units and Tanh activations, and the covariance as a learned diagonal matrix independent of state.
+
+To allow the latent representation $z$ to provide commands for the goal-conditioned policy, we separately train a reconstruction network $\psi$ which minimizes the loss function $\mathbb { E } _ { s } [ \lVert \psi ( \phi ( s ) ) - s \rVert _ { 2 } ]$ . For any latent $z$ , we can now use $\psi ( z )$ as an input into the goal-conditioned policy. Note that an alternative method of providing commands in latent space is to train a new goal-conditioned policy $\pi _ { \phi }$ , which is trained to minimize the loss $\mathbb { E } _ { s , g } [ D _ { K L } ( \pi _ { \phi } ( \cdot | s , \phi ( g ) ) \| \pi ( \cdot | s , g ) ) ]$ , however to maintain abstraction between the representation and the goal-conditioned policy, we choose the former approach.
+
+# A.6 HIERARCHICAL REINFORCEMENT LEARNING IN CLUSTER SPACE
+
+We provide comparisons on using the learned representation to direct a goal-conditioned policy in cluster space, as described in Section 4. We consider navigation through a sequence of rooms in order in the rooms and wheeled rooms environment, as visualized in Figure 4. A sequence of 50 checkpoints are sampled uniformly from the four rooms with the extra constraint that the same room is never repeated two checkpoints in a row (that is, each checkpoint is chosen to be any of the four rooms), and held fixed for the entirety of the experiment. The agent is tasked with going through these rooms in order, receiving a $+ 1$ reward every time it enters the appropriate room. The policy receives as input the state of the agent, and which number checkpoint the agent is currently trying to reach (encoded as a 50-dimensional one-hot vector).
+
+After having learned a representation $\phi$ using some set of trajectory data, as described in Appendix A.2, we run $k$ -means clustering on states in the trajectory data to cluster latent states in the representation into $k$ components. We then consider learning a high-level policy $\pi _ { h } ( c _ { h } \vert s )$ which outputs a cluster between $\{ 1 \ldots k \}$ . Given a cluster number $c _ { h }$ from the high-level policy, the low-level policy samples a latent state $z _ { h }$ uniformly from the cluster, and then proceeds to command a learnt goal-conditioned policy exactly as described in Appendix A.5.
+
+Specifically, we learn a high-level policy of the form $\pi _ { h } ( c _ { h } | s ) \sim \mathrm { C a t e g o r i c a l } ( p _ { \theta } ( s ) )$ using TRPO where the probabilities for each cluster are specified by a neural network $\pi _ { \theta }$ which has two layers of 50 units each, with Tanh activations, and a final Softmax activation to normalize outputs into the probability simplex.
+
+We performed hyperparameter sweeps over $k$ - the number of clusters - for each representation method.
+
+# B BENCHMARK REPRESENTATIONS
+
+We provide the loss functions that are used to train each of the representations evaluated in our work. All representations are trained on a dataset of trajectories $\mathcal { D } \stackrel { - } { = } \{ \tau _ { i } \} _ { i = 1 } ^ { n }$ . We use the notation $s \sim \mathcal { D }$ to denote sampling a state uniformly at random from a trajectory uniformly at random from the dataset. We use the notation $s _ { t } , s _ { t + 1 } \sim \mathcal { D }$ to denote sampling a state and the state right after it according to the same uniform sampling scheme.
+
+• ARC - After precomputing $D _ { a c t }$ :a matrix of actionable distances, we train a neural network $\phi$ to minimize
+
+$$
+\begin{array} { r l r } & { } & { D _ { a c t } \big ( s _ { i } , s _ { j } \big ) = \mathbb { E } _ { s \sim \mathcal { D } } \left[ D _ { K L } \big ( \pi ( a | s , s _ { i } ) \| \pi ( a | s , s _ { j } ) \big ) \right] } \\ & { } & { \mathcal { L } ( \phi ) = \mathbb { E } _ { s \sim \mathcal { D } } \left[ \mathbb { E } _ { s ^ { \prime } \sim \mathcal { D } } \left[ \| \| \phi ( s ) - \phi ( s ^ { \prime } ) \| - D _ { a c t } ( s , s ^ { \prime } ) \| ^ { 2 } \right] \right] } \end{array}
+$$
+
+• VAE (Kingma & Welling, 2013) - Given $q _ { \phi } ( z | x ) = \mathcal { N } ( \mu _ { \phi } ( x ) , \sigma _ { \phi } ( x ) ) , p _ { \theta } ( x | z ) = $ $\mathcal { N } ( \psi _ { \theta } ( z ) , \bar { I } )$ , and $p ( z ) = \bar { \mathcal { N } } ( 0 , I )$
+
+$$
+\mathcal { L } ( \phi , \theta ) = \mathbb { E } _ { s \sim \mathcal { D } } \left[ \mathbb { E } _ { z \sim q _ { \phi } ( x ) } \left[ \log p _ { \theta } ( x | z ) - \beta D _ { K L } \big ( q _ { \theta } ( z | x ) \| p ( z ) \big ) \right] \right]
+$$
+
+Here $\mu _ { \phi } , \sigma _ { \phi } , \psi _ { \theta }$ are all neural networks, and $\beta$ is a tunable hyperparameter. The loglikelihood term is equivalent to minimizing mean squared error.
+
+• Slowness (Jonschkowski & Brock, 2015) - Given $q _ { \phi } ( z | x ) = \mathcal { N } ( \mu _ { \phi } ( x ) , \sigma _ { \phi } ( x ) ) , p _ { \theta } ( x | z ) =$ $\mathcal { N } ( \psi _ { \theta } ( z ) , I )$ , and $p ( z ) = \mathcal { N } ( 0 , I )$
+
+$$
+\begin{array} { r } { \mathcal { L } ( \phi , \theta ) = \mathbb { E } _ { ( s _ { t } , s _ { t + 1 } ) \sim \mathcal { D } } \left[ \mathbb { E } _ { z \sim q _ { \phi } ( s _ { t } ) } \left[ \log p \theta ( s _ { t } | z ) - \beta D _ { K L } ( q _ { \theta } ( z | x ) \| p ( z ) ) - \alpha \| \mu _ { \theta } ( s _ { t + 1 } - \mu _ { \theta } ( s _ { t } ) \| ] \right] \right] } \end{array}
+$$
+
+Here $\mu _ { \phi } , \sigma _ { \phi } , \psi _ { \theta }$ are all neural networks, and $\alpha , \beta$ are tunable hyperparameters. The loglikelihood terms are equivalent to minimizing mean squared error.
+
+• Predictive Model (Oh et al., 2015) - Given $z = \phi ( s _ { t } )$ , $\hat { z } _ { t + 1 } = f ( z , a )$ and $\psi ( z ) = \hat { s }$ , we
+
+$$
+\mathcal { L } ( \phi , f , \psi ) = \mathbb { E } _ { ( s _ { t } , a _ { t } , s _ { t + 1 } ) \sim \mathcal { D } } \left[ \| s _ { t + 1 } - \psi ( f ( \phi ( s _ { t } ) , a _ { t } ) ) \| _ { 2 } ^ { 2 } \right]
+$$
+
+where $\phi$ is the learnt representation, $f$ a model in representation space, and $\psi$ a reconstruction network retrieving are all neural networks.
+
+• Inverse Model (Burda et al., 2018a) - Given $z _ { t } ~ = ~ \phi ( s _ { t } ) , \hat { z } _ { t + 1 } ~ = ~ f ( z _ { t } , a ) , \hat { a } _ { t + 1 } ~ = ~$ $g ( z _ { t } , z _ { t + 1 } )$
+
+$$
+\mathcal { L } ( \phi , f , g ) = \mathbb { E } _ { ( s _ { t } , a _ { t } , a _ { t + 1 } ) \sim \mathcal { D } } \left[ \| a _ { t } - g ( \phi ( s _ { t } ) , \phi ( s _ { t + 1 } ) ) \| _ { 2 } ^ { 2 } + \beta \| \phi ( s _ { t + 1 } ) - f ( \phi ( s _ { t } ) , a _ { t } ) \| _ { 2 } ^ { 2 } \right]
+$$
+
+Here, $\phi$ is the learnt representation, $f$ is a learnt model in the representation space, and $g$ is a learnt inverse dynamics model in the representation space. $\beta$ is a hyperparameter which controls how forward prediction error is balanced with inverse prediction error.
+
+# C TASK DESCRIPTIONS
+
+• 2D Navigation This environment consists of an agent navigating to points in an environment, either with a wall as in Figure 4a or with four rooms, as in Figure 4b. The state space is 2-dimensional, consisting of the Cartesian coordinates of the agent. The agent has acceleration control, so the action space is 2-dimensional. Downstream tasks for this environment include reaching target locations in the environment and navigating through a sequence of 50 rooms.
+
+• Wheeled Navigation This environment consists of a car navigating to locations in an empty region, or with four rooms, as illustrated in Figure 4. The state space is 6-dimensional, consisting of the Cartesian coordinates, heading, forward velocity, and angular velocity of the car. The agent controls the velocity of both of its wheels, resulting in a 2-dimensional action space. Goal-conditioned policies are trained within a $3 \times 3$ meter square.
+
+Downstream tasks for wheeled navigation include reaching target locations in the environment, navigating through sequences of rooms, and navigating through sequences of waypoints.
+
+• Ant This task requires a quadrupedal ant robot navigating in free space.The state space is 15-dimensional, consisting of the Cartesian coordinates of the ant, body orientation as a quaternion, and all the joint angles of the ant. The agent must use torque control to control it’s joints, resulting in an 8-dimensional action space. Goal conditioned policies are trained within a $2 \times 2$ meter square.
+
+Downstream tasks for the ant include reaching target locations in the environment, navigating through sequences of waypoints, and reaching target locations while avoiding other locations.
+
+• Sawyer This environment involves a Sawyer manipulator and a freely moving block on a table-top. The state space is 6-dimensional, consisting of the Cartesian coordinates of the end-effector of the Sawyer, and the Cartesian coordinates of the block. The Sawyer is controlled via end-effector position control with a 3-dimensional action space.
+
+Because training a goal-conditioned policy takes an inordinate number of samples for the Sawyer environment, we instead use the following shaped reward to train the GCVF where $h ( s ) { \overline { { } } }$ is the position of the hand and $o ( s )$ is the position of the object
+
+$$
+r _ { s h a p e d } ( s , g ) = r _ { s p a r s e } ( s , g ) - \| h ( s ) - o ( s ) \| - 2 \| o ( s ) - o ( g ) \|
+$$
+
+# D HYPERPARAMETER TUNING
+
+We perform hyperparameter tuning on three ends: one to discover appropriate parameters for each representation for each environment which are then held constant for the experimental analysis, then on the downstream applications, to choose a scaling factor for reward shaping (see Appendix A.3), and to choose the number of clusters for the hierarchical RL experiments in cluster space (see Appendix A.6).
+
+To discover appropriate parameters for each representation for the legged and wheeled locomotion environments, we evaluate representations on the downstream reward-shaping task, performing a hyperparameter sweep on latent dimension and the parameters which weigh the various terms in the representation learning objectives. We keep the network architecture fixed for each representation and each task. We emphasize carefully here that the ARC representation requires no parameters to tune beyond the size of the latent dimension, and we perform a hyperparameter sweep on the penalty terms to ensure that other methods aren’t improperly penalized. On the size of the latent dimension, we sweep over $\{ 2 , 3 , 4 \}$ for wheeled locomotion and $\{ 3 , 5 , 7 , 9 , 1 1 \}$ for the ant. For the relative weighting for the penalty terms for the comparison representations (defined by $\beta$ in Appendix B), we evaluate possible values $\beta \in \{ 4 ^ { - 2 } , 4 ^ { - 1 } , \dot { 1 } , 4 ^ { 1 } , 4 ^ { 2 } \}$ . These representations are then fixed and used for all the downstream applications.
+
+For reward shaping, we tune the relative scales between the sparse reward and the shaping term, (denoted by $\alpha$ in Appendix A.3) over possible values $\alpha \in \{ 1 , 4 ^ { 1 } , 4 ^ { 2 } , 4 ^ { 3 } , 4 ^ { 4 } \}$ for each representation on both the legged and wheeled locomotion environments. Tuning for $\alpha$ is required because the representations may have different latent dimensions and different scales, and chose to perform this hyperparameter sweep instead of adding a term to the representation learning objectives to ensure uniformity in scale. For performing $k$ -means clustering on the HRL cluster experiments, we sweep over possible values $k \in \{ 4 , 5 , 6 , 7 , 8 \}$ for each representation on the room navigation tasks for 2D and wheeled navigation, but however found that most representations were robust to choice of the number of clusters.

parse/train/MQQeeDiO5vv/MQQeeDiO5vv.md

@@ -0,0 +1,303 @@
+# Combiner: Full Attention Transformer with Sparse Computation Cost
+
+∗Hongyu Ren†, ∗Hanjun Dai⋄, ∗Zihang Dai⋄ Mengjiao Yang⋄, Jure Leskovec†, Dale Schuurmans⋄,‡, Bo Dai⋄ †Stanford University, {hyren,jure}@cs.stanford.edu ⋄Google Research, Brain Team, {hadai,zihangd,sherryy,schuurmans,bodai}@google.com ‡University of Alberta
+
+# Abstract
+
+Transformers provide a class of expressive architectures that are extremely effective for sequence modeling. However, the key limitation of transformers is their quadratic memory and time complexity $\mathcal { O } ( L ^ { 2 } )$ with respect to the sequence length in attention layers, which restricts application in extremely long sequences. Most existing approaches leverage sparsity or low-rank assumptions in the attention matrix to reduce cost, but sacrifice expressiveness. Instead, we propose Combiner, which provides full attention capability in each attention head while maintaining low computation and memory complexity. The key idea is to treat the self-attention mechanism as a conditional expectation over embeddings at each location, and approximate the conditional distribution with a structured factorization. Each location can attend to all other locations, either via direct attention, or through indirect attention to abstractions, which are again conditional expectations of embeddings from corresponding local regions. We show that most sparse attention patterns used in existing sparse transformers are able to inspire the design of such factorization for full attention, resulting in the same sub-quadratic cost $( { \mathcal { O } } ( L \log ( L ) )$ or $\mathcal { O } ( L \sqrt { L } ) )$ . Combiner is a drop-in replacement for attention layers in existing transformers and can be easily implemented in common frameworks. An experimental evaluation on both autoregressive and bidirectional sequence tasks demonstrates the effectiveness of this approach, yielding state-of-the-art results on several image and text modeling tasks.
+
+# 1 Introduction
+
+The Transformer [1] is a powerful neural network architecture that has demonstrated state-of-the-art performance in machine translation [2] and many other natural language processing (NLP) tasks via pretraining, using either unidirectional language modeling [3] or bidirectional language modeling [4–8]. It has also achieved excellent results in other domains like image recognition [9], code understanding [10], speech recognition [11], protein [12], music [13] and image [14] generative modeling. The core component of Transformer is the attention mechanism, which computes dependencies between all pairs of positions in a sequence. However, for a sequence of length $L$ , the expressiveness of pairwise attention comes at a quadratic cost $\mathcal { O } ( L ^ { 2 } )$ in both time and memory consumption. This makes the vanilla Transformer [1] prohibitive for applications that involve long sequences, including high-resolution images, protein sequences, or raw speech signals [15], where the sequence length $L$ is often larger than 10, 000 [14].
+
+Recently, there have been several attempts to scale up attention to long sequences. A popular class of methods sparsifies the attention matrix with different sparsity patterns, including local window [16, 17], local+stride [14], log-sparse [18], axial [19, 20], or learnable patterns through hashing [21] or clustering [22]. Sparse attention enjoys sub-quadratic cost, but is lossy in capturing all-pair relationships. Generally, sparse attention requires more layers [14, 20, 23] to achieve full autoregressive or bidirectional dependencies (or receptive fields [20]) for each location in a long sequence.
+
+Alternatively, another line of research has tried to achieve scalability with an explicit low-rank assumption [24, 25] on the attention matrix or by using explicit feature maps of some kernels [26]. However these explicit low dimensional approximations might be too restricted for the potentially full rank attention matrix, which uses exponential kernels that are effectively infinite dimensional [27]. The Performer [28] is among the first works that attempts to approximate regular full-rank attention with the random feature trick [29]. However such random-feature based approaches [30] require many more bases to better approximate the exponential kernel [27], and empirically we found it produces inferior results in some sequence modeling tasks, such as density estimation.
+
+In this paper we propose Combiner, a drop-in replacement for the vanilla quadratic attention mechanism with sub-quadratic computation and memory cost. Combiner still achieves full attention capability within each head of Multi-Head Attention, unlike approaches that adopt sparse or low-rank approximations. As we will discuss, the standard attention computed at each location can be seen as the conditional expectation of the value embeddings at all feasible locations given the current location. Based on such an understanding, Combiner explicitly approximates the conditional distribution in through a structured factorization of the probability space. Specifically, given a location $x$ , the probability of attending to location $y$ can be either directly calculated via the query vector of $x$ and key vector of $y$ , or indirectly through a local abstraction where $x$ first attends to the key vector that represents a group of locations containing $y$ , and multiplying the probability of choosing $y$ within that group. We refer to this model as Combiner since the conditional distributions in attention become a combination between several local attentions and direct attentions. This structured decomposition enables Combiner to take existing sparse attention patterns and convert them into corresponding design choices for probability factorizations that achieve full attention. As shown in Figure 1, Combiner achieves full attention with the same asymptotic complexity as sparse variants. Combiner can be easily implemented in most existing deep learning frameworks without the need for specialized hardware implementation, and is GPU/TPU friendly. In fact, both the fixed and learnable sparse attention patterns from many existing Transformer variants [14, 18, 20, 22] can be enhanced with such structured factorizations, with the same order of time or memory cost.
+
+We validate Combiner on both autoregressive and bidirectional sequence modeling tasks over a variety of domains including text and images. We show that Combiner can achieve better perplexity and accuracy when using the same transformer architectures while being much faster in terms of runtime, and achieves state of the art performance on density estimation on standard datasets CIFAR-10 (2.77 bits/dim) and ImageNet-64 (3.42 bits/dim), as well as the Long-Range Arena [31]. The implementation of Combiner can be found at https://github.com/google-research/googleresearch/tree/master/combiner.
+
+# 2 Attention as Conditional Expectation
+
+In this section, we revisit the formulation of the standard Transformer [1] from the perspective of conditional expectation, which inspires the derivation of Combiner.
+
+Without loss of generality, we use a single sequence in the self-attention scenario. Given a sequence of $L$ embeddings $X = \bar { [ { x _ { 1 } } , { x _ { 2 } } , { . . . , \bar { { x _ { L } } } } ] }$ , where $X \in \mathbb { R } ^ { L \times d }$ and each embedding $x _ { i } \in \mathbb { R } ^ { d }$ is a $d$ -dimensional vector, the core component of Transformer is the multi-head attention, where each head $h$ is a scaled dot-product attention:
+
+$$
+A _ { h } ( X ) = \operatorname { s o f } \operatorname { t m a x } \left( \frac { Q _ { h } } { \sqrt { d } } K _ { h } ^ { \top } \right) V _ { h } , \left\{ Q _ { h } = X W _ { h } ^ { Q } , K _ { h } = X W _ { h } ^ { K } , V _ { h } = X W _ { h } ^ { V } \right\} \in \mathbb { R } ^ { L \times d } ,
+$$
+
+and the attention vector from each head $A _ { h } ( X )$ is concatenated and projected:
+
+$$
+\mathrm { i } \mathrm { H e a d A t t u } ( X ) = [ A _ { 1 } ( X ) , A _ { 2 } ( X ) , \ldots , A _ { H } ( X ) ] W ^ { o } , W ^ { o } \in \mathbb { R } ^ { H \times d } .
+$$
+
+Mult (2) Here $H$ is the total number of heads per Transformer layer. In this paper, we focus on how to approximate full attention within each head of multi-head attention. For ease of notation, we drop the head index $h$ whenever possible, and use lower-case letters $x _ { i } , q _ { i } , k _ { i } , v _ { i } \in \mathbb { R } ^ { d }$ to denote rows in
+
+![](images/d44efb34f4d107361bc8899c32aa28458e1c5d06ab6ad955f245d4864635aec0.jpg)  
+Figure 1: Attention matrices of several instantiations of Combiner in the autoregressive setting. We transform several sparse attention patterns: Fixed (A) [14], Logsparse (B) [18] and Axial (C) [20] to Combiner-Fixed (D), Combiner-Logsparse (E) and Combiner-Axial (F). Combiner approximates the conditional expectation (3) with a combination of direct expectation (blue) and local expectation (yellow). Our instantiations (D)(E)(F) achieves full attention with the same sub-quadratic complexity.
+
+$X , Q , K , V$ respectively, which corresponds to a location $i$ in the original sequence of length $L$ . We use $[ n ]$ to denote the set of positive integers $\{ 1 , 2 , \ldots , n \}$ .
+
+For a position $i \in [ L ]$ , the attention formulation (1) can be viewed as conditional expectation of rows in $V$ . Specifically, since softmax outputs a probability distribution, we can rewrite (1) as
+
+$$
+A ( x _ { i } ) = \mathbb { E } _ { p ( j | i ) } \left[ v _ { j } \right] , \qquad p ( j | i ) = \frac { 1 } { Z \left( x _ { i } \right) } \exp \left( \frac { q _ { i } } { \sqrt { d } } k _ { j } ^ { \top } \right) ,
+$$
+
+where $p ( j | i )$ denotes the conditional probability at position $j$ given the token at position $i$ and the partition function $\begin{array} { r } { Z \left( x _ { i } \right) = \sum _ { j \in \Omega _ { i } } \exp \left( \frac { q _ { i } } { \sqrt { d } } k _ { j } ^ { \top } \right) } \end{array}$ over support $\Omega _ { i }$ . The support $\Omega _ { i }$ of $p \left( j | i \right)$ defines the set of valid locations that the $i$ -th token can attend to. For instance, the support set in autoregressive language modeling (LM) consists of all previous tokens, i.e., $\Omega _ { i } ^ { \mathrm { L M } } = [ i ] ^ { 2 }$ ; in masked i language modeling (MLM) the support consists of all tokens in the sequence, i.e., $\Omega _ { i } ^ { \mathrm { M L M } } = [ L ]$ . That is, $\bar { \Omega } _ { i } ^ { \mathrm { L M } }$ and $\Omega _ { i } ^ { \mathrm { M L M } }$ represent the full attention capability respectively in the LM and MLM setting.
+
+# 3 Combiner: Full Attention via Structured Conditional Expectation
+
+The complexity of $p \left( j | i \right)$ is the bottleneck of the computation for $A \left( x _ { i } \right)$ . Generally, in existing sparse transformers, the support of $p \left( j | i \right)$ is sparsified to reduce the computation and memory complexity, e.g., $\Omega _ { i } ^ { \mathrm { S p a r s e } } \subsetneq \Omega _ { i } ^ { \mathrm { L M } }$ for LM and $\Omega _ { i } ^ { \mathrm { S p a r s e } } \subsetneq \Omega _ { i } ^ { \mathrm { M L M } }$ for MLM, but this can lead to either reduced capacity or limited applicability. We defer detailed discussion of the full capacity of the model to Appendix A. In this section we introduce the Combiner, which achieves $\Omega _ { i } ^ { \mathrm { C o m b i n e r } } = \Omega _ { i } ^ { \mathrm { L M } }$ for LM and $\Omega _ { i } ^ { \mathrm { C o m b i n e r } } = \Omega _ { i } ^ { \mathrm { M L M } }$ for MLM, while still maintaining sub-quadratic computation and memory cost. Below we denote $\Omega _ { i }$ as the support for full attention if there is no ambiguity or need to distinguish between LM or MLM. We introduce the main design framework in Section 3.1 and possible parameterizations in Section 3.2. Then in Section 3.3 we analyze the trade-off of Combiner.
+
+# 3.1 Local Factorization for Conditional Expectation
+
+The main idea of Combiner is to exploit a hierarchical structure for conditional probability modeling in (3), which provides the opportunity for reducing computation complexity while maintaining the
+
+same support. Specifically, we introduce support variables $\Omega _ { i } ^ { r }$ , for $r = 0 , \ldots , n _ { i }$ and $i \in [ L ]$ . The support variables are disjoint, i.e., $\Omega _ { i } ^ { r } \cap \Omega _ { i } ^ { s } = \emptyset , \forall r \neq s$ , and $\cup _ { r = 0 } ^ { n _ { i } } \Omega _ { i } ^ { r } = \Omega _ { i }$ . Then we can factorize $p ( j | i )$ as
+
+$$
+p ( j | i ) = \sum _ { r = 0 } ^ { n _ { i } } p ( j , \Omega _ { i } ^ { r } | i ) = \sum _ { r = 0 } ^ { n _ { i } } p ( j | \Omega _ { i } ^ { r } , i ) p ( \Omega _ { i } ^ { r } | i ) = p ( j | \Omega _ { i } ^ { r _ { j } } , i ) p ( \Omega _ { i } ^ { r _ { j } } | i ) ,
+$$
+
+where $r _ { j }$ denotes the index of the support to which $j$ belongs. The last equation arises from the fact that the $\Omega _ { i } ^ { r }$ are disjoint from each other $( \Omega _ { i } ^ { r } \cap \Omega _ { i } ^ { s } = \emptyset , \forall r \neq s )$ . Therefore, there is only one support, $\Omega _ { i } ^ { r _ { j } }$ , containing $j$ . The remaining terms, where $j \notin \Omega _ { i } ^ { r }$ for $r \neq r _ { j }$ , are all zero since $p \left( j | \Omega _ { i } ^ { r } , i \right) = 0$
+
+Furthermore, assume $\Omega _ { i } ^ { r _ { j } }$ is a sufficient statistic, i.e., $j$ and $i$ are independent given $\Omega _ { i } ^ { r _ { j } }$ , we obtain
+
+$$
+p ( j | i ) = p ( j | \Omega _ { i } ^ { r _ { j } } ) p ( \Omega _ { i } ^ { r _ { j } } | i ) .
+$$
+
+Given the partition $\left\{ \Omega _ { i } ^ { r } \right\} _ { r = 0 } ^ { n _ { i } }$ , the attention form in (3) can be rewritten as
+
+$$
+\begin{array} { r c l } { A \left( \boldsymbol { x } _ { i } \right) } & { = } & { \mathbb { E } _ { p ( \boldsymbol { j } | i ) } \left[ \boldsymbol { v } _ { j } \right] = \displaystyle \sum _ { r = 0 } ^ { n _ { i } } \sum _ { j \in \Omega _ { i } ^ { r } } p \left( \boldsymbol { j } , \Omega _ { i } ^ { r } | i \right) \boldsymbol { v } _ { j } } \\ & { = } & { \displaystyle \sum _ { j \in \Omega _ { i } ^ { 0 } } \tilde { p } ( \boldsymbol { j } | i ) \boldsymbol { v } _ { j } + \sum _ { r = 1 } ^ { n _ { i } } p ( \Omega _ { i } ^ { r } | i ) \underbrace { \left( \sum _ { j \in \Omega _ { i } ^ { r } } p ( \boldsymbol { j } | \Omega _ { i } ^ { r } ) \boldsymbol { v } _ { j } \right) } _ { j \in \Omega _ { i } ^ { r } } , } \end{array}
+$$
+
+where we consider direct attention in partition $\Omega _ { i } ^ { 0 }$ and apply the local factorization (5) to the partition $r = 1 , \ldots , n _ { i }$ . Here $\tilde { p } ( j | i ) \propto p ( j | i )$ but with different normalization constants, which will be explained below. We refer to this model as Combiner since the structured attention (7) combines the direct expectation of $\Omega _ { i } ^ { 0 }$ and multiple local expectations via $p ( j | \Omega _ { i } ^ { r } )$ and $p ( \Omega _ { i } ^ { r } | i )$ to form the final conditional expectation.
+
+Equivalently, we can also rewrite the structured attention (7) as
+
+$$
+\begin{array} { r } { A ( x _ { i } ) = \sum _ { j \in \Omega _ { i } } \left[ \mathbb { I } ( j \in \Omega _ { i } ^ { 0 } ) \tilde { p } ( j | i ) + \displaystyle \sum _ { r = 1 } ^ { n _ { i } } \mathbb { I } ( j \in \Omega _ { i } ^ { r } ) { p } ( j | \Omega _ { i } ^ { r } ) { p } ( \Omega _ { i } ^ { r } | i ) \right] v _ { j } , } \end{array}
+$$
+
+where $\mathbb { I } ( \cdot )$ is a binary indicator function. After reordering, one can see from (8) that we obtain the effective conditional probability $q ( j | i )$ that tries to approximate the original $p ( j | i )$ . Each probability term depends on both current location $i$ and other location $j$ , and the expectation is still obtained with respect to a valid conditional probability (non-negative and sums up to 1 over $\Omega _ { i }$ ).
+
+Requirement for Sub-quadratic Cost. We can immediately see the benefit of this formulation from the fact that the local expectation in (7) is independent of the position $i$ . The full dependence is achieved via the multiplier $p ( \Omega _ { i } ^ { r } | i )$ where $j \in \Omega _ { i } ^ { r }$ . If we can design the local factorization such that:
+
+1. the order of number of terms in (7) for $p ( \cdot | i )$ $i ) , \forall i \in [ L ] \colon \sum _ { i = 1 } ^ { L } ( n _ { i } + | \Omega _ { i } ^ { 0 } | )$ is sub-quadratic; and 2. let $\mathcal { U } = \{ \Omega _ { i } ^ { r } \} _ { i \in [ L ] , r \in [ 1 , n _ { i } ] }$ be the unique set of partitions used for local expectation calculation, then the order of $| \mathcal { U } |$ (i.e., the number of unique partitions in $\mathcal { U }$ ) is sub-quadratic; 3. the order of total number of unique calculations of local expectation across all locations in (7), $\textstyle \sum _ { \Omega \in { \mathcal { U } } } | \Omega |$ is sub-quadratic;
+
+then one can see that the overall computation and memory cost will be sub-quadratic with full attention support $\Omega _ { i } ^ { \mathrm { C o m b i n e r } } = \Omega _ { i } , \forall i \in \overline { { [ L ] } }$ . We will discuss in detail in Section 4 how to instantiate such a principle by drawing inspiration from existing sparse transformers, and how to convert them into a full attention model almost for free with identical asymptotic complexity.
+
+Remark (Further Hierarchical Decomposition): We introduce the local decomposition with a one   
+layer partition of support of $p ( \cdot | i )$ for simplicity. In fact, such local decompositions can be stacked   
+furthesubset roduces a partition tre, and consider local dec Specificmposition $\Omega _ { i } ^ { r }$ $\left\{ \Omega _ { i } ^ { r k } \right\} _ { k = 1 } ^ { n _ { r } }$ $p ( j , \Omega _ { i } ^ { r } | i ) = p ( j | \Omega _ { i } ^ { r k _ { j } } , i ) p ( \Omega _ { i } ^ { r k _ { j } } | \Omega _ { i } ^ { r } , i ) p ( \Omega _ { i } ^ { r } | i )$ $k _ { j }$ $j$ of $p ( j | i )$ , which can also be plugged to (6) and yield a new full attention formulation.
+
+# 3.2 Parameterizing Conditional Probabilities
+
+While we obtained a possible way to speed up the standard Transformer via a combination of direct expectation and local expectations, it is also important to have an efficient design choice for the probability terms in (7), namely $\tilde { p } ( j | i )$ from direct expectation, $p ( j | \Omega _ { i } ^ { r } )$ from local expectation and $\bar { \boldsymbol { p } } ( \Omega _ { i } ^ { r } | i )$ for $r \in [ 1 , n _ { i } ]$ . For simplicity we use the scaled dot-product, which means that we will associate positions $i , j$ and variable sets $\Omega _ { i } ^ { r }$ with the corresponding embedding representation, and thus the probability is proportional to the exponential of the embedding inner products. Specifically:
+
+• $\tilde { p } ( j | i )$ : As this term is for the direct expectation, we can let $\begin{array} { r } { \tilde { p } ( j | i ) \propto \exp ( \frac { q _ { i } } { \sqrt { d } } k _ { j } ^ { \top } ) } \end{array}$ , which is the same as vanilla attention (3) but with different normalizations, which will be explained in Equation 9. • $p ( \Omega _ { i } ^ { r } | i )$ : This term aims to capture the joint event probability, i.e., $\begin{array} { r } { p ( \Omega _ { i } ^ { r } | i ) \propto \exp \left( \frac { q _ { i } } { \sqrt { d } } k _ { \Omega _ { i } ^ { r } } ^ { \top } \right) } \end{array}$ . Thus the design choice of $k _ { \Omega _ { i } ^ { r } }$ should make an abstraction of the corresponding support $\Omega _ { i } ^ { r }$ . We find $k _ { \Omega _ { i } ^ { r } } = \operatorname* { m a x } { \mathrm { p o o l i n g } _ { j \in \Omega _ { i } ^ { r } } k _ { j } }$ already provides good empirical results without introducing additional parameters; we can also use DeepSets [32] to obtain such abstraction. • $\bar { p } ( j | \Omega _ { i } ^ { r } )$ : This term is the probability of getting $j$ within this local span $\Omega _ { i } ^ { r }$ . We make $p ( j | \Omega _ { i } ^ { r } ) \propto$ $\exp \left( \frac { q _ { \Omega _ { i } ^ { r } } } { \sqrt { d } } k _ { j } ^ { \top } \right)$ , where we use max pooling or DeepSets over $\{ q _ { j } \} _ { j \in \Omega _ { i } ^ { r } }$ to obtain ${ { q } \ o \Omega _ { i } ^ { r } }$ similarly.
+
+Normalizing Probability Terms. The terms in each local expectation $p ( j | \Omega _ { i } ^ { r } )$ , $\forall j \in \Omega _ { i } ^ { r }$ can be normalized within the local span; the direct expectation $\tilde { p } ( j | i )$ and the terms in $p ( \Omega _ { i } ^ { r } | i )$ should be normalized together,
+
+$$
+Z ( x _ { i } ) = \sum _ { j \in \Omega _ { i } ^ { ( 0 ) } } \exp \left( \frac { q _ { i } } { \sqrt { d } } k _ { j } ^ { \top } \right) + \sum _ { r = 1 } ^ { n _ { i } } \exp \left( \frac { q _ { i } } { \sqrt { d } } k _ { \Omega _ { i } ^ { r } } ^ { \top } \right) ,
+$$
+
+and $Z ( x _ { i } )$ is the normalizing constant when calculating $\tilde { p } ( j | i )$ and $p ( \Omega _ { i } ^ { r } | i )$ .
+
+# 3.3 Trade-offs in Combiner
+
+Combiner achieves full attention with reduced cost without making explicit sparsity or low-rank assumptions over the attention matrix. However this efficiency gain is not free. In this section we discuss the limitations of the simplification made by Combiner, and provide a simple workaround.
+
+Structured Attention Approximation. We obtain the local decomposition (5) under the conditional independence assumption. Therefore, the local expectation in (7) is independent of the position $i$ , this suggests that any two locations $i _ { 1 }$ and $i _ { 2 }$ with $\Omega _ { i _ { 1 } } ^ { r } = \Omega _ { i _ { 2 } } ^ { r } = \Omega$ would have linearly dependent attention scores over the region $\Omega$ . Formally, the probabilities formed by the effective conditional distribution $\begin{array} { r } { \vec { a } ( \Omega ) _ { i _ { 1 } } = \left[ q ( j _ { 1 } | i _ { 1 } ) , q ( j _ { 2 } | i _ { 1 } ) , \dots , q ( j _ { | \Omega _ { i _ { 1 } } ^ { r } | } | i _ { 1 } ) \right] = \frac { p ( \Omega _ { i _ { 1 } } ^ { r } | i _ { 1 } ) } { p ( \Omega _ { i _ { 2 } } ^ { r } | i _ { 2 } ) } \vec { a } ( \Omega ) _ { i _ { 2 } } } \end{array}$ . In other words, the rank of the sub-matrix over the same partition in the resulting attention matrix is 1, therefore, the attention matrix is locally low-rank based on the partition. On the other hand, the direct expectation fully attends to each position in sub-support $\Omega _ { 0 }$ , which ensures the full-rank block. These two attention schemes make the attention matrix of Combiner structured. Compared with the low-rank approximation for attention [26, 28, 30], which is inspired from random features [29] in the kernel community, a structured approximation that exploits both the locally low-rank and full-rank blocks has been proved more powerful theoretically and empirically in large-scale kernel machines [27].
+
+Improving Expressiveness Using a Mixture Model. One way to further improve the expressiveness of the local factorization is to use a mixture model. This idea is adapted from the mixture of softmaxs [33] to obtain high-rank softmax layer in language modeling. Let $\omega$ be a certain partition of the support (i.e., collection of $\Omega _ { i } ^ { r }$ ) of $\Omega _ { i }$ , then one can easily use $\begin{array} { r } { A ( x _ { i } ) = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } A ( x _ { i } ; \omega _ { m } ) } \end{array}$ to compute the attention, where each component of the mixture $A ( x _ { i } ; \omega _ { m } )$ is the term (7) using a specific factorization plan $\omega _ { m }$ . Empirically we find two components are already sufficient to improve performance.
+
+# 4 Combiner Instantiations
+
+In this section we show several local factorization schemes satisfying the requirements in Section 3.1. As we will see, Combiner is able to convert several sparse transformers [14, 18, 20–22] into full
+
+attention, with the same order of computation and memory consumption. One can also design other factorization patterns, which can be easily instantiated in Combiner.
+
+# 4.1 Combiner-Fixed
+
+The Sparse Transformer [14] is one of the most representative variants that can achieve $\mathcal { O } ( L \sqrt { L } )$ computation and memory cost with sparse attention. Here we show how to convert this fixed pattern proposed in [14] (Figure 1(A)) into a factorization plan, and instantiate a full attention variant named the Combiner-Fixed (Figure 1(D)).
+
+In the fixed-sparse attention, the support is Ωsparse MLMi $\Omega _ { i } ^ { \mathrm { s p a r s e M L M } } = \{ j : j$ mod $s = 0 \} \cup \{ j : j \equiv i \left( \operatorname { d i v } s \right) \}$ where $s$ is a hyper-parameter, div is integer division, and $j \equiv i$ (div $s$ ) denotes that the quotients of $i$ and $j$ w.r.t. $s$ are the same. In the autoregressive case, $\Omega _ { i } ^ { \mathrm { s p a r s e L M } } = \Omega _ { i } ^ { \mathrm { s p a r s e M L M } } \cap [ i ]$ . Please refer to Figure 1(A) for an illustration of the LM version.
+
+Our design of $\omega _ { \mathrm { f i x e d } } ^ { \mathrm { M L M } }$ has the following form:
+
+$\Omega _ { i } ^ { 0 } = \left\{ j : j \equiv i \left( \mathrm { d i v } s \right) \right\} , \Omega _ { i } ^ { r } = \left\{ j : j \equiv \left( \mathrm { d i v } s \right) \right\} ,$ div $s = r , j \notin \Omega _ { i } ^ { 0 } \} , \forall r \in [ L \operatorname { d i v } s ] , \forall i \in [ L ]$ (10) where each local expectation is performed in each span of size $s$ , and there are totally $L$ div $s$ spans across all locations. For each position $i \in [ L ]$ , there are $( s + ( L \dim s ) )$ terms in (7) ; the local expectation has (√ $L$ div $s$ ) terms . The overall complexity is √ $\mathcal { O } ( L \cdot ( s + 2 ( L \operatorname { d i v } s ) ) _ { } ^ { }$ ). The optimal $s$ is $\mathcal { O } ( \sqrt { L } )$ , and we can achieve $\mathcal { O } ( L \sqrt { L } )$ computation and memory complexity, which is the same as [14] but here we gain full attention capability in each attention head. For the LM case, we can√ simply have $\omega _ { \mathrm { f i x e d } } ^ { \mathrm { L M } } : \{ \Omega _ { i } ^ { r } \cap [ i ] | \Omega _ { i } ^ { r } \in \omega _ { \mathrm { f i x e d } } ^ { \mathrm { M L M } } \}$ , which has the same $\mathcal { O } ( L \sqrt { L } )$ optimal complexity.
+
+# 4.2 Combiner-Logsparse
+
+The Logsparse Transformer is proposed in [18] and can theoretically achieve $\mathcal { O } ( L \log L )$ cost. The general idea is to make the size of support $\Omega _ { i } ^ { \mathrm { s p a r s e } }$ no larger than $\lceil \log _ { 2 } i \rceil$ . For the ease of notation, we first define $\mathbf { b i t s } ( n ) = [ b _ { 1 } , b _ { 2 } , \ldots , { \bar { b } } _ { \lceil \log _ { 2 } n \rceil } ]$ to be the binary representation of integer $n$ , with $b _ { t } \in \{ 0 , 1 \}$ the coefficient of basis $2 ^ { t }$ . Thus we have P⌈log2 n⌉t=1 bt ∗ 2t. One of the possible design choices to make Logsparse in the LM case is Ωsparse LMi $\Omega _ { i } ^ { \mathrm { s p a r s e L M } } = \left\{ \mathrm { s u f f } _ { t } : = \sum _ { \tau = t } ^ { \left\lceil \log _ { 2 } i - 1 \right\rceil } b _ { \tau } * 2 ^ { \tau } \right\} _ { t = 1 } ^ { \left\lceil \log _ { 2 } i - 1 \right\rceil } \cup$ $\{ i \}$ , i.e., attend to the location indices that equal to the suffix sum of the weighted bits $( i - 1 )$ , as well as location $i$ itself. This serves as our base sparse version as shown in Figure 1(B).
+
+To exploit this scheme in the Combiner framework, we can define $\lceil \log _ { 2 } n \rceil$ non-overlapping supports, where $\Omega _ { i } ^ { r } = [ \mathrm { s u f f } _ { r } ] \setminus [ \mathrm { s u f f } _ { r + 1 } ]$ with the boundary case $\big [ \mathrm { s u f f } _ { [ \log _ { 2 } i - 1 ] + 1 } \big ] = \emptyset$ . Note that for the ease of notation, some of the $\Omega _ { i } ^ { r }$ are empty which will be ignored. In this case, the direct attention set $\Omega _ { i } ^ { 0 }$ includes $\{ i \}$ , as well as $\{ i - 1 \}$ when $i$ is an even number. Such a factorization leads to Combiner-Logsparse, as shown in Figure 1(E). From the Figure, we observe that in total we will have span summaries for every $2 , 4 , 8 , \ldots , 2 ^ { \lfloor \log _ { 2 } L \rfloor }$ locations, resulting in total $\scriptstyle \sum _ { t = 1 } ^ { \lfloor \log _ { 2 } L \rfloor } \lfloor { \frac { L } { 2 ^ { t } } } \rfloor$ or $i$ will select at most non-overlapping spans to cover the full support $\Omega _ { i }$ , and thus, the total cost will be $\mathcal { O } \left( L \log L \right)$ . We leave the design of MLM case to Appendix B.
+
+# 4.3 Combiner-Axial
+
+The Axial Transformer [20] builds the attention along each axis of the input data. Without loss of generality, we focus on 2D case where the input sequence is reshaped into a matrix of size $n \times m = L$ . Specifically, the location $i$ in original sequence will be in $r o w _ { i } = ( i - 1 )$ div $m + 1$ and $c o l _ { i } = ( i - 1 )$ mod $m + 1$ . We show how to simply enable full attention with factorization on 2D matrix, hence Combiner-Axial.
+
+The sparse axial has Ωsparse MLM $\Omega _ { i } ^ { \mathrm { s p a r s e ~ M L M } } = \{ j : j - 1 \equiv i - 1 ( { \bmod { m } } ) \} \cup \{ j : j - 1 \equiv i - 1 ( \operatorname { d i v } m ) \}$ , and Ωsparse LMi $\Omega _ { i } ^ { \mathrm { s p a r s e L M } } = \Omega _ { i } ^ { \mathrm { s p a r s e M L M } } \cap [ i ]$ , which all have at most $O ( m + n )$ entries for each $i$ , as illustrated in Figure 1(C). We propose several factorization schemes to make it an attention with full support.
+
+$\omega _ { \mathrm { a x i a l - v e r t i c a l } } ^ { \mathrm { L M } }$ : $\Omega _ { i } ^ { 0 } = \Omega _ { i } ^ { \mathrm { s p a r s e L M } }$ , and corre $\Omega _ { i } ^ { r } = \left\{ j : j \equiv r ( { \bmod { m } } ) \right\} \cap \left[ i - c o l _ { i } \right]$ , for ere w $r \in [ m ] \setminus \{ c o l _ { i } \}$ . Asg to $\Omega _ { i } ^ { r }$ $r$ $r o w _ { i }$
+
+![](images/f21eec83f1fc7887399610c4e25456057ef79db8490d3c8ed0c89963791a36c5.jpg)  
+Figure 2: Attention matrices and sequence being attended (e.g., a $3 { \tt x } 4$ image) of vertical and horizontal variants of Combiner-Axial. Blue and yellow correspond to direct and local attention respectively for location $i$ (purple). Locations connected by arrows correspond to the same support $\Omega ^ { r }$ .
+
+obtain the abstraction. To obtain such abstraction for all the locations, we can leverage the cummax operator for each column to efficiently obtain the prefix-max.
+
+$\omega _ { \mathrm { a x i a l - h o r i z o n t a l } } ^ { \mathrm { L M } }$ : similar as $\omega _ { \mathrm { a x i a l } }$ -vertical except that each $\Omega _ { i } ^ { r }$ summarizes the row $r$ before $r o w _ { i }$ and   
+excludes $c o l _ { i }$ (Figure 2(B)).   
+• $\omega _ { \mathrm { a x i a l - r o w m a j o r } } ^ { \mathrm { L M } }$ : $\Omega _ { i } ^ { 0 } = \{ j : j - 1 \equiv i - 1 ( \operatorname { d i v } m ) \} \cap [ i ]$ , i.e., elements in the same row are directly   
+attended, while $\Omega _ { i } ^ { r } = \{ j : j \equiv r ( \operatorname { d i v } m ) \} \cap [ i - c o l _ { i } ]$ captures the rows before $r o w _ { i }$ . This structure is similar to Combiner-Fixed, except for the way that the abstraction (and thus the local expectation) is computed. Combiner-Fixed computes the abstraction only based on $r$ of partition $\Omega _ { i } ^ { r }$ , where   
+ωaxial-rowmajor depends on both $r$ and the column $c o l _ { i }$ (Figure 1(F)).
+
+In all cases above, the cost is similar to the Axial Transformer [20], which is $O ( L \sqrt { L } )$ if we reshape the sequence to a 2D matrix with $n , m = O ( { \sqrt { L } } )$ . We defer the MLM case to Appendix C.
+
+# 4.4 Combiner-Learnable
+
+Inspired by the Reformer [21] and Routing Transformer [22], we can also learn the factorization plan $\omega$ from the data. We illustrate this with Routing Transformer and provide a way to enable full attention in Routing Transformer following the Combiner principle.
+
+For a specific layer, suppose we have a learned disjoint region (or cluster in Routing Transformer) $\left\{ \Omega ^ { r } \right\} _ { r = 1 } ^ { n }$ where $\cup _ { r } \Omega ^ { r } = [ L ]$ . In Routing Transformer, we simply have $\Omega _ { i } ^ { \mathrm { s p a r s e M L M } } = \mathbf { \bar { \Omega } } ^ { r _ { i } }$ where $\Omega ^ { r _ { i } }$ denotes the region where position $i$ belongs to. To define the Combiner factorization, we let
+
+$$
+\begin{array} { r } { \omega _ { \mathrm { r o u t i n g ~ M L M } } : \Omega _ { i } ^ { 0 } = \Omega ^ { r _ { i } } , \quad \Omega _ { i } ^ { r } = \Omega ^ { r } \setminus \Omega _ { i } ^ { 0 } , \quad \forall r \in [ n _ { i } ] . } \end{array}
+$$
+
+Note that $n _ { i } = n$ (i.e., number of learned clusters) for all locations. The above factorization can only work for MLM. LM requires the following definition:
+
+$$
+\omega _ { \mathrm { r o u t i n g \ L M } } : \Omega _ { i } ^ { 0 } = \Omega ^ { r _ { i } } \cap [ i ] , \quad \Omega _ { i } ^ { r } = \left( \Omega ^ { r } \setminus \Omega _ { i } ^ { 0 } \right) \cap [ i ] , \quad \forall r \in [ n _ { i } ] .
+$$
+
+In general, both LM and MLM can have sub-quadratic cost when $n = O ( { \sqrt { L } } )$ . However, routing variants (including the Routing Transformer) require a gather operation, which can be slow on TPUs (see illustration in Appendix D).
+
+# 5 Experimental Evaluation
+
+We evaluate Combiner with different full attention patterns on both autoregressive and bidirectional sequence modeling tasks, covering a wide range of input data from images to texts. All tasks considered involve long sequences for up to 12,000 in length, some of which prevent the applicability of the vanilla transformer. We compare Combiner with state-of-the-art Transformers. We also perform a series of ablation studies where all of the models being compared use the exact same architecture that only differ in the attention module, avoiding individual tricks employed in the original works (e.g., using both learnable and fixed patterns in Routing Transformer [22]). Details to reproducing all experimental results can be found in Appendix E.
+
+Table 1: Ablation results in Bits per Dimension (Bits/Dim) on CIFAR-10 and ImageNet-64.   
+
+<table><tr><td>Model</td><td>Layers</td><td>CIFAR-10</td><td>ImageNet-64</td></tr><tr><td>Reformer [21]</td><td>6</td><td>1</td><td>3.740</td></tr><tr><td>Performer [28]</td><td>6</td><td>3.335</td><td>3.719</td></tr><tr><td>Logsparse [18]</td><td>6</td><td>4.253</td><td>4.351</td></tr><tr><td>Combiner-Logsparse (Ours)</td><td>6</td><td>3.366</td><td>3.795</td></tr><tr><td>Fixed [14]</td><td>6</td><td>3.408</td><td>3.696</td></tr><tr><td>Combiner-Fixed (Ours)</td><td>6</td><td>3.321</td><td>3.654</td></tr><tr><td>Axial [20]</td><td>6</td><td>3.666</td><td>4.032</td></tr><tr><td>Combiner-Axial (Ours)</td><td>6</td><td>3.050</td><td>3.585</td></tr><tr><td>Combiner-Mixture (Ours)</td><td>6</td><td>3.040</td><td>3.585</td></tr><tr><td>Reformer r[21]</td><td>12</td><td>1</td><td>3.710</td></tr><tr><td>Performer [28]</td><td>12</td><td>3.310</td><td>3.636</td></tr><tr><td>Routing Transformer [22]</td><td>12</td><td>2.950</td><td>1</td></tr><tr><td>Combiner-Mixture (Ours)</td><td>12</td><td>2.885</td><td>3.504</td></tr></table>
+
+# 5.1 Autoregressive Sequence Modeling
+
+In this subsection, we first perform density estimation on text and image using Combiner.
+
+# 5.1.1 Language Modeling
+
+For language modeling, we focus on the Wiki-40BEn dataset [34], which consists of clean Wikipedia pages in English. We use a sentence piece model with vocabulary size 32K to tokenize the text and measure the perplexity at the sentence piece level. To ensure fair comparison, all models being compared again have the same number of layers and hidden sizes, are are implemented under the same code base.
+
+Table 2 shows the results of the comparison. As we can see, under $2 \mathrm { k }$ sequence length, Combiner variants are consistently better than their corresponding baselines, and are very close to the standard Transformer. When sequence length goes to 8k, the standard Transformer runs out of memory, whereas Combiner continues to achieve improved perplexity, surpassing the result of Transformer- $2 \mathrm { k }$ . If we further use DeepSets to calculate the summarization terms ${ { q } \Omega _ { i } ^ { r } }$ and $k _ { \Omega _ { i } ^ { r } }$ , we may further achieve lower perplexity as shown in Table 3.
+
+Table 2: LM Perplexity on Wiki-40B (Main).   
+
+<table><tr><td>Model</td><td>Perplexity</td></tr><tr><td>Transformer-2k[1]</td><td>17.26</td></tr><tr><td>Performer-2k [28]</td><td>19.66</td></tr><tr><td>Routing-2k [22]</td><td>20.85</td></tr><tr><td>Fixed-2k [14] Combiner-Fixed-2k (Ours)</td><td>18.04 17.70</td></tr><tr><td>Axial-2k [20] Combiner-Axial-2k (Ours)</td><td>20.82</td></tr><tr><td>Combiner-Fixed-8k (Ours) Combiner-Axial-8k (Ours)</td><td>17.56 16.60</td></tr></table>
+
+Table 3: LM Perplexity on Wiki-40B (Ablation).   
+
+<table><tr><td>Model</td><td>Perplexity</td></tr><tr><td>Transformer-2k [1]</td><td>17.26</td></tr><tr><td>Combiner-DeepSets-Max-8k (Ours)</td><td>16.29</td></tr><tr><td>Combiner-DeepSets-Mean-8k(Ours)</td><td>16.48</td></tr><tr><td>Combiner-Max-8k(Ours)</td><td>16.60</td></tr><tr><td>Combiner-Mean-8k (Ours)</td><td>16.54</td></tr></table>
+
+# 5.1.2 Image Generative Models
+
+CIFAR-10. We first perform a sanity check where we compare sparse attention baselines against Combiner with full attention under the same architecture on the CIFAR-10 dataset. The sequence length is 3072. For all the methods, we use a same 6-layer transformer with 8 attention heads and 512 embedding dimensions. We train all models for $5 0 0 \mathrm { k }$ iterations using batch size 32 on TPU v2. As shown in Table 1, given the same model architecture, Combiner-X performs significantly better than the base model $\mathbf { X }$ under the bits per dimension (BPD) metric on the 10,000 test images. In particular, Combiner significantly decreases BPD by 0.887, 0.087, and 0.626 compared to the base models Logsparse, Fixed and Axial, respectively. Note that all of the Combiner variants achieve better performance than the best of the base models. This demonstrates the advantage of Combiner over the baselines given the same 6-layer architecture. We observe a similar trend under a 12-layer architecture.
+
+<table><tr><td>CIFAR-10</td><td>Bits/Dim</td></tr><tr><td>PixelCNN[15]</td><td>3.03</td></tr><tr><td>PixelCNN++ [36]</td><td>2.92</td></tr><tr><td>Image Transformer [16]</td><td>2.90</td></tr><tr><td>PixelSNAIL [37]</td><td>2.85</td></tr><tr><td>Sparse Transformer [14]</td><td>2.80</td></tr><tr><td>Combiner-Axial (ours)</td><td>2.77</td></tr></table>
+
+Table 4: Bits per Dimension (Bits/Dim) on CIFAR-10 and ImageNet-64.   
+
+<table><tr><td>ImageNet 64x64</td><td>Bits/Dim</td></tr><tr><td>PixelCNN[15]</td><td>3.57</td></tr><tr><td>Parallel Multiscale [38] Glow [39]</td><td>3.70 3.81</td></tr><tr><td>SPN [40]</td><td>3.52</td></tr><tr><td>Sparse Transformer[14]</td><td>3.44</td></tr><tr><td>Axial Transformer[20]</td><td>3.44</td></tr><tr><td>Routing Transformer [22] Combiner-Axial (ours)</td><td>3.43 3.42</td></tr></table>
+
+Following the 128-layer architecture in Child et al. [14], we apply Combiner-Axial and achieve state-of-the-art performance, 2.77 BPD on CIFAR-10, as listed in Table 4. We run all of the models in Table 4 without data augmentation [35].
+
+ImageNet-64. We also evaluate performance under the autoregressive setting on ImageNet-64, where sequence length is 12,288. We first perform the same analysis as CIFAR-10 and compare Combiner-X with the baselines using the same model architecture. As shown in Table 1, Combiner consistently outperforms the baselines with the same attention pattern. We further apply CombinerAxial to a 30-layer Transformer, which achieves state-of-the-art performance on density estimation on ImageNet-64, demonstrating the effectiveness of full attention achieved by Combiner.
+
+# 5.2 Bidirectional Sequence Modeling
+
+Besides autoregressive tasks, we also evaluate Combiner on a set of standard bidirectional tasks to show the general applicability of the method.
+
+# 5.2.1 Long-Range Arena
+
+Long-Range Arena (LRA) is a unified benchmark [31] for probing the capability of efficient transformers on handling long sequences. We evaluate our models on five tasks from LRA: ListOps, Text Classification, Retrieval, Image Classification and Pathfinder. All of the tasks are sequence-level multi-class classification. Please refer to the original LRA paper for more details.
+
+Table 5: Experimental results on Long-Range Arena benchmark.   
+
+<table><tr><td>Model</td><td>ListOps</td><td>Text</td><td>Retrieval</td><td>Image</td><td>Pathfinder</td><td>Avg</td></tr><tr><td>Chance Transformer Local Attention</td><td>10.00 36.38 15.95</td><td>50.00 64.27 52.98</td><td>50.00 57.46 53.39</td><td>10.00 42.44 41.46</td><td>50.00 88.81 84.64</td><td>34.00 57.87 49.68</td></tr><tr><td>Sparse TRans. Longformer Linformer Reformer Sinkhorn Trans. Synthesizer BigBird Linear Trans.</td><td>35.78 36.03 35.49 36.30 34.20 36.50 37.08 17.15</td><td>63.58 62.85 53.94 56.10 61.20 61.68 64.02 65.90</td><td>59.59 56.89 52.27 53.40 53.83 54.67 59.29 53.09</td><td>44.24 42.22 38.56 38.07 41.23 41.61 40.83 42.34</td><td>83.90 86.68 86.17 79.18 73.36 81.61 86.75 88.13</td><td>57.42 56.93 53.28 52.61 52.76 55.21 57.59 53.32</td></tr><tr><td>Performer Combiner-Fixed Combiner-Axial</td><td>36.00 36.65 36.15</td><td>65.40 64.99 64.36</td><td>53.82 59.81 56.10</td><td>42.77 41.67 41.33</td><td>88.76 88.59 88.43</td><td>57.35 58.34 57.27</td></tr></table>
+
+As shown in Table 5, Combiner is able to match the performance of vanilla Transformer and achieves even better performance in some tasks. Following the protocol of LRA, all methods use the same architecture and hyperparameters for a controllable comparison. We use the numbers from Tay et al. [31] for all tasks except for Pathfinder. Since we were unable to reproduce the original Pathfinder results using the default setup in LRA Github repository, we rerun all the baselines using Pathfinderinter configuration to conduct fair comparison. However, as the benchmark is still of small-scale and the LRA official website discourages hyperparameter tuning, Table 5 should be treated as results for the test bench of expressiveness compared to vanilla Transformer.
+
+Table 6: MLM perplexity on C4 dataset.   
+
+<table><tr><td>Model</td><td>Perplexity</td></tr><tr><td>Transformer-2k [1]</td><td>4.552</td></tr><tr><td>BigBird-2k [41]</td><td>4.696</td></tr><tr><td>Performer-2k [28]</td><td>10.940</td></tr><tr><td>Fixed-2k [14]</td><td>5.279</td></tr><tr><td>Combiner-Fixed-2k (Ours)</td><td>5.170</td></tr><tr><td>Axial-2k [20]</td><td>5.370</td></tr><tr><td>Combiner-Axial-2k (Ours)</td><td>4.809</td></tr><tr><td>Routing-2k [22] Combiner-Routing-2k (Ours)</td><td>6.703 6.539</td></tr><tr><td>BigBird-8k [41]</td><td></td></tr><tr><td></td><td>4.542</td></tr><tr><td>Combiner-Axial-8k(Ours)</td><td>4.190</td></tr><tr><td>Combiner-Fixed-8k (Ours)</td><td>4.139</td></tr></table>
+
+![](images/2a35b8f15325e3c385ba7b653ee59e16f7f468d8a1d936c355072ec94c8eca34.jpg)  
+Figure 3: We measure the inference runtime and memory usage for eight models. Overall Combiner has similar speed with Performer and its sparse counterpart but Vanilla Transformer quickly goes OOM when sequence length grows.
+
+# 5.2.2 Masked Language Modeling
+
+As the core element of BERT langauge pretraining [5], masked language modeling (MLM) refers to the task of reconstructing tokens that are randomly masked out in the input sequence. As with the LM task, we use perplexity as the main metric, which correlates relatively well with down-stream task performance. Specifically, we use the large scale C4 dataset [8] for training and evaluation, and consider different sequence lengths. Following the original BERT setup, we mask out $15 \%$ of the tokens in each input sequence. The comparison is summarized in Table 6. Similar to the LM result, different Combiner variants consistently outperform their corresponding baselines under $2 \mathrm { k }$ sequence length. However, apart from the standard Transformer, Combiner-2k also falls behind BigBird- $2 \mathrm { k }$ . We conjecture that this is related to the special design in BigBird such as all tokens can always attend to the $< \mathsf { c l s } >$ token directly, which is only applicable in non-causal problems. That said, when we further increase sequence length to 8k, the standard Transformer runs into OOM issue, whereas Combiner not only outperforms BigBird but also substantially surpasses Transformer-2k. This suggests that Combiner can truly benefit from scaling learning to longer sequence lengths.
+
+# 5.3 Runtime and Memory Usage of Combiner
+
+Here we evaluate the inference runtime and memory usage of five baselines – Transformer, Performer, BigBird, Sparse-Fixed and Sparse-Axial, as well as three variants of Combiner– Combiner-Fixed, Combiner-Axial and Combiner-Mixture. We run inference of all the models on a TPU v3-16 (16 cores $\mathrm { ~ x ~ } 1 6 6 \mathrm { B }$ ) with batch size 16, and we test sequences of length from $2 ^ { 1 0 }$ to $2 ^ { 1 4 }$ . As shown in Figure 3, Combiner instantiations achieve comparable runtime and memory usage with their sparse counterpart and Performer. Note Combiner achieves much better empirical performance than the sparse models and Performer. Combiner-Mixture has the same asymptotic complexity with Combiner-Fixed and Combiner-Axial, however, since it requires running two partition plans, it is slower than Combiner-Fixed and Combiner-Axial. Due to the gather operation required by the random attention which is not very TPU/GPU friendly, BigBird is very computationally expensive. And the Transformer model quickly runs out of memory when sequence length increases.
+
+# 6 Conclusion
+
+Inspired by the conditional expectation view of attention mechanism, we propose Combiner, a drop-in replacement of the attention module. By introducing structured decomposition to the conditional probability, Combiner achieves full attention capability while maintaining sub-quadratic computational and memory cost. We instantiate several Combiner variants converting existing sparse transformers to full attention. Combiner achieves state-of-the-art performance on both autoregressive and bidirectional tasks for image and text modeling, showing benefits in both modeling effectiveness and runtime efficiency. Future work includes additional factorization pattern designs, as well as applications of Combiner in domains like bioinformatics and speech.
+
+# Acknowledgments and Disclosure of Funding
+
+We would like to thank Richard Song and David Dohan for the help on introducing Performer codebase and experiment configurations, Yi Tay and Mostafa Dehghani for clarifications on the LRA benchmark, James Lee-Thorp, Joshua Ainslie, and Ilya Eckstein for clarification on their LRA experiment results, Adams Yu for performing internal paper review and helpful suggestions. We also gratefully acknowledge the support of DARPA under Nos. HR00112190039 (TAMI), N660011924033 (MCS); ARO under Nos. W911NF-16-1-0342 (MURI), W911NF-16-1-0171 (DURIP); NSF under Nos. OAC-1835598 (CINES), OAC-1934578 (HDR), CCF-1918940 (Expeditions), IIS-2030477 (RAPID), NIH under No. R56LM013365; Stanford Data Science Initiative, Wu Tsai Neurosciences Institute, Chan Zuckerberg Biohub, Amazon, JPMorgan Chase, Docomo, Hitachi, Intel, JD.com, KDDI, NVIDIA, Dell, Toshiba, Visa, and UnitedHealth Group. Hongyu Ren is supported by the Masason Foundation Fellowship and the Apple PhD Fellowship. Jure Leskovec is a Chan Zuckerberg Biohub investigator.
+
+# References
+
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+[2] Mia Xu Chen, Orhan Firat, Ankur Bapna, Melvin Johnson, Wolfgang Macherey, George Foster, Llion Jones, Niki Parmar, Mike Schuster, Zhifeng Chen, et al. The best of both worlds: Combining recent advances in neural machine translation. In Annual Meeting of the Association for Computational Linguistics (ACL), 2018.   
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parse/train/MQQeeDiO5vv/MQQeeDiO5vv_content_list.json

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+    {
+        "type": "text",
+        "text": "Combiner: Full Attention Transformer with Sparse Computation Cost ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "∗Hongyu Ren†, ∗Hanjun Dai⋄, ∗Zihang Dai⋄ Mengjiao Yang⋄, Jure Leskovec†, Dale Schuurmans⋄,‡, Bo Dai⋄ †Stanford University, {hyren,jure}@cs.stanford.edu ⋄Google Research, Brain Team, {hadai,zihangd,sherryy,schuurmans,bodai}@google.com ‡University of Alberta ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Transformers provide a class of expressive architectures that are extremely effective for sequence modeling. However, the key limitation of transformers is their quadratic memory and time complexity $\\mathcal { O } ( L ^ { 2 } )$ with respect to the sequence length in attention layers, which restricts application in extremely long sequences. Most existing approaches leverage sparsity or low-rank assumptions in the attention matrix to reduce cost, but sacrifice expressiveness. Instead, we propose Combiner, which provides full attention capability in each attention head while maintaining low computation and memory complexity. The key idea is to treat the self-attention mechanism as a conditional expectation over embeddings at each location, and approximate the conditional distribution with a structured factorization. Each location can attend to all other locations, either via direct attention, or through indirect attention to abstractions, which are again conditional expectations of embeddings from corresponding local regions. We show that most sparse attention patterns used in existing sparse transformers are able to inspire the design of such factorization for full attention, resulting in the same sub-quadratic cost $( { \\mathcal { O } } ( L \\log ( L ) )$ or $\\mathcal { O } ( L \\sqrt { L } ) )$ . Combiner is a drop-in replacement for attention layers in existing transformers and can be easily implemented in common frameworks. An experimental evaluation on both autoregressive and bidirectional sequence tasks demonstrates the effectiveness of this approach, yielding state-of-the-art results on several image and text modeling tasks. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
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+        "type": "text",
+        "text": "The Transformer [1] is a powerful neural network architecture that has demonstrated state-of-the-art performance in machine translation [2] and many other natural language processing (NLP) tasks via pretraining, using either unidirectional language modeling [3] or bidirectional language modeling [4–8]. It has also achieved excellent results in other domains like image recognition [9], code understanding [10], speech recognition [11], protein [12], music [13] and image [14] generative modeling. The core component of Transformer is the attention mechanism, which computes dependencies between all pairs of positions in a sequence. However, for a sequence of length $L$ , the expressiveness of pairwise attention comes at a quadratic cost $\\mathcal { O } ( L ^ { 2 } )$ in both time and memory consumption. This makes the vanilla Transformer [1] prohibitive for applications that involve long sequences, including high-resolution images, protein sequences, or raw speech signals [15], where the sequence length $L$ is often larger than 10, 000 [14]. ",
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+        "type": "text",
+        "text": "Recently, there have been several attempts to scale up attention to long sequences. A popular class of methods sparsifies the attention matrix with different sparsity patterns, including local window [16, 17], local+stride [14], log-sparse [18], axial [19, 20], or learnable patterns through hashing [21] or clustering [22]. Sparse attention enjoys sub-quadratic cost, but is lossy in capturing all-pair relationships. Generally, sparse attention requires more layers [14, 20, 23] to achieve full autoregressive or bidirectional dependencies (or receptive fields [20]) for each location in a long sequence. ",
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+    {
+        "type": "text",
+        "text": "Alternatively, another line of research has tried to achieve scalability with an explicit low-rank assumption [24, 25] on the attention matrix or by using explicit feature maps of some kernels [26]. However these explicit low dimensional approximations might be too restricted for the potentially full rank attention matrix, which uses exponential kernels that are effectively infinite dimensional [27]. The Performer [28] is among the first works that attempts to approximate regular full-rank attention with the random feature trick [29]. However such random-feature based approaches [30] require many more bases to better approximate the exponential kernel [27], and empirically we found it produces inferior results in some sequence modeling tasks, such as density estimation. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this paper we propose Combiner, a drop-in replacement for the vanilla quadratic attention mechanism with sub-quadratic computation and memory cost. Combiner still achieves full attention capability within each head of Multi-Head Attention, unlike approaches that adopt sparse or low-rank approximations. As we will discuss, the standard attention computed at each location can be seen as the conditional expectation of the value embeddings at all feasible locations given the current location. Based on such an understanding, Combiner explicitly approximates the conditional distribution in through a structured factorization of the probability space. Specifically, given a location $x$ , the probability of attending to location $y$ can be either directly calculated via the query vector of $x$ and key vector of $y$ , or indirectly through a local abstraction where $x$ first attends to the key vector that represents a group of locations containing $y$ , and multiplying the probability of choosing $y$ within that group. We refer to this model as Combiner since the conditional distributions in attention become a combination between several local attentions and direct attentions. This structured decomposition enables Combiner to take existing sparse attention patterns and convert them into corresponding design choices for probability factorizations that achieve full attention. As shown in Figure 1, Combiner achieves full attention with the same asymptotic complexity as sparse variants. Combiner can be easily implemented in most existing deep learning frameworks without the need for specialized hardware implementation, and is GPU/TPU friendly. In fact, both the fixed and learnable sparse attention patterns from many existing Transformer variants [14, 18, 20, 22] can be enhanced with such structured factorizations, with the same order of time or memory cost. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We validate Combiner on both autoregressive and bidirectional sequence modeling tasks over a variety of domains including text and images. We show that Combiner can achieve better perplexity and accuracy when using the same transformer architectures while being much faster in terms of runtime, and achieves state of the art performance on density estimation on standard datasets CIFAR-10 (2.77 bits/dim) and ImageNet-64 (3.42 bits/dim), as well as the Long-Range Arena [31]. The implementation of Combiner can be found at https://github.com/google-research/googleresearch/tree/master/combiner. ",
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+    {
+        "type": "text",
+        "text": "2 Attention as Conditional Expectation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In this section, we revisit the formulation of the standard Transformer [1] from the perspective of conditional expectation, which inspires the derivation of Combiner. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Without loss of generality, we use a single sequence in the self-attention scenario. Given a sequence of $L$ embeddings $X = \\bar { [ { x _ { 1 } } , { x _ { 2 } } , { . . . , \\bar { { x _ { L } } } } ] }$ , where $X \\in \\mathbb { R } ^ { L \\times d }$ and each embedding $x _ { i } \\in \\mathbb { R } ^ { d }$ is a $d$ -dimensional vector, the core component of Transformer is the multi-head attention, where each head $h$ is a scaled dot-product attention: ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/a009de902554041509ad676b8a8c81855a749d98054743239e43e1969557d83c.jpg",
+        "text": "$$\nA _ { h } ( X ) = \\operatorname { s o f } \\operatorname { t m a x } \\left( \\frac { Q _ { h } } { \\sqrt { d } } K _ { h } ^ { \\top } \\right) V _ { h } , \\left\\{ Q _ { h } = X W _ { h } ^ { Q } , K _ { h } = X W _ { h } ^ { K } , V _ { h } = X W _ { h } ^ { V } \\right\\} \\in \\mathbb { R } ^ { L \\times d } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "and the attention vector from each head $A _ { h } ( X )$ is concatenated and projected: ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/4d3d083eb3d1c216ede426625e297cd0607155f5989e74366807c62d19d7db79.jpg",
+        "text": "$$\n\\mathrm { i } \\mathrm { H e a d A t t u } ( X ) = [ A _ { 1 } ( X ) , A _ { 2 } ( X ) , \\ldots , A _ { H } ( X ) ] W ^ { o } , W ^ { o } \\in \\mathbb { R } ^ { H \\times d } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Mult (2) Here $H$ is the total number of heads per Transformer layer. In this paper, we focus on how to approximate full attention within each head of multi-head attention. For ease of notation, we drop the head index $h$ whenever possible, and use lower-case letters $x _ { i } , q _ { i } , k _ { i } , v _ { i } \\in \\mathbb { R } ^ { d }$ to denote rows in ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d44efb34f4d107361bc8899c32aa28458e1c5d06ab6ad955f245d4864635aec0.jpg",
+        "image_caption": [
+            "Figure 1: Attention matrices of several instantiations of Combiner in the autoregressive setting. We transform several sparse attention patterns: Fixed (A) [14], Logsparse (B) [18] and Axial (C) [20] to Combiner-Fixed (D), Combiner-Logsparse (E) and Combiner-Axial (F). Combiner approximates the conditional expectation (3) with a combination of direct expectation (blue) and local expectation (yellow). Our instantiations (D)(E)(F) achieves full attention with the same sub-quadratic complexity. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$X , Q , K , V$ respectively, which corresponds to a location $i$ in the original sequence of length $L$ . We use $[ n ]$ to denote the set of positive integers $\\{ 1 , 2 , \\ldots , n \\}$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a position $i \\in [ L ]$ , the attention formulation (1) can be viewed as conditional expectation of rows in $V$ . Specifically, since softmax outputs a probability distribution, we can rewrite (1) as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b00c825cc831f57f67f59204c99e0da57691eea75eda9f874e160b233ff5b8c2.jpg",
+        "text": "$$\nA ( x _ { i } ) = \\mathbb { E } _ { p ( j | i ) } \\left[ v _ { j } \\right] , \\qquad p ( j | i ) = \\frac { 1 } { Z \\left( x _ { i } \\right) } \\exp \\left( \\frac { q _ { i } } { \\sqrt { d } } k _ { j } ^ { \\top } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            489,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $p ( j | i )$ denotes the conditional probability at position $j$ given the token at position $i$ and the partition function $\\begin{array} { r } { Z \\left( x _ { i } \\right) = \\sum _ { j \\in \\Omega _ { i } } \\exp \\left( \\frac { q _ { i } } { \\sqrt { d } } k _ { j } ^ { \\top } \\right) } \\end{array}$ over support $\\Omega _ { i }$ . The support $\\Omega _ { i }$ of $p \\left( j | i \\right)$ defines the set of valid locations that the $i$ -th token can attend to. For instance, the support set in autoregressive language modeling (LM) consists of all previous tokens, i.e., $\\Omega _ { i } ^ { \\mathrm { L M } } = [ i ] ^ { 2 }$ ; in masked i language modeling (MLM) the support consists of all tokens in the sequence, i.e., $\\Omega _ { i } ^ { \\mathrm { M L M } } = [ L ]$ . That is, $\\bar { \\Omega } _ { i } ^ { \\mathrm { L M } }$ and $\\Omega _ { i } ^ { \\mathrm { M L M } }$ represent the full attention capability respectively in the LM and MLM setting. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Combiner: Full Attention via Structured Conditional Expectation ",
+        "text_level": 1,
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The complexity of $p \\left( j | i \\right)$ is the bottleneck of the computation for $A \\left( x _ { i } \\right)$ . Generally, in existing sparse transformers, the support of $p \\left( j | i \\right)$ is sparsified to reduce the computation and memory complexity, e.g., $\\Omega _ { i } ^ { \\mathrm { S p a r s e } } \\subsetneq \\Omega _ { i } ^ { \\mathrm { L M } }$ for LM and $\\Omega _ { i } ^ { \\mathrm { S p a r s e } } \\subsetneq \\Omega _ { i } ^ { \\mathrm { M L M } }$ for MLM, but this can lead to either reduced capacity or limited applicability. We defer detailed discussion of the full capacity of the model to Appendix A. In this section we introduce the Combiner, which achieves $\\Omega _ { i } ^ { \\mathrm { C o m b i n e r } } = \\Omega _ { i } ^ { \\mathrm { L M } }$ for LM and $\\Omega _ { i } ^ { \\mathrm { C o m b i n e r } } = \\Omega _ { i } ^ { \\mathrm { M L M } }$ for MLM, while still maintaining sub-quadratic computation and memory cost. Below we denote $\\Omega _ { i }$ as the support for full attention if there is no ambiguity or need to distinguish between LM or MLM. We introduce the main design framework in Section 3.1 and possible parameterizations in Section 3.2. Then in Section 3.3 we analyze the trade-off of Combiner. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Local Factorization for Conditional Expectation ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            818,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The main idea of Combiner is to exploit a hierarchical structure for conditional probability modeling in (3), which provides the opportunity for reducing computation complexity while maintaining the ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "same support. Specifically, we introduce support variables $\\Omega _ { i } ^ { r }$ , for $r = 0 , \\ldots , n _ { i }$ and $i \\in [ L ]$ . The support variables are disjoint, i.e., $\\Omega _ { i } ^ { r } \\cap \\Omega _ { i } ^ { s } = \\emptyset , \\forall r \\neq s$ , and $\\cup _ { r = 0 } ^ { n _ { i } } \\Omega _ { i } ^ { r } = \\Omega _ { i }$ . Then we can factorize $p ( j | i )$ as ",
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+        "type": "equation",
+        "img_path": "images/14731fe323d20043ea31ef10702788faae124e1d94215c40853cd8463d628929.jpg",
+        "text": "$$\np ( j | i ) = \\sum _ { r = 0 } ^ { n _ { i } } p ( j , \\Omega _ { i } ^ { r } | i ) = \\sum _ { r = 0 } ^ { n _ { i } } p ( j | \\Omega _ { i } ^ { r } , i ) p ( \\Omega _ { i } ^ { r } | i ) = p ( j | \\Omega _ { i } ^ { r _ { j } } , i ) p ( \\Omega _ { i } ^ { r _ { j } } | i ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $r _ { j }$ denotes the index of the support to which $j$ belongs. The last equation arises from the fact that the $\\Omega _ { i } ^ { r }$ are disjoint from each other $( \\Omega _ { i } ^ { r } \\cap \\Omega _ { i } ^ { s } = \\emptyset , \\forall r \\neq s )$ . Therefore, there is only one support, $\\Omega _ { i } ^ { r _ { j } }$ , containing $j$ . The remaining terms, where $j \\notin \\Omega _ { i } ^ { r }$ for $r \\neq r _ { j }$ , are all zero since $p \\left( j | \\Omega _ { i } ^ { r } , i \\right) = 0$ ",
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+        "type": "text",
+        "text": "Furthermore, assume $\\Omega _ { i } ^ { r _ { j } }$ is a sufficient statistic, i.e., $j$ and $i$ are independent given $\\Omega _ { i } ^ { r _ { j } }$ , we obtain ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d192d04a26e380033bdc1c19f9b03cbb190e8276dd4da40f4b1f384faddf68ef.jpg",
+        "text": "$$\np ( j | i ) = p ( j | \\Omega _ { i } ^ { r _ { j } } ) p ( \\Omega _ { i } ^ { r _ { j } } | i ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Given the partition $\\left\\{ \\Omega _ { i } ^ { r } \\right\\} _ { r = 0 } ^ { n _ { i } }$ , the attention form in (3) can be rewritten as ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/5420973101c6688cab715081384385d741bb267a8cd0f301cd068c3aac746119.jpg",
+        "text": "$$\n\\begin{array} { r c l } { A \\left( \\boldsymbol { x } _ { i } \\right) } & { = } & { \\mathbb { E } _ { p ( \\boldsymbol { j } | i ) } \\left[ \\boldsymbol { v } _ { j } \\right] = \\displaystyle \\sum _ { r = 0 } ^ { n _ { i } } \\sum _ { j \\in \\Omega _ { i } ^ { r } } p \\left( \\boldsymbol { j } , \\Omega _ { i } ^ { r } | i \\right) \\boldsymbol { v } _ { j } } \\\\ & { = } & { \\displaystyle \\sum _ { j \\in \\Omega _ { i } ^ { 0 } } \\tilde { p } ( \\boldsymbol { j } | i ) \\boldsymbol { v } _ { j } + \\sum _ { r = 1 } ^ { n _ { i } } p ( \\Omega _ { i } ^ { r } | i ) \\underbrace { \\left( \\sum _ { j \\in \\Omega _ { i } ^ { r } } p ( \\boldsymbol { j } | \\Omega _ { i } ^ { r } ) \\boldsymbol { v } _ { j } \\right) } _ { j \\in \\Omega _ { i } ^ { r } } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we consider direct attention in partition $\\Omega _ { i } ^ { 0 }$ and apply the local factorization (5) to the partition $r = 1 , \\ldots , n _ { i }$ . Here $\\tilde { p } ( j | i ) \\propto p ( j | i )$ but with different normalization constants, which will be explained below. We refer to this model as Combiner since the structured attention (7) combines the direct expectation of $\\Omega _ { i } ^ { 0 }$ and multiple local expectations via $p ( j | \\Omega _ { i } ^ { r } )$ and $p ( \\Omega _ { i } ^ { r } | i )$ to form the final conditional expectation. ",
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+    {
+        "type": "text",
+        "text": "Equivalently, we can also rewrite the structured attention (7) as ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/b345f5cf1117f455079051c230a2741542f05695183530876b0ae5d5e15e8526.jpg",
+        "text": "$$\n\\begin{array} { r } { A ( x _ { i } ) = \\sum _ { j \\in \\Omega _ { i } } \\left[ \\mathbb { I } ( j \\in \\Omega _ { i } ^ { 0 } ) \\tilde { p } ( j | i ) + \\displaystyle \\sum _ { r = 1 } ^ { n _ { i } } \\mathbb { I } ( j \\in \\Omega _ { i } ^ { r } ) { p } ( j | \\Omega _ { i } ^ { r } ) { p } ( \\Omega _ { i } ^ { r } | i ) \\right] v _ { j } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbb { I } ( \\cdot )$ is a binary indicator function. After reordering, one can see from (8) that we obtain the effective conditional probability $q ( j | i )$ that tries to approximate the original $p ( j | i )$ . Each probability term depends on both current location $i$ and other location $j$ , and the expectation is still obtained with respect to a valid conditional probability (non-negative and sums up to 1 over $\\Omega _ { i }$ ). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Requirement for Sub-quadratic Cost. We can immediately see the benefit of this formulation from the fact that the local expectation in (7) is independent of the position $i$ . The full dependence is achieved via the multiplier $p ( \\Omega _ { i } ^ { r } | i )$ where $j \\in \\Omega _ { i } ^ { r }$ . If we can design the local factorization such that: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1. the order of number of terms in (7) for $p ( \\cdot | i )$ $i ) , \\forall i \\in [ L ] \\colon \\sum _ { i = 1 } ^ { L } ( n _ { i } + | \\Omega _ { i } ^ { 0 } | )$ is sub-quadratic; and 2. let $\\mathcal { U } = \\{ \\Omega _ { i } ^ { r } \\} _ { i \\in [ L ] , r \\in [ 1 , n _ { i } ] }$ be the unique set of partitions used for local expectation calculation, then the order of $| \\mathcal { U } |$ (i.e., the number of unique partitions in $\\mathcal { U }$ ) is sub-quadratic; 3. the order of total number of unique calculations of local expectation across all locations in (7), $\\textstyle \\sum _ { \\Omega \\in { \\mathcal { U } } } | \\Omega |$ is sub-quadratic; ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "then one can see that the overall computation and memory cost will be sub-quadratic with full attention support $\\Omega _ { i } ^ { \\mathrm { C o m b i n e r } } = \\Omega _ { i } , \\forall i \\in \\overline { { [ L ] } }$ . We will discuss in detail in Section 4 how to instantiate such a principle by drawing inspiration from existing sparse transformers, and how to convert them into a full attention model almost for free with identical asymptotic complexity. ",
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+    {
+        "type": "text",
+        "text": "Remark (Further Hierarchical Decomposition): We introduce the local decomposition with a one   \nlayer partition of support of $p ( \\cdot | i )$ for simplicity. In fact, such local decompositions can be stacked   \nfurthesubset roduces a partition tre, and consider local dec Specificmposition $\\Omega _ { i } ^ { r }$ $\\left\\{ \\Omega _ { i } ^ { r k } \\right\\} _ { k = 1 } ^ { n _ { r } }$ $p ( j , \\Omega _ { i } ^ { r } | i ) = p ( j | \\Omega _ { i } ^ { r k _ { j } } , i ) p ( \\Omega _ { i } ^ { r k _ { j } } | \\Omega _ { i } ^ { r } , i ) p ( \\Omega _ { i } ^ { r } | i )$ $k _ { j }$ $j$ of $p ( j | i )$ , which can also be plugged to (6) and yield a new full attention formulation. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "3.2 Parameterizing Conditional Probabilities ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While we obtained a possible way to speed up the standard Transformer via a combination of direct expectation and local expectations, it is also important to have an efficient design choice for the probability terms in (7), namely $\\tilde { p } ( j | i )$ from direct expectation, $p ( j | \\Omega _ { i } ^ { r } )$ from local expectation and $\\bar { \\boldsymbol { p } } ( \\Omega _ { i } ^ { r } | i )$ for $r \\in [ 1 , n _ { i } ]$ . For simplicity we use the scaled dot-product, which means that we will associate positions $i , j$ and variable sets $\\Omega _ { i } ^ { r }$ with the corresponding embedding representation, and thus the probability is proportional to the exponential of the embedding inner products. Specifically: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• $\\tilde { p } ( j | i )$ : As this term is for the direct expectation, we can let $\\begin{array} { r } { \\tilde { p } ( j | i ) \\propto \\exp ( \\frac { q _ { i } } { \\sqrt { d } } k _ { j } ^ { \\top } ) } \\end{array}$ , which is the same as vanilla attention (3) but with different normalizations, which will be explained in Equation 9. • $p ( \\Omega _ { i } ^ { r } | i )$ : This term aims to capture the joint event probability, i.e., $\\begin{array} { r } { p ( \\Omega _ { i } ^ { r } | i ) \\propto \\exp \\left( \\frac { q _ { i } } { \\sqrt { d } } k _ { \\Omega _ { i } ^ { r } } ^ { \\top } \\right) } \\end{array}$ . Thus the design choice of $k _ { \\Omega _ { i } ^ { r } }$ should make an abstraction of the corresponding support $\\Omega _ { i } ^ { r }$ . We find $k _ { \\Omega _ { i } ^ { r } } = \\operatorname* { m a x } { \\mathrm { p o o l i n g } _ { j \\in \\Omega _ { i } ^ { r } } k _ { j } }$ already provides good empirical results without introducing additional parameters; we can also use DeepSets [32] to obtain such abstraction. • $\\bar { p } ( j | \\Omega _ { i } ^ { r } )$ : This term is the probability of getting $j$ within this local span $\\Omega _ { i } ^ { r }$ . We make $p ( j | \\Omega _ { i } ^ { r } ) \\propto$ $\\exp \\left( \\frac { q _ { \\Omega _ { i } ^ { r } } } { \\sqrt { d } } k _ { j } ^ { \\top } \\right)$ , where we use max pooling or DeepSets over $\\{ q _ { j } \\} _ { j \\in \\Omega _ { i } ^ { r } }$ to obtain ${ { q } \\ o \\Omega _ { i } ^ { r } }$ similarly. ",
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+    {
+        "type": "text",
+        "text": "Normalizing Probability Terms. The terms in each local expectation $p ( j | \\Omega _ { i } ^ { r } )$ , $\\forall j \\in \\Omega _ { i } ^ { r }$ can be normalized within the local span; the direct expectation $\\tilde { p } ( j | i )$ and the terms in $p ( \\Omega _ { i } ^ { r } | i )$ should be normalized together, ",
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+    {
+        "type": "equation",
+        "img_path": "images/477bd1b2d1419f6b5ab8a2bc431a6b1d60782aa1c0549c53476208807e82647d.jpg",
+        "text": "$$\nZ ( x _ { i } ) = \\sum _ { j \\in \\Omega _ { i } ^ { ( 0 ) } } \\exp \\left( \\frac { q _ { i } } { \\sqrt { d } } k _ { j } ^ { \\top } \\right) + \\sum _ { r = 1 } ^ { n _ { i } } \\exp \\left( \\frac { q _ { i } } { \\sqrt { d } } k _ { \\Omega _ { i } ^ { r } } ^ { \\top } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "and $Z ( x _ { i } )$ is the normalizing constant when calculating $\\tilde { p } ( j | i )$ and $p ( \\Omega _ { i } ^ { r } | i )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 Trade-offs in Combiner ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Combiner achieves full attention with reduced cost without making explicit sparsity or low-rank assumptions over the attention matrix. However this efficiency gain is not free. In this section we discuss the limitations of the simplification made by Combiner, and provide a simple workaround. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Structured Attention Approximation. We obtain the local decomposition (5) under the conditional independence assumption. Therefore, the local expectation in (7) is independent of the position $i$ , this suggests that any two locations $i _ { 1 }$ and $i _ { 2 }$ with $\\Omega _ { i _ { 1 } } ^ { r } = \\Omega _ { i _ { 2 } } ^ { r } = \\Omega$ would have linearly dependent attention scores over the region $\\Omega$ . Formally, the probabilities formed by the effective conditional distribution $\\begin{array} { r } { \\vec { a } ( \\Omega ) _ { i _ { 1 } } = \\left[ q ( j _ { 1 } | i _ { 1 } ) , q ( j _ { 2 } | i _ { 1 } ) , \\dots , q ( j _ { | \\Omega _ { i _ { 1 } } ^ { r } | } | i _ { 1 } ) \\right] = \\frac { p ( \\Omega _ { i _ { 1 } } ^ { r } | i _ { 1 } ) } { p ( \\Omega _ { i _ { 2 } } ^ { r } | i _ { 2 } ) } \\vec { a } ( \\Omega ) _ { i _ { 2 } } } \\end{array}$ . In other words, the rank of the sub-matrix over the same partition in the resulting attention matrix is 1, therefore, the attention matrix is locally low-rank based on the partition. On the other hand, the direct expectation fully attends to each position in sub-support $\\Omega _ { 0 }$ , which ensures the full-rank block. These two attention schemes make the attention matrix of Combiner structured. Compared with the low-rank approximation for attention [26, 28, 30], which is inspired from random features [29] in the kernel community, a structured approximation that exploits both the locally low-rank and full-rank blocks has been proved more powerful theoretically and empirically in large-scale kernel machines [27]. ",
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+    {
+        "type": "text",
+        "text": "Improving Expressiveness Using a Mixture Model. One way to further improve the expressiveness of the local factorization is to use a mixture model. This idea is adapted from the mixture of softmaxs [33] to obtain high-rank softmax layer in language modeling. Let $\\omega$ be a certain partition of the support (i.e., collection of $\\Omega _ { i } ^ { r }$ ) of $\\Omega _ { i }$ , then one can easily use $\\begin{array} { r } { A ( x _ { i } ) = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } A ( x _ { i } ; \\omega _ { m } ) } \\end{array}$ to compute the attention, where each component of the mixture $A ( x _ { i } ; \\omega _ { m } )$ is the term (7) using a specific factorization plan $\\omega _ { m }$ . Empirically we find two components are already sufficient to improve performance. ",
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+    {
+        "type": "text",
+        "text": "4 Combiner Instantiations ",
+        "text_level": 1,
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "In this section we show several local factorization schemes satisfying the requirements in Section 3.1. As we will see, Combiner is able to convert several sparse transformers [14, 18, 20–22] into full ",
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+    {
+        "type": "text",
+        "text": "attention, with the same order of computation and memory consumption. One can also design other factorization patterns, which can be easily instantiated in Combiner. ",
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+    {
+        "type": "text",
+        "text": "4.1 Combiner-Fixed ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The Sparse Transformer [14] is one of the most representative variants that can achieve $\\mathcal { O } ( L \\sqrt { L } )$ computation and memory cost with sparse attention. Here we show how to convert this fixed pattern proposed in [14] (Figure 1(A)) into a factorization plan, and instantiate a full attention variant named the Combiner-Fixed (Figure 1(D)). ",
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+    {
+        "type": "text",
+        "text": "In the fixed-sparse attention, the support is Ωsparse MLMi $\\Omega _ { i } ^ { \\mathrm { s p a r s e M L M } } = \\{ j : j$ mod $s = 0 \\} \\cup \\{ j : j \\equiv i \\left( \\operatorname { d i v } s \\right) \\}$ where $s$ is a hyper-parameter, div is integer division, and $j \\equiv i$ (div $s$ ) denotes that the quotients of $i$ and $j$ w.r.t. $s$ are the same. In the autoregressive case, $\\Omega _ { i } ^ { \\mathrm { s p a r s e L M } } = \\Omega _ { i } ^ { \\mathrm { s p a r s e M L M } } \\cap [ i ]$ . Please refer to Figure 1(A) for an illustration of the LM version. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our design of $\\omega _ { \\mathrm { f i x e d } } ^ { \\mathrm { M L M } }$ has the following form: ",
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+    },
+    {
+        "type": "text",
+        "text": "$\\Omega _ { i } ^ { 0 } = \\left\\{ j : j \\equiv i \\left( \\mathrm { d i v } s \\right) \\right\\} , \\Omega _ { i } ^ { r } = \\left\\{ j : j \\equiv \\left( \\mathrm { d i v } s \\right) \\right\\} ,$ div $s = r , j \\notin \\Omega _ { i } ^ { 0 } \\} , \\forall r \\in [ L \\operatorname { d i v } s ] , \\forall i \\in [ L ]$ (10) where each local expectation is performed in each span of size $s$ , and there are totally $L$ div $s$ spans across all locations. For each position $i \\in [ L ]$ , there are $( s + ( L \\dim s ) )$ terms in (7) ; the local expectation has (√ $L$ div $s$ ) terms . The overall complexity is √ $\\mathcal { O } ( L \\cdot ( s + 2 ( L \\operatorname { d i v } s ) ) _ { } ^ { }$ ). The optimal $s$ is $\\mathcal { O } ( \\sqrt { L } )$ , and we can achieve $\\mathcal { O } ( L \\sqrt { L } )$ computation and memory complexity, which is the same as [14] but here we gain full attention capability in each attention head. For the LM case, we can√ simply have $\\omega _ { \\mathrm { f i x e d } } ^ { \\mathrm { L M } } : \\{ \\Omega _ { i } ^ { r } \\cap [ i ] | \\Omega _ { i } ^ { r } \\in \\omega _ { \\mathrm { f i x e d } } ^ { \\mathrm { M L M } } \\}$ , which has the same $\\mathcal { O } ( L \\sqrt { L } )$ optimal complexity. ",
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+    {
+        "type": "text",
+        "text": "4.2 Combiner-Logsparse ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The Logsparse Transformer is proposed in [18] and can theoretically achieve $\\mathcal { O } ( L \\log L )$ cost. The general idea is to make the size of support $\\Omega _ { i } ^ { \\mathrm { s p a r s e } }$ no larger than $\\lceil \\log _ { 2 } i \\rceil$ . For the ease of notation, we first define $\\mathbf { b i t s } ( n ) = [ b _ { 1 } , b _ { 2 } , \\ldots , { \\bar { b } } _ { \\lceil \\log _ { 2 } n \\rceil } ]$ to be the binary representation of integer $n$ , with $b _ { t } \\in \\{ 0 , 1 \\}$ the coefficient of basis $2 ^ { t }$ . Thus we have P⌈log2 n⌉t=1 bt ∗ 2t. One of the possible design choices to make Logsparse in the LM case is Ωsparse LMi $\\Omega _ { i } ^ { \\mathrm { s p a r s e L M } } = \\left\\{ \\mathrm { s u f f } _ { t } : = \\sum _ { \\tau = t } ^ { \\left\\lceil \\log _ { 2 } i - 1 \\right\\rceil } b _ { \\tau } * 2 ^ { \\tau } \\right\\} _ { t = 1 } ^ { \\left\\lceil \\log _ { 2 } i - 1 \\right\\rceil } \\cup$ $\\{ i \\}$ , i.e., attend to the location indices that equal to the suffix sum of the weighted bits $( i - 1 )$ , as well as location $i$ itself. This serves as our base sparse version as shown in Figure 1(B). ",
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+    {
+        "type": "text",
+        "text": "To exploit this scheme in the Combiner framework, we can define $\\lceil \\log _ { 2 } n \\rceil$ non-overlapping supports, where $\\Omega _ { i } ^ { r } = [ \\mathrm { s u f f } _ { r } ] \\setminus [ \\mathrm { s u f f } _ { r + 1 } ]$ with the boundary case $\\big [ \\mathrm { s u f f } _ { [ \\log _ { 2 } i - 1 ] + 1 } \\big ] = \\emptyset$ . Note that for the ease of notation, some of the $\\Omega _ { i } ^ { r }$ are empty which will be ignored. In this case, the direct attention set $\\Omega _ { i } ^ { 0 }$ includes $\\{ i \\}$ , as well as $\\{ i - 1 \\}$ when $i$ is an even number. Such a factorization leads to Combiner-Logsparse, as shown in Figure 1(E). From the Figure, we observe that in total we will have span summaries for every $2 , 4 , 8 , \\ldots , 2 ^ { \\lfloor \\log _ { 2 } L \\rfloor }$ locations, resulting in total $\\scriptstyle \\sum _ { t = 1 } ^ { \\lfloor \\log _ { 2 } L \\rfloor } \\lfloor { \\frac { L } { 2 ^ { t } } } \\rfloor$ or $i$ will select at most non-overlapping spans to cover the full support $\\Omega _ { i }$ , and thus, the total cost will be $\\mathcal { O } \\left( L \\log L \\right)$ . We leave the design of MLM case to Appendix B. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 Combiner-Axial ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The Axial Transformer [20] builds the attention along each axis of the input data. Without loss of generality, we focus on 2D case where the input sequence is reshaped into a matrix of size $n \\times m = L$ . Specifically, the location $i$ in original sequence will be in $r o w _ { i } = ( i - 1 )$ div $m + 1$ and $c o l _ { i } = ( i - 1 )$ mod $m + 1$ . We show how to simply enable full attention with factorization on 2D matrix, hence Combiner-Axial. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The sparse axial has Ωsparse MLM $\\Omega _ { i } ^ { \\mathrm { s p a r s e ~ M L M } } = \\{ j : j - 1 \\equiv i - 1 ( { \\bmod { m } } ) \\} \\cup \\{ j : j - 1 \\equiv i - 1 ( \\operatorname { d i v } m ) \\}$ , and Ωsparse LMi $\\Omega _ { i } ^ { \\mathrm { s p a r s e L M } } = \\Omega _ { i } ^ { \\mathrm { s p a r s e M L M } } \\cap [ i ]$ , which all have at most $O ( m + n )$ entries for each $i$ , as illustrated in Figure 1(C). We propose several factorization schemes to make it an attention with full support. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "$\\omega _ { \\mathrm { a x i a l - v e r t i c a l } } ^ { \\mathrm { L M } }$ : $\\Omega _ { i } ^ { 0 } = \\Omega _ { i } ^ { \\mathrm { s p a r s e L M } }$ , and corre $\\Omega _ { i } ^ { r } = \\left\\{ j : j \\equiv r ( { \\bmod { m } } ) \\right\\} \\cap \\left[ i - c o l _ { i } \\right]$ , for ere w $r \\in [ m ] \\setminus \\{ c o l _ { i } \\}$ . Asg to $\\Omega _ { i } ^ { r }$ $r$ $r o w _ { i }$ ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/f21eec83f1fc7887399610c4e25456057ef79db8490d3c8ed0c89963791a36c5.jpg",
+        "image_caption": [
+            "Figure 2: Attention matrices and sequence being attended (e.g., a $3 { \\tt x } 4$ image) of vertical and horizontal variants of Combiner-Axial. Blue and yellow correspond to direct and local attention respectively for location $i$ (purple). Locations connected by arrows correspond to the same support $\\Omega ^ { r }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            209,
+            87,
+            787,
+            258
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "obtain the abstraction. To obtain such abstraction for all the locations, we can leverage the cummax operator for each column to efficiently obtain the prefix-max. ",
+        "bbox": [
+            183,
+            315,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "$\\omega _ { \\mathrm { a x i a l - h o r i z o n t a l } } ^ { \\mathrm { L M } }$ : similar as $\\omega _ { \\mathrm { a x i a l } }$ -vertical except that each $\\Omega _ { i } ^ { r }$ summarizes the row $r$ before $r o w _ { i }$ and   \nexcludes $c o l _ { i }$ (Figure 2(B)).   \n• $\\omega _ { \\mathrm { a x i a l - r o w m a j o r } } ^ { \\mathrm { L M } }$ : $\\Omega _ { i } ^ { 0 } = \\{ j : j - 1 \\equiv i - 1 ( \\operatorname { d i v } m ) \\} \\cap [ i ]$ , i.e., elements in the same row are directly   \nattended, while $\\Omega _ { i } ^ { r } = \\{ j : j \\equiv r ( \\operatorname { d i v } m ) \\} \\cap [ i - c o l _ { i } ]$ captures the rows before $r o w _ { i }$ . This structure is similar to Combiner-Fixed, except for the way that the abstraction (and thus the local expectation) is computed. Combiner-Fixed computes the abstraction only based on $r$ of partition $\\Omega _ { i } ^ { r }$ , where   \nωaxial-rowmajor depends on both $r$ and the column $c o l _ { i }$ (Figure 1(F)). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In all cases above, the cost is similar to the Axial Transformer [20], which is $O ( L \\sqrt { L } )$ if we reshape the sequence to a 2D matrix with $n , m = O ( { \\sqrt { L } } )$ . We defer the MLM case to Appendix C. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.4 Combiner-Learnable ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Inspired by the Reformer [21] and Routing Transformer [22], we can also learn the factorization plan $\\omega$ from the data. We illustrate this with Routing Transformer and provide a way to enable full attention in Routing Transformer following the Combiner principle. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For a specific layer, suppose we have a learned disjoint region (or cluster in Routing Transformer) $\\left\\{ \\Omega ^ { r } \\right\\} _ { r = 1 } ^ { n }$ where $\\cup _ { r } \\Omega ^ { r } = [ L ]$ . In Routing Transformer, we simply have $\\Omega _ { i } ^ { \\mathrm { s p a r s e M L M } } = \\mathbf { \\bar { \\Omega } } ^ { r _ { i } }$ where $\\Omega ^ { r _ { i } }$ denotes the region where position $i$ belongs to. To define the Combiner factorization, we let ",
+        "bbox": [
+            174,
+            593,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a39cc45c05c70368040dcaec6558d875a01c40ff0bb40fa0ef8d8d4c35ec2972.jpg",
+        "text": "$$\n\\begin{array} { r } { \\omega _ { \\mathrm { r o u t i n g ~ M L M } } : \\Omega _ { i } ^ { 0 } = \\Omega ^ { r _ { i } } , \\quad \\Omega _ { i } ^ { r } = \\Omega ^ { r } \\setminus \\Omega _ { i } ^ { 0 } , \\quad \\forall r \\in [ n _ { i } ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that $n _ { i } = n$ (i.e., number of learned clusters) for all locations. The above factorization can only work for MLM. LM requires the following definition: ",
+        "bbox": [
+            173,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/16c72f70ab0b05492a2ba8389db918e3c50dee4ab7d83c20b2d746b28ee6ddd8.jpg",
+        "text": "$$\n\\omega _ { \\mathrm { r o u t i n g \\ L M } } : \\Omega _ { i } ^ { 0 } = \\Omega ^ { r _ { i } } \\cap [ i ] , \\quad \\Omega _ { i } ^ { r } = \\left( \\Omega ^ { r } \\setminus \\Omega _ { i } ^ { 0 } \\right) \\cap [ i ] , \\quad \\forall r \\in [ n _ { i } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In general, both LM and MLM can have sub-quadratic cost when $n = O ( { \\sqrt { L } } )$ . However, routing variants (including the Routing Transformer) require a gather operation, which can be slow on TPUs (see illustration in Appendix D). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experimental Evaluation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            767,
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+            786
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We evaluate Combiner with different full attention patterns on both autoregressive and bidirectional sequence modeling tasks, covering a wide range of input data from images to texts. All tasks considered involve long sequences for up to 12,000 in length, some of which prevent the applicability of the vanilla transformer. We compare Combiner with state-of-the-art Transformers. We also perform a series of ablation studies where all of the models being compared use the exact same architecture that only differ in the attention module, avoiding individual tricks employed in the original works (e.g., using both learnable and fixed patterns in Routing Transformer [22]). Details to reproducing all experimental results can be found in Appendix E. ",
+        "bbox": [
+            173,
+            800,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1bf0e8e10cb19b323f3ca715fe7ef9bd20970818bc7d290a4326be8d23302dc1.jpg",
+        "table_caption": [
+            "Table 1: Ablation results in Bits per Dimension (Bits/Dim) on CIFAR-10 and ImageNet-64. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Layers</td><td>CIFAR-10</td><td>ImageNet-64</td></tr><tr><td>Reformer [21]</td><td>6</td><td>1</td><td>3.740</td></tr><tr><td>Performer [28]</td><td>6</td><td>3.335</td><td>3.719</td></tr><tr><td>Logsparse [18]</td><td>6</td><td>4.253</td><td>4.351</td></tr><tr><td>Combiner-Logsparse (Ours)</td><td>6</td><td>3.366</td><td>3.795</td></tr><tr><td>Fixed [14]</td><td>6</td><td>3.408</td><td>3.696</td></tr><tr><td>Combiner-Fixed (Ours)</td><td>6</td><td>3.321</td><td>3.654</td></tr><tr><td>Axial [20]</td><td>6</td><td>3.666</td><td>4.032</td></tr><tr><td>Combiner-Axial (Ours)</td><td>6</td><td>3.050</td><td>3.585</td></tr><tr><td>Combiner-Mixture (Ours)</td><td>6</td><td>3.040</td><td>3.585</td></tr><tr><td>Reformer r[21]</td><td>12</td><td>1</td><td>3.710</td></tr><tr><td>Performer [28]</td><td>12</td><td>3.310</td><td>3.636</td></tr><tr><td>Routing Transformer [22]</td><td>12</td><td>2.950</td><td>1</td></tr><tr><td>Combiner-Mixture (Ours)</td><td>12</td><td>2.885</td><td>3.504</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1 Autoregressive Sequence Modeling ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            334,
+            455,
+            349
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this subsection, we first perform density estimation on text and image using Combiner. ",
+        "bbox": [
+            173,
+            363,
+            759,
+            377
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1.1 Language Modeling ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            400,
+            364,
+            415
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For language modeling, we focus on the Wiki-40BEn dataset [34], which consists of clean Wikipedia pages in English. We use a sentence piece model with vocabulary size 32K to tokenize the text and measure the perplexity at the sentence piece level. To ensure fair comparison, all models being compared again have the same number of layers and hidden sizes, are are implemented under the same code base. ",
+        "bbox": [
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+            426,
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+            551
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2 shows the results of the comparison. As we can see, under $2 \\mathrm { k }$ sequence length, Combiner variants are consistently better than their corresponding baselines, and are very close to the standard Transformer. When sequence length goes to 8k, the standard Transformer runs out of memory, whereas Combiner continues to achieve improved perplexity, surpassing the result of Transformer- $2 \\mathrm { k }$ . If we further use DeepSets to calculate the summarization terms ${ { q } \\Omega _ { i } ^ { r } }$ and $k _ { \\Omega _ { i } ^ { r } }$ , we may further achieve lower perplexity as shown in Table 3. ",
+        "bbox": [
+            174,
+            558,
+            503,
+            709
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/54b6db801c17327fc8192edcbe5486948f37b2c33b1a3e7058e72d5fc17169d3.jpg",
+        "table_caption": [
+            "Table 2: LM Perplexity on Wiki-40B (Main). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Perplexity</td></tr><tr><td>Transformer-2k[1]</td><td>17.26</td></tr><tr><td>Performer-2k [28]</td><td>19.66</td></tr><tr><td>Routing-2k [22]</td><td>20.85</td></tr><tr><td>Fixed-2k [14] Combiner-Fixed-2k (Ours)</td><td>18.04 17.70</td></tr><tr><td>Axial-2k [20] Combiner-Axial-2k (Ours)</td><td>20.82</td></tr><tr><td>Combiner-Fixed-8k (Ours) Combiner-Axial-8k (Ours)</td><td>17.56 16.60</td></tr></table>",
+        "bbox": [
+            519,
+            417,
+            818,
+            594
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/a4272b3334e67916579f0179b84edb96f0ac0dd47bed6c69c54c416d38a19fb9.jpg",
+        "table_caption": [
+            "Table 3: LM Perplexity on Wiki-40B (Ablation). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Perplexity</td></tr><tr><td>Transformer-2k [1]</td><td>17.26</td></tr><tr><td>Combiner-DeepSets-Max-8k (Ours)</td><td>16.29</td></tr><tr><td>Combiner-DeepSets-Mean-8k(Ours)</td><td>16.48</td></tr><tr><td>Combiner-Max-8k(Ours)</td><td>16.60</td></tr><tr><td>Combiner-Mean-8k (Ours)</td><td>16.54</td></tr></table>",
+        "bbox": [
+            519,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1.2 Image Generative Models ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            732,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "CIFAR-10. We first perform a sanity check where we compare sparse attention baselines against Combiner with full attention under the same architecture on the CIFAR-10 dataset. The sequence length is 3072. For all the methods, we use a same 6-layer transformer with 8 attention heads and 512 embedding dimensions. We train all models for $5 0 0 \\mathrm { k }$ iterations using batch size 32 on TPU v2. As shown in Table 1, given the same model architecture, Combiner-X performs significantly better than the base model $\\mathbf { X }$ under the bits per dimension (BPD) metric on the 10,000 test images. In particular, Combiner significantly decreases BPD by 0.887, 0.087, and 0.626 compared to the base models Logsparse, Fixed and Axial, respectively. Note that all of the Combiner variants achieve better performance than the best of the base models. This demonstrates the advantage of Combiner over the baselines given the same 6-layer architecture. We observe a similar trend under a 12-layer architecture. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/76a30a18bdeedf67f4562f5e7b0b5c55792438076f3820a0d126ddbeecb9bf07.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>CIFAR-10</td><td>Bits/Dim</td></tr><tr><td>PixelCNN[15]</td><td>3.03</td></tr><tr><td>PixelCNN++ [36]</td><td>2.92</td></tr><tr><td>Image Transformer [16]</td><td>2.90</td></tr><tr><td>PixelSNAIL [37]</td><td>2.85</td></tr><tr><td>Sparse Transformer [14]</td><td>2.80</td></tr><tr><td>Combiner-Axial (ours)</td><td>2.77</td></tr></table>",
+        "bbox": [
+            205,
+            121,
+            465,
+            234
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/5455f51ffc1a5ae14c03f3df131468bd96a9bb99dae30a5687425dace723457e.jpg",
+        "table_caption": [
+            "Table 4: Bits per Dimension (Bits/Dim) on CIFAR-10 and ImageNet-64. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>ImageNet 64x64</td><td>Bits/Dim</td></tr><tr><td>PixelCNN[15]</td><td>3.57</td></tr><tr><td>Parallel Multiscale [38] Glow [39]</td><td>3.70 3.81</td></tr><tr><td>SPN [40]</td><td>3.52</td></tr><tr><td>Sparse Transformer[14]</td><td>3.44</td></tr><tr><td>Axial Transformer[20]</td><td>3.44</td></tr><tr><td>Routing Transformer [22] Combiner-Axial (ours)</td><td>3.43 3.42</td></tr></table>",
+        "bbox": [
+            527,
+            112,
+            789,
+            246
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Following the 128-layer architecture in Child et al. [14], we apply Combiner-Axial and achieve state-of-the-art performance, 2.77 BPD on CIFAR-10, as listed in Table 4. We run all of the models in Table 4 without data augmentation [35]. ",
+        "bbox": [
+            176,
+            251,
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+            294
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ImageNet-64. We also evaluate performance under the autoregressive setting on ImageNet-64, where sequence length is 12,288. We first perform the same analysis as CIFAR-10 and compare Combiner-X with the baselines using the same model architecture. As shown in Table 1, Combiner consistently outperforms the baselines with the same attention pattern. We further apply CombinerAxial to a 30-layer Transformer, which achieves state-of-the-art performance on density estimation on ImageNet-64, demonstrating the effectiveness of full attention achieved by Combiner. ",
+        "bbox": [
+            173,
+            300,
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+            383
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2 Bidirectional Sequence Modeling ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            401,
+            442,
+            416
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Besides autoregressive tasks, we also evaluate Combiner on a set of standard bidirectional tasks to show the general applicability of the method. ",
+        "bbox": [
+            178,
+            428,
+            823,
+            457
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.2.1 Long-Range Arena ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            473,
+            357,
+            488
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Long-Range Arena (LRA) is a unified benchmark [31] for probing the capability of efficient transformers on handling long sequences. We evaluate our models on five tasks from LRA: ListOps, Text Classification, Retrieval, Image Classification and Pathfinder. All of the tasks are sequence-level multi-class classification. Please refer to the original LRA paper for more details. ",
+        "bbox": [
+            174,
+            497,
+            825,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/816223c0034a49c1bc9e8b70080a1cc343e69afb0852e3370bdf14ea22e176e9.jpg",
+        "table_caption": [
+            "Table 5: Experimental results on Long-Range Arena benchmark. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>ListOps</td><td>Text</td><td>Retrieval</td><td>Image</td><td>Pathfinder</td><td>Avg</td></tr><tr><td>Chance Transformer Local Attention</td><td>10.00 36.38 15.95</td><td>50.00 64.27 52.98</td><td>50.00 57.46 53.39</td><td>10.00 42.44 41.46</td><td>50.00 88.81 84.64</td><td>34.00 57.87 49.68</td></tr><tr><td>Sparse TRans. Longformer Linformer Reformer Sinkhorn Trans. Synthesizer BigBird Linear Trans.</td><td>35.78 36.03 35.49 36.30 34.20 36.50 37.08 17.15</td><td>63.58 62.85 53.94 56.10 61.20 61.68 64.02 65.90</td><td>59.59 56.89 52.27 53.40 53.83 54.67 59.29 53.09</td><td>44.24 42.22 38.56 38.07 41.23 41.61 40.83 42.34</td><td>83.90 86.68 86.17 79.18 73.36 81.61 86.75 88.13</td><td>57.42 56.93 53.28 52.61 52.76 55.21 57.59 53.32</td></tr><tr><td>Performer Combiner-Fixed Combiner-Axial</td><td>36.00 36.65 36.15</td><td>65.40 64.99 64.36</td><td>53.82 59.81 56.10</td><td>42.77 41.67 41.33</td><td>88.76 88.59 88.43</td><td>57.35 58.34 57.27</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "As shown in Table 5, Combiner is able to match the performance of vanilla Transformer and achieves even better performance in some tasks. Following the protocol of LRA, all methods use the same architecture and hyperparameters for a controllable comparison. We use the numbers from Tay et al. [31] for all tasks except for Pathfinder. Since we were unable to reproduce the original Pathfinder results using the default setup in LRA Github repository, we rerun all the baselines using Pathfinderinter configuration to conduct fair comparison. However, as the benchmark is still of small-scale and the LRA official website discourages hyperparameter tuning, Table 5 should be treated as results for the test bench of expressiveness compared to vanilla Transformer. ",
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+    {
+        "type": "table",
+        "img_path": "images/bd3d5f73584ac4bc85e8d75f657d388e767927177eb9189e4d315f9d6fb6e021.jpg",
+        "table_caption": [
+            "Table 6: MLM perplexity on C4 dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Perplexity</td></tr><tr><td>Transformer-2k [1]</td><td>4.552</td></tr><tr><td>BigBird-2k [41]</td><td>4.696</td></tr><tr><td>Performer-2k [28]</td><td>10.940</td></tr><tr><td>Fixed-2k [14]</td><td>5.279</td></tr><tr><td>Combiner-Fixed-2k (Ours)</td><td>5.170</td></tr><tr><td>Axial-2k [20]</td><td>5.370</td></tr><tr><td>Combiner-Axial-2k (Ours)</td><td>4.809</td></tr><tr><td>Routing-2k [22] Combiner-Routing-2k (Ours)</td><td>6.703 6.539</td></tr><tr><td>BigBird-8k [41]</td><td></td></tr><tr><td></td><td>4.542</td></tr><tr><td>Combiner-Axial-8k(Ours)</td><td>4.190</td></tr><tr><td>Combiner-Fixed-8k (Ours)</td><td>4.139</td></tr></table>",
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+    {
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+        "image_caption": [
+            "Figure 3: We measure the inference runtime and memory usage for eight models. Overall Combiner has similar speed with Performer and its sparse counterpart but Vanilla Transformer quickly goes OOM when sequence length grows. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.2.2 Masked Language Modeling ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "As the core element of BERT langauge pretraining [5], masked language modeling (MLM) refers to the task of reconstructing tokens that are randomly masked out in the input sequence. As with the LM task, we use perplexity as the main metric, which correlates relatively well with down-stream task performance. Specifically, we use the large scale C4 dataset [8] for training and evaluation, and consider different sequence lengths. Following the original BERT setup, we mask out $15 \\%$ of the tokens in each input sequence. The comparison is summarized in Table 6. Similar to the LM result, different Combiner variants consistently outperform their corresponding baselines under $2 \\mathrm { k }$ sequence length. However, apart from the standard Transformer, Combiner-2k also falls behind BigBird- $2 \\mathrm { k }$ . We conjecture that this is related to the special design in BigBird such as all tokens can always attend to the $< \\mathsf { c l s } >$ token directly, which is only applicable in non-causal problems. That said, when we further increase sequence length to 8k, the standard Transformer runs into OOM issue, whereas Combiner not only outperforms BigBird but also substantially surpasses Transformer-2k. This suggests that Combiner can truly benefit from scaling learning to longer sequence lengths. ",
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+    {
+        "type": "text",
+        "text": "5.3 Runtime and Memory Usage of Combiner ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Here we evaluate the inference runtime and memory usage of five baselines – Transformer, Performer, BigBird, Sparse-Fixed and Sparse-Axial, as well as three variants of Combiner– Combiner-Fixed, Combiner-Axial and Combiner-Mixture. We run inference of all the models on a TPU v3-16 (16 cores $\\mathrm { ~ x ~ } 1 6 6 \\mathrm { B }$ ) with batch size 16, and we test sequences of length from $2 ^ { 1 0 }$ to $2 ^ { 1 4 }$ . As shown in Figure 3, Combiner instantiations achieve comparable runtime and memory usage with their sparse counterpart and Performer. Note Combiner achieves much better empirical performance than the sparse models and Performer. Combiner-Mixture has the same asymptotic complexity with Combiner-Fixed and Combiner-Axial, however, since it requires running two partition plans, it is slower than Combiner-Fixed and Combiner-Axial. Due to the gather operation required by the random attention which is not very TPU/GPU friendly, BigBird is very computationally expensive. And the Transformer model quickly runs out of memory when sequence length increases. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Inspired by the conditional expectation view of attention mechanism, we propose Combiner, a drop-in replacement of the attention module. By introducing structured decomposition to the conditional probability, Combiner achieves full attention capability while maintaining sub-quadratic computational and memory cost. We instantiate several Combiner variants converting existing sparse transformers to full attention. Combiner achieves state-of-the-art performance on both autoregressive and bidirectional tasks for image and text modeling, showing benefits in both modeling effectiveness and runtime efficiency. Future work includes additional factorization pattern designs, as well as applications of Combiner in domains like bioinformatics and speech. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We would like to thank Richard Song and David Dohan for the help on introducing Performer codebase and experiment configurations, Yi Tay and Mostafa Dehghani for clarifications on the LRA benchmark, James Lee-Thorp, Joshua Ainslie, and Ilya Eckstein for clarification on their LRA experiment results, Adams Yu for performing internal paper review and helpful suggestions. We also gratefully acknowledge the support of DARPA under Nos. HR00112190039 (TAMI), N660011924033 (MCS); ARO under Nos. W911NF-16-1-0342 (MURI), W911NF-16-1-0171 (DURIP); NSF under Nos. OAC-1835598 (CINES), OAC-1934578 (HDR), CCF-1918940 (Expeditions), IIS-2030477 (RAPID), NIH under No. R56LM013365; Stanford Data Science Initiative, Wu Tsai Neurosciences Institute, Chan Zuckerberg Biohub, Amazon, JPMorgan Chase, Docomo, Hitachi, Intel, JD.com, KDDI, NVIDIA, Dell, Toshiba, Visa, and UnitedHealth Group. Hongyu Ren is supported by the Masason Foundation Fellowship and the Apple PhD Fellowship. Jure Leskovec is a Chan Zuckerberg Biohub investigator. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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