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+# RETHINKING NETWORK DESIGN AND LOCAL GEOMETRY IN POINT CLOUD: A SIMPLE RESIDUAL MLP FRAMEWORK
+
+Xu $\mathbf { M a } ^ { 1 }$ , Can ${ \bf { Q } i n } ^ { 1 }$ , Haoxuan $\mathbf { Y o u } ^ { 2 }$ , Haoxi Ran1, Yun Fu1 1Northeastern University, Boston, MA, USA 2Columbia University, New York, NY, USA {ma.xu1,qin.ca,ran.h}@northeastern.edu {haoxuanyou,ranhaoxi}@gmail.com yunfu@ece.neu.edu
+
+# ABSTRACT
+
+Point cloud analysis is challenging due to irregularity and unordered data structure. To capture the 3D geometries, prior works mainly rely on exploring sophisticated local geometric extractors using convolution, graph, or attention mechanisms. These methods, however, incur unfavorable latency during inference, and the performance saturates over the past few years. In this paper, we present a novel perspective on this task. We notice that detailed local geometrical information probably is not the key to point cloud analysis – we introduce a pure residual MLP network, called PointMLP, which integrates no “sophisticated” local geometrical extractors but still performs very competitively. Equipped with a proposed lightweight geometric affine module, PointMLP delivers the new stateof-the-art on multiple datasets. On the real-world ScanObjectNN dataset, our method even surpasses the prior best method by $3 . 3 \%$ accuracy. We emphasize that PointMLP achieves this strong performance without any sophisticated operations, hence leading to a superior inference speed. Compared to most recent CurveNet, PointMLP trains $2 \times$ faster, tests $7 \times$ faster, and is more accurate on ModelNet40 benchmark. We hope our PointMLP may help the community towards a better understanding of point cloud analysis. The code is available at https://github.com/ma-xu/pointMLP-pytorch.
+
+# 1 INTRODUCTION
+
+Lately, point cloud analysis has emerged as a popular topic in 3D understanding, attracting attention from academia and industry (Qi et al., 2017a; Shi et al., 2019; Xu et al., 2020). Different from 2D images represented by regular dense pixels, point clouds are composed of unordered and irregular sets of points $\mathcal { P } \in \bar { \mathbb { R } ^ { N \times \tilde { 3 } } }$ , making it infeasible to apply image processing methods to point cloud analysis directly. Meanwhile, the nature of sparseness and the presence of noises further restrict the performance. In the past few years, endowing with neural networks, point cloud analysis has seen a great improvement in various applications, including 3D shape classification (Qi et al., 2017a), semantic segmentation (Hu et al., 2020) and object detection (Shi & Rajkumar, 2020), etc.
+
+Recent efforts have shown promising results for point cloud analysis by exploring local geometric information, using convolution (Li et al., 2021a), graph (Li et al., 2021a), or attention mechanism (Guo et al., 2021) (see Section 2 for details). These methods, despite their gratifying results, have mainly relied on the premise that an elaborate local extractor is essential for point cloud analysis, leading to the competition for careful designs that explore fine local geometric properties. Nevertheless, sophisticated extractors are not without drawbacks. On the one hand, due to prohibitive computations and the overhead of memory access, these sophisticated extractors hamper the efficiency of applications in natural scenes. As an example, until now, most 3D point cloud applications are still based on the simple PointNet ( and PointNet++) or the voxel-based methods (Liu et al., 2021; Li et al., 2021b; Zhang et al., 2021). However, applications that employ the aforementioned advanced methods are rare in literature. On the other hand, the booming sophisticated extractors saturate the performance since they already describe the local geometric properties well. A more complicated design is no longer to improve the performance further. These phenomena suggest that we may need to stop the race of local feature extraction designing, rethinking the necessity of elaborate local feature extractors and further revisiting the succinct design philosophy in point cloud analysis.
+
+In this paper, we aim at the ambitious goal of building a deep network for point cloud analysis using only residual feed-forward MLPs, without any delicate local feature explorations. By doing so, we eschew the prohibitive computations and continued memory access caused by the sophisticated local geometric extractors and enjoy the advantage of efficiency from the highly-optimized MLPs. To further improve the performance and generalization ability, We introduce a lightweight local geometric affine module that adaptively transforms the point feature in a local region. We term our new network architecture as PointMLP. In the sense of MLPbased design philosophy, our PointMLP is similar to PointNet and PointNet+ $^ +$ (Qi et al., 2017a;b). However, our model is more generic and exhibits promising performance. Different from the models with sophisticated local geometric extractors (e.g., DeepGCNs (Li et al., 2019), RSCNN (Liu et al.,
+
+![](images/635ed58352f670fed8bd190e81fdebc6d88479acd157fa0f7e28451bbf4cf84e.jpg)  
+Figure 1: Accuracy-speed tradeoff on ModelNet40. Our PointMLP performs best. Please refer to Section 4 for details.
+
+2019b), RPNet (Ran et al., 2021).), our PointMLP is conceptually simpler and achieves results on par or even better than these state-of-the-art methods (see Figure 1). Keep in mind that we did not challenge the advantages of these local geometric extractors and we acknowledge their contributions; however, a more succinct framework should be studied considering both the efficiency and accuracy. In Table 1, we systemically compare our PointMLP with some representative methods.
+
+Even though the design philosophy is simple, PointMLP (as well as the elite version) exhibits superior performance on 3D point cloud analysis. Specifically, we achieve the state-of-the-art classification performance, $9 4 . 5 \%$ , on the ModelNet40 benchmark, and we outperform related works by $3 . 3 \%$ accuracy on the real-world ScanObjectNN dataset, with a significantly higher inference speed.
+
+# 2 RELATED WORK
+
+Point cloud analysis. There are mainly two streams to process point cloud. Since the point cloud data structure is irregular and unordered, some works consider projecting the original point clouds to intermediate voxels (Maturana & Scherer, 2015; Shi et al., 2020) or images (You et al., 2018; Li et al., 2020), translating the challenging 3D task into a well-explored 2D image problem. In this regime, point clouds understanding is largely boosted and enjoys the fast processing speed from 2D images or voxels. Albeit efficient, information loss caused by projection degrades the representational quality of details for point clouds (Yang et al., 2019). To this end, some methods are proposed to process the original point cloud sets directly. PointNet (Qi et al., 2017a) is a pioneering work that directly consumes unordered point sets as inputs using shared MLPs. Based on PointNet, PointNet+ $^ { - + }$ (Qi et al., 2017b) further introduced a hierarchical feature learning paradigm to capture the local geometric structures recursively. Owing to the local point representation (and multi-scale information), PointNet $^ { + + }$ exhibits promising results and has been the cornerstone of modern point cloud methods (Wang et al., 2019; Fan et al., 2021; Xu et al., 2021a). Our PointMLP also follows the design philosophy of PointNet++ but explores a simpler yet much deeper network architecture.
+
+Local geometry exploration. As PointNet+ $^ +$ built the generic point cloud analysis network framework, the recent research focus is shifted to how to generate better regional points representation. Predominantly, the explorations of local points representation can be divided into three categories: convolution-, graph-, and attention-based methods. One of the most distinguished convolution-based methods is PointConv (Wu et al., 2019). By approximating continuous weight and density functions in convolutional filters using an MLP, PointConv is able to extend the dynamic filter to a new convolution operation. Also, PAConv (Xu et al., 2021a) constructs the convolution kernel by dynamically assembling basic weight matrices stored in a weight bank. Without modifying network configurations, PAConv can be seamlessly integrated into classical MLP-based pipelines. Unlike convolution-based methods, Graph-based methods investigate mutually correlated relationships among points with a graph. In Wang et al. (2019), an EdgeConv is proposed to generate edge features that describe the relationships between a point and its neighbors. By doing so, a local graph is built, and the point relationships are well preserved. In 3D-GCN (Lin et al., 2021), authors aim at deriving deformable 3D kernels using a 3D Graph Convolution Network. Closely related to graphbased methods, the attention-based methods exhibit excellent ability on relationship exploration as well, like PCT (Guo et al., 2021) and Point Transformer (Zhao et al., 2021; Engel et al., 2020). With the development of local geometry exploration, the performances on various tasks appear to be saturated. Continuing on this track would bring minimal improvements. In this paper, we showcase that even without the carefully designed operations for local geometry exploration, a pure deep hierarchical MLP architecture is able to exhibit gratifying performances and even better results.
+
+Deep network architecture for point cloud. Interestingly, the development of point cloud analysis is closely related to the evolution of the image processing network. In the early era, works in the image processing field simply stack several learning layers to probe the performance limitations (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; Dong et al., 2014). Then, the great success of deep learning was significantly promoted by deep neural architectures like ResNet (He et al., 2016), which brings a profound impact to various research fields. Recently, attention-based models, including atten
+
+Table 1: Systematic comparison among some representative methods. “Deep” indicates that a model is expandable along depth. “Opt.” stands for the principal operator.   
+
+<table><tr><td>Method</td><td>hierarchy</td><td>locality</td><td>deep</td><td>opt.</td></tr><tr><td>PointNet</td><td>x／</td><td>X</td><td></td><td>MLP</td></tr><tr><td>PointNet++</td><td></td><td></td><td></td><td>MLP</td></tr><tr><td>DGCNN</td><td>X</td><td></td><td></td><td>GCN</td></tr><tr><td>DeepGCNs</td><td></td><td></td><td>xxx/</td><td>GCN</td></tr><tr><td>PointConv</td><td></td><td></td><td>X</td><td>Conv.</td></tr><tr><td>Point Trans.</td><td></td><td></td><td></td><td>Atten.</td></tr><tr><td>PointMLP</td><td></td><td></td><td></td><td>MLP</td></tr></table>
+
+tion blocks (Wang et al., 2018) and Transformer architectures (Dosovitskiy et al., 2021), further flesh out the community. Most recently, the succinct deep MLP architectures have attracted a lot of attention due to their efficiency and generality. Point cloud analysis follows the same develop history as well, from MLP-based PointNet (Qi et al., 2017a), deep hierarchical PointNet+ $^ +$ (Qi et al., 2017b), convolution-/graph-/relation- based methods (Wu et al., 2019; Wang et al., 2019; Ran et al., 2021), to state-of-the-art Transformer-based models (Guo et al., 2021; Zhao et al., 2021). In this paper, we abandon sophisticated details and present a simple yet effective deep residual MLP network for point cloud analysis. Instead of following the tendency in the vision community deliberately, we are in pursuit of an inherently simple and empirically powerful architecture for point cloud analysis.
+
+# 3 DEEP RESIDUAL MLP FOR POINT CLOUD
+
+We propose to learn the point cloud representation by a simple feed-forward residual MLP network (named PointMLP), which hierarchically aggregates the local features extracted by MLPs, and abandons the use of delicate local geometric extractors. To further improve the robustness and improve the performance, we also introduce a lightweight geometric affine module to transform the local points to a normal distribution. The detailed framework of our method is illustrated in Figure 2.
+
+# 3.1 REVISITING POINT-BASED METHODS
+
+The design of point-based methods for point cloud analysis dates back to the PointNet and Point$\mathrm { N e t } { + + }$ papers (Qi et al., 2017a;b), if not earlier. The motivation behind this direction is to directly consume point clouds from the beginning and avoid unnecessary rendering processes.
+
+Given a set of points $\mathcal P = \{ p _ { i } | i = 1 , \cdots , N \} \in \mathbb { R } ^ { N \times 3 }$ , where $N$ indicates the number of points in a $( x , y , z )$ Cartesian space, point-based methods aims to directly learn the underlying representation $f$ of $\mathcal { P }$ using neural networks. One of the most pioneering works is PointNet $^ { + + }$ , which learns hierarchical features by stacking multiple learning stages. In each stage, $N _ { s }$ points are re-sampled by the farthest point sampling (FPS) algorithm where $s$ indexes the stage and $K$ neighbors are employed for each sampled point and aggregated by max-pooling to capture local structures. Conceptually, the kernel operation of PointNet++ can be formulated as:
+
+![](images/295efa483234887acc5f296fe8f2504323ab06afe3fc5017c4d57094223bdd15.jpg)  
+Figure 2: Overview of one stage in PointMLP. Given an input point cloud, PointMLP progressively extracts local features using residual point MLP blocks. In each stage, we first transform the local points using a geometric affine module, then they are extracted before and after the aggregation operation, respectively. PointMLP progressively enlarges the receptive field and models complete point cloud geometric information by repeating multiple stages.
+
+$$
+g _ { i } = \mathcal { A } \left( \Phi \left( f _ { i , j } \right) \middle | j = 1 , \cdot \cdot \ , K \right) ,
+$$
+
+where $\boldsymbol { \mathcal { A } } \left( \cdot \right)$ means aggregation function (max-pooling in PointNet++), $\Phi \left( \cdot \right)$ denotes the local feature extraction function (MLP in PointNet $^ { + + }$ ), and $f _ { i , j }$ is the $j$ -th neighbor point feature of $i$ -th sampled point. By doing so, PointNet+ $^ +$ is able to effectively capture local geometric information and progressively enlarge the receptive fields by repeating the operation.
+
+In the sense of network architecture design, PointNet $^ { + + }$ exhibits a universal pipeline for point cloud analysis. Following this pipeline, some plug-and-play methods have been proposed, mainly focusing on the local feature extractor $\Phi \left( \cdot \right)$ (Xu et al., 2021a; Liu et al., 2019b; Thomas et al., 2019; Zhao et al., 2021). Generally, these local feature extractors thoroughly explore the local geometric information using convolution, graph, or self-attention mechanisms. In RSCNN (Liu et al., 2019b), the extractor is mainly achieved by exploring point relations as follow:
+
+$$
+\begin{array} { r } { \Phi \left( f _ { i , j } \right) = \mathrm { M L P } \left( \left[ \left. x _ { i , j } - x _ { i } \right. _ { 2 } , x _ { i , j } - x _ { i } , x _ { i , j } , x _ { i } \right] \right) * f _ { i , j } , \forall j \in \left\{ 1 , \cdots , K \right\} , } \end{array}
+$$
+
+where $[ \cdot ]$ is the concatenation operation and MLP is a small network composed of a Fully-connected (FC) layer, Batch Normalization layer, and activation function. Unlike RSCNN, Point Transformer introduces the self-attention mechanism into point cloud analysis and considers the similarities between pair-wise points in a local region. To this end, it re-formulates the extractor as:
+
+$$
+\Phi \left( f _ { i } \right) = \sum _ { j = 1 } ^ { k } { { \rho \left( { \gamma \left( { \varphi \left( { { f _ { i } } } \right) - \psi \left( { { f _ { i , j } } } \right) + \delta } \right) } \right) } \odot \left( { \alpha \left( { { f _ { i , j } } + \delta } \right) } \right) } ,
+$$
+
+where $\gamma , \varphi , \psi$ and $\alpha$ are linear mapping function, “ $\odot$ ” is a Hadamard product, and $\rho$ is a softmax normalization. In particular, Point Transformer introduces a relative position encoding, $\delta \ = \ \theta \left( x _ { i } - x _ { i , j } \right)$ , where the relative position is encoded by two FC layers with a ReLU nonlinearity layer, into both attention weights and features. The lightweight positional encoder largely improves the performance of Point Transformer.
+
+While these methods can easily take the advantage of detailed local geometric information and usually exhibit promising results, two issues limit their development. First, with the introduction of delicate extractors, the computational complexity is largely increased, leading to prohibitive inference latency 1. For example, the FLOPs of Equation 3 in Point Transformer would be $1 4 K d ^ { 2 }$ , ignoring the summation and subtraction operations. Compared with the conventional FC layer that enjoys $2 K d ^ { 2 }$ FLOPs, it increases the computations by times. Notice that the memory access cost is not considered yet. Second, with the development of local feature extractors, the performance gain has started to saturate on popular benchmarks. Moreover, empirical analysis in Liu et al. (2020) reveals that most sophisticated local extractors make surprisingly similar contributions to the network performance under the same network input. Both limitations encourage us to develop a new method that circumvents the employment of sophisticated local extractors, and provides gratifying results.
+
+# 3.2 FRAMEWORK OF POINTMLP
+
+In order to get rid of the restrictions mentioned above, we present a simple yet effective MLP-based network for point cloud analysis that no sophisticated or heavy operations are introduced. The key operation of our PointMLP can be formulated as:
+
+$$
+g _ { i } = \Phi _ { p o s } ( A ( \Phi _ { p r e } ( f _ { i , j } ) , | j = 1 , \cdot \cdot \cdot , K ) ) ,
+$$
+
+where $\Phi _ { p r e }$ (·) and $\Phi _ { p o s } \left( \cdot \right)$ are residual point MLP blocks: the shared $\Phi _ { p r e } \left( \cdot \right)$ is designed to learn shared weights from a local region while the $\Phi _ { p o s } \left( \cdot \right)$ is leveraged to extract deep aggregated features. In detail, the mapping function can be written as a series of homogeneous residual MLP blocks, MLP $( x ) + x$ , in which MLP is combined by FC, normalization and activation layers (repeated two times). Following Qi et al. (2017a), we consider the aggregation function $\boldsymbol { \mathcal { A } } \left( \cdot \right)$ as max-pooling operation. Equation 4 describes one stage of of PointMLP. For a hierarchical and deep network, we recursively repeat the operation by $s$ stages. Albeit the framework of PointMLP is succinct, it exhibits some prominent merits. 1) Since PointMLP only leverages MLPs, it is naturally invariant to permutation, which perfectly fits the characteristic of point cloud. 2) By incorporating residual connections, PointMLP can be easily extended to dozens layers, resulting deep feature representations. 3) In addition, since there is no sophisticated extractors included and the main operation is only highly optimized feed-forward MLPs, even we introduce more layers, our PointMLP still performs efficiently. Unless explicitly stated, the networks in our experiments use four stages, and two residual blocks in both $\Phi _ { p r e }$ (·) and $\Phi _ { p o s }$ (·). We employ $\mathbf { k }$ -nearest neighbors algorithm (kNN) to select the neighbors and set the number $K$ to 24.
+
+# 3.3 GEOMETRIC AFFINE MODULE
+
+While it may be easy to simply increase the depth by considering more stages or stacking more blocks in $\Phi _ { p r e }$ and $\Phi _ { p o s }$ , we notice that a simple deep MLP structure will decrease the accuracy and stability, making the model less robust. This is perhaps caused by the sparse and irregular geometric structures in local regions. Diverse geometric structures among different local regions may require different extractors but shared residual MLPs struggle at achieving this. We flesh out this intuition and develop a lightweight geometric affine module to tackle this problem. Let $\{ f _ { i , j } \} _ { j = 1 , \cdots , k } \in$ $\mathbb { R } ^ { k \times d }$ be the grouped local neighbors of $f _ { i } \in \mathbb { R } ^ { d }$ containing $k$ points, and each neighbor point $f _ { i , j }$ is a $d$ -dimensional vector. We transform the local neighbor points by the following formulation:
+
+$$
+\{ f _ { i , j } \} = \alpha \odot \frac { \{ f _ { i , j } \} - f _ { i } } { \sigma + \epsilon } + \beta , ~ \sigma = \sqrt { \frac { 1 } { k \times n \times d } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { k } { ( f _ { i , j } - f _ { i } ) ^ { 2 } } } ,
+$$
+
+where $\alpha \in \mathbb { R } ^ { d }$ and $\beta \in \mathbb { R } ^ { d }$ are learnable parameters, $\odot$ indicates Hadamard production, and $\epsilon =$ $1 e ^ { - 5 }$ is a small number for numerical stability (Ioffe & Szegedy, 2015; Wu & He, 2018; Dixon & Massey Jr, 1951). Note that $\sigma$ is a scalar describes the feature deviation across all local groups and channels. By doing so, we transform the local points to a normal distribution while maintaining original geometric properties.
+
+# 3.4 COMPUTATIONAL COMPLEXITY AND ELITE VERSION
+
+Although the FC layer is highly optimized by mainstream deep learning framework, the theoretical number of parameters and computational complexity are still high. To further improve the efficiency, we introduce a lightweight version of PointMLP named as pointMLP-elite, with less than 0.7M parameters and prominent inference speed (176 samples/second on ModelNet40 benchmark).
+
+Table 2: Classification results on ModelNet40 dataset. With only 1k points, our method achieves state-of-the-art results on both class mean accuracy (mAcc) and overall accuracy (OA) metrics. We also report the speed of some open-sourced methods by samples/second tested on one Tesla V100- pcie GPU and four cores AMD EPYC $7 3 5 1 @ 2 . 6 0 \mathrm { G H z }$ CPU. \* For KPConv, we take the results from the original paper. The best is marked in bold and second best is in blue.   
+
+<table><tr><td rowspan=1 colspan=3>Method</td><td rowspan=1 colspan=1>Inputs  mAcc(%)  OA(%)</td><td rowspan=1 colspan=1>Train   TestParam.speed speed</td></tr><tr><td rowspan=1 colspan=3>PointNet (Qi et al., 2017a)PointNet++ (Qi et al., 2017b)PointNet++ (Qi et al., 2017b)</td><td rowspan=1 colspan=1>1k P      86.0      89.21kP        -        90.75k P+N      -        91.9</td><td rowspan=1 colspan=1>1.41M  223.8  308.51.41M</td></tr><tr><td rowspan=10 colspan=3>PointCNN (Li et al., 2018b)PointConv (Wu et al., 2019)KPConv (Thomas et al., 2019)DGCNN (Wang et al., 2019)RS-CNN (Liu et al., 2019b)DensePoint (Liu et al., 2019a)PointASNL (Yan et al., 2020)PosPool (Liu et al., 2020)Point Trans. (Engel et al., 2020)</td><td rowspan=2 colspan=1>1k P      88.1      92.5</td><td rowspan=4 colspan=1>18.6M  17.9   10.215.2M  31.0*  80.0*</td></tr><tr><td rowspan=2 colspan=1>18.6M  17.915.2M  31.0*</td></tr><tr><td rowspan=1 colspan=1>1k P+N       -        92.5</td></tr><tr><td rowspan=1 colspan=1>7kP        -        92.9</td><td rowspan=1 colspan=1>15.2M  31.0*</td></tr><tr><td rowspan=1 colspan=1>1k P      90.2      92.9</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>1k P        -        92.9</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=1 colspan=1>1k P        1        93.2</td><td rowspan=5 colspan=1>8.39M   16.3   112</td></tr><tr><td rowspan=1 colspan=1>1k P        =        92.9</td></tr><tr><td rowspan=1 colspan=1>5k P        =        93.2</td></tr><tr><td rowspan=1 colspan=2>020）</td><td rowspan=1 colspan=1>1k P        =        92.8</td></tr><tr><td rowspan=7 colspan=3>GBNet (Qiu et al., 2021b)GDANet (Xu et al., 2021b)PA-DGC (Xu et al., 2021a)MLMSPT (Han et al., 2021)PCT (Guo et al., 2021)Point Trans. (Zhao et al.,2021)CurveNet (Xiang et al., 2021)</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1k P      91.0      93.8</td></tr><tr><td rowspan=1 colspan=1>1k P        =        93.8</td><td rowspan=6 colspan=1>0.93M   26.3   14.02.04M  20.8   15.0</td></tr><tr><td rowspan=1 colspan=1>1k P                  93.9</td></tr><tr><td rowspan=1 colspan=1>1k P                  92.9</td></tr><tr><td rowspan=1 colspan=1>1k P        =        93.2</td></tr><tr><td rowspan=1 colspan=1>1k P      90.6      93.7</td></tr><tr><td rowspan=1 colspan=1>1k P        -       94.2</td></tr><tr><td rowspan=4 colspan=3>PointMLP w/o vot.PointMLP w/ vot.PointMLP-elite w/o vot.PointMLP-elite w/ vot.</td><td rowspan=1 colspan=1>1k P      91.3      94.1</td><td rowspan=1 colspan=1>12.6M  47.1    112</td></tr><tr><td rowspan=1 colspan=1>1k P      91.4      94.5</td><td rowspan=1 colspan=1>12.6M  47.1    112</td></tr><tr><td rowspan=1 colspan=1>1k P      90.9      93.6</td><td rowspan=1 colspan=1>0.68M   116    176</td></tr><tr><td rowspan=1 colspan=1>1k P      90.7      94.0</td><td rowspan=1 colspan=1>0.68M   116    176</td></tr></table>
+
+Inspired by He et al. (2016); Hu et al. (2018), we present a bottleneck structure for the mapping function $\Phi _ { p r e }$ and $\Phi _ { p o s }$ . We opt to reduce the channel number of the intermediate FC layer by a factor of $r$ and increase the channel number as the original feature map. This strategy is opposite to the design in Vaswani et al. (2017); Touvron et al. (2021) which increases the intermediate feature dimensions. Empirically, we do not observe a significant performance drop. This method reduce the parameters of residual MLP blocks from $2 d ^ { 2 }$ to $\scriptstyle { \frac { 2 } { r } } d ^ { 2 }$ . By default, we set $r$ to 4 in PointMLPelite. Besides, we also slightly adjust the network architecture, reducing both the MLP blocks and embedding dimension number (see appendix for details). Inspired by Xie et al. (2017), we also investigated a grouped FC operation in the network that divides one FC layer into $g$ groups of subFC layers, like group convolution layer. However, we empirically found that this strategy would largely hamper the performance. As a result, we did not consider it in our implementation.
+
+# 4 EXPERIMENTS
+
+In this section, we comprehensively evaluate PointMLP on several benchmarks. Detailed ablation studies demonstrate the effectiveness of PointMLP with both quantitative and qualitative analysis.
+
+# 4.1 SHAPE CLASSIFICATION ON MODELNET40
+
+We first evaluate PointMLP on the ModelNet40 (Wu et al., 2015) benchmark, which contains 9,843 training and 2,468 testing meshed CAD models belonging to 40 categories. Following the standard practice in the community, we report the class-average accuracy (mAcc) and overall accuracy (OA) on the testing set. We train all models for 300 epochs using SGD optimizer.
+
+![](images/071941c75a0797a644a63c156f0143a299fb274fe0ca60d40c36dc15a5458c0d.jpg)  
+Figure 3: Four run results (mean $\pm$ std) of PointMLP with/without our geometric affine module on ScanObjectNN test set. We zoom in on the details of PointMLP40 to show the stability difference.
+
+Experimental results are presented in Table 2. Among these methods, our PointMLP clearly outperforms state-of-the-art method CurveNet by $0 . 3 \%$ $9 4 . 5 \%$ vs. $9 4 . 2 \%$ ) overall accuracy with only 1k points. Note that this improvement could be considered as a promising achievement since the results on ModelNet40 recent methods have been saturated around $94 \%$ for a long time. Even without the voting strategy (Liu et al., 2019b), our PointMLP still performs on par or even better than other methods that are tested with voting strategy.
+
+Despite having better accuracy, our method is much faster than the methods with sophisticated local geometric extractors. We compare PointMLP to several open-sourced methods and report the parameters, classification accuracy, training, and testing speed. As we stated previously, a key intuition behind this experiment is that model complexity can not directly reflect efficiency. For example, CurveNet is lightweight and delivers a strong result, whereas the inference cost is prohibitive (15 samples/second). On the contrary, our PointMLP presents a high inference speed $( \mathbf { 1 1 2 \ s a m } .$ - ples/second). To further reduce the model size and speed up the inference, we present a lightweight PointMLP-elite, which significantly reduces the number of parameters to 0.68M, while maintaining high-performance $9 0 . 9 \%$ mAcc and $9 4 . 0 \%$ OA on ModelNet40. With PointMLP-elite, we further speed up the inference to 176 samples/second.
+
+# 4.2 SHAPE CLASSIFICATION ON SCANOBJECTNN
+
+While ModelNet40 is the de-facto canonical benchmark for point cloud analysis, it may not meet the requirement of modern methods due to its synthetic nature and the fast development of point cloud analysis. To this end, we also conduct experiments on the ScanObjectNN benchmark (Uy et al., 2019).
+
+ScanObjectNN is a recently released point cloud benchmark that contains 15,000 objects that are categorized into 15 classes with 2,902 unique object instances in the real world. Due to the existence of background, noise, and occlusions, this benchmark poses significant challenges to existing point cloud analysis methods. We consider the hard
+
+Table 3: Classification results on ScanObjectNN dataset. We examine all methods on the most challenging variant $( \mathrm { P B } _ { - } \mathrm { T } 5 0 \mathrm { \_ R S } )$ . For our pointMLP and PointMLP-elite, we train and test for four runs and report mean $\pm$ std results.   
+
+<table><tr><td>Method</td><td>mAcc(%)</td><td>OA(%)</td></tr><tr><td>3DmFV PointNet (Qi et al., 2017a) SpiderCNN (Xu et al., 2018) PointNet++ (Qi et al., 2017b)</td><td>58.1 63.4 69.8</td><td>63 68.2 73.7</td></tr><tr><td>DGCNN (Wang et al., 2019) PointCNN (Li et al., 2018b) BGA-DGCNN (Uy et al., 2019) BGA-PN++ (Uy et al., 2019)</td><td>75.4 73.6 75.1 75.7 77.5</td><td>77.9 78.1 78.5 79.7</td></tr><tr><td>DRNet (Qiu et al.,2021a) GBNet (Qiu et al., 2021b) SimpleView (Goyal et al., 2021) PRANet (Cheng et al., 2021)</td><td>78.0 77.8 =</td><td>80.2 80.3 80.5 80.5±0.3</td></tr><tr><td>MVTN (Hamdi et al., 2021) PointMLP (ours) PointMLP-elite (ours)</td><td>79.1 - 83.9±0.5</td><td>82.1 82.8 85.4±0.3</td></tr></table>
+
+est perturbed variant (PB T50 RS) in our experiments. We train our model using an SGD optimizer for 200 epochs with a batch size of 32. For a better illustration, we train and test our method for four runs and report the mean $\pm$ standard deviation in Table 3.
+
+Table 4: Classification accuracy of pointMLP on ScanObjectNN test set using 24, 40, and 56 layers, respectively.   
+
+<table><tr><td>Depth</td><td>mAcc(%)</td><td>0A(%)</td></tr><tr><td>24 layers</td><td>83.4±0.4</td><td>84.8±0.5</td></tr><tr><td>40 layers</td><td>83.9±0.5</td><td>85.4±0.3</td></tr><tr><td>56 layers</td><td>83.2±0.2</td><td>85.0±0.1</td></tr></table>
+
+Table 5: Component ablation studies on ScanObjectNN test set.   
+
+<table><tr><td>Tpre</td><td>Tpos</td><td>Affine</td><td>mAcc(%)</td><td>0A(%)</td></tr><tr><td></td><td></td><td></td><td>80.8±0.4</td><td>82.8±0.0</td></tr><tr><td>x/</td><td>×</td><td>√</td><td>83.3±0.3</td><td>84.7±0.2</td></tr><tr><td></td><td>√</td><td>×</td><td>79.1±1.7</td><td>81.5±1.4</td></tr><tr><td></td><td></td><td>「</td><td>83.9±0.5</td><td>85.4±0.3</td></tr></table>
+
+Empirically, our PointMLP surpasses all methods by a significant improvement on both class mean accuracy (mAcc) and the overall accuracy (OA). For example, we outperform PRANet by $4 . 8 \%$ mAcc and $3 . 3 \%$ OA. Even compared with the heavy multi-view projection method MVTN (12 views), our PointMLP still performs much better $( 8 5 . 3 9 \% 8 2 . 8 \% )$ . Notice that we achieve this by fewer training epochs and did not consider the voting strategy. Moreover, we notice that our method achieves the smallest gap between class mean accuracy and overall accuracy. This phenomenon indicates that PointMLP did not bias to a particular category, showing decent robustness.
+
+# 4.3 ABLATION STUDIES
+
+Network Depth. Network depth has been exploited in many tasks but is rare in point cloud analysis. We first investigate the performance of PointMLP with different depths in Table 4. We vary the network depth by setting the number of homogeneous residual MLP blocks to 1, 2, and 3, respectively, resulting in 24, 40, and 56-layers PointMLP variants. Detailed depth formulation can be found in Appendix D. At first glance, we notice that simply increasing the depth would not always bring better performance; an appropriate depth would be a good solution. Additionally, the model gets stable with more layers introduced, as demonstrated by the decreasing standard deviation. When the depth is set to 40, we achieve the best tradeoff between accuracy and stability $( 8 5 . 4 \%$ mean accuracy and 0.3 standard deviations). Remarkably, PointMLP consistently achieves gratifying results that outperform recent methods, regardless of the depth.
+
+Geometric Affine Module. Other work provides sophisticated local geometric extractors to explore geometric structures. Instead, our PointMLP discards these burdensome modules and introduces a lightweight geometric affine module. Figure 3 presents the results of PointMLP with/without the geometric affine module. By integrating the module, we systematically improve the performance of PointMLP by about $3 \%$ for all variants. The reasons for this large improvement are two-fold. First, the geometric affine module maps local input features to a normal distribution, which eases the training of PointMLP. Second, the geometric affine module implicitly encodes the local geometrical information by the channel-wise distance to local centroid and variance, remedying the deficiency of geometric information. Besides the gratifying improvements, the geometric affine module also largely boosts the stability of PointMLP, suggesting better robustness.
+
+Component ablation study. Table 5 reports the results on ScanObjectNN of removing each individual component in PointMLP. Consistent with Figure 3, geometric affine module plays an important role in PointMLP, improving the base architecture by $3 . 9 \%$ . Remarkably, even without this module, which is an unfair setting for PointMLP, our base network stills achieves
+
+![](images/64f9834890660778d301ec0a08af23475a0ca068c31e381a3a3c9289448af64f.jpg)  
+Figure 4: Loss landscape along two rand directions. By introducing residual connection, we ease the optimization of PointMLP and achieve a flat landscape like a simple shallow network intuitively.
+
+Table 6: Part segmentation results on the ShapeNetPart dataset. Empirically, our method is much faster than the best method KPConv, and presents a competitive performance.   
+
+<table><tr><td rowspan="2">Method</td><td rowspan="2">Cls. mIoU</td><td rowspan="2">Inst. mIoU</td><td rowspan="2">aero</td><td rowspan="2">bag</td><td rowspan="2">cap</td><td rowspan="2">car</td><td rowspan="2">chair</td><td rowspan="2">aerp- hone</td><td rowspan="2">guitar</td><td rowspan="2">knife</td><td rowspan="2">lamp</td><td rowspan="2">laptop</td><td rowspan="2">motor- bike</td><td rowspan="2">mug pistol</td><td rowspan="2"></td><td rowspan="2">rocket skate- board</td><td rowspan="2"></td><td rowspan="2">table</td></tr><tr><td></td></tr><tr><td>PointNet</td><td>80.4</td><td>83.7</td><td>83.4</td><td>78.7</td><td>82.5</td><td>74.9</td><td>89.6</td><td>73.0</td><td>91.5</td><td>85.9</td><td>80.8</td><td>95.3</td><td>65.2</td><td>93.0</td><td>81.2</td><td>57.9</td><td>72.8</td><td>80.6</td></tr><tr><td>PointNet++</td><td>81.9</td><td>85.1</td><td>82.4</td><td>79.0</td><td>87.7</td><td>77.3</td><td>90.8</td><td>71.8</td><td>91.0</td><td>85.9</td><td>83.7</td><td>95.3</td><td>71.6</td><td>94.1</td><td>81.3</td><td>58.7</td><td>76.4</td><td>82.6</td></tr><tr><td>Kd-Net</td><td>-</td><td>82.3</td><td>80.1</td><td>74.6</td><td>74.3</td><td>70.3</td><td>88.6</td><td>73.5</td><td>90.2</td><td>87.2</td><td>81.0</td><td>94.9</td><td>57.4</td><td>86.7</td><td>78.1</td><td>51.8</td><td>69.9</td><td>80.3</td></tr><tr><td>SO-Net</td><td>-</td><td>84.9</td><td>82.8</td><td>77.8</td><td>88.0</td><td>77.3</td><td>90.6</td><td>73.5</td><td>90.7</td><td>83.9</td><td>82.8</td><td>94.8</td><td>69.1</td><td>94.2</td><td>80.9</td><td>53.1</td><td>72.9</td><td>83.0</td></tr><tr><td>PCNN</td><td>81.8</td><td>85.1</td><td>82.4</td><td>80.1</td><td>85.5</td><td>79.5</td><td>90.8</td><td>73.2</td><td>91.3</td><td>86.0</td><td>85.0</td><td>95.7</td><td>73.2</td><td>94.8</td><td>83.3</td><td>51.0</td><td>75.0</td><td>81.8</td></tr><tr><td>DGCNN</td><td>82.3</td><td>85.2</td><td>84.0</td><td>83.4</td><td>86.7</td><td>77.8</td><td>90.6</td><td>74.7</td><td>91.2</td><td>87.5</td><td>82.8</td><td>95.7</td><td>66.3</td><td>94.9</td><td>81.1</td><td>63.5</td><td>74.5</td><td>82.6</td></tr><tr><td>P2Sequence</td><td>-</td><td>85.2</td><td>82.6</td><td>81.8</td><td>87.5</td><td>77.3</td><td>90.8</td><td>77.1</td><td>91.1</td><td>86.9</td><td>83.9</td><td>95.7</td><td>70.8</td><td>94.6</td><td>79.3</td><td>58.1</td><td>75.2</td><td>82.8</td></tr><tr><td>PointCNN</td><td>84.6</td><td>86.1</td><td>84.1</td><td>86.5</td><td>86.0</td><td>80.8</td><td>90.6</td><td>79.7</td><td>92.3</td><td>88.4</td><td>85.3</td><td>96.1</td><td>77.2</td><td>95.2</td><td>84.2</td><td>64.2</td><td>80.0</td><td>83.0</td></tr><tr><td>PointASNL</td><td>-</td><td>86.1</td><td>84.1</td><td>84.7</td><td>87.9</td><td>79.7</td><td>92.2</td><td>73.7</td><td>91.0</td><td>87.2</td><td>84.2</td><td>95.8</td><td>74.4</td><td>95.2</td><td>81.0</td><td>63.0</td><td>76.3</td><td>83.2</td></tr><tr><td>RS-CNN</td><td>84.0</td><td>86.2</td><td>83.5</td><td>84.8</td><td>88.8</td><td>79.6</td><td>91.2</td><td>81.1</td><td>91.6</td><td>88.4</td><td>86.0</td><td>96.0</td><td>73.7</td><td>94.1</td><td>83.4</td><td>60.5</td><td>77.7</td><td>83.6</td></tr><tr><td>SynSpec</td><td>82.0</td><td>84.7</td><td>81.6</td><td>81.7</td><td>81.9</td><td>75.2</td><td>90.2</td><td>74.9</td><td>93.0</td><td>86.1</td><td>84.7</td><td>95.6</td><td>66.7</td><td>92.7</td><td>81.6</td><td>60.6</td><td>82.9</td><td>82.1</td></tr><tr><td>SPLATNet</td><td>83.7</td><td>85.4</td><td>83.2</td><td>84.3</td><td>89.1</td><td>80.3</td><td>90.7</td><td>75.5</td><td>92.1</td><td>87.1</td><td>83.9</td><td>96.3</td><td>75.6</td><td>95.8</td><td>83.8</td><td>64.0</td><td>75.5</td><td>81.8</td></tr><tr><td>SpiderCNN</td><td>82.4 85.1</td><td>85.3 86.4</td><td>83.5</td><td>81.0</td><td>87.2</td><td>77.5</td><td>90.7</td><td>76.8</td><td>91.1 92.6</td><td>87.3 88.4</td><td>83.3</td><td>95.8</td><td>70.2</td><td>93.5</td><td>82.7</td><td>59.7</td><td>75.8</td><td>82.8</td></tr><tr><td>KPConv PA-DGC</td><td>84.6</td><td>86.1</td><td>84.6</td><td>86.3</td><td>87.2</td><td>81.1</td><td>91.1 90.6</td><td>77.8 80.8</td><td>92.0</td><td>88.7</td><td>82.7</td><td>96.2</td><td>78.1 73.9</td><td>95.8 94.7</td><td>85.4 84.7</td><td>69.0</td><td>82.0</td><td>83.6 84.0</td></tr><tr><td></td><td></td><td></td><td>84.3</td><td>85.0</td><td>90.4</td><td>79.7</td><td></td><td></td><td></td><td></td><td>82.2</td><td>95.9</td><td></td><td></td><td></td><td>65.9</td><td>81.4</td><td></td></tr><tr><td>PointMLP</td><td>84.6</td><td>86.1</td><td>83.5</td><td>83.4</td><td>87.5</td><td>80.54</td><td>90.3</td><td>78.2</td><td>92.2</td><td>88.1</td><td>82.6</td><td>96.2</td><td>77.5</td><td>95.8</td><td>85.4</td><td>64.6</td><td>83.3</td><td>84.3</td></tr></table>
+
+$8 1 . 5 \pm 1 . 4 \%$ OA, outperforming most related methods (see Table 3). Removing $\Phi _ { p r e }$ function (MLPs before aggregator $\mathcal { A }$ ), the performance drops $2 . 6 \%$ overall accuracy. Combining all these components together, we achieve the best result $8 5 . 4 \%$ OA. See Appendix C for more ablations.
+
+Loss landscape. We depict the 3D loss landscape (Li et al., 2018a) in Figure 4. Simply increasing the network depth may not achieve a better representation and even hamper the results. When removing the residual connection in PointMLP, the loss landscape turns sharp, and the performance plummets to $8 8 . 1 \%$ $6 \%$ drop) on ModelNet40. With residual connection, we greatly ease the optimization course of PointMLP and make it possible to train a deep network.
+
+# 4.4 PART SEGMENTATION
+
+Our PointMLP can also be generalized to other 3D point cloud tasks. We next test PointMLP for 3D shape part segmentation task on the ShapeNetPart benchmark (Yi et al., 2016). The shapeNetPart dataset consists of 16,881 shapes with 16 classes belonging to 50 parts labels in total. In each class, the number of parts is between 2 and 6. We follow the settings from Qi et al. (2017b) that randomly select 2048 points as input for a fair comparison. We compare our methods with several recent works,
+
+![](images/937de1c2245f3fbccbf26ca1813370b71e9c2b3909037a38cd4c4f9c3749559f.jpg)  
+Figure 5: Part segmentation results on ShapeNetPart. Top line is ground truth and bottom line is our prediction.
+
+including SyncSpecCNN (Yi et al., 2017), SPLATNet (Su et al., 2018), etc. We also visualize the segmentation ground truths and predictions in Figure 5. Intuitively, the predictions of our PointMLP are close to the ground truth. Best viewed in color.
+
+# 5 CONCLUSION
+
+In this paper, we propose a simple yet powerful architecture named PointMLP for point cloud analysis. The key insight behind PointMLP is that a sophisticated local geometric extractor may not be crucial for performance. We begin with representing local points with simple residual MLPs as they are permutation-invariant and straightforward. Then we introduce a lightweight geometric affine module to boost the performance. To improve efficiency further, we also introduce a lightweight counterpart, dubbed as PointMLP-elite. Experimental results have shown that PointMLP outperforms related work on different benchmarks beyond simplicity and efficiency. We hope this novel idea will inspire the community to rethink the network design and local geometry in point cloud.
+
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+Yifan Xu, Tianqi Fan, Mingye Xu, Long Zeng, and Yu Qiao. Spidercnn: Deep learning on point sets with parameterized convolutional filters. In ECCV, pp. 87–102, 2018.
+
+Xu Yan, Chaoda Zheng, Zhen Li, Sheng Wang, and Shuguang Cui. Pointasnl: Robust point clouds processing using nonlocal neural networks with adaptive sampling. In CVPR, pp. 5589–5598, 2020.
+
+Zetong Yang, Yanan Sun, Shu Liu, Xiaoyong Shen, and Jiaya Jia. Std: Sparse-to-dense 3d object detector for point cloud. In ICCV, pp. 1951–1960, 2019.
+
+Li Yi, Vladimir G Kim, Duygu Ceylan, I-Chao Shen, Mengyan Yan, Hao Su, Cewu Lu, Qixing Huang, Alla Sheffer, and Leonidas Guibas. A scalable active framework for region annotation in 3d shape collections. ACM Transactions on Graphics (ToG), 35(6):1–12, 2016.
+
+Li Yi, Hao Su, Xingwen Guo, and Leonidas J Guibas. Syncspeccnn: Synchronized spectral cnn for 3d shape segmentation. In CVPR, pp. 2282–2290, 2017.
+
+Haoxuan You, Yifan Feng, Rongrong Ji, and Yue Gao. Pvnet: A joint convolutional network of point cloud and multi-view for 3d shape recognition. In ACM MM, pp. 1310–1318, 2018.
+
+Hang Zhang, Chongruo Wu, Zhongyue Zhang, Yi Zhu, Haibin Lin, Zhi Zhang, Yue Sun, Tong He, Jonas Mueller, R Manmatha, et al. Resnest: Split-attention networks. arXiv preprint arXiv:2004.08955, 2020.
+
+Zaiwei Zhang, Rohit Girdhar, Armand Joulin, and Ishan Misra. Self-supervised pretraining of 3d features on any point-cloud. arXiv preprint arXiv:2101.02691, 2021.
+
+Hengshuang Zhao, Li Jiang, Jiaya Jia, Philip Torr, and Vladlen Koltun. Point transformer. In ICCV, 2021.
+
+# A POINTMLP DETAIL
+
+We detail the architecture of PointMLP in Figure 6 (as well as PointMLP-elite in Figure 7) for a better understanding. Compared with PointMLP, the elite version mainly adjusts three configurations: 1) it reduces the number of residual point (Resp) MLP blocks; 2) it reduces the embedding dimension from 64 to 32, hence the overall model overhead is significantly alleviated; 3) by introducing a bottleneck structure, PointMLP further reduces the parameters by four times.
+
+For part segmentation task, we use the framework presented in PointNet (Qi et al., 2017a) and replace the backbone to our PointMLP. With the only modification, we improve the performance from 85.1 to 86.1 Instance mIoU.
+
+![](images/61378541d7d7ab1bfa651ab48ae34b0ce8784c4821b4f81db8424ae8c2f88c60.jpg)  
+Figure 6: Detail architecture of PointMLP for classification.
+
+![](images/29e776509eac6ea6a31083e3cafa6b828bf28a0aa1fe7f9f5e9a64cef2650dab.jpg)  
+Figure 7: Detail architecture of PointMLP-elite for classification.
+
+# B DETAIL EXPERIMENTAL SETTING
+
+# B.1 MODELNET40 AND SCANOBJECTNN
+
+Our implementations are based on PyTorch. For ModelNet40, we train models for 300 epochs on one Tesla V100 GPU with a batch size of 32. All our models are trained using synchronous SGD with a Nesterov momentum of 0.9 and a weight decay of 0.0002. The learning rate is set to 0.1 initially. We use the cosine annealing scheduler (Loshchilov & Hutter, 2017) to adjust the learning rate. For each sample, we randomly select 1024 points and consider the same augmentation strategy as Qi et al. (2017b). The setting for ScanObjectNN is similar to ModelNet40, except we train all models for only 200 epochs.
+
+For the reported speed in Table 2, we test the open-source code on a Tesla V100-pcie GPU. All the source codes we used are listed2 in the footnote.
+
+# B.2 SHAPENETPART
+
+Our setting for part segmentation task is following PointNet (Qi et al., 2017a). We randomly sample 2048 points for each sample and re-scale the input in a range of [0.67, 1.5]. Note that we did not test the result using a multi-scale testing strategy, which could further improve the performance, but is not realizable in real-world applications. Hence, we only report the single-scale results. Even the comparison is unfair, we still achieve competitive performance.
+
+# C MORE DETAILED ABLATION STUDIES
+
+Skip connection. Figure 4 shows the loss landscapes of our PointMLP with and without skip connections. We also consider adding skip connections to PointNet+ $^ +$ to validate the effectiveness of skip connections. Due to the structure of PointNe $^ { + + }$ , only two skip connections could be added without modifying the original architecture of PointNet++. By adding the skip connections, we achieve a classification accuracy of $9 2 . 7 \%$ on ModelNet40 in our re-implementation.
+
+Pre-MLP block vs. Pos-MLP block. we also modified the configuration of our PointMLP and retrained the model to investigate the importance of Pre-MLP and Pos-MLP blocks. In our original implementation, we set the pre-MLP block list to [2, 2, 2, 2] and the pos-MLP blocks list to [2, 2, 2, 2]. Here, we remove the pos-MLP blocks and change the pre-MLP blocks to [4, 4, 4, 4] to match the block number. The 3-layer classifier can be considered as the MLP at the end of the last stage. We trained the models two times and got an average OA of $8 4 . 1 3 \%$ $8 3 . 8 7 \%$ and $8 4 . 3 9 \%$ ), which is lower than vanilla PointMLP $8 5 . 4 \%$ , and even the result in Table 5 second-row $8 4 . 7 \%$ . This result indicates that pos-MLP does benefit our PointMLP, and simply adding more pre-MLP blocks does not help. We acknowledge that the effect of pos-MLP is not as strong as other components and believe that a detailed fine-tuning of the configurations would deliver an even better performance-efficiency balance.
+
+Geometric Affine Module Applications. Geometric affine module plays an essential role in our PointMLP, exhibiting promising performance improvements. While this module can be considered as a plug-and-play method, the overlap with some local geometric extractors in other methods may limit its application. Here we integrate the module to two popular methods, PointNet $^ { - + }$ and DGCNN, for illustration and experiment on the ModelNet40 benchmark. By integrating the geometric affine module, we improve the performance of PointNet $^ { + + }$ to $9 3 . 3 \%$ , achieving an improvement of $1 . 4 \%$ . However, when integrating the module to DGCNN, we get a performance of $9 2 . 8 \%$ , which is slightly lower than the original results $( 9 2 . 9 \% )$ . Note that both results are tested without voting.
+
+# D POINTMLP DEPTH
+
+Here we format the detailed formulation of layer number in our PointMLP. For the sake of clarity, we ignore Batch Normalization layers and activation functions. Let $\mathrm { P r e } _ { i }$ and $\mathrm { P o s } _ { i }$ indicate the repeating number of the $\Phi _ { p r e }$ block (which includes 3 layers) and $\Phi _ { p o s }$ block (which includes 2 layers) in $i$ -th stage, respectively. Note that we have one layer in feature embedding in the beginning, one layer for channel number matching in each stage, and three layers in the classifier. Hence, the total number of learnable layers $L$ would be
+
+$$
+L = 1 + \sum _ { i = 1 } ^ { 4 } \left( 1 + 2 \times \mathrm { P r e } _ { i } + 2 \times \mathrm { P o s } _ { i } \right) + 3 .
+$$
+
+As a result, the depth configuration of our network (24, 40, and 56) can be summarized as:
+
+<table><tr><td></td><td></td><td>Depth|[Pre1,Pre2,Pre3,Pre4]|[Pos1,Pos2,Pos3,Pos4]</td></tr><tr><td>24</td><td>[1,1,1,1]</td><td>[1,1,1,1]</td></tr><tr><td>40</td><td>[2,2,2,2]</td><td>[2,2,2,2]</td></tr><tr><td>56</td><td>[3,3,3,3]</td><td>[3,3,3,3]</td></tr></table>

parse/dev/OtxyysUdBE/OtxyysUdBE.md

@@ -0,0 +1,294 @@
+# FedRolex: Model-Heterogeneous Federated Learning with Rolling Sub-Model Extraction
+
+Samiul Alam1,2, Luyang $\mathbf { L i u ^ { 3 } }$ , Ming Yan4,1, Mi Zhang2,1 1Michigan State University, 2The Ohio State University, 3Google Research, 4The Chinese University of Hong Kong, Shenzhen alamsami@msu.edu, luyangliu@google.com, yanming@cuhk.edu.cn, mizhang.1@osu.edu
+
+# Abstract
+
+Most cross-device federated learning (FL) studies focus on the model-homogeneous setting where the global server model and local client models are identical. However, such constraint not only excludes low-end clients who would otherwise make unique contributions to model training but also restrains clients from training large models due to on-device resource bottlenecks. In this work, we propose FedRolex, a partial training (PT)-based approach that enables model-heterogeneous FL and can train a global server model larger than the largest client model. At its core, FedRolex employs a rolling sub-model extraction scheme that allows different parts of the global server model to be evenly trained, which mitigates the client drift induced by the inconsistency between individual client models and server model architectures. We show that FedRolex outperforms state-of-the-art PTbased model-heterogeneous FL methods (e.g. Federated Dropout) and reduces the gap between model-heterogeneous and model-homogeneous FL, especially under the large-model large-dataset regime. In addition, we provide theoretical statistical analysis on its advantage over Federated Dropout and evaluate FedRolex on an emulated real-world device distribution to show that FedRolex can enhance the inclusiveness of FL and boost the performance of low-end devices that would otherwise not benefit from FL. Our code is available at: https://github.com/AIoTMLSys-Lab/FedRolex.
+
+# 1 Introduction
+
+Federated learning (FL) is a machine learning paradigm that trains models from distributed clients with private data under the coordination of a central server [1, 2]. In this work, we focus on crossdevice FL where clients are usually resource-constrained edge devices. The majority of existing cross-device FL studies focus on the model-homogeneous setting [3–6], in which the server model and the client models across all the participating client devices are identical. However, modelhomogeneous FL are confronted with two fundamental constraints: (1) device heterogeneity is a more realistic consideration when deploying FL systems in real-world applications: different client devices could have very diverse on-device resources and are only capable of training models with capacities that match their on-device resources. Having the same model on all the devices would, unfortunately, exclude clients with low-end devices who would otherwise make unique contributions to model training from their own local data; (2) state-of-the-art machine learning has moved towards large models [7] such as Transformer [8]. Restricting server and client models to be the same inevitably causes model-homogeneous FL to fail to train such large models due to the resource constraint of client devices.
+
+To relax the fundamental constraints of model-homogeneous FL, model-heterogeneous FL was proposed where heterogeneous models with different capacities across the server and the clients are trained during the federated training process. One primary challenge in model-heterogeneous $\mathrm { F L }$ is the aggregation of heterogeneous client models. To address this challenge, knowledge distillation (KD)-based approaches have been proposed [9–12], in which the client models serve as teachers and the server ensembles the knowledge distilled from the individual client models. However, KD-based approaches in general require public data on the server to achieve competitive model accuracy, whereas the desired public data may not be always available in practice. Moreover, since KD-based approaches need the individual client models (whole models or prediction layers) or their outputs to be sent to the server, they are incompatible with secure aggregation protocols [16], which limits their privacy guarantee. To remove the dependency on public data and ensure compatibility with secure aggregation, partial training (PT)-based approaches such as random sub-model extraction (Federated Dropout [13]) and static sub-model extraction (HeteroFL [14], FjORD [15]) were proposed. In these approaches, each client trains a smaller sub-model extracted from the larger global server model, and the server model is updated by aggregating those trained sub-models. However, the fundamental issue of existing PT-based methods is that the sub-models are extracted in ways (either random or static) such that the parameters of the global server model are not evenly trained. This makes the server model vulnerable to client drift1 induced by the inconsistency between individual client model and server model architectures – a unique challenge of model-heterogeneous FL.
+
+Table 1: Comparison of FedRolex with model-homogeneous and model-heterogeneous FL methods.   
+
+<table><tr><td></td><td>Model Heterogeneity</td><td>Aggregation Scheme</td><td>Sub-model Extraction Scheme</td><td>Need of Public Data</td><td>Server Model Size</td><td>Compatibility with Secure Aggregation</td></tr><tr><td>FedAvg[3]</td><td rowspan="4">No</td><td rowspan="4"></td><td rowspan="4"></td><td>No</td><td>= Client Model</td><td>Yes</td></tr><tr><td>FedProx [4]</td><td>No</td><td>= Client Model</td><td>Yes</td></tr><tr><td>SCAFFOLD [5]</td><td>No</td><td>= Client Model</td><td>Yes</td></tr><tr><td>FedBE[6]</td><td>Unlabeled</td><td>= Client Model</td><td>No</td></tr><tr><td>FedGKT[9]</td><td rowspan="4">Yes</td><td rowspan="4">Knowledge Distillation</td><td rowspan="4"></td><td>No</td><td>≥ Largest Client Model</td><td>No</td></tr><tr><td>FedDF[10]</td><td>Unlabeled</td><td>= Largest Client Model</td><td>No</td></tr><tr><td>DS-FL [11] Fed-ET [12]</td><td>Unlabeled</td><td>= Largest Client Model</td><td>No</td></tr><tr><td></td><td>Unlabeled</td><td>≥ Largest Client Model</td><td>No</td></tr><tr><td>Federated Dropout [13]</td><td rowspan="4">Yes</td><td rowspan="4">Partial Training</td><td>Random Static</td><td>No</td><td>≥ Largest Client Model</td><td>Yes</td></tr><tr><td>HeteroFL [14]</td><td>Static</td><td>No No</td><td>=Largest Client Model</td><td>Yes</td></tr><tr><td>FjORD [15]</td><td></td><td></td><td>=Largest Client Model</td><td>Yes</td></tr><tr><td>FedRolex (Our Approach)</td><td>Rolling</td><td>No</td><td>≥Largest Client Model</td><td>Yes</td></tr></table>
+
+In this work, we propose a PT-based model-heterogeneous FL approach named FedRolex to tackle the fundamental issue of existing methods. The key difference between FedRolex and existing PT-based methods is how the sub-models are extracted for each client over communication rounds in the federated training process. Specifically, instead of extracting sub-models in either random or static manner, FedRolex proposes a rolling sub-model extraction scheme, where the sub-model is extracted from the global server model using a rolling window that advances in each communication round. Since the window is rolling, sub-models from different parts of the global model are extracted in sequence in different rounds. As a result, all the parameters of the global server model are evenly trained over the local data of client devices.
+
+The proposed rolling sub-model extraction scheme, though simple, has equipped FedRolex with multifold merits compared to prior arts (Table 1): (1) FedRolex enables different parts of the global server model to be evenly trained, which mitigates the client drift induced by model heterogeneity. (2) Contrary to static sub-model extraction approaches (HeteroFL, FjORD), FedRolex is able to train a global server model that is larger than the largest client model, enabling FL to benefit from the superior performance brought by large models. It echoes some concurrent efforts in developing FL primitives to support training large server models in cross-device settings, e.g. Federated Select [17]. (3) Compared to random sub-model extraction (Federated Dropout), as we show in our theoretical statistical analysis in Section 3 and Appendix A.1, the global server model is trained more evenly by FedRolex as the expected number of rounds for FedRolex going through all the parameters of the global model for at least certain times is smaller than that of Federated Dropout. (4) FedRolex only needs to transmit the sub-model that is needed by a given client instead of the full server model to the client. This allows clients to contribute to federated training under resource constraints and reduces communication overheads (Appendix A.6). (5) Lastly, FedRolex is fully compatible with existing secure aggregation protocols that enhance the privacy properties of FL systems.
+
+We evaluate the performance of FedRolex under two regimes: i) small-model small-dataset regime (most existing cross-device FL studies use this combination), and ii) large-model large-dataset regime (this combination echos recent efforts on pushing the frontier of cross-device FL towards training large server models on large-scale datasets [18–21]). We highlight five of our findings: (1) FedRolex consistently outperforms state-of-the-art PT-based model-heterogeneous FL methods under both small-model small-dataset and large-model large-dataset regimes $( \ S 4 . 1 )$ . (2) FedRolex reduces the gap between model-heterogeneous and model-homogeneous FL, especially under large-model large-dataset regime (§4.2). (3) With FedRolex, under both regimes, having a small fraction of large-capacity models could significantly boost the global model accuracy (§4.3). (4) FedRolex is able to train a global server model that is larger than the largest client model and outperforms Federated Dropout in terms of global model accuracy $( \ S 4 . 4 )$ . (5) Using an emulated real-world device distribution, we show that FedRolex enhances the inclusiveness of FL and boosts the performance of low-end devices that would otherwise not benefit from FL (§4.5).
+
+# 2 Related Work
+
+Knowledge Distillation (KD)-based Model-Heterogeneous FL. One primary approach for modelheterogeneous FL in cross-device settings is based on knowledge distillation (KD) [22]. In particular, FedDF [10] distills knowledge from a set of classifiers trained with private data from a federation of client devices. The logit outputs of each classifier against an unlabeled public dataset are then used to train a student model at the server with KD. Similarly, DS-FL [11] utilized an unlabeled public dataset at the server and proposed a distillation-based semi-supervised FL approach to enhance performance by pseudo-labeling the public data. FedGKT [9] proposed group knowledge transfer in which knowledge is transferred to a large model in the server from clients without public data. Fed-ET [12] proposed a weighted consensus distillation scheme with diversity regularization that enables the training of a large server model with smaller client models. KD-based approaches, however, have several limitations: they often require public data to achieve competitive model accuracy. This is because model accuracy is dependent on the size of public data as well as the domain similarity of public data with client data [10, 12, 23]. Furthermore, as KD-based methods use client model weights partially or entirely as teachers to transfer knowledge to the server, they are incompatible with secure aggregation protocols, making them vulnerable to backdoor attacks [20].
+
+Partial Training (PT)-based Model-Heterogeneous FL. To address the limitations of KD-based approaches, partial training (PT) has emerged as another solution for model-heterogeneous FL. Depending on how the sub-models are extracted from the global server model, existing PT-based methods can be in general categorized into two groups: random sub-model extraction and static sub-model extraction. Specifically, inspired by the dropout technique commonly used in centralized training [24], Federated Dropout [13] proposed to randomly extract sub-models from the global model. Though easy to be integrated into existing FL frameworks, as reported in [25], Federated Dropout becomes less effective when the data heterogeneity is high and the client cohort is small due to its randomness in selecting sub-models. In contrast, HeteroFL [14] and FjORD [15] proposed static extraction schemes where sub-models are always extracted from a designated part of the global server model. However, such a static extraction strategy has two primary drawbacks. First, the global server model is restricted to the same size as the largest client model. As such, the size and capability of the global model are implicitly restricted by the resources of client devices, making it not able to train large models due to resource bottlenecks at client devices. Second and more importantly, under static extraction, depending on their resource demands, different sub-models can only be trained on clients whose on-device resources are matched. As a consequence, part of the global server model cannot be trained on data at low-end client devices, causing different parts of the global model to be trained on data with different distributions. This would degrade the performance of the global model, especially under high data heterogeneity. In this work, we propose a rolling sub-model extraction scheme that tackles the issues of both random and static sub-model extraction methods.
+
+# 3 Methodology
+
+# 3.1 Formulation of Model-Heterogeneous FL
+
+Let $\mathcal { N }$ denote $N$ client devices with non-IID (non-identically and independently distributed) local data $\boldsymbol { D } = \{ D _ { 1 } , D _ { 2 } , . . . , D _ { N } \}$ . Model-homogeneous $\mathrm { F L }$ trains a global model of parameter $\theta$ by solving the following optimization problem:
+
+$$
+\operatorname* { m i n } _ { \theta } F \left( \theta \right) \triangleq \sum _ { n = 1 } ^ { N } p _ { n } F _ { n } ( \theta )
+$$
+
+with
+
+$$
+F _ { n } ( \theta ) \triangleq { \frac { 1 } { m _ { n } } } \sum _ { k = 1 } ^ { m _ { n } } l ( \theta ; d _ { n , k } ) ,
+$$
+
+where $D _ { n } \triangleq \{ d _ { n , 1 } , d _ { n , 2 } , d _ { n , 3 } . . . d _ { n , m _ { n } } \}$ is the set of local data samples of client $n$ and $p _ { n }$ is its corresponding weight such that $p _ { n } \geq 0$ and $\textstyle \sum _ { n = 1 } ^ { N } p _ { n } = 1$ .
+
+In comparison, in model-heterogeneous $\mathrm { F L }$ , clients train local models with heterogeneous capacities $\beta = \{ \bar { \beta } _ { 1 } , \beta _ { 2 } , . . . , \beta _ { N } \}$ , and the local objective function of the $n ^ { t h }$ client becomes
+
+$$
+F _ { n } ^ { \prime } ( \theta _ { n } ) \triangleq { \frac { 1 } { m _ { n } } } \sum _ { k = 1 } ^ { m _ { n } } l ( \theta _ { n } ; d _ { n , k } ) .
+$$
+
+Here, $\beta _ { n }$ denote the model capacity of client $n$ , and we define it as the proportion of nodes extracted from each layer in $\theta$ for client $n$ . The size of $\theta _ { n }$ depends on $\beta _ { n }$ , and the parameter $\theta _ { n }$ is obtained by selecting a sub-model from the global model $\theta$ , which can change from one round to another. If $\theta _ { n }$ changes, the objective function also changes. For simplicity, we use the same notation $l$ for the loss function for all clients and rounds, though they differ between clients and rounds. The key to model-heterogeneous $\mathrm { F L }$ is selecting $\theta _ { n }$ from the global model $\theta$ given model capacity $\beta _ { n }$ .
+
+# 3.2 FedRolex: Model-Heterogeneous FL with Rolling Sub-Model Extraction
+
+As a partial training (PT)-based approach, at each client, FedRolex trains only a sub-model extracted from the global server model and sends the corresponding submodel updates back to the server for update aggregation. To help understand how FedRolex works, for simplicity, Figure 1 illustrates three rounds of federated training of FedRolex on two participating heterogeneous clients, where one trains a large-capacity sub-model (left) and the other trains a small-capacity one (right). At the high level, at each round, the server extracts sub-models of different capacities from the global model and separately broadcasts them to the clients that have the corresponding capabilities. The clients train the received sub-models on their local data and transmit their heterogeneous sub-model updates to the server. Lastly, the server aggregates those updates, and the result of the aggregation is used to update the global model for the next round. The pseudocode of FedRolex is in Algorithm 1.
+
+The key to the design of FedRolex involves two design choices. In the following, we describe them in detail.
+
+(1) What sub-models to be extracted for each client across different rounds? At the server, FedRolex utilizes a rolling window to extract the sub-model from the global model. The rolling window advances in each round, and loops over all parts of the global model in sequence across different rounds. This process iterates such that the global model is evenly trained until convergence.
+
+![](images/279cece0e5a7046b19d662526f8fb00c8c1e4993e9851453b67c2a2f55b155a2.jpg)  
+Figure 1: Overview of the rolling submodel extraction scheme in FedRolex.
+
+Taking Figure 1 as an example: in round $j$ , the large-capacity and small-capacity client model extracted from the global model is $\{ a , b , c , d \}$ and $\{ c , \bar { d } , e \}$ , respectively. In round $j + 1$ , the rolling window advances one step2, the large-capacity and small-capacity client model becomes $\{ b , c , d , e \}$ and $\{ d , e , a \}$ , respectively. Similarly, in round $j + 2$ , the rolling window advances one step further, and the large-capacity and small-capacity client model becomes $\{ c , d , e , a \}$ and $\{ e , a , b \}$ , respectively.
+
+Such a rolling sub-model extraction scheme can be formalized as follows. Let $\theta _ { n } ^ { ( j ) }$ denote the parameters of the sub-model extracted from the global model for client $n$ in round $j$ , $K _ { i }$ denote the total number of nodes in layer $i$ of the global model, and $S _ { n , i } ^ { ( j ) }$ denote the node indices of layer $i$ of the global model that belongs to the extracted sub-model for client $n$ in round $j$ . Then the layer $i$ of the sub-model extracted by the rolling sub-model extraction scheme for client $n$ in round $j$ is given by:
+
+$$
+S _ { n , i } ^ { ( j ) } = \left\{ \begin{array} { l l } { \{ \hat { j } , \hat { j } + 1 , \dots , \hat { j } + \lfloor \beta _ { n } K _ { i } \rfloor - 1 \} } & { \mathrm { ~ i f ~ } \hat { j } + \lfloor \beta _ { n } K _ { i } \rfloor \le K _ { i } , } \\ { \{ \hat { j } , \hat { j } + 1 , \dots , K _ { i } - 1 \} \cup \{ 0 , 1 , \dots , \hat { j } + \lfloor \beta _ { n } K _ { i } \rfloor - 1 - K _ { i } \} } & { \mathrm { ~ e l s e . ~ } } \end{array} \right.
+$$
+
+where ${ \hat { j } } = j$ mod $K _ { i }$
+
+(2) How to aggregate heterogeneous sub-model updates to update the global model? FedRolex employs a straightforward selective averaging scheme with no client weighting to aggregate heterogeneous sub-model updates sent from the clients to update the global model3. Specifically, it computes the average of the updates for each parameter of the global model separately based on how many clients in a round updated that parameter. The parameter remains unchanged if no clients updated it.
+
+Taking Figure 1 again as an example: in round $j$ , the updates for $a$ and $b$ are obtained from the large-capacity model and the update for $e$ is from the small-capacity model only. In contrast, since $c$ and $d$ are part of both models, the update is computed by taking the average from both models.
+
+# Algorithm 1: FedRolex
+
+1 Initialization ; $\theta ^ { ( 0 ) }$ , N 12 Subroutine clientStep(θ(j)n , Dn)   
+Input : $: D _ { n } \beta _ { n } \forall n \in \mathcal N ,$ 13 mn ←− len(Dn)   
+Output : 14 for k ← 0 to mn do   
+2 Server Executes 15 $\begin{array} { r l } { \boxed { } } & { { } \theta _ { n } \longleftarrow \theta _ { n } - \eta \nabla l ( \theta _ { n } ; d _ { n , k } ) } \end{array}$   
+3 for $j  0$ to $J - 1$ do 16 end   
+4 Sample subset $\mathcal { M }$ from $\mathcal { N }$ 17 return $\theta _ { n }$   
+5 Broadcast θ(j) $\theta _ { m , S _ { m , i } ^ { ( j ) } } ^ { ( j ) }$ to client $m \in \mathcal { M }$   
+6 $\forall i , \boldsymbol { S } _ { m , i } ^ { ( j ) }$ from Equation (4)   
+7 for each client $m \in \mathcal { M }$ do   
+8 clientStep(θ(j)m , Dm)   
+9 end   
+10 Aggregate θ(j+1)[i,k] according to Equation (10)   
+11 end
+
+# 3.3 Comparison with Random and Static Sub-model Extraction Schemes
+
+Existing sub-model extraction schemes can be grouped as random-based (Federated Dropout) and static-based (HeteroFL, FjORD) methods. In this section, we describe the differences between them and the proposed rolling-based scheme employed in FedRolex. For comparison purpose, the pseudocodes of both Federated Dropout and HeteroFL are included in Appendix A.8.
+
+# 3.3.1 Comparison with Random Sub-Model Extraction Scheme
+
+In random sub-model extraction scheme, in each round, the sub-models are extracted from the global model in a random manner. As such, the layer $i$ of the sub-model extracted by the random sub-model extraction scheme for client $n$ in round $j$ is given by:
+
+$$
+\begin{array} { r } { S _ { n , i } ^ { ( j ) } = \{ k _ { c } ~ | ~ \mathrm { i n t e g e r } ~ k _ { c } \in [ 0 , K _ { i } - 1 ] \mathrm { ~ f o r } ~ 1 \leq c \leq \lfloor \beta _ { n } K _ { i } \rfloor \} , } \end{array}
+$$
+
+where a total number of $\lfloor \beta _ { n } K _ { i } \rfloor$ nodes are randomly chosen from the global model.
+
+Discussion: As shown in Figure 2(left), similar to the proposed rolling-based scheme, the sub-models extracted across different rounds by the random-based scheme have different architectures. However, due to its randomness in selecting sub-models in each round, the global model is trained less evenly, making it more vulnerable to client drift. In short, although the expected value of the frequency for updating each index is the same for all the indices, their exact frequencies are not the same due to randomness. Consequently, the random-based scheme cannot balance the update frequencies of different parts of the global model, and it inevitably takes more rounds to update the whole global model. Moreover, as we show in Appendix A.1, the expected number of rounds for Federated Dropout selecting all $I$ sub-models at least $m$ times is in the order of $I \log ( I ) + I ( m - 1 ) \log \log I$ , which is larger than that of FedRolex, $m I$ .
+
+![](images/e165594baec07c04a727b4d8fe9fbf59ae3c69c99c899f262efc0e7c6ec1f5ba.jpg)  
+Figure 2: Illustration of how sub-models are extracted by random sub-model extraction scheme (Left) and static sub-model extraction scheme (Right) over two rounds.
+
+# 3.3.2 Comparison with Static Sub-Model Extraction Scheme
+
+In static sub-model extraction scheme, in each round, the sub-models are always extracted from a designated part of the global model. As such, the layer $i$ of the sub-model extracted by the static sub-model extraction scheme for client $n$ in round $j$ is given by:
+
+$$
+\mathcal { S } _ { n , i } ^ { ( j ) } = \{ 0 , 1 , 2 , \ldots , \lfloor \beta _ { n } K _ { i } \rfloor - 1 \} .
+$$
+
+Note that $S _ { n , i } ^ { ( j ) }$ does not depend on $j$ . In other words, as shown in Figure $2 ( \mathrm { r i g h t } )$ , the same sub-model is extracted for each client in every round. Moreover, the client model with smaller capacity and client model with larger capacity are not independent. As shown in Figure 2(right), the small-capacity model $\{ a , b , c \}$ is a part of the large-capacity model $\{ a , b , c , d \}$ , which again, is a part of the globalcapacity model $\{ a , b , c , d , e \}$ . These are the two key differences between both the random-based and the proposed rolling-based scheme.
+
+Discussion: Given that, the static-based scheme, however, has two primary drawbacks. First, to cover the whole global model, there must be clients to train the full-size global model $\{ a , b , c , d , e \}$ . As such, the global model is restricted to the same size as the largest client model. Second, as shown in Figure 2(right), while $a , b$ and $c$ will be trained on data on all three types of clients, $d$ will not be trained on data on small-capacity clients, and $e$ will only be trained on data on global-model-capacity clients. As a consequence, different parts of the global model are trained on data with different distributions, which inevitably degrades the global model training quality.
+
+# 4 Experiments
+
+Datasets and Models. We evaluate the performance of FedRolex under two regimes. Under small-model small-dataset regime, we train pre-activated ResNet18 (PreResNet18) models [26] on CIFAR-10 and CIFAR-100 [27]. We replace the batch Normalization in PreResNet18 with static batch normalization [14, 28] and add a scalar module after each convolution layer [14]. Under largemodel large-dataset regime, we use Stack Overflow [29] and followed [2] to train a modified 3-layer Transformer [8] with a vocabulary of 10, 000 words, where the dimension of token embeddings is
+
+128, and the hidden dimension of the feed-forward network (FFN) block is 2048. We use ReLU activation and use 8 heads for the multi-head attention where each head is based on 12-dimensional (query, key, value) vectors. The statistics of the datasets are listed in Table 2.
+
+Table 2: Dataset statistics.   
+
+<table><tr><td>Dataset CIFAR-10</td><td>Train Clients</td><td>Train Examples 50.000</td><td>Validation Clients</td><td>Validation Examples</td><td>Test Clients</td><td>Test Examples 10,000</td></tr><tr><td>CIFAR-100</td><td>100 100</td><td>50.000</td><td>N/A N/A</td><td>N/A N/A</td><td>N/A N/A</td><td>10.000</td></tr><tr><td>Stack Overflow</td><td>342,477</td><td>135,818,730</td><td>38,758</td><td>16,491,230</td><td>204,088</td><td>16,586,035</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Data Heterogeneity. For CIFAR-10 and CIFAR-100, we followed HeteroFL [14] to model non-IID distributions by restricting each client to have $L$ labels. In our evaluation, we consider two levels of data heterogeneity. For CIFAR-10, we define $L = 2$ as high data heterogeneity and $L = 5$ as low data heterogeneity. For CIFAR-100, we use $L = 2 0$ as high data heterogeneity and $L = 5 0$ as low data heterogeneity. These two levels roughly correspond to Dirichlet distribution $D i r _ { K } ( \alpha )$ with $\alpha$ equal to 0.1 and 0.5, respectively. For Stack Overflow, the dataset is partitioned over user IDs, making the dataset naturally non-IID distributed.
+
+Model Heterogeneity. Without loss of generality, in our evaluation, we consider five different client model capacities $\beta = \{ 1 , ^ { 1 / 2 } , ^  1 / 4 , 1 / 8 , 1 / \bar { 1 } 6 \}$ where for instance, $^ 1 / 2$ means the client model capacity is half of the largest client model capacity (full model). To generate these client models, for ResNet18, we vary the number of kernels in convolution layers and keep the nodes in the output layers the same. For Transformer, we vary the number of nodes in the hidden layer of the attention heads.
+
+Baselines. We compare FedRolex against both state-of-the-art PT-based model-heterogeneous FL methods including Federated Dropout [13] and HeteroFL $[ 1 4 ] ^ { 4 }$ as well as state-of-the-art KD-based model-heterogeneous $\mathrm { F L }$ methods including FedDF [10], DS-FL [11] and Fed-ET [12] 5. To ensure a fair comparison, all the PT-based baselines are trained using the same learning rate, number of communication rounds, and multi-step learning rate decay schedule. The details of the schedule for each dataset and experiment are described in Appendix A.7.
+
+Configurations and Platform. For CIFAR-10 and CIFAR-100, we apply bounding box crop [30] to augment the images. In each communication round, $1 0 \%$ of the clients are randomly selected from a pool of 100 clients. For Stack Overflow, we followed [2] to use a $1 0 \%$ dropout rate to prevent overfitting, and 200 clients are randomly selected from a pool of 342, 477 clients in each communication round. The details of the hyper-parameters for model training are included in Appendix A.7. We implemented FedRolex and PT-based baselines using PyTorch [31] and Ray [32], and conducted our experiments on 8 NVIDIA A6000 GPUs.
+
+Evaluation Metrics. We use global and local model accuracy as our evaluation metrics. Specifically, global model accuracy is defined as the server model accuracy on the test set; and local model accuracy is defined as the accuracy of the server model on each of the client’s local datasets. For CIFAR-10 and CIFAR-100, we report the classification accuracy. For Stack Overflow, we report the next word prediction accuracy which includes both out-of-vocabulary (OOV) and end-of-sentence (EOS) tokens. We run our experiments using five different seeds for CIFAR-10 and CIFAR-100 and using three different seeds for Stack Overflow.
+
+# 4.1 Performance Comparison with State-of-the-Art Model-Heterogeneous FL Methods
+
+First, we compare the performance of FedRolex with state-of-the-art PT and KD-based modelheterogeneous FL methods. For a fair comparison, we followed the experimental settings used in prior arts where the distributions of client model capacities are uniform and the global server model is the same as the largest client model.
+
+Evaluation Results: Table 3 summarizes our results. We have two observations. (1) In comparison with state-of-the-art PT-based methods, under the small-model small-dataset regime, FedRolex consistently outperforms HeteroFL and Federated Dropout under both low and more challenging high data heterogeneity scenarios. In particular, under high data heterogeneity, Federated Dropout which extracts sub-model randomly has worse performance than FedRolex and HeteroFL which both extract sub-models in a deterministic manner. Under large-model large-dataset regime, FedRolex also outperforms both HeteroFL and Federated Dropout. These results together demonstrate the superiority of FedRolex under both regimes. (2) In comparison with state-of-the-art KD-based methods, FedRolex only performs worse than Fed-ET and FedDF on CIFAR-10 under high data heterogeneity, but outperforms all the KD-based methods on the more challenging CIFAR-100 which has a larger number classes than CIFAR-10 under both low and high data heterogeneity scenarios. It is important to note that KD-based methods leverage public data to boost their model accuracy while FedRolex does not.
+
+Table 3: Global model accuracy comparison between FedRolex, PT and KD-based modelheterogeneous FL methods, and model-homogeneous FL methods. Note that the results of KD-based methods were obtained from [12]. For Stack Overflow, since KD-based methods cannot be directly used for language modeling tasks, their results are marked as N/A.   
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Method</td><td colspan="2">High Data Heterogeneity</td><td colspan="2">Low Data Heterogeneity</td><td rowspan="2">Stack Overflow</td></tr><tr><td>CIFAR-10</td><td>CIFAR-100</td><td>CIFAR-10</td><td>CIFAR-100</td></tr><tr><td rowspan="3">KD-based</td><td>FedDF</td><td>73.81 (± 0.42)</td><td>31.87 (± 0.46)</td><td>76.55 (± 0.32)</td><td>37.87 (± 0.31)</td><td>N/A</td></tr><tr><td>DS-FL</td><td>65.27 (±0.53)</td><td>29.12 (± 0.51)</td><td>68.44 (± 0.47)</td><td>33.56 (± 0.55)</td><td>N/A</td></tr><tr><td>Fed-ET</td><td>78.66 (± 0.31)</td><td>35.78 (± 0.45)</td><td>81.13 (± 0.28)</td><td>41.58 (± 0.36)</td><td>N/A</td></tr><tr><td rowspan="3">PT-based</td><td>HeteroFL</td><td>63.90 (± 2.74)</td><td>52.38 (± 0.80)</td><td>73.19 (± 1.71)</td><td>57.44 (± 0.42)</td><td>27.21 (± 0.22)</td></tr><tr><td>Federated Dropout</td><td>46.64 (± 3.05)</td><td>45.07 (± 0.07)</td><td>76.20 (± 2.53)</td><td>46.40 (± 0.21)</td><td>23.46 (± 0.12)</td></tr><tr><td>FedRolex</td><td>69.44 (± 1.50)</td><td>56.57 (± 0.15)</td><td>84.45 (± 0.36)</td><td>58.73 (± 0.33)</td><td>29.22 (± 0.24)</td></tr><tr><td rowspan="2"></td><td>Homogeneous (smallest)</td><td>38.82(± 0.88)</td><td>12.69 (± 0.50)</td><td>46.86 (± 0.54)</td><td>19.70 (± 0.34)</td><td>27.32 (± 0.12)</td></tr><tr><td>Homogeneous (largest)</td><td>75.74 (± 0.42)</td><td>60.89 (± 0.60)</td><td>84.48 (± 0.58)</td><td>62.51 (± 0.20)</td><td>29.79 (± 0.32)</td></tr></table>
+
+# 4.2 Performance Comparison with Model-Homogeneous FL Methods
+
+We also compare the global model accuracy of FedRolex with two model-homogeneous cases where all the clients have the largest capacity model $\beta = \{ 1 \}$ ) and the smallest capacity model $\begin{array} { r } { \left( \beta = \sqrt { 1 / 1 6 } \right) } \end{array}$ ), representing the upper and lower-bound performance, respectively.
+
+Evaluation Results: As listed in Table 3, compared with other PT-based methods, FedRolex reduces the gap in global model accuracy between model-heterogeneous and upper-bound modelhomogeneous settings. In particular, FedRolex is on par with the upper-bound model-homogeneous case for Stack Overflow, whereas both HeteroFL and Federated Dropout perform even worse than the model homogeneous case using the smallest model. This result indicates that with FedRolex, we will not be constrained to only using high-end devices to achieve competitive global model accuracy. Note that Fed-ET achieves a higher global model accuracy than the model-homogeneous upper bound on CIFAR-10 under high data heterogeneity, which showcases the advantage of using public data.
+
+# 4.3 Impact of Client Model Heterogeneity Distribution
+
+In our previous experiments, the distributions of model capacities across client devices are set to be uniform. In this experiment, we aim to understand the impact of the client model heterogeneity distribution. To do so, without loss of generality, we use two client model capacities $\beta = \{ \bar { 1 } , { ^ { 1 } / _ { 1 6 } } \}$ and vary the distribution ratio between the two (denoted as $\rho \mathrm { \hbar }$ ) where $\rho = 1$ represents the case in which all the clients have the largest capacity model $\begin{array} { r } { \beta = \{ 1 \} } \end{array}$ ) and $\rho = 0$ represents the case in which all the clients have the smallest capacity model $( \beta = \left\{ { 1 } / { 1 6 } \right\}$ ).
+
+Evaluation Results: Figure 3 shows how global model accuracy changes when $\rho$ varies from 0 to 1 for CIFAR-10, CIFAR-100 and Stack Overflow. We have three observations. (1) For CIFAR-10 (Figure 3(i)), there is a large gap in global model accuracy between high and low data heterogeneity for a wide range of $\rho$ (from 0.1 to 1). This is because CIFAR-10 is a relatively simple task and hence the global model accuracy is bottlenecked by the level of data heterogeneity instead of model capacity. This result indicates that having more high-capacity models in the cohort has only limited contribution to global model accuracy. (2) For the more challenging CIFAR-100 (Figure 3(ii)), the gap in global model accuracy is much lower between high and low data heterogeneity. In contrast to CIFAR-10, the global model accuracy is bottlenecked by the highest capacity of the models rather than the level of data heterogeneity. (3) For both regimes (Figure 3(i)(ii) vs. Figure 3(iii)), we observe that having a small fraction of large-capacity models significantly boosts the global model accuracy, but keeping increasing the ratio of large-capacity models has limited contribution to the accuracy.
+
+![](images/54a8526c0ffbaff84447015f2bec64690c39e542670cf9bce67091bc7c1df68c.jpg)  
+Figure 3: Impact of client model heterogeneity distribution on global model accuracy for (i) CIFAR10, (ii) CIFAR-100, and (iii) Stack Overflow.
+
+# 4.4 Performance on Training Larger Server Model
+
+Similar to Federated Dropout, one advantage of FedRolex over static sub-model extraction methods (HeteroFL and FjORD) is that FedRolex is able to train a global server model that is larger than the largest client model. In this experiment, we aim to evaluate the performance of FedRolex on training larger server models. To do so, we consider the case where the size of the global server model is $\gamma \overset { \cdot } { = } \{ 2 , 4 , 8 , 1 6 \}$ times the size of client models. For simplicity, all client models have the same size.
+
+Evaluation Results: Figure 4(i) and Figure 4(ii) compare FedRolex with Federated Dropout in terms of global model accuracy when $\gamma$ for CIFAR-10 and CIFAR-100, respectively. As shown, although the global model accuracy drops for both FedRolex and Federated Dropout when $\gamma$ increases, especially from 1 to 4, FedRolex consistently achieves higher global model accuracy than Federated Dropout across $\gamma = \{ 2 , 4 , 8 , 1 6 \}$ under both low and high data heterogeneity. For Stack Overflow (Figure 4(iii)), the global model accuracy has a much smaller drop when $\gamma$ increases. This demonstrates the superiority of using large models on large-scale datasets for training larger server models.
+
+![](images/dc26fa907235d516ac65f0e07501364b48fc28547d09a1bd772a601c68306ae5.jpg)  
+Figure 4: Performance on training larger server model when the server model is $\gamma$ times the size of the client model for (i) CIFAR-10, (ii) CIFAR-100, and (iii) Stack Overflow.
+
+# 4.5 Enhance Inclusiveness of FL in Real-world Distribution
+
+A primary vision of FedRolex is to enhance the inclusiveness of FL. To demonstrate this, in this experiment, we use real-world household income distribution to emulate real-world device distribution. Specifically, we retrieve household income distribution information from Bureau [33]. We map $\beta _ { n } = { ^ { 1 } } / { _ { 1 6 } }$ with the income group with earning less than $\$ 75,000$ and assign proportions of remaining groups in $\$ 25$ , 000 increments with increasing values of $\beta _ { n }$ . Detailed mapping of this distribution to the corresponding income distribution is provided in Figure 7 in Appendix A.7.
+
+![](images/3a5cc1ceb24d9af6e3ccb5412cfe9ff89a92852da8096a9afd3da5980d14c226.jpg)  
+Figure 5: Local model accuracy distribution of FedRolex (orange color) vs. the smallest modelhomogeneous case (blue color) for CIFAR-10 and CIFAR-100 under low and high data heterogeneity.
+
+Evaluation Results: Table 4 shows both the global and local model accuracies of FedRolex for CIFAR-10 and CIFAR-100 as well as the global model accuracy on Stack Overflow under the emulated real-world device distribution. Again, we compare with two model-homogeneous cases where all clients have the smallest and largest model capacities, representing lower and upper-bound accuracy, respectively. We make two observations. (1) Looking at the global model accuracy, FedRolex consistently outperforms the lower-bound model-homogeneous case across CIFAR-10, CIFAR-100, and Stack Overflow. This result indicates that FedRolex enhances the inclusiveness of FL and improves the accuracy of the global model, which would otherwise not be able to achieve. (2) Looking at the local model accuracy, FedRolex significantly outperforms the lower-bound modelhomogeneous case on CIFAR-10 and CIFAR-100 under both low and high data heterogeneity. This result indicates that FedRolex effectively boosts the performance of low-end devices, which would otherwise not benefit from FL. A detailed illustration of how local model accuracy distribution of individual clients shifts when FedRolex is used compared to the smallest model-homogeneous case with the same client outreach is shown in Figure 5.
+
+# 5 Conclusion
+
+We presented FedRolex, a partial training (PT)-based model-heterogeneous FL approach that is able to train a global server model larger than the largest client model. FedRolex proposed a rolling sub-model extraction scheme that enables parameters of the global server model to be evenly trained to mitigate client drift induced by model heterogeneity. We provided a theoretical statistical analysis of its advantage over Federated Dropout. Our experimental results show that FedRolex consistently outperforms state-of-the-art PT-based methods across models and datasets at both small and large scales. Moreover, we demonstrated its performance on an emulated real-world device distribution and show FedRolex contributes to making FL more inclusive.
+
+Limitations and Future works. In this work, we provided a statistical analysis of FedRolex. Full convergence analysis of FedRolex is not trivial and is left for future work. In addition, the goal of this work is to train a global server model using a federation of heterogeneous client models. Determining what models to deploy onto each client after the global server model is trained is a separate task, especially when the global server model is large. We will pursue it as our future work.
+
+# 6 Acknowledgement
+
+We thank the reviewers for their helpful comments. This work was partially supported by NSF PFI:BIC-1632051, CNS-1814551, DMS-2012439, and a Google Computing Platform (GCP) grant.
+
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+[33] U.S. Census Bureau. Percentage distribution of household income in the u.s. in 2020. In Statista, September 2021. Retrieved May 18, 2022, from https://www.statista.com/statistics/ 203183/percentage-distribution-of-household-income-in-the-us.
+
+[34] Sheldon M. Ross. Introduction to Probability Models. Academic Press, San Diego, CA, USA, 11th edition, 2014.
+
+[35] Donald J Newman. The double dixie cup problem. The American Mathematical Monthly, 67 (1):58–61, 1960.
+
+[36] Manoj Ghuhan Arivazhagan, Vinay Aggarwal, Aaditya Kumar Singh, and Sunav Choudhary. Federated learning with personalization layers. arXiv preprint arXiv:1912.00818, 2019.
+
+# 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]   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(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? [Yes] (b) Did you include complete proofs of all theoretical results? [Yes]
+
+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]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   
+(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]
+
+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]   
+(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/dev/OtxyysUdBE/OtxyysUdBE_content_list.json

@@ -0,0 +1,1433 @@
+[
+    {
+        "type": "text",
+        "text": "FedRolex: Model-Heterogeneous Federated Learning with Rolling Sub-Model Extraction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Samiul Alam1,2, Luyang $\\mathbf { L i u ^ { 3 } }$ , Ming Yan4,1, Mi Zhang2,1 1Michigan State University, 2The Ohio State University, 3Google Research, 4The Chinese University of Hong Kong, Shenzhen alamsami@msu.edu, luyangliu@google.com, yanming@cuhk.edu.cn, mizhang.1@osu.edu ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            335
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Most cross-device federated learning (FL) studies focus on the model-homogeneous setting where the global server model and local client models are identical. However, such constraint not only excludes low-end clients who would otherwise make unique contributions to model training but also restrains clients from training large models due to on-device resource bottlenecks. In this work, we propose FedRolex, a partial training (PT)-based approach that enables model-heterogeneous FL and can train a global server model larger than the largest client model. At its core, FedRolex employs a rolling sub-model extraction scheme that allows different parts of the global server model to be evenly trained, which mitigates the client drift induced by the inconsistency between individual client models and server model architectures. We show that FedRolex outperforms state-of-the-art PTbased model-heterogeneous FL methods (e.g. Federated Dropout) and reduces the gap between model-heterogeneous and model-homogeneous FL, especially under the large-model large-dataset regime. In addition, we provide theoretical statistical analysis on its advantage over Federated Dropout and evaluate FedRolex on an emulated real-world device distribution to show that FedRolex can enhance the inclusiveness of FL and boost the performance of low-end devices that would otherwise not benefit from FL. Our code is available at: https://github.com/AIoTMLSys-Lab/FedRolex. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Federated learning (FL) is a machine learning paradigm that trains models from distributed clients with private data under the coordination of a central server [1, 2]. In this work, we focus on crossdevice FL where clients are usually resource-constrained edge devices. The majority of existing cross-device FL studies focus on the model-homogeneous setting [3–6], in which the server model and the client models across all the participating client devices are identical. However, modelhomogeneous FL are confronted with two fundamental constraints: (1) device heterogeneity is a more realistic consideration when deploying FL systems in real-world applications: different client devices could have very diverse on-device resources and are only capable of training models with capacities that match their on-device resources. Having the same model on all the devices would, unfortunately, exclude clients with low-end devices who would otherwise make unique contributions to model training from their own local data; (2) state-of-the-art machine learning has moved towards large models [7] such as Transformer [8]. Restricting server and client models to be the same inevitably causes model-homogeneous FL to fail to train such large models due to the resource constraint of client devices. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "To relax the fundamental constraints of model-homogeneous FL, model-heterogeneous FL was proposed where heterogeneous models with different capacities across the server and the clients are trained during the federated training process. One primary challenge in model-heterogeneous $\\mathrm { F L }$ is the aggregation of heterogeneous client models. To address this challenge, knowledge distillation (KD)-based approaches have been proposed [9–12], in which the client models serve as teachers and the server ensembles the knowledge distilled from the individual client models. However, KD-based approaches in general require public data on the server to achieve competitive model accuracy, whereas the desired public data may not be always available in practice. Moreover, since KD-based approaches need the individual client models (whole models or prediction layers) or their outputs to be sent to the server, they are incompatible with secure aggregation protocols [16], which limits their privacy guarantee. To remove the dependency on public data and ensure compatibility with secure aggregation, partial training (PT)-based approaches such as random sub-model extraction (Federated Dropout [13]) and static sub-model extraction (HeteroFL [14], FjORD [15]) were proposed. In these approaches, each client trains a smaller sub-model extracted from the larger global server model, and the server model is updated by aggregating those trained sub-models. However, the fundamental issue of existing PT-based methods is that the sub-models are extracted in ways (either random or static) such that the parameters of the global server model are not evenly trained. This makes the server model vulnerable to client drift1 induced by the inconsistency between individual client model and server model architectures – a unique challenge of model-heterogeneous FL. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "table",
+        "img_path": "images/a31005b611b97633a2e26199175d66c6e288638f43a05594141f3a9b10b90885.jpg",
+        "table_caption": [
+            "Table 1: Comparison of FedRolex with model-homogeneous and model-heterogeneous FL methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Model Heterogeneity</td><td>Aggregation Scheme</td><td>Sub-model Extraction Scheme</td><td>Need of Public Data</td><td>Server Model Size</td><td>Compatibility with Secure Aggregation</td></tr><tr><td>FedAvg[3]</td><td rowspan=\"4\">No</td><td rowspan=\"4\"></td><td rowspan=\"4\"></td><td>No</td><td>= Client Model</td><td>Yes</td></tr><tr><td>FedProx [4]</td><td>No</td><td>= Client Model</td><td>Yes</td></tr><tr><td>SCAFFOLD [5]</td><td>No</td><td>= Client Model</td><td>Yes</td></tr><tr><td>FedBE[6]</td><td>Unlabeled</td><td>= Client Model</td><td>No</td></tr><tr><td>FedGKT[9]</td><td rowspan=\"4\">Yes</td><td rowspan=\"4\">Knowledge Distillation</td><td rowspan=\"4\"></td><td>No</td><td>≥ Largest Client Model</td><td>No</td></tr><tr><td>FedDF[10]</td><td>Unlabeled</td><td>= Largest Client Model</td><td>No</td></tr><tr><td>DS-FL [11] Fed-ET [12]</td><td>Unlabeled</td><td>= Largest Client Model</td><td>No</td></tr><tr><td></td><td>Unlabeled</td><td>≥ Largest Client Model</td><td>No</td></tr><tr><td>Federated Dropout [13]</td><td rowspan=\"4\">Yes</td><td rowspan=\"4\">Partial Training</td><td>Random Static</td><td>No</td><td>≥ Largest Client Model</td><td>Yes</td></tr><tr><td>HeteroFL [14]</td><td>Static</td><td>No No</td><td>=Largest Client Model</td><td>Yes</td></tr><tr><td>FjORD [15]</td><td></td><td></td><td>=Largest Client Model</td><td>Yes</td></tr><tr><td>FedRolex (Our Approach)</td><td>Rolling</td><td>No</td><td>≥Largest Client Model</td><td>Yes</td></tr></table>",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a PT-based model-heterogeneous FL approach named FedRolex to tackle the fundamental issue of existing methods. The key difference between FedRolex and existing PT-based methods is how the sub-models are extracted for each client over communication rounds in the federated training process. Specifically, instead of extracting sub-models in either random or static manner, FedRolex proposes a rolling sub-model extraction scheme, where the sub-model is extracted from the global server model using a rolling window that advances in each communication round. Since the window is rolling, sub-models from different parts of the global model are extracted in sequence in different rounds. As a result, all the parameters of the global server model are evenly trained over the local data of client devices. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The proposed rolling sub-model extraction scheme, though simple, has equipped FedRolex with multifold merits compared to prior arts (Table 1): (1) FedRolex enables different parts of the global server model to be evenly trained, which mitigates the client drift induced by model heterogeneity. (2) Contrary to static sub-model extraction approaches (HeteroFL, FjORD), FedRolex is able to train a global server model that is larger than the largest client model, enabling FL to benefit from the superior performance brought by large models. It echoes some concurrent efforts in developing FL primitives to support training large server models in cross-device settings, e.g. Federated Select [17]. (3) Compared to random sub-model extraction (Federated Dropout), as we show in our theoretical statistical analysis in Section 3 and Appendix A.1, the global server model is trained more evenly by FedRolex as the expected number of rounds for FedRolex going through all the parameters of the global model for at least certain times is smaller than that of Federated Dropout. (4) FedRolex only needs to transmit the sub-model that is needed by a given client instead of the full server model to the client. This allows clients to contribute to federated training under resource constraints and reduces communication overheads (Appendix A.6). (5) Lastly, FedRolex is fully compatible with existing secure aggregation protocols that enhance the privacy properties of FL systems. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We evaluate the performance of FedRolex under two regimes: i) small-model small-dataset regime (most existing cross-device FL studies use this combination), and ii) large-model large-dataset regime (this combination echos recent efforts on pushing the frontier of cross-device FL towards training large server models on large-scale datasets [18–21]). We highlight five of our findings: (1) FedRolex consistently outperforms state-of-the-art PT-based model-heterogeneous FL methods under both small-model small-dataset and large-model large-dataset regimes $( \\ S 4 . 1 )$ . (2) FedRolex reduces the gap between model-heterogeneous and model-homogeneous FL, especially under large-model large-dataset regime (§4.2). (3) With FedRolex, under both regimes, having a small fraction of large-capacity models could significantly boost the global model accuracy (§4.3). (4) FedRolex is able to train a global server model that is larger than the largest client model and outperforms Federated Dropout in terms of global model accuracy $( \\ S 4 . 4 )$ . (5) Using an emulated real-world device distribution, we show that FedRolex enhances the inclusiveness of FL and boosts the performance of low-end devices that would otherwise not benefit from FL (§4.5). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            255,
+            323,
+            271
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Knowledge Distillation (KD)-based Model-Heterogeneous FL. One primary approach for modelheterogeneous FL in cross-device settings is based on knowledge distillation (KD) [22]. In particular, FedDF [10] distills knowledge from a set of classifiers trained with private data from a federation of client devices. The logit outputs of each classifier against an unlabeled public dataset are then used to train a student model at the server with KD. Similarly, DS-FL [11] utilized an unlabeled public dataset at the server and proposed a distillation-based semi-supervised FL approach to enhance performance by pseudo-labeling the public data. FedGKT [9] proposed group knowledge transfer in which knowledge is transferred to a large model in the server from clients without public data. Fed-ET [12] proposed a weighted consensus distillation scheme with diversity regularization that enables the training of a large server model with smaller client models. KD-based approaches, however, have several limitations: they often require public data to achieve competitive model accuracy. This is because model accuracy is dependent on the size of public data as well as the domain similarity of public data with client data [10, 12, 23]. Furthermore, as KD-based methods use client model weights partially or entirely as teachers to transfer knowledge to the server, they are incompatible with secure aggregation protocols, making them vulnerable to backdoor attacks [20]. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Partial Training (PT)-based Model-Heterogeneous FL. To address the limitations of KD-based approaches, partial training (PT) has emerged as another solution for model-heterogeneous FL. Depending on how the sub-models are extracted from the global server model, existing PT-based methods can be in general categorized into two groups: random sub-model extraction and static sub-model extraction. Specifically, inspired by the dropout technique commonly used in centralized training [24], Federated Dropout [13] proposed to randomly extract sub-models from the global model. Though easy to be integrated into existing FL frameworks, as reported in [25], Federated Dropout becomes less effective when the data heterogeneity is high and the client cohort is small due to its randomness in selecting sub-models. In contrast, HeteroFL [14] and FjORD [15] proposed static extraction schemes where sub-models are always extracted from a designated part of the global server model. However, such a static extraction strategy has two primary drawbacks. First, the global server model is restricted to the same size as the largest client model. As such, the size and capability of the global model are implicitly restricted by the resources of client devices, making it not able to train large models due to resource bottlenecks at client devices. Second and more importantly, under static extraction, depending on their resource demands, different sub-models can only be trained on clients whose on-device resources are matched. As a consequence, part of the global server model cannot be trained on data at low-end client devices, causing different parts of the global model to be trained on data with different distributions. This would degrade the performance of the global model, especially under high data heterogeneity. In this work, we propose a rolling sub-model extraction scheme that tackles the issues of both random and static sub-model extraction methods. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Formulation of Model-Heterogeneous FL ",
+        "text_level": 1,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Let $\\mathcal { N }$ denote $N$ client devices with non-IID (non-identically and independently distributed) local data $\\boldsymbol { D } = \\{ D _ { 1 } , D _ { 2 } , . . . , D _ { N } \\}$ . Model-homogeneous $\\mathrm { F L }$ trains a global model of parameter $\\theta$ by solving the following optimization problem: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c760ff9c5878308a635eab7dba1f201e4b34b028c520da22fb0a38b196438a04.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } F \\left( \\theta \\right) \\triangleq \\sum _ { n = 1 } ^ { N } p _ { n } F _ { n } ( \\theta )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "with ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9b195b4ac313e9254b5a5378bdfb92c523315005e0ad56f670136d95e8380f7.jpg",
+        "text": "$$\nF _ { n } ( \\theta ) \\triangleq { \\frac { 1 } { m _ { n } } } \\sum _ { k = 1 } ^ { m _ { n } } l ( \\theta ; d _ { n , k } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            108,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $D _ { n } \\triangleq \\{ d _ { n , 1 } , d _ { n , 2 } , d _ { n , 3 } . . . d _ { n , m _ { n } } \\}$ is the set of local data samples of client $n$ and $p _ { n }$ is its corresponding weight such that $p _ { n } \\geq 0$ and $\\textstyle \\sum _ { n = 1 } ^ { N } p _ { n } = 1$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In comparison, in model-heterogeneous $\\mathrm { F L }$ , clients train local models with heterogeneous capacities $\\beta = \\{ \\bar { \\beta } _ { 1 } , \\beta _ { 2 } , . . . , \\beta _ { N } \\}$ , and the local objective function of the $n ^ { t h }$ client becomes ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f5a2357a12965746cef3397a0263f31d647e736ac8770900a5695e8b3d9d37b.jpg",
+        "text": "$$\nF _ { n } ^ { \\prime } ( \\theta _ { n } ) \\triangleq { \\frac { 1 } { m _ { n } } } \\sum _ { k = 1 } ^ { m _ { n } } l ( \\theta _ { n } ; d _ { n , k } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\beta _ { n }$ denote the model capacity of client $n$ , and we define it as the proportion of nodes extracted from each layer in $\\theta$ for client $n$ . The size of $\\theta _ { n }$ depends on $\\beta _ { n }$ , and the parameter $\\theta _ { n }$ is obtained by selecting a sub-model from the global model $\\theta$ , which can change from one round to another. If $\\theta _ { n }$ changes, the objective function also changes. For simplicity, we use the same notation $l$ for the loss function for all clients and rounds, though they differ between clients and rounds. The key to model-heterogeneous $\\mathrm { F L }$ is selecting $\\theta _ { n }$ from the global model $\\theta$ given model capacity $\\beta _ { n }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 FedRolex: Model-Heterogeneous FL with Rolling Sub-Model Extraction ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As a partial training (PT)-based approach, at each client, FedRolex trains only a sub-model extracted from the global server model and sends the corresponding submodel updates back to the server for update aggregation. To help understand how FedRolex works, for simplicity, Figure 1 illustrates three rounds of federated training of FedRolex on two participating heterogeneous clients, where one trains a large-capacity sub-model (left) and the other trains a small-capacity one (right). At the high level, at each round, the server extracts sub-models of different capacities from the global model and separately broadcasts them to the clients that have the corresponding capabilities. The clients train the received sub-models on their local data and transmit their heterogeneous sub-model updates to the server. Lastly, the server aggregates those updates, and the result of the aggregation is used to update the global model for the next round. The pseudocode of FedRolex is in Algorithm 1. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The key to the design of FedRolex involves two design choices. In the following, we describe them in detail. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "(1) What sub-models to be extracted for each client across different rounds? At the server, FedRolex utilizes a rolling window to extract the sub-model from the global model. The rolling window advances in each round, and loops over all parts of the global model in sequence across different rounds. This process iterates such that the global model is evenly trained until convergence. ",
+        "bbox": [
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+    {
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+        "img_path": "images/279cece0e5a7046b19d662526f8fb00c8c1e4993e9851453b67c2a2f55b155a2.jpg",
+        "image_caption": [
+            "Figure 1: Overview of the rolling submodel extraction scheme in FedRolex. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            550,
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+        ],
+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Taking Figure 1 as an example: in round $j$ , the large-capacity and small-capacity client model extracted from the global model is $\\{ a , b , c , d \\}$ and $\\{ c , \\bar { d } , e \\}$ , respectively. In round $j + 1$ , the rolling window advances one step2, the large-capacity and small-capacity client model becomes $\\{ b , c , d , e \\}$ and $\\{ d , e , a \\}$ , respectively. Similarly, in round $j + 2$ , the rolling window advances one step further, and the large-capacity and small-capacity client model becomes $\\{ c , d , e , a \\}$ and $\\{ e , a , b \\}$ , respectively. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Such a rolling sub-model extraction scheme can be formalized as follows. Let $\\theta _ { n } ^ { ( j ) }$ denote the parameters of the sub-model extracted from the global model for client $n$ in round $j$ , $K _ { i }$ denote the total number of nodes in layer $i$ of the global model, and $S _ { n , i } ^ { ( j ) }$ denote the node indices of layer $i$ of the global model that belongs to the extracted sub-model for client $n$ in round $j$ . Then the layer $i$ of the sub-model extracted by the rolling sub-model extraction scheme for client $n$ in round $j$ is given by: ",
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+        "type": "equation",
+        "img_path": "images/9049f6d5815d235fc61830a577a88b845e4fd5f6a3655151b3c93db2c880402e.jpg",
+        "text": "$$\nS _ { n , i } ^ { ( j ) } = \\left\\{ \\begin{array} { l l } { \\{ \\hat { j } , \\hat { j } + 1 , \\dots , \\hat { j } + \\lfloor \\beta _ { n } K _ { i } \\rfloor - 1 \\} } & { \\mathrm { ~ i f ~ } \\hat { j } + \\lfloor \\beta _ { n } K _ { i } \\rfloor \\le K _ { i } , } \\\\ { \\{ \\hat { j } , \\hat { j } + 1 , \\dots , K _ { i } - 1 \\} \\cup \\{ 0 , 1 , \\dots , \\hat { j } + \\lfloor \\beta _ { n } K _ { i } \\rfloor - 1 - K _ { i } \\} } & { \\mathrm { ~ e l s e . ~ } } \\end{array} \\right.\n$$",
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+    {
+        "type": "text",
+        "text": "where ${ \\hat { j } } = j$ mod $K _ { i }$ ",
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+    {
+        "type": "text",
+        "text": "(2) How to aggregate heterogeneous sub-model updates to update the global model? FedRolex employs a straightforward selective averaging scheme with no client weighting to aggregate heterogeneous sub-model updates sent from the clients to update the global model3. Specifically, it computes the average of the updates for each parameter of the global model separately based on how many clients in a round updated that parameter. The parameter remains unchanged if no clients updated it. ",
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+        "type": "text",
+        "text": "Taking Figure 1 again as an example: in round $j$ , the updates for $a$ and $b$ are obtained from the large-capacity model and the update for $e$ is from the small-capacity model only. In contrast, since $c$ and $d$ are part of both models, the update is computed by taking the average from both models. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: FedRolex ",
+        "text_level": 1,
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+        "text": "1 Initialization ; $\\theta ^ { ( 0 ) }$ , N 12 Subroutine clientStep(θ(j)n , Dn)   \nInput : $: D _ { n } \\beta _ { n } \\forall n \\in \\mathcal N ,$ 13 mn ←− len(Dn)   \nOutput : 14 for k ← 0 to mn do   \n2 Server Executes 15 $\\begin{array} { r l } { \\boxed { } } & { { } \\theta _ { n } \\longleftarrow \\theta _ { n } - \\eta \\nabla l ( \\theta _ { n } ; d _ { n , k } ) } \\end{array}$   \n3 for $j  0$ to $J - 1$ do 16 end   \n4 Sample subset $\\mathcal { M }$ from $\\mathcal { N }$ 17 return $\\theta _ { n }$   \n5 Broadcast θ(j) $\\theta _ { m , S _ { m , i } ^ { ( j ) } } ^ { ( j ) }$ to client $m \\in \\mathcal { M }$   \n6 $\\forall i , \\boldsymbol { S } _ { m , i } ^ { ( j ) }$ from Equation (4)   \n7 for each client $m \\in \\mathcal { M }$ do   \n8 clientStep(θ(j)m , Dm)   \n9 end   \n10 Aggregate θ(j+1)[i,k] according to Equation (10)   \n11 end ",
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+        "type": "text",
+        "text": "3.3 Comparison with Random and Static Sub-model Extraction Schemes ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Existing sub-model extraction schemes can be grouped as random-based (Federated Dropout) and static-based (HeteroFL, FjORD) methods. In this section, we describe the differences between them and the proposed rolling-based scheme employed in FedRolex. For comparison purpose, the pseudocodes of both Federated Dropout and HeteroFL are included in Appendix A.8. ",
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+        "type": "text",
+        "text": "3.3.1 Comparison with Random Sub-Model Extraction Scheme ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In random sub-model extraction scheme, in each round, the sub-models are extracted from the global model in a random manner. As such, the layer $i$ of the sub-model extracted by the random sub-model extraction scheme for client $n$ in round $j$ is given by: ",
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+    {
+        "type": "equation",
+        "img_path": "images/18348aaf6912f49ff4fe835364ccd70bc5f406ef7a838d27b93f342b7b717051.jpg",
+        "text": "$$\n\\begin{array} { r } { S _ { n , i } ^ { ( j ) } = \\{ k _ { c } ~ | ~ \\mathrm { i n t e g e r } ~ k _ { c } \\in [ 0 , K _ { i } - 1 ] \\mathrm { ~ f o r } ~ 1 \\leq c \\leq \\lfloor \\beta _ { n } K _ { i } \\rfloor \\} , } \\end{array}\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where a total number of $\\lfloor \\beta _ { n } K _ { i } \\rfloor$ nodes are randomly chosen from the global model. ",
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+    {
+        "type": "text",
+        "text": "Discussion: As shown in Figure 2(left), similar to the proposed rolling-based scheme, the sub-models extracted across different rounds by the random-based scheme have different architectures. However, due to its randomness in selecting sub-models in each round, the global model is trained less evenly, making it more vulnerable to client drift. In short, although the expected value of the frequency for updating each index is the same for all the indices, their exact frequencies are not the same due to randomness. Consequently, the random-based scheme cannot balance the update frequencies of different parts of the global model, and it inevitably takes more rounds to update the whole global model. Moreover, as we show in Appendix A.1, the expected number of rounds for Federated Dropout selecting all $I$ sub-models at least $m$ times is in the order of $I \\log ( I ) + I ( m - 1 ) \\log \\log I$ , which is larger than that of FedRolex, $m I$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/e165594baec07c04a727b4d8fe9fbf59ae3c69c99c899f262efc0e7c6ec1f5ba.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of how sub-models are extracted by random sub-model extraction scheme (Left) and static sub-model extraction scheme (Right) over two rounds. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "3.3.2 Comparison with Static Sub-Model Extraction Scheme ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In static sub-model extraction scheme, in each round, the sub-models are always extracted from a designated part of the global model. As such, the layer $i$ of the sub-model extracted by the static sub-model extraction scheme for client $n$ in round $j$ is given by: ",
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+        "text": "$$\n\\mathcal { S } _ { n , i } ^ { ( j ) } = \\{ 0 , 1 , 2 , \\ldots , \\lfloor \\beta _ { n } K _ { i } \\rfloor - 1 \\} .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Note that $S _ { n , i } ^ { ( j ) }$ does not depend on $j$ . In other words, as shown in Figure $2 ( \\mathrm { r i g h t } )$ , the same sub-model is extracted for each client in every round. Moreover, the client model with smaller capacity and client model with larger capacity are not independent. As shown in Figure 2(right), the small-capacity model $\\{ a , b , c \\}$ is a part of the large-capacity model $\\{ a , b , c , d \\}$ , which again, is a part of the globalcapacity model $\\{ a , b , c , d , e \\}$ . These are the two key differences between both the random-based and the proposed rolling-based scheme. ",
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+    {
+        "type": "text",
+        "text": "Discussion: Given that, the static-based scheme, however, has two primary drawbacks. First, to cover the whole global model, there must be clients to train the full-size global model $\\{ a , b , c , d , e \\}$ . As such, the global model is restricted to the same size as the largest client model. Second, as shown in Figure 2(right), while $a , b$ and $c$ will be trained on data on all three types of clients, $d$ will not be trained on data on small-capacity clients, and $e$ will only be trained on data on global-model-capacity clients. As a consequence, different parts of the global model are trained on data with different distributions, which inevitably degrades the global model training quality. ",
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+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Datasets and Models. We evaluate the performance of FedRolex under two regimes. Under small-model small-dataset regime, we train pre-activated ResNet18 (PreResNet18) models [26] on CIFAR-10 and CIFAR-100 [27]. We replace the batch Normalization in PreResNet18 with static batch normalization [14, 28] and add a scalar module after each convolution layer [14]. Under largemodel large-dataset regime, we use Stack Overflow [29] and followed [2] to train a modified 3-layer Transformer [8] with a vocabulary of 10, 000 words, where the dimension of token embeddings is ",
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+    {
+        "type": "text",
+        "text": "128, and the hidden dimension of the feed-forward network (FFN) block is 2048. We use ReLU activation and use 8 heads for the multi-head attention where each head is based on 12-dimensional (query, key, value) vectors. The statistics of the datasets are listed in Table 2. ",
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+    {
+        "type": "table",
+        "img_path": "images/aa7c72531a93bb351ff83184ed6df8ead096711f1c3930ba3621d6ef590f22e2.jpg",
+        "table_caption": [
+            "Table 2: Dataset statistics. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset CIFAR-10</td><td>Train Clients</td><td>Train Examples 50.000</td><td>Validation Clients</td><td>Validation Examples</td><td>Test Clients</td><td>Test Examples 10,000</td></tr><tr><td>CIFAR-100</td><td>100 100</td><td>50.000</td><td>N/A N/A</td><td>N/A N/A</td><td>N/A N/A</td><td>10.000</td></tr><tr><td>Stack Overflow</td><td>342,477</td><td>135,818,730</td><td>38,758</td><td>16,491,230</td><td>204,088</td><td>16,586,035</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Data Heterogeneity. For CIFAR-10 and CIFAR-100, we followed HeteroFL [14] to model non-IID distributions by restricting each client to have $L$ labels. In our evaluation, we consider two levels of data heterogeneity. For CIFAR-10, we define $L = 2$ as high data heterogeneity and $L = 5$ as low data heterogeneity. For CIFAR-100, we use $L = 2 0$ as high data heterogeneity and $L = 5 0$ as low data heterogeneity. These two levels roughly correspond to Dirichlet distribution $D i r _ { K } ( \\alpha )$ with $\\alpha$ equal to 0.1 and 0.5, respectively. For Stack Overflow, the dataset is partitioned over user IDs, making the dataset naturally non-IID distributed. ",
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+        "type": "text",
+        "text": "Model Heterogeneity. Without loss of generality, in our evaluation, we consider five different client model capacities $\\beta = \\{ 1 , ^ { 1 / 2 } , ^  1 / 4 , 1 / 8 , 1 / \\bar { 1 } 6 \\}$ where for instance, $^ 1 / 2$ means the client model capacity is half of the largest client model capacity (full model). To generate these client models, for ResNet18, we vary the number of kernels in convolution layers and keep the nodes in the output layers the same. For Transformer, we vary the number of nodes in the hidden layer of the attention heads. ",
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+    {
+        "type": "text",
+        "text": "Baselines. We compare FedRolex against both state-of-the-art PT-based model-heterogeneous FL methods including Federated Dropout [13] and HeteroFL $[ 1 4 ] ^ { 4 }$ as well as state-of-the-art KD-based model-heterogeneous $\\mathrm { F L }$ methods including FedDF [10], DS-FL [11] and Fed-ET [12] 5. To ensure a fair comparison, all the PT-based baselines are trained using the same learning rate, number of communication rounds, and multi-step learning rate decay schedule. The details of the schedule for each dataset and experiment are described in Appendix A.7. ",
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+        "type": "text",
+        "text": "Configurations and Platform. For CIFAR-10 and CIFAR-100, we apply bounding box crop [30] to augment the images. In each communication round, $1 0 \\%$ of the clients are randomly selected from a pool of 100 clients. For Stack Overflow, we followed [2] to use a $1 0 \\%$ dropout rate to prevent overfitting, and 200 clients are randomly selected from a pool of 342, 477 clients in each communication round. The details of the hyper-parameters for model training are included in Appendix A.7. We implemented FedRolex and PT-based baselines using PyTorch [31] and Ray [32], and conducted our experiments on 8 NVIDIA A6000 GPUs. ",
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+        "type": "text",
+        "text": "Evaluation Metrics. We use global and local model accuracy as our evaluation metrics. Specifically, global model accuracy is defined as the server model accuracy on the test set; and local model accuracy is defined as the accuracy of the server model on each of the client’s local datasets. For CIFAR-10 and CIFAR-100, we report the classification accuracy. For Stack Overflow, we report the next word prediction accuracy which includes both out-of-vocabulary (OOV) and end-of-sentence (EOS) tokens. We run our experiments using five different seeds for CIFAR-10 and CIFAR-100 and using three different seeds for Stack Overflow. ",
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+    {
+        "type": "text",
+        "text": "4.1 Performance Comparison with State-of-the-Art Model-Heterogeneous FL Methods ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "First, we compare the performance of FedRolex with state-of-the-art PT and KD-based modelheterogeneous FL methods. For a fair comparison, we followed the experimental settings used in prior arts where the distributions of client model capacities are uniform and the global server model is the same as the largest client model. ",
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+    {
+        "type": "text",
+        "text": "Evaluation Results: Table 3 summarizes our results. We have two observations. (1) In comparison with state-of-the-art PT-based methods, under the small-model small-dataset regime, FedRolex consistently outperforms HeteroFL and Federated Dropout under both low and more challenging high data heterogeneity scenarios. In particular, under high data heterogeneity, Federated Dropout which extracts sub-model randomly has worse performance than FedRolex and HeteroFL which both extract sub-models in a deterministic manner. Under large-model large-dataset regime, FedRolex also outperforms both HeteroFL and Federated Dropout. These results together demonstrate the superiority of FedRolex under both regimes. (2) In comparison with state-of-the-art KD-based methods, FedRolex only performs worse than Fed-ET and FedDF on CIFAR-10 under high data heterogeneity, but outperforms all the KD-based methods on the more challenging CIFAR-100 which has a larger number classes than CIFAR-10 under both low and high data heterogeneity scenarios. It is important to note that KD-based methods leverage public data to boost their model accuracy while FedRolex does not. ",
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+    {
+        "type": "table",
+        "img_path": "images/c1297d2829b7612f0753dec4c5e5a96eb8553bd65ecccc8c127878563b9d609b.jpg",
+        "table_caption": [
+            "Table 3: Global model accuracy comparison between FedRolex, PT and KD-based modelheterogeneous FL methods, and model-homogeneous FL methods. Note that the results of KD-based methods were obtained from [12]. For Stack Overflow, since KD-based methods cannot be directly used for language modeling tasks, their results are marked as N/A. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td rowspan=\"2\">Method</td><td colspan=\"2\">High Data Heterogeneity</td><td colspan=\"2\">Low Data Heterogeneity</td><td rowspan=\"2\">Stack Overflow</td></tr><tr><td>CIFAR-10</td><td>CIFAR-100</td><td>CIFAR-10</td><td>CIFAR-100</td></tr><tr><td rowspan=\"3\">KD-based</td><td>FedDF</td><td>73.81 (± 0.42)</td><td>31.87 (± 0.46)</td><td>76.55 (± 0.32)</td><td>37.87 (± 0.31)</td><td>N/A</td></tr><tr><td>DS-FL</td><td>65.27 (±0.53)</td><td>29.12 (± 0.51)</td><td>68.44 (± 0.47)</td><td>33.56 (± 0.55)</td><td>N/A</td></tr><tr><td>Fed-ET</td><td>78.66 (± 0.31)</td><td>35.78 (± 0.45)</td><td>81.13 (± 0.28)</td><td>41.58 (± 0.36)</td><td>N/A</td></tr><tr><td rowspan=\"3\">PT-based</td><td>HeteroFL</td><td>63.90 (± 2.74)</td><td>52.38 (± 0.80)</td><td>73.19 (± 1.71)</td><td>57.44 (± 0.42)</td><td>27.21 (± 0.22)</td></tr><tr><td>Federated Dropout</td><td>46.64 (± 3.05)</td><td>45.07 (± 0.07)</td><td>76.20 (± 2.53)</td><td>46.40 (± 0.21)</td><td>23.46 (± 0.12)</td></tr><tr><td>FedRolex</td><td>69.44 (± 1.50)</td><td>56.57 (± 0.15)</td><td>84.45 (± 0.36)</td><td>58.73 (± 0.33)</td><td>29.22 (± 0.24)</td></tr><tr><td rowspan=\"2\"></td><td>Homogeneous (smallest)</td><td>38.82(± 0.88)</td><td>12.69 (± 0.50)</td><td>46.86 (± 0.54)</td><td>19.70 (± 0.34)</td><td>27.32 (± 0.12)</td></tr><tr><td>Homogeneous (largest)</td><td>75.74 (± 0.42)</td><td>60.89 (± 0.60)</td><td>84.48 (± 0.58)</td><td>62.51 (± 0.20)</td><td>29.79 (± 0.32)</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "4.2 Performance Comparison with Model-Homogeneous FL Methods ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We also compare the global model accuracy of FedRolex with two model-homogeneous cases where all the clients have the largest capacity model $\\beta = \\{ 1 \\}$ ) and the smallest capacity model $\\begin{array} { r } { \\left( \\beta = \\sqrt { 1 / 1 6 } \\right) } \\end{array}$ ), representing the upper and lower-bound performance, respectively. ",
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+    {
+        "type": "text",
+        "text": "Evaluation Results: As listed in Table 3, compared with other PT-based methods, FedRolex reduces the gap in global model accuracy between model-heterogeneous and upper-bound modelhomogeneous settings. In particular, FedRolex is on par with the upper-bound model-homogeneous case for Stack Overflow, whereas both HeteroFL and Federated Dropout perform even worse than the model homogeneous case using the smallest model. This result indicates that with FedRolex, we will not be constrained to only using high-end devices to achieve competitive global model accuracy. Note that Fed-ET achieves a higher global model accuracy than the model-homogeneous upper bound on CIFAR-10 under high data heterogeneity, which showcases the advantage of using public data. ",
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+        "type": "text",
+        "text": "4.3 Impact of Client Model Heterogeneity Distribution ",
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+        "text": "In our previous experiments, the distributions of model capacities across client devices are set to be uniform. In this experiment, we aim to understand the impact of the client model heterogeneity distribution. To do so, without loss of generality, we use two client model capacities $\\beta = \\{ \\bar { 1 } , { ^ { 1 } / _ { 1 6 } } \\}$ and vary the distribution ratio between the two (denoted as $\\rho \\mathrm { \\hbar }$ ) where $\\rho = 1$ represents the case in which all the clients have the largest capacity model $\\begin{array} { r } { \\beta = \\{ 1 \\} } \\end{array}$ ) and $\\rho = 0$ represents the case in which all the clients have the smallest capacity model $( \\beta = \\left\\{ { 1 } / { 1 6 } \\right\\}$ ). ",
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+        "type": "text",
+        "text": "Evaluation Results: Figure 3 shows how global model accuracy changes when $\\rho$ varies from 0 to 1 for CIFAR-10, CIFAR-100 and Stack Overflow. We have three observations. (1) For CIFAR-10 (Figure 3(i)), there is a large gap in global model accuracy between high and low data heterogeneity for a wide range of $\\rho$ (from 0.1 to 1). This is because CIFAR-10 is a relatively simple task and hence the global model accuracy is bottlenecked by the level of data heterogeneity instead of model capacity. This result indicates that having more high-capacity models in the cohort has only limited contribution to global model accuracy. (2) For the more challenging CIFAR-100 (Figure 3(ii)), the gap in global model accuracy is much lower between high and low data heterogeneity. In contrast to CIFAR-10, the global model accuracy is bottlenecked by the highest capacity of the models rather than the level of data heterogeneity. (3) For both regimes (Figure 3(i)(ii) vs. Figure 3(iii)), we observe that having a small fraction of large-capacity models significantly boosts the global model accuracy, but keeping increasing the ratio of large-capacity models has limited contribution to the accuracy. ",
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+        "img_path": "images/54a8526c0ffbaff84447015f2bec64690c39e542670cf9bce67091bc7c1df68c.jpg",
+        "image_caption": [
+            "Figure 3: Impact of client model heterogeneity distribution on global model accuracy for (i) CIFAR10, (ii) CIFAR-100, and (iii) Stack Overflow. "
+        ],
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+        "type": "text",
+        "text": "4.4 Performance on Training Larger Server Model ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Similar to Federated Dropout, one advantage of FedRolex over static sub-model extraction methods (HeteroFL and FjORD) is that FedRolex is able to train a global server model that is larger than the largest client model. In this experiment, we aim to evaluate the performance of FedRolex on training larger server models. To do so, we consider the case where the size of the global server model is $\\gamma \\overset { \\cdot } { = } \\{ 2 , 4 , 8 , 1 6 \\}$ times the size of client models. For simplicity, all client models have the same size. ",
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+    {
+        "type": "text",
+        "text": "Evaluation Results: Figure 4(i) and Figure 4(ii) compare FedRolex with Federated Dropout in terms of global model accuracy when $\\gamma$ for CIFAR-10 and CIFAR-100, respectively. As shown, although the global model accuracy drops for both FedRolex and Federated Dropout when $\\gamma$ increases, especially from 1 to 4, FedRolex consistently achieves higher global model accuracy than Federated Dropout across $\\gamma = \\{ 2 , 4 , 8 , 1 6 \\}$ under both low and high data heterogeneity. For Stack Overflow (Figure 4(iii)), the global model accuracy has a much smaller drop when $\\gamma$ increases. This demonstrates the superiority of using large models on large-scale datasets for training larger server models. ",
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+        "img_path": "images/dc26fa907235d516ac65f0e07501364b48fc28547d09a1bd772a601c68306ae5.jpg",
+        "image_caption": [
+            "Figure 4: Performance on training larger server model when the server model is $\\gamma$ times the size of the client model for (i) CIFAR-10, (ii) CIFAR-100, and (iii) Stack Overflow. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "4.5 Enhance Inclusiveness of FL in Real-world Distribution ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "A primary vision of FedRolex is to enhance the inclusiveness of FL. To demonstrate this, in this experiment, we use real-world household income distribution to emulate real-world device distribution. Specifically, we retrieve household income distribution information from Bureau [33]. We map $\\beta _ { n } = { ^ { 1 } } / { _ { 1 6 } }$ with the income group with earning less than $\\$ 75,000$ and assign proportions of remaining groups in $\\$ 25$ , 000 increments with increasing values of $\\beta _ { n }$ . Detailed mapping of this distribution to the corresponding income distribution is provided in Figure 7 in Appendix A.7. ",
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+    {
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+        "img_path": "images/3a5cc1ceb24d9af6e3ccb5412cfe9ff89a92852da8096a9afd3da5980d14c226.jpg",
+        "image_caption": [
+            "Figure 5: Local model accuracy distribution of FedRolex (orange color) vs. the smallest modelhomogeneous case (blue color) for CIFAR-10 and CIFAR-100 under low and high data heterogeneity. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Evaluation Results: Table 4 shows both the global and local model accuracies of FedRolex for CIFAR-10 and CIFAR-100 as well as the global model accuracy on Stack Overflow under the emulated real-world device distribution. Again, we compare with two model-homogeneous cases where all clients have the smallest and largest model capacities, representing lower and upper-bound accuracy, respectively. We make two observations. (1) Looking at the global model accuracy, FedRolex consistently outperforms the lower-bound model-homogeneous case across CIFAR-10, CIFAR-100, and Stack Overflow. This result indicates that FedRolex enhances the inclusiveness of FL and improves the accuracy of the global model, which would otherwise not be able to achieve. (2) Looking at the local model accuracy, FedRolex significantly outperforms the lower-bound modelhomogeneous case on CIFAR-10 and CIFAR-100 under both low and high data heterogeneity. This result indicates that FedRolex effectively boosts the performance of low-end devices, which would otherwise not benefit from FL. A detailed illustration of how local model accuracy distribution of individual clients shifts when FedRolex is used compared to the smallest model-homogeneous case with the same client outreach is shown in Figure 5. ",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We presented FedRolex, a partial training (PT)-based model-heterogeneous FL approach that is able to train a global server model larger than the largest client model. FedRolex proposed a rolling sub-model extraction scheme that enables parameters of the global server model to be evenly trained to mitigate client drift induced by model heterogeneity. We provided a theoretical statistical analysis of its advantage over Federated Dropout. Our experimental results show that FedRolex consistently outperforms state-of-the-art PT-based methods across models and datasets at both small and large scales. Moreover, we demonstrated its performance on an emulated real-world device distribution and show FedRolex contributes to making FL more inclusive. ",
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+    {
+        "type": "text",
+        "text": "Limitations and Future works. In this work, we provided a statistical analysis of FedRolex. Full convergence analysis of FedRolex is not trivial and is left for future work. In addition, the goal of this work is to train a global server model using a federation of heterogeneous client models. Determining what models to deploy onto each client after the global server model is trained is a separate task, especially when the global server model is large. We will pursue it as our future work. ",
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+    {
+        "type": "text",
+        "text": "6 Acknowledgement ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We thank the reviewers for their helpful comments. This work was partially supported by NSF PFI:BIC-1632051, CNS-1814551, DMS-2012439, and a Google Computing Platform (GCP) grant. ",
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+        "type": "text",
+        "text": "References   \n[1] Peter Kairouz, H Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Kallista Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, et al. Advances and open problems in federated learning. Foundations and Trends® in Machine Learning, 14(1–2):1–210, 2021.   \n[2] Jianyu Wang, Zachary Charles, Zheng Xu, Gauri Joshi, H. Brendan McMahan, Blaise Aguera y Arcas, Maruan Al-Shedivat, Galen Andrew, Salman Avestimehr, Katharine Daly, Deepesh Data, Suhas Diggavi, Hubert Eichner, Advait Gadhikar, Zachary Garrett, Antonious M. Girgis, Filip Hanzely, Andrew Hard, Chaoyang He, Samuel Horvath, Zhouyuan Huo, Alex Ingerman, Martin Jaggi, Tara Javidi, Peter Kairouz, Satyen Kale, Sai Praneeth Karimireddy, Jakub Konecny, Sanmi Koyejo, Tian Li, Luyang Liu, Mehryar Mohri, Hang Qi, Sashank J. Reddi, Peter Richtarik, Karan Singhal, Virginia Smith, Mahdi Soltanolkotabi, Weikang Song, Ananda Theertha Suresh, Sebastian U. Stich, Ameet Talwalkar, Hongyi Wang, Blake Woodworth, Shanshan Wu, Felix X. Yu, Honglin Yuan, Manzil Zaheer, Mi Zhang, Tong Zhang, Chunxiang Zheng, Chen Zhu, and Wennan Zhu. A field guide to federated optimization, 2021.   \n[3] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas. Communication-efficient learning of deep networks from decentralized data. In Artificial intelligence and statistics, pages 1273–1282. PMLR, 2017.   \n[4] Tian Li, Anit Kumar Sahu, Manzil Zaheer, Maziar Sanjabi, Ameet Talwalkar, and Virginia Smith. Federated optimization in heterogeneous networks. Proceedings of Machine Learning and Systems, 2:429–450, 2020.   \n[5] Sai Praneeth Karimireddy, Satyen Kale, Mehryar Mohri, Sashank Reddi, Sebastian Stich, and Ananda Theertha Suresh. Scaffold: Stochastic controlled averaging for federated learning. In International Conference on Machine Learning, pages 5132–5143. PMLR, 2020.   \n[6] Hong-You Chen and Wei-Lun Chao. Fedbe: Making bayesian model ensemble applicable to federated learning. arXiv preprint arXiv:2009.01974, 2020.   \n[7] Rishi Bommasani, Drew A Hudson, Ehsan Adeli, Russ Altman, Simran Arora, Sydney von Arx, Michael S Bernstein, Jeannette Bohg, Antoine Bosselut, Emma Brunskill, et al. On the opportunities and risks of foundation models. arXiv preprint arXiv:2108.07258, 2021.   \n[8] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. Advances in neural information processing systems, 30, 2017.   \n[9] Chaoyang He, Murali Annavaram, and Salman Avestimehr. Group knowledge transfer: Federated learning of large cnns at the edge. Advances in Neural Information Processing Systems, 33: 14068–14080, 2020.   \n[10] Tao Lin, Lingjing Kong, Sebastian U Stich, and Martin Jaggi. Ensemble distillation for robust model fusion in federated learning. Advances in Neural Information Processing Systems, 33: 2351–2363, 2020.   \n[11] Sohei Itahara, Takayuki Nishio, Yusuke Koda, Masahiro Morikura, and Koji Yamamoto. Distillation-based semi-supervised federated learning for communication-efficient collaborative training with non-iid private data. arXiv preprint arXiv:2008.06180, 2020.   \n[12] Yae Jee Cho, Andre Manoel, Gauri Joshi, Robert Sim, and Dimitrios Dimitriadis. Heterogeneous ensemble knowledge transfer for training large models in federated learning. International Joint Conference on Artificial Intelligence (IJCAI), 2022.   \n[13] Sebastian Caldas, Jakub Konecny, H Brendan McMahan, and Ameet Talwalkar. Expanding ˇ the reach of federated learning by reducing client resource requirements. arXiv preprint arXiv:1812.07210, 2018.   \n[14] Enmao Diao, Jie Ding, and Vahid Tarokh. Heterofl: Computation and communication efficient federated learning for heterogeneous clients. arXiv preprint arXiv:2010.01264, 2020. ",
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+    {
+        "type": "text",
+        "text": "Checklist ",
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+        "type": "text",
+        "text": "1. For all authors... ",
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+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [No]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "type": "text",
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parse/dev/q-LMlivZrV/q-LMlivZrV_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "CLOOB: Modern Hopfield Networks with InfoLOOB Outperform CLIP ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Andreas Fürst ⇤1 Elisabeth Rumetshofer ⇤1 Johannes Lehner 1 Viet Tran 1 Fei Tang 3 Hubert Ramsauer 1 David Kreil 2 Michael Kopp 2 Günter Klambauer 1 Angela Bitto-Nemling 1 Sepp Hochreiter 1 2 ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 ELLIS Unit Linz and LIT AI Lab, Institute for Machine Learning, Johannes Kepler University, Linz, Austria 2 Institute of Advanced Research in Artificial Intelligence (IARAI), Vienna, Austria 3 HERE Technologies, Zurich, Switzerland ⇤ Equal contribution ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "CLIP yielded impressive results on zero-shot transfer learning tasks and is considered as a foundation model like BERT or GPT3. CLIP vision models that have a rich representation are pre-trained using the InfoNCE objective and natural language supervision before they are fine-tuned on particular tasks. Though CLIP excels at zero-shot transfer learning, it suffers from an explaining away problem, that is, it focuses on one or few features, while neglecting other relevant features. This problem is caused by insufficiently extracting the covariance structure in the original multi-modal data. We suggest to use modern Hopfield networks to tackle the problem of explaining away. Their retrieved embeddings have an enriched covariance structure derived from co-occurrences of features in the stored embeddings. However, modern Hopfield networks increase the saturation effect of the InfoNCE objective which hampers learning. We propose to use the InfoLOOB objective to mitigate this saturation effect. We introduce the novel “Contrastive Leave One Out Boost” (CLOOB), which uses modern Hopfield networks for covariance enrichment together with the InfoLOOB objective. In experiments we compare CLOOB to CLIP after pre-training on the Conceptual Captions and the YFCC dataset with respect to their zero-shot transfer learning performance on other datasets. CLOOB consistently outperforms CLIP at zero-shot transfer learning across all considered architectures and datasets. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Contrastive Language-Image Pre-training (CLIP) showed spectacular performance at zero-shot transfer learning (Radford et al., 2021). CLIP learns expressive image embeddings directly from raw text, thereby leverages a much richer source of supervision than just labels. The CLIP model is considered as an important foundation model (Bommasani et al., 2021), therefore a plethora of follow-up work has been published (see Appendix Section A.4). CLIP as a contrastive learning method has two simultaneous goals, namely (i) increasing the similarity of matched language-image pairs and (ii) decreasing the similarity of unmatched language-image pairs. Though CLIP yielded striking zero-shot transfer learning results, it still suffers from “explaining away”. Explaining away is known in reasoning as the concept that the confirmation of one cause of an observed event dismisses alternative causes (Pearl, 1988; Wellman & Henrion, 1993). CLIP’s explaining away problem is its focus on one or few features while neglecting other relevant features. This problem is caused by insufficiently extracting feature co-occurrences and covariance structures in the original multimodal data. Humans extract co-occurrences and covariances by associating current perceptions with memories (Bonner & Epstein, 2021; Potter, 2012). In analogy to these human cognitive processes, we suggest to use modern Hopfield networks to amplify co-occurrences and covariance structures of the original data. ",
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+    {
+        "type": "text",
+        "text": "Hopfield networks are energy-based, binary associative memories, which popularized artificial neural networks in the 1980s (Amari, 1972; Hopfield, 1982, 1984). Associative memory networks have been designed to store and retrieve samples. Their storage capacity can be considerably increased by polynomial terms in the energy function (Chen et al., 1986; Psaltis & Cheol, 1986; Baldi & Venkatesh, 1987; Gardner, 1987; Abbott & Arian, 1987; Horn & Usher, 1988; Caputo & Niemann, 2002; Krotov & Hopfield, 2016). In contrast to these binary memory networks, we use continuous associative memory networks with very high storage capacity. These modern Hopfield networks for deep learning architectures have an energy function with continuous states and can retrieve samples with only one update (Ramsauer et al., 2021). Modern Hopfield networks have already been successfully applied to immune repertoire classification (Widrich et al., 2020), chemical reaction prediction (Seidl et al., 2022) and reinforcement learning (Widrich et al., 2021; Paischer et al., 2022). Modern Hopfield networks are a novel concept for contrastive learning to extract more covariance structure. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "However, modern Hopfield networks lead to a higher similarity of retrieved samples. The increased similarity exacerbates the saturation of CLIP’s InfoNCE objective (van den Oord et al., 2018). InfoNCE saturates because it contains terms of the form $a / ( a + b )$ . In analogy to Wang & Isola (2020), $a$ is called the “alignment score” that measures the similarity of matched pairs and $b$ is called the “uniformity penalty” that measures the similarity of unmatched pairs. The saturation problem becomes more severe for retrieved samples of the modern Hopfield network since the alignment score $a$ increases. Saturation of InfoNCE hampers the decrease of the uniformity penalty $b$ (see also Yeh et al. (2021)). Contrary to InfoNCE, the “InfoLOOB” (LOOB for “Leave One Out Bound”) objective (Poole et al., 2019) contains only terms of the form $a / b$ which do not saturate. Thus, even for a large alignment score $a$ , learning still decreases the uniformity penalty $b$ by distributing samples more uniformly. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We introduce “Contrastive Leave One Out Boost” (CLOOB) which combines modern Hopfield networks with the “InfoLOOB” objective. Our contributions are: ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "(a) we propose CLOOB, a new contrastive learning method,   \n(b) we propose modern Hopfield networks to reinforce covariance structures,   \n(c) we propose InfoLOOB as an objective to avoid saturation as observed with InfoNCE, and provide theoretical justifications for optimizing InfoLOOB. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 CLOOB: Modern Hopfield Networks with InfoLOOB ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our novel contrastive learning method CLOOB can be seen as a replacement of CLIP and therefore be used in any method which builds upon CLIP. Figure 1 sketches the CLOOB architecture for image-text pairs. The training set consists of $N$ pairs of embeddings $\\left\\{ ( \\pmb { x } _ { 1 } , \\pmb { y } _ { 1 } ) , \\dotsc , ( \\pmb { x } _ { N } , \\pmb { y } _ { N } ) \\right\\}$ with $\\pmb { X } = \\mathsf { \\bar { \\Psi } } ( \\pmb { x } _ { 1 } , \\dots , \\pmb { x } _ { N } )$ and $\\pmb { Y } = ( \\pmb { y } _ { 1 } , \\dots , \\pmb { y } _ { N } )$ , $M$ stored embeddings $\\pmb { U } = ( \\pmb { u } _ { 1 } , \\dots , \\pmb { u } _ { M } )$ , and $K$ stored embeddings $V = \\left( \\pmb { v } _ { 1 } , \\dots , \\pmb { v } _ { K } \\right)$ . The state or query embeddings $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ and $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }$ retrieve $U _ { { \\pmb x } _ { i } }$ and $U _ { y _ { i } }$ , respectively, from $U$ — analog for retrievals from $V$ . The samples are normalized: $\\| \\pmb { x } _ { i } \\| = \\| \\pmb { y } _ { i } \\| = \\| \\pmb { u } _ { i } \\| = \\| \\pmb { v } _ { i } \\| = 1$ . $U _ { { \\pmb x } _ { i } }$ denotes an image-retrieved image embedding, $U _ { y _ { i } }$ a text-retrieved image embedding, $V _ { { \\pmb x } _ { i } }$ an image-retrieved text embedding and $V _ { y _ { i } }$ a text-retrieved text embedding. These retrievals from modern Hopfield networks are computed as follows (Ramsauer et al., 2021): ",
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+        "img_path": "images/182c4d903ae228b2867e8d011df1b3141f2dc5d32b0275117c14663f2028b017.jpg",
+        "text": "$$\n\\begin{array} { r l } { U _ { x _ { i } } \\ = \\ U \\ \\mathrm { s o f t m a x } ( \\beta U ^ { T } x _ { i } ) , \\qquad ( 1 ) \\qquad } & { V _ { x _ { i } } \\ = V \\ \\mathrm { s o f t m a x } ( \\beta V ^ { T } x _ { i } ) , } \\\\ { U _ { y _ { i } } \\ = \\ U \\ \\mathrm { s o f t m a x } ( \\beta U ^ { T } y _ { i } ) , \\qquad ( 2 ) \\qquad } & { V _ { y _ { i } } \\ = V \\ \\mathrm { s o f t m a x } ( \\beta V ^ { T } y _ { i } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The hyperparameter $\\beta$ corresponds to the inverse temperature: $\\beta = 0$ retrieves the average of the stored pattern, while large $\\beta$ retrieves the stored pattern that is most similar to the state pattern (query). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/679b9f810d55197d9e18c57c22a0bbd4bf5eab576111477284add4731034c9cc.jpg",
+        "image_caption": [
+            "Figure 1: The CLOOB architecture for image-text pairs. The image embedding $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ and the text embedding $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\mathbf { \\psi } _ { 2 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\qquad \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 }$ retrieve the embeddings $U _ { { \\pmb x } _ { i } }$ and $U _ { y _ { i } }$ , respectively, from a modern Hopfield network that stores image embeddings $U = ( \\pmb { u } _ { 1 } , \\dots , \\pmb { u } _ { M } )$ (green boxes at the left). The image-retrieved image embedding $U _ { { \\pmb x } _ { i } }$ serves as anchor in order to contrast the positive text-retrieved image embedding $U _ { y _ { i } }$ with the negative text-retrieved image embedding $U _ { y _ { j } }$ for $j \\neq i$ . Analogously, for the second modern Hopfield network that stores text embeddings $V = \\left( \\pmb { v } _ { 1 } , \\dots , \\pmb { v } _ { K } \\right)$ (green boxes at the right). "
+        ],
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the InfoLOOB loss Eq. (8), CLOOB substitutes the embedded samples $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ and $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }$ by the normalized retrieved embedded samples. In the first term, $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ and $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\mathbf { \\psi } _ { 2 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\qquad \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\qquad \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 3 } \\mathbf { \\psi } _ { 4 } \\mathbf { \\psi } _ { 4 }$ are substituted by $U _ { { \\pmb x } _ { i } }$ and $U _ { y _ { i } }$ , respectively, while in the second term they are substituted by $V _ { { \\pmb x } _ { i } }$ and $V _ { y _ { i } }$ . After retrieval, the samples are re-normalized to ensure $\\| U _ { x _ { i } } \\| = \\| U _ { y _ { i } } \\| = \\| V _ { x _ { i } } \\| = \\| V _ { y _ { i } } \\| = \\mathrm { \\bar { 1 } }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We obtain the CLOOB loss function: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b3e853f42bd792eed2dc532c41dbeb0e7df300d7b8419d3f8e27d2d1bf37968e.jpg",
+        "text": "$$\n\\mathrm { L } _ { \\mathrm { I n f o L O O B } } = - \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\ln \\frac { \\exp ( \\tau ^ { - 1 } U _ { { \\alpha } _ { i } } ^ { T } U _ { y _ { i } } ) } { \\sum _ { j \\neq i } ^ { N } \\exp ( \\tau ^ { - 1 } U _ { { \\alpha } _ { i } } ^ { T } U _ { y _ { j } } ) } \\ - \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\ln \\frac { \\exp ( \\tau ^ { - 1 } V _ { { \\alpha } _ { i } } ^ { T } V _ { y _ { i } } ) } { \\sum _ { j \\neq i } ^ { N } \\exp ( \\tau ^ { - 1 } V _ { { \\alpha } _ { j } } ^ { T } V _ { y _ { i } } ) } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "By default, we store the minibatch in the modern Hopfield networks, that is, $U = X$ and $V = Y$ . Thus, in Eq. (1) $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ can retrieve itself from $U = X$ , but in Eq. (3) it can not retrieve itself from $V = Y$ . Analogously, in Eq. (4) $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }$ can retrieve itself from $V = Y$ , but in Eq. (2) it can not retrieve itself from $U = X$ . By storing the embeddings of the mini-batch examples in the Hopfield memory, we do not require to compute the embeddings of additional samples via text and image encoders. However, the modern Hopfield networks can also store prototypes, templates, or proprietary samples to amplify particular embedding features via the stored samples. Either the original embeddings $_ { \\textbf { \\em x } }$ and $\\textbf {  { y } }$ or the retrieved embeddings $U _ { x }$ , $U _ { y }$ , $V _ { x }$ , and $V _ { y }$ may serve for the downstream tasks, e.g. for zero-shot transfer learning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Pseudocode 1 shows CLOOB in a PyTorch-like style. CLOOB has two major components: (i) modern Hopfield networks that alleviate CLIP’s problem of insufficiently exploiting the covariance structure in the data and (ii) the InfoLOOB objective that does not suffer from InfoNCE’s saturation problem. The next two sections analyze CLOOB’s major components. ",
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+    {
+        "type": "text",
+        "text": "3 Modern Hopfield Networks for Enriching the Covariance Structure ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use modern Hopfield networks to amplify co-occurrences and the covariance structure. Replacing the original embeddings by retrieved embeddings reinforces features that frequently occur together in stored embeddings. Additionally, spurious co-occurrences that are peculiar to a sample are averaged out. By this means, the covariance structure is reinforced by the retrieved embeddings $U _ { { \\pmb x } _ { i } } ^ { T } \\breve { U } _ { { \\pmb y } _ { i } }$ and $V _ { \\pmb { x } _ { i } } ^ { T } V _ { \\pmb { y } _ { i } }$ . The Jacobian J of the softmax $\\pmb { p } = \\mathrm { s o f t m a x } ( \\beta \\pmb { a } )$ is $\\mathrm { J } ( \\beta \\pmb { a } ) = \\beta$ $\\left( \\mathrm { d i a g } ( \\pmb { p } ) - \\pmb { p } \\pmb { p } ^ { T } \\right)$ . We define the weighted covariance $\\operatorname { C o v } ( U )$ , where sample $\\mathbf { \\delta } \\mathbf { u } _ { i }$ is drawn with probability $p _ { i }$ , as $\\begin{array} { r } { \\left[ \\operatorname { C o v } ( \\pmb { U } ) \\right] _ { k l } = \\left[ \\pmb { U } \\mathbf { J } ( \\beta a ) \\pmb { U } ^ { T } \\right] _ { k l } = \\beta ( \\sum _ { i = 1 } ^ { M } p _ { i } u _ { i k } u _ { i l } - \\sum _ { i = 1 } ^ { M } p _ { i } u _ { i k } \\sum _ { i = 1 } ^ { M } p _ { i } u _ { i l } ) . } \\end{array}$ . The formula of the ",
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+    {
+        "type": "text",
+        "text": "1 # image_encoder - ResNet 2 3 # text_encoder - Text Transformer # I[n, h, w, c] - minibatch of images # T[n, l] - minibatch of texts 5 6 # W_i[d_i, d_e] - image projection # W_t[d_t, d_e] - text projection # beta - inverse temperature Hopfield retrieval 8 # tau - temperature InfoLOOB 9 10 # extract feature representations 11 I_f $=$ image_encoder(I) #[n, d_i] 12 T_f $=$ text_encoder(T) #[n, d_t] 13 14 # joint multimodal embedding 15 $\\texttt { x } =$ l2_normalize(I_f @ W_i) #[n, d_e] 16 $\\ y \\ = \\ 1 2$ _normalize(T_f @ W_t) #[n, d_e] 17 18 # Hopfield retrieval H with batch stored ",
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/28aeff6cbdebc6c33784904647e2954bc408388958ddead1711c3f9e8d74942c.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "weighted covariance differs from the standard empirical covariance, since the factor $1 / M$ is replaced by $p _ { i }$ . Thus, $\\mathbf { \\Delta } \\mathbf { u } _ { i }$ is sampled with probability $p _ { i }$ instead with probability $1 / M$ (uniformly). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We apply the mean value theorem to the softmax function with mean Jacobian matrix $\\mathrm { J } ^ { \\mathrm { m } } ( \\beta \\pmb { a } ) =$ $\\begin{array} { r } { \\int _ { 0 } ^ { 1 } \\mathrm { J } ( \\lambda \\beta \\pmb { a } ) ~ \\mathrm { d } \\lambda } \\end{array}$ . The mean Jacobian $\\mathrm { J } ^ { \\mathrm { m } } ( \\beta \\mathbf { a } )$ is a symmetric, diagonally dominant, positive semidefinite matrix with one eigenvalue of zero for eigenvector 1 and spectral norm bounded by $\\| \\mathbf { J } ^ { \\mathrm { m } } \\| _ { 2 } \\leqslant$ $0 . 5 \\beta$ (see Appendix Lemma A1). According to Appendix Theorem A3, we can express $U _ { { \\pmb x } _ { i } } ^ { T } { \\pmb U } _ { { \\pmb y } _ { i } }$ as: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9aca29fb36a00c6a65d06671ddd85e171ad6cb6e4010dc1dee11e4cd550d2a41.jpg",
+        "text": "$$\n\\left( \\bar { \\pmb u } + \\mathrm { C o v } ( { \\pmb U } , { \\pmb x } _ { i } ) { \\pmb x } _ { i } \\right) ^ { T } \\left( \\bar { \\pmb u } + \\mathrm { C o v } ( { \\pmb U } , { \\pmb y } _ { i } ) { \\pmb y } _ { i } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            331,
+            445,
+            663,
+            467
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the mean is $\\bar { \\pmb { u } } = 1 / M U { \\bf 1 }$ and the weighted covariances are $\\operatorname { C o v } ( \\pmb { U } , \\pmb { x } _ { i } ) = \\pmb { U } \\mathrm { J ^ { m } } ( \\beta \\pmb { U } ^ { T } \\pmb { x } _ { i } ) \\pmb { U } ^ { T }$ and $\\operatorname { C o v } ( \\pmb { U } , \\pmb { y } _ { i } ) = \\pmb { U } \\mathbb { J } ^ { \\mathrm { m } } ( \\beta \\pmb { U } ^ { T } \\pmb { y } _ { i } ) \\pmb { U } ^ { T }$ . The weighted covariance $\\operatorname { C o v } ( U , . )$ is the covariance if the stored pattern $\\mathbf { \\Delta } \\mathbf { u } _ { i }$ is drawn according to an averaged $p _ { i }$ given by $\\mathrm { J ^ { m } ( . ) }$ . Maximizing the dot product $U _ { { \\pmb x } _ { i } } ^ { T } { \\pmb U } _ { { \\pmb y } _ { i } } ^ { \\dagger }$ forces the normalized vectors $\\mathbf { \\boldsymbol { x } } _ { i }$ and $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { j } } \\mathbf { \\sigma } _ { \\mathbf { \\psi } _ { j } }$ to agree on drawing the patterns $\\mathbf { \\Delta } \\mathbf { u } _ { i }$ with the same probability $p _ { i }$ in order to generate similar weighted covariance matrices $\\operatorname { C o v } ( U , . )$ . If subsets of $U$ have a strong covariance structure, then it can be exploited to produce large weighted covariances and, in turn, large dot products of $U _ { \\pmb { x } _ { i } } ^ { T } { U } _ { \\pmb { y } _ { i } }$ . Furthermore, for a large dot product $\\bar { U } _ { { \\pmb x } _ { i } } ^ { T } U _ { { \\pmb y } _ { i } }$ , $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ and $\\mathbf { \\nabla } _ { \\mathbf { \\psi } _ { 3 } } \\psi _ { i }$ have to be similar to each other to extract the same direction from the covariance matrices. The above considerations for $U _ { { \\pmb x } _ { i } } ^ { T } { \\pmb U } _ { { \\pmb y } _ { i } }$ analogously apply to $V _ { { \\pmb x } _ { i } } ^ { T } V _ { { \\pmb y } _ { i } }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We did not use a loss function that contains dot products like $U _ { { \\pmb x } _ { i } } ^ { T } V _ { { \\pmb y } _ { i } }$ , because they have higher variance than the ones we have used. The dot product $U _ { { \\pmb x } _ { i } } ^ { T } V _ { { \\pmb y } _ { i } }$ has higher variance, since it uses $M + K$ stored patterns, whereas $U _ { \\pmb { x } _ { i } } ^ { T } { U } _ { \\pmb { y } _ { i } }$ and $V _ { \\pmb { x } _ { i } } ^ { T } V _ { \\pmb { y } _ { i } }$ use $M$ and $K$ stored patterns, respectively. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Modern Hopfield networks enable to extract more covariance structure. To demonstrate the effect of modern Hopfield networks, we computed the eigenvalues of the covariance matrix of the image and text embeddings. We counted the number of effective eigenvalues, that is, the number of eigenvalues needed to obtain $9 9 \\%$ of the total sum of eigenvalues. Figure 2 shows the relative change of the number of effective eigenvalues compared to the respective reference epoch (the epoch before the first learning rate restart). Modern Hopfield networks consistently increase the number of effective eigenvalues during learning. Consequently, modern Hopfield networks enable to extract more covariance structure during learning, i.e. enrich the embeddings by covariances that are already in the raw multi-modal data. This enrichment of embeddings mitigates explaining away. Further details can be found in Appendix Section A.2.7. ",
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+    {
+        "type": "text",
+        "text": "4 InfoLOOB for Contrastive Learning ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Modern Hopfield networks lead to a higher similarity of retrieved samples. The increased similarity exacerbates the saturation of the InfoNCE objective. To avoid the saturation of InfoNCE, CLOOB uses the “InfoLOOB” objective. The “InfoLOOB” objective is called “Leave one out upper bound” in Poole et al. (2019) and “L1Out” in Cheng et al. (2020). InfoLOOB is not established as a contrastive objective, although it is a known bound. Recently, InfoLOOB was independently introduced as objective for image-to-image contrastive learning (Yeh et al., 2021). ",
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+    {
+        "type": "image",
+        "img_path": "images/19162b5fca6ca7274704fbf4cd66bb4200a50fba241b8fe6ea6b5f712e8f5c2a.jpg",
+        "image_caption": [
+            "Figure 2: Relative change in the number of the effective eigenvalues of the embedding covariance matrices, which were obtained from image and text encoders at two different training points. Models with modern Hopfield networks steadily extract more covariance structure during learning. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "InfoNCE and InfoLOOB loss functions. $N$ samples are drawn iid from $p ( { \\pmb x } , { \\pmb y } )$ giving the training set $\\left\\{ ( \\pmb { x } _ { 1 } , \\pmb { y } _ { 1 } ) , \\dotsc , ( \\pmb { x } _ { N } , \\pmb { y } _ { N } ) \\right\\}$ . For the sample $\\mathbf { \\pmb { y } } _ { 1 }$ , InfoNCE uses for the matrix of negative samples $\\pmb { X } = ( \\pmb { x } _ { 1 } , \\dots , \\pmb { x } _ { N } )$ , while InfoLOOB uses $\\tilde { \\pmb { X } } = ( \\pmb { x } _ { 2 } , \\dots , \\pmb { x } _ { N } )$ . The matrices differ by the positive sample $\\scriptstyle { \\mathbf { { \\vec { x } } } } _ { 1 }$ . For the score function $f ( { \\pmb x } , { \\pmb y } )$ , we use $f ( \\pmb { x } , \\pmb { y } ) = \\mathrm { e x p } ( \\tau ^ { - 1 } \\mathrm { s i m } ( \\pmb { x } , \\pmb { y } ) )$ with the similarity $\\mathrm { s i m } ( { \\pmb x } , { \\pmb y } ) = { \\pmb y } ^ { T } { \\pmb x }$ and $\\tau$ as the temperature. We have the InfoNCE and InfoLOOB loss functions: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b8c1d2a7fdcf6653a50886123681aae76c1ab1bf7f6e615c059e43c023a8d324.jpg",
+        "text": "$$\n\\mathrm { L } _ { \\mathrm { I n f o N C E } } \\ = \\ - \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\ln \\frac { \\exp ( \\tau ^ { - 1 } x _ { i } ^ { T } y _ { i } ) } { \\sum _ { j = 1 } ^ { N } \\exp ( \\tau ^ { - 1 } x _ { i } ^ { T } y _ { j } ) } \\ - \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\ln \\frac { \\exp ( \\tau ^ { - 1 } x _ { i } ^ { T } y _ { i } ) } { \\sum _ { j = 1 } ^ { N } \\exp ( \\tau ^ { - 1 } x _ { j } ^ { T } y _ { i } ) } \\ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/222ae46196ee482ad5c1f8d0d1e2053b6cc51b242ca0b76412e733cb9fe3ac58.jpg",
+        "text": "$$\n\\mathrm { L } _ { \\mathrm { I n f o L O O B } } \\ = \\ - \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\ln \\frac { \\exp ( \\tau ^ { - 1 } x _ { i } ^ { T } y _ { i } ) } { \\sum _ { j \\ne i } ^ { N } \\exp ( \\tau ^ { - 1 } x _ { i } ^ { T } y _ { j } ) } \\ - \\ \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\ln \\frac { \\exp ( \\tau ^ { - 1 } x _ { i } ^ { T } y _ { i } ) } { \\sum _ { j \\ne i } ^ { N } \\exp ( \\tau ^ { - 1 } x _ { j } ^ { T } y _ { i } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            191,
+            460,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We abbreviate $\\mathbf { \\mu } _ { y } = \\mathbf { \\mu } _ { y _ { 1 } }$ leading to the pair $( \\pmb { x } _ { 1 } , \\pmb { y } )$ and the negatives $\\tilde { \\pmb { X } } = ( \\pmb { x } _ { 2 } , \\dots , \\pmb { x } _ { N } )$ . In the second sum of the losses in Eq. 7 and Eq. 8, we consider only the first term, respectively: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7079be87ef79e42cae553b92e713fb885af0905e5c0b88027442fc8f74f77b76.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { L } _ { \\mathrm { I n f o N C E } } ( y ) ~ { = } ~ - ~ \\ln \\frac { { \\overbrace { \\exp ( \\tau ^ { - 1 } x _ { 1 } ^ { T } y ) } ^ { a } } } { \\underbrace { \\exp ( \\tau ^ { - 1 } x _ { 1 } ^ { T } y ) } _ { a } + \\underbrace { \\sum _ { j = 2 } ^ { N } \\exp ( \\tau ^ { - 1 } x _ { j } ^ { T } y ) } _ { b } } , } \\\\ & { \\mathrm { L } _ { \\mathrm { I n f o L O O B } } ( y ) ~ { = } ~ - ~ \\ln \\frac { { \\overbrace { \\exp ( \\tau ^ { - 1 } x _ { 1 } ^ { T } y ) } ^ { a } } } { \\underbrace { \\sum _ { j = 2 } ^ { N } \\exp ( \\tau ^ { - 1 } x _ { j } ^ { T } y ) } _ { b } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In analogy to Wang & Isola (2020), $a$ is called the “alignment score” that measures the similarity of matched pairs and $b$ the “uniformity penalty” that measures the similarity of unmatched pairs. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Gradients of InfoNCE and InfoLOOB loss functions. Eq. (9) and Eq. (10) are equal to ",
+        "bbox": [
+            169,
+            736,
+            759,
+            752
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d466bf8430f497f6f01d3fe3ed30a5290277e404f9b6796400dc4e2cddcfd2aa.jpg",
+        "text": "$$\n- \\tau ^ { - 1 } y ^ { T } x _ { 1 } + \\tau ^ { - 1 } \\mathrm { l s e } ( \\tau ^ { - 1 } , X ^ { T } y ) \\ , \\qquad - \\tau ^ { - 1 } y ^ { T } x _ { 1 } + \\tau ^ { - 1 } \\mathrm { l s e } ( \\tau ^ { - 1 } , \\tilde { X } ^ { T } y ) \\ ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            240,
+            768,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where lse is the log-sum-exp function (see Eq. (A73) in the Appendix). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The gradients of Eq. (9) and Eq. (10) with respect to $\\textbf {  { y } }$ are ",
+        "bbox": [
+            168,
+            816,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/58629228b524eb89f0cd45116243fb3d2f06583a4b5e58ced9b5655579d71a1d.jpg",
+        "text": "$$\n- \\tau ^ { - 1 } x _ { 1 } + \\tau ^ { - 1 } X \\mathrm { s o f t m a x } ( \\tau ^ { - 1 } X ^ { T } y ) , \\qquad - \\tau ^ { - 1 } x _ { 1 } + \\tau ^ { - 1 } \\tilde { X } \\mathrm { s o f t m a x } ( \\tau ^ { - 1 } \\tilde { X } ^ { T } y ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Using $\\pmb { p } = ( p _ { 1 } , \\dots , p _ { N } ) ^ { T } = \\mathrm { s o f t m a x } ( \\tau ^ { - 1 } \\pmb { X } ^ { T } \\pmb { y } )$ , the gradient of InfoNCE with respect to $\\textbf {  { y } }$ is ",
+        "bbox": [
+            179,
+            861,
+            799,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f8ee7549317efbd83bcbefa2f325c3dd011a023931b0e5d31c5b0726fe4a6eb.jpg",
+        "text": "$$\n\\frac { \\partial \\mathrm { L } _ { \\mathrm { I n f o N C E } } ( y ) } { \\partial y } = - \\tau ^ { - 1 } ( 1 - p _ { 1 } ) ( x _ { 1 } - \\tilde { X } \\mathrm { s o f t m a x } ( \\tau ^ { - 1 } \\tilde { X } ^ { T } y ) ) = ( 1 - p _ { 1 } ) \\frac { \\partial \\mathrm { L } _ { \\mathrm { I n f o L O O B } } ( y ) } { \\partial y } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/5935c45b41395f5ce243b1e7cf801ae0acca5edc4cf4c542ca8d466a36c3cf59.jpg",
+        "image_caption": [
+            "Figure 3: Ajne uniformity test statistics for image and text embeddings for two different epochs during training. A high test statistic indicates low uniformity of an embedding. Models trained with the InfoLOOB objective develop more uniform image and text embeddings on the hypersphere. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            88,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "By and large, the gradient of InfoNCE is scaled by $( 1 - p _ { 1 } )$ compared to the gradient of InfoLOOB, where $p _ { 1 }$ is the softmax similarity between the anchor $\\textbf {  { y } }$ and the positive sample $\\scriptstyle { \\mathbf { \\mathscr { x } } } _ { 1 }$ . Consequently, InfoNCE is saturating with increasing similarity between the anchor and the positive sample. For more details we refer to Appendix Section A.1.4. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This saturation of InfoNCE motivated the use of the InfoLOOB objective in order to decrease the uniformity penalty even for large alignment scores. The uniformity penalty decreases since learning does not stall and the most prominent features become down-scaled which makes negative examples less similar to the anchor sample. The InfoNCE objective Eq. 9 has the form $a / ( a + b )$ , while the InfoLOOB objective Eq. 10 has the form $a / b$ . InfoLOOB does not saturate and keeps decreasing the uniformity penalty $b$ . Figure 3 shows how InfoLOOB leads to an increase in the uniformity of image and text embeddings on the sphere, which is described by the statistics of the uniformity test of Ajne that was extended by Prentice (Ajne, 1968; Prentice, 1978). Higher uniformity on the sphere correlates with a lower uniformity penalty $b$ . For more details we refer to Appendix Section A.2.7. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Theoretical justification for optimizing InfoLOOB. The InfoNCE information is a lower bound on the mutual information, which was proven by Poole et al. (2019). In the Appendix Section A.1, we elaborate more on theoretical properties of the bounds and properties of the objective functions. Specifically, we show that InfoLOOB with neural networks is not an upper bound on the mutual information. Thus, unlike hitherto approaches to contrastive learning we use InfoLOOB as an objective, since it does not suffer from saturation effects as InfoNCE. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "CLOOB is compared to CLIP with respect to zero-shot transfer learning performance on two pretraining datasets. The first dataset, Conceptual Captions (CC) (Sharma et al., 2018), has a very rich textual description of images but only three million image-text pairs. The second dataset, a subset of YFCC100M (Thomee et al., 2016), has 15 million image-text pairs but the textual description is less rich than for CC and often vacuous. For both pre-training datasets, the downstream zero-shot transfer learning performance is tested on seven image classification datasets. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 Conceptual Captions Pre-training ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Pre-training dataset. The Conceptual Captions (CC) (Sharma et al., 2018) dataset contains 2.9 million images with high-quality captions. Images and their captions have been gathered from the web via an automated process and have a wide variety of content. Raw descriptions of images are from the alt-text HTML attribute. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Methods. The CLOOB implementation is based on OpenCLIP (Ilharco et al., 2021), which achieves results equivalent to CLIP on the YFCC dataset (see Section 5.2). OpenCLIP also reports results on the CC dataset. As CLIP does not train models on CC, we report results from this reimplementation as baseline. Analogously to Radford et al. (2021, Section 2.4), we used the modified ResNet (He et al., 2016) and BERT (Devlin et al., 2018, 2019) architectures to encode image and text input. We used the ResNet encoder ResNet-50 for experiments on CC. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/efcc1405c473a4243d6a6aa566786ee283c0fa1b832ecbf73b69f4417312be08.jpg",
+        "table_caption": [
+            "Table 1: Zero-shot results for models trained on CC with ResNet-50 vision encoders for two different checkpoints. Results are given as mean accuracy over 5 runs. Statistically significant results are shown in bold. CLIP and CLOOB were trained for 31 epochs while $\\mathrm { { C L I P ^ { * } } }$ and $\\mathrm { C L O O B ^ { * } }$ were trained for 128 epochs. In the majority of tasks CLOOB significantly outperforms CLIP. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>CLIP RN-50</td><td>CLOOB RN-50</td><td>CLIP* RN-50</td><td>CLOOB* RN-50</td></tr><tr><td>Birdsnap</td><td>2.26 ± 0.20</td><td>3.06 ± 0.30</td><td>2.8 ± 0.16</td><td>3.24 ± 0.31</td></tr><tr><td>Country211</td><td>0.67 ± 0.11</td><td>0.67 ± 0.05</td><td>0.7 ± 0.04</td><td>0.73 ± 0.05</td></tr><tr><td>Flowers102</td><td>12.56 ± 0.38</td><td>13.45 ± 1.19</td><td>13.32 ± 0.43</td><td>14.36 ± 1.17</td></tr><tr><td>GTSRB</td><td>7.66 ± 1.07</td><td>6.38 ± 2.11</td><td>8.96 ± 1.70</td><td>7.03 ±1.22</td></tr><tr><td>UCF101</td><td>20.98 ± 1.55</td><td>22.26 ± 0.72</td><td>21.63 ± 0.65</td><td>23.03 ± 0.85</td></tr><tr><td>Stanford Cars</td><td>0.91 ± 0.10</td><td>1.23 ± 0.10</td><td>0.99 ± 0.16</td><td>1.41 ± 0.32</td></tr><tr><td>ImageNet</td><td>20.33 ± 0.28</td><td>23.97 ± 0.15</td><td>21.3 ± 0.42</td><td>25.67 ± 0.22</td></tr><tr><td>ImageNet V2</td><td>20.24 ± 0.50</td><td>23.59 ± 0.15</td><td>21.24 ± 0.22</td><td>25.49 ± 0.11</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameter selection and learning schedule. The hyperparameter values of OpenCLIP were used as default, concretely, a learning rate of $1 \\times 1 0 ^ { - 3 }$ and a weight decay of 0.1 for the Adam optimizer (Kingma et al., 2014) with decoupled weight decay regularization (Loshchilov & Hutter, 2019). Deviating from OpenCLIP, we used a batch size of 512 due to computational restraints, which did not change the performance. The learning rate scheduler for all experiments was cosine annealing with warmup and hard restarts (Loshchilov & Hutter, 2017). We report the hyperparameter $\\tau$ (default 0.07) from CLIP as $\\tau ^ { - 1 }$ of 14.3 to be in the same regime as the hyperparameter $\\beta$ for the modern Hopfield networks. The main hyperparameter search for CLOOB (also for YFCC pre-training in the next section) was done with ResNet-50 as the vision encoder. Learnable $\\tau ^ { - 1 }$ in combination with the InfoLOOB loss results in undesired learning behavior (see Appendix Section A.1.4). Therefore, we set $\\tau ^ { - 1 }$ to a fixed value of 30, which was determined via a hyperparameter search (see Appendix Section A.2.2). For modern Hopfield networks, the hyperparameter $\\beta$ was set to 8. Further we scaled the loss $\\mathrm { L _ { I n f o L O O B } }$ with $\\tau$ to remove the factor $\\tau ^ { - 1 }$ from the gradients (see Appendix Section A.1.4) resulting in the loss function $\\tau _ { \\mathrm { L _ { I n f o L O O B } } }$ . ",
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+    {
+        "type": "text",
+        "text": "Evaluation metrics: Zero-shot transfer learning. We evaluated and compared both CLIP and CLOOB on their zero-shot transfer learning capabilities on the following downstream image classification tasks. Birdsnap (Berg et al., 2014) contains images of 500 different North American bird species. The Country211 (Radford et al., 2021) dataset consists of photos across 211 countries and is designed to test the geolocalization capability of visual representations. Flowers102 (Nilsback & Zisserman, 2008) is a dataset containing images of 102 flower species. GTSRB (Stallkamp et al., 2011) contains images for classification of German traffic signs. UCF101 (Soomro et al., 2012) is a video dataset with short clips for action recognition. For UCF101 we followed the procedure reported in CLIP and extract the middle frame of every video to assemble the dataset. Stanford Cars (Krause et al., 2013) contains images of 196 types of cars. ImageNet (Deng et al., 2009) is a large scale image classification dataset with images across 1,000 classes. ImageNetv2 (Recht et al., 2019) consists of three new test sets with 10,000 images each for the ImageNet benchmark. For further details see Appendix Section A.2.3. ",
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+        "type": "text",
+        "text": "Results. We employed the same evaluation strategy and used the prompts as published in CLIP (see Appendix Section A.2.3). We report zero-shot results from two checkpoints in Table 1. CLIP and CLOOB were trained for a comparable number of epochs used in CLIP (see Appendix Section A.2.2) while ${ \\mathrm { C L I P } } ^ { * }$ and $\\mathrm { C L O O B ^ { \\ast } }$ were trained until evaluation performance plateaued (epoch 128). In both cases CLOOB significantly outperforms CLIP on the majority of tasks or matches its performance. Statistical significance of these results was assessed by an unpaired Wilcoxon test on a $5 \\%$ level. ",
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+        "text": "5.2 YFCC Pre-training ",
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+        "text": "Pre-training dataset. To be comparable to the CLIP results, we used the same subset of 15 million samples from the YFCC100M dataset (Thomee et al., 2016) as in Radford et al. (2021), which we refer to as YFCC. YFCC was created by filtering YFCC100M for images which contain natural language descriptions and/or titles in English. It was not filtered by quality of the captions, therefore the textual descriptions are less rich and contain superfluous information. The dataset with 400 million samples used to train the CLIP models in Radford et al. (2021) has not been released and, thus, is not available for comparison. Due to limited computational resources we were unable to compare CLOOB to CLIP on other datasets of this size. ",
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+        "type": "text",
+        "text": "Methods. Besides experiments with a ResNet-50 image encoder, we additionally conducted experiments with the larger ResNet variants ResNet-101 and ResNet-50x4. In addition to the comparison of CLOOB and CLIP based on the OpenCLIP reimplementation (Ilharco et al., 2021), we include the original CLIP results (Radford et al., 2021, Table 12). ",
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+        "type": "text",
+        "text": "Hyperparameter selection. Hyperparameters were the same as for the Conceptual Captions dataset, except learning rate, batch size, and $\\beta$ . For modern Hopfield networks, the hyperparameter $\\beta$ was set to 14.3, which is default for $\\tau ^ { - 1 }$ in Radford et al. (2021). Furthermore, the learning rate was set to $5 \\times 1 0 ^ { - 4 }$ and the batch size to 1024 as used in OpenCLIP of Ilharco et al. (2021). All models were trained for 28 epochs. For further details see Appendix Section A.2.2. ",
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+    {
+        "type": "text",
+        "text": "Evaluation metrics. As in the previous experiment, methods were again evaluated at their zero-shot transfer learning capabilities on downstream tasks. ",
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+        "table_caption": [
+            "Table 2: Results of CLIP and CLOOB trained on YFCC with ResNet-50 encoder. Except for one linear probing dataset, CLOOB consistently outperforms CLIP at all tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Linear Probing</td><td colspan=\"2\">Zero-Shot</td></tr><tr><td>Dataset</td><td>CLIP (OpenAI)</td><td>CLOOB (ours)</td><td>CLIP (OpenAI)</td><td>CLOOB (ours)</td></tr><tr><td>Birdsnap</td><td>47.4</td><td>56.2</td><td>19.9</td><td>28.9</td></tr><tr><td>Country211</td><td>23.1</td><td>20.6</td><td>5.2</td><td>7.9</td></tr><tr><td>Flowers102</td><td>94.4</td><td>96.1</td><td>48.6</td><td>55.1</td></tr><tr><td>GTSRB</td><td>66.8</td><td>78.9</td><td>6.9</td><td>8.1</td></tr><tr><td>UCF101</td><td>69.2</td><td>72.3</td><td>22.9</td><td>25.3</td></tr><tr><td>Stanford Cars</td><td>31.4</td><td>37.7</td><td>3.8</td><td>4.1</td></tr><tr><td>ImageNet</td><td>62.0</td><td>65.7</td><td>31.3</td><td>35.7</td></tr><tr><td></td><td></td><td>58.7</td><td>1</td><td>34.6</td></tr><tr><td>ImageNet V2</td><td>1</td><td></td><td></td><td></td></tr></table>",
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+        "table_caption": [
+            "Table 3: Zero-shot results for the CLIP reimplementation and CLOOB using different ResNet architectures trained on YFCC. CLOOB outperforms CLIP in 7 out of 8 tasks using ResNet-50 encoders. With larger ResNet encoders CLOOB outperforms CLIP on all tasks. The performance of CLOOB scales with increased encoder size. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">RN-50</td><td colspan=\"2\">RN-101</td><td colspan=\"2\">RN-50x4</td></tr><tr><td>Dataset</td><td>CLIP</td><td>CLOOB</td><td>CLIP</td><td>CLOOB</td><td>CLIP</td><td>CLOOB</td></tr><tr><td>Birdsnap</td><td>21.8</td><td>28.9</td><td>22.6</td><td>30.3</td><td>20.8</td><td>32.0</td></tr><tr><td>Country211</td><td>6.9</td><td>7.9</td><td>7.8</td><td>8.5</td><td>8.1</td><td>9.3</td></tr><tr><td>Flowers102</td><td>48.0</td><td>55.1</td><td>48.0</td><td>55.3</td><td>50.1</td><td>54.3</td></tr><tr><td>GTSRB</td><td>7.9</td><td>8.1</td><td>7.4</td><td>11.6</td><td>9.4</td><td>11.8</td></tr><tr><td>UCF101</td><td>27.2</td><td>25.3</td><td>28.6</td><td>28.8</td><td>31.0</td><td>31.9</td></tr><tr><td>Stanford Cars</td><td>3.7</td><td>4.1</td><td>3.8</td><td>5.5</td><td>3.5</td><td>6.1</td></tr><tr><td>ImageNet</td><td>34.6</td><td>35.7</td><td>35.3</td><td>37.1</td><td>37.7</td><td>39.0</td></tr><tr><td>ImageNet V2</td><td>33.4</td><td>34.6</td><td>34.1</td><td>35.6</td><td>35.9</td><td>37.3</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Results. Table 2 provides results of the original CLIP and CLOOB trained on YFCC. Results on zero-shot downstream tasks show that CLOOB outperforms CLIP on all 7 tasks (ImageNet V2 results have not been reported in Radford et al. (2021)). Similarly, CLOOB outperforms CLIP on 6 out of 7 tasks for linear probing. Results of CLOOB and the CLIP reimplementation of OpenCLIP are given in Table 3. CLOOB exceeds the CLIP reimplementation in 7 out of 8 tasks for zero-shot classification using ResNet-50 encoders. With larger ResNet encoders, CLOOB outperforms CLIP on all tasks. Furthermore, the experiments with larger vision encoder networks show that CLOOB performance increases with network size. Results of CLOOB zero-shot classification on all datasets are shown in Appendix Section A.2.4. ",
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+        "type": "text",
+        "text": "5.3 Ablation studies ",
+        "text_level": 1,
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+        "text": "CLOOB has two new major components compared to CLIP: (1) modern Hopfield networks and (2) the InfoLOOB objective instead of the InfoNCE objective. To assess effects of the new major components of CLOOB, we performed ablation studies. ",
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+        "text": "Modern Hopfield networks. Modern Hopfield networks amplify the covariance structure in the data via the retrievals. Ablation studies confirm this amplification as modern Hopfield networks consistently increase the number of effective eigenvalues of the embedding covariance matrices during learning. Figure 2 shows the relative change of the number of effective eigenvalues compared to the respective reference epoch, which is the epoch before the first learning rate restart. These results indicate that modern Hopfield networks steadily extract more covariance structure during learning. Modern Hopfield networks induce higher similarity of retrieved samples, which in turn leads to stronger saturation of the InfoNCE objective. As a result, we observe low uniformity (see Figure 3) and a small number of effective eigenvalues (see Appendix Figure A1). ",
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+        "type": "text",
+        "text": "Modern Hopfield networks with InfoLOOB. CLOOB counters the saturation of InfoNCE by using the InfoLOOB objective. The effectiveness of InfoLOOB manifests in an increased uniformity measure of image and text embeddings on the sphere, as shown in Figure 3. The ablation study verifies that modern Hopfield networks together with InfoLOOB have a strong synergistic effect. ",
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+    {
+        "type": "text",
+        "text": "InfoLOOB. However, using solely InfoLOOB results in overfitting of the alignment score. This overfitting occasionally leads to high similarities of unmatched pairs (see Figure 4), which may decreases the zero-shot downstream performance. The reason for this is that the top-1 evaluation metric is very sensitive to occasionally high similarities of prompts of the incorrect class. Yeh et al. (2021) and Zhang et al. (2022) reported similar observations of overfitting. ",
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+        "type": "image",
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+        "image_caption": [
+            "Figure 4: Distribution of the cosine similarity of matched pairs and the cosine similarity of the 10 unmatched pairs that have the highest similarity score with the anchor. Modern Hopfield networks lead to higher values of both matched and unmatched pairs. InfoLOOB without Hopfield has high similarity scores of the matched pairs which correlate with high similarity scores of the top-10 unmatched pairs. In contrast, InfoLOOB with Hopfield does not suffer from this overfitting problem. "
+        ],
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+    {
+        "type": "text",
+        "text": "CLOOB balances the overfitting of InfoLOOB with the underfitting of modern Hopfield networks and remains in effective learning regimes. For more details and further ablation studies see Appendix Section A.2.1. ",
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+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We have introduced “Contrastive Leave One Out Boost” (CLOOB), which combines modern Hopfield networks with the InfoLOOB objective. Modern Hopfield networks enable CLOOB to extract additional covariance structure in the data. This allows for building more relevant features in the embedding space, mitigating the explaining away problem. We show that InfoLOOB avoids the saturation problem of InfoNCE. Additionally, we theoretically justify the use of the InfoLOOB objective for contrastive learning and suggest it as an alternative to InfoNCE. At seven zero-shot transfer learning tasks, the novel CLOOB was compared to CLIP after pre-training on the Conceptual Captions and the YFCC dataset. CLOOB consistently outperforms CLIP at zero-shot transfer learning across all considered architectures and datasets. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The ELLIS Unit Linz, the LIT AI Lab, the Institute for Machine Learning, are supported by the Federal State Upper Austria. IARAI is supported by Here Technologies. We thank the projects AIMOTION (LIT-2018-6-YOU-212), AI-SNN (LIT-2018-6-YOU-214), DeepFlood (LIT-2019-8-YOU213), Medical Cognitive Computing Center (MC3), INCONTROL-RL (FFG-881064), PRIMAL (FFG-873979), S3AI (FFG-872172), DL for GranularFlow (FFG-871302), AIRI FG 9-N (FWF36284, FWF-36235), ELISE (H2020-ICT-2019-3 ID: 951847). We thank Audi.JKU Deep Learning Center, TGW LOGISTICS GROUP GMBH, Silicon Austria Labs (SAL), FILL Gesellschaft mbH, Anyline GmbH, Google, ZF Friedrichshafen AG, Robert Bosch GmbH, UCB Biopharma SRL, Merck Healthcare KGaA, Verbund AG, Software Competence Center Hagenberg GmbH, TÜV Austria, Frauscher Sensonic and the NVIDIA Corporation. ",
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+        "type": "text",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] Our method is currently limited to natural images and short text prompts as inputs, and, thus its use for other types of images, such as medical or biological images, is unexplored. While we hypothesize that our approach could also be useful for similar data in other domains, this has not been assessed.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] One potential danger arises from users that overly rely on systems built on our method. For example in the domain of self-driving cars, users might start paying less attention to the traffic because of the AI-based driving system. Finally, our method might also be used to automate various simple tasks, which might lead to reduced need for particular jobs in production systems. As for almost all machine learning methods, our proposed method relies on human-annotated training data and thereby human decisions, which are usually strongly biased. Therefore, the responsible use of our method requires the careful selection of the training data and awareness of potential biases within those.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] For the model implementations we used PyTorch (Paszke et al., 2017, BSD license) and for monitoring the runs we used Weights & Biases (Biewald, 2020, MIT license).   \n(b) Did you mention the license of the assets? [Yes] See above.   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We provide the code as supplementary material.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] We only use public datasets that can be used for research purposes, such as the YFCC dataset which was published under the Creative Commons licence.   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] As almost all computer vision and natural language datasets, the data suffers from many biases including social biases. We refer to Yang et al. (2020) for a detailed analysis of biases in such datasets. ",
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+        "text": "(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
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