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+# A UNIVERSAL REPRESENTATION TRANSFORMER LAYER FOR FEW-SHOT IMAGE CLASSIFICATION
+
+Lu $\mathbf { L i u ^ { 1 , 2 * } }$ , William Hamilton $^ { 1 , 3 }$ †, Guodong Long2, Jing Jiang2, Hugo Larochelle1,4† 1 Mila, 2 Australian AI Institute, UTS, 3 McGill University, 4 Google Research, Brain Team Correspondence to lu.liu.cs@icloud.com
+
+# ABSTRACT
+
+Few-shot classification aims to recognize unseen classes when presented with only a small number of samples. We consider the problem of multi-domain few-shot image classification, where unseen classes and examples come from diverse data sources. This problem has seen growing interest and has inspired the development of benchmarks such as Meta-Dataset. A key challenge in this multi-domain setting is to effectively integrate the feature representations from the diverse set of training domains. Here, we propose a Universal Representation Transformer (URT) layer, that meta-learns to leverage universal features for few-shot classification by dynamically re-weighting and composing the most appropriate domain-specific representations. In experiments, we show that URT sets a new state-of-the-art result on Meta-Dataset. Specifically, it achieves top-performance on the highest number of data sources compared to competing methods. We analyze variants of URT and present a visualization of the attention score heatmaps that sheds light on how the model performs cross-domain generalization. Our code is available at https://github.com/liulu112601/URT.
+
+# 1 INTRODUCTION
+
+Learning tasks from small data remains a challenge for machine learning systems, which show a noticeable gap compared to the ability of humans to understand new concepts from few examples. A promising direction to address this challenge is developing methods that are capable of performing transfer learning across the collective data of many tasks. Since machine learning systems generally improve with the availability of more data, a natural assumption is that few-shot learning systems should benefit from leveraging data across many different tasks and domains—even if each individual task has limited training data available.
+
+This research direction is well captured by the problem of multi-domain few-shot classification. In this setting, training and test data spans a number of different domains, each represented by a different source dataset. A successful approach in this multi-domain setting must not only address the regular challenge of few-shot classification—i.e., the challenge of having only a handful of examples per class. It must also discover how to leverage (or ignore) what is learned from different domains, achieving generalization and avoiding cross-domain interference.
+
+Recently, Triantafillou et al. (2020) proposed a benchmark for multi-domain few-shot classification, Meta-Dataset, and highlighted some of the challenges that current methods face when training data is heterogeneous. Crucially, they found that methods which trained on all available domains would normally obtain improved performance on some domains at the expense of others. Following on their work, progress has been made, which includes the design of adapted hyper-parameter optimization strategies (Saikia et al., 2020) and more flexible meta-learning algorithms (Requeima et al., 2019). Most notable is SUR (Selecting Universal Representation) (Dvornik et al., 2020), a method that relies on a so-called universal representation, extracting from a collection of pre-trained and domain-specific neural network backbones. SUR prescribes a hand-crafted feature-selection procedure to infer how to weight each backbone for each task at hand, and produces an adapted representation for each task. This was shown to lead to some of the best performances on Meta-Dataset.
+
+In SUR, the classification procedure for each task is fixed and not learned. Thus, except for the underlying universal representation, there is no transfer learning performed with regards to how classification rules are inferred across tasks and domains. Yet, cross-domain generalization might be beneficial in that area as well, in particular when tasks have only few examples per class.
+
+Present work. To explore this question, we propose a Universal Representation Transformer (URT) layer, which can effectively learn to transform a universal representation into task-adapted representations. The URT layer is inspired from Transformer (Vaswani et al., 2017) and uses an attention mechanism to learn to retrieve or blend the appropriate backbones to use for each task. By training this layer across few-shot tasks from many domains, it can support transfer across these tasks.
+
+We show that our URT layer on top of a universal representation’s pre-trained backbones sets a new state-of-the-art performance on Meta-Dataset. It succeeds at outperforming SUR on 4 dataset sources without impairing accuracy on the others. This leads to top performance on 7 dataset sources when comparing to a set of competing methods. To interpret the strategy that URT learns to weigh the backbones from different domains, we visualize the attention scores for both seen and unseen domains and find that our model generates meaningful weights for the pre-trained domains. A comprehensive analysis on variants and ablations of the URT layer is provided to show the importance of various components of URT, notably the number of attention heads.
+
+# 2 FEW-SHOT CLASSIFICATION
+
+# 2.1 PROBLEM SETTING
+
+In this section, we will introduce the problem setting for few-shot classification and the formulation of meta-learning for few-shot classification. Few-shot classification aims to classify samples where only few examples are available for each class. We describe a few-shot learning classification task as the pair of examples, comprising of a support set $S$ to define the classification task and the query set $Q$ of samples to be classified.
+
+Meta-learning is a technique that aims to model the problem of few-shot classification as learning to learn from instances of few-shot classification tasks. The most popular way to train a meta-learning model is with episodic training. Here, tasks $T = ( Q , S )$ are sampled from a larger dataset by taking subsets of the dataset to build a support set $S$ and a query set $Q$ for the task. A common approach is to sample $N$ -way- $K$ -shot tasks, each time selecting a random subset of $N$ classes from the original dataset and choosing only $K$ examples for each class to add to the support set $S$ .
+
+The meta-learning problem can then be formulated by the following optimization:
+
+$$
+\operatorname* { m i n } _ { \Theta } \mathbb { E } _ { ( S , Q ) \sim p ( T ) } \left[ \mathcal { L } ( S , Q , \Theta ) \right] , \ \mathcal { L } ( S , Q , \Theta ) = \frac { 1 } { | Q | } \sum _ { { ( x , y ) } \sim Q } - \log p ( y | x , S ; \Theta ) + \lambda \Omega ( \Theta ) ,
+$$
+
+where $p ( T )$ is the distribution of tasks, $\Theta$ are the parameters of the model and $p ( \boldsymbol { y } | \boldsymbol { x } , S ; \Theta )$ is the probability assigned by the model to label $y$ of query example $_ { \textbf { \em x } }$ (given the support set $S$ ), and $\Omega ( \Theta )$ is an optional regularization term on the model parameters with factor $\lambda$ .
+
+Conventional few-shot classification targets the setting of $N$ -way- $K$ -shot, where the number of classes and examples are fixed in each episode. Popular benchmarks following this approach include Omniglot (Lake et al., 2015) or benchmarks made of subsets of ImageNet, such as miniImageNet (Vinyals et al., 2016) and tieredImageNet (Ren et al., 2018). In such benchmarks, the tasks for training cover a set of classes that is disjoint from the classes in the test set of tasks. However, with the training and test sets tasks coming from a single dataset/domain, the distribution of tasks found in either sets is similar and lacks variability, which may be unrealistic in practice.
+
+It is in this context that Triantafillou et al. (2020) proposed Meta-Dataset, as a further step towards large-scale, multi-domain few shot classification. Meta-Dataset includes ten datasets (domains), with eight of them available for training. Additionally, each task sampled in the benchmark varies in the number of classes $N$ , with each class also varying in the number of shots $K$ . As in all few-shot learning benchmarks, the classes used for training and testing do not overlap.
+
+# 2.2 BACKGROUND AND RELATED WORK
+
+Meta-Learning A promising approach for few-shot classification is to use meta-learning to more directly train a model to learn to perform few-shot classification, in an end-to-end way. The two most popular methods are Prototypical Networks (Snell et al., 2017) and Model Agnostic Meta-Learning (MAML) (Finn et al., 2017). Triantafillou et al. (2020) showed that prototypical networks and MAML could be combined by leveraging prototypes for the initialization of the output weights value in the inner loop. Requeima et al. (2019) also proposed Conditional Neural Adaptive Processes (CNAPs) for few-shot classification, which can be seen as extending prototypical networks with a more sophisticated architecture that allows for improved task adaptation. This architecture was later improved further by Bateni et al. (2020) with Simple CNAPS, leading to one of the current best methods on Meta-Dataset. Another line of work which leverages the idea of “transfer by fine-tuning” can be found in Appendix B.
+
+Universal Representations In contrast, our work instead builds on that of Dvornik et al. (2020) and their method SUR (Selecting from Universal Representations). Bilen & Vedaldi (2017) introduced the term universal representation to refer to a representation that supports good performance in multiple domains. One proposal towards such a representation is to train different neural networks backbones separately on the data of each available domain, then simply to concatenate the representation learned by each. Another is to introduce some parameter sharing between the backbones, by having a single network conditioned on the domain of the provenance of each batch of training data (Rebuffi et al., 2018), e.g. using Feature-wise Linear Modulate (FiLM) (Perez et al., 2018). SUR proposes to leverage a universal representation in few-shot learning tasks with a feature selection procedure that assigns different weights to each of the domain-specific subvectors of the universal representation. The objective is to assign high weights only to the domain-specific representations that are specifically useful for each few-shot task at hand. The weights are inferred by optimizing a loss on the support set that encourages high accuracy of a nearest-centroid classifier. As such, the method does not involve any meta-learning—a choice motivated by the concern that meta-learning may struggle in generalizing to domains that are dissimilar to the training domains. SUR achieved some of the best performances on Meta-Dataset. However, a contribution of our work is to provide evidence that meta-learning can actually be used to replace SUR’s hand-designed inference procedure and improve performance further.
+
+Task Adaptive Representations Another line of work tries to retrieve task adaptive representations for each task. Task specific representations can be conditioned on a representation of the current task (Oreshkin et al., 2018; Wang et al., 2019), projected to another space (Yoon et al., 2019), or masked based on inter-class commonality and inter-class uniqueness (Li et al., 2019). While the representation extracted from URT is also task adaptive, it is adaptive to a set of pretrained backbones and can be applied to more complicated multi-domain scenarios. Wang & Hebert (2016) proposed to improve a CNN by adding extra layers and train it using unsupervised data while our contribution mainly lies in composing representations instead of an improved CNN. Alet et al. (2018) introduced a modular meta-learning method, which learns a repertoire of modules that serves as nodes to construct a tree structure to solve a new robotic-related task. Comparatively, URT is a one-for-all layer which doesn’t need to construct different module structures for each task.
+
+Transformer Networks Our meta-learning approach to leverage universal representations is inspired directly from Transformer networks (Vaswani et al., 2017). Our model structure is inspired by the structure of the dot-product self-attention in the Transformer, which we adapted here to multidomain few-shot learning by designing appropriate parametrizations for queries, keys and values. Self-attention was explored in the single-domain training regime by Ye et al. (2020); Liu et al. (2019b;a; 2020), however for a different purpose, where each representation of individual examples in a task support set is influenced by all other examples. Rather than using self-attention between individual examples in the support set, our model uses self-attention to select between different domain-specific backbones.
+
+# 3 UNIVERSAL REPRESENTATION TRANSFORMER LAYER
+
+In this section, we describe our proposed URT layer, which uses meta-learning episodic training to learn how to combine the domain-specific backbones of a universal representation for any given fewshot learning classification task. URT layer can be built on top of any set of pretrained backbones without further costly fine-tuning of the backbones. More details on how to train multiple domainspecific backbones can be found in Appendix C.
+
+![](images/b8ec3fe34a1bbf140191b9bf5722ec5773c3834b441b3d9391ea604b4b89789b.jpg)  
+Figure 1: Illustration of how a single-head URT layer uses a universal representation to produce a task-specific representation. This example assumes the use of four backbones, with each color illustrating their domain-specific sub-vector representation in the universal representation.
+
+Conceptually, the proposed model views the support set $S$ of a task as providing information on how to query and retrieve from the set $\{ r _ { i } \}$ of $m$ pre-trained backbones the most appropriate backbone to build an adapted representation $\phi$ for the task.
+
+We would like the model to support a variety of strategies on how to retrieve backbones. For example, it might be beneficial for the model to retrieve a single backbone from the set, especially if the domain of the given task matches perfectly that of a domain found in the training set. Alternatively, if some of the training domains benefit from much more training data than others, a better strategy might be to attempt some cross-domain generalization towards the few-shot learning task by blending many backbones together, even if none matches the domain of the task perfectly.
+
+This motivates us to use dot-product self-attention, inspired by layers of Transformer networks (Vaswani et al., 2017). For this reason, we refer to our model as a Universal Representation Transformer (URT) layer. Additionally, since each class of the support set might require a different strategy, we perform attention separately for each class and their support set $S _ { c } = \{ { \pmb x } | ( { \pmb x } , y ) \in S$ and $y = c \}$ .
+
+# 3.1 SINGLE-HEAD URT LAYER
+
+We start by describing an URT layer consisting of a single attention head. An illustration of a singlehead URT layer is shown in Figure 1. Let $r _ { i } ( { \pmb x } )$ be the output vector of the backbone for domain $i$ . We then write the universal representation as
+
+$$
+r ( \mathbf x ) = \mathrm { c o n c a t } ( r _ { 1 } ( \mathbf x ) , \hdots , r _ { m } ( \mathbf x ) ) .
+$$
+
+This representation provides a natural starting point to obtain a representation of a support set class. Specifically, we will note
+
+$$
+r ( S _ { c } ) = \frac { 1 } { \left| S _ { c } \right| } \sum _ { { \pmb x } \in S _ { c } } r ( { \pmb x } )
+$$
+
+as the representation for the set $S _ { c }$ . From this, we can describe the URT layer by defining the queries1, keys, the attention mechanism and output of the layer:
+
+Queries $\mathbf { q } _ { c }$ : For each class $c$ , we obtain a query through $\mathbf { q } _ { c } = \mathbf { W } ^ { q } r ( S _ { c } ) + \mathbf { b } ^ { q }$ , where we have a learnable query linear transformation represented by matrix $\mathbf { W } ^ { q }$ and bias $\mathbf { b } ^ { q }$ .
+
+Keys $\mathbf { k } _ { i , c }$ : For each domain $i$ and class $c$ , we define keys as $\mathbf { k } _ { i , c } = \mathbf { W } ^ { k } r _ { i } ( S _ { c } ) + \mathbf { b } ^ { k }$ , using a learnable linear transformation $\mathbf { W } ^ { k }$ and $\mathbf { b } ^ { k }$ and where $r _ { i } ( S _ { c } ) = 1 / | S _ { c } | \textstyle \sum _ { { \pmb x } \in S _ { c } } r _ { i } ( { \pmb x } )$ , using a similar notation as for $r ( S _ { c } )$ .
+
+# Algorithm 1 Training of URT layer
+
+Input: Number of tasks $\tau _ { t o t a l }$ , $m$ pre-trained backbones ;
+
+1: for $\tau \in \{ 1 , \cdots , \tau _ { t o t a l } \}$ do   
+2: Sample a few-shot task $T$ with support set $S$ and query set $Q$ ;   
+3: # Infer adapted representation for task from $S$   
+4: For each class, obtain representation using $m$ pre-trained backbones as in Eq. (3);   
+5: Obtain attention scores using Eq. (4,5) for each head using support set $S$ ;   
+6: # Use adapted representation to predict labels in $Q$ from support set $S$   
+7: Compute adapted representation of examples in $S$ and $Q$ as in Eq. (6,7);   
+8: Compute probabilities of label of examples in $Q$ using Prototypical Network as in Eq. (9);   
+9: Compute loss as in Eq. (1,8) and perform gradient descent step on URT parameters $\Theta$ ;
+
+10: end for
+
+Attention scores $\alpha _ { i }$ : as for regular Transformer layers, we use scaled dot-product attention
+
+$$
+\alpha _ { i , c } = \frac { \exp ( \beta _ { i , c } ) } { \sum _ { i ^ { \prime } } \exp ( \beta _ { i ^ { \prime } , c } ) } , \beta _ { i , c } = \frac { { \bf q } _ { c } \mathrm {  ~ } ^ { \top } { \bf k } _ { i , c } } { \sqrt { l } } ,
+$$
+
+where $l$ is the dimensionality of the keys and queries. Then, these per-class scores are aggregated to obtain scores for the full support set by averaging
+
+$$
+\alpha _ { i } = \frac { \sum _ { c } \alpha _ { i , c } } { N } .
+$$
+
+Equipped with these attention scores, the URT layer can now produce an adapted representation for the task (for the support and query set examples) by computing
+
+$$
+\phi ( { \bf x } ) = \sum _ { i } \alpha _ { i } r _ { i } ( { \bf x } ) .
+$$
+
+As we can see, this approach has the flexibility of either selecting a single domain-specific backbone (by assigning $\alpha _ { i } = 1$ for a single domain) or blending different domains together (by having $\alpha _ { i } > >$ 0 for multiple backbones).
+
+# 3.2 MULTI-HEAD URT LAYER
+
+The URT layer described so far can only learn to retrieve a single backbone (or blending of backbones). Yet, it might be beneficial to retrieve multiple different (blended) backbones, especially for a few-shot task that would include many classes of varying complexity.
+
+Thus, to achieve such diversity in the adapted representation, we also consider URT layers with multiple heads, i.e. where each head corresponds to the calculation of Equation 6 and each head has its own set of parameters $( \mathbf { W } ^ { q } , \mathbf { b } ^ { q } , \mathbf { W } ^ { k } , \bar { \mathbf { b } ^ { k } } )$ . Denoting each head now as $\phi _ { h }$ , a multi-head URT layer then produces as its output the concatenation of all of its heads:
+
+$$
+\phi ( { \bf x } ) = \mathrm { c o n c a t } ( \phi _ { 1 } ( { \bf x } ) , \ldots , \phi _ { \mathrm { H } } ( { \bf x } ) ) .
+$$
+
+Empirically we found that the randomness in the initialization of head weights alone did not lead to uniqueness and being complimentary between the heads, so inspired by Lin et al. (2017), we add a regularizer to avoid duplication of the attention scores:
+
+$$
+\Omega ( \Theta ) = \| ( \mathbf { A } \mathbf { A } ^ { \top } - \mathbf { I } ) \| _ { F } ^ { 2 } ,
+$$
+
+where $\| \cdot \| _ { F }$ is the Frobenius norm of a matrix and $\mathbf { A } \in \mathbb { R } ^ { n \times m }$ is the matrix for attention scores, with $\mathbf { A } _ { h }$ being the vector of all scores $\alpha _ { i }$ for head $h$ . The identity matrix I regularizes each set of attention scores to be more focused so that multiple heads can attend to different domain-specific backbones.
+
+# 3.3 TRAINING STRATEGY
+
+We train representations produced by the URT layer by following the approach of Prototypical Networks (Snell et al., 2017), where the probability of a label $y$ for a query example $_ { \textbf { \em x } }$ given the
+
+Table 1: Test accuracy (mean $\pm \mathrm { C I } \% 9 5 $ ) over 600 few-shot tasks. URT and the most recent methods, which are listed in the first column, are compared on Meta-Dataset (Triantafillou et al., 2020), which are listed in the first row. The numbers in bold have intersecting confidence intervals with the most accurate method.   
+
+<table><tr><td></td><td>ILSVRC</td><td>Omniglot</td><td>Aircraft</td><td>Birds</td><td>Textures</td><td>QuickDraw</td><td>Fungi</td><td>VGGFlower</td><td>TrafficSigns</td><td></td><td>MSCOCO</td><td>avg.rank</td></tr><tr><td>MAML</td><td></td><td>37.8±1.0 83.9±1.0 76.4±0.7 62.4±1.1(</td><td></td><td></td><td></td><td>64.1±0.8</td><td>59.7±1.1</td><td>33.5±1.1</td><td>79.9±0.8</td><td>42.9±1.3</td><td>29.4±1.1</td><td>6.4</td></tr><tr><td>ProtoNet</td><td></td><td>44.5±1.1 79.6±1.1 71.1±0.9 67.0±1.0 65.2±0.8</td><td></td><td></td><td></td><td></td><td>64.9±0.9</td><td>40.3±1.1</td><td>86.9±0.7</td><td>46.5±1.0</td><td>39.9±1.1</td><td>5.7</td></tr><tr><td>ProtoMAML</td><td></td><td>46.5±1.1 82.7±1.0 75.2±0.8 69.9±1.0 68.3±0.8</td><td></td><td></td><td></td><td></td><td>66.8±0.9</td><td>42.0±1.2</td><td>88.7±0.7</td><td>52.4±1.1</td><td>41.7±1.1</td><td>4.1</td></tr><tr><td>CNAPs</td><td>50.8±1.1</td><td>91.7±0.5 83.7±0.6 73.6±0.9</td><td></td><td></td><td>59.5±0.7</td><td></td><td>74.7±0.8</td><td>50.2±1.1</td><td>88.9±0.5</td><td>56.5±1.1</td><td>39.4±1.1</td><td>3.6</td></tr><tr><td>SUR</td><td>56.1±1.1 </td><td>93.1±0.5 84.6±0.7 70.6±1.0 71.0±0.8</td><td></td><td></td><td></td><td></td><td>81.3±0.6</td><td>64.2±1.0</td><td>82.8±0.7</td><td>53.4±1.0</td><td>50.1±1.0</td><td>2.3</td></tr><tr><td>SimpleCNAPS 56.5±1.1 91.9±0.6 83.8±0.7 76.1±0.870.0±0.8</td><td></td><td></td><td></td><td></td><td></td><td></td><td>78.3±0.7</td><td>49.1±1.2</td><td>91.3±0.6</td><td>59.2±1.0</td><td>42.4±1.1</td><td>2.2</td></tr><tr><td>URT (Ours)</td><td></td><td>55.7±1.0 94.4±0.4 85.8±0.6 76.3±0.871.8±0.7</td><td></td><td></td><td></td><td></td><td>82.5±0.6</td><td>63.5±1.0</td><td>88.2±0.6</td><td>51.1±1.1</td><td>52.2±1.1</td><td>1.5</td></tr></table>
+
+support set of a task is modeled as:
+
+$$
+p ( y = c | \pmb { x } , S ; \Theta ) = \frac { \exp ( - d ( \phi ( \pmb { x } ) - \pmb { p _ { c } } ) ) } { \sum _ { c ^ { \prime } = 1 } ^ { N } \exp ( - d ( \phi ( \pmb { x } ) - \pmb { p _ { c ^ { \prime } } } ) ) } ,
+$$
+
+where $d$ is a distance metric and $\pmb { p } _ { c } = 1 / | S _ { c } | \sum _ { \pmb { x } \in S _ { c } } \phi ( \pmb { x } )$ corresponds to the centroid of class $c _ { \cdot }$ referred to as its prototype. We use (negative) cosine similarity as the distance. The full training algorithm is presented in Algorithm 1.
+
+# 4 EXPERIMENTS
+
+In this section, we seek to answer three key experimental questions:
+
+Q1 How does URT compare with previous state-of-the-art on Meta-Dataset for multi-domain fewshot classification?   
+Q2 Do the URT attention heads generate interpretable and meaningful attention scores?   
+Q3 Does the URT layer provide consistent benefits, even when pre-trained backbones are trained in different ways?
+
+In addition, we investigate architectural choices made, such as our models for keys/queries and their regularization, and study their contribution to achieving strong performance with URT.
+
+# 4.1 DATASETS AND SETUP
+
+We test our methods on the large-scale few-shot learning benchmark Meta-Dataset (Triantafillou et al., 2020). It consists of ten datasets with various data distributions across different domains, including natural images (Birds, Fungi, VGG Flower), hand-written characters (Omniglot, Quick Draw), and human created objects (Traffic Signs, Aircraft). Among the ten datasets, eight provide data that can be used during either training, validation and testing (with each class assigned to only one of those sets), while two datasets are solely used for testing. Following Bateni et al. (2020); Requeima et al. (2019), we also report results on MNIST (LeCun et al., 1998), CIFAR10 and CIFAR100 (Krizhevsky et al., 2009) as additional unseen test datasets. Following Triantafillou et al. (2020), few-shot tasks are sampled with varying number of classes $N$ , varying number of shots $K$ and class imbalance. The performance is reported as the average accuracy over 600 sampled tasks. More details of Meta-Dataset can be found in Triantafillou et al. (2020).
+
+The domain-specific backbones are pre-trained following the setup in (Dvornik et al., 2020). Then, we freeze the backbone and train the URT layer for 10,000 episodes, with an initial learning rate of 0.01 and a cosine learning rate scheduler. Following Chen et al. (2020), the training episodes have $50 \%$ probability coming from the ImageNet data source. Since different pre-trained backbones may produce representations with different vector norms, we normalize the outputs of the backbones as in Dvornik et al. (2020). URT is trained with parameter weight decay of 1e-5 and with a regularization factor $\lambda = 0 . 1$ . The number of heads ( $H$ in Equation 7), is set to 2 and the dimension of the keys and queries (l in Equation 4) is set to 1024. We choose the hyper-parameters based on the performance of the validation set. Details of the hyper-parameter selection and how the performance is influenced by them are outlined in Section 4.5. We find that the bottleneck of training URT is extracting features from CNN. Since we freeze the CNN when training the URT, we find dumping the extracted feature episodes can significantly speed up the training procedure from days to around 2 hours.
+
+![](images/c10077fb200921d3bb6bc7d24d0c40b0a826b01959da838218a14b2d49389f42.jpg)  
+Figure 2: Average attention scores generated by URT with two heads. Rows correspond to the domain of the test tasks and the columns correspond to the pre-trained backbones $r _ { i } ( { \pmb x } )$ trained on the eight training domains.
+
+# 4.2 COMPARISON WITH PREVIOUS APPROACHES
+
+Table 1 presents a comparison of URT with SUR, as well as other baselines based on transfer learning by fine-tuning (Saikia et al., 2020) or meta-learning (Prototypical Networks (Snell et al., 2017), first-order MAML (Finn et al., 2017), ProtoMAML (Triantafillou et al., 2020), CNAPs (Requeima et al., 2019)) and Simple CNAPS(Bateni et al., 2020).
+
+We observe in Table 1 that URT establishes a new state-of-the-art on Meta-Dataset, by achieving the top performance on 8 out of the 10 dataset sources. When comparing to its predecessor, URT outperforms SUR on 4 datasets without compromising performance on others, which is challenging to achieve in the multi-domain setting. Of note, the average inference time for URT is 0.04 second per task, compared to 0.43 for SUR, on a single V100. Thus, getting rid of the optimization procedure for every episode with our meta-trained URT layer also significantly increases the latency, by more than $1 0 \times$ . More results on additional datasets can be found in Appendix A.
+
+# 4.3 INTERPRETING AND VISUALIZING ATTENTION BY URT
+
+To better understand how the URT model of Section 4.2 uses its two heads to build adapted representations, we visualize the attention scores produced on the test tasks of Meta-Dataset in Figure 2.
+
+The blue (first head) and orange (second head) heatmaps summarize the values of the attention scores (Equation 5), averaged across several tasks for each test domain. Specifically, the element on row $t$ and column $i$ is the averaged attention scores $\alpha _ { i }$ computed on test set domain $t$ for the backbone from domain $i$ . Note that the last two rows are the two unseen domain datasets. We found that for datasets from the seen domains, i.e. the first eight rows, one head (right, orange) consistently puts most of its weight on the backbone pre-trained on the same domain, while the other head (left, blue) learns relatively smoother weight distributions that blend other related domains. For unseen datasets, the right head puts half of its weight on ImageNet and the left head learned to blend the representations from four backbones.
+
+Table 2: Test accuracy (mean $\pm \mathrm { C I } \% 9 5$ ) over 600 few-shot tasks. All methods use parametric network family (pf) backbones.   
+
+<table><tr><td></td><td> SUR-pf</td><td>URT-pf</td><td>VS.</td></tr><tr><td>ILSVRC</td><td>56.0 ± 1.1</td><td>55.5 ± 1.1</td><td>二</td></tr><tr><td>Omniglot</td><td>90.0 ± 0.8</td><td>90.2 ± 0.6</td><td>二</td></tr><tr><td>Aircraft</td><td>79.7 ± 0.8</td><td>79.8 ± 0.7</td><td>=</td></tr><tr><td>Birds</td><td>75.9 ± 0.9</td><td>77.5 ± 0.8</td><td>三</td></tr><tr><td>Textures</td><td>72.5 ± 0.7</td><td>73.5 ± 0.7</td><td></td></tr><tr><td>Quick Draw</td><td>76.7 ± 0.7</td><td>75.8 ± 0.7</td><td>三 二</td></tr><tr><td>Fungi</td><td>49.8 ± 1.1</td><td>48.1 ± 0.9</td><td>+</td></tr><tr><td>VGG Flower</td><td>90.0 ± 0.6</td><td>91.9 ± 0.5</td><td>二</td></tr><tr><td>Traffic Signs</td><td>52.2 ± 0.8</td><td>52.0 ± 1.4</td><td></td></tr><tr><td>MSCOCO</td><td>50.2 ± 1.1</td><td>52.1 ± 1.0</td><td></td></tr><tr><td>MNIST</td><td>93.2 ± 0.4</td><td>93.9 ± 0.4</td><td>=</td></tr><tr><td>CIFAR10</td><td>66.4 ± 0.8</td><td>66.1 ± 0.8</td><td>三</td></tr><tr><td>CIFAR100</td><td>57.1 ± 1.0</td><td>57.3 ± 1.0</td><td>二</td></tr></table>
+
+# 4.4 URT USING FILM MODULATED BACKBONES
+
+As additional evidence of the benefit of URT on universal representations, we also present experiments based on a different set of backbone architectures. Following SUR (Dvornik et al., 2020), we consider the backbones from a parametric network family, obtained by training a base backbone on one dataset (ILSVRC) and then learning separate FiLM layers (Perez et al., 2018) for each other dataset, to modulate the backbone so it is adapted to the other domains. These backbones collectively have only $0 . 5 \%$ more parameters than a single backbone. More details of the backbones can be found in Appendix C.
+
+A comparison between SUR and URT using these backbones (referred to as SUR-pf and URT-pf) is presented in Table 2. Once again, URT can improve the performance on VGG Flower without sacrificing performance on others.
+
+# 4.5 HYPER-PARAMETER AND ABLATION STUDIES
+
+We analyze the importance of the various components of URT’s attention mechanism structure and training strategy in Table 3. First we analyze the importance of using the support set to model queries and/or keys. To this end, we consider setting the matrices $\mathbf { W } ^ { q } / \mathbf { W } ^ { k }$ of the query / key linear transformation to 0, which only leaves the bias term. We found that the support set representation is most crucial for building the keys (row w/o $\mathbf { W } ^ { k }$ in the table) and has minor benefits for queries (row $\mathbf { w } / \mathbf { o } \mathbf { W } ^ { q } )$ in the table. This observation is possibly related to the success of attention-based models with learnable constant queries (Liu et al., 2016; Lin et al., 2017). We also found that adding a regularizer $\Omega ( \Theta )$ as in Equation 8 is important for some datasets, specifically VGG Flower and Birds.
+
+Table 3: Meta-Dataset performance variation on ablations of elements of the URT layer.   
+
+<table><tr><td colspan="11">ILSVRC Omniglot Aircraft Birds Textures Draw Fungi Flower Signs MSCOCO</td></tr><tr><td>w/o Wq</td><td>+0.2</td><td>-0.2</td><td>-0.6</td><td>-0.1</td><td>-0.3</td><td>-0.2</td><td>0.0</td><td>-0.2</td><td>-0.8</td><td>-0.1</td></tr><tr><td>w/o Wk</td><td>-14.2</td><td>-2.8</td><td>-10.7</td><td>-18.1</td><td>-7.6</td><td>-9.3</td><td>-22.4</td><td>-3.6</td><td>-0.26</td><td>-10.9</td></tr><tr><td>w/or(Sc)</td><td>-14.2</td><td>-2.8</td><td>-10.7</td><td>-18.1</td><td>-7.6</td><td>-9.2</td><td>-22.4</td><td>-3.6</td><td>-0.26</td><td>-10.9</td></tr><tr><td>w/0 Ω(0)</td><td>0.0</td><td>-0.9</td><td>-0.4</td><td>-3.3</td><td>-1.2</td><td>-0.2</td><td>+0.3</td><td>-9.0</td><td>-2.0</td><td>0.0</td></tr></table>
+
+An important hyper-parameter in URT is the number of heads $H$ . We chose this hyper-parameter based on the performance on validation set of tasks in Meta-Dataset. In Table 4, we show the validation performance of URT for varying number of heads. As suggested by Triantafillou et al. (2020), we considered looking at the rank of the performance achieved by each choice of $H$ for each validation domains, and taking the average across domains as a validation metric. However, since the performances when using two to four heads are similar and yield the same average rank, we instead simply consider the average accuracy as the selection criteria.
+
+Table 4: Validation performance on Meta-Dataset using different number of heads   
+
+<table><tr><td></td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td>Average Accuracy</td><td>74.605</td><td>77.145</td><td>76.943</td><td>76.984</td><td>76.602</td><td>75.906</td><td>75.454</td><td>74.473</td></tr><tr><td>Average Rank</td><td>2.875</td><td>1.000</td><td>1.000</td><td>1.000</td><td>2.250</td><td>2.250</td><td>2.25</td><td>2.50</td></tr></table>
+
+In general, we observe a large jump in performance when using multiple heads instead of just one. However, since the number of heads controls the capacity, predictably we also observe that having too many heads leads to overfitting.
+
+# 5 CONCLUSION
+
+We proposed the URT layer to effectively integrate representations from multiple domains and demonstrated improved performance in multi-domain few-shot classification. Notably, our URT approach was able to set a new state-of-the-art on Meta-Dataset, and never performs worse than its predecessor (SUR) while also being $1 0 \times$ more efficient at inference. This work suggests that combining meta-learning with pre-trained universal representations is a promising direction for new few-shot learning methods. Specifically, we hope that future work can investigate the design of richer forms of universal representations that go beyond simply pre-training a single backbone for each domain, and developing meta-learners adapted to those settings.
+
+# REFERENCES
+
+Ferran Alet, Tomas Lozano-P ´ erez, and Leslie P Kaelbling. Modular meta-learning. In ´ Conference on Robot Learning (CoRL), 2018.
+
+Peyman Bateni, Raghav Goyal, Vaden Masrani, Frank Wood, and Leonid Sigal. Improved few-shot visual classification. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2020.
+
+Hakan Bilen and Andrea Vedaldi. Universal representations: The missing link between faces, text, planktons, and cat breeds. arXiv preprint arXiv:1701.07275, 2017.
+
+Wei-Yu Chen, Yen-Cheng Liu, Zsolt Liu, Yu-Chiang Frank Wang, and Jia-Bin Huang. A closer look at few-shot classification. In International Conference on Learning Representations (ICLR), 2019.
+
+Yinbo Chen, Xiaolong Wang, Zhuang Liu, Huijuan Xu, and Trevor Darrell. A new meta-baseline for few-shot learning. arXiv preprint arXiv:2003.04390, 2020.
+
+Guneet Singh Dhillon, Pratik Chaudhari, Avinash Ravichandran, and Stefano Soatto. A baseline for few-shot image classification. In International Conference on Learning Representations (ICLR), 2020.
+
+Nikita Dvornik, Cordelia Schmid, and Julien Mairal. Selecting relevant features from a universal representation for few-shot classification. arXiv preprint arXiv:2003.09338, 2020.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In The International Conference on Machine Learning (ICML), 2017.
+
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016.
+
+Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009.
+
+Brenden M Lake, Ruslan Salakhutdinov, and Joshua B Tenenbaum. Human-level concept learning through probabilistic program induction. Science, 350(6266):1332–1338, 2015.
+
+Yann LeCun, Leon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to ´ document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+
+Hongyang Li, David Eigen, Samuel Dodge, Matthew Zeiler, and Xiaogang Wang. Finding taskrelevant features for few-shot learning by category traversal. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019.
+
+Zhouhan Lin, Minwei Feng, Cicero Nogueira dos Santos, Mo Yu, Bing Xiang, Bowen Zhou, and Yoshua Bengio. A structured self-attentive sentence embedding. arXiv preprint arXiv:1703.03130, 2017.
+
+Lu Liu, Tianyi Zhou, Guodong Long, Jing Jiang, Lina Yao, and Chengqi Zhang. Prototype propagation networks (PPN) for weakly-supervised few-shot learning on category graph. In International Joint Conferences on Artificial Intelligence (IJCAI), 2019a.
+
+Lu Liu, Tianyi Zhou, Guodong Long, Jing Jiang, and Chengqi Zhang. Learning to propagate for graph meta-learning. In Neural Information Processing Systems (NeurIPS), 2019b.
+
+Lu Liu, Tianyi Zhou, Guodong Long, Jing Jiang, and Chengqi Zhang. Attribute propagation network for graph zero-shot learning. In AAAI Conference on Artificial Intelligence (AAAI), 2020.
+
+Yang Liu, Chengjie Sun, Lei Lin, and Xiaolong Wang. Learning natural language inference using bidirectional lstm model and inner-attention. arXiv preprint arXiv:1605.09090, 2016.
+
+Boris Oreshkin, Pau Rodr´ıguez Lopez, and Alexandre Lacoste. Tadam: Task dependent adaptive ´ metric for improved few-shot learning. In The Conference on Neural Information Processing Systems (NeurIPS), 2018.
+
+Ethan Perez, Florian Strub, Harm de Vries, Vincent Dumoulin, and Aaron C. Courville. Film: Visual reasoning with a general conditioning layer. In AAAI Conference on Artificial Intelligence (AAAI), 2018.
+
+Sylvestre-Alvise Rebuffi, Hakan Bilen, and Andrea Vedaldi. Efficient parametrization of multidomain deep neural networks. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2018.
+
+Mengye Ren, Eleni Triantafillou, Sachin Ravi, Jake Snell, Kevin Swersky, Joshua B Tenenbaum, Hugo Larochelle, and Richard S Zemel. Meta-learning for semi-supervised few-shot classification. In International Conference on Learning Representations (ICLR), 2018.
+
+James Requeima, Jonathan Gordon, John Bronskill, Sebastian Nowozin, and Richard E Turner. Fast and flexible multi-task classification using conditional neural adaptive processes. In The Conference on Neural Information Processing Systems (NeurIPS), pp. 7957–7968, 2019.
+
+Tonmoy Saikia, Thomas Brox, and Cordelia Schmid. Optimized generic feature learning for fewshot classification across domains. arXiv preprint arXiv:2001.07926, 2020.
+
+Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical networks for few-shot learning. In The Conference on Neural Information Processing Systems (NeurIPS), 2017.
+
+Eleni Triantafillou, Tyler Zhu, Vincent Dumoulin, Pascal Lamblin, Utku Evci, Kelvin Xu, Ross Goroshin, Carles Gelada, Kevin Swersky, Pierre-Antoine Manzagol, et al. Meta-dataset: A dataset of datasets for learning to learn from few examples. In International Conference on Learning Representations (ICLR), 2020.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In The Conference on Neural Information Processing Systems (NeurIPS), 2017.
+
+Oriol Vinyals, Charles Blundell, Tim Lillicrap, Daan Wierstra, et al. Matching networks for one shot learning. In The Conference on Neural Information Processing Systems (NeurIPS), 2016.
+
+Xin Wang, Fisher Yu, Ruth Wang, Trevor Darrell, and Joseph E Gonzalez. Tafe-net: Task-aware feature embeddings for low shot learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019.
+
+Yu-Xiong Wang and Martial Hebert. Learning from small sample sets by combining unsupervised meta-training with cnns. In The Conference on Neural Information Processing Systems (NeurIPS), 2016.
+
+Han-Jia Ye, Hexiang Hu, De-Chuan Zhan, and Fei Sha. Few-shot learning via embedding adaptation with set-to-set functions. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2020.
+
+Sung Whan Yoon, Jun Seo, and Jaekyun Moon. Tapnet: Neural network augmented with taskadaptive projection for few-shot learning. In The International Conference on Machine Learning (ICML), 2019.
+
+# A EXPERIMENTS ON MORE DATASETS
+
+We also report performances on the MNIST, CIFAR-10 and CIFAR-100 dataset sources in Table 5, and compare with the subset of methods that have reported on these datasets. There, URT neither improves nor gets worse performance than SUR, yeilding top performance on the MNIST domain but not on the CIFAR-10/CIFAR-100 domain, on which Simple CNAPS has the best performance.
+
+Table 5: Test performance (mean $1 { + } \mathrm { C I } \% 9 5$ ) over 600 few-shot tasks on additional datasets.   
+
+<table><tr><td></td><td>MNIST CIFAR10</td><td>CIFAR100</td><td> avg. rank</td></tr><tr><td>CNAPs</td><td>92.7 ± 0.4 61.5 ± 0.7</td><td>50.1 ± 1.0</td><td>4.7</td></tr><tr><td>TaskNorm</td><td>92.3 ± 0.4 69.3 ± 0.8</td><td>54.6 ± 1.1</td><td>3.3</td></tr><tr><td>SUR</td><td>94.3 ± 0.4 66.8 ± 0.9</td><td>56.6 ± 1.0</td><td>2.3</td></tr><tr><td>SimpleCNAPS</td><td>93.9 ± 0.4 74.3± 0.7</td><td>60.5 ± 1.0</td><td>1.7</td></tr><tr><td>URT (Ours)</td><td>94.8 ± 0.4 67.3 ± 0.8</td><td>56.9 ± 1.0</td><td>2.0</td></tr></table>
+
+# B MORE RELATED WORKS
+
+Transfer by fine-tuning A simple and effective method for few-shot classification is to perform transfer learning by first learning a neural network classifier on all data available for training and using its representation to initialize and then fine-tune neural networks on the few-shot classification tasks found at test time (Chen et al., 2019; Triantafillou et al., 2020; Dhillon et al., 2020; Saikia et al., 2020). Specifically, Saikia et al. (2020) have shown that competitive performance can be reached using a strong hyper-parameter optimization method applied on a carefully designed validation metric appropriate for few-shot learning.
+
+# C STRUCTURES AND TRAINING STRATEGY OF BACKBONES
+
+Our URT layer can be built on top of a set of pretrained backbones. The structures of the backbones follow the approach in Dvornik et al. (2020). For Table 1, we use the ResNet18 architecture (He et al., 2016) for the backbones, where a separate backbone is pretrained on each domain separately using the corresponding training data in meta-dataset. The training domains include ImageNet, Omniglot, Aircraft, CU-Birds, Textures, Quick Draw, Fungi and VGG-Flower. For the parametric network family in Table 2, a base ResNet18 is first trained on ImageNet. Then a small number of modulating parameters are trained on the other domains, using their domain-specific training data, while the rest of the base backbone’s weights stay fixed. Specifically, FiLM feature modulation (Perez et al., 2018) is used. This type of parametric network family thus allows for a much reduced number of learnable parameters, by reusing the weights from the base network. In experiments, we use the pretrained backbones released by Dvornik et al. (2020) for both cases, without any further finetuning. End-to-end training of a set of backbones and the URT layers requires unaffordable computational cost, so we fix the pretrained backbones and only train the URT layer.
+
+Implementation details for training backbones The training details of the backbones come from Dvornik et al. (2020). For optimization, SGD with momentum was used, using cosine learning rate annealing. Since the datasets come from different domains, the starting learning rate, the maximum number of training iterations and annealing frequency are set individually for each dataset. Data augmentation is applied and a constant weight decay of $\mathrm { \dot { 7 } } \times 1 0 ^ { - 4 }$ is set. For each dataset, a grid search over batch size in [8, 16, 32, 64] was run and the one that maximizes accuracy on the validation set was picked. For the parametric network family, the base ResNet18 trained on ImageNet is the same. For other backbones, cosine annealing as learning rate policy is also used, weight decay and data augmentation employed as above. Please refer to Table 4 and Table 5 in the original SUR paper (Dvornik et al., 2020) for the specific values of the hyperparameters for the individual feature networks and the parametric network family, respectively.
+
+# D RESULTS ON TRAFFIC SIGNS
+
+The shuffle buffer bug described in meta-dataset issue #54 (https://github.com/googleresearch/meta-dataset/issues/54) has been propagated to previous works such as CNAPs (Requeima et al., 2019). The results for the Traffic Signs dataset are considerably worse after fixing this bug. For instance, URT degrades from $6 9 . 4 { \pm } 0 . 8 $ to $5 1 . 1 { \pm } 1 . 1$ . Please visit the official meta-dataset GitHub repo for more details. The main paper shows the corrected results (for URT and competing approaches). For completeness, we also provide the tables as they were in the initial version of this paper for your reference in this Appendix. Both sets of results overall support the advantageous performance of URT over previous work.
+
+Table 6: Test accuracy ( $\mathrm { m e a n \pm C I \% 9 5 } ,$ ) over 600 few-shot tasks. URT and the most recent methods, which are listed in the first column, are compared on Meta-Dataset (Triantafillou et al., 2020), which are listed in the first row. The numbers in bold have intersecting confidence intervals with the most accurate method.   
+
+<table><tr><td></td><td>ILSVRC</td><td>Omniglot</td><td>Aircraft</td><td>Birds</td><td>Textures</td><td>QuickDraw</td><td>Fungi</td><td>VGGFlower</td><td>TrafficSigns</td><td>MSCOCO</td><td>avg.rank</td></tr><tr><td>MAML</td><td>37.8±1.0</td><td>83.9±1.0 76.4±0.7</td><td></td><td>62.4±1.1</td><td>64.1±0.8</td><td>59.7±1.1</td><td>33.5±1.1</td><td>79.9±0.8</td><td>42.9±1.3</td><td>29.4±1.1</td><td>8.0</td></tr><tr><td>ProtoNet</td><td>44.5±1.1</td><td>79.6±1.1</td><td>71.1±0.9 67.0±1.0 65.2±0.8</td><td></td><td></td><td>64.9±0.9</td><td>40.3±1.1</td><td>86.9±0.7</td><td>46.5±1.0</td><td>39.9±1.1</td><td>7.3</td></tr><tr><td>ProtoMAML</td><td>46.5±1.1</td><td>82.7±1.0 75.2±0.8 69.9±1.0 68.3±0.8</td><td></td><td></td><td></td><td>66.8±0.9</td><td>42.0±1.2</td><td>88.7±0.7</td><td>52.4±1.1</td><td>41.7±1.1</td><td>5.4</td></tr><tr><td>CNAPs</td><td>52.3±1.0</td><td>88.4±0.7 80.5±0.6 72.2±0.9 58.3±0.7</td><td></td><td></td><td></td><td>72.5±0.8</td><td>47.4±1.0</td><td>86.0±0.5</td><td>60.2±0.9</td><td>42.6±1.1</td><td>5.1</td></tr><tr><td>BOHB-E</td><td>55.4±1.1</td><td>77.5±1.1</td><td>60.9±0.9</td><td>73.6±0.8</td><td>72.8±0.7</td><td>61.2±0.9</td><td>44.5±1.1</td><td>90.6±0.6</td><td>57.5±1.0</td><td>51.9±1.0</td><td>4.4</td></tr><tr><td>TaskNorm</td><td>50.6±1.1</td><td>90.7±0.6 83.8±0.6</td><td></td><td>574.6±0.8</td><td>62.1±0.7</td><td>74.8±0.7</td><td>48.7±1.0</td><td>89.6±0.6</td><td>67.0±0.7</td><td>43.4±1.0</td><td>3.8</td></tr><tr><td>SUR</td><td>56.3±1.1</td><td>93.1±0.5 85.4±0.7 71.4±1.0 71.5±0.8</td><td></td><td></td><td></td><td>81.3±0.6</td><td>63.1±1.0</td><td>82.8±0.7</td><td>70.4±0.8</td><td>52.4±1.1</td><td>2.5</td></tr><tr><td>SimpleCNAPS 58.6±1.1 9</td><td></td><td>91.7±0.6 82.4±0.7 74.9±0.8 67.8±0.8</td><td></td><td></td><td></td><td>77.7±0.7</td><td>46.9±1.0</td><td>90.7±0.5</td><td>73.5±0.7</td><td>46.2±1.1</td><td>2.4</td></tr><tr><td>URT(Ours)</td><td>55.7±1.0</td><td>94.4±0.4 85.8±0.6 76.3±0.8 71.8±0.7</td><td></td><td></td><td></td><td>82.5±0.6</td><td>63.5±1.0</td><td>88.2±0.6</td><td>69.4±0.8</td><td>52.2±1.1</td><td>1.6</td></tr></table>
+
+Table 7: Test accuracy (mean $\pm \mathrm { C I } \% 9 5 $ ) over 600 few-shot tasks. All methods use parametric network family (pf) backbones.   
+
+<table><tr><td></td><td>SUR-pf</td><td>URT-pf</td><td>VS.</td></tr><tr><td>ILSVRC</td><td>56.4 ± 1.2</td><td>55.5 ± 1.1</td><td>=</td></tr><tr><td>Omniglot</td><td>88.5±0.8</td><td>90.2 ± 0.6</td><td>+</td></tr><tr><td>Aircraft</td><td>79.5 ± 0.8</td><td>79.8 ± 0.7</td><td>=</td></tr><tr><td>Birds</td><td>76.4 ± 0.9</td><td>77.5 ± 0.8</td><td>=</td></tr><tr><td>Textures</td><td>73.1 ± 0.7</td><td>73.5± 0.7</td><td>=</td></tr><tr><td>Quick Draw</td><td>75.7 ± 0.7</td><td>75.8 ± 0.7</td><td>=</td></tr><tr><td>Fungi</td><td>48.2 ± 0.9</td><td>48.1 ± 0.9</td><td>=</td></tr><tr><td>VGG Flower</td><td>90.6 ± 0.5</td><td>91.9 ± 0.5</td><td>+</td></tr><tr><td>Traffic Signs</td><td>65.1 ± 0.8</td><td>67.5 ± 0.8</td><td>+</td></tr><tr><td>MSCOCO</td><td>52.1 ± 1.0</td><td>52.1 ± 1.0</td><td>=</td></tr><tr><td>MNIST</td><td>93.2 ± 0.4</td><td>93.9 ± 0.4</td><td>=</td></tr><tr><td>CIFAR10</td><td>66.4 ± 0.8</td><td>66.1 ± 0.8</td><td>=</td></tr><tr><td>CIFAR100</td><td>57.1 ± 1.0</td><td>57.3 ±1.0</td><td>=</td></tr></table>

parse/train/04cII6MumYV/04cII6MumYV_content_list.json

@@ -0,0 +1,1604 @@
+[
+    {
+        "type": "text",
+        "text": "A UNIVERSAL REPRESENTATION TRANSFORMER LAYER FOR FEW-SHOT IMAGE CLASSIFICATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
+            763,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lu $\\mathbf { L i u ^ { 1 , 2 * } }$ , William Hamilton $^ { 1 , 3 }$ †, Guodong Long2, Jing Jiang2, Hugo Larochelle1,4† 1 Mila, 2 Australian AI Institute, UTS, 3 McGill University, 4 Google Research, Brain Team Correspondence to lu.liu.cs@icloud.com ",
+        "bbox": [
+            184,
+            169,
+            785,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            265
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Few-shot classification aims to recognize unseen classes when presented with only a small number of samples. We consider the problem of multi-domain few-shot image classification, where unseen classes and examples come from diverse data sources. This problem has seen growing interest and has inspired the development of benchmarks such as Meta-Dataset. A key challenge in this multi-domain setting is to effectively integrate the feature representations from the diverse set of training domains. Here, we propose a Universal Representation Transformer (URT) layer, that meta-learns to leverage universal features for few-shot classification by dynamically re-weighting and composing the most appropriate domain-specific representations. In experiments, we show that URT sets a new state-of-the-art result on Meta-Dataset. Specifically, it achieves top-performance on the highest number of data sources compared to competing methods. We analyze variants of URT and present a visualization of the attention score heatmaps that sheds light on how the model performs cross-domain generalization. Our code is available at https://github.com/liulu112601/URT. ",
+        "bbox": [
+            233,
+            280,
+            764,
+            486
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            511,
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+    {
+        "type": "text",
+        "text": "Learning tasks from small data remains a challenge for machine learning systems, which show a noticeable gap compared to the ability of humans to understand new concepts from few examples. A promising direction to address this challenge is developing methods that are capable of performing transfer learning across the collective data of many tasks. Since machine learning systems generally improve with the availability of more data, a natural assumption is that few-shot learning systems should benefit from leveraging data across many different tasks and domains—even if each individual task has limited training data available. ",
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+        "text": "This research direction is well captured by the problem of multi-domain few-shot classification. In this setting, training and test data spans a number of different domains, each represented by a different source dataset. A successful approach in this multi-domain setting must not only address the regular challenge of few-shot classification—i.e., the challenge of having only a handful of examples per class. It must also discover how to leverage (or ignore) what is learned from different domains, achieving generalization and avoiding cross-domain interference. ",
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+        "text": "Recently, Triantafillou et al. (2020) proposed a benchmark for multi-domain few-shot classification, Meta-Dataset, and highlighted some of the challenges that current methods face when training data is heterogeneous. Crucially, they found that methods which trained on all available domains would normally obtain improved performance on some domains at the expense of others. Following on their work, progress has been made, which includes the design of adapted hyper-parameter optimization strategies (Saikia et al., 2020) and more flexible meta-learning algorithms (Requeima et al., 2019). Most notable is SUR (Selecting Universal Representation) (Dvornik et al., 2020), a method that relies on a so-called universal representation, extracting from a collection of pre-trained and domain-specific neural network backbones. SUR prescribes a hand-crafted feature-selection procedure to infer how to weight each backbone for each task at hand, and produces an adapted representation for each task. This was shown to lead to some of the best performances on Meta-Dataset. ",
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+        "type": "text",
+        "text": "In SUR, the classification procedure for each task is fixed and not learned. Thus, except for the underlying universal representation, there is no transfer learning performed with regards to how classification rules are inferred across tasks and domains. Yet, cross-domain generalization might be beneficial in that area as well, in particular when tasks have only few examples per class. ",
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+        "text": "Present work. To explore this question, we propose a Universal Representation Transformer (URT) layer, which can effectively learn to transform a universal representation into task-adapted representations. The URT layer is inspired from Transformer (Vaswani et al., 2017) and uses an attention mechanism to learn to retrieve or blend the appropriate backbones to use for each task. By training this layer across few-shot tasks from many domains, it can support transfer across these tasks. ",
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+        "type": "text",
+        "text": "We show that our URT layer on top of a universal representation’s pre-trained backbones sets a new state-of-the-art performance on Meta-Dataset. It succeeds at outperforming SUR on 4 dataset sources without impairing accuracy on the others. This leads to top performance on 7 dataset sources when comparing to a set of competing methods. To interpret the strategy that URT learns to weigh the backbones from different domains, we visualize the attention scores for both seen and unseen domains and find that our model generates meaningful weights for the pre-trained domains. A comprehensive analysis on variants and ablations of the URT layer is provided to show the importance of various components of URT, notably the number of attention heads. ",
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+        "type": "text",
+        "text": "2 FEW-SHOT CLASSIFICATION ",
+        "text_level": 1,
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+        "text": "2.1 PROBLEM SETTING ",
+        "text_level": 1,
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+        "text": "In this section, we will introduce the problem setting for few-shot classification and the formulation of meta-learning for few-shot classification. Few-shot classification aims to classify samples where only few examples are available for each class. We describe a few-shot learning classification task as the pair of examples, comprising of a support set $S$ to define the classification task and the query set $Q$ of samples to be classified. ",
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+        "text": "Meta-learning is a technique that aims to model the problem of few-shot classification as learning to learn from instances of few-shot classification tasks. The most popular way to train a meta-learning model is with episodic training. Here, tasks $T = ( Q , S )$ are sampled from a larger dataset by taking subsets of the dataset to build a support set $S$ and a query set $Q$ for the task. A common approach is to sample $N$ -way- $K$ -shot tasks, each time selecting a random subset of $N$ classes from the original dataset and choosing only $K$ examples for each class to add to the support set $S$ . ",
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+        "type": "text",
+        "text": "The meta-learning problem can then be formulated by the following optimization: ",
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+        "type": "equation",
+        "img_path": "images/4cf0074922d261504e06a69f6956545689d9a7e1a60382178e60d7c096702ad3.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\Theta } \\mathbb { E } _ { ( S , Q ) \\sim p ( T ) } \\left[ \\mathcal { L } ( S , Q , \\Theta ) \\right] , \\ \\mathcal { L } ( S , Q , \\Theta ) = \\frac { 1 } { | Q | } \\sum _ { { ( x , y ) } \\sim Q } - \\log p ( y | x , S ; \\Theta ) + \\lambda \\Omega ( \\Theta ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $p ( T )$ is the distribution of tasks, $\\Theta$ are the parameters of the model and $p ( \\boldsymbol { y } | \\boldsymbol { x } , S ; \\Theta )$ is the probability assigned by the model to label $y$ of query example $_ { \\textbf { \\em x } }$ (given the support set $S$ ), and $\\Omega ( \\Theta )$ is an optional regularization term on the model parameters with factor $\\lambda$ . ",
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+    {
+        "type": "text",
+        "text": "Conventional few-shot classification targets the setting of $N$ -way- $K$ -shot, where the number of classes and examples are fixed in each episode. Popular benchmarks following this approach include Omniglot (Lake et al., 2015) or benchmarks made of subsets of ImageNet, such as miniImageNet (Vinyals et al., 2016) and tieredImageNet (Ren et al., 2018). In such benchmarks, the tasks for training cover a set of classes that is disjoint from the classes in the test set of tasks. However, with the training and test sets tasks coming from a single dataset/domain, the distribution of tasks found in either sets is similar and lacks variability, which may be unrealistic in practice. ",
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+    {
+        "type": "text",
+        "text": "It is in this context that Triantafillou et al. (2020) proposed Meta-Dataset, as a further step towards large-scale, multi-domain few shot classification. Meta-Dataset includes ten datasets (domains), with eight of them available for training. Additionally, each task sampled in the benchmark varies in the number of classes $N$ , with each class also varying in the number of shots $K$ . As in all few-shot learning benchmarks, the classes used for training and testing do not overlap. ",
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+    {
+        "type": "text",
+        "text": "2.2 BACKGROUND AND RELATED WORK ",
+        "text_level": 1,
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+        "text": "Meta-Learning A promising approach for few-shot classification is to use meta-learning to more directly train a model to learn to perform few-shot classification, in an end-to-end way. The two most popular methods are Prototypical Networks (Snell et al., 2017) and Model Agnostic Meta-Learning (MAML) (Finn et al., 2017). Triantafillou et al. (2020) showed that prototypical networks and MAML could be combined by leveraging prototypes for the initialization of the output weights value in the inner loop. Requeima et al. (2019) also proposed Conditional Neural Adaptive Processes (CNAPs) for few-shot classification, which can be seen as extending prototypical networks with a more sophisticated architecture that allows for improved task adaptation. This architecture was later improved further by Bateni et al. (2020) with Simple CNAPS, leading to one of the current best methods on Meta-Dataset. Another line of work which leverages the idea of “transfer by fine-tuning” can be found in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "Universal Representations In contrast, our work instead builds on that of Dvornik et al. (2020) and their method SUR (Selecting from Universal Representations). Bilen & Vedaldi (2017) introduced the term universal representation to refer to a representation that supports good performance in multiple domains. One proposal towards such a representation is to train different neural networks backbones separately on the data of each available domain, then simply to concatenate the representation learned by each. Another is to introduce some parameter sharing between the backbones, by having a single network conditioned on the domain of the provenance of each batch of training data (Rebuffi et al., 2018), e.g. using Feature-wise Linear Modulate (FiLM) (Perez et al., 2018). SUR proposes to leverage a universal representation in few-shot learning tasks with a feature selection procedure that assigns different weights to each of the domain-specific subvectors of the universal representation. The objective is to assign high weights only to the domain-specific representations that are specifically useful for each few-shot task at hand. The weights are inferred by optimizing a loss on the support set that encourages high accuracy of a nearest-centroid classifier. As such, the method does not involve any meta-learning—a choice motivated by the concern that meta-learning may struggle in generalizing to domains that are dissimilar to the training domains. SUR achieved some of the best performances on Meta-Dataset. However, a contribution of our work is to provide evidence that meta-learning can actually be used to replace SUR’s hand-designed inference procedure and improve performance further. ",
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+    {
+        "type": "text",
+        "text": "Task Adaptive Representations Another line of work tries to retrieve task adaptive representations for each task. Task specific representations can be conditioned on a representation of the current task (Oreshkin et al., 2018; Wang et al., 2019), projected to another space (Yoon et al., 2019), or masked based on inter-class commonality and inter-class uniqueness (Li et al., 2019). While the representation extracted from URT is also task adaptive, it is adaptive to a set of pretrained backbones and can be applied to more complicated multi-domain scenarios. Wang & Hebert (2016) proposed to improve a CNN by adding extra layers and train it using unsupervised data while our contribution mainly lies in composing representations instead of an improved CNN. Alet et al. (2018) introduced a modular meta-learning method, which learns a repertoire of modules that serves as nodes to construct a tree structure to solve a new robotic-related task. Comparatively, URT is a one-for-all layer which doesn’t need to construct different module structures for each task. ",
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+    {
+        "type": "text",
+        "text": "Transformer Networks Our meta-learning approach to leverage universal representations is inspired directly from Transformer networks (Vaswani et al., 2017). Our model structure is inspired by the structure of the dot-product self-attention in the Transformer, which we adapted here to multidomain few-shot learning by designing appropriate parametrizations for queries, keys and values. Self-attention was explored in the single-domain training regime by Ye et al. (2020); Liu et al. (2019b;a; 2020), however for a different purpose, where each representation of individual examples in a task support set is influenced by all other examples. Rather than using self-attention between individual examples in the support set, our model uses self-attention to select between different domain-specific backbones. ",
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+    {
+        "type": "text",
+        "text": "3 UNIVERSAL REPRESENTATION TRANSFORMER LAYER ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we describe our proposed URT layer, which uses meta-learning episodic training to learn how to combine the domain-specific backbones of a universal representation for any given fewshot learning classification task. URT layer can be built on top of any set of pretrained backbones without further costly fine-tuning of the backbones. More details on how to train multiple domainspecific backbones can be found in Appendix C. ",
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+        "img_path": "images/b8ec3fe34a1bbf140191b9bf5722ec5773c3834b441b3d9391ea604b4b89789b.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of how a single-head URT layer uses a universal representation to produce a task-specific representation. This example assumes the use of four backbones, with each color illustrating their domain-specific sub-vector representation in the universal representation. "
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+        "type": "text",
+        "text": "Conceptually, the proposed model views the support set $S$ of a task as providing information on how to query and retrieve from the set $\\{ r _ { i } \\}$ of $m$ pre-trained backbones the most appropriate backbone to build an adapted representation $\\phi$ for the task. ",
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+        "type": "text",
+        "text": "We would like the model to support a variety of strategies on how to retrieve backbones. For example, it might be beneficial for the model to retrieve a single backbone from the set, especially if the domain of the given task matches perfectly that of a domain found in the training set. Alternatively, if some of the training domains benefit from much more training data than others, a better strategy might be to attempt some cross-domain generalization towards the few-shot learning task by blending many backbones together, even if none matches the domain of the task perfectly. ",
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+        "type": "text",
+        "text": "This motivates us to use dot-product self-attention, inspired by layers of Transformer networks (Vaswani et al., 2017). For this reason, we refer to our model as a Universal Representation Transformer (URT) layer. Additionally, since each class of the support set might require a different strategy, we perform attention separately for each class and their support set $S _ { c } = \\{ { \\pmb x } | ( { \\pmb x } , y ) \\in S$ and $y = c \\}$ . ",
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+        "type": "text",
+        "text": "3.1 SINGLE-HEAD URT LAYER",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We start by describing an URT layer consisting of a single attention head. An illustration of a singlehead URT layer is shown in Figure 1. Let $r _ { i } ( { \\pmb x } )$ be the output vector of the backbone for domain $i$ . We then write the universal representation as ",
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+        "img_path": "images/20cd0cfe3b8789b6c61ebda051b620bcef0b8eb26d523e37e71f43158da5018b.jpg",
+        "text": "$$\nr ( \\mathbf x ) = \\mathrm { c o n c a t } ( r _ { 1 } ( \\mathbf x ) , \\hdots , r _ { m } ( \\mathbf x ) ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This representation provides a natural starting point to obtain a representation of a support set class. Specifically, we will note ",
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+        "img_path": "images/cfc05fdbf3c29339a67c56cfba43d5a73aa2af44caeed29ee04f50f303a1b70c.jpg",
+        "text": "$$\nr ( S _ { c } ) = \\frac { 1 } { \\left| S _ { c } \\right| } \\sum _ { { \\pmb x } \\in S _ { c } } r ( { \\pmb x } )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "as the representation for the set $S _ { c }$ . From this, we can describe the URT layer by defining the queries1, keys, the attention mechanism and output of the layer: ",
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+        "text": "Queries $\\mathbf { q } _ { c }$ : For each class $c$ , we obtain a query through $\\mathbf { q } _ { c } = \\mathbf { W } ^ { q } r ( S _ { c } ) + \\mathbf { b } ^ { q }$ , where we have a learnable query linear transformation represented by matrix $\\mathbf { W } ^ { q }$ and bias $\\mathbf { b } ^ { q }$ . ",
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+        "type": "text",
+        "text": "Keys $\\mathbf { k } _ { i , c }$ : For each domain $i$ and class $c$ , we define keys as $\\mathbf { k } _ { i , c } = \\mathbf { W } ^ { k } r _ { i } ( S _ { c } ) + \\mathbf { b } ^ { k }$ , using a learnable linear transformation $\\mathbf { W } ^ { k }$ and $\\mathbf { b } ^ { k }$ and where $r _ { i } ( S _ { c } ) = 1 / | S _ { c } | \\textstyle \\sum _ { { \\pmb x } \\in S _ { c } } r _ { i } ( { \\pmb x } )$ , using a similar notation as for $r ( S _ { c } )$ . ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Training of URT layer ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Input: Number of tasks $\\tau _ { t o t a l }$ , $m$ pre-trained backbones ; ",
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+        "type": "text",
+        "text": "1: for $\\tau \\in \\{ 1 , \\cdots , \\tau _ { t o t a l } \\}$ do   \n2: Sample a few-shot task $T$ with support set $S$ and query set $Q$ ;   \n3: # Infer adapted representation for task from $S$   \n4: For each class, obtain representation using $m$ pre-trained backbones as in Eq. (3);   \n5: Obtain attention scores using Eq. (4,5) for each head using support set $S$ ;   \n6: # Use adapted representation to predict labels in $Q$ from support set $S$   \n7: Compute adapted representation of examples in $S$ and $Q$ as in Eq. (6,7);   \n8: Compute probabilities of label of examples in $Q$ using Prototypical Network as in Eq. (9);   \n9: Compute loss as in Eq. (1,8) and perform gradient descent step on URT parameters $\\Theta$ ; ",
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+    {
+        "type": "text",
+        "text": "10: end for ",
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+    {
+        "type": "text",
+        "text": "Attention scores $\\alpha _ { i }$ : as for regular Transformer layers, we use scaled dot-product attention ",
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+    {
+        "type": "equation",
+        "img_path": "images/f80c80697eff8820793887cfcca412325b4c43c2fbce6e2b68dcbdd78749ce86.jpg",
+        "text": "$$\n\\alpha _ { i , c } = \\frac { \\exp ( \\beta _ { i , c } ) } { \\sum _ { i ^ { \\prime } } \\exp ( \\beta _ { i ^ { \\prime } , c } ) } , \\beta _ { i , c } = \\frac { { \\bf q } _ { c } \\mathrm {  ~ } ^ { \\top } { \\bf k } _ { i , c } } { \\sqrt { l } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $l$ is the dimensionality of the keys and queries. Then, these per-class scores are aggregated to obtain scores for the full support set by averaging ",
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+        "type": "equation",
+        "img_path": "images/6b3b04ffee3d3c7737ff3c611447aeddbc0b7d65be875d006a1bbe7804ca2a29.jpg",
+        "text": "$$\n\\alpha _ { i } = \\frac { \\sum _ { c } \\alpha _ { i , c } } { N } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Equipped with these attention scores, the URT layer can now produce an adapted representation for the task (for the support and query set examples) by computing ",
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+        "type": "equation",
+        "img_path": "images/4b471f4840dafae6714d583b6640d1699e665ab5102cf95e435838ac722fa7ee.jpg",
+        "text": "$$\n\\phi ( { \\bf x } ) = \\sum _ { i } \\alpha _ { i } r _ { i } ( { \\bf x } ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "As we can see, this approach has the flexibility of either selecting a single domain-specific backbone (by assigning $\\alpha _ { i } = 1$ for a single domain) or blending different domains together (by having $\\alpha _ { i } > >$ 0 for multiple backbones). ",
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+    {
+        "type": "text",
+        "text": "3.2 MULTI-HEAD URT LAYER ",
+        "text_level": 1,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The URT layer described so far can only learn to retrieve a single backbone (or blending of backbones). Yet, it might be beneficial to retrieve multiple different (blended) backbones, especially for a few-shot task that would include many classes of varying complexity. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Thus, to achieve such diversity in the adapted representation, we also consider URT layers with multiple heads, i.e. where each head corresponds to the calculation of Equation 6 and each head has its own set of parameters $( \\mathbf { W } ^ { q } , \\mathbf { b } ^ { q } , \\mathbf { W } ^ { k } , \\bar { \\mathbf { b } ^ { k } } )$ . Denoting each head now as $\\phi _ { h }$ , a multi-head URT layer then produces as its output the concatenation of all of its heads: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/51aea894444eb2153de34ba6f12e8f75aa9476e1977cb126028888a229c28be9.jpg",
+        "text": "$$\n\\phi ( { \\bf x } ) = \\mathrm { c o n c a t } ( \\phi _ { 1 } ( { \\bf x } ) , \\ldots , \\phi _ { \\mathrm { H } } ( { \\bf x } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Empirically we found that the randomness in the initialization of head weights alone did not lead to uniqueness and being complimentary between the heads, so inspired by Lin et al. (2017), we add a regularizer to avoid duplication of the attention scores: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b7481abfe5e37468fee9ee66f1bf848baad911a832252c6fa1167cff9a59805.jpg",
+        "text": "$$\n\\Omega ( \\Theta ) = \\| ( \\mathbf { A } \\mathbf { A } ^ { \\top } - \\mathbf { I } ) \\| _ { F } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            770,
+            588,
+            791
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\| \\cdot \\| _ { F }$ is the Frobenius norm of a matrix and $\\mathbf { A } \\in \\mathbb { R } ^ { n \\times m }$ is the matrix for attention scores, with $\\mathbf { A } _ { h }$ being the vector of all scores $\\alpha _ { i }$ for head $h$ . The identity matrix I regularizes each set of attention scores to be more focused so that multiple heads can attend to different domain-specific backbones. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.3 TRAINING STRATEGY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We train representations produced by the URT layer by following the approach of Prototypical Networks (Snell et al., 2017), where the probability of a label $y$ for a query example $_ { \\textbf { \\em x } }$ given the ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/5fa2ce867a1ab77004f325f1dc712c61a6528c3fc1d8efaa69f9ffd99cd7ff20.jpg",
+        "table_caption": [
+            "Table 1: Test accuracy (mean $\\pm \\mathrm { C I } \\% 9 5 $ ) over 600 few-shot tasks. URT and the most recent methods, which are listed in the first column, are compared on Meta-Dataset (Triantafillou et al., 2020), which are listed in the first row. The numbers in bold have intersecting confidence intervals with the most accurate method. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>ILSVRC</td><td>Omniglot</td><td>Aircraft</td><td>Birds</td><td>Textures</td><td>QuickDraw</td><td>Fungi</td><td>VGGFlower</td><td>TrafficSigns</td><td></td><td>MSCOCO</td><td>avg.rank</td></tr><tr><td>MAML</td><td></td><td>37.8±1.0 83.9±1.0 76.4±0.7 62.4±1.1(</td><td></td><td></td><td></td><td>64.1±0.8</td><td>59.7±1.1</td><td>33.5±1.1</td><td>79.9±0.8</td><td>42.9±1.3</td><td>29.4±1.1</td><td>6.4</td></tr><tr><td>ProtoNet</td><td></td><td>44.5±1.1 79.6±1.1 71.1±0.9 67.0±1.0 65.2±0.8</td><td></td><td></td><td></td><td></td><td>64.9±0.9</td><td>40.3±1.1</td><td>86.9±0.7</td><td>46.5±1.0</td><td>39.9±1.1</td><td>5.7</td></tr><tr><td>ProtoMAML</td><td></td><td>46.5±1.1 82.7±1.0 75.2±0.8 69.9±1.0 68.3±0.8</td><td></td><td></td><td></td><td></td><td>66.8±0.9</td><td>42.0±1.2</td><td>88.7±0.7</td><td>52.4±1.1</td><td>41.7±1.1</td><td>4.1</td></tr><tr><td>CNAPs</td><td>50.8±1.1</td><td>91.7±0.5 83.7±0.6 73.6±0.9</td><td></td><td></td><td>59.5±0.7</td><td></td><td>74.7±0.8</td><td>50.2±1.1</td><td>88.9±0.5</td><td>56.5±1.1</td><td>39.4±1.1</td><td>3.6</td></tr><tr><td>SUR</td><td>56.1±1.1 </td><td>93.1±0.5 84.6±0.7 70.6±1.0 71.0±0.8</td><td></td><td></td><td></td><td></td><td>81.3±0.6</td><td>64.2±1.0</td><td>82.8±0.7</td><td>53.4±1.0</td><td>50.1±1.0</td><td>2.3</td></tr><tr><td>SimpleCNAPS 56.5±1.1 91.9±0.6 83.8±0.7 76.1±0.870.0±0.8</td><td></td><td></td><td></td><td></td><td></td><td></td><td>78.3±0.7</td><td>49.1±1.2</td><td>91.3±0.6</td><td>59.2±1.0</td><td>42.4±1.1</td><td>2.2</td></tr><tr><td>URT (Ours)</td><td></td><td>55.7±1.0 94.4±0.4 85.8±0.6 76.3±0.871.8±0.7</td><td></td><td></td><td></td><td></td><td>82.5±0.6</td><td>63.5±1.0</td><td>88.2±0.6</td><td>51.1±1.1</td><td>52.2±1.1</td><td>1.5</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "support set of a task is modeled as: ",
+        "bbox": [
+            174,
+            323,
+            403,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3d5cde7247b3781a24ad56599fc1582b79232feb4132d2997ca4f23f7ac2e95.jpg",
+        "text": "$$\np ( y = c | \\pmb { x } , S ; \\Theta ) = \\frac { \\exp ( - d ( \\phi ( \\pmb { x } ) - \\pmb { p _ { c } } ) ) } { \\sum _ { c ^ { \\prime } = 1 } ^ { N } \\exp ( - d ( \\phi ( \\pmb { x } ) - \\pmb { p _ { c ^ { \\prime } } } ) ) } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            343,
+            668,
+            382
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $d$ is a distance metric and $\\pmb { p } _ { c } = 1 / | S _ { c } | \\sum _ { \\pmb { x } \\in S _ { c } } \\phi ( \\pmb { x } )$ corresponds to the centroid of class $c _ { \\cdot }$ referred to as its prototype. We use (negative) cosine similarity as the distance. The full training algorithm is presented in Algorithm 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+            450,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we seek to answer three key experimental questions: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Q1 How does URT compare with previous state-of-the-art on Meta-Dataset for multi-domain fewshot classification?   \nQ2 Do the URT attention heads generate interpretable and meaningful attention scores?   \nQ3 Does the URT layer provide consistent benefits, even when pre-trained backbones are trained in different ways? ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition, we investigate architectural choices made, such as our models for keys/queries and their regularization, and study their contribution to achieving strong performance with URT. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASETS AND SETUP ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We test our methods on the large-scale few-shot learning benchmark Meta-Dataset (Triantafillou et al., 2020). It consists of ten datasets with various data distributions across different domains, including natural images (Birds, Fungi, VGG Flower), hand-written characters (Omniglot, Quick Draw), and human created objects (Traffic Signs, Aircraft). Among the ten datasets, eight provide data that can be used during either training, validation and testing (with each class assigned to only one of those sets), while two datasets are solely used for testing. Following Bateni et al. (2020); Requeima et al. (2019), we also report results on MNIST (LeCun et al., 1998), CIFAR10 and CIFAR100 (Krizhevsky et al., 2009) as additional unseen test datasets. Following Triantafillou et al. (2020), few-shot tasks are sampled with varying number of classes $N$ , varying number of shots $K$ and class imbalance. The performance is reported as the average accuracy over 600 sampled tasks. More details of Meta-Dataset can be found in Triantafillou et al. (2020). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The domain-specific backbones are pre-trained following the setup in (Dvornik et al., 2020). Then, we freeze the backbone and train the URT layer for 10,000 episodes, with an initial learning rate of 0.01 and a cosine learning rate scheduler. Following Chen et al. (2020), the training episodes have $50 \\%$ probability coming from the ImageNet data source. Since different pre-trained backbones may produce representations with different vector norms, we normalize the outputs of the backbones as in Dvornik et al. (2020). URT is trained with parameter weight decay of 1e-5 and with a regularization factor $\\lambda = 0 . 1$ . The number of heads ( $H$ in Equation 7), is set to 2 and the dimension of the keys and queries (l in Equation 4) is set to 1024. We choose the hyper-parameters based on the performance of the validation set. Details of the hyper-parameter selection and how the performance is influenced by them are outlined in Section 4.5. We find that the bottleneck of training URT is extracting features from CNN. Since we freeze the CNN when training the URT, we find dumping the extracted feature episodes can significantly speed up the training procedure from days to around 2 hours. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/c10077fb200921d3bb6bc7d24d0c40b0a826b01959da838218a14b2d49389f42.jpg",
+        "image_caption": [
+            "Figure 2: Average attention scores generated by URT with two heads. Rows correspond to the domain of the test tasks and the columns correspond to the pre-trained backbones $r _ { i } ( { \\pmb x } )$ trained on the eight training domains. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2 COMPARISON WITH PREVIOUS APPROACHES",
+        "text_level": 1,
+        "bbox": [
+            174,
+            508,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Table 1 presents a comparison of URT with SUR, as well as other baselines based on transfer learning by fine-tuning (Saikia et al., 2020) or meta-learning (Prototypical Networks (Snell et al., 2017), first-order MAML (Finn et al., 2017), ProtoMAML (Triantafillou et al., 2020), CNAPs (Requeima et al., 2019)) and Simple CNAPS(Bateni et al., 2020). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We observe in Table 1 that URT establishes a new state-of-the-art on Meta-Dataset, by achieving the top performance on 8 out of the 10 dataset sources. When comparing to its predecessor, URT outperforms SUR on 4 datasets without compromising performance on others, which is challenging to achieve in the multi-domain setting. Of note, the average inference time for URT is 0.04 second per task, compared to 0.43 for SUR, on a single V100. Thus, getting rid of the optimization procedure for every episode with our meta-trained URT layer also significantly increases the latency, by more than $1 0 \\times$ . More results on additional datasets can be found in Appendix A. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 INTERPRETING AND VISUALIZING ATTENTION BY URT ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To better understand how the URT model of Section 4.2 uses its two heads to build adapted representations, we visualize the attention scores produced on the test tasks of Meta-Dataset in Figure 2. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The blue (first head) and orange (second head) heatmaps summarize the values of the attention scores (Equation 5), averaged across several tasks for each test domain. Specifically, the element on row $t$ and column $i$ is the averaged attention scores $\\alpha _ { i }$ computed on test set domain $t$ for the backbone from domain $i$ . Note that the last two rows are the two unseen domain datasets. We found that for datasets from the seen domains, i.e. the first eight rows, one head (right, orange) consistently puts most of its weight on the backbone pre-trained on the same domain, while the other head (left, blue) learns relatively smoother weight distributions that blend other related domains. For unseen datasets, the right head puts half of its weight on ImageNet and the left head learned to blend the representations from four backbones. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/f01506e1b72bf9bb95aa270a7cf98ba226d36150c671d167afce474318d305bd.jpg",
+        "table_caption": [
+            "Table 2: Test accuracy (mean $\\pm \\mathrm { C I } \\% 9 5$ ) over 600 few-shot tasks. All methods use parametric network family (pf) backbones. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td> SUR-pf</td><td>URT-pf</td><td>VS.</td></tr><tr><td>ILSVRC</td><td>56.0 ± 1.1</td><td>55.5 ± 1.1</td><td>二</td></tr><tr><td>Omniglot</td><td>90.0 ± 0.8</td><td>90.2 ± 0.6</td><td>二</td></tr><tr><td>Aircraft</td><td>79.7 ± 0.8</td><td>79.8 ± 0.7</td><td>=</td></tr><tr><td>Birds</td><td>75.9 ± 0.9</td><td>77.5 ± 0.8</td><td>三</td></tr><tr><td>Textures</td><td>72.5 ± 0.7</td><td>73.5 ± 0.7</td><td></td></tr><tr><td>Quick Draw</td><td>76.7 ± 0.7</td><td>75.8 ± 0.7</td><td>三 二</td></tr><tr><td>Fungi</td><td>49.8 ± 1.1</td><td>48.1 ± 0.9</td><td>+</td></tr><tr><td>VGG Flower</td><td>90.0 ± 0.6</td><td>91.9 ± 0.5</td><td>二</td></tr><tr><td>Traffic Signs</td><td>52.2 ± 0.8</td><td>52.0 ± 1.4</td><td></td></tr><tr><td>MSCOCO</td><td>50.2 ± 1.1</td><td>52.1 ± 1.0</td><td></td></tr><tr><td>MNIST</td><td>93.2 ± 0.4</td><td>93.9 ± 0.4</td><td>=</td></tr><tr><td>CIFAR10</td><td>66.4 ± 0.8</td><td>66.1 ± 0.8</td><td>三</td></tr><tr><td>CIFAR100</td><td>57.1 ± 1.0</td><td>57.3 ± 1.0</td><td>二</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 URT USING FILM MODULATED BACKBONES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As additional evidence of the benefit of URT on universal representations, we also present experiments based on a different set of backbone architectures. Following SUR (Dvornik et al., 2020), we consider the backbones from a parametric network family, obtained by training a base backbone on one dataset (ILSVRC) and then learning separate FiLM layers (Perez et al., 2018) for each other dataset, to modulate the backbone so it is adapted to the other domains. These backbones collectively have only $0 . 5 \\%$ more parameters than a single backbone. More details of the backbones can be found in Appendix C. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "A comparison between SUR and URT using these backbones (referred to as SUR-pf and URT-pf) is presented in Table 2. Once again, URT can improve the performance on VGG Flower without sacrificing performance on others. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.5 HYPER-PARAMETER AND ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            574,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We analyze the importance of the various components of URT’s attention mechanism structure and training strategy in Table 3. First we analyze the importance of using the support set to model queries and/or keys. To this end, we consider setting the matrices $\\mathbf { W } ^ { q } / \\mathbf { W } ^ { k }$ of the query / key linear transformation to 0, which only leaves the bias term. We found that the support set representation is most crucial for building the keys (row w/o $\\mathbf { W } ^ { k }$ in the table) and has minor benefits for queries (row $\\mathbf { w } / \\mathbf { o } \\mathbf { W } ^ { q } )$ in the table. This observation is possibly related to the success of attention-based models with learnable constant queries (Liu et al., 2016; Lin et al., 2017). We also found that adding a regularizer $\\Omega ( \\Theta )$ as in Equation 8 is important for some datasets, specifically VGG Flower and Birds. ",
+        "bbox": [
+            174,
+            601,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/2090b55d4176e9ae8806b9fc8b8c05ff9aa2f421aea5280c51e4ad19c6101fb1.jpg",
+        "table_caption": [
+            "Table 3: Meta-Dataset performance variation on ablations of elements of the URT layer. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"11\">ILSVRC Omniglot Aircraft Birds Textures Draw Fungi Flower Signs MSCOCO</td></tr><tr><td>w/o Wq</td><td>+0.2</td><td>-0.2</td><td>-0.6</td><td>-0.1</td><td>-0.3</td><td>-0.2</td><td>0.0</td><td>-0.2</td><td>-0.8</td><td>-0.1</td></tr><tr><td>w/o Wk</td><td>-14.2</td><td>-2.8</td><td>-10.7</td><td>-18.1</td><td>-7.6</td><td>-9.3</td><td>-22.4</td><td>-3.6</td><td>-0.26</td><td>-10.9</td></tr><tr><td>w/or(Sc)</td><td>-14.2</td><td>-2.8</td><td>-10.7</td><td>-18.1</td><td>-7.6</td><td>-9.2</td><td>-22.4</td><td>-3.6</td><td>-0.26</td><td>-10.9</td></tr><tr><td>w/0 Ω(0)</td><td>0.0</td><td>-0.9</td><td>-0.4</td><td>-3.3</td><td>-1.2</td><td>-0.2</td><td>+0.3</td><td>-9.0</td><td>-2.0</td><td>0.0</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "An important hyper-parameter in URT is the number of heads $H$ . We chose this hyper-parameter based on the performance on validation set of tasks in Meta-Dataset. In Table 4, we show the validation performance of URT for varying number of heads. As suggested by Triantafillou et al. (2020), we considered looking at the rank of the performance achieved by each choice of $H$ for each validation domains, and taking the average across domains as a validation metric. However, since the performances when using two to four heads are similar and yield the same average rank, we instead simply consider the average accuracy as the selection criteria. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/4d8ba408c40e08bd536f77a1c70e98f29f2746946a99fcec59d5d330eba77bb8.jpg",
+        "table_caption": [
+            "Table 4: Validation performance on Meta-Dataset using different number of heads "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td></tr><tr><td>Average Accuracy</td><td>74.605</td><td>77.145</td><td>76.943</td><td>76.984</td><td>76.602</td><td>75.906</td><td>75.454</td><td>74.473</td></tr><tr><td>Average Rank</td><td>2.875</td><td>1.000</td><td>1.000</td><td>1.000</td><td>2.250</td><td>2.250</td><td>2.25</td><td>2.50</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In general, we observe a large jump in performance when using multiple heads instead of just one. However, since the number of heads controls the capacity, predictably we also observe that having too many heads leads to overfitting. ",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We proposed the URT layer to effectively integrate representations from multiple domains and demonstrated improved performance in multi-domain few-shot classification. Notably, our URT approach was able to set a new state-of-the-art on Meta-Dataset, and never performs worse than its predecessor (SUR) while also being $1 0 \\times$ more efficient at inference. This work suggests that combining meta-learning with pre-trained universal representations is a promising direction for new few-shot learning methods. Specifically, we hope that future work can investigate the design of richer forms of universal representations that go beyond simply pre-training a single backbone for each domain, and developing meta-learners adapted to those settings. ",
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+    {
+        "type": "text",
+        "text": "A EXPERIMENTS ON MORE DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            329,
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+        ],
+        "page_idx": 10
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+    {
+        "type": "text",
+        "text": "We also report performances on the MNIST, CIFAR-10 and CIFAR-100 dataset sources in Table 5, and compare with the subset of methods that have reported on these datasets. There, URT neither improves nor gets worse performance than SUR, yeilding top performance on the MNIST domain but not on the CIFAR-10/CIFAR-100 domain, on which Simple CNAPS has the best performance. ",
+        "bbox": [
+            174,
+            359,
+            825,
+            416
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "table",
+        "img_path": "images/f864a6d9019eee14fa98fbebc1ff5f990d48ca424245b876b9c9c75238f69d0b.jpg",
+        "table_caption": [
+            "Table 5: Test performance (mean $1 { + } \\mathrm { C I } \\% 9 5$ ) over 600 few-shot tasks on additional datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>MNIST CIFAR10</td><td>CIFAR100</td><td> avg. rank</td></tr><tr><td>CNAPs</td><td>92.7 ± 0.4 61.5 ± 0.7</td><td>50.1 ± 1.0</td><td>4.7</td></tr><tr><td>TaskNorm</td><td>92.3 ± 0.4 69.3 ± 0.8</td><td>54.6 ± 1.1</td><td>3.3</td></tr><tr><td>SUR</td><td>94.3 ± 0.4 66.8 ± 0.9</td><td>56.6 ± 1.0</td><td>2.3</td></tr><tr><td>SimpleCNAPS</td><td>93.9 ± 0.4 74.3± 0.7</td><td>60.5 ± 1.0</td><td>1.7</td></tr><tr><td>URT (Ours)</td><td>94.8 ± 0.4 67.3 ± 0.8</td><td>56.9 ± 1.0</td><td>2.0</td></tr></table>",
+        "bbox": [
+            284,
+            455,
+            710,
+            563
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B MORE RELATED WORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            589,
+            411,
+            606
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Transfer by fine-tuning A simple and effective method for few-shot classification is to perform transfer learning by first learning a neural network classifier on all data available for training and using its representation to initialize and then fine-tune neural networks on the few-shot classification tasks found at test time (Chen et al., 2019; Triantafillou et al., 2020; Dhillon et al., 2020; Saikia et al., 2020). Specifically, Saikia et al. (2020) have shown that competitive performance can be reached using a strong hyper-parameter optimization method applied on a carefully designed validation metric appropriate for few-shot learning. ",
+        "bbox": [
+            174,
+            621,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "C STRUCTURES AND TRAINING STRATEGY OF BACKBONES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            739,
+            681,
+            756
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Our URT layer can be built on top of a set of pretrained backbones. The structures of the backbones follow the approach in Dvornik et al. (2020). For Table 1, we use the ResNet18 architecture (He et al., 2016) for the backbones, where a separate backbone is pretrained on each domain separately using the corresponding training data in meta-dataset. The training domains include ImageNet, Omniglot, Aircraft, CU-Birds, Textures, Quick Draw, Fungi and VGG-Flower. For the parametric network family in Table 2, a base ResNet18 is first trained on ImageNet. Then a small number of modulating parameters are trained on the other domains, using their domain-specific training data, while the rest of the base backbone’s weights stay fixed. Specifically, FiLM feature modulation (Perez et al., 2018) is used. This type of parametric network family thus allows for a much reduced number of learnable parameters, by reusing the weights from the base network. In experiments, we use the pretrained backbones released by Dvornik et al. (2020) for both cases, without any further finetuning. End-to-end training of a set of backbones and the URT layers requires unaffordable computational cost, so we fix the pretrained backbones and only train the URT layer. ",
+        "bbox": [
+            173,
+            770,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
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+            132
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Implementation details for training backbones The training details of the backbones come from Dvornik et al. (2020). For optimization, SGD with momentum was used, using cosine learning rate annealing. Since the datasets come from different domains, the starting learning rate, the maximum number of training iterations and annealing frequency are set individually for each dataset. Data augmentation is applied and a constant weight decay of $\\mathrm { \\dot { 7 } } \\times 1 0 ^ { - 4 }$ is set. For each dataset, a grid search over batch size in [8, 16, 32, 64] was run and the one that maximizes accuracy on the validation set was picked. For the parametric network family, the base ResNet18 trained on ImageNet is the same. For other backbones, cosine annealing as learning rate policy is also used, weight decay and data augmentation employed as above. Please refer to Table 4 and Table 5 in the original SUR paper (Dvornik et al., 2020) for the specific values of the hyperparameters for the individual feature networks and the parametric network family, respectively. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D RESULTS ON TRAFFIC SIGNS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            313,
+            442,
+            327
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The shuffle buffer bug described in meta-dataset issue #54 (https://github.com/googleresearch/meta-dataset/issues/54) has been propagated to previous works such as CNAPs (Requeima et al., 2019). The results for the Traffic Signs dataset are considerably worse after fixing this bug. For instance, URT degrades from $6 9 . 4 { \\pm } 0 . 8 $ to $5 1 . 1 { \\pm } 1 . 1$ . Please visit the official meta-dataset GitHub repo for more details. The main paper shows the corrected results (for URT and competing approaches). For completeness, we also provide the tables as they were in the initial version of this paper for your reference in this Appendix. Both sets of results overall support the advantageous performance of URT over previous work. ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/21a17eab8e4c44baa2686bff8cddf7e972c790331fe2f93b79cf117289fa1d6d.jpg",
+        "table_caption": [
+            "Table 6: Test accuracy ( $\\mathrm { m e a n \\pm C I \\% 9 5 } ,$ ) over 600 few-shot tasks. URT and the most recent methods, which are listed in the first column, are compared on Meta-Dataset (Triantafillou et al., 2020), which are listed in the first row. The numbers in bold have intersecting confidence intervals with the most accurate method. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>ILSVRC</td><td>Omniglot</td><td>Aircraft</td><td>Birds</td><td>Textures</td><td>QuickDraw</td><td>Fungi</td><td>VGGFlower</td><td>TrafficSigns</td><td>MSCOCO</td><td>avg.rank</td></tr><tr><td>MAML</td><td>37.8±1.0</td><td>83.9±1.0 76.4±0.7</td><td></td><td>62.4±1.1</td><td>64.1±0.8</td><td>59.7±1.1</td><td>33.5±1.1</td><td>79.9±0.8</td><td>42.9±1.3</td><td>29.4±1.1</td><td>8.0</td></tr><tr><td>ProtoNet</td><td>44.5±1.1</td><td>79.6±1.1</td><td>71.1±0.9 67.0±1.0 65.2±0.8</td><td></td><td></td><td>64.9±0.9</td><td>40.3±1.1</td><td>86.9±0.7</td><td>46.5±1.0</td><td>39.9±1.1</td><td>7.3</td></tr><tr><td>ProtoMAML</td><td>46.5±1.1</td><td>82.7±1.0 75.2±0.8 69.9±1.0 68.3±0.8</td><td></td><td></td><td></td><td>66.8±0.9</td><td>42.0±1.2</td><td>88.7±0.7</td><td>52.4±1.1</td><td>41.7±1.1</td><td>5.4</td></tr><tr><td>CNAPs</td><td>52.3±1.0</td><td>88.4±0.7 80.5±0.6 72.2±0.9 58.3±0.7</td><td></td><td></td><td></td><td>72.5±0.8</td><td>47.4±1.0</td><td>86.0±0.5</td><td>60.2±0.9</td><td>42.6±1.1</td><td>5.1</td></tr><tr><td>BOHB-E</td><td>55.4±1.1</td><td>77.5±1.1</td><td>60.9±0.9</td><td>73.6±0.8</td><td>72.8±0.7</td><td>61.2±0.9</td><td>44.5±1.1</td><td>90.6±0.6</td><td>57.5±1.0</td><td>51.9±1.0</td><td>4.4</td></tr><tr><td>TaskNorm</td><td>50.6±1.1</td><td>90.7±0.6 83.8±0.6</td><td></td><td>574.6±0.8</td><td>62.1±0.7</td><td>74.8±0.7</td><td>48.7±1.0</td><td>89.6±0.6</td><td>67.0±0.7</td><td>43.4±1.0</td><td>3.8</td></tr><tr><td>SUR</td><td>56.3±1.1</td><td>93.1±0.5 85.4±0.7 71.4±1.0 71.5±0.8</td><td></td><td></td><td></td><td>81.3±0.6</td><td>63.1±1.0</td><td>82.8±0.7</td><td>70.4±0.8</td><td>52.4±1.1</td><td>2.5</td></tr><tr><td>SimpleCNAPS 58.6±1.1 9</td><td></td><td>91.7±0.6 82.4±0.7 74.9±0.8 67.8±0.8</td><td></td><td></td><td></td><td>77.7±0.7</td><td>46.9±1.0</td><td>90.7±0.5</td><td>73.5±0.7</td><td>46.2±1.1</td><td>2.4</td></tr><tr><td>URT(Ours)</td><td>55.7±1.0</td><td>94.4±0.4 85.8±0.6 76.3±0.8 71.8±0.7</td><td></td><td></td><td></td><td>82.5±0.6</td><td>63.5±1.0</td><td>88.2±0.6</td><td>69.4±0.8</td><td>52.2±1.1</td><td>1.6</td></tr></table>",
+        "bbox": [
+            173,
+            536,
+            823,
+            683
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/f7ad129999878812ce8ff86a91bf5be271f0274521d82cf801a02d27060ef6b1.jpg",
+        "table_caption": [
+            "Table 7: Test accuracy (mean $\\pm \\mathrm { C I } \\% 9 5 $ ) over 600 few-shot tasks. All methods use parametric network family (pf) backbones. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>SUR-pf</td><td>URT-pf</td><td>VS.</td></tr><tr><td>ILSVRC</td><td>56.4 ± 1.2</td><td>55.5 ± 1.1</td><td>=</td></tr><tr><td>Omniglot</td><td>88.5±0.8</td><td>90.2 ± 0.6</td><td>+</td></tr><tr><td>Aircraft</td><td>79.5 ± 0.8</td><td>79.8 ± 0.7</td><td>=</td></tr><tr><td>Birds</td><td>76.4 ± 0.9</td><td>77.5 ± 0.8</td><td>=</td></tr><tr><td>Textures</td><td>73.1 ± 0.7</td><td>73.5± 0.7</td><td>=</td></tr><tr><td>Quick Draw</td><td>75.7 ± 0.7</td><td>75.8 ± 0.7</td><td>=</td></tr><tr><td>Fungi</td><td>48.2 ± 0.9</td><td>48.1 ± 0.9</td><td>=</td></tr><tr><td>VGG Flower</td><td>90.6 ± 0.5</td><td>91.9 ± 0.5</td><td>+</td></tr><tr><td>Traffic Signs</td><td>65.1 ± 0.8</td><td>67.5 ± 0.8</td><td>+</td></tr><tr><td>MSCOCO</td><td>52.1 ± 1.0</td><td>52.1 ± 1.0</td><td>=</td></tr><tr><td>MNIST</td><td>93.2 ± 0.4</td><td>93.9 ± 0.4</td><td>=</td></tr><tr><td>CIFAR10</td><td>66.4 ± 0.8</td><td>66.1 ± 0.8</td><td>=</td></tr><tr><td>CIFAR100</td><td>57.1 ± 1.0</td><td>57.3 ±1.0</td><td>=</td></tr></table>",
+        "bbox": [
+            372,
+            732,
+            627,
+            905
+        ],
+        "page_idx": 11
+    }
+]

parse/train/YCqx6zhEzRp/YCqx6zhEzRp.md

@@ -0,0 +1,332 @@
+# Federated Multi-Task Learning under a Mixture of Distributions
+
+Othmane Marfoq1,3, Giovanni Neglia1, Aurélien Bellet2, Laetitia Kameni3, and Richard Vidal3
+
+1Inria, Université Côte d’Azur, France, {othmane.marfoq, giovanni.neglia} $@$ inria.fr 2Inria, Université de Lille, France, aurelien.bellet $@$ inria.fr 3Accenture Labs, France, {richard.vidal, laetitia.kameni} $@$ accenture.com
+
+# Abstract
+
+The increasing size of data generated by smartphones and IoT devices motivated the development of Federated Learning (FL), a framework for on-device collaborative training of machine learning models. First efforts in FL focused on learning a single global model with good average performance across clients, but the global model may be arbitrarily bad for a given client, due to the inherent heterogeneity of local data distributions. Federated multi-task learning (MTL) approaches can learn personalized models by formulating an opportune penalized optimization problem. The penalization term can capture complex relations among personalized models, but eschews clear statistical assumptions about local data distributions.
+
+In this work, we propose to study federated MTL under the flexible assumption that each local data distribution is a mixture of unknown underlying distributions. This assumption encompasses most of the existing personalized FL approaches and leads to federated EM-like algorithms for both client-server and fully decentralized settings. Moreover, it provides a principled way to serve personalized models to clients not seen at training time. The algorithms’ convergence is analyzed through a novel federated surrogate optimization framework, which can be of general interest. Experimental results on FL benchmarks show that our approach provides models with higher accuracy and fairness than state-of-the-art methods.
+
+# 1 Introduction
+
+Federated Learning (FL) [28] allows a set of clients to collaboratively train models without sharing their local data. Standard FL approaches train a unique model for all clients [47, 32, 38, 29, 48]. However, as discussed in [56], the existence of such a global model suited for all clients is at odds with the statistical heterogeneity observed across different clients [37, 28]. Indeed, clients can have non-iid data and varying preferences. Consider for example a language modeling task: given the sequence of tokens $^ { 6 6 } I$ love eating,” the next word can be arbitrarily different from one client to another. Thus, having personalized models for each client is a necessity in many FL applications.
+
+Previous work on personalized FL. A naive approach for FL personalization consists in learning first a global model and then fine-tuning its parameters at each client via a few iterations of stochastic gradient descent [58]. In this case, the global model plays the role of a meta-model to be used as initialization for few-shot adaptation at each client. In particular, the connection between FL and Model Agnostic Meta Learning (MAML) [27] has been studied in [19, 30, 1] in order to build a more suitable meta-model for local personalization. Unfortunately, these methods can fail to build a model with low generalization error (as exemplified by LEAF synthetic dataset [7, App. 1]). An alternative approach is to jointly train a global model and one local model per client and then let each client build a personalized model by interpolating them [14, 9, 44]. However, if local distributions are far from the average distribution, a relevant global model does not exist and this approach boils down to every client learning only on its own local data. This issue is formally captured by the generalization bound in [14, Theorem 1].
+
+Clustered FL [56, 20, 44] addresses the potential lack of a global model by assuming that clients can be partitioned into several clusters. Clients belonging to the same cluster share the same optimal model, but those models can be arbitrarily different across clusters (see [56, Assumption 2] for a rigorous formulation). During training, clients learn the cluster to which they belong as well as the cluster model. The Clustered FL assumption is also quite limiting, as no knowledge transfer is possible across clusters. In the extreme case where each client has its own optimal local model (recall the example on language modeling), the number of clusters coincides with the number of clients and no federated learning is possible.
+
+Multi-Task Learning (MTL) has recently emerged as an alternative approach to learn personalized models in the federated setting and allows for more nuanced relations among clients’ models [59, 63, 67, 24, 16]. The authors of [59, 63] were the first to frame FL personalization as a MTL problem. In particular, they defined federated MTL as a penalized optimization problem, where the penalization term models relationships among tasks (clients). The work [59] proposed the MOCHA algorithm for the client-server scenario, while [63, 67] presented decentralized algorithms for the same problem. Unfortunately, these algorithms can only learn simple models (linear models or linear combination of pre-trained models), because of the complex penalization term. Other MTL-based approaches [24, 23, 16, 26, 36] are able to train more general models at the cost of considering simpler penalization terms (e.g., the distance to the average model), thereby losing the capability to capture complex relations among tasks. Moreover, a general limitation of this line of work is that the penalization term is justified qualitatively and not on the basis of clear statistical assumptions on local data distributions.
+
+More recently, [57] proposed pFedHN. pFedHN feeds local clients’ representations to a global (across clients) hypernetwork, which can output personalized heterogeneous models. Unfortunately, the hypernetwork has a large memory footprint already for small clients’ models (e.g., the hypernetwork in the experiments in [57] has 100 more parameters than the output model). Hence, it is not clear if pFedHN can scale to more complex models. Moreover, pFedHN requires each client to communicate multiple times for the server to learn meaningful representations. Therefore, its performance is likely to deteriorate when clients participate only once (or few times) to training, as it is the case for large-scale cross-device FL training. Furthermore, even once the hypernetwork parameters have been learned, training personalized models for new clients still requires multiple client-server communication rounds. More similar to our approach, FedFOMO [68] lets each client interpolate other clients’ local models with opportune weights learned during training. However, this method lacks both theoretical justifications for such linear combinations and convergence guarantees. Moreover, FedFOMO requires the presence of a powerful server able to 1) store all individual local models and 2) learn for each client—through repeated interactions—which other clients’ local models may be useful. Therefore, FedFOMO is not suited for cross-device FL where the number of clients may be very large (e.g., $\mathrm { 1 0 ^ { 5 } - 1 0 ^ { 7 } }$ participating clients [28, Table 2]) and a given client may only participate in a single training round.
+
+Overall, although current personalization approaches can lead to superior empirical performance in comparison to a shared global model or individually trained local models, it is still not well understood whether and under which conditions clients are guaranteed to benefit from collaboration.
+
+Our contributions. In this work, we first show that federated learning is impossible without assumptions on local data distributions. Motivated by this negative result, we formulate a general and flexible assumption: the data distribution of each client is a mixture of $M$ underlying distributions. The proposed formulation has the advantage that each client can benefit from knowledge distilled from all other clients’ datasets (even if any two clients can be arbitrarily different from each other). We also show that this assumption encompasses most of the personalized FL approaches previously proposed in the literature.
+
+In our framework, a personalized model is a linear combination of $M$ shared component models. All clients jointly learn the $M$ components, while each client learns its personalized mixture weights. We show that federated EM-like algorithms can be used for training. In particular, we propose FedEM and D-FedEM for the client-server and the fully decentralized settings, respectively, and we prove convergence guarantees. Our approach also provides a principled and efficient way to infer personalized models for clients unseen at training time. Our algorithms can easily be adapted to solve more general problems in a novel framework, which can be seen as a federated extension of the centralized surrogate optimization approach in [43]. To the best of our knowledge, our paper is the first work to propose federated surrogate optimization algorithms with convergence guarantees.
+
+Through extensive experiments on FL benchmark datasets, we show that our approach generally yields models that 1) are on average more accurate, 2) are fairer across clients, and 3) generalize better to unseen clients than state-of-the-art personalized and non-personalized FL approaches.
+
+Paper outline. The rest of the paper is organized as follows. In Section 2 we provide our impossibility result, introduce our main assumptions, and show that several popular personalization approaches can be obtained as special cases of our framework. Section 3 describes our algorithms, states their convergence results, and presents our general federated surrogate optimization framework. Finally, we provide experimental results in Section 4 before concluding in Section 5.
+
+# 2 Problem Formulation
+
+We consider a (countable) set $\tau$ of classification (or regression) tasks which represent the set of possible clients. We will use the terms task and client interchangeably. Data at client $t \in \mathcal { T }$ is generated according to a local distribution $\mathcal { D } _ { t }$ over $\mathcal { X } \times \mathcal { V }$ . Local data distributions $\{ \mathcal { D } _ { t } \} _ { t \in \mathcal { T } }$ are in general different, thus it is natural to fit a separate model (hypothesis) $h _ { t } \in { \mathcal { H } }$ to each data distribution $\mathcal { D } _ { t }$ . The goal is then to solve (in parallel) the following optimization problems
+
+$$
+\forall t \in { \mathcal { T } } , \quad \operatorname* { m i n i m i z e } _ { h _ { t } \in { \mathcal { H } } } { \mathcal { L } } _ { { \mathcal { D } } _ { t } } ( h _ { t } ) ,
+$$
+
+where $h _ { t } : \mathcal { X } \mapsto \Delta ^ { | \mathcal { V } | }$ $\Delta ^ { D }$ denoting the unitary simplex of dimension $D$ ), $l : \Delta ^ { | y | } \times \mathcal { Y } \mapsto \mathbb { R } ^ { + }$ is a loss function,1 and $\mathcal { L } _ { \mathcal { D } _ { t } } ( h _ { t } ) = \mathbb { E } _ { ( \mathbf { x } , y ) \sim \mathcal { D } _ { t } } ^ { - } \left[ l ( \bar { h } _ { t } ( \mathbf { x } ) , \bar { y } ) \right]$ is the true risk of a model $h _ { t }$ under data distribution $\mathcal { D } _ { t }$ . For $( \mathbf { x } , y ) \in \mathcal { X } \times \mathcal { Y }$ , we will denote the joint distribution density associated to $\mathcal { D } _ { t }$ by $p _ { t } ( \mathbf { x } , y )$ , and the marginal densities by $p _ { t } ( \mathbf { x } )$ and $p _ { t } ( y )$ .
+
+A set of $T$ clients $[ T ] \triangleq \{ 1 , 2 , . . . T \} \subseteq \mathcal { T }$ participate to the initial training phase; other clients may e system in a late drawn i.i.d. from age. We, and by $S _ { t } = \{ s _ { t } ^ { ( i ) } = ( \mathbf { x } _ { t } ^ { ( i ) } , \ y _ { t } ^ { ( i ) } ) \} _ { i = 1 } ^ { n _ { t } }$ the dataset at client $t \in [ T ]$ $\mathcal { D } _ { t }$ $\begin{array} { r } { n = \sum _ { t = 1 } ^ { T } n _ { t } } \end{array}$
+
+The idea of federated learning is to enable each client to benefit from data samples available at other clients in order to get a better estimation of $\mathcal { L } _ { \mathcal { D } _ { t } }$ , and therefore get a model with a better generalization ability to unseen examples.
+
+# 2.1 An Impossibility Result
+
+We start by showing that some assumptions on the local distributions $p _ { t } ( \mathbf x , y ) , t \in \mathcal T$ are needed for federated learning to be possible, i.e., for each client to be able to take advantage of the data at other clients. This holds even if all clients participate to the initial training phase (i.e., $\mathcal { T } = [ T ] )$ .
+
+We consider the classic PAC learning framework where we fix a class of models $\mathcal { H }$ and seek a learning algorithm which is guaranteed, for all possible data distributions over $\mathcal { X } \times \mathcal { V }$ , to return with high probability a model with expected error $\epsilon$ -close to the best possible error in the class $\mathcal { H }$ . The worst-case sample complexity then refers to the minimum amount of labeled data required by any algorithm to reach a given $\epsilon$ -approximation.
+
+Our impossibility result for FL is based on a reduction to an impossibility result for Semi-Supervised Learning (SSL), which is the problem of learning from a training set with only a small amount of labeled data. The authors of [4] conjectured that, when the quantity of unlabeled data goes to infinity, the worst-case sample complexity of SSL improves over supervised learning at most by a constant factor that only depends on the hypothesis class [4, Conjecture 4]. This conjecture was later proved for the realizable case and hypothesis classes of finite VC dimension [13, Theorem 1], even when the marginal distribution over the domain set $\mathcal { X }$ is known [21, Theorem 2]. 2
+
+In the context of $\mathrm { F L }$ , if the marginal distributions $p _ { t } \left( \mathbf { x } \right)$ are identical, but the conditional distributions $p _ { t } \left( y | \mathbf { x } \right)$ can be arbitrarily different, then each client $t$ can learn using: 1) its own local labeled dataset, and 2) the other clients’ datasets, but only as unlabeled ones (because their labels have no relevance for $t$ ). The FL problem, with $T$ clients, then reduces to $T$ parallel SSL problems, or more precisely, it is at least as difficult as $T$ parallel SSL problems (because client $t$ has no direct access to the other local datasets but can only learn through the communication exchanges allowed by the FL algorithm). The SSL impossibility result implies that, without any additional assumption on the local distributions $p _ { t } \left( \mathbf { x } , y \right)$ , $t \in [ T ]$ , any FL algorithm can reduce the sample complexity of client- $\mathbf { \nabla } \cdot t$ ’s problem in (1) only by a constant in comparison to local learning, independently of how many other clients participate to training and how large their datasets’ sizes are.
+
+# 2.2 Learning under a Mixture Model
+
+Motivated by the above impossibility result, in this work we propose to consider that each local data distribution $\mathcal { D } _ { t }$ is a mixture of $M$ underlying distributions $\tilde { \mathcal { D } } _ { m }$ , $1 \leq m \leq M$ , as formalized below.
+
+Assumption 1. There exist $M$ underlying (independent) distributions $\tilde { \mathcal { D } } _ { m }$ , $1 \leq m \leq M$ , such that for $t \in \tau$ , $\mathcal { D } _ { t }$ is mixture of the distributions $\{ \tilde { \mathcal { D } } _ { m } \} _ { m = 1 } ^ { M }$ with weights $\pi _ { t } ^ { * } = [ \pi _ { t 1 } ^ { * } , \ldots , \pi _ { t M } ^ { * } ] \in \Delta ^ { M }$ , i.e.
+
+$$
+z _ { t } \sim \mathcal { M } ( \pi _ { t } ^ { * } ) , \quad ( ( \mathbf { x } _ { t } , y _ { t } ) | z _ { t } = m ) \sim \tilde { \mathcal { D } } _ { m } , \quad \forall t \in \mathcal { T } ,
+$$
+
+where $\mathcal { M } ( \pi )$ is a multinomial (categorical) distribution with parameters $\pi$ .
+
+Similarly to what was done above, we use $p _ { m } ( \mathbf { x } , y )$ , $p _ { m } ( \mathbf { x } )$ , and $p _ { m } ( y )$ to denote the probability distribution densities associated to $\tilde { \mathcal { D } } _ { m }$ . We further assume that marginals over $\mathcal { X }$ are identical.
+
+Assumption 2. For all $m \in [ M ]$ , we have $p _ { m } ( \mathbf { x } ) = p ( \mathbf { x } )$ .
+
+Assumption 2 is not strictly required for our analysis to hold, but, in the most general case, solving Problem (1) requires to learn generative models. Instead, under Assumption 2 we can restrict our attention to discriminative models (e.g., neural networks). 3 More specifically, we consider a parameterized set of models $\tilde { \mathcal { H } }$ with the following properties.
+
+Assumption 3. $\tilde { \mathcal { H } } = \{ h _ { \theta } \} _ { \theta \in \mathbb { R } ^ { d } }$ is a set of hypotheses parameterized by $\theta \in \mathbb { R } ^ { d }$ , whose convex hull is in $\mathcal { H }$ . For each distribution $\tilde { \mathcal { D } } _ { m }$ with $m \in [ M ]$ , there exists a hypothesis $h _ { \theta _ { m } ^ { * } }$ , such that
+
+$$
+\begin{array} { r } { l \left( h _ { \theta _ { m } ^ { * } } ( \mathbf { x } ) , y \right) = - \log p _ { m } ( y | \mathbf { x } ) + c , } \end{array}
+$$
+
+where $c \in \mathbb { R }$ is a normalization constant. The function $l ( \cdot , \cdot )$ is then the log-loss associated to $p _ { m } ( y | \mathbf { x } )$ .
+
+We refer to the hypotheses in $\tilde { \mathcal { H } }$ as component models or simply components. We denote by $\Theta ^ { * } \in \mathbb { R } ^ { M \times d }$ the matrix whose $m$ -th row is $\theta _ { m } ^ { * }$ , and by $\Pi ^ { * } \in \Delta ^ { T \times } { \bar { M } }$ the matrix whose $t$ -th row is $\pi _ { t } ^ { * } \in \Delta ^ { M }$ . Similarly, we will use $\Theta$ and $\Pi$ to denote arbitrary parameters.
+
+Remark 1. Assumptions 2–3 are mainly technical and are not required for our approach to work in practice. Experiments in Section 4 show that our algorithms perform well on standard $F L$ benchmark datasets, for which these assumptions do not hold in general.
+
+Note that, under the above assumptions, $p _ { t } ( \mathbf { x } , y )$ depends on $\Theta ^ { * }$ and $\pi _ { t } ^ { * }$ . Moreover, we can prove (see App. A) that the optimal local model $h _ { t } ^ { * } \in \mathcal { H }$ for client $t$ is a weighted average of models in $\tilde { \mathcal { H } }$ . Proposition 2.1. Let $l ( \cdot , \cdot )$ be the mean squared error loss, the logistic loss or the cross-entropy loss, and $\breve { \Theta }$ and $\breve { \Pi }$ be a solution of the following optimization problem:
+
+$$
+\underset { \Theta , \Pi } { \mathrm { m i n i m i z e } } \underset { t \sim D _ { T } } { \mathbb { E } } \underset { ( \mathbf { x } , y ) \sim D _ { t } } { \mathbb { E } } \left[ - \log p _ { t } ( \mathbf { x } , y | \Theta , \pi _ { t } ) \right] ,
+$$
+
+where $D _ { \mathcal { T } }$ is any distribution with support $\tau$ . Under Assumptions $^ { l }$ , 2, and $^ 3$ , the predictors
+
+$$
+h _ { t } ^ { * } = \sum _ { m = 1 } ^ { M } \breve { \pi } _ { t m } h _ { { \breve { \theta } } _ { m } } \left( \mathbf { x } \right) , \quad \forall t \in \mathcal { T }
+$$
+
+minimize $\mathbb { E } _ { ( \mathbf { x } , y ) \sim \mathcal { D } _ { t } } \left[ l ( h _ { t } ( \mathbf { x } ) , y ) \right]$ and thus solve Problem (1).
+
+Proposition 2.1 suggests the following approach to solve Problem (1). First, we estimate the parameters $\breve { \Theta }$ and $\breve { \pi } _ { t }$ , $1 \leq t \leq T$ , by minimizing the empirical version of Problem (4) on the training data, i.e., minimizing:
+
+$$
+f ( \Theta , \Pi ) \triangleq - \frac { \log p ( \mathcal { S } _ { 1 : T } | \Theta , \Pi ) } { n } \triangleq - \frac { 1 } { n } \sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { n _ { t } } \log p ( s _ { t } ^ { ( i ) } | \Theta , \pi _ { t } ) ,
+$$
+
+which is the (negative) likelihood of the probabilistic model (2). 4 Second, we use (5) to get the client predictor for the $T$ clients present at training time. Finally, to deal with a client $t _ { \mathrm { n e w } } \notin [ T ]$ not seen during training, we keep the mixture component models fixed and simply choose the weights $\pi _ { t _ { \mathrm { n e w } } }$ that maximize the likelihood of the client data and get the client predictor via (5).
+
+# 2.3 Generalizing Existing Frameworks
+
+Before presenting our federated learning algorithms in Section 3, we show that the generative model in Assumption 1 extends some popular multi-task/personalized FL formulations in the literature.
+
+Clustered Federated Learning [56, 20] assumes that each client belongs to one among $C$ clusters and proposes that all clients in the same cluster learn the same model. Our framework recovers this scenario considering $M = C$ and $\pi _ { t c } ^ { * } = 1$ if task (client) $t$ is in cluster $c$ and $\pi _ { t c } ^ { * } = 0$ otherwise.
+
+Personalization via model interpolation [44, 14] relies on learning a global model $h _ { \mathrm { g l o b } }$ and $T$ local models $h _ { \mathrm { l o c } , t }$ , and then using at each client the linear interpolation $h _ { t } = \alpha _ { t } h _ { \mathrm { l o c } , t } + ( 1 - \alpha _ { t } ) h _ { \mathrm { g l o b } }$ . Each client model can thus be seen as a linear combination of $M = T + 1$ models $h _ { m } = h _ { { \mathrm { l o c } } , m }$ for $m \in [ T ]$ and $h _ { 0 } = h _ { \mathrm { g l o b } }$ with specific weights $\pi _ { t t } ^ { * } = \alpha _ { t }$ , $\pi _ { t 0 } ^ { * } = 1 - \alpha _ { t }$ , and $\pi _ { t t ^ { \prime } } ^ { * } = 0$ for $t ^ { \prime } \in [ T ] \backslash \{ t \}$ .
+
+Federated MTL via task relationships. The authors of [59] proposed to learn personalized models by solving the following optimization problem inspired from classic MTL formulations:
+
+$$
+\operatorname* { m i n } _ { W , \Omega } \ \sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { n _ { t } } l ( h _ { w _ { t } } ( \mathbf { x } _ { t } ^ { ( i ) } ) , y _ { t } ^ { ( i ) } ) + \lambda \mathrm { t r } \left( W \Omega W ^ { \intercal } \right) ,
+$$
+
+where $h _ { w _ { t } }$ are linear predictors parameterized by the rows of matrix $W$ and the matrix $\Omega$ captures task relationships (similarity). This formulation is motivated by the alternating structure optimization method (ASO) [2, 70]. In App. B, we show that, when predictors $h _ { \theta _ { m } ^ { * } }$ are linear and have bounded norm, our framework leads to the same ASO formulation that motivated Problem (7). Problem (7) can also be justified by probabilistic priors [69] or graphical models [35] (see [59, App. B.1]). Similar considerations hold for our framework (see again App. B). Reference [67] extends the approach in [59] by letting each client learn a personalized model as a weighted combination of $M$ known hypotheses. Our approach is more general and flexible as clients learn both the weights and the hypotheses. Finally, other personalized FL algorithms, like pFedMe [16], FedU [17], and those studied in [24] and in [23], can be framed as special cases of formulation (7). Their assumptions can thus also be seen as a particular case of our framework.
+
+# 3 Federated Expectation-Maximization
+
+# 3.1 Centralized Expectation-Maximization
+
+Our goal is to estimate the optimal components’ parameters $\Theta ^ { * } \ = \ ( \theta _ { m } ^ { * } ) _ { 1 \leq m \leq M }$ and mixture weights $\Pi ^ { * } = ( \pi _ { t } ^ { * } ) _ { 1 \leq t \leq T }$ by minimizing the negative log-likelihood $f ( \Theta , \bar { \Pi } )$ in (6). A natural approach to solve such non-convex problems is the Expectation-Maximization algorithm (EM), which alternates between two steps. Expectation steps update the distribution (denoted by $q _ { t }$ ) over the latent variables $\boldsymbol { z } _ { t } ^ { ( i ) }$ for every data point $s _ { t } ^ { ( i ) } = ( \mathbf { x } _ { t } ^ { ( i ) } , y _ { t } ^ { ( i ) } )$ given the current estimates of the parameters $\{ \Theta , \Pi \}$ . Maximization steps update the parameters $\{ \Theta , \Pi \}$ by maximizing the expected log-likelihood, where the expectation is computed according to the current latent variables’ distributions.
+
+The following proposition provides the EM updates for our problem (proof in App. C).
+
+Proposition 3.1. Under Assumptions $I$ and 2, at the $k$ -th iteration the EM algorithm updates parameter estimates through the following steps:
+
+$$
+\begin{array} { r l r } {  { q _ { t } ^ { k + 1 } ( z _ { t } ^ { ( i ) } = m ) \propto \pi _ { t m } ^ { k } \cdot \exp ( - l ( h _ { \theta _ { m } ^ { k } } ( \mathbf { x } _ { t } ^ { ( i ) } ) , y _ { t } ^ { ( i ) } ) ) , } } & { t \in [ T ] , m \in [ M ] , i \in [ n _ { t } ] } \\ & { } & \\ & { \pi _ { t m } ^ { k + 1 } = \frac { \sum _ { i = 1 } ^ { n _ { t } } q _ { t } ^ { k + 1 } ( z _ { t } ^ { ( i ) } = m ) } { n _ { t } } , } & { t \in [ T ] , m \in [ M ] } \\ & { } & \\ & { \theta _ { m } ^ { k + 1 } \in \arg \operatorname* { m i n } _ { \theta \in \mathbb { R } ^ { d } } \sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { n _ { t } } q _ { t } ^ { k + 1 } ( z _ { t } ^ { ( i ) } = m ) l \big ( h _ { \theta } ( \mathbf { x } _ { t } ^ { ( i ) } ) , y _ { t } ^ { ( i ) } \big ) , } & { m \in [ M ] } \end{array}
+$$
+
+The EM updates in Proposition 3.1 have a natural interpretation. In the $\mathrm { E }$ -step, given current component models $\Theta ^ { k }$ and mixture weights $\Pi ^ { k }$ , (8) updates the a-posteriori probability $q _ { t } ^ { k + 1 } ( z _ { t } ^ { ( i ) } =$ $m$ ) that point $s _ { t } ^ { ( i ) }$ of client $t$ was drawn from the $m$ -th distribution based on the current mixture weight $\pi _ { t m } ^ { k }$ and on how well the corresponding component $\theta _ { m } ^ { k }$ classifies $s _ { t } ^ { ( i ) }$ . The M-step consists of two updates under fixed probabilities $q _ { t } ^ { k + 1 }$ . First, (9) updates the mixture weights $\pi _ { t } ^ { k + 1 }$ to reflect the prominence of each distribution $\tilde { \mathcal { D } } _ { m }$ in $S _ { t }$ as given by $q _ { t } ^ { k + 1 }$ . Finally, (10) updates the components’ parameters $\Theta ^ { k + 1 }$ by solving $M$ independent, weighted empirical risk minimization problems with weights given by $q _ { t } ^ { k + 1 }$ . These weights aim to construct an unbiased estimate of the true risk over each underlying distribution $\tilde { \mathcal { D } } _ { m }$ using only points sampled from the client mixtures, similarly to importance sampling strategies used to learn from data with sample selection bias [61, 11, 10, 64].
+
+# 3.2 Client-Server Algorithm
+
+Federated learning aims to train machine learning models directly on the clients, without exchanging raw data, and thus we should run EM while assuming that only client $t$ has access to dataset $S _ { t }$ . The E-step (8) and the $\Pi$ update (9) in the M-step operate separately on each local dataset $S _ { t }$ and can thus be performed locally at each client $t$ . On the contrary, the $\Theta$ update (10) requires interaction with other clients, since the computation spans all data samples $\mathcal { S } _ { 1 : T }$ .
+
+In this section, we consider a client-server setting, in which each client $t$ can communicate only with a centralized server (the orchestrator) and wants to learn components’ parameters $\Theta ^ { * } = \left( \theta _ { m } ^ { * } \right) _ { 1 \leq m \leq M }$ and its own mixture weights $\pi _ { t } ^ { * }$ .
+
+We propose the algorithm FedEM for Federated Expectation-Maximization (Alg. 1). FedEM proceeds through communication rounds similarly to most FL algorithms including FedAvg [47], FedProx [38], SCAFFOLD [29], and pFedMe [16]. At each round, 1) the central server broadcasts the (shared) component models to the clients, 2) each client locally updates components and its personalized mixture weights, and 3) sends the updated components back to the server, 4) the server aggregates the updates. The local update performed at client $t$ consists in performing the steps in (8) and (9) and updating the local estimates of $\theta _ { m }$ through a solver which approximates the exact minimization in (10) using only the local dataset $S _ { t }$ (see line 7). FedEM can operate with different local solvers—even different across clients—as far as they satisfy some local improvement guarantees (see the discussion in App. H). In what follows, we restrict our focus on the practically important case where the local solver performs multiple stochastic gradient descent updates (local SGD [60]). Under the following standard assumptions (see e.g., [66]), FedEM converges to a stationary point of $f$ . Below, we use the more compact notation $l ( \boldsymbol { \theta } ; s _ { t } ^ { ( i ) } ) \triangleq l ( h _ { \boldsymbol { \theta } } ( \mathbf { x } _ { t } ^ { ( i ) } ) , y _ { t } ^ { ( i ) } )$ .
+
+Assumption 4. The negative log-likelihood $f$ is bounded below by $f ^ { * } \in \mathbb { R }$ .
+
+Assumption 5. (Smoothness) For all $t \in [ T ]$ and $i \in [ n _ { t } ]$ , the function $\theta \mapsto l ( \theta ; s _ { t } ^ { ( i ) } )$ is $L$ -smooth and twice continuously differentiable.
+
+Assumption 6. (Unbiased gradients and bounded variance) Each client $t \in [ T ]$ can sample a random batch $\xi$ from $S _ { t }$ and compute an unbiased estimator $g _ { t } ( \theta , \xi )$ of the local gradient with bounded variance, i.e., $\begin{array} { r } { \mathbb { E } _ { \xi } [ g _ { t } ( \theta , \xi ) ] = \frac { 1 } { n _ { t } } \sum _ { i = 1 } ^ { n _ { t } } \nabla _ { \theta } l ( \theta ; s _ { t } ^ { ( i ) } ) } \end{array}$ and $\begin{array} { r } { \mathbb { E } _ { \xi } \| g _ { t } ( \theta , \xi ) - \frac { 1 } { n _ { t } } \sum _ { i = 1 } ^ { n _ { t } } \nabla _ { \theta } l ( \theta ; s _ { t } ^ { ( i ) } ) \| ^ { 2 } \leq \sigma ^ { 2 } } \end{array}$
+
+Assumption 7. (Bounded dissimilarity) There exist $\beta$ and $G$ such that for any set of weights $\alpha \in \Delta ^ { M }$ :
+
+$$
+\sum _ { t = 1 } ^ { T } \frac { n _ { t } } { n } \Big \| \frac { 1 } { n _ { t } } \sum _ { i = 1 } ^ { n _ { t } } \sum _ { m = 1 } ^ { M } \alpha _ { m } \cdot l ( \theta ; s _ { t } ^ { ( i ) } ) \Big \| ^ { 2 } \leq G ^ { 2 } + \beta ^ { 2 } \Big \| \frac { 1 } { n } \sum _ { t = 1 } ^ { T } \sum _ { i = 1 } ^ { n _ { t } } \sum _ { m = 1 } ^ { M } \alpha _ { m } \cdot l ( \theta ; s _ { t } ^ { ( i ) } ) \Big \| ^ { 2 } .
+$$
+
+Input : Data $\overline { { S _ { 1 : T } } }$ ; number of mixture distributions $M$ ; number of communication rounds $K$   
+Output : $\theta _ { m } ^ { K }$ , $m \in [ M ]$   
+1 for iterations $k = 1 , \ldots , K$ do   
+2 server broadcasts $\theta _ { m } ^ { k - 1 }$ , $1 \leq m \leq M$ , to the $T$ clients;   
+3 for tasks $t = 1 , \dots , T$ in parallel over $T$ clients do   
+4 for component $m = 1 , \ldots , M$ do   
+5 update $q _ { t } ^ { k } ( z _ { t } ^ { ( i ) } = m )$ as in (8), $\forall i \in \left\{ 1 , \ldots , n _ { t } \right\}$ ;   
+6 update $\pi _ { t m } ^ { k }$ as in (9);   
+7 $\theta _ { m , t } ^ { k } \gets \mathtt { L o c a l S o l v e r } ( m , \theta _ { m } ^ { k - 1 } , q _ { t } ^ { k } , S _ { t } )$ ;   
+8 client $t$ sends $\theta _ { m , t } ^ { k }$ , $1 \leq m \leq M$ , to the server;   
+9 for component $m = 1 , \ldots , M$ do   
+10 $\begin{array} { r } { \theta _ { m } ^ { k }  \sum _ { t = 1 } ^ { T } \frac { n _ { t } } { n } \times \theta _ { m , t } ^ { k } } \end{array}$ ;
+
+Assumption 7 limits the level of dissimilarity of the different tasks, similarly to what is done in [66].
+
+Theorem 3.2. Under Assumptions $^ { 1 - 7 , }$ , when clients use $S G D$ as local solver with learning rate $\begin{array} { r } { \eta = \frac { a _ { 0 } } { \sqrt { K } } } \end{array}$ , after a large enough number of communication rounds $K$ , FedEM’s iterates satisfy:
+
+$$
+\frac { 1 } { K } \sum _ { k = 1 } ^ { K } \mathbb { E } \left\| \nabla _ { \Theta } f \left( \Theta ^ { k } , \Pi ^ { k } \right) \right\| _ { F } ^ { 2 } \leq \mathcal { O } \bigg ( \frac { 1 } { \sqrt { K } } \bigg ) , \qquad \frac { 1 } { K } \sum _ { k = 1 } ^ { K } \Delta _ { \Pi } f ( \Theta ^ { k } , \Pi ^ { k } ) \leq \mathcal { O } \bigg ( \frac { 1 } { K ^ { 3 / 4 } } \bigg ) ,
+$$
+
+where the expectation is over the random batches samples, and $\Delta _ { \Pi } f ( \Theta ^ { k } , \Pi ^ { k } ) \triangleq f \left( \Theta ^ { k } , \Pi ^ { k } \right) -$ $f \left( \Theta ^ { k } , \Pi ^ { k + 1 } \right) \ge 0$ .
+
+Theorem 3.2 (proof in App. G.1) expresses the convergence of both sets of parameters $\Theta$ and $\Pi$ ) to a stationary point of $f$ . Indeed, the gradient of $f$ with respect to $\Theta$ becomes arbitrarily small (left inequality in (11)) and the update in Eq. (9) leads to arbitrarily small improvements of $f$ (right inequality in (11)).
+
+We conclude this section observing that FedEM allows an unseen client, i.e., a client $t _ { \mathrm { n e w } } \notin [ T ]$ arriving after the distributed training procedure, to learn its personalized model. The client simply retrieves the learned components’ parameters $\Theta ^ { K }$ and computes its personalized weights $\pi _ { t _ { \mathrm { n e w } } }$ (starting for example from a uniform initialization) through one $\mathrm { E }$ -step (8) and the first update in the M-step (9).
+
+# 3.3 Fully Decentralized Algorithm
+
+In some cases, clients may want to communicate directly in a peer-to-peer fashion instead of relying on the central server mediation [see 28, Section 2.1]. In fact, fully decentralized schemes may provide stronger privacy guarantees [12] and speed-up training as they better use communication resources [41, 46] and reduce the effect of stragglers [50]. For these reasons, they have attracted significant interest recently in the machine learning community [41, 63, 42, 62, 3, 51, 46, 31]. We refer to [49] for a comprehensive survey of fully decentralized optimization (also known as consensus-based optimization), and to [31] for a unified theoretical analysis of decentralized SGD.
+
+We propose D-FedEM (Alg. 4 in App. D.2), a fully decentralized version of our federated expectation maximization algorithm. As in FedEM, the M-step for $\Theta$ update is replaced by an approximate maximization step consisting of local updates. The global aggregation step in FedEM (Alg. 1, line 10) is replaced by a partial aggregation step, where each client computes a weighted average of its current components and those of a subset of clients (its neighborhood), which may vary over time. The convergence of decentralized optimization schemes requires certain assumptions to guarantee that each client can influence the estimates of other clients over time. In our paper, we consider the general assumption in [31, Assumption 4] (restated as Assumption 8 in App. E for completeness). For instance, this assumption is satisfied if the graph of clients’ communications is strongly connected every $\tau$ rounds.
+
+D-FedEM converges to a stationary point of $f$ (formal statement in App. E and proof in App. G.2).
+
+Theorem 3.3 (Informal). In the same setting of Theorem 3.2 and under the additional Assumption 8, D-FedEM’s individual estimates $( \Theta _ { t } ^ { k } ) _ { 1 \leq t \leq T }$ converge to a common value $\bar { \Theta } ^ { k }$ . Moreover, $\bar { \Theta } ^ { k }$ and $\Pi ^ { k }$ converge to a stationary point of $f$ .
+
+# 3.4 Federated Surrogate Optimization
+
+FedEM and D-FedEM can be seen as particular instances of a more general framework—of potential interest for other applications—that we call federated surrogate optimization.
+
+The standard majorization-minimization principle [34] iteratively minimizes, at each iteration $k$ , a surrogate function $g ^ { k }$ majorizing the objective function $f$ . The work [43] studied this approach when each $\mathbf { \bar { \boldsymbol { g } } ^ { k } }$ is a first-order surrogate of $f$ (the formal definition from [43] is given in App. F.1).
+
+Our novel federated surrogate optimization framework considers that the objective function $f$ is a weighted sum $\begin{array} { r } { f = \sum _ { t = 1 } ^ { T } \hat { \omega } _ { t } f _ { t } } \end{array}$ of $T$ functions and iteratively minimizes $f$ in a distributed fashion using partial first-order surrogates $g _ { t } ^ { k }$ for each function $f _ { t }$ . “Partial” refers to the fact that $g _ { t } ^ { k }$ is not required to be a first order surrogate wrt the whole set of parameters, as defined formally below.
+
+Definition 1 (Partial first-order surrogate). A function $g ( \mathbf { u } , \mathbf { v } ) : \mathbb { R } ^ { d _ { u } } \times \mathcal { V }  \mathbb { R }$ is a partial-first-order surrogate of $f ( { \bf u } , { \bf v } )$ wrt u near $( { \bf u } _ { 0 } , { \bf v } _ { 0 } ) \in R ^ { d _ { u } } \times \mathcal { V }$ when the following conditions are satisfied:
+
+1. $g ( \mathbf { u } , \mathbf { v } ) \geq f ( \mathbf { u } , \mathbf { v } ) .$ for all $\mathbf { u } \in \mathbb { R } ^ { d _ { u } }$ and $\mathbf { v } \in \mathcal { V }$ ;   
+2. $r ( \mathbf { u } , \mathbf { v } ) \triangleq g ( \mathbf { u } , \mathbf { v } ) - f ( \mathbf { u } , \mathbf { v } )$ is differentiable and $L$ -smooth with respect to u. Moreover, we have $r ( { \bf u } _ { 0 } , { \bf v } _ { 0 } ) = 0$ and $\begin{array} { r } { \nabla _ { \mathbf { u } } r ( \mathbf { u } _ { 0 } , \mathbf { v } _ { 0 } ) = 0 . } \end{array}$ .   
+3. $g ( \mathbf { u } , \mathbf { v } _ { 0 } ) - g ( \mathbf { u } , \mathbf { v } ) = d _ { \mathcal { V } } ( \mathbf { v } _ { 0 } , \mathbf { v } ) .$ for all $\mathbf { u } \in \mathbb { R } ^ { d _ { u } }$ and $\begin{array} { r } { \mathbf { v } \in \arg \operatorname* { m i n } _ { \mathbf { v } ^ { \prime } \in \mathcal { V } } g ( \mathbf { u } , \mathbf { v } ^ { \prime } ) } \end{array}$ , where $d _ { \nu }$ is non-negative and $d _ { \mathcal { V } } ( \mathbf { v } , \mathbf { v } ^ { \prime } ) = 0 \iff \mathbf { v } = \mathbf { v } ^ { \prime }$ .
+
+Under the assumption that each client $t$ can compute a partial first-order surrogate of $f _ { t }$ , we propose algorithms for federated surrogate optimization in both the client-server setting (Alg. 3) and the fully decentralized one (Alg. 5) and prove their convergence under mild conditions (App. G.1 and G.2). FedEM and D-FedEM can be seen as particular instances of these algorithms and Theorem. 3.2 and Theorem. 3.3 follow from the more general convergence results for federated surrogate optimization. We can also use our framework to analyze the convergence of other FL algorithms such as pFedMe [16], as we illustrate in App. F.3.
+
+# 4 Experiments
+
+Datasets and models. We evaluated our method on five federated benchmark datasets spanning a wide range of machine learning tasks: image classification (CIFAR10 and CIFAR100 [33]), handwritten character recognition (EMNIST [8] and FEMNIST [7]),5 and language modeling (Shakespeare [7, 47]). Shakespeare dataset (resp. FEMNIST) was naturally partitioned by assigning all lines from the same characters (resp. all images from the same writer) to the same client. We created federated versions of CIFAR10 and EMNIST by distributing samples with the same label across the clients according to a symmetric Dirichlet distribution with parameter 0.4, as in [65]. For CIFAR100, we exploited the availability of “coarse” and “fine” labels, using a two-stage Pachinko allocation method [39] to assign 600 sample to each of the 100 clients, as in [54]. We also evaluated our method on a synthetic dataset verifying Assumptions 1–3. For all tasks, we randomly split each local dataset into training $( 6 0 \% )$ , validation $( 2 0 \% )$ and test $( 2 0 \% )$ sets. Table 1 summarizes datasets, models, and number of clients (more details can be found in App. I.1). Code is available at https://github.com/omarfoq/FedEM.
+
+Other FL approaches. We compared our algorithms with global models trained with FedAvg [47] and FedProx [38] as well as different personalization approaches: a personalized model trained only on the local dataset, FedAvg with local tuning $( \mathtt { F e d A v g } + )$ [27], Clustered FL [56] and pFedMe [16]. For each method and each task, the learning rate and the other hyperparameters were tuned via grid search (details in App. I.2). FedAvg+ updated the local model through a single pass on the local dataset. Unless otherwise stated, the number of components considered by FedEM was $M = 3$ , training occurred over 80 communication rounds for Shakespeare and 200 rounds for all other datasets.
+
+Table 1: Datasets and models (details in App. I.1).   
+
+<table><tr><td>Dataset</td><td>Task</td><td>Clients</td><td>Total samples</td><td>Model</td></tr><tr><td>FEMNIST[7]</td><td>Handwriten character recognition</td><td>539</td><td>120,772</td><td>2-layer CNN + 2-layer FFN</td></tr><tr><td>EMNIST[8]</td><td>Handwritten character recognition</td><td>100</td><td>81,425</td><td>2-layer CNN + 2-layer FFN</td></tr><tr><td>CIFAR10 [33]</td><td>Image classification</td><td>80</td><td>60,000</td><td>MobileNet-v2 [55]</td></tr><tr><td>CIFAR100 [33]</td><td>Image classification</td><td>100</td><td>60,000</td><td>MobileNet-v2 [55]</td></tr><tr><td>Shakespeare [7,47]</td><td>Next-Character Prediction</td><td>778</td><td>4,226,158</td><td>Stacked-LSTM[25]</td></tr><tr><td>Synthetic</td><td>Binary Classification</td><td>300</td><td>1,570,507</td><td>Linear model</td></tr></table>
+
+Table 2: Test accuracy: average across clients / bottom decile.   
+
+<table><tr><td>Dataset</td><td>Local</td><td>FedAvg[47]</td><td>FedProx [38]</td><td>FedAvg+ [27]</td><td>Clustered FL [56]</td><td>pFedMe [16]</td><td>FedEM (Ours)</td></tr><tr><td>FEMNIST</td><td>71.0/57.5</td><td>78.6/63.9</td><td>78.9 /64.0</td><td>75.3/53.0</td><td>73.5/55.1</td><td>74.9/57.6</td><td>79.9/64.8</td></tr><tr><td>EMNIST</td><td>71.9/64.3</td><td>82.6/75.0</td><td>83.0/75.4</td><td>83.1/75.8</td><td>82.7/75.0</td><td>83.3/76.4</td><td>83.5/76.6</td></tr><tr><td>CIFAR10</td><td>70.2/48.7</td><td>78.2/72.4</td><td>78.0/70.8</td><td>82.3/70.6</td><td>78.6/71.2</td><td>81.7/73.6</td><td>84.3 /78.1</td></tr><tr><td>CIFAR100</td><td>31.5/19.9</td><td>40.9/33.2</td><td>41.0/33.2</td><td>39.0/28.3</td><td>41.5/34.1</td><td>41.8/32.5</td><td>44.1/35.0</td></tr><tr><td>Shakespeare</td><td>32.0/16.6</td><td>46.7/42.8</td><td>45.7/41.9</td><td>40.0/25.5</td><td>46.6/42.7</td><td>41.2/36.8</td><td>46.7/43.0</td></tr><tr><td>Synthetic</td><td>65.7/58.4</td><td>68.2/58.9</td><td>68.2/59.0</td><td>68.9/60.2</td><td>69.1/59.0</td><td>69.2/61.2</td><td>74.7/66.7</td></tr></table>
+
+At each round, clients train for one epoch. Results for D-FedEM are in App. J.1. A comparison with MOCHA [59], which can only train linear models, is presented in App. J.2.
+
+Average performance of personalized models. The performance of each personalized model (which is the same for all clients in the case of FedAvg and FedProx) is evaluated on the local test dataset (unseen at training). Table 2 shows the average weighted accuracy with weights proportional to local dataset sizes. We observe that FedEM obtains the best performance across all datasets.
+
+Fairness across clients. FedEM’s improvement in terms of average accuracy could be the result of learning particularly good models for some clients at the expense of bad models for other clients. Table 2 shows the bottom decile of the accuracy of local models, i.e., the $( T / 1 0 )$ -th worst accuracy (the minimum accuracy is particularly noisy, notably because some local test datasets are very small). Even clients with the worst personalized models are still better off when FedEM is used for training.
+
+Clients sampling. In cross-device federated learning, only a subset of clients may be available at each round. We ran CIFAR10 experiments with different levels of participation: at each round a given fraction of all clients were sampled uniformly without replacement. We restrict the comparison to FedEM and ${ \mathrm { F e d A v g } } +$ , as 1) ${ \mathrm { F e d A v g } } +$ performed better than FedProx and FedAvg in the previous CIFAR10 experiments, 2) it is not clear how to extend pFedMe and Clustered FL to handle client sampling. Results in Fig. 1 (left) show that FedEM is more robust to low clients’ participation levels. We provide additional results on client sampling, including a comparison with APFL [14], in App. J.6.
+
+Generalization to unseen clients. As discussed in Section 3.2, FedEM allows new clients arriving after the distributed training to easily learn their personalized models. With the exception of ${ \mathrm { F e d A v g } } +$ , it is not clear how the other personalized FL algorithms should be extended to tackle the same goal (see discussion in App. J.3). In order to evaluate the quality of new clients’ personalized models, we performed an experiment where only $80 \%$ of the clients (“old” clients) participate to the training. The remaining $20 \%$ join the system in a second phase and use their local training datasets to learn their personalized weights. Table 3 shows that FedEM allows new clients to learn a personalized model at least as good as FedAvg’s global one and always better than FedAvg+’s one. Unexpectedly, new clients achieve sometimes a significantly higher test accuracy than old clients (e.g., $4 7 . 5 \%$ against $4 4 . 1 \%$ on CIFAR100). Our investigation in App. J.3 suggests that, by selecting their mixture weights on local datasets that were not used to train the components, new clients can compensate for potential overfitting in the initial training phase. We also investigate in App. J.3 the effect of the local dataset size on the accuracy achieved by unseen clients, showing that personalization is effective even when unseen clients have small datasets.
+
+Effect of $M$ . A limitation of FedEM is that each client needs to update and transmit $M$ components at each round, requiring roughly $M$ times more computation and $M$ times larger messages. Nevertheless, the number of components to consider in practice is quite limited. We used $M = 3$ in our previous experiments, and Fig. 1 (right) shows that larger values do not yield much improvement and $M = 2$ already provides a significant level of personalization. In all experiments above, the number of communication rounds allowed all approaches to converge. As a consequence, even if other methods trained over $M = 3$ times more rounds—in order to have as much computation and communication as FedEM—the conclusions would not change. As a final experiment, we considered a time-constrained setting, where FedEM is limited to run one third $( = 1 / M )$ ) of the rounds (Table 7 in App. J.5). Even if FedEM does not reach its maximum accuracy, it still outperforms the other methods on 3 datasets.
+
+Table 3: Average test accuracy across clients unseen at training (train accuracy in parenthesis).   
+
+<table><tr><td>Dataset</td><td>FedAvg [47]</td><td>FedAvg+[27]</td><td>FedEM (Ours)</td></tr><tr><td>FEMNIST</td><td>78.3(80.9)</td><td>74.2 (84.2)</td><td>79.1 (81.5)</td></tr><tr><td>EMNIST</td><td>83.4(82.7)</td><td>83.7(92.9)</td><td>84.0(83.3)</td></tr><tr><td>CIFAR10</td><td>77.3(77.5)</td><td>80.4 (80.5)</td><td>85.9 (90.7)</td></tr><tr><td>CIFAR100</td><td>41.1 (42.1)</td><td>36.5 (55.3)</td><td>47.5 (46.6)</td></tr><tr><td>Shakespeare</td><td>46.7 (47.1)</td><td>40.2 (93.0)</td><td>46.7 (46.6)</td></tr><tr><td>Synthetic</td><td>68.6(70.0)</td><td>69.1(72.1)</td><td>73.0 (74.1)</td></tr></table>
+
+![](images/eeb1beb065cef85e985d5f9666ab0cf36430081c9d877b100da90cd7e8b773ff.jpg)  
+Figure 1: Effect of client sampling rate (left) and FedEM number of mixture components $M$ (right) on the test accuracy for CIFAR10 [33].
+
+# 5 Conclusion
+
+In this paper, we proposed a novel federated MTL approach based on the flexible assumption that local data distributions are mixtures of underlying distributions. Our EM-like algorithms allow clients to jointly learn shared component models and personalized mixture weights in client-server and fully decentralized settings. We proved convergence guarantees for our algorithms through a general federated surrogate optimization framework which can be used to analyze other FL formulations. Extensive empirical evaluation shows that our approach learns models with higher accuracy and fairness than state-of-the-art FL algorithms, even for clients not present at training time.
+
+In future work, we aim to reduce the local computation and communication of our algorithms. Aside from standard compression schemes [22], a promising direction is to limit the number of component models that a client updates/transmits at each step. This could be done in an adaptive manner based on the client’s current mixture weights. A simultaneously published work [15] proposes a federated EM algorithm (also called FedEM), which does not address personalization but reduces communication requirements by compressing appropriately defined complete data sufficient statistics.
+
+A second interesting research direction is to study personalized FL approaches under privacy constraints (quite unexplored until now with the notable exception of [3]). Some features of our algorithms may be beneficial for privacy (e.g., the fact that personalized weights are kept locally and that all users contribute to all shared models). We hope to design differentially private versions of our algorithms and characterize their privacy-utility trade-offs.
+
+# 6 Acknowledgements
+
+This work has been supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002, and through grants ANR-16-CE23-0016 (Project PAMELA) and ANR-20- CE23-0015 (Project PRIDE). The authors are grateful to the OPAL infrastructure from Université Côte d’Azur for providing computational resources and technical support.
+
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+        "text": "Federated Multi-Task Learning under a Mixture of Distributions ",
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+        "text": "Othmane Marfoq1,3, Giovanni Neglia1, Aurélien Bellet2, Laetitia Kameni3, and Richard Vidal3 ",
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+        "text": "1Inria, Université Côte d’Azur, France, {othmane.marfoq, giovanni.neglia} $@$ inria.fr 2Inria, Université de Lille, France, aurelien.bellet $@$ inria.fr 3Accenture Labs, France, {richard.vidal, laetitia.kameni} $@$ accenture.com ",
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+        "text": "Abstract ",
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+        "text": "The increasing size of data generated by smartphones and IoT devices motivated the development of Federated Learning (FL), a framework for on-device collaborative training of machine learning models. First efforts in FL focused on learning a single global model with good average performance across clients, but the global model may be arbitrarily bad for a given client, due to the inherent heterogeneity of local data distributions. Federated multi-task learning (MTL) approaches can learn personalized models by formulating an opportune penalized optimization problem. The penalization term can capture complex relations among personalized models, but eschews clear statistical assumptions about local data distributions. ",
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+        "text": "In this work, we propose to study federated MTL under the flexible assumption that each local data distribution is a mixture of unknown underlying distributions. This assumption encompasses most of the existing personalized FL approaches and leads to federated EM-like algorithms for both client-server and fully decentralized settings. Moreover, it provides a principled way to serve personalized models to clients not seen at training time. The algorithms’ convergence is analyzed through a novel federated surrogate optimization framework, which can be of general interest. Experimental results on FL benchmarks show that our approach provides models with higher accuracy and fairness than state-of-the-art methods. ",
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+        "text": "1 Introduction ",
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+        "text": "Federated Learning (FL) [28] allows a set of clients to collaboratively train models without sharing their local data. Standard FL approaches train a unique model for all clients [47, 32, 38, 29, 48]. However, as discussed in [56], the existence of such a global model suited for all clients is at odds with the statistical heterogeneity observed across different clients [37, 28]. Indeed, clients can have non-iid data and varying preferences. Consider for example a language modeling task: given the sequence of tokens $^ { 6 6 } I$ love eating,” the next word can be arbitrarily different from one client to another. Thus, having personalized models for each client is a necessity in many FL applications. ",
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+        "text": "Previous work on personalized FL. A naive approach for FL personalization consists in learning first a global model and then fine-tuning its parameters at each client via a few iterations of stochastic gradient descent [58]. In this case, the global model plays the role of a meta-model to be used as initialization for few-shot adaptation at each client. In particular, the connection between FL and Model Agnostic Meta Learning (MAML) [27] has been studied in [19, 30, 1] in order to build a more suitable meta-model for local personalization. Unfortunately, these methods can fail to build a model with low generalization error (as exemplified by LEAF synthetic dataset [7, App. 1]). An alternative approach is to jointly train a global model and one local model per client and then let each client build a personalized model by interpolating them [14, 9, 44]. However, if local distributions are far from the average distribution, a relevant global model does not exist and this approach boils down to every client learning only on its own local data. This issue is formally captured by the generalization bound in [14, Theorem 1]. ",
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+        "text": "Clustered FL [56, 20, 44] addresses the potential lack of a global model by assuming that clients can be partitioned into several clusters. Clients belonging to the same cluster share the same optimal model, but those models can be arbitrarily different across clusters (see [56, Assumption 2] for a rigorous formulation). During training, clients learn the cluster to which they belong as well as the cluster model. The Clustered FL assumption is also quite limiting, as no knowledge transfer is possible across clusters. In the extreme case where each client has its own optimal local model (recall the example on language modeling), the number of clusters coincides with the number of clients and no federated learning is possible. ",
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+        "text": "Multi-Task Learning (MTL) has recently emerged as an alternative approach to learn personalized models in the federated setting and allows for more nuanced relations among clients’ models [59, 63, 67, 24, 16]. The authors of [59, 63] were the first to frame FL personalization as a MTL problem. In particular, they defined federated MTL as a penalized optimization problem, where the penalization term models relationships among tasks (clients). The work [59] proposed the MOCHA algorithm for the client-server scenario, while [63, 67] presented decentralized algorithms for the same problem. Unfortunately, these algorithms can only learn simple models (linear models or linear combination of pre-trained models), because of the complex penalization term. Other MTL-based approaches [24, 23, 16, 26, 36] are able to train more general models at the cost of considering simpler penalization terms (e.g., the distance to the average model), thereby losing the capability to capture complex relations among tasks. Moreover, a general limitation of this line of work is that the penalization term is justified qualitatively and not on the basis of clear statistical assumptions on local data distributions. ",
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+        "text": "More recently, [57] proposed pFedHN. pFedHN feeds local clients’ representations to a global (across clients) hypernetwork, which can output personalized heterogeneous models. Unfortunately, the hypernetwork has a large memory footprint already for small clients’ models (e.g., the hypernetwork in the experiments in [57] has 100 more parameters than the output model). Hence, it is not clear if pFedHN can scale to more complex models. Moreover, pFedHN requires each client to communicate multiple times for the server to learn meaningful representations. Therefore, its performance is likely to deteriorate when clients participate only once (or few times) to training, as it is the case for large-scale cross-device FL training. Furthermore, even once the hypernetwork parameters have been learned, training personalized models for new clients still requires multiple client-server communication rounds. More similar to our approach, FedFOMO [68] lets each client interpolate other clients’ local models with opportune weights learned during training. However, this method lacks both theoretical justifications for such linear combinations and convergence guarantees. Moreover, FedFOMO requires the presence of a powerful server able to 1) store all individual local models and 2) learn for each client—through repeated interactions—which other clients’ local models may be useful. Therefore, FedFOMO is not suited for cross-device FL where the number of clients may be very large (e.g., $\\mathrm { 1 0 ^ { 5 } - 1 0 ^ { 7 } }$ participating clients [28, Table 2]) and a given client may only participate in a single training round. ",
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+        "type": "text",
+        "text": "Overall, although current personalization approaches can lead to superior empirical performance in comparison to a shared global model or individually trained local models, it is still not well understood whether and under which conditions clients are guaranteed to benefit from collaboration. ",
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+        "type": "text",
+        "text": "Our contributions. In this work, we first show that federated learning is impossible without assumptions on local data distributions. Motivated by this negative result, we formulate a general and flexible assumption: the data distribution of each client is a mixture of $M$ underlying distributions. The proposed formulation has the advantage that each client can benefit from knowledge distilled from all other clients’ datasets (even if any two clients can be arbitrarily different from each other). We also show that this assumption encompasses most of the personalized FL approaches previously proposed in the literature. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In our framework, a personalized model is a linear combination of $M$ shared component models. All clients jointly learn the $M$ components, while each client learns its personalized mixture weights. We show that federated EM-like algorithms can be used for training. In particular, we propose FedEM and D-FedEM for the client-server and the fully decentralized settings, respectively, and we prove convergence guarantees. Our approach also provides a principled and efficient way to infer personalized models for clients unseen at training time. Our algorithms can easily be adapted to solve more general problems in a novel framework, which can be seen as a federated extension of the centralized surrogate optimization approach in [43]. To the best of our knowledge, our paper is the first work to propose federated surrogate optimization algorithms with convergence guarantees. ",
+        "bbox": [
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Through extensive experiments on FL benchmark datasets, we show that our approach generally yields models that 1) are on average more accurate, 2) are fairer across clients, and 3) generalize better to unseen clients than state-of-the-art personalized and non-personalized FL approaches. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Paper outline. The rest of the paper is organized as follows. In Section 2 we provide our impossibility result, introduce our main assumptions, and show that several popular personalization approaches can be obtained as special cases of our framework. Section 3 describes our algorithms, states their convergence results, and presents our general federated surrogate optimization framework. Finally, we provide experimental results in Section 4 before concluding in Section 5. ",
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+    {
+        "type": "text",
+        "text": "2 Problem Formulation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We consider a (countable) set $\\tau$ of classification (or regression) tasks which represent the set of possible clients. We will use the terms task and client interchangeably. Data at client $t \\in \\mathcal { T }$ is generated according to a local distribution $\\mathcal { D } _ { t }$ over $\\mathcal { X } \\times \\mathcal { V }$ . Local data distributions $\\{ \\mathcal { D } _ { t } \\} _ { t \\in \\mathcal { T } }$ are in general different, thus it is natural to fit a separate model (hypothesis) $h _ { t } \\in { \\mathcal { H } }$ to each data distribution $\\mathcal { D } _ { t }$ . The goal is then to solve (in parallel) the following optimization problems ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e8f811405be16a3a761f678ec9fae1ce2d3573bcdd5d650ee94e3f09584de98a.jpg",
+        "text": "$$\n\\forall t \\in { \\mathcal { T } } , \\quad \\operatorname* { m i n i m i z e } _ { h _ { t } \\in { \\mathcal { H } } } { \\mathcal { L } } _ { { \\mathcal { D } } _ { t } } ( h _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $h _ { t } : \\mathcal { X } \\mapsto \\Delta ^ { | \\mathcal { V } | }$ $\\Delta ^ { D }$ denoting the unitary simplex of dimension $D$ ), $l : \\Delta ^ { | y | } \\times \\mathcal { Y } \\mapsto \\mathbb { R } ^ { + }$ is a loss function,1 and $\\mathcal { L } _ { \\mathcal { D } _ { t } } ( h _ { t } ) = \\mathbb { E } _ { ( \\mathbf { x } , y ) \\sim \\mathcal { D } _ { t } } ^ { - } \\left[ l ( \\bar { h } _ { t } ( \\mathbf { x } ) , \\bar { y } ) \\right]$ is the true risk of a model $h _ { t }$ under data distribution $\\mathcal { D } _ { t }$ . For $( \\mathbf { x } , y ) \\in \\mathcal { X } \\times \\mathcal { Y }$ , we will denote the joint distribution density associated to $\\mathcal { D } _ { t }$ by $p _ { t } ( \\mathbf { x } , y )$ , and the marginal densities by $p _ { t } ( \\mathbf { x } )$ and $p _ { t } ( y )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A set of $T$ clients $[ T ] \\triangleq \\{ 1 , 2 , . . . T \\} \\subseteq \\mathcal { T }$ participate to the initial training phase; other clients may e system in a late drawn i.i.d. from age. We, and by $S _ { t } = \\{ s _ { t } ^ { ( i ) } = ( \\mathbf { x } _ { t } ^ { ( i ) } , \\ y _ { t } ^ { ( i ) } ) \\} _ { i = 1 } ^ { n _ { t } }$ the dataset at client $t \\in [ T ]$ $\\mathcal { D } _ { t }$ $\\begin{array} { r } { n = \\sum _ { t = 1 } ^ { T } n _ { t } } \\end{array}$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The idea of federated learning is to enable each client to benefit from data samples available at other clients in order to get a better estimation of $\\mathcal { L } _ { \\mathcal { D } _ { t } }$ , and therefore get a model with a better generalization ability to unseen examples. ",
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+        "type": "text",
+        "text": "2.1 An Impossibility Result ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We start by showing that some assumptions on the local distributions $p _ { t } ( \\mathbf x , y ) , t \\in \\mathcal T$ are needed for federated learning to be possible, i.e., for each client to be able to take advantage of the data at other clients. This holds even if all clients participate to the initial training phase (i.e., $\\mathcal { T } = [ T ] )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider the classic PAC learning framework where we fix a class of models $\\mathcal { H }$ and seek a learning algorithm which is guaranteed, for all possible data distributions over $\\mathcal { X } \\times \\mathcal { V }$ , to return with high probability a model with expected error $\\epsilon$ -close to the best possible error in the class $\\mathcal { H }$ . The worst-case sample complexity then refers to the minimum amount of labeled data required by any algorithm to reach a given $\\epsilon$ -approximation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our impossibility result for FL is based on a reduction to an impossibility result for Semi-Supervised Learning (SSL), which is the problem of learning from a training set with only a small amount of labeled data. The authors of [4] conjectured that, when the quantity of unlabeled data goes to infinity, the worst-case sample complexity of SSL improves over supervised learning at most by a constant factor that only depends on the hypothesis class [4, Conjecture 4]. This conjecture was later proved for the realizable case and hypothesis classes of finite VC dimension [13, Theorem 1], even when the marginal distribution over the domain set $\\mathcal { X }$ is known [21, Theorem 2]. 2 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the context of $\\mathrm { F L }$ , if the marginal distributions $p _ { t } \\left( \\mathbf { x } \\right)$ are identical, but the conditional distributions $p _ { t } \\left( y | \\mathbf { x } \\right)$ can be arbitrarily different, then each client $t$ can learn using: 1) its own local labeled dataset, and 2) the other clients’ datasets, but only as unlabeled ones (because their labels have no relevance for $t$ ). The FL problem, with $T$ clients, then reduces to $T$ parallel SSL problems, or more precisely, it is at least as difficult as $T$ parallel SSL problems (because client $t$ has no direct access to the other local datasets but can only learn through the communication exchanges allowed by the FL algorithm). The SSL impossibility result implies that, without any additional assumption on the local distributions $p _ { t } \\left( \\mathbf { x } , y \\right)$ , $t \\in [ T ]$ , any FL algorithm can reduce the sample complexity of client- $\\mathbf { \\nabla } \\cdot t$ ’s problem in (1) only by a constant in comparison to local learning, independently of how many other clients participate to training and how large their datasets’ sizes are. ",
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+    {
+        "type": "text",
+        "text": "2.2 Learning under a Mixture Model ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Motivated by the above impossibility result, in this work we propose to consider that each local data distribution $\\mathcal { D } _ { t }$ is a mixture of $M$ underlying distributions $\\tilde { \\mathcal { D } } _ { m }$ , $1 \\leq m \\leq M$ , as formalized below. ",
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+        "type": "text",
+        "text": "Assumption 1. There exist $M$ underlying (independent) distributions $\\tilde { \\mathcal { D } } _ { m }$ , $1 \\leq m \\leq M$ , such that for $t \\in \\tau$ , $\\mathcal { D } _ { t }$ is mixture of the distributions $\\{ \\tilde { \\mathcal { D } } _ { m } \\} _ { m = 1 } ^ { M }$ with weights $\\pi _ { t } ^ { * } = [ \\pi _ { t 1 } ^ { * } , \\ldots , \\pi _ { t M } ^ { * } ] \\in \\Delta ^ { M }$ , i.e. ",
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+    {
+        "type": "equation",
+        "img_path": "images/cc3c420a261ad6c50863639630a2d5e8c2608641518f304bec26ecedcf026ae4.jpg",
+        "text": "$$\nz _ { t } \\sim \\mathcal { M } ( \\pi _ { t } ^ { * } ) , \\quad ( ( \\mathbf { x } _ { t } , y _ { t } ) | z _ { t } = m ) \\sim \\tilde { \\mathcal { D } } _ { m } , \\quad \\forall t \\in \\mathcal { T } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { M } ( \\pi )$ is a multinomial (categorical) distribution with parameters $\\pi$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similarly to what was done above, we use $p _ { m } ( \\mathbf { x } , y )$ , $p _ { m } ( \\mathbf { x } )$ , and $p _ { m } ( y )$ to denote the probability distribution densities associated to $\\tilde { \\mathcal { D } } _ { m }$ . We further assume that marginals over $\\mathcal { X }$ are identical. ",
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+    {
+        "type": "text",
+        "text": "Assumption 2. For all $m \\in [ M ]$ , we have $p _ { m } ( \\mathbf { x } ) = p ( \\mathbf { x } )$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "Assumption 2 is not strictly required for our analysis to hold, but, in the most general case, solving Problem (1) requires to learn generative models. Instead, under Assumption 2 we can restrict our attention to discriminative models (e.g., neural networks). 3 More specifically, we consider a parameterized set of models $\\tilde { \\mathcal { H } }$ with the following properties. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Assumption 3. $\\tilde { \\mathcal { H } } = \\{ h _ { \\theta } \\} _ { \\theta \\in \\mathbb { R } ^ { d } }$ is a set of hypotheses parameterized by $\\theta \\in \\mathbb { R } ^ { d }$ , whose convex hull is in $\\mathcal { H }$ . For each distribution $\\tilde { \\mathcal { D } } _ { m }$ with $m \\in [ M ]$ , there exists a hypothesis $h _ { \\theta _ { m } ^ { * } }$ , such that ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/606bc6112d74a803717a3cbc8e9e7bab02855d4a3b6f5e928b483d8a06d9e441.jpg",
+        "text": "$$\n\\begin{array} { r } { l \\left( h _ { \\theta _ { m } ^ { * } } ( \\mathbf { x } ) , y \\right) = - \\log p _ { m } ( y | \\mathbf { x } ) + c , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $c \\in \\mathbb { R }$ is a normalization constant. The function $l ( \\cdot , \\cdot )$ is then the log-loss associated to $p _ { m } ( y | \\mathbf { x } )$ . ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We refer to the hypotheses in $\\tilde { \\mathcal { H } }$ as component models or simply components. We denote by $\\Theta ^ { * } \\in \\mathbb { R } ^ { M \\times d }$ the matrix whose $m$ -th row is $\\theta _ { m } ^ { * }$ , and by $\\Pi ^ { * } \\in \\Delta ^ { T \\times } { \\bar { M } }$ the matrix whose $t$ -th row is $\\pi _ { t } ^ { * } \\in \\Delta ^ { M }$ . Similarly, we will use $\\Theta$ and $\\Pi$ to denote arbitrary parameters. ",
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+    {
+        "type": "text",
+        "text": "Remark 1. Assumptions 2–3 are mainly technical and are not required for our approach to work in practice. Experiments in Section 4 show that our algorithms perform well on standard $F L$ benchmark datasets, for which these assumptions do not hold in general. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that, under the above assumptions, $p _ { t } ( \\mathbf { x } , y )$ depends on $\\Theta ^ { * }$ and $\\pi _ { t } ^ { * }$ . Moreover, we can prove (see App. A) that the optimal local model $h _ { t } ^ { * } \\in \\mathcal { H }$ for client $t$ is a weighted average of models in $\\tilde { \\mathcal { H } }$ . Proposition 2.1. Let $l ( \\cdot , \\cdot )$ be the mean squared error loss, the logistic loss or the cross-entropy loss, and $\\breve { \\Theta }$ and $\\breve { \\Pi }$ be a solution of the following optimization problem: ",
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+    {
+        "type": "equation",
+        "img_path": "images/30e495234e39211e76c83e167f42f959a934bec2385d2488f4811926f025f7ef.jpg",
+        "text": "$$\n\\underset { \\Theta , \\Pi } { \\mathrm { m i n i m i z e } } \\underset { t \\sim D _ { T } } { \\mathbb { E } } \\underset { ( \\mathbf { x } , y ) \\sim D _ { t } } { \\mathbb { E } } \\left[ - \\log p _ { t } ( \\mathbf { x } , y | \\Theta , \\pi _ { t } ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $D _ { \\mathcal { T } }$ is any distribution with support $\\tau$ . Under Assumptions $^ { l }$ , 2, and $^ 3$ , the predictors ",
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+        "type": "equation",
+        "img_path": "images/1e8170c696c7f3bbf8f288b25c6e3947ee1bc6c4128b596d68dfc2ef24f3e804.jpg",
+        "text": "$$\nh _ { t } ^ { * } = \\sum _ { m = 1 } ^ { M } \\breve { \\pi } _ { t m } h _ { { \\breve { \\theta } } _ { m } } \\left( \\mathbf { x } \\right) , \\quad \\forall t \\in \\mathcal { T }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "minimize $\\mathbb { E } _ { ( \\mathbf { x } , y ) \\sim \\mathcal { D } _ { t } } \\left[ l ( h _ { t } ( \\mathbf { x } ) , y ) \\right]$ and thus solve Problem (1). ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Proposition 2.1 suggests the following approach to solve Problem (1). First, we estimate the parameters $\\breve { \\Theta }$ and $\\breve { \\pi } _ { t }$ , $1 \\leq t \\leq T$ , by minimizing the empirical version of Problem (4) on the training data, i.e., minimizing: ",
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/36625e7bc57423255b4176cfb6f6fa9a9720fbc6ea0c9daacdaa6acaafb5220b.jpg",
+        "text": "$$\nf ( \\Theta , \\Pi ) \\triangleq - \\frac { \\log p ( \\mathcal { S } _ { 1 : T } | \\Theta , \\Pi ) } { n } \\triangleq - \\frac { 1 } { n } \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n _ { t } } \\log p ( s _ { t } ^ { ( i ) } | \\Theta , \\pi _ { t } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "which is the (negative) likelihood of the probabilistic model (2). 4 Second, we use (5) to get the client predictor for the $T$ clients present at training time. Finally, to deal with a client $t _ { \\mathrm { n e w } } \\notin [ T ]$ not seen during training, we keep the mixture component models fixed and simply choose the weights $\\pi _ { t _ { \\mathrm { n e w } } }$ that maximize the likelihood of the client data and get the client predictor via (5). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.3 Generalizing Existing Frameworks ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Before presenting our federated learning algorithms in Section 3, we show that the generative model in Assumption 1 extends some popular multi-task/personalized FL formulations in the literature. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Clustered Federated Learning [56, 20] assumes that each client belongs to one among $C$ clusters and proposes that all clients in the same cluster learn the same model. Our framework recovers this scenario considering $M = C$ and $\\pi _ { t c } ^ { * } = 1$ if task (client) $t$ is in cluster $c$ and $\\pi _ { t c } ^ { * } = 0$ otherwise. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Personalization via model interpolation [44, 14] relies on learning a global model $h _ { \\mathrm { g l o b } }$ and $T$ local models $h _ { \\mathrm { l o c } , t }$ , and then using at each client the linear interpolation $h _ { t } = \\alpha _ { t } h _ { \\mathrm { l o c } , t } + ( 1 - \\alpha _ { t } ) h _ { \\mathrm { g l o b } }$ . Each client model can thus be seen as a linear combination of $M = T + 1$ models $h _ { m } = h _ { { \\mathrm { l o c } } , m }$ for $m \\in [ T ]$ and $h _ { 0 } = h _ { \\mathrm { g l o b } }$ with specific weights $\\pi _ { t t } ^ { * } = \\alpha _ { t }$ , $\\pi _ { t 0 } ^ { * } = 1 - \\alpha _ { t }$ , and $\\pi _ { t t ^ { \\prime } } ^ { * } = 0$ for $t ^ { \\prime } \\in [ T ] \\backslash \\{ t \\}$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Federated MTL via task relationships. The authors of [59] proposed to learn personalized models by solving the following optimization problem inspired from classic MTL formulations: ",
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/436643fd2c8b17156e10849bfa53423a742a55fdfafe0ae42218567f1cd852df.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { W , \\Omega } \\ \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n _ { t } } l ( h _ { w _ { t } } ( \\mathbf { x } _ { t } ^ { ( i ) } ) , y _ { t } ^ { ( i ) } ) + \\lambda \\mathrm { t r } \\left( W \\Omega W ^ { \\intercal } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            463,
+            661,
+            507
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $h _ { w _ { t } }$ are linear predictors parameterized by the rows of matrix $W$ and the matrix $\\Omega$ captures task relationships (similarity). This formulation is motivated by the alternating structure optimization method (ASO) [2, 70]. In App. B, we show that, when predictors $h _ { \\theta _ { m } ^ { * } }$ are linear and have bounded norm, our framework leads to the same ASO formulation that motivated Problem (7). Problem (7) can also be justified by probabilistic priors [69] or graphical models [35] (see [59, App. B.1]). Similar considerations hold for our framework (see again App. B). Reference [67] extends the approach in [59] by letting each client learn a personalized model as a weighted combination of $M$ known hypotheses. Our approach is more general and flexible as clients learn both the weights and the hypotheses. Finally, other personalized FL algorithms, like pFedMe [16], FedU [17], and those studied in [24] and in [23], can be framed as special cases of formulation (7). Their assumptions can thus also be seen as a particular case of our framework. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Federated Expectation-Maximization ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 Centralized Expectation-Maximization ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to estimate the optimal components’ parameters $\\Theta ^ { * } \\ = \\ ( \\theta _ { m } ^ { * } ) _ { 1 \\leq m \\leq M }$ and mixture weights $\\Pi ^ { * } = ( \\pi _ { t } ^ { * } ) _ { 1 \\leq t \\leq T }$ by minimizing the negative log-likelihood $f ( \\Theta , \\bar { \\Pi } )$ in (6). A natural approach to solve such non-convex problems is the Expectation-Maximization algorithm (EM), which alternates between two steps. Expectation steps update the distribution (denoted by $q _ { t }$ ) over the latent variables $\\boldsymbol { z } _ { t } ^ { ( i ) }$ for every data point $s _ { t } ^ { ( i ) } = ( \\mathbf { x } _ { t } ^ { ( i ) } , y _ { t } ^ { ( i ) } )$ given the current estimates of the parameters $\\{ \\Theta , \\Pi \\}$ . Maximization steps update the parameters $\\{ \\Theta , \\Pi \\}$ by maximizing the expected log-likelihood, where the expectation is computed according to the current latent variables’ distributions. ",
+        "bbox": [
+            173,
+            737,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The following proposition provides the EM updates for our problem (proof in App. C). ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 3.1. Under Assumptions $I$ and 2, at the $k$ -th iteration the EM algorithm updates parameter estimates through the following steps: ",
+        "bbox": [
+            173,
+            90,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4c6ce84cc6741df84535fa5b7ae5ad6b57edbbb9b9fdfabcbce3634fef444ce0.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { q _ { t } ^ { k + 1 } ( z _ { t } ^ { ( i ) } = m ) \\propto \\pi _ { t m } ^ { k } \\cdot \\exp ( - l ( h _ { \\theta _ { m } ^ { k } } ( \\mathbf { x } _ { t } ^ { ( i ) } ) , y _ { t } ^ { ( i ) } ) ) , } } & { t \\in [ T ] , m \\in [ M ] , i \\in [ n _ { t } ] } \\\\ & { } & \\\\ & { \\pi _ { t m } ^ { k + 1 } = \\frac { \\sum _ { i = 1 } ^ { n _ { t } } q _ { t } ^ { k + 1 } ( z _ { t } ^ { ( i ) } = m ) } { n _ { t } } , } & { t \\in [ T ] , m \\in [ M ] } \\\\ & { } & \\\\ & { \\theta _ { m } ^ { k + 1 } \\in \\arg \\operatorname* { m i n } _ { \\theta \\in \\mathbb { R } ^ { d } } \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n _ { t } } q _ { t } ^ { k + 1 } ( z _ { t } ^ { ( i ) } = m ) l \\big ( h _ { \\theta } ( \\mathbf { x } _ { t } ^ { ( i ) } ) , y _ { t } ^ { ( i ) } \\big ) , } & { m \\in [ M ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The EM updates in Proposition 3.1 have a natural interpretation. In the $\\mathrm { E }$ -step, given current component models $\\Theta ^ { k }$ and mixture weights $\\Pi ^ { k }$ , (8) updates the a-posteriori probability $q _ { t } ^ { k + 1 } ( z _ { t } ^ { ( i ) } =$ $m$ ) that point $s _ { t } ^ { ( i ) }$ of client $t$ was drawn from the $m$ -th distribution based on the current mixture weight $\\pi _ { t m } ^ { k }$ and on how well the corresponding component $\\theta _ { m } ^ { k }$ classifies $s _ { t } ^ { ( i ) }$ . The M-step consists of two updates under fixed probabilities $q _ { t } ^ { k + 1 }$ . First, (9) updates the mixture weights $\\pi _ { t } ^ { k + 1 }$ to reflect the prominence of each distribution $\\tilde { \\mathcal { D } } _ { m }$ in $S _ { t }$ as given by $q _ { t } ^ { k + 1 }$ . Finally, (10) updates the components’ parameters $\\Theta ^ { k + 1 }$ by solving $M$ independent, weighted empirical risk minimization problems with weights given by $q _ { t } ^ { k + 1 }$ . These weights aim to construct an unbiased estimate of the true risk over each underlying distribution $\\tilde { \\mathcal { D } } _ { m }$ using only points sampled from the client mixtures, similarly to importance sampling strategies used to learn from data with sample selection bias [61, 11, 10, 64]. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.2 Client-Server Algorithm ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            416,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Federated learning aims to train machine learning models directly on the clients, without exchanging raw data, and thus we should run EM while assuming that only client $t$ has access to dataset $S _ { t }$ . The E-step (8) and the $\\Pi$ update (9) in the M-step operate separately on each local dataset $S _ { t }$ and can thus be performed locally at each client $t$ . On the contrary, the $\\Theta$ update (10) requires interaction with other clients, since the computation spans all data samples $\\mathcal { S } _ { 1 : T }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we consider a client-server setting, in which each client $t$ can communicate only with a centralized server (the orchestrator) and wants to learn components’ parameters $\\Theta ^ { * } = \\left( \\theta _ { m } ^ { * } \\right) _ { 1 \\leq m \\leq M }$ and its own mixture weights $\\pi _ { t } ^ { * }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We propose the algorithm FedEM for Federated Expectation-Maximization (Alg. 1). FedEM proceeds through communication rounds similarly to most FL algorithms including FedAvg [47], FedProx [38], SCAFFOLD [29], and pFedMe [16]. At each round, 1) the central server broadcasts the (shared) component models to the clients, 2) each client locally updates components and its personalized mixture weights, and 3) sends the updated components back to the server, 4) the server aggregates the updates. The local update performed at client $t$ consists in performing the steps in (8) and (9) and updating the local estimates of $\\theta _ { m }$ through a solver which approximates the exact minimization in (10) using only the local dataset $S _ { t }$ (see line 7). FedEM can operate with different local solvers—even different across clients—as far as they satisfy some local improvement guarantees (see the discussion in App. H). In what follows, we restrict our focus on the practically important case where the local solver performs multiple stochastic gradient descent updates (local SGD [60]). Under the following standard assumptions (see e.g., [66]), FedEM converges to a stationary point of $f$ . Below, we use the more compact notation $l ( \\boldsymbol { \\theta } ; s _ { t } ^ { ( i ) } ) \\triangleq l ( h _ { \\boldsymbol { \\theta } } ( \\mathbf { x } _ { t } ^ { ( i ) } ) , y _ { t } ^ { ( i ) } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 4. The negative log-likelihood $f$ is bounded below by $f ^ { * } \\in \\mathbb { R }$ . ",
+        "bbox": [
+            176,
+            752,
+            669,
+            768
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 5. (Smoothness) For all $t \\in [ T ]$ and $i \\in [ n _ { t } ]$ , the function $\\theta \\mapsto l ( \\theta ; s _ { t } ^ { ( i ) } )$ is $L$ -smooth and twice continuously differentiable. ",
+        "bbox": [
+            176,
+            772,
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+            801
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 6. (Unbiased gradients and bounded variance) Each client $t \\in [ T ]$ can sample a random batch $\\xi$ from $S _ { t }$ and compute an unbiased estimator $g _ { t } ( \\theta , \\xi )$ of the local gradient with bounded variance, i.e., $\\begin{array} { r } { \\mathbb { E } _ { \\xi } [ g _ { t } ( \\theta , \\xi ) ] = \\frac { 1 } { n _ { t } } \\sum _ { i = 1 } ^ { n _ { t } } \\nabla _ { \\theta } l ( \\theta ; s _ { t } ^ { ( i ) } ) } \\end{array}$ and $\\begin{array} { r } { \\mathbb { E } _ { \\xi } \\| g _ { t } ( \\theta , \\xi ) - \\frac { 1 } { n _ { t } } \\sum _ { i = 1 } ^ { n _ { t } } \\nabla _ { \\theta } l ( \\theta ; s _ { t } ^ { ( i ) } ) \\| ^ { 2 } \\leq \\sigma ^ { 2 } } \\end{array}$ ",
+        "bbox": [
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+            804,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Assumption 7. (Bounded dissimilarity) There exist $\\beta$ and $G$ such that for any set of weights $\\alpha \\in \\Delta ^ { M }$ : ",
+        "bbox": [
+            173,
+            856,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2dd1297d9e05485a9181ef09509836cf750b7ad27333413507a0f0233de90ec7.jpg",
+        "text": "$$\n\\sum _ { t = 1 } ^ { T } \\frac { n _ { t } } { n } \\Big \\| \\frac { 1 } { n _ { t } } \\sum _ { i = 1 } ^ { n _ { t } } \\sum _ { m = 1 } ^ { M } \\alpha _ { m } \\cdot l ( \\theta ; s _ { t } ^ { ( i ) } ) \\Big \\| ^ { 2 } \\leq G ^ { 2 } + \\beta ^ { 2 } \\Big \\| \\frac { 1 } { n } \\sum _ { t = 1 } ^ { T } \\sum _ { i = 1 } ^ { n _ { t } } \\sum _ { m = 1 } ^ { M } \\alpha _ { m } \\cdot l ( \\theta ; s _ { t } ^ { ( i ) } ) \\Big \\| ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input : Data $\\overline { { S _ { 1 : T } } }$ ; number of mixture distributions $M$ ; number of communication rounds $K$   \nOutput : $\\theta _ { m } ^ { K }$ , $m \\in [ M ]$   \n1 for iterations $k = 1 , \\ldots , K$ do   \n2 server broadcasts $\\theta _ { m } ^ { k - 1 }$ , $1 \\leq m \\leq M$ , to the $T$ clients;   \n3 for tasks $t = 1 , \\dots , T$ in parallel over $T$ clients do   \n4 for component $m = 1 , \\ldots , M$ do   \n5 update $q _ { t } ^ { k } ( z _ { t } ^ { ( i ) } = m )$ as in (8), $\\forall i \\in \\left\\{ 1 , \\ldots , n _ { t } \\right\\}$ ;   \n6 update $\\pi _ { t m } ^ { k }$ as in (9);   \n7 $\\theta _ { m , t } ^ { k } \\gets \\mathtt { L o c a l S o l v e r } ( m , \\theta _ { m } ^ { k - 1 } , q _ { t } ^ { k } , S _ { t } )$ ;   \n8 client $t$ sends $\\theta _ { m , t } ^ { k }$ , $1 \\leq m \\leq M$ , to the server;   \n9 for component $m = 1 , \\ldots , M$ do   \n10 $\\begin{array} { r } { \\theta _ { m } ^ { k }  \\sum _ { t = 1 } ^ { T } \\frac { n _ { t } } { n } \\times \\theta _ { m , t } ^ { k } } \\end{array}$ ; ",
+        "bbox": [
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+            113,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Assumption 7 limits the level of dissimilarity of the different tasks, similarly to what is done in [66]. ",
+        "bbox": [
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+            330,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2. Under Assumptions $^ { 1 - 7 , }$ , when clients use $S G D$ as local solver with learning rate $\\begin{array} { r } { \\eta = \\frac { a _ { 0 } } { \\sqrt { K } } } \\end{array}$ , after a large enough number of communication rounds $K$ , FedEM’s iterates satisfy: ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f514bc1dbdf3766409b31226919d487757d074a954186aba7649da1cac75f41.jpg",
+        "text": "$$\n\\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\mathbb { E } \\left\\| \\nabla _ { \\Theta } f \\left( \\Theta ^ { k } , \\Pi ^ { k } \\right) \\right\\| _ { F } ^ { 2 } \\leq \\mathcal { O } \\bigg ( \\frac { 1 } { \\sqrt { K } } \\bigg ) , \\qquad \\frac { 1 } { K } \\sum _ { k = 1 } ^ { K } \\Delta _ { \\Pi } f ( \\Theta ^ { k } , \\Pi ^ { k } ) \\leq \\mathcal { O } \\bigg ( \\frac { 1 } { K ^ { 3 / 4 } } \\bigg ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            196,
+            391,
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+            435
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where the expectation is over the random batches samples, and $\\Delta _ { \\Pi } f ( \\Theta ^ { k } , \\Pi ^ { k } ) \\triangleq f \\left( \\Theta ^ { k } , \\Pi ^ { k } \\right) -$ $f \\left( \\Theta ^ { k } , \\Pi ^ { k + 1 } \\right) \\ge 0$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.2 (proof in App. G.1) expresses the convergence of both sets of parameters $\\Theta$ and $\\Pi$ ) to a stationary point of $f$ . Indeed, the gradient of $f$ with respect to $\\Theta$ becomes arbitrarily small (left inequality in (11)) and the update in Eq. (9) leads to arbitrarily small improvements of $f$ (right inequality in (11)). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conclude this section observing that FedEM allows an unseen client, i.e., a client $t _ { \\mathrm { n e w } } \\notin [ T ]$ arriving after the distributed training procedure, to learn its personalized model. The client simply retrieves the learned components’ parameters $\\Theta ^ { K }$ and computes its personalized weights $\\pi _ { t _ { \\mathrm { n e w } } }$ (starting for example from a uniform initialization) through one $\\mathrm { E }$ -step (8) and the first update in the M-step (9). ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.3 Fully Decentralized Algorithm ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            622,
+            426,
+            638
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In some cases, clients may want to communicate directly in a peer-to-peer fashion instead of relying on the central server mediation [see 28, Section 2.1]. In fact, fully decentralized schemes may provide stronger privacy guarantees [12] and speed-up training as they better use communication resources [41, 46] and reduce the effect of stragglers [50]. For these reasons, they have attracted significant interest recently in the machine learning community [41, 63, 42, 62, 3, 51, 46, 31]. We refer to [49] for a comprehensive survey of fully decentralized optimization (also known as consensus-based optimization), and to [31] for a unified theoretical analysis of decentralized SGD. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We propose D-FedEM (Alg. 4 in App. D.2), a fully decentralized version of our federated expectation maximization algorithm. As in FedEM, the M-step for $\\Theta$ update is replaced by an approximate maximization step consisting of local updates. The global aggregation step in FedEM (Alg. 1, line 10) is replaced by a partial aggregation step, where each client computes a weighted average of its current components and those of a subset of clients (its neighborhood), which may vary over time. The convergence of decentralized optimization schemes requires certain assumptions to guarantee that each client can influence the estimates of other clients over time. In our paper, we consider the general assumption in [31, Assumption 4] (restated as Assumption 8 in App. E for completeness). For instance, this assumption is satisfied if the graph of clients’ communications is strongly connected every $\\tau$ rounds. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "D-FedEM converges to a stationary point of $f$ (formal statement in App. E and proof in App. G.2). ",
+        "bbox": [
+            169,
+            896,
+            813,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 3.3 (Informal). In the same setting of Theorem 3.2 and under the additional Assumption 8, D-FedEM’s individual estimates $( \\Theta _ { t } ^ { k } ) _ { 1 \\leq t \\leq T }$ converge to a common value $\\bar { \\Theta } ^ { k }$ . Moreover, $\\bar { \\Theta } ^ { k }$ and $\\Pi ^ { k }$ converge to a stationary point of $f$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.4 Federated Surrogate Optimization ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            150,
+            452,
+            165
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "FedEM and D-FedEM can be seen as particular instances of a more general framework—of potential interest for other applications—that we call federated surrogate optimization. ",
+        "bbox": [
+            173,
+            176,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The standard majorization-minimization principle [34] iteratively minimizes, at each iteration $k$ , a surrogate function $g ^ { k }$ majorizing the objective function $f$ . The work [43] studied this approach when each $\\mathbf { \\bar { \\boldsymbol { g } } ^ { k } }$ is a first-order surrogate of $f$ (the formal definition from [43] is given in App. F.1). ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Our novel federated surrogate optimization framework considers that the objective function $f$ is a weighted sum $\\begin{array} { r } { f = \\sum _ { t = 1 } ^ { T } \\hat { \\omega } _ { t } f _ { t } } \\end{array}$ of $T$ functions and iteratively minimizes $f$ in a distributed fashion using partial first-order surrogates $g _ { t } ^ { k }$ for each function $f _ { t }$ . “Partial” refers to the fact that $g _ { t } ^ { k }$ is not required to be a first order surrogate wrt the whole set of parameters, as defined formally below. ",
+        "bbox": [
+            174,
+            258,
+            825,
+            319
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Partial first-order surrogate). A function $g ( \\mathbf { u } , \\mathbf { v } ) : \\mathbb { R } ^ { d _ { u } } \\times \\mathcal { V }  \\mathbb { R }$ is a partial-first-order surrogate of $f ( { \\bf u } , { \\bf v } )$ wrt u near $( { \\bf u } _ { 0 } , { \\bf v } _ { 0 } ) \\in R ^ { d _ { u } } \\times \\mathcal { V }$ when the following conditions are satisfied: ",
+        "bbox": [
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+            323,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "1. $g ( \\mathbf { u } , \\mathbf { v } ) \\geq f ( \\mathbf { u } , \\mathbf { v } ) .$ for all $\\mathbf { u } \\in \\mathbb { R } ^ { d _ { u } }$ and $\\mathbf { v } \\in \\mathcal { V }$ ;   \n2. $r ( \\mathbf { u } , \\mathbf { v } ) \\triangleq g ( \\mathbf { u } , \\mathbf { v } ) - f ( \\mathbf { u } , \\mathbf { v } )$ is differentiable and $L$ -smooth with respect to u. Moreover, we have $r ( { \\bf u } _ { 0 } , { \\bf v } _ { 0 } ) = 0$ and $\\begin{array} { r } { \\nabla _ { \\mathbf { u } } r ( \\mathbf { u } _ { 0 } , \\mathbf { v } _ { 0 } ) = 0 . } \\end{array}$ .   \n3. $g ( \\mathbf { u } , \\mathbf { v } _ { 0 } ) - g ( \\mathbf { u } , \\mathbf { v } ) = d _ { \\mathcal { V } } ( \\mathbf { v } _ { 0 } , \\mathbf { v } ) .$ for all $\\mathbf { u } \\in \\mathbb { R } ^ { d _ { u } }$ and $\\begin{array} { r } { \\mathbf { v } \\in \\arg \\operatorname* { m i n } _ { \\mathbf { v } ^ { \\prime } \\in \\mathcal { V } } g ( \\mathbf { u } , \\mathbf { v } ^ { \\prime } ) } \\end{array}$ , where $d _ { \\nu }$ is non-negative and $d _ { \\mathcal { V } } ( \\mathbf { v } , \\mathbf { v } ^ { \\prime } ) = 0 \\iff \\mathbf { v } = \\mathbf { v } ^ { \\prime }$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Under the assumption that each client $t$ can compute a partial first-order surrogate of $f _ { t }$ , we propose algorithms for federated surrogate optimization in both the client-server setting (Alg. 3) and the fully decentralized one (Alg. 5) and prove their convergence under mild conditions (App. G.1 and G.2). FedEM and D-FedEM can be seen as particular instances of these algorithms and Theorem. 3.2 and Theorem. 3.3 follow from the more general convergence results for federated surrogate optimization. We can also use our framework to analyze the convergence of other FL algorithms such as pFedMe [16], as we illustrate in App. F.3. ",
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+        "text": "4 Experiments ",
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+        "type": "text",
+        "text": "Datasets and models. We evaluated our method on five federated benchmark datasets spanning a wide range of machine learning tasks: image classification (CIFAR10 and CIFAR100 [33]), handwritten character recognition (EMNIST [8] and FEMNIST [7]),5 and language modeling (Shakespeare [7, 47]). Shakespeare dataset (resp. FEMNIST) was naturally partitioned by assigning all lines from the same characters (resp. all images from the same writer) to the same client. We created federated versions of CIFAR10 and EMNIST by distributing samples with the same label across the clients according to a symmetric Dirichlet distribution with parameter 0.4, as in [65]. For CIFAR100, we exploited the availability of “coarse” and “fine” labels, using a two-stage Pachinko allocation method [39] to assign 600 sample to each of the 100 clients, as in [54]. We also evaluated our method on a synthetic dataset verifying Assumptions 1–3. For all tasks, we randomly split each local dataset into training $( 6 0 \\% )$ , validation $( 2 0 \\% )$ and test $( 2 0 \\% )$ sets. Table 1 summarizes datasets, models, and number of clients (more details can be found in App. I.1). Code is available at https://github.com/omarfoq/FedEM. ",
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+        "type": "text",
+        "text": "Other FL approaches. We compared our algorithms with global models trained with FedAvg [47] and FedProx [38] as well as different personalization approaches: a personalized model trained only on the local dataset, FedAvg with local tuning $( \\mathtt { F e d A v g } + )$ [27], Clustered FL [56] and pFedMe [16]. For each method and each task, the learning rate and the other hyperparameters were tuned via grid search (details in App. I.2). FedAvg+ updated the local model through a single pass on the local dataset. Unless otherwise stated, the number of components considered by FedEM was $M = 3$ , training occurred over 80 communication rounds for Shakespeare and 200 rounds for all other datasets. ",
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+        "table_caption": [
+            "Table 1: Datasets and models (details in App. I.1). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Task</td><td>Clients</td><td>Total samples</td><td>Model</td></tr><tr><td>FEMNIST[7]</td><td>Handwriten character recognition</td><td>539</td><td>120,772</td><td>2-layer CNN + 2-layer FFN</td></tr><tr><td>EMNIST[8]</td><td>Handwritten character recognition</td><td>100</td><td>81,425</td><td>2-layer CNN + 2-layer FFN</td></tr><tr><td>CIFAR10 [33]</td><td>Image classification</td><td>80</td><td>60,000</td><td>MobileNet-v2 [55]</td></tr><tr><td>CIFAR100 [33]</td><td>Image classification</td><td>100</td><td>60,000</td><td>MobileNet-v2 [55]</td></tr><tr><td>Shakespeare [7,47]</td><td>Next-Character Prediction</td><td>778</td><td>4,226,158</td><td>Stacked-LSTM[25]</td></tr><tr><td>Synthetic</td><td>Binary Classification</td><td>300</td><td>1,570,507</td><td>Linear model</td></tr></table>",
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+        "table_caption": [
+            "Table 2: Test accuracy: average across clients / bottom decile. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Local</td><td>FedAvg[47]</td><td>FedProx [38]</td><td>FedAvg+ [27]</td><td>Clustered FL [56]</td><td>pFedMe [16]</td><td>FedEM (Ours)</td></tr><tr><td>FEMNIST</td><td>71.0/57.5</td><td>78.6/63.9</td><td>78.9 /64.0</td><td>75.3/53.0</td><td>73.5/55.1</td><td>74.9/57.6</td><td>79.9/64.8</td></tr><tr><td>EMNIST</td><td>71.9/64.3</td><td>82.6/75.0</td><td>83.0/75.4</td><td>83.1/75.8</td><td>82.7/75.0</td><td>83.3/76.4</td><td>83.5/76.6</td></tr><tr><td>CIFAR10</td><td>70.2/48.7</td><td>78.2/72.4</td><td>78.0/70.8</td><td>82.3/70.6</td><td>78.6/71.2</td><td>81.7/73.6</td><td>84.3 /78.1</td></tr><tr><td>CIFAR100</td><td>31.5/19.9</td><td>40.9/33.2</td><td>41.0/33.2</td><td>39.0/28.3</td><td>41.5/34.1</td><td>41.8/32.5</td><td>44.1/35.0</td></tr><tr><td>Shakespeare</td><td>32.0/16.6</td><td>46.7/42.8</td><td>45.7/41.9</td><td>40.0/25.5</td><td>46.6/42.7</td><td>41.2/36.8</td><td>46.7/43.0</td></tr><tr><td>Synthetic</td><td>65.7/58.4</td><td>68.2/58.9</td><td>68.2/59.0</td><td>68.9/60.2</td><td>69.1/59.0</td><td>69.2/61.2</td><td>74.7/66.7</td></tr></table>",
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+        "type": "text",
+        "text": "At each round, clients train for one epoch. Results for D-FedEM are in App. J.1. A comparison with MOCHA [59], which can only train linear models, is presented in App. J.2. ",
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+        "type": "text",
+        "text": "Average performance of personalized models. The performance of each personalized model (which is the same for all clients in the case of FedAvg and FedProx) is evaluated on the local test dataset (unseen at training). Table 2 shows the average weighted accuracy with weights proportional to local dataset sizes. We observe that FedEM obtains the best performance across all datasets. ",
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+    {
+        "type": "text",
+        "text": "Fairness across clients. FedEM’s improvement in terms of average accuracy could be the result of learning particularly good models for some clients at the expense of bad models for other clients. Table 2 shows the bottom decile of the accuracy of local models, i.e., the $( T / 1 0 )$ -th worst accuracy (the minimum accuracy is particularly noisy, notably because some local test datasets are very small). Even clients with the worst personalized models are still better off when FedEM is used for training. ",
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+    {
+        "type": "text",
+        "text": "Clients sampling. In cross-device federated learning, only a subset of clients may be available at each round. We ran CIFAR10 experiments with different levels of participation: at each round a given fraction of all clients were sampled uniformly without replacement. We restrict the comparison to FedEM and ${ \\mathrm { F e d A v g } } +$ , as 1) ${ \\mathrm { F e d A v g } } +$ performed better than FedProx and FedAvg in the previous CIFAR10 experiments, 2) it is not clear how to extend pFedMe and Clustered FL to handle client sampling. Results in Fig. 1 (left) show that FedEM is more robust to low clients’ participation levels. We provide additional results on client sampling, including a comparison with APFL [14], in App. J.6. ",
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+    {
+        "type": "text",
+        "text": "Generalization to unseen clients. As discussed in Section 3.2, FedEM allows new clients arriving after the distributed training to easily learn their personalized models. With the exception of ${ \\mathrm { F e d A v g } } +$ , it is not clear how the other personalized FL algorithms should be extended to tackle the same goal (see discussion in App. J.3). In order to evaluate the quality of new clients’ personalized models, we performed an experiment where only $80 \\%$ of the clients (“old” clients) participate to the training. The remaining $20 \\%$ join the system in a second phase and use their local training datasets to learn their personalized weights. Table 3 shows that FedEM allows new clients to learn a personalized model at least as good as FedAvg’s global one and always better than FedAvg+’s one. Unexpectedly, new clients achieve sometimes a significantly higher test accuracy than old clients (e.g., $4 7 . 5 \\%$ against $4 4 . 1 \\%$ on CIFAR100). Our investigation in App. J.3 suggests that, by selecting their mixture weights on local datasets that were not used to train the components, new clients can compensate for potential overfitting in the initial training phase. We also investigate in App. J.3 the effect of the local dataset size on the accuracy achieved by unseen clients, showing that personalization is effective even when unseen clients have small datasets. ",
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+    {
+        "type": "text",
+        "text": "Effect of $M$ . A limitation of FedEM is that each client needs to update and transmit $M$ components at each round, requiring roughly $M$ times more computation and $M$ times larger messages. Nevertheless, the number of components to consider in practice is quite limited. We used $M = 3$ in our previous experiments, and Fig. 1 (right) shows that larger values do not yield much improvement and $M = 2$ already provides a significant level of personalization. In all experiments above, the number of communication rounds allowed all approaches to converge. As a consequence, even if other methods trained over $M = 3$ times more rounds—in order to have as much computation and communication as FedEM—the conclusions would not change. As a final experiment, we considered a time-constrained setting, where FedEM is limited to run one third $( = 1 / M )$ ) of the rounds (Table 7 in App. J.5). Even if FedEM does not reach its maximum accuracy, it still outperforms the other methods on 3 datasets. ",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Average test accuracy across clients unseen at training (train accuracy in parenthesis). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>FedAvg [47]</td><td>FedAvg+[27]</td><td>FedEM (Ours)</td></tr><tr><td>FEMNIST</td><td>78.3(80.9)</td><td>74.2 (84.2)</td><td>79.1 (81.5)</td></tr><tr><td>EMNIST</td><td>83.4(82.7)</td><td>83.7(92.9)</td><td>84.0(83.3)</td></tr><tr><td>CIFAR10</td><td>77.3(77.5)</td><td>80.4 (80.5)</td><td>85.9 (90.7)</td></tr><tr><td>CIFAR100</td><td>41.1 (42.1)</td><td>36.5 (55.3)</td><td>47.5 (46.6)</td></tr><tr><td>Shakespeare</td><td>46.7 (47.1)</td><td>40.2 (93.0)</td><td>46.7 (46.6)</td></tr><tr><td>Synthetic</td><td>68.6(70.0)</td><td>69.1(72.1)</td><td>73.0 (74.1)</td></tr></table>",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/eeb1beb065cef85e985d5f9666ab0cf36430081c9d877b100da90cd7e8b773ff.jpg",
+        "image_caption": [
+            "Figure 1: Effect of client sampling rate (left) and FedEM number of mixture components $M$ (right) on the test accuracy for CIFAR10 [33]. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this paper, we proposed a novel federated MTL approach based on the flexible assumption that local data distributions are mixtures of underlying distributions. Our EM-like algorithms allow clients to jointly learn shared component models and personalized mixture weights in client-server and fully decentralized settings. We proved convergence guarantees for our algorithms through a general federated surrogate optimization framework which can be used to analyze other FL formulations. Extensive empirical evaluation shows that our approach learns models with higher accuracy and fairness than state-of-the-art FL algorithms, even for clients not present at training time. ",
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+    {
+        "type": "text",
+        "text": "In future work, we aim to reduce the local computation and communication of our algorithms. Aside from standard compression schemes [22], a promising direction is to limit the number of component models that a client updates/transmits at each step. This could be done in an adaptive manner based on the client’s current mixture weights. A simultaneously published work [15] proposes a federated EM algorithm (also called FedEM), which does not address personalization but reduces communication requirements by compressing appropriately defined complete data sufficient statistics. ",
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+    {
+        "type": "text",
+        "text": "A second interesting research direction is to study personalized FL approaches under privacy constraints (quite unexplored until now with the notable exception of [3]). Some features of our algorithms may be beneficial for privacy (e.g., the fact that personalized weights are kept locally and that all users contribute to all shared models). We hope to design differentially private versions of our algorithms and characterize their privacy-utility trade-offs. ",
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+    {
+        "type": "text",
+        "text": "6 Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    },
+    {
+        "type": "text",
+        "text": "This work has been supported by the French government, through the 3IA Côte d’Azur Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-19-P3IA-0002, and through grants ANR-16-CE23-0016 (Project PAMELA) and ANR-20- CE23-0015 (Project PRIDE). The authors are grateful to the OPAL infrastructure from Université Côte d’Azur for providing computational resources and technical support. ",
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+    {
+        "type": "text",
+        "text": "References ",
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In: Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence, UAI 2010. 2010, p. 733.   \n[70] Jiayu Zhou, Jianhui Chen, and Jieping Ye. “Clustered Multi-Task Learning Via Alternating Structure Optimization”. In: Advances in Neural Information Processing Systems. Ed. by J. Shawe-Taylor, R. Zemel, P. Bartlett, F. Pereira, and K. Q. Weinberger. Vol. 24. Curran Associates, Inc., 2011. URL: https://proceedings.neurips.cc/paper/2011/file/ a516a87cfcaef229b342c437fe2b95f7-Paper.pdf. ",
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parse/train/rklFh34Kwr/rklFh34Kwr.md

@@ -0,0 +1,261 @@
+# Bayesian Inference for Large Scale Image Classification
+
+Anonymous authors Paper under double-blind review
+
+# Abstract
+
+Bayesian inference promises to ground and improve the performance of deep neural networks. It promises to be robust to overfitting, to simplify the training procedure and the space of hyperparameters, and to provide a calibrated measure of uncertainty that can enhance decision making, agent exploration and prediction fairness. Markov Chain Monte Carlo (MCMC) methods enable Bayesian inference by generating samples from the posterior distribution over model parameters. Despite the theoretical advantages of Bayesian inference and the similarity between MCMC and optimization methods, the performance of sampling methods has so far lagged behind optimization methods for large scale deep learning tasks. We aim to fill this gap and introduce ATMC, an adaptive noise MCMC algorithm that estimates and is able to sample from the posterior of a neural network. ATMC dynamically adjusts the amount of momentum and noise applied to each parameter update in order to compensate for the use of stochastic gradients. We use a ResNet architecture without batch normalization to test ATMC on the Cifar10 benchmark and the large scale ImageNet benchmark and show that, despite the absence of batch normalization, ATMC outperforms a strong optimization baseline in terms of both classification accuracy and test log-likelihood. We show that ATMC is intrinsically robust to overfitting on the training data and that ATMC provides a better calibrated measure of uncertainty compared to the optimization baseline.
+
+# 1 Introduction
+
+In contrast to optimization approaches in machine learning that derive a single estimate for the weights of a neural network, Bayesian inference aims at deriving a posterior distribution over the weights of the network. This makes it possible to sample model instances from the distribution over the weights and offers unique advantages. Multiple model instances can be aggregated to obtain robust uncertainty estimates over the network’s predictions; uncertainty estimates are crucial in domains such as medical diagnosis and autonomous driving where following a model’s incorrect predictions can result in catastrophe (Kendall & Gal, 2017). Sampling a distribution, as opposed to optimizing a loss, is less prone to overfitting and more training doesn’t decrease test performance. Bayesian inference can also be applied to differential privacy, where each individual sample has increased privacy guarantees (Wang et al., 2015), and to reinforcement learning, where one can leverage model uncertainty to balance between exploration and exploitation (Osband & Van Roy, 2017).
+
+Traditional Markov Chain Monte Carlo (MCMC) methods like HMC (Neal et al., 2011) are a standard class of methods for generating samples from the posterior distribution over model parameters. These methods are seldom applied in deep learning because they have traditionally failed to scale well with large datasets and many parameters (Rajaratnam $\&$ Sparks, 2015). Stochastic Gradient MCMC (SG-MCMC) methods have fared somewhat better in scaling to large datasets due to their close relationship to stochastic optimization methods. For example the SGLD sampler (Welling & Teh, 2011) amounts to performing stochastic gradient descent while adding Gaussian noise to each parameter update. Despite these improvements, samplers like SGLD are only guaranteed to converge to the correct distribution when the step size is annealed to zero; additional control variates have been developed to mitigate this to some extent (Ahn et al., 2012; Ding et al., 2014).
+
+Algorithm 1 The ATMC sampler. The algorithm accepts the initialized model parameters $\pmb { \theta } _ { 0 }$ , step size $h$ , pre-conditioner $m$ , and momentum noise $D _ { c }$ .   
+
+<table><tr><td colspan="4">1: procedure ATMC_TRAINING(0o,h,m,D)</td></tr><tr><td>2:</td><td>Po←0</td><td></td><td></td></tr><tr><td>3:</td><td>m←0</td><td></td><td></td></tr><tr><td>4:</td><td>whilet&lt;Tdo</td><td></td><td></td></tr><tr><td>5:</td><td></td><td>Gt ← minibatch_gradient(0t)</td><td></td></tr><tr><td>6:</td><td>nt ← random_normal()</td><td></td><td></td></tr><tr><td>7:</td><td>αt ←max(D-εt,0)</td><td></td><td></td></tr><tr><td>8:</td><td>βt←αt+t</td><td></td><td></td></tr><tr><td>9:</td><td>Pt+h←eβth</td><td>exp[Bth]-1Gt+ Pt-</td><td>exp[2βth]-1 αtnt</td></tr><tr><td>10:</td><td>0t+h←0t+hPt+h</td><td>βt</td><td>βt</td></tr><tr><td>11:</td><td>St+h←t+h</td><td>-1]</td><td></td></tr><tr><td>12:</td><td></td><td>m</td><td></td></tr><tr><td></td><td>t←t+h</td><td></td><td></td></tr></table>
+
+The objective of this work is to make Bayesian inference practical for deep learning by making SG-MCMC methods scale to large models and datasets. The contributions described in this work fall in three categories. We first propose the Adaptive Thermostat Monte Carlo (ATMC) sampler that offers improved convergence and stability. ATMC dynamically adjusts the amount of momentum and noise applied to each model parameter. Secondly, we improve an existing second order numerical integration method that is needed for the ATMC sampler. Third, since ATMC, like other SG-MCMC samplers, is not directly compatible with stochastic regularization methods such as batch normalization (BatchNorm) and Dropout (see Sect. 4), we construct the ResNet++ network by taking the original ResNet architecture (He et al., 2016), removing BatchNorm and introducing SELUs (Klambauer et al., 2017), Fixup initialization (Zhang et al., 2019a) and weight normalization (Salimans & Kingma, 2016). We design ResNet $^ { + + }$ so that its parameters are easy to sample from and the gradients are well-behaved even in the absence of BatchNorm.
+
+We show that the ATMC sampler is able to outperform optimization methods in terms of accuracy, log-likelihood and uncertainty calibration in the following settings. First, when using the ResNet $^ { + + }$ architecture for both the ATMC sampler and the optimization baseline, the ATMC sampler significantly outperforms the optimization baseline on both Cifar-10 and ImageNet. Secondly, when using the standard ResNet for the optimization baseline and the ResNet++ for the ATMC sampler, multiple samples of the ATMC that approximate the predictive posterior of the model are still able to outperform the optimization baseline on ImageNet. Using the ResNet $^ { + + }$ architecture, the ATMC sampler reduces the need for hyper-parameter tuning since it does not require early stopping, does not use stochastic regularization, is not prone to over-fitting on the training data and avoids a carefully tuned learning rate decay schedule.
+
+# 2 ATMC Sampler
+
+In this section we define the Stochastic Differential Equation (SDE) that gives rise to the ATMC sampler described in Algorithm 1. A detailed background and framework for constructing SDEs that converge to a target distribution can be found in (Ma et al., 2015).
+
+# 2.1 General form of the SDE
+
+Our starting point for constructing the ATMC sampler is the framework of Stochastic Differential Equations. We are interested in SDEs that converge to a distribution $p ( z )$ over the vector $z \in \mathbb { R } ^ { d }$ for which we can evaluate $\nabla \log p ( z )$ . Because only the gradient of $\log p ( z )$
+
+is required, it is sufficient to define an energy function $H ( z ) = - \log p ( z ) + C$ up to a constant $C$ . As a consequence, we can sample from the posterior distribution $p ( \theta | x )$ by only evaluating the energy function gradient $\nabla H ( \theta ) = - \nabla \log p ( x , \theta )$ . The general form of SDEs converging to $p ( z )$ for which only the gradient of $p ( z )$ is required is as follows (Ma et al., 2015):
+
+$$
+\small { \begin{array} { l } { \displaystyle { d z = - \left[ D ( z ) + Q ( z ) \right] \nabla H ( z ) d t + \Gamma ( z ) d t + \sqrt { 2 D ( z ) } d W _ { t } , \quad \Gamma _ { i } ( z ) = \sum _ { j } ^ { d } \frac { \partial \left[ D _ { i j } ( z ) + Q _ { i j } ( z ) \right] } { \partial z _ { j } } , } } \end{array} }
+$$
+
+where $D ( z )$ is a positive-definite matrix that determines the amount of noise, $Q ( z )$ is a skew-symmetric matrix that mixes energy between variables, $W _ { t }$ is a Wiener process, and $\Gamma ( z )$ is a correction factor that compensates for dynamics that depend on the current state $z$ . The ATMC sampler that we propose is an instance of (1) for specific definitions of $H ( z )$ , $D ( z )$ , and $Q ( z )$ .
+
+# 2.2 Energy Function
+
+We start by defining the energy function $H ( z )$ . The energy function for the model posterior $p ( \theta | x )$ is defined by the loss function $\mathcal { L } ( \theta ) = - \log p ( x , \theta )$ . Because the dataset $x$ is generally large, we would like to only evaluate a mini-batch loss $\ddot { \mathcal { L } } ( \boldsymbol { \theta } )$ . However, naively using a stochastic gradient in (1) will result in significant bias (Chen et al., 2014). Motivated by the Central Limit Theorem, the stochastic gradient is assumed to follow a Gaussian distribution $\nabla \tilde { \mathcal { L } } ( \theta ) \sim \mathcal { N } ( \nabla \mathcal { L } ( \theta ) , B )$ where the covariance $B$ is additionally assumed to be diagonal and constant w.r.t. $\theta$ . The energy function for the ATMC sampler is defined as:
+
+$$
+H ( \theta , p , \xi ) = \mathcal { L } ( \theta ) + K ( p ) + \frac { 1 } { 2 } \left( \xi - \frac { \mathrm { d i a g } ( B ) } { 2 m } \right) ^ { 2 } ,
+$$
+
+where $p$ is the momentum, $K ( p )$ defines the momentum distribution, and $\xi$ is a control variate referred to as the temperature. Both $p$ and $\xi$ have the same dimensionality as $\theta$ . The hyper-parameter $m$ controls the strength of the coupling between $\xi$ and $p$ . The distribution of the control variate $p ( \xi )$ depends on the amount of noise $B$ in the stochastic gradient estimate $\ddot { \mathcal { L } } ( \boldsymbol { \theta } )$ .
+
+# 2.3 Noise robust dynamics
+
+Next we define the dynamics $Q ( z )$ and $D ( z )$ such that the SDE that results from (1) can be simulated without the need to evaluate $B$ :
+
+$$
+\begin{array} { r } { D ( \theta , p , \xi ) = \left( \begin{array} { c c c } { 0 } & { 0 } & { 0 } \\ { 0 } & { \alpha ( \xi ) m + \frac { 1 } { 2 } B } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} \right) , \quad Q ( \theta , p , \xi ) = \left( \begin{array} { c c c } { 0 } & { - I } & { 0 } \\ { I } & { 0 } & { m \nabla K ( p ) } \\ { 0 } & { - m \nabla K ( p ) } & { 0 } \end{array} \right) , } \end{array}
+$$
+
+where $\alpha ( \xi )$ is a non-negative function that determines how the temperature $\xi$ affects the amount of noise added to the momentum update.
+
+We first illustrate the resulting SDE by using a simpler Gaussian momentum distribution $K ( p ) = \left\| p \right\| ^ { 2 } / ( 2 m )$ . Note that the variance of the momentum $\operatorname { V a r } ( p ) = m$ is reused in (2) and (3) to control the strength of the coupling between $\xi$ and $p$ . This will result in a temperature control with a momentum friction term proportional to $\xi$ , unlike previously reported thermostat MCMC methods (Ding et al., 2014; Lu et al., 2016) where the friction term is proportional to $\xi / m$ . We substitute the dynamics $Q ( z )$ and $D ( z )$ defined in (3) and energy function $H ( z )$ defined in (2) into (1):
+
+$$
+\binom { d \theta } { d \xi } = \left( - \nabla \tilde { \mathcal { L } } ( \theta ) - \beta ( \xi ) p \right) d t + \left( \begin{array} { c c c } { 0 } & { 0 } & { 0 } \\ { 0 } & { \sqrt { 2 \alpha ( \xi ) m } } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} \right) d W _ { t } , \beta ( \xi ) = \alpha ( \xi ) + \xi ,
+$$
+
+where we use $\nabla \tilde { \mathcal { L } } ( \boldsymbol { \theta } ) d t = \nabla \mathcal { L } ( \boldsymbol { \theta } ) d t + \sqrt { B } d W _ { t }$ to replace the gradient of the loss with the minibatch estimate. The momentum $p$ is dampened by a friction term $\beta ( \xi )$ that depends on the choice of $\alpha ( \xi )$ . The stochastic gradient noise $B$ does not show up in (4) due to the particular choice of energy function $H ( z )$ and dynamics $Q ( z ) , D ( z )$ . Note however this analysis relies on the assumption that the covariance of the stochastic gradient noise $B$ is constant in $\theta$ and a single temperature variable per parameter can only correct for a diagonal covariance $B$ . We do not expect that this assumption will hold in practice and the approximation will therefore lead to bias in the samples. However, annealing the step size $h$ will reduce the error due to mini-batching together with other sources of discretization error (Welling $\&$ Teh, 2011).
+
+# 2.4 Adaptive Noise Thermostat
+
+Finally, we must choose a function $\alpha ( \xi )$ which controls the amount of noise and momentum damping $\beta ( \xi )$ . Previous work uses the Nos´e-Hoover thermostat that is defined by $\alpha ( \xi ) = D _ { c }$ where $D _ { c }$ is a constant determining the amount of noise added to the momentum update (Ding et al., 2014). Although the Nos´e-Hoover thermostat is able to correct the stochastic gradient noise $B$ , the correction comes at the cost of slower convergence because additional friction $\beta ( \xi )$ is applied as $B$ increases. Another drawback of the Nos´e-Hoover thermostat is that it causes negative friction when $\xi < - D _ { c }$ . In the negative friction phase $\beta ( \xi ) < 0$ , previous gradient terms are amplified rather than dampened. Although this behavior is mathematically sound we find that it can cause exploding momentum variables.
+
+Our choice of $\alpha ( \xi )$ is based on the idea that negative friction should not occur and convergence speed should not be reduced by the stochastic gradient noise. Based on this intuition, we define the ATMC sampler by $\alpha ( \xi ) = \operatorname* { m a x } ( D - \xi , 0 )$ . The ATMC sampler is best characterized by the various temperature stages. For $0 < \xi < D _ { c }$ the total amount of noise added to the momentum is $D _ { c }$ and the friction coefficient $\beta ( \xi ) = D _ { c }$ . At this stage, the stochastic gradient noise is compensated for by adding less noise to the momentum update. If $B \gg D _ { c }$ the dominant stage will be $\xi > D$ resulting in $\beta ( \xi ) < D _ { c }$ and zero noise being added to the momentum. Finally, when $\xi < 0$ the friction coefficient $\beta ( \xi ) = D _ { c }$ and the noise added to the momentum is proportional to $D _ { c } - \xi$ . Thus, the momentum always experiences a minimum amount of friction $\beta ( \xi ) \ge D _ { c }$ determined by the hyper-parameter $D _ { c }$ and the noise added to the momentum update is automatically adjusted based on the amount of noise present in the stochastic gradients.
+
+# 2.5 Momentum energy function
+
+Following (Lu et al., 2016), we generalize the momentum energy function $K ( p )$ to the symmetric hyperbolic distribution which is defined as follows (Lu et al., 2016):
+
+$$
+K ( p ) = \sum _ { i } m c ^ { 2 } \left[ \sqrt { \frac { p _ { i } ^ { 2 } } { m ^ { 2 } c ^ { 2 } } + 1 } - 1 \right] ,
+$$
+
+where $m$ and $c$ are hyper-parameters. The Gaussian kinetic energy $K ( p ) = \left\| p \right\| ^ { 2 } / ( 2 m )$ is a special case obtained by taking the limit $c  \infty$ . The magnitude of parameter updates $\| \Delta \theta \|$ is determined by the gradient of the momentum:
+
+$$
+\| \Delta \theta \| = \| \nabla K ( p ) \| = \left\| \frac { p } { M ( p ) } \right\| , \quad M ( p ) = m \sqrt { \frac { p ^ { 2 } } { m ^ { 2 } c ^ { 2 } } + 1 } .
+$$
+
+Hence, the hyperbolic distribution results in relativistic momentum dynamics where the parameter updates are upper bounded by $c$ and the pre-conditioner √ $M ( p )$ depends on $p$ . The average update magnitude $\mathcal { E } [ | \nabla K ( p ) | | ] \approx 1 / \sqrt { m }$ for $c \gg m$ . Consequently, the parameters $m$ and $c$ are interpretable hyper-parameters controlling the average and maximum parameter update per step together with the step size $h$ .
+
+The SDE we derive in (4) and integrate in Sec. 3 uses a Gaussian momentum energy function for clarity. Deriving ATMC with a different momentum distribution like the hyperbolic distribution amounts to substituting (2), (3), and the alternative momentum distribution into (1). For the hyperbolic distribution, the dynamic friction coefficient $\beta ( \xi )$ will also depend on $p$ . For the numerical integration of (4) with a hyperbolic momentum distribution we assume $\beta ( \xi )$ to be constant in $p$ .
+
+# 3 Improved numerical integrator for MCMC samplers
+
+In this section we construct the numerical integrator required to numerically approximate the ATMC sampler defined in (4). An efficient numerical integrator can be constructed by splitting the SDE into two terms:
+
+$$
+\binom { d \theta } { d \xi } = \underbrace { \binom { p / m } { 0 } } _ { \displaystyle { p ^ { 2 } / m - 1 } } d t + \underbrace { \left( - \nabla \tilde { \mathcal { L } } ( \theta ) - \beta ( p , \xi ) p \right) d t + \left( 0 \begin{array} { c c c } { 0 } & { 0 } & { 0 } \\ { 0 } & { \sqrt { 2 \alpha ( \xi ) m } } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} \right) d W _ { t } } _ { B } .
+$$
+
+Hence, we obtain a linear ODE in part (A) that updates the parameters $\theta$ and the thermostats $\xi$ and a linear SDE in part (B) that updates the momentum $p$ . The operators that simulate these dynamics exactly for a time step $h$ are denoted $\phi _ { A } ^ { h }$ and $\phi _ { B } ^ { h }$ , respectively. Using the Strang splitting scheme yields a second order method (Chen et al., 2015):
+
+$$
+\phi ^ { h } = \phi _ { B } ^ { h / 2 } \circ \phi _ { A } ^ { h } \circ \phi _ { B } ^ { h / 2 } .
+$$
+
+The first operator $\phi _ { A } ^ { h }$ is given by
+
+$$
+\begin{array} { r } { \phi _ { A } ^ { h } ( z _ { t } ) = \left( \theta _ { t } + h \frac { p _ { t } } { m } \quad p _ { t } \quad \xi _ { t } + h \left[ \frac { p _ { t } ^ { 2 } } { m } - 1 \right] \right) ^ { T } . } \end{array}
+$$
+
+The second operator $\phi _ { B }$ is an instance of the Ornstein–Uhlenbeck process which can also be computed analytically as follows (Nelson, 1967):
+
+$$
+\begin{array} { r l } { \phi _ { B } ^ { h } ( z _ { t } ) = \Big ( \theta _ { t } } & { e ^ { \beta ( \xi _ { t } ) h } \Big [ p _ { t } - \gamma _ { 1 } ( \xi _ { t } ) \nabla \tilde { \mathcal { L } } ( \theta _ { t } ) + \sqrt { \gamma _ { 2 } ( \xi _ { t } ) \alpha ( \xi _ { t } ) } ~ \eta _ { t } \Big ] \quad \xi _ { t } \Big ) ^ { T } , } \\ & { \quad \quad \gamma _ { a } ( \xi _ { t } ) = \frac { \exp [ a \beta ( \xi _ { t } ) h ] - 1 } { \beta ( \xi _ { t } ) } , } \end{array}
+$$
+
+where $\eta _ { t }$ is isotropic Gaussian noise. Previous work (Chen et al., 2015) on higher order integrators for samplers splits the SDE into three parts where the third term is obtained from separating the friction term from the other terms in the momentum update $\phi _ { B }$ . By integrating (10) exactly the gradient step and the noise and gradient term are directly affected by the friction. An exact momentum update provides additional robustness to large gradients because the temperature will increase in order to compensate for momentum updates that would lead to excessively large steps. Another advantage of a two-way split integrator is that the first and last steps in (8) can be fused together such that only a momentum update is performed per iteration. Algorithm 1 shows the pseudocode for the ATMC sampler with the split integrator defined in (9) and (10).
+
+# 4 The ResNet++ Architecture
+
+The generalization performance of large neural nets trained using optimization depend on stochastic regularization methods like Dropout (Srivastava et al., 2014) and BatchNorm (Ioffe & Szegedy, 2015). These methods implicitly add noise into the model parameters (Kingma et al., 2015; Teye et al., 2018) and significantly boost training performance and generalization for image classifiers. These methods can be interpreted as a coarse approximation of Bayesian Inference (Kingma et al., 2015; Teye et al., 2018). But a stochastic gradient sampler like ATMC already adds the necessary amount of noise and combined with BatchNorm or Dropout it leads to underfitting. We thus define a BatchNorm free version of ResNet called ResNet++ that includes SELUs (Klambauer et al., 2017), Fixup initialization (Zhang et al., 2019a) and weight normalization (Salimans & Kingma, 2016) (see Fig. 1). We use ATMC to fill the significant gap in performance due to the absence of BatchNorm in ResNet++.
+
+Figure 1: Residual blocks in respectively the ResNet and $\mathrm { R e s N e t { + + } }$ architectures.   
+
+<table><tr><td rowspan=1 colspan=1>x</td><td rowspan=1 colspan=1>Conv 1x1</td><td rowspan=1 colspan=1>BatchNorm</td><td rowspan=1 colspan=1>ReLU</td><td rowspan=1 colspan=1>Conv 3x3</td><td rowspan=1 colspan=1>BatchNorm</td><td rowspan=1 colspan=1>ReLU</td><td rowspan=1 colspan=1>Conv 1x1</td><td rowspan=1 colspan=1>BatchNorm</td><td rowspan=1 colspan=1>+x</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>x</td><td rowspan=1 colspan=1>Conv 1x1</td><td rowspan=1 colspan=1>SeLU</td><td rowspan=1 colspan=1>Conv 3x3</td><td rowspan=1 colspan=1>SeLU</td><td rowspan=1 colspan=1>Conv 1x1</td><td rowspan=1 colspan=1>+x</td><td rowspan=1 colspan=1>SeLU</td><td></td><td></td><td></td></tr></table>
+
+![](images/3581dff74a2bd9e2f91fa7c34ce15d9be1bcc8162cfbad8a4e171f8cab32d0ff.jpg)  
+Figure 2: Calibration plot for Cifar10
+
+![](images/f0139509c48ef718aaaace052fe20b41da0beb0225dc95165077a1b68ecb2edf.jpg)  
+Figure 3: Calibration plot for ImageNet
+
+# 4.1 SELU
+
+We find the SELU activation to work well in BatchNorm free networks. SELU forces the statistics of the activations towards zero mean and unit variance (Klambauer et al., 2017). The SELU activation function additionally has a non-zero gradient everywhere which could improve the mixing of the sampler by providing a more informative gradient.
+
+# 4.2 Fixup initialization
+
+ResNets are known to scale well with depth (He et al., 2016). However, the additive effect of the residual branch causes the magnitudes of the activations to increase with the number of residual connections. Fixup is a recently proposed initialization method that mitigates the exploding residual branch problem without using BatchNorm (Zhang et al., 2019a). We use a simplified version of Fixup by initializing the scales of the final layer in each residual branch to a small constant.
+
+# 4.3 Weight normalization
+
+We use weight normalization (Salimans $\&$ Kingma, 2016) to separate the direction and scale of each linear feature vector
+
+$$
+\theta ^ { ( i ) } = \phi _ { s } ^ { ( i ) } \frac { \phi _ { d } ^ { ( i ) } } { \left\| \phi _ { d } ^ { ( i ) } \right\| } ,
+$$
+
+Table 1: Performance on Cifar10 with ResNet-56 model. The posterior predictive is estimate using a sample of the posterior parameters at the end of each learning rate cycle.   
+
+<table><tr><td>Setup</td><td>Top 1 acc. [%]</td></tr><tr><td>SGD</td><td>91.5</td></tr><tr><td>SGD + BatchNorm</td><td>94.4</td></tr><tr><td>ATMC (single sample)</td><td></td></tr><tr><td>ATMC (Posterior predictive)</td><td></td></tr><tr><td>SGNHT (single sample)</td><td></td></tr><tr><td>SGNHT (Posterior predictive)</td><td></td></tr></table>
+
+where φ(i)d is the direction vector and $\phi _ { s } ^ { ( i ) }$ is the magnitude of a feature vector $\theta ^ { ( i ) }$ . Weight normalization does not depend on batch statistics and is compatible with MCMC methods.
+
+The scale of the direction vector does not affect the outputs of the model. It does however affect the effective step size (Wu et al., 2018). Therefore the prior on the direction vector $\phi _ { d } ^ { ( i ) }$ is chosen such that it is forced to unit length
+
+$$
+p ( \phi _ { d } ^ { ( i ) } ) \propto \exp \left[ - \frac { d } { 2 } \left( \left\| \phi _ { d } ^ { ( i ) } \right\| ^ { 2 } - 1 \right) ^ { 2 } \right] .
+$$
+
+The prior on the scales $p ( \phi _ { s } )$ is problem-specific and can for example be chosen to encode a preference for structurally sparse models.
+
+# 5 Experiments
+
+The experiments presented here aim to demonstrate that the ATMC sampler is competitive with a well-tuned optimization baseline for large-scale datasets and models. We use the TensorFlow official implementation of ResNet-56 and ResNet-50 on Cifar10 and ImageNet, respectively. We compare our ATMC sampler to an optimization baseline with and without BatchNorm. For the optimization baseline without BatchNorm we use the ResNet $^ { + + }$ architecture as described in Sec. 4. For the baseline with BatchNorm we found standard ResNet with Xavier initialization and the ReLU non-linearity to work better.
+
+For the ATMC sampler we report both the performance of a single sample and the estimated posterior predictive based on a finite number of samples. Similar to earlier work (Zhang et al., 2019b) we found that many fewer samples are needed when a cyclic step size $h _ { t } =$ $h _ { 0 } * \frac { 1 } { 2 } [ 1 + \cos ( \pi \bmod [ t , n ] ) ]$ with cycle length $n$ is used. The final sample in each cycle is used to estimate the posterior predictive.
+
+For $\mathrm { R e s N e t { + + } }$ we further use a group Laplace prior $p ( \theta _ { i } ) \propto \exp ( - \left\| \theta _ { i } \right\| / b )$ with $b = 5$ to regularize the scales of each linear feature in ResNet $^ { + + }$ . The momentum noise is chosen as $D _ { c } = - \log ( 0 . 9 ) / h _ { 0 }$ such that the friction applied to the momentum is at least 0.9.
+
+# 5.1 Cifar 10
+
+For Cifar10 we choose the step size $h _ { 0 } = 0 . 0 0 1$ and the cycle length is set to 50 epochs. The momentum hyper-parameters are $m = ( 0 . 0 0 0 3 / h _ { 0 } ) ^ { - 2 }$ and $c = 0 . 0 0 1 / h _ { 0 }$ such that the average speed and maximum speed per step are 0.0003 and 0.001, respectively. The number of convolution filters is doubled to 32 compared to the original ResNet-56 implementation. We use a single V100 GPU with a batch size of 128. The sampler runs for 1000 epochs and we start collecting samples for the posterior predictive after 150 epochs. The optimization baseline converges in 180 epochs. We also report the results of sampling with a sampler based Nos´e-Hoover thermostats (SGNHT) (Ding et al., 2014; Lu et al., 2016) applied to the ResNet++ architecture.
+
+Table 2: Performance on ImageNet with ResNet-50 model. The posterior predictive is estimated using a sample of the posterior parameters at the end of each learning rate cycle.   
+
+<table><tr><td>Setup</td><td>Top 1 acc. [%]</td></tr><tr><td>SGD</td><td>70.9</td></tr><tr><td>SGD + BatchNorm</td><td>76.2</td></tr><tr><td>ATMC (single sample)</td><td>74.2</td></tr><tr><td>ATMC (Posterior predictive)</td><td></td></tr><tr><td>SGNHT (single sample)</td><td></td></tr><tr><td>SGNHT ( (Posterior predictive)</td><td></td></tr></table>
+
+Table 1 lists the test set performance for Cifar10. A single sample from the posterior already outperforms the baseline without BatchNorm by a significant margin in both test accuracy and log-likelihood. Using BatchNorm significantly improves the generalization of the optimization baseline. It outperforms the estimate of the posterior predictive in accuracy yet it does not have a better test log-likelihood.
+
+To further analyze the quality of the uncertainty estimates, we group each model’s prediction in 8 equally sized bins based on the confidence $p ( \hat { \omega } _ { i } | x _ { i } )$ where $\hat { \omega } _ { i }$ is the maximum probability class for example $x _ { i }$ . If the probabilities are well-calibrated, the average confidence should be close to the average accuracy. Figure 2 shows the calibration of the uncertainty estimates for the posterior predictive and optimization baselines. The posterior predictive is calibrated for the least confident predictions $p ( \hat { \omega } _ { i } | x _ { i } ) < 0 . 9$ and shows less bias towards overconfidence compared to the models trained with SGD.
+
+# 5.2 ImageNet
+
+For the ImageNet experiments we use an initial step size $h _ { 0 } = 0 . 0 0 0 5$ and a cycle length of 20 epochs. The other hyper-parameters for the sampler are the same as for the Cifar10 experiments. We use a a single Google Cloud TPUv3 with a batch size of 1024. We did not observe a significant difference in wall clock time per training step between SGD and ATMC on the same model. Each training step using ResNet $^ +$ BatchNorm model takes $2 0 \%$ longer in wall clock time compared to a single train step using ResNet $^ { + + }$ . Samples for the posterior predictive are collected after 150 epochs and the sampler runs for 1000 epochs. The optimization baseline converges in 90 epochs.
+
+Table 2 lists the results for ImageNet classification. A single sample from the posterior outperforms the optimization baseline without BatchNorm. The posterior predictive based on ATMC outperforms the optimizer with BatchNorm by a wide margin in both accuracy and test log-likelihood. We note that the sampler runs significantly longer (10x) compared to the optimization baseline because it takes a long time for the posterior predictive estimate to converge. However, the posterior predictive of ATMC matches the accuracy of the optimization baseline with BatchNorm (76.2%) after 240 epochs.
+
+Figure 3 shows the quality of the uncertainty for various levels of confidence. Again, the ATMC based posterior predictive produces much better calibrated predictions and is almost perfectly calibrated for low confidence predictions $p ( \hat { \omega } _ { i } | x _ { i } ) < 0 . 9$ and shows less bias towards overconfidence compared to the optimization baseline.
+
+# 6 Discussion
+
+The empirical results show it is possible to sample the posterior distribution of neural networks on large scale image classification problems like ImageNet. A major obstacle for sampling the posterior of ResNets in particular is the lack of compatibility with BatchNorm. Using recent advances in initialization and the SELU activation function we are able to stabilize and speed up training of ResNets without resorting to BatchNorm. Nonetheless, we observe that BatchNorm still offers a unique advantage in terms of generalization performance. We hope that future work will allow the implicit inductive bias that BatchNorm has to be transferred into an explicit prior that is compatible with sampling methods.
+
+Multiple posterior samples provide a much more accurate estimate of the posterior predictive, and consequently much better accuracy and uncertainty estimates. For inference, making predictions using a large ensemble of models sampled from the posterior can be costly. Variational Inference methods can be used to quickly characterize a local mode of the posterior (Blundell et al., 2015). More recent work shows that a running estimate of the mean and variance of the parameters during training can also be used to approximate a mode of the posterior (Maddox et al., 2019). Methods like distillation could potentially be used to compress a high-quality ensemble into a single network with a limited computational budget (Balan et al., 2015).
+
+Although the form in (4) is very general, alternative methods for dealing with stochastic gradients have been proposed in the literature. One approach is to estimate the covariance of the stochastic gradient noise $B$ explicitly and use it correct and pre-condition the sampling dynamics (Ahn et al., 2012; Li et al., 2016).
+
+Other sampling methods are not based on an SDE that converges to the target distribution. Under some conditions stochastic optimization methods can be interpreted as such a biased sampling method (Mandt et al., 2017). Predictions based on multiple samples from the trajectory of SGD have been used successfully for obtaining uncertainty estimates in large scale Deep Learning (Maddox et al., 2019). However, these methods rely on tuning hyperparameters in such a way that just the right amount of noise is inserted.
+
+# 7 Conclusion
+
+This work introduces the ATMC sampler, a robust posterior sampling method that scales to large deep learning problems. To the best of our knowledge, we are the first to successfully train neural networks using MCMC on ImageNet. In a BatchNorm free setting, a single sample from the posterior generated by ATMC outperforms the optimization baseline. A posterior predictive estimate outperforms the optimization baseline with BatchNorm on ImageNet. Based on these empirical results we hope the ATMC sampler will enable new applications of Bayesian inference in deep learning.
+
+# References
+
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+
+Anoop Korattikara Balan, Vivek Rathod, Kevin P Murphy, and Max Welling. Bayesian dark knowledge. In Advances in Neural Information Processing Systems, pp. 3438–3446, 2015.
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+Charles Blundell, Julien Cornebise, Koray Kavukcuoglu, and Daan Wierstra. Weight uncertainty in neural network. In International Conference on Machine Learning, pp. 1613– 1622, 2015.
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+Radford M Neal et al. Mcmc using hamiltonian dynamics. Handbook of markov chain monte carlo, 2(11):2, 2011.
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+Edward Nelson. Dynamical theories of Brownian motion, volume 3. Princeton university press, 1967.
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+Ian Osband and Benjamin Van Roy. Why is posterior sampling better than optimism for reinforcement learning? In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 2701–2710. JMLR. org, 2017.
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+Bala Rajaratnam and Doug Sparks. Mcmc-based inference in the era of big data: A fundamental analysis of the convergence complexity of high-dimensional chains. arXiv preprint arXiv:1508.00947, 2015.
+
+Tim Salimans and Durk P Kingma. Weight normalization: A simple reparameterization to accelerate training of deep neural networks. In Advances in Neural Information Processing Systems, pp. 901–909, 2016.
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+        "text": "Bayesian Inference for Large Scale Image Classification ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "Bayesian inference promises to ground and improve the performance of deep neural networks. It promises to be robust to overfitting, to simplify the training procedure and the space of hyperparameters, and to provide a calibrated measure of uncertainty that can enhance decision making, agent exploration and prediction fairness. Markov Chain Monte Carlo (MCMC) methods enable Bayesian inference by generating samples from the posterior distribution over model parameters. Despite the theoretical advantages of Bayesian inference and the similarity between MCMC and optimization methods, the performance of sampling methods has so far lagged behind optimization methods for large scale deep learning tasks. We aim to fill this gap and introduce ATMC, an adaptive noise MCMC algorithm that estimates and is able to sample from the posterior of a neural network. ATMC dynamically adjusts the amount of momentum and noise applied to each parameter update in order to compensate for the use of stochastic gradients. We use a ResNet architecture without batch normalization to test ATMC on the Cifar10 benchmark and the large scale ImageNet benchmark and show that, despite the absence of batch normalization, ATMC outperforms a strong optimization baseline in terms of both classification accuracy and test log-likelihood. We show that ATMC is intrinsically robust to overfitting on the training data and that ATMC provides a better calibrated measure of uncertainty compared to the optimization baseline. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "In contrast to optimization approaches in machine learning that derive a single estimate for the weights of a neural network, Bayesian inference aims at deriving a posterior distribution over the weights of the network. This makes it possible to sample model instances from the distribution over the weights and offers unique advantages. Multiple model instances can be aggregated to obtain robust uncertainty estimates over the network’s predictions; uncertainty estimates are crucial in domains such as medical diagnosis and autonomous driving where following a model’s incorrect predictions can result in catastrophe (Kendall & Gal, 2017). Sampling a distribution, as opposed to optimizing a loss, is less prone to overfitting and more training doesn’t decrease test performance. Bayesian inference can also be applied to differential privacy, where each individual sample has increased privacy guarantees (Wang et al., 2015), and to reinforcement learning, where one can leverage model uncertainty to balance between exploration and exploitation (Osband & Van Roy, 2017). ",
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+        "type": "text",
+        "text": "Traditional Markov Chain Monte Carlo (MCMC) methods like HMC (Neal et al., 2011) are a standard class of methods for generating samples from the posterior distribution over model parameters. These methods are seldom applied in deep learning because they have traditionally failed to scale well with large datasets and many parameters (Rajaratnam $\\&$ Sparks, 2015). Stochastic Gradient MCMC (SG-MCMC) methods have fared somewhat better in scaling to large datasets due to their close relationship to stochastic optimization methods. For example the SGLD sampler (Welling & Teh, 2011) amounts to performing stochastic gradient descent while adding Gaussian noise to each parameter update. Despite these improvements, samplers like SGLD are only guaranteed to converge to the correct distribution when the step size is annealed to zero; additional control variates have been developed to mitigate this to some extent (Ahn et al., 2012; Ding et al., 2014). ",
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+        "table_caption": [
+            "Algorithm 1 The ATMC sampler. The algorithm accepts the initialized model parameters $\\pmb { \\theta } _ { 0 }$ , step size $h$ , pre-conditioner $m$ , and momentum noise $D _ { c }$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"4\">1: procedure ATMC_TRAINING(0o,h,m,D)</td></tr><tr><td>2:</td><td>Po←0</td><td></td><td></td></tr><tr><td>3:</td><td>m←0</td><td></td><td></td></tr><tr><td>4:</td><td>whilet&lt;Tdo</td><td></td><td></td></tr><tr><td>5:</td><td></td><td>Gt ← minibatch_gradient(0t)</td><td></td></tr><tr><td>6:</td><td>nt ← random_normal()</td><td></td><td></td></tr><tr><td>7:</td><td>αt ←max(D-εt,0)</td><td></td><td></td></tr><tr><td>8:</td><td>βt←αt+t</td><td></td><td></td></tr><tr><td>9:</td><td>Pt+h←eβth</td><td>exp[Bth]-1Gt+ Pt-</td><td>exp[2βth]-1 αtnt</td></tr><tr><td>10:</td><td>0t+h←0t+hPt+h</td><td>βt</td><td>βt</td></tr><tr><td>11:</td><td>St+h←t+h</td><td>-1]</td><td></td></tr><tr><td>12:</td><td></td><td>m</td><td></td></tr><tr><td></td><td>t←t+h</td><td></td><td></td></tr></table>",
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+        "text": "The objective of this work is to make Bayesian inference practical for deep learning by making SG-MCMC methods scale to large models and datasets. The contributions described in this work fall in three categories. We first propose the Adaptive Thermostat Monte Carlo (ATMC) sampler that offers improved convergence and stability. ATMC dynamically adjusts the amount of momentum and noise applied to each model parameter. Secondly, we improve an existing second order numerical integration method that is needed for the ATMC sampler. Third, since ATMC, like other SG-MCMC samplers, is not directly compatible with stochastic regularization methods such as batch normalization (BatchNorm) and Dropout (see Sect. 4), we construct the ResNet++ network by taking the original ResNet architecture (He et al., 2016), removing BatchNorm and introducing SELUs (Klambauer et al., 2017), Fixup initialization (Zhang et al., 2019a) and weight normalization (Salimans & Kingma, 2016). We design ResNet $^ { + + }$ so that its parameters are easy to sample from and the gradients are well-behaved even in the absence of BatchNorm. ",
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+        "text": "We show that the ATMC sampler is able to outperform optimization methods in terms of accuracy, log-likelihood and uncertainty calibration in the following settings. First, when using the ResNet $^ { + + }$ architecture for both the ATMC sampler and the optimization baseline, the ATMC sampler significantly outperforms the optimization baseline on both Cifar-10 and ImageNet. Secondly, when using the standard ResNet for the optimization baseline and the ResNet++ for the ATMC sampler, multiple samples of the ATMC that approximate the predictive posterior of the model are still able to outperform the optimization baseline on ImageNet. Using the ResNet $^ { + + }$ architecture, the ATMC sampler reduces the need for hyper-parameter tuning since it does not require early stopping, does not use stochastic regularization, is not prone to over-fitting on the training data and avoids a carefully tuned learning rate decay schedule. ",
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+    {
+        "type": "text",
+        "text": "2 ATMC Sampler ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "In this section we define the Stochastic Differential Equation (SDE) that gives rise to the ATMC sampler described in Algorithm 1. A detailed background and framework for constructing SDEs that converge to a target distribution can be found in (Ma et al., 2015). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 General form of the SDE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Our starting point for constructing the ATMC sampler is the framework of Stochastic Differential Equations. We are interested in SDEs that converge to a distribution $p ( z )$ over the vector $z \\in \\mathbb { R } ^ { d }$ for which we can evaluate $\\nabla \\log p ( z )$ . Because only the gradient of $\\log p ( z )$ ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "is required, it is sufficient to define an energy function $H ( z ) = - \\log p ( z ) + C$ up to a constant $C$ . As a consequence, we can sample from the posterior distribution $p ( \\theta | x )$ by only evaluating the energy function gradient $\\nabla H ( \\theta ) = - \\nabla \\log p ( x , \\theta )$ . The general form of SDEs converging to $p ( z )$ for which only the gradient of $p ( z )$ is required is as follows (Ma et al., 2015): ",
+        "bbox": [
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+            102,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c48fae0a891e4dd94dbb8437589c7a18be73b39018fe253810f579bf8adddd60.jpg",
+        "text": "$$\n\\small { \\begin{array} { l } { \\displaystyle { d z = - \\left[ D ( z ) + Q ( z ) \\right] \\nabla H ( z ) d t + \\Gamma ( z ) d t + \\sqrt { 2 D ( z ) } d W _ { t } , \\quad \\Gamma _ { i } ( z ) = \\sum _ { j } ^ { d } \\frac { \\partial \\left[ D _ { i j } ( z ) + Q _ { i j } ( z ) \\right] } { \\partial z _ { j } } , } } \\end{array} }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            205,
+            820,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $D ( z )$ is a positive-definite matrix that determines the amount of noise, $Q ( z )$ is a skew-symmetric matrix that mixes energy between variables, $W _ { t }$ is a Wiener process, and $\\Gamma ( z )$ is a correction factor that compensates for dynamics that depend on the current state $z$ . The ATMC sampler that we propose is an instance of (1) for specific definitions of $H ( z )$ , $D ( z )$ , and $Q ( z )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 Energy Function ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We start by defining the energy function $H ( z )$ . The energy function for the model posterior $p ( \\theta | x )$ is defined by the loss function $\\mathcal { L } ( \\theta ) = - \\log p ( x , \\theta )$ . Because the dataset $x$ is generally large, we would like to only evaluate a mini-batch loss $\\ddot { \\mathcal { L } } ( \\boldsymbol { \\theta } )$ . However, naively using a stochastic gradient in (1) will result in significant bias (Chen et al., 2014). Motivated by the Central Limit Theorem, the stochastic gradient is assumed to follow a Gaussian distribution $\\nabla \\tilde { \\mathcal { L } } ( \\theta ) \\sim \\mathcal { N } ( \\nabla \\mathcal { L } ( \\theta ) , B )$ where the covariance $B$ is additionally assumed to be diagonal and constant w.r.t. $\\theta$ . The energy function for the ATMC sampler is defined as: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6d009852acea78c9b6ddf49998bac412e8e87eb39ec4a79d429621a8c015cc54.jpg",
+        "text": "$$\nH ( \\theta , p , \\xi ) = \\mathcal { L } ( \\theta ) + K ( p ) + \\frac { 1 } { 2 } \\left( \\xi - \\frac { \\mathrm { d i a g } ( B ) } { 2 m } \\right) ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $p$ is the momentum, $K ( p )$ defines the momentum distribution, and $\\xi$ is a control variate referred to as the temperature. Both $p$ and $\\xi$ have the same dimensionality as $\\theta$ . The hyper-parameter $m$ controls the strength of the coupling between $\\xi$ and $p$ . The distribution of the control variate $p ( \\xi )$ depends on the amount of noise $B$ in the stochastic gradient estimate $\\ddot { \\mathcal { L } } ( \\boldsymbol { \\theta } )$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 Noise robust dynamics ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Next we define the dynamics $Q ( z )$ and $D ( z )$ such that the SDE that results from (1) can be simulated without the need to evaluate $B$ : ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/43b950835b031456323437e1729455cdd14f9c8163e828d35d1315acd42b8175.jpg",
+        "text": "$$\n\\begin{array} { r } { D ( \\theta , p , \\xi ) = \\left( \\begin{array} { c c c } { 0 } & { 0 } & { 0 } \\\\ { 0 } & { \\alpha ( \\xi ) m + \\frac { 1 } { 2 } B } & { 0 } \\\\ { 0 } & { 0 } & { 0 } \\end{array} \\right) , \\quad Q ( \\theta , p , \\xi ) = \\left( \\begin{array} { c c c } { 0 } & { - I } & { 0 } \\\\ { I } & { 0 } & { m \\nabla K ( p ) } \\\\ { 0 } & { - m \\nabla K ( p ) } & { 0 } \\end{array} \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            197,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\alpha ( \\xi )$ is a non-negative function that determines how the temperature $\\xi$ affects the amount of noise added to the momentum update. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We first illustrate the resulting SDE by using a simpler Gaussian momentum distribution $K ( p ) = \\left\\| p \\right\\| ^ { 2 } / ( 2 m )$ . Note that the variance of the momentum $\\operatorname { V a r } ( p ) = m$ is reused in (2) and (3) to control the strength of the coupling between $\\xi$ and $p$ . This will result in a temperature control with a momentum friction term proportional to $\\xi$ , unlike previously reported thermostat MCMC methods (Ding et al., 2014; Lu et al., 2016) where the friction term is proportional to $\\xi / m$ . We substitute the dynamics $Q ( z )$ and $D ( z )$ defined in (3) and energy function $H ( z )$ defined in (2) into (1): ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            102,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5c73f5f97fafff2b844a166980c52b7697f95365af690744f7f86ed6941f05c4.jpg",
+        "text": "$$\n\\binom { d \\theta } { d \\xi } = \\left( - \\nabla \\tilde { \\mathcal { L } } ( \\theta ) - \\beta ( \\xi ) p \\right) d t + \\left( \\begin{array} { c c c } { 0 } & { 0 } & { 0 } \\\\ { 0 } & { \\sqrt { 2 \\alpha ( \\xi ) m } } & { 0 } \\\\ { 0 } & { 0 } & { 0 } \\end{array} \\right) d W _ { t } , \\beta ( \\xi ) = \\alpha ( \\xi ) + \\xi ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            217,
+            148,
+            779,
+            198
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where we use $\\nabla \\tilde { \\mathcal { L } } ( \\boldsymbol { \\theta } ) d t = \\nabla \\mathcal { L } ( \\boldsymbol { \\theta } ) d t + \\sqrt { B } d W _ { t }$ to replace the gradient of the loss with the minibatch estimate. The momentum $p$ is dampened by a friction term $\\beta ( \\xi )$ that depends on the choice of $\\alpha ( \\xi )$ . The stochastic gradient noise $B$ does not show up in (4) due to the particular choice of energy function $H ( z )$ and dynamics $Q ( z ) , D ( z )$ . Note however this analysis relies on the assumption that the covariance of the stochastic gradient noise $B$ is constant in $\\theta$ and a single temperature variable per parameter can only correct for a diagonal covariance $B$ . We do not expect that this assumption will hold in practice and the approximation will therefore lead to bias in the samples. However, annealing the step size $h$ will reduce the error due to mini-batching together with other sources of discretization error (Welling $\\&$ Teh, 2011). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.4 Adaptive Noise Thermostat ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            364,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, we must choose a function $\\alpha ( \\xi )$ which controls the amount of noise and momentum damping $\\beta ( \\xi )$ . Previous work uses the Nos´e-Hoover thermostat that is defined by $\\alpha ( \\xi ) = D _ { c }$ where $D _ { c }$ is a constant determining the amount of noise added to the momentum update (Ding et al., 2014). Although the Nos´e-Hoover thermostat is able to correct the stochastic gradient noise $B$ , the correction comes at the cost of slower convergence because additional friction $\\beta ( \\xi )$ is applied as $B$ increases. Another drawback of the Nos´e-Hoover thermostat is that it causes negative friction when $\\xi < - D _ { c }$ . In the negative friction phase $\\beta ( \\xi ) < 0$ , previous gradient terms are amplified rather than dampened. Although this behavior is mathematically sound we find that it can cause exploding momentum variables. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Our choice of $\\alpha ( \\xi )$ is based on the idea that negative friction should not occur and convergence speed should not be reduced by the stochastic gradient noise. Based on this intuition, we define the ATMC sampler by $\\alpha ( \\xi ) = \\operatorname* { m a x } ( D - \\xi , 0 )$ . The ATMC sampler is best characterized by the various temperature stages. For $0 < \\xi < D _ { c }$ the total amount of noise added to the momentum is $D _ { c }$ and the friction coefficient $\\beta ( \\xi ) = D _ { c }$ . At this stage, the stochastic gradient noise is compensated for by adding less noise to the momentum update. If $B \\gg D _ { c }$ the dominant stage will be $\\xi > D$ resulting in $\\beta ( \\xi ) < D _ { c }$ and zero noise being added to the momentum. Finally, when $\\xi < 0$ the friction coefficient $\\beta ( \\xi ) = D _ { c }$ and the noise added to the momentum is proportional to $D _ { c } - \\xi$ . Thus, the momentum always experiences a minimum amount of friction $\\beta ( \\xi ) \\ge D _ { c }$ determined by the hyper-parameter $D _ { c }$ and the noise added to the momentum update is automatically adjusted based on the amount of noise present in the stochastic gradients. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.5 Momentum energy function ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Following (Lu et al., 2016), we generalize the momentum energy function $K ( p )$ to the symmetric hyperbolic distribution which is defined as follows (Lu et al., 2016): ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f8f09436596883b38c54dd975a0cbf91a93b012eea4fbc2bb55deca25b01494d.jpg",
+        "text": "$$\nK ( p ) = \\sum _ { i } m c ^ { 2 } \\left[ \\sqrt { \\frac { p _ { i } ^ { 2 } } { m ^ { 2 } c ^ { 2 } } + 1 } - 1 \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            777,
+            625,
+            819
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $m$ and $c$ are hyper-parameters. The Gaussian kinetic energy $K ( p ) = \\left\\| p \\right\\| ^ { 2 } / ( 2 m )$ is a special case obtained by taking the limit $c  \\infty$ . The magnitude of parameter updates $\\| \\Delta \\theta \\|$ is determined by the gradient of the momentum: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a2eec3c8f2c04bbe17d98d00139c4402621f62cc44598fa68a89dfeeb86d8364.jpg",
+        "text": "$$\n\\| \\Delta \\theta \\| = \\| \\nabla K ( p ) \\| = \\left\\| \\frac { p } { M ( p ) } \\right\\| , \\quad M ( p ) = m \\sqrt { \\frac { p ^ { 2 } } { m ^ { 2 } c ^ { 2 } } + 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            892,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Hence, the hyperbolic distribution results in relativistic momentum dynamics where the parameter updates are upper bounded by $c$ and the pre-conditioner √ $M ( p )$ depends on $p$ . The average update magnitude $\\mathcal { E } [ | \\nabla K ( p ) | | ] \\approx 1 / \\sqrt { m }$ for $c \\gg m$ . Consequently, the parameters $m$ and $c$ are interpretable hyper-parameters controlling the average and maximum parameter update per step together with the step size $h$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The SDE we derive in (4) and integrate in Sec. 3 uses a Gaussian momentum energy function for clarity. Deriving ATMC with a different momentum distribution like the hyperbolic distribution amounts to substituting (2), (3), and the alternative momentum distribution into (1). For the hyperbolic distribution, the dynamic friction coefficient $\\beta ( \\xi )$ will also depend on $p$ . For the numerical integration of (4) with a hyperbolic momentum distribution we assume $\\beta ( \\xi )$ to be constant in $p$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Improved numerical integrator for MCMC samplers ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In this section we construct the numerical integrator required to numerically approximate the ATMC sampler defined in (4). An efficient numerical integrator can be constructed by splitting the SDE into two terms: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3306f692019ffe4bee61d8ce562ed4285fdda4ca4908221a55790b8018cb75cd.jpg",
+        "text": "$$\n\\binom { d \\theta } { d \\xi } = \\underbrace { \\binom { p / m } { 0 } } _ { \\displaystyle { p ^ { 2 } / m - 1 } } d t + \\underbrace { \\left( - \\nabla \\tilde { \\mathcal { L } } ( \\theta ) - \\beta ( p , \\xi ) p \\right) d t + \\left( 0 \\begin{array} { c c c } { 0 } & { 0 } & { 0 } \\\\ { 0 } & { \\sqrt { 2 \\alpha ( \\xi ) m } } & { 0 } \\\\ { 0 } & { 0 } & { 0 } \\end{array} \\right) d W _ { t } } _ { B } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Hence, we obtain a linear ODE in part (A) that updates the parameters $\\theta$ and the thermostats $\\xi$ and a linear SDE in part (B) that updates the momentum $p$ . The operators that simulate these dynamics exactly for a time step $h$ are denoted $\\phi _ { A } ^ { h }$ and $\\phi _ { B } ^ { h }$ , respectively. Using the Strang splitting scheme yields a second order method (Chen et al., 2015): ",
+        "bbox": [
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+            453,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39adc6a7ada1306ba2e1b895f27512cba327ba539bf66761ccb5fd6368f187ed.jpg",
+        "text": "$$\n\\phi ^ { h } = \\phi _ { B } ^ { h / 2 } \\circ \\phi _ { A } ^ { h } \\circ \\phi _ { B } ^ { h / 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            512,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The first operator $\\phi _ { A } ^ { h }$ is given by ",
+        "bbox": [
+            176,
+            534,
+            411,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5e87a3a0a55940d0bed7c545c17a306e67c4c1d829e9ee738d21c4ed25120585.jpg",
+        "text": "$$\n\\begin{array} { r } { \\phi _ { A } ^ { h } ( z _ { t } ) = \\left( \\theta _ { t } + h \\frac { p _ { t } } { m } \\quad p _ { t } \\quad \\xi _ { t } + h \\left[ \\frac { p _ { t } ^ { 2 } } { m } - 1 \\right] \\right) ^ { T } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            553,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The second operator $\\phi _ { B }$ is an instance of the Ornstein–Uhlenbeck process which can also be computed analytically as follows (Nelson, 1967): ",
+        "bbox": [
+            171,
+            589,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/52ea47c130f45795297214da91996a2fe17fbd83d9f10fbe910f8ad4792fe5a1.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\phi _ { B } ^ { h } ( z _ { t } ) = \\Big ( \\theta _ { t } } & { e ^ { \\beta ( \\xi _ { t } ) h } \\Big [ p _ { t } - \\gamma _ { 1 } ( \\xi _ { t } ) \\nabla \\tilde { \\mathcal { L } } ( \\theta _ { t } ) + \\sqrt { \\gamma _ { 2 } ( \\xi _ { t } ) \\alpha ( \\xi _ { t } ) } ~ \\eta _ { t } \\Big ] \\quad \\xi _ { t } \\Big ) ^ { T } , } \\\\ & { \\quad \\quad \\gamma _ { a } ( \\xi _ { t } ) = \\frac { \\exp [ a \\beta ( \\xi _ { t } ) h ] - 1 } { \\beta ( \\xi _ { t } ) } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            635,
+            743,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\eta _ { t }$ is isotropic Gaussian noise. Previous work (Chen et al., 2015) on higher order integrators for samplers splits the SDE into three parts where the third term is obtained from separating the friction term from the other terms in the momentum update $\\phi _ { B }$ . By integrating (10) exactly the gradient step and the noise and gradient term are directly affected by the friction. An exact momentum update provides additional robustness to large gradients because the temperature will increase in order to compensate for momentum updates that would lead to excessively large steps. Another advantage of a two-way split integrator is that the first and last steps in (8) can be fused together such that only a momentum update is performed per iteration. Algorithm 1 shows the pseudocode for the ATMC sampler with the split integrator defined in (9) and (10). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 The ResNet++ Architecture ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            864,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The generalization performance of large neural nets trained using optimization depend on stochastic regularization methods like Dropout (Srivastava et al., 2014) and BatchNorm (Ioffe & Szegedy, 2015). These methods implicitly add noise into the model parameters (Kingma et al., 2015; Teye et al., 2018) and significantly boost training performance and generalization for image classifiers. These methods can be interpreted as a coarse approximation of Bayesian Inference (Kingma et al., 2015; Teye et al., 2018). But a stochastic gradient sampler like ATMC already adds the necessary amount of noise and combined with BatchNorm or Dropout it leads to underfitting. We thus define a BatchNorm free version of ResNet called ResNet++ that includes SELUs (Klambauer et al., 2017), Fixup initialization (Zhang et al., 2019a) and weight normalization (Salimans & Kingma, 2016) (see Fig. 1). We use ATMC to fill the significant gap in performance due to the absence of BatchNorm in ResNet++. ",
+        "bbox": [
+            174,
+            895,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/ac3e7271eab15513a931e184806164ab19b90c8adb69f1de675035c45b556885.jpg",
+        "table_caption": [
+            "Figure 1: Residual blocks in respectively the ResNet and $\\mathrm { R e s N e t { + + } }$ architectures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>x</td><td rowspan=1 colspan=1>Conv 1x1</td><td rowspan=1 colspan=1>BatchNorm</td><td rowspan=1 colspan=1>ReLU</td><td rowspan=1 colspan=1>Conv 3x3</td><td rowspan=1 colspan=1>BatchNorm</td><td rowspan=1 colspan=1>ReLU</td><td rowspan=1 colspan=1>Conv 1x1</td><td rowspan=1 colspan=1>BatchNorm</td><td rowspan=1 colspan=1>+x</td><td rowspan=1 colspan=1>ReLU</td></tr><tr><td rowspan=1 colspan=1>x</td><td rowspan=1 colspan=1>Conv 1x1</td><td rowspan=1 colspan=1>SeLU</td><td rowspan=1 colspan=1>Conv 3x3</td><td rowspan=1 colspan=1>SeLU</td><td rowspan=1 colspan=1>Conv 1x1</td><td rowspan=1 colspan=1>+x</td><td rowspan=1 colspan=1>SeLU</td><td></td><td></td><td></td></tr></table>",
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+    {
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+        "image_caption": [
+            "Figure 2: Calibration plot for Cifar10 "
+        ],
+        "image_footnote": [],
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+        "image_caption": [
+            "Figure 3: Calibration plot for ImageNet "
+        ],
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+    {
+        "type": "text",
+        "text": "4.1 SELU ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We find the SELU activation to work well in BatchNorm free networks. SELU forces the statistics of the activations towards zero mean and unit variance (Klambauer et al., 2017). The SELU activation function additionally has a non-zero gradient everywhere which could improve the mixing of the sampler by providing a more informative gradient. ",
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+    {
+        "type": "text",
+        "text": "4.2 Fixup initialization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "ResNets are known to scale well with depth (He et al., 2016). However, the additive effect of the residual branch causes the magnitudes of the activations to increase with the number of residual connections. Fixup is a recently proposed initialization method that mitigates the exploding residual branch problem without using BatchNorm (Zhang et al., 2019a). We use a simplified version of Fixup by initializing the scales of the final layer in each residual branch to a small constant. ",
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+        "type": "text",
+        "text": "4.3 Weight normalization ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We use weight normalization (Salimans $\\&$ Kingma, 2016) to separate the direction and scale of each linear feature vector ",
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+    {
+        "type": "equation",
+        "img_path": "images/1bc39962fa60ed8db3039a6c3d4a1c20305e20631042530249ec27236197eebf.jpg",
+        "text": "$$\n\\theta ^ { ( i ) } = \\phi _ { s } ^ { ( i ) } \\frac { \\phi _ { d } ^ { ( i ) } } { \\left\\| \\phi _ { d } ^ { ( i ) } \\right\\| } ,\n$$",
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+        "img_path": "images/119835f31ca38b2f1ddca65a16037fbefbf8d134ed39b34fa832a0519ef08469.jpg",
+        "table_caption": [
+            "Table 1: Performance on Cifar10 with ResNet-56 model. The posterior predictive is estimate using a sample of the posterior parameters at the end of each learning rate cycle. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Setup</td><td>Top 1 acc. [%]</td></tr><tr><td>SGD</td><td>91.5</td></tr><tr><td>SGD + BatchNorm</td><td>94.4</td></tr><tr><td>ATMC (single sample)</td><td></td></tr><tr><td>ATMC (Posterior predictive)</td><td></td></tr><tr><td>SGNHT (single sample)</td><td></td></tr><tr><td>SGNHT (Posterior predictive)</td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where φ(i)d is the direction vector and $\\phi _ { s } ^ { ( i ) }$ is the magnitude of a feature vector $\\theta ^ { ( i ) }$ . Weight normalization does not depend on batch statistics and is compatible with MCMC methods. ",
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+    {
+        "type": "text",
+        "text": "The scale of the direction vector does not affect the outputs of the model. It does however affect the effective step size (Wu et al., 2018). Therefore the prior on the direction vector $\\phi _ { d } ^ { ( i ) }$ is chosen such that it is forced to unit length ",
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+        "type": "equation",
+        "img_path": "images/fb2dba7c5eb01d9a61c356c954e4a813f28745e797793e9cec38087fffdc0b66.jpg",
+        "text": "$$\np ( \\phi _ { d } ^ { ( i ) } ) \\propto \\exp \\left[ - \\frac { d } { 2 } \\left( \\left\\| \\phi _ { d } ^ { ( i ) } \\right\\| ^ { 2 } - 1 \\right) ^ { 2 } \\right] .\n$$",
+        "text_format": "latex",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The prior on the scales $p ( \\phi _ { s } )$ is problem-specific and can for example be chosen to encode a preference for structurally sparse models. ",
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+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The experiments presented here aim to demonstrate that the ATMC sampler is competitive with a well-tuned optimization baseline for large-scale datasets and models. We use the TensorFlow official implementation of ResNet-56 and ResNet-50 on Cifar10 and ImageNet, respectively. We compare our ATMC sampler to an optimization baseline with and without BatchNorm. For the optimization baseline without BatchNorm we use the ResNet $^ { + + }$ architecture as described in Sec. 4. For the baseline with BatchNorm we found standard ResNet with Xavier initialization and the ReLU non-linearity to work better. ",
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+    {
+        "type": "text",
+        "text": "For the ATMC sampler we report both the performance of a single sample and the estimated posterior predictive based on a finite number of samples. Similar to earlier work (Zhang et al., 2019b) we found that many fewer samples are needed when a cyclic step size $h _ { t } =$ $h _ { 0 } * \\frac { 1 } { 2 } [ 1 + \\cos ( \\pi \\bmod [ t , n ] ) ]$ with cycle length $n$ is used. The final sample in each cycle is used to estimate the posterior predictive. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For $\\mathrm { R e s N e t { + + } }$ we further use a group Laplace prior $p ( \\theta _ { i } ) \\propto \\exp ( - \\left\\| \\theta _ { i } \\right\\| / b )$ with $b = 5$ to regularize the scales of each linear feature in ResNet $^ { + + }$ . The momentum noise is chosen as $D _ { c } = - \\log ( 0 . 9 ) / h _ { 0 }$ such that the friction applied to the momentum is at least 0.9. ",
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+    {
+        "type": "text",
+        "text": "5.1 Cifar 10",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For Cifar10 we choose the step size $h _ { 0 } = 0 . 0 0 1$ and the cycle length is set to 50 epochs. The momentum hyper-parameters are $m = ( 0 . 0 0 0 3 / h _ { 0 } ) ^ { - 2 }$ and $c = 0 . 0 0 1 / h _ { 0 }$ such that the average speed and maximum speed per step are 0.0003 and 0.001, respectively. The number of convolution filters is doubled to 32 compared to the original ResNet-56 implementation. We use a single V100 GPU with a batch size of 128. The sampler runs for 1000 epochs and we start collecting samples for the posterior predictive after 150 epochs. The optimization baseline converges in 180 epochs. We also report the results of sampling with a sampler based Nos´e-Hoover thermostats (SGNHT) (Ding et al., 2014; Lu et al., 2016) applied to the ResNet++ architecture. ",
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+    {
+        "type": "table",
+        "img_path": "images/df9bd8a7638e6d9b5322767bd18d0225eab3e1f11b837d8a653b4c4aa194e9b3.jpg",
+        "table_caption": [
+            "Table 2: Performance on ImageNet with ResNet-50 model. The posterior predictive is estimated using a sample of the posterior parameters at the end of each learning rate cycle. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Setup</td><td>Top 1 acc. [%]</td></tr><tr><td>SGD</td><td>70.9</td></tr><tr><td>SGD + BatchNorm</td><td>76.2</td></tr><tr><td>ATMC (single sample)</td><td>74.2</td></tr><tr><td>ATMC (Posterior predictive)</td><td></td></tr><tr><td>SGNHT (single sample)</td><td></td></tr><tr><td>SGNHT ( (Posterior predictive)</td><td></td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 1 lists the test set performance for Cifar10. A single sample from the posterior already outperforms the baseline without BatchNorm by a significant margin in both test accuracy and log-likelihood. Using BatchNorm significantly improves the generalization of the optimization baseline. It outperforms the estimate of the posterior predictive in accuracy yet it does not have a better test log-likelihood. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To further analyze the quality of the uncertainty estimates, we group each model’s prediction in 8 equally sized bins based on the confidence $p ( \\hat { \\omega } _ { i } | x _ { i } )$ where $\\hat { \\omega } _ { i }$ is the maximum probability class for example $x _ { i }$ . If the probabilities are well-calibrated, the average confidence should be close to the average accuracy. Figure 2 shows the calibration of the uncertainty estimates for the posterior predictive and optimization baselines. The posterior predictive is calibrated for the least confident predictions $p ( \\hat { \\omega } _ { i } | x _ { i } ) < 0 . 9$ and shows less bias towards overconfidence compared to the models trained with SGD. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 ImageNet ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "For the ImageNet experiments we use an initial step size $h _ { 0 } = 0 . 0 0 0 5$ and a cycle length of 20 epochs. The other hyper-parameters for the sampler are the same as for the Cifar10 experiments. We use a a single Google Cloud TPUv3 with a batch size of 1024. We did not observe a significant difference in wall clock time per training step between SGD and ATMC on the same model. Each training step using ResNet $^ +$ BatchNorm model takes $2 0 \\%$ longer in wall clock time compared to a single train step using ResNet $^ { + + }$ . Samples for the posterior predictive are collected after 150 epochs and the sampler runs for 1000 epochs. The optimization baseline converges in 90 epochs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2 lists the results for ImageNet classification. A single sample from the posterior outperforms the optimization baseline without BatchNorm. The posterior predictive based on ATMC outperforms the optimizer with BatchNorm by a wide margin in both accuracy and test log-likelihood. We note that the sampler runs significantly longer (10x) compared to the optimization baseline because it takes a long time for the posterior predictive estimate to converge. However, the posterior predictive of ATMC matches the accuracy of the optimization baseline with BatchNorm (76.2%) after 240 epochs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Figure 3 shows the quality of the uncertainty for various levels of confidence. Again, the ATMC based posterior predictive produces much better calibrated predictions and is almost perfectly calibrated for low confidence predictions $p ( \\hat { \\omega } _ { i } | x _ { i } ) < 0 . 9$ and shows less bias towards overconfidence compared to the optimization baseline. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 Discussion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The empirical results show it is possible to sample the posterior distribution of neural networks on large scale image classification problems like ImageNet. A major obstacle for sampling the posterior of ResNets in particular is the lack of compatibility with BatchNorm. Using recent advances in initialization and the SELU activation function we are able to stabilize and speed up training of ResNets without resorting to BatchNorm. Nonetheless, we observe that BatchNorm still offers a unique advantage in terms of generalization performance. We hope that future work will allow the implicit inductive bias that BatchNorm has to be transferred into an explicit prior that is compatible with sampling methods. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Multiple posterior samples provide a much more accurate estimate of the posterior predictive, and consequently much better accuracy and uncertainty estimates. For inference, making predictions using a large ensemble of models sampled from the posterior can be costly. Variational Inference methods can be used to quickly characterize a local mode of the posterior (Blundell et al., 2015). More recent work shows that a running estimate of the mean and variance of the parameters during training can also be used to approximate a mode of the posterior (Maddox et al., 2019). Methods like distillation could potentially be used to compress a high-quality ensemble into a single network with a limited computational budget (Balan et al., 2015). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Although the form in (4) is very general, alternative methods for dealing with stochastic gradients have been proposed in the literature. One approach is to estimate the covariance of the stochastic gradient noise $B$ explicitly and use it correct and pre-condition the sampling dynamics (Ahn et al., 2012; Li et al., 2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Other sampling methods are not based on an SDE that converges to the target distribution. Under some conditions stochastic optimization methods can be interpreted as such a biased sampling method (Mandt et al., 2017). Predictions based on multiple samples from the trajectory of SGD have been used successfully for obtaining uncertainty estimates in large scale Deep Learning (Maddox et al., 2019). However, these methods rely on tuning hyperparameters in such a way that just the right amount of noise is inserted. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            439,
+            320,
+            455
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work introduces the ATMC sampler, a robust posterior sampling method that scales to large deep learning problems. To the best of our knowledge, we are the first to successfully train neural networks using MCMC on ImageNet. In a BatchNorm free setting, a single sample from the posterior generated by ATMC outperforms the optimization baseline. A posterior predictive estimate outperforms the optimization baseline with BatchNorm on ImageNet. Based on these empirical results we hope the ATMC sampler will enable new applications of Bayesian inference in deep learning. ",
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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