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+# INCREMENTAL FEW-SHOT LEARNING VIA VECTOR QUANTIZATION IN DEEP EMBEDDED SPACE
+
+# Kuilin Chen
+
+# Chi-Guhn Lee
+
+Department of Mechanical and Industrial Engineering   
+University of Toronto   
+Toronto, Ontario, Canada   
+kuilin.chen@mail.utoronto.ca   
+Department of Mechanical and Industrial Engineering   
+University of Toronto   
+Toronto, Ontario, Canada   
+cglee@mie.utoronto.ca
+
+# ABSTRACT
+
+The capability of incrementally learning new tasks without forgetting old ones is a challenging problem due to catastrophic forgetting. This challenge becomes greater when novel tasks contain very few labelled training samples. Currently, most methods are dedicated to class-incremental learning and rely on sufficient training data to learn additional weights for newly added classes. Those methods cannot be easily extended to incremental regression tasks and could suffer from severe overfitting when learning few-shot novel tasks. In this study, we propose a nonparametric method in deep embedded space to tackle incremental few-shot learning problems. The knowledge about the learned tasks is compressed into a small number of quantized reference vectors. The proposed method learns new tasks sequentially by adding more reference vectors to the model using few-shot samples in each novel task. For classification problems, we employ the nearest neighbor scheme to make classification on sparsely available data and incorporate intra-class variation, less forgetting regularization and calibration of reference vectors to mitigate catastrophic forgetting. In addition, the proposed learning vector quantization (LVQ) in deep embedded space can be customized as a kernel smoother to handle incremental few-shot regression tasks. Experimental results demonstrate that the proposed method outperforms other state-of-the-art methods in incremental learning.
+
+# 1 INTRODUCTION
+
+Incremental learning is a learning paradigm that allows the model to continually learn new tasks on novel data, without forgetting how to perform previously learned tasks (Cauwenberghs & Poggio, 2001; Kuzborskij et al., 2013; Mensink et al., 2013). The capability of incremental learning becomes more important in real-world applications, in which the deployed models are exposed to possible out-of-sample data. Typically, hundreds of thousands of labelled samples in new tasks are required to re-train or fine-tune the model (Rebuffi et al., 2017). Unfortunately, it is impractical to gather sufficient samples of new tasks in real applications. In contrast, humans can learn new concepts from just one or a few examples, without losing old knowledge. Therefore, it is desirable to develop algorithms to support incremental learning from very few samples.
+
+While a natural approach for incremental few-shot learning is to fine-tune part of the base model using novel training data (Donahue et al., 2014; Girshick et al., 2014), the model could suffer from severe over-fitting on new tasks due to a limited number of training samples. Moreover, simple fine-tuning also leads to significant performance drop on previously learned tasks, termed as catastrophic forgetting (Goodfellow et al., 2014). Recent attempts to mitigate the catastrophic forgetting are generally categorized into two streams: memory relay of old training samples (Rebuffi et al., 2017; Shin et al., 2017; Kemker & Kanan, 2018) and regularization on important model parameters (Kirkpatrick et al., 2017; Zenke et al., 2017). However, those incremental learning approaches are developed and tested on unrealistic scenarios where sufficient training samples are available in novel tasks. They may not work well when the training samples in novel tasks are few (Tao et al., 2020b).
+
+To the best of our knowledge, the majority of incremental learning methodologies focus on classification problems and they cannot be extended to regression problems easily. In class-incremental learning, the model has to expand output dimensions to learn $N ^ { \prime }$ novel classes while keeping the knowledge of existing $N$ classes. Parametric models estimate additional classification weights for novel classes, while nonparametric methods compute the class centroids for novel classes. In comparison, output dimensions in regression problems do not change in incremental learning as neither additional weights nor class centroids are applicable to regression problems.
+
+Besides, we find that catastrophic forgetting in incremental few-shot classification can be attributed to three reasons. First, the model is biased towards new classes and forgets old classes because the model is fine-tuned on new data only (Hou et al., 2019; Zhao et al., 2020). Meanwhile, the prediction accuracy on novel classes is not good due to over-fitting on few-shot training samples. Second, features of novel samples could overlap with those of old classes in the feature space, leading to ambiguity among classes in the feature space. Finally, features of old classes and classification weights are no longer compatible after the model is fine-tuned with new data.
+
+In this paper, we investigate the problem of incremental few-shot learning, where only a few training samples are available in new tasks. A unified model is learned sequentially to jointly recognize all classes or regression targets that have been encountered in previous tasks (Rebuffi et al., 2017; Wu et al., 2019). To tackle aforementioned problems, we propose a nonparametric method to handle incremental few-shot learning based on learning vector quantization (LVQ) (Sato & Yamada, 1996) in deep embedded space. As such, the adverse effects of imbalanced weights in a parametric classifier can be completely avoided (Mensink et al., 2013; Snell et al., 2017; Yu et al., 2020). Our contributions are three fold. First, a unified framework is developed, termed as incremental deep learning vector quantization (IDLVQ), to handle both incremental classification (IDLVQ-C) and regression (IDLVQ-R) problems. Second, we develop intra-class variance regularization, less forgetting constraints and calibration factors to mitigate catastrophic forgetting in class-incremental learning. Finally, the proposed methods achieve state-of-the-art performance on incremental fewshot classification and regression datasets.
+
+# 2 RELATED WORK
+
+Incremental learning: Some incremental learning approaches rely on memory replay of old exemplars to prevent forgetting previously learned knowledge. Old exemplars can be saved in memory (Rebuffi et al., 2017; Castro et al., 2018; Prabhu et al., 2020) or sampled from generative models (Shin et al., 2017; Kemker & Kanan, 2018; van de Ven et al., 2020). However, explicit storage of training samples is not scalable if the number of classes is large. Furthermore, it is difficult to train a reliable generative model for all classes from very few training samples. In parallel, regularization approaches do not require old exemplars and impose regularization on network weights or outputs to minimize the change of parameters that are important to old tasks (Kirkpatrick et al., 2017; Zenke et al., 2017). To avoid quick performance deterioration after learning a sequence of novel tasks in regularization approaches, semantic drift compensation (SDC) is developed by learning an embedding network via triplet loss (Schroff et al., 2015) and compensates the drift of class centroids using novel data only (Yu et al., 2020). In comparison, IDLVQ-C saves only one exemplar per class and uses saved exemplars to regularize the change in feature extractor and calibrate the change in reference vectors.
+
+Few-shot learning: Few-shot learning attempts to obtain models for classification or regression tasks with only a few labelled samples. Few-shot models are trained on widely-varying episodes of fake few-shot tasks with labelled samples drawn from a large-scale meta-training dataset (Vinyals et al., 2016; Finn et al., 2017; Ravi & Larochelle, 2017; Snell et al., 2017; Sung et al., 2018). Meanwhile, recent works attempt to handle novel few-shot tasks while retraining the knowledge of the base task. These methods are referred to as dynamic few-shot learning (Gidaris & Komodakis, 2018; Ren et al., 2019a; Gidaris & Komodakis, 2019). However, dynamic few-shot learning is different from incremental few-shot learning, because they rely on the entire base training dataset and an extra meta-training dataset during meta-training. In addition, dynamic few-shot learning does not accumulate knowledge for multiple novel tasks sequentially.
+
+Incremental few-shot learning: Prior works on incremental few-shot learning focus on classification problems by computing the weights for novel classes in parametric classifiers, without iterative gradient descent. For instance, the weights of novel classes can be imprinted by normalized prototypes of novel classes, while keeping the feature extractor fixed (Qi et al., 2018). Since novel weights are computed only with the samples of novel classes, the fixed feature extractor may not be compatible with novel classification weights. More recently, neural gas network is employed to construct an undirected graph to represent knowledge of old classes (Tao et al., 2020b;a). The vertices in the graph are constructed in an unsupervised manner using competitive Hebbian learning (Fritzke, 1995), while the feature embedding is fixed. In contrast, IDLVQ learns both feature extractor and reference vectors concurrently in a supervised manner.
+
+# 3 BACKGROUND
+
+# 3.1 INCREMENTAL FEW-SHOT LEARNING
+
+In this paper, incremental few-shot learning is studied for both classification and regression tasks. For classification tasks, we consider the standard class-incremental setup in literature. After the model is trained on a base task $\mathit { t } = 1$ ) with sufficient data, the model learns novel tasks sequentially. Each novel task contains a number of novel classes with only a few training samples per class. Learning a novel task $( t > 1 )$ ) is referred to as an incremental learning session. In task $t$ , we have access only to training data $\mathcal { D } ^ { t }$ in the current task and previously saved exemplars (one exemplar per class in this study). Each task has a set of classes $\hat { C } ^ { t } = \{ c _ { 1 } ^ { t } , . . . , c _ { n ^ { t } } ^ { t } \}$ , where $n _ { t }$ is the number of classes in task $t$ . In addition, it is assumed that there is no overlap between classes in different tasks $C ^ { t } \bigcap C ^ { s } = \emptyset$ for $t \neq s$ . After an incremental learning session, the performance of the model is evaluated on a test set that contains all previously seen classes $C = \cap _ { i } { \bar { C } } ^ { i }$ . Note that our focus is not multi-task scenario, where a task ID is exposed to the model during test phase and the model is only required to perform a given task one time (van de Ven & Tolias, 2019). Our model is evaluated in a task-agnostic setting, where task ID is not exposed to the model at test time.
+
+For regression tasks, we follow a similar setting with a notable difference that the target is realvalued $y \in \mathbb { R }$ . In addition, the target values in different tasks do not have to be mutually exclusive, unlike the class-incremental setup.
+
+# 3.2 LEARNING VECTOR QUANTIZATION
+
+Traditional nonparametric methods, such as nearest neighbors, represent knowledge and make predictions by storing the entire training set. Despite the simplicity and effectiveness, they are not scalable to a large-scale base dataset. Typically, incremental learning methods are only allowed to store a small number of exemplars to preserve the knowledge of previously learned tasks. However, randomly selected exemplars may not well present the knowledge in old tasks. LVQ is a classical data compression method that represents the knowledge through a few learned reference vectors (Sato & Yamada, 1996; Seo & Obermayer, 2003; Biehl et al., 2007). A new sample is classified to the same label as the nearest reference vector in the input space. LVQ has been combined with deep feature extractors as an alternative to standard neural networks for better interpretability (De Vries et al., 2016; Villmann et al., 2017; Saralajew et al., 2018). The combinations of LVQ and deep feature extractors have been applied to natural language processing (NLP), facial recognition and biometrics (Variani et al., 2015; Wang et al., 2016; Ren et al., 2019b; Leng et al., 2015). We notice that LVQ is a nonparametric method which is well suited for incremental few-shot learning because the model capacity grows by incorporating more reference vectors to learn new knowledge. For example, incremental learning vector quantization (ILVQ) has been developed to learn classification models adaptively from raw features (Xu et al., 2012). In this study, we present the knowledge by learning reference vectors in the feature space through LVQ and adapt them in incremental few-shot learning. Compared with ILVQ by Xu et al. (2012), our method does not rely on predefined rules to update reference vectors and can be learned along with deep neural networks in an end-to-end fashion. Besides, our method uses a single reference vector for each class, while ILVQ automatically assigns different numbers of prototypes for different classes.
+
+# 4 METHODOLOGY
+
+# 4.1 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION
+
+The general framework of IDLVQ for both classification and regression can be derived from a Gaussian mixture perspective (Ghahramani $\&$ Jordan, 1994), with a simplified covariance structure and supervised deep representation learning. In the base dataset $( t = 1 )$ ), a raw input $\mathbf { x }$ is projected into a feature space ${ \bar { \mathcal { F } } } ^ { 1 }$ by a deep neural network $f _ { \theta ^ { 1 } }$ , where $\theta ^ { 1 }$ denotes the parameters in neural networks. In addition, $N ^ { 1 }$ reference vectors $\mathbf { M } ^ { 1 } = \{ \mathbf { m } _ { 1 } ^ { 1 } , . . . , \mathbf { m } _ { N ^ { 1 } } ^ { 1 } \}$ are placed in the feature space ${ \mathcal { F } } ^ { 1 }$ , which can be learned to capture the representation of the base dataset. More reference vectors will be added incrementally while learning novel tasks. The marginal distribution $p ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) )$ of feature vector can be described by a Gaussian mixture model $\begin{array} { r } { p ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) ) = \sum _ { i = 1 } ^ { N ^ { 1 } } p ( i ) p ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) | i ) } \end{array}$ of $N ^ { 1 }$ components, where the prior $p ( i ) = 1 / N ^ { 1 }$ and the component distribution $p ( f _ { \boldsymbol { \theta } ^ { 1 } } ( \mathbf { x } ) | i )$ is Gaussian. By assuming that each component distribution $p ( f _ { \boldsymbol { \theta } ^ { 1 } } ( \mathbf { x } ) | i )$ is isotropic Gaussian centered at $\mathbf { m } _ { i } ^ { 1 }$ with the same covariance, the posterior distribution of a component given the input is
+
+$$
+p ^ { 1 } ( i | \mathbf { x } ) = \frac { \kappa ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) , \mathbf { m } _ { i } ^ { 1 } ) } { \sum _ { j = 1 } ^ { N ^ { 1 } } \kappa ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) , \mathbf { m } _ { j } ^ { 1 } ) } ,
+$$
+
+where $\kappa ( f _ { \theta ^ { 1 } } ( \mathbf { x } ) , \mathbf { m } _ { i } ^ { 1 } ) = \exp ( - \| f _ { \theta ^ { 1 } } ( \mathbf { x } ) - \mathbf { m } _ { i } ^ { 1 } \| ^ { 2 } / \gamma )$ is a Gaussian kernel and $\gamma$ is a scale factor. The conditional expectation of the output from a Gaussian mixture is $\begin{array} { r } { \hat { y } = \sum _ { i = 1 } ^ { N ^ { 1 } } p ^ { 1 } ( i | \mathbf { x } ) q _ { i } ^ { 1 } } \end{array}$ , where $q _ { i } ^ { 1 }$ is the reference target associated with reference vector $\mathbf { m } _ { i } ^ { 1 }$ . In classification problems, $q _ { i } ^ { 1 }$ is either 0 or 1 indicating whether $\mathbf { m } _ { i } ^ { 1 }$ and $\mathbf { x }$ have the same label. Since each reference vector is assigned to a class at initialization, $q _ { i } ^ { 1 }$ is fixed and does not require learning. Meanwhile, $q _ { i } ^ { 1 }$ in regression problems is real-valued and has to be learned. The weights in neural networks $\theta ^ { \mathrm { { I } } }$ , reference vectors ${ \bf { M } } ^ { 1 }$ , reference targets $q _ { i } ^ { 1 }$ (in regression problems only) and the scale factor $\gamma$ are learned concurrently by minimizing a loss function between the true label $y$ and the predicted label $\hat { y }$ .
+
+The proposed IDLVQ is a nonparametric method as it makes prediction based on similarity to reference vectors, instead of using any regression or classification weights. The capacity of the model grows naturally by adding more reference vectors to learn novel tasks, while the old knowledge is preserved in existing reference vectors.
+
+# 4.2 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION FOR CLASSIFICATION
+
+For classification problems, one reference vector is assigned to each class in our study. Thus, $\hat { y }$ represents the predicted probability that an input belongs to a class. The model can be trained to classify data correctly by minimizing the cross-entropy loss $\mathcal { L } _ { C E }$ between the predicted probability $\hat { y }$ and the true label $y$ . Although the cross-entropy loss encourages separability of features in base classes, it does not guarantee compact intra-class variation in the feature space. Specifically, in an incremental learning session, features of novel classes could overlap with those of previously learned classes. As a result, the overall classification accuracy could deteriorate after incremental learning sessions. A desirable feature embedding leaves large margin between classes to mitigate overlap in features across old and new classes. Inspired by center loss (Wen et al., 2016) to enhance discriminative capability in facial recognition, a regularization term on intra-class distance to reference vectors is added to get compact intra-class variation.
+
+$$
+\mathcal { L } _ { i n t r a } = \sum _ { \forall ( \mathbf { x } , y ) , y = i } \left\| f _ { \theta ^ { 1 } } ( \mathbf { x } ) - \mathbf { m } _ { i } ^ { 1 } \right\| ^ { 2 }
+$$
+
+As such, $f _ { \theta ^ { 1 } } ( \mathbf { x } )$ is forced to stay close to the reference vector with the same label and naturally moves away from other reference vectors. Consequently, features of new classes are more likely to lie in the margin between old classes to mitigate ambiguity in features across different classes. The total loss in training the base task is given by $\mathcal { L } = \mathcal { L } _ { C E } + \lambda _ { i n t r a } \mathcal { L } _ { i n t r a }$ , where $\lambda _ { i n t r a }$ is a hyper-parameter to control the weight for intra-class variation loss. The total loss is differentiable w.r.t. neural network parameters $\theta ^ { \mathrm { { i } } }$ , reference vectors $\mathbf { M } ^ { 1 } = \{ \mathbf { m } _ { 1 } ^ { 1 } , . . . , \mathbf { m } _ { n ^ { 1 } } ^ { 1 } \}$ and scaling factor $\gamma$ . All parameters in the model can be trained jointly in an end-to-end fashion.
+
+In an incremental session $( t > 1 )$ ), a novel dataset $\mathcal { D } ^ { t }$ contains $n ^ { t }$ classes and $K ^ { t }$ samples per class ( $\cdot \boldsymbol { n } ^ { t }$ -way $K ^ { t }$ -shot). $n ^ { t }$ new reference vectors are added and each reference vector is initialized as the centroid of features in a class mti = 1Kt PKtk=1 fθt (xk). The new reference vectors along with the neural network parameters are fine-tuned on $\mathcal { D } ^ { t }$ to learn new knowledge in task $t$ . To preserve the knowledge from the old tasks during incremental learning, the model should be updated only when necessary. Therefore, cross-entropy loss is not used in incremental learning sessions because it always updates model parameters even if the sample is correctly classified. Let $\mathbf { m } _ { + } ^ { t }$ be the reference vector with the correct label and $\mathbf { m } _ { - } ^ { t }$ be the nearest reference vector with a wrong label. For a training sample $\left( \mathbf { x } , y \right)$ in $\mathcal { D } ^ { t }$ , the sample is classified correctly if $\left\| f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { + } ^ { t } \right\| ^ { 2 } <$ $\left\| f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { - } ^ { t } \right\| ^ { 2 }$ . In this case, the loss should be 0. When $\left\| f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { + } ^ { t } \right\| ^ { 2 } > \left\| f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { - } ^ { t } \right\| ^ { 2 }$ , the sample is misclassified. We adapt the margin based loss function ${ \mathcal { L } } _ { M }$ from De Vries et al. (2016) with a minor modification
+
+$$
+\mathcal { L } _ { M } = \mathrm { R e L U } \left( \frac { \left. f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { + } ^ { t } \right. ^ { 2 } - \left. f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { - } ^ { t } \right. ^ { 2 } } { \left. f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { + } ^ { t } \right. ^ { 2 } + \left. f _ { \theta ^ { t } } ( \mathbf { x } ) - \mathbf { m } _ { - } ^ { t } \right. ^ { 2 } } \right) ,
+$$
+
+where $\mathrm { R e L U } ( \cdot )$ stands for the rectified linear unit function. The margin based loss leads to slow training convergence because it only updates two reference vectors one time. However, the adapted margin based loss is well suited in learning from few-shot samples while avoids unnecessary parameter updates.
+
+Features for an old class could deviate away from the corresponding reference vector due to changes in $\theta ^ { t }$ during incremental learning, leading to catastrophic forgetting. A forgetting loss $\mathcal { L } _ { F }$ is developed to regularize the drift in the feature space
+
+$$
+\mathcal { L } _ { F } = \sum _ { i = 1 } ^ { N ^ { t - 1 } } \| f _ { \theta ^ { t } } ( \mathbf { x } _ { i } ^ { ' } ) - f _ { \theta ^ { t - 1 } } ( \mathbf { x } _ { i } ^ { ' } ) \| ^ { 2 } ,
+$$
+
+where $\mathbf { x } _ { i } ^ { ' }$ is the selected exemplar for class $i$ and $N ^ { t - 1 }$ denotes the total number of classes in the base task and all previous novel tasks. Note that the exemplar $\mathbf { x } _ { i } ^ { ' }$ for class $i \in [ N ^ { t - 1 } , N ^ { t } ]$ is picked from $\mathcal { D } ^ { t }$ whose feature is nearest to $\mathbf { m } _ { i } ^ { t }$ at the end of each learning session. The total loss in the incremental learning session $t$ is $\mathcal { L } = \mathcal { L } _ { M } + \lambda _ { F } \mathcal { L } _ { F } + \lambda _ { i n t r a } \mathcal { L } _ { i n t r a }$ , where $\lambda _ { F }$ and $\lambda _ { i n t r a }$ are weights for forgetting loss and intra-class variation loss, respectively. The total loss is optimized w.r.t. neural network parameters $\theta ^ { t }$ and new reference vectors $\{ \mathbf { m } _ { N ^ { t - 1 } + 1 } ^ { t } , . . . , \mathbf { m } _ { N ^ { t } } ^ { t } \}$ .
+
+The reference vectors for previously learned tasks are not updated by novel data to prevent catastrophic forgetting. However, they may not be well suited to represent knowledge and make classification in the new feature space $\mathcal { F } ^ { t }$ as feature embedding is changed with updated $\theta ^ { t }$ . Although the true optimal location of those reference vectors are difficult to estimate without using the entire data from all tasks, they can be calculated approximately using the shift in features of exemplars. Considering that features of an exemplar $\mathbf { x } _ { i } ^ { ' }$ are close to $\mathbf { m } _ { i }$ in the feature space, the shift of a reference vector $\delta _ { i } ^ { t }$ in the new feature space can be approximated by the shift of the exemplar’s features $\delta _ { i } ^ { t } = f _ { \theta ^ { t } } ( \mathbf { x } _ { i } ^ { ' } ) { - } f _ { \theta ^ { t - 1 } } ( \mathbf { x } _ { i } ^ { ' } )$ . Therefore, the reference vectors for previously learned tasks are calibrated $\mathbf { m } _ { i } ^ { t } = \mathbf { m } _ { i } ^ { t - 1 } + \delta _ { i } ^ { t }$ , where $\mathbf { m } _ { i } ^ { t - 1 }$ is the uncalibrated reference vector for class $i \in [ 1 , N ^ { t - 1 } ]$ . A test sample, which could be from any seen classes, is classified according to the distance to reference vectors $\{ \mathbf { m } _ { 1 } ^ { t } , . . . , \mathbf { m } _ { N ^ { t } } ^ { t } \}$ . The pseudo code for IDLVQ-C is presented in the appendix.
+
+# 4.3 INCREMENTAL DEEP LEARNING VECTOR QUANTIZATION FOR REGRESSION
+
+For regression problems, the model is trained to recognize regression targets by the minimizing mean squared error (MSE) loss $\mathcal { L } _ { M S E } = ( y - \hat { y } ) ^ { 2 }$ , where $y$ is the real-valued target in training dataset. The MSE loss function is differentiable w.r.t. neural network weights, reference vectors and targets, and scale factor. Therefore, all parameters can be trained jointly in an end-to-end manner. The proposed IDLVQ-R can also be interpreted as a kernel smoother in deep embedded space. Compared with traditional kernel smoother, such as Nadaraya-Watson estimator (Nadaraya, 1964), IDLVQ-R is sparse and hence more scalable as it only relies on a few reference vectors and targets.
+
+In an incremental learning session $( t > 1 )$ ), we have access to data $\mathcal { D } ^ { t }$ that contains $K ^ { t }$ pairs of training samples $( \mathbf { x } _ { i } ^ { t } , y _ { i } ^ { t } )$ . ${ \bar { n } } ^ { t }$ new reference vectors $( n ^ { t } \leq K ^ { t } )$ along with corresponding targets are added to the model to learn new knowledge in the novel task $t$ . We randomly select $n ^ { t }$ samples from $\mathcal { D } ^ { t }$ to initialize reference vectors and targets as follows
+
+$$
+\begin{array} { c } { { { \bf m } _ { i + N ^ { t - 1 } } = f _ { \theta } ( { \bf x } _ { i } ^ { t } ) , } } \\ { { { \bf \ q } _ { i + N ^ { t - 1 } } = y _ { i } ^ { t } , } } \end{array}
+$$
+
+where $N ^ { t - 1 }$ is the total number of reference vectors in all previous tasks. The new reference vectors and targets are fine-tuned by minimizing MSE on $\mathcal { D } ^ { t }$ while keeping other parameters frozen. After new reference vectors and targets are fine-tuned with novel data $\mathcal { D } ^ { t }$ , the model makes prediction by smoothing targets of all reference vectors
+
+$$
+\hat { y } = \frac { \sum _ { i = 1 } ^ { N ^ { t } } \kappa ( f _ { \theta } ( \mathbf { x } ) , \mathbf { m } _ { i } ) q _ { i } } { \sum _ { i = 1 } ^ { N ^ { t } } \kappa ( f _ { \theta } ( \mathbf { x } ) , \mathbf { m } _ { i } ) } ,
+$$
+
+where $N ^ { t }$ is the current total number of reference vectors.
+
+# 5 EXPERIMENTS
+
+We first describe the overall protocols, then we present the results on incremental few-shot classification and regression problems.
+
+# 5.1 INCREMENTAL FEW-SHOT CLASSIFICATION
+
+We empirically evaluate the performance of IDLVQ-C on incremental few-shot classification on CUB200-2011 (Welinder et al., 2010) and miniImageNet datasets (Vinyals et al., 2016). The dataset is split into base classes and multiple groups of novel classes. We apply standard data augmentation, including random crop, horizontal flip and color jitter, on all training images. After each training session, the model performance is evaluated on a test set, which contains all classes that the model has been trained on.
+
+CUB dataset is composed of 200 fine-grained bird species with 11,788 images. We split the dataset into 5894 training images, 2947 validation images and 2947 test images. All images are resized to $2 2 4 \times 2 2 4$ . In addition, the first 100 classes are chosen as base classes, where all training samples in base classes are used to train the base model. The remaining 100 classes are treated as novel categories and split into 10 incremental learning sessions. Each incremental learning session contains 10 novel classes and 5 randomly selected training samples per class (10-way 5-shot).
+
+miniImageNet dataset is a 100-class subset of the original ImageNet dataset (Deng et al., 2009). Each class contains 500 training images, 50 validation images, and 50 test images. The images are in RGB format of the size $8 4 \times 8 4$ . We choose 60 and 40 classes for base and novel classes, respectfully. The 40 novel classes are divided into 8 sessions and each session contains 5 novel classes with 5 randomly selected training samples per class (5-way 5-shot).
+
+ResNet18 (He et al., 2016) is used as the feature extractor for incremental classification problems. The learning process for each dataset is repeated 10 times and the average test accuracy is reported. The proposed method is compared with six methods for few-shot class-incremental learning: fineturning using $\mathcal { D } ^ { t }$ , joint training using the entire training set from all encountered classes, iCaRL (Rebuffi et al., 2017), Rebalancing (Hou et al., 2019), ProtoNet (Snell et al., 2017), incremental learning vector quantization (ILVQ) (Xu et al., 2012), SDC (Yu et al., 2020), and Imprint (Qi et al., 2018). Note that ILVQ is applied to the features extracted by neural networks in our experiment.
+
+The incremental few-shot learning results on CUB and miniImageNet are shown in Table 1 and 2, respectively. Our method outperforms fine-tuning, iCaRL (Rebuffi et al., 2017), and ProtoNet (Snell et al., 2017) by a large margin. Simply fine-tuning the weights in classifier with few-shot training samples for novel classes significantly deteriorates the prediction accuracy. Although iCaRL alleviates catastrophic forgetting by tuning the model with a mix of old exemplars and novel few-shot data, the prediction accuracy still drops quickly because iCaRL requires sufficient samples per class to achieve satisfactory performance. The ProtoNet relies on distance to prototypes (the mean of features within a class) to make classification but the fixed feature extractor may not be able to well separate novel classes. ILVQ is slightly better than ProtoNet because prototypes can be learned adaptively when more classes are available in incremental learning sessions. Some prototypes in
+
+ILVQ are close to the border of a class, which are more effective than class centroids in ProtoNet. However, ILVQ does not achieve the best performance because the feature extractor is fixed and cannot be learned along with the prototypes. IDLVQ-C has a small gain in the first couple of incremental few-shot learning sessions compared with SDC (Yu et al., 2020) and Imprint (Qi et al., 2018). Similar to ProtoNet, SDC also relies on prototypes to make classification. The performance of SDC is better than that of ProtoNet because SDC fine-tunes the feature extractor with novel dataset and compensates the drift in prototypes. However, the compensation for the drift of old-class prototypes can be less accurate in SDC because it is approximated by samples in novel classes. In parallel, the imprint method directly computes the normalized classification weights from the average of normalized features within a novel class. The imprint method avoids imbalanced classification weights and circumvents the overfitting in few-shot class-incremental learning through weight normalization. Nevertheless, the fixed feature extractor in the imprint method may not be well suited for novel classes. In contrast, IDLVQ-C updates the feature extractor only when necessary and compensates the shift of old reference vectors more accurately using exemplars from old classes. That is why the gain of IDLVQ-C increases with more incremental few-shot learning sessions. The performance of SDC, Imprint and IDLVQ-C is better than offline joint training in early sessions of incremental few-shot learning. Offline joint training may not result in oracle performance due to extremely imbalanced samples between base classes and novel classes.
+
+Table 1: Prediction accuracy on CUB all classes using the 10-way 5-shot incremental setting.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="10">sessions</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>Fine-tune</td><td>77.30</td><td>46.23</td><td>34.71</td><td>25.35</td><td>23.16</td><td>20.65</td><td>16.21</td><td>13.32</td><td>11.98</td><td>11.17</td><td>10.76</td></tr><tr><td>Joint train</td><td>77.30</td><td>73.28</td><td>68.80</td><td>65.34</td><td>63.75</td><td>62.00</td><td>60.81</td><td>59.71</td><td>59.06</td><td>58.69</td><td>58.23</td></tr><tr><td>iCaRL (Rebuffi et al.,2017)</td><td>77.30</td><td>57.18</td><td>54.67</td><td>48.11</td><td>40.76</td><td>36.85</td><td>33.12</td><td>30.42</td><td>28.22</td><td>26.84</td><td>25.23</td></tr><tr><td>Rebalancing (Hou et al., 2019)</td><td>77.30</td><td>64.53</td><td>56.14</td><td>47.29</td><td>38.92</td><td>34.39</td><td>31.04</td><td>27.93</td><td>27.12</td><td>24.46</td><td>23.61</td></tr><tr><td>ProtoNet (Snell et al., 2017)</td><td>77.30</td><td>69.76</td><td>66.01</td><td>62.29</td><td>59.58</td><td>57.10</td><td>55.13</td><td>54.09</td><td>52.40</td><td>51.65</td><td>50.36</td></tr><tr><td>ILVQ (Xu et al., 2012)</td><td>77.30</td><td>71.50</td><td>66.79</td><td>62.71</td><td>60.20</td><td>57.84</td><td>55.27</td><td>55.06</td><td>52.42</td><td>51.72</td><td>50.47</td></tr><tr><td>SDC (Yu et al., 2020)</td><td>77.34</td><td>74.45</td><td>69.45</td><td>65.27</td><td>61.81</td><td>58.26</td><td>56.14</td><td>55.71</td><td>53.31</td><td>52.79</td><td>51.52</td></tr><tr><td>Imprint (Qi et al., 2018)</td><td>77.02</td><td>73.39</td><td>69.50</td><td>65.61</td><td>62.81</td><td>60.74</td><td>59.39</td><td>58.61</td><td>56.85</td><td>55.93</td><td>54.82</td></tr><tr><td>IDLVQ-C</td><td>77.37</td><td>74.72</td><td>70.28</td><td>67.13</td><td>65.34</td><td>63.52</td><td>62.10</td><td>61.54</td><td>59.04</td><td>58.68</td><td>57.81</td></tr></table>
+
+Table 2: Prediction accuracy on miniImageNet all classes using the 5-way 5-shot incremental setting.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="9">sessions</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td>Fine-tune</td><td>64.25</td><td>30.11</td><td>18.53</td><td>6.31</td><td>2.86</td><td>2.68</td><td>1.87</td><td>1.56</td><td>1.42</td></tr><tr><td>Joint train</td><td>64.25</td><td>58.80</td><td>55.26</td><td>52.38</td><td>49.71</td><td>48.37</td><td>45.91</td><td>44.68</td><td>43.38</td></tr><tr><td>iCaRL (Rebuffi et al., 2017)</td><td>64.25</td><td>48.04</td><td>43.13</td><td>38.28</td><td>30.01</td><td>24.46</td><td>21.85</td><td>19.84</td><td>17.76</td></tr><tr><td>Rebalancing (Hou et al.,2019)</td><td>64.25</td><td>49.21</td><td>44.17</td><td>37.71</td><td>30.11</td><td>22.92</td><td>19.99</td><td>17.96</td><td>16.25</td></tr><tr><td>ProtoNet (Snell et al., 2017)</td><td>64.25</td><td>55.12</td><td>51.67</td><td>48.91</td><td>46.52</td><td>44.25</td><td>41.91</td><td>40.07</td><td>38.42</td></tr><tr><td>ILVQ (Xu et al., 2012)</td><td>64.25</td><td>56.01</td><td>52.43</td><td>49.31</td><td>46.98</td><td>44.37</td><td>42.06</td><td>40.11</td><td>38.43</td></tr><tr><td>SDC (Yu et al., 2020)</td><td>64.62</td><td>59.63</td><td>55.39</td><td>50.92</td><td>48.30</td><td>45.28</td><td>42.97</td><td>42.51</td><td>41.24</td></tr><tr><td>Imprint (Qi et al., 2018)</td><td>64.71</td><td>59.85</td><td>55.71</td><td>52.47</td><td>49.90</td><td>47.31</td><td>44.57</td><td>42.57</td><td>41.26</td></tr><tr><td>IDLVQ-C</td><td>64.77</td><td>59.87</td><td>55.93</td><td>52.62</td><td>49.88</td><td>47.55</td><td>44.83</td><td>43.14</td><td>41.84</td></tr></table>
+
+Ablation studies are conducted to analyze how individual components affect the performance of incremental few-shot learning. We study five variants of our methods: (a) new reference vectors are initialized as class centroids and no tuning is done for feature extractor or old reference vectors; (b) $\mathcal { L } _ { i n t r a }$ is not used in incremental learning sessions; (c) $\mathcal { L } _ { F }$ is not used in the incremental learning sessions; (d) shift in old reference vectors are not compensated; (e) replace the margin based loss ${ \mathcal { L } } _ { M }$ with the cross entropy loss $\mathcal { L } _ { C E }$ . Table 3 shows the results of our ablation studies on CUB dataset. Without any fine-tuning, the initial reference vectors for novel classes lead to descent accuracy in incremental few-shot classification. It demonstrates the robustness of nonparametric classifier. $\mathcal { L } _ { i n t r a }$ leads to $0 . 5 7 \%$ gain due to tight intra-class variation. The less forgetting regularization $\mathcal { L } _ { F }$ is proved to prevent forgetting old classes and achieves a performance boost by $2 . 3 5 \%$ . The shift compensation for reference vectors effectively adapts old reference vector in new embedding spaces with a gain round of $1 \%$ . The margin based loss is more effective in preventing forgetting in early sessions of incremental learning.
+
+The effect of the number of training samples per class. The proposed method is evaluated under different few-shot settings on CUB dataset to investigate the effect of different numbers of training samples in novel classes, including 5-shot, 10-shot and 20-shot settings. One reference vector is assigned to each class in all few-shot settings. As shown in Fig. 2 in the appendix, the performance of incremental learning improves as the number of samples per class increases. When the training samples are scarce, the training samples may not well present the generative distribution of training data. Therefore, the learned reference vectors could be biased and classification accuracy is low. With more training samples, the learned reference vectors could well present the center of the distribution and classification accuracy is improved. The gap in performance becomes more obvious as the number of incremental learning sessions grows. The detailed results are reported in Table 7 and 8.
+
+Table 3: Ablation study on CUB using the 10-way 5-shot incremental setting.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="10">sessions</td></tr><tr><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>No tuning</td><td>71.93</td><td>67.14</td><td>64.21</td><td>62.61</td><td>60.13</td><td>59.04</td><td>58.47</td><td>55.64</td><td>54.25</td><td>53.66</td></tr><tr><td>W.o.Lintra</td><td>74.75</td><td>70.26</td><td>66.89</td><td>65.05</td><td>63.18</td><td>61.84</td><td>61.36</td><td>58.61</td><td>58.14</td><td>57.24</td></tr><tr><td>w.0. LF</td><td>73.85</td><td>69.54</td><td>66.21</td><td>64.02</td><td>62.74</td><td>60.28</td><td>59.49</td><td>56.97</td><td>56.38</td><td>55.46</td></tr><tr><td>w.o. δ</td><td>74.67</td><td>70.01</td><td>66.74</td><td>64.81</td><td>63.90</td><td>61.42</td><td>60.73</td><td>58.16</td><td>57.62</td><td>56.79</td></tr><tr><td>LM→LCE</td><td>73.22</td><td>69.41</td><td>66.03</td><td>63.93</td><td>63.07</td><td>61.14</td><td>60.98</td><td>58.67</td><td>58.11</td><td>57.32</td></tr><tr><td>IDLVQ-C</td><td>74.72</td><td>70.28</td><td>67.13</td><td>65.34</td><td>63.52</td><td>62.10</td><td>61.54</td><td>59.04</td><td>58.68</td><td>57.81</td></tr></table>
+
+# 5.2 INCREMENTAL FEW-SHOT REGRESSION
+
+IDLVQ-R is tested on two regression datasets: sinusoidal wave and 3D spatial data. Considering that there is no state-of-the-art method for incremental few-shot regression, we compare IDLVQ-R against three alternative methods: fine-tuning using novel task data only, fine-tuning using novel task data along with exemplars and offline training using the entire training dataset from all tasks.
+
+Sinusoidal wave is defined by a function $y = \sin ( 3 \pi x ) + 0 . 3 \cos ( 9 \pi x ) + 0 . 5 \sin ( 7 \pi x ) + \epsilon ,$ , where $\epsilon$ is white noise with a standard deviation of 0.1. 1000 training samples in the first task (bas task) are generated by sampling $x \in [ - 1 . 0 , 1 . 0 ]$ uniformly. 5-shot training samples in two novel tasks are generated by sampling $x \in [ 1 . 0 , 1 . 5 ]$ and $x \in [ 1 . 5 , 2 . 0 ]$ , respectively.
+
+As shown in Fig. 1(a), IDLVQ-R achieves comparable performance to offline neural networks in Fig. 1(d) which are trained using the entire training set from all tasks. In comparison, neural networks trained sequentially with few-shot training samples show catastrophic forgetting on old tasks in Fig. 1(b). With the addition of exemplars during training, the networks perform better but still suffer from catastrophic forgetting on the base task in Fig. 1(c). In conclusion, IDLVQ-R preserves old knowledge and adapts to new knowledge quickly using a few reference vectors and achieves satisfactory performance on incremental few-shot regression tasks. The experiment details can be found in the appendix.
+
+![](images/83a97f4c3e066abb502771d10b9d6938bf3934093e0891e51d8029367598a481.jpg)  
+Figure 1: Comparison of performance for incremental few-shot regression. Red dots denote test samples for base task, purple dots denotes test sample for novel tasks, grey lines denote model predictions, and black crosses denotes few-shot training samples in novel tasks. (a) IDLVQ-R; (b) neural networks incrementally fine-tuned with novel data only in each session; (c) neural networks incrementally fine-tuned with exemplars and novel training samples; (d) offline neural networks trained with training samples from all tasks.
+
+3D spatial data1 is collected in North Jutland, Denmark. The inputs are longitude $x _ { 1 }$ and latitude $x _ { 2 }$ , and the output is altitude y. 2482 training samples in the 1st task are collected in the area where $x _ { 1 } \in [ 9 . 9 8 , 9 . 9 9 5 ]$ and $x _ { 2 } \in [ 5 7 . 0 , 5 7 . 0 5 ] .$ 20 training samples in the 2nd task are collected in the area where $x _ { 1 } \ \in \ [ 9 . 9 9 5 , 1 0 . 0 ]$ and $x _ { 2 } ~ \in ~ [ 5 7 . 0 2$ , 57.03]. 20 training samples in the 3rd task are collected in the area where $x _ { 1 } \in [ 9 . 9 9 5 , 1 0 . 0 ]$ and $x _ { 2 } ~ \overset { \cdot } { \in } ~ [ 5 7 . 0 3 , 5 7 . 0 4 ]$ . Approximately 300 test samples are collected for each task.
+
+Table 4: Normalized RMSE of incremental few-shot regression on 3D spatial data   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">sessions</td></tr><tr><td>1</td><td>2</td><td>3</td></tr><tr><td>Joint train offline</td><td>0.02174(2e-4)</td><td>0.02232(2e-4)</td><td>0.02296(2e-4)</td></tr><tr><td>Fine-tune w.novel data</td><td>0.02174(2e-4)</td><td>0.08462(4e-4)</td><td>0.11870(6e-4)</td></tr><tr><td>Fine-tune w. exemplars</td><td>0.02174(2e-4)</td><td>0.02988(2e-4)</td><td>0.03128(2e-4)</td></tr><tr><td>IDLVQ-R</td><td>0.02181(2e-4)</td><td>0.02641(2e-4)</td><td>0.02817(2e-4)</td></tr></table>
+
+The normalized root mean squared errors (RMSE) between actual and predicted altitude in the test set are listed in Table 4. The prediction accuracy drops significantly when the model is finetuned with novel data only. Catastrophic forgetting can be alleviated using exemplars from previous tasks. IDLVQ-R achieves better results than fine tuning with exemplars. The good performance of IDLVQ-R can be attributed to two reasons. First, IDLVQ-R learns a number of reference vectors and targets to preserve the knowledge in encountered tasks. Compared with a linear layer on top of neural networks, a number of reference vectors represent richer information about the training data. Second, IDLVQ-R is nonparametric and can represent local and nonlinear relationship without learning any regression coefficient from few-shot data.
+
+# 6 CONCLUSIONS
+
+A new incremental few-shot learning approach is developed to harmonize old knowledge preserving and new knowledge adaptation through quantized vector in deep embedded space. Prediction is made in a nonparametric way using similarity to learned reference vectors, which circumvents biased weights in a parametric classification layer during incremental few-shot learning. For classification problems, additional mechanisms are developed to mitigate the forgetting in old classes and improve representation learning for few-shot novel classes. For regression problems, the proposed approach has been reinterpreted as a kernel smoother to predict real-valued target over novel domain.
+
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+
+# A APPENDIX
+
+A.1 PSEUDO CODE FOR IDLVQ-C
+
+# Algorithm 1 IDLVQ-C
+
+<table><tr><td>In the base task(t = 1)</td><td></td></tr><tr><td>Initialize 01,{m𝑖,.,mN1} and γ</td><td></td></tr><tr><td>Minimize L = LcE + XintraLintra W.r.t. θ1,{ml,.,mN1} and γ</td><td></td></tr><tr><td>Pick exemplars from D1 for classes in the base task: xi = arg minxeD1 |lft-1(x) - m1|l²</td><td></td></tr><tr><td>for novel task t = 2,3,... do</td><td></td></tr><tr><td>Initialize {mt-1+1.*., mNt}</td><td></td></tr><tr><td>Minimize L=LM +XFLF + XintraLintra W.r.t. 0t and {mNt-1+1,.,mNt}</td><td></td></tr><tr><td></td><td>t-1 t-1 +</td></tr><tr><td></td><td></td></tr><tr><td></td><td>Pick exemplars from Dt for classes in the novel task t: x&#x27; = arg minx∈Dt |lfet-1(x) - m ll²</td></tr><tr><td>end for</td><td></td></tr></table>
+
+# A.2 PSEUDO CODE FOR IDLVQ-R
+
+# Algorithm 2 IDLVQ-R
+
+<table><tr><td>In the base task (t =1) Initializeθ,{m1,..,mN1},{q],..,qN1} and γ</td></tr><tr><td>Minimize Lmse w.r.t. θ,{ml,..,mN1},{ql,.,qN1} and γ</td></tr><tr><td>for novel task t = 2,3,.. do</td></tr><tr><td></td></tr><tr><td>Initialize {mNt-1+1,, mNt} and {qNt-1+1.…,qNt}</td></tr><tr><td>Minimize LmsE w.r.t. {mNt-1+1..*,mNt} and {qNt-1+1,.,qNt}</td></tr><tr><td>end for</td></tr><tr><td></td></tr></table>
+
+# A.3 EXPERIMENT DETAILS FOR INCREMENTAL FEW-SHOT CLASSIFICATIO
+
+The base model is trained by the SGD optimizer (momentum of 0.9 and weight decay of 1e-4) with a mini-batch size of 64. For CUB dataset, the initial learning rate is 0.01 and is decayed by 0.1 after 60 and 120 epochs (200 epochs in total). For miniImageNet, the learning rate also starts from 0.01 and is decayed by 0.1 every 200 epochs (600 epochs in total). In an incremental learning session $( t > 1 )$ ), the model is fine-tuned with with $\mathcal { D } ^ { t }$ with a learning rate of 0.01 for 100 epochs. Since novel data $\mathcal { D } ^ { t }$ $t > 1$ ) contains very few training samples, all training samples in $\mathcal { D } ^ { t }$ are included in one mini-batch. In addition, we use $\lambda _ { i n t r a } = 1 . 0$ and $\lambda _ { F } = 0 . 5$ for both datasets. Empirically, larger $\lambda _ { i n t r a }$ leads to more compact intra-class variation. However, convergence could be slow if $\lambda _ { i n t r a }$ is too large. In addition, larger $\lambda _ { F }$ results in less forgetting in old classes but makes learning novel classes more difficult.
+
+# A.4 ADDITIONAL RESULTS FOR INCREMENTAL FEW-SHOT CLASSIFICATION
+
+The accuracies for base and novel classes are reported separately in Table 5 and 6 for CUB amd miniImageNet, respectively. The prediction accuracy of novel classes is calculated upon all novel classes the model has been trained on. Note that the accuracy in Table 1 and 2 is calculated upon all classes (including base and novel classes) that the model has been trained on. The proposed IDLVQC demonstrates strong capability of preserving old knowledge by achieving the best performance on old classes across all learning sessions. In parallel, Imprint method performs slightly better on novel classes than IDLVQ-C in early incremental learning sessions, while IDLVQ-C outperforms Imprint method in longer incremental learning sessions. The advantage of IDLVQ-C can be attributed to the adaptive feature extractor, which is tuned in each learning session.
+
+Table 5: Prediction accuracy on CUB base and novel classes using the 10-way 5-shot incremental setting.   
+
+<table><tr><td rowspan="2">Base classes</td><td colspan="10">sessions</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>Fine-tune</td><td>77.30</td><td>44.23</td><td>36.28</td><td>27.52</td><td>25.96</td><td>23.05</td><td>17.68</td><td>13.07</td><td>11.78</td><td>10.99</td><td>10.71</td></tr><tr><td>Joint train</td><td>77.30</td><td>75.83</td><td>75.25</td><td>74.51</td><td>74.58</td><td>73.74</td><td>73.95</td><td>73.25</td><td>73.11</td><td>73.25</td><td>73.18</td></tr><tr><td>iCaRL (Rebuffi et al., 2017)</td><td>77.30</td><td>59.38</td><td>58.81</td><td>54.43</td><td>48.27</td><td>43.28</td><td>39.17</td><td>34.91</td><td>32.43</td><td>29.36</td><td>25.87</td></tr><tr><td>Rebalancing (Hou et al.,2019)</td><td>77.30</td><td>66.43</td><td>60.32</td><td>55.36</td><td>46.39</td><td>41.76</td><td>37.12</td><td>32.58</td><td>31.26</td><td>27.03</td><td>24.25</td></tr><tr><td>ProtoNet (Snell et al., 2017)</td><td>77.30</td><td>72.55</td><td>72.21</td><td>72.06</td><td>71.64</td><td>71.29</td><td>71.02</td><td>70.94</td><td>70.67</td><td>70.60</td><td>70.53</td></tr><tr><td>ILVQ (Xu et al., 2012)</td><td>77.30</td><td>74.18</td><td>73.57</td><td>72.66</td><td>72.56</td><td>71.57</td><td>71.14</td><td>71.12</td><td>71.02</td><td>70.98</td><td>70.85</td></tr><tr><td>SDC (Yu et al., 2020)</td><td>77.34</td><td>76.05</td><td>75.21</td><td>74.12</td><td>72.36</td><td>71.81</td><td>71.68</td><td>71.43</td><td>71.25</td><td>71.27</td><td>70.96</td></tr><tr><td>Imprint (Qi et al., 2018)</td><td>77.02</td><td>74.76</td><td>74.57</td><td>73.69</td><td>72.69</td><td>70.88</td><td>70.34</td><td>70.12</td><td>70.07</td><td>69.84</td><td>69.27</td></tr><tr><td>IDLVQ-C</td><td>77.37</td><td>76.32</td><td>75.90</td><td>75.91</td><td>75.49</td><td>74.86</td><td>74.58</td><td>74.37</td><td>74.02</td><td>73.39</td><td>73.32</td></tr><tr><td rowspan="2">Novel classes</td><td></td><td></td><td></td><td></td><td></td><td>sessions</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>Fine-tune</td><td>=</td><td>66.23</td><td>26.86</td><td>18.12</td><td>16.16</td><td>15.85</td><td>13.76</td><td>13.68</td><td>12.23</td><td>11.37</td><td>10.81</td></tr><tr><td>Joint train</td><td>=</td><td>47.78</td><td>36.55</td><td>34.77</td><td>36.68</td><td>38.52</td><td>38.91</td><td>40.37</td><td>41.50</td><td>42.51</td><td>43.28</td></tr><tr><td>iCaRL (Rebuffi et al., 2017)</td><td>-</td><td>35.18</td><td>33.97</td><td>27.04</td><td>21.99</td><td>23.99</td><td>23.04</td><td>24.01</td><td>22.96</td><td>24.04</td><td>24.59</td></tr><tr><td>Rebalancing (Hou et al.,2019)</td><td>=</td><td>45.53</td><td>35.24</td><td>20.39</td><td>20.25</td><td>19.65</td><td>20.91</td><td>21.29</td><td>21.95</td><td>21.60</td><td>22.97</td></tr><tr><td>ProtoNet (Snell et al., 2017)</td><td></td><td>41.86</td><td>35.01</td><td>29.72</td><td>29.43</td><td>28.72</td><td>28.65</td><td>30.02</td><td>29.56</td><td>30.59</td><td>30.19</td></tr><tr><td>ILVQ (Xu et al., 2012)</td><td></td><td>40.74</td><td>32.89</td><td>29.54</td><td>29.30</td><td>30.38</td><td>28.82</td><td>32.12</td><td>29.17</td><td>30.32</td><td>30.09</td></tr><tr><td>SDC (Yu et al., 2020)</td><td></td><td>58.45</td><td>40.65</td><td>35.77</td><td>33.23</td><td>31.16</td><td>30.24</td><td>33.25</td><td>30.89</td><td>32.26</td><td>32.08</td></tr><tr><td>Imprint (Qi et al.,2018)</td><td></td><td>59.69</td><td>44.15</td><td>38.68</td><td>38.11</td><td>40.46</td><td>41.14</td><td>42.17</td><td>40.33</td><td>40.47</td><td>40.37</td></tr><tr><td>IDLVQ-C</td><td>=</td><td>58.72</td><td>42.18</td><td>37.86</td><td>39.97</td><td>40.84</td><td>41.30</td><td>43.21</td><td>40.32</td><td>42.34</td><td>42.30</td></tr></table>
+
+The test accuracy on CUB dataset using 10-way 10-shot and 10-way 20-shot incremental settings are reported in Table 7 and 8, respectively. The prediction accuracy improves in all methods with more training samples per class. The iCaRL and Rebalancing methods show the most significant improvement when the number of training samples increases. The proposed IDLVQ-C is effective in different incremental few-shot scenarios as it achieves the best performance on 5-shot, 10-shot and 20-shot settings.
+
+# A.5 EXPERIMENT DETAILS FOR INCREMENTAL FEW-SHOT REGRESSION
+
+Sinusoidal wave: A six-layer feedforward neural network with ReLU nonlinear activation is used as the feature extractor. IDLVQ-R learns 10 reference vectors and targets from the base task. In each incremental learning sessions, 5 pairs of reference vectors and targets are added. After new reference vectors and targets are fine-tuned, the model is capable of make prediction for all seen tasks. 10 exemplars are selected uniformly from the training set in the base task. In the incremental learning session of the 2nd task, the model is fine-tuned on 10 exemplars and 5 novel training samples. After the training converges, 5 novel training samples in the current task are added to the exemplar set. In the incremental learning session of the 3rd task, the model is fine-tuned with 15 exemplars from old tasks and 5 novel training samples.
+
+Table 6: Prediction accuracy on miniImageNet base and novel classes using the 5-way 5-shot incremental setting.   
+
+<table><tr><td rowspan="2">Base classes</td><td colspan="9">sessions</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td>Fine-tune</td><td>64.25</td><td>32.28</td><td>20.87</td><td>6.95</td><td>3.17</td><td>3.16</td><td>1.92</td><td>1.53</td><td>1.46</td></tr><tr><td>Joint train</td><td>64.25</td><td>63.30</td><td>62.83</td><td>62.16</td><td>62.18</td><td>62.68</td><td>61.86</td><td>61.87</td><td>61.89</td></tr><tr><td>iCaRL (Rebuffi et al., 2017)</td><td>64.25</td><td>51.66</td><td>48.97</td><td>45.62</td><td>37.39</td><td>30.86</td><td>28.68</td><td>26.83</td><td>24.47</td></tr><tr><td>Rebalancing (Hou et al., 2019)</td><td>64.25</td><td>52.87</td><td>50.16</td><td>44.78</td><td>37.48</td><td>28.75</td><td>25.58</td><td>22.97</td><td>21.57</td></tr><tr><td>ProtoNet (Snell et al., 2017)</td><td>64.25</td><td>59.27</td><td>58.88</td><td>58.69</td><td>58.22</td><td>57.63</td><td>57.03</td><td>56.80</td><td>56.47</td></tr><tr><td>ILVQ (Xu et al.,2012)</td><td>64.25</td><td>60.24</td><td>59.62</td><td>59.02</td><td>58.61</td><td>57.71</td><td>57.16</td><td>56.83</td><td>56.49</td></tr><tr><td>SDC (Yu et al., 2020)</td><td>64.62</td><td>63.58</td><td>62.78</td><td>61.12</td><td>60.29</td><td>59.37</td><td>59.05</td><td>59.97</td><td>59.87</td></tr><tr><td>Imprint (Qi et al.,2018)</td><td>64.71</td><td>63.52</td><td>62.96</td><td>62.13</td><td>61.17</td><td>61.27</td><td>60.63</td><td>59.86</td><td>59.64</td></tr><tr><td>IDLVQ-C</td><td>64.77 63.77</td><td></td><td>63.22</td><td>62.44</td><td>61.22</td><td>61.47</td><td>60.97</td><td>60.66</td><td>60.44</td></tr><tr><td rowspan="2">Novel classes</td><td></td><td></td><td></td><td></td><td>sessions</td><td></td><td></td><td></td><td></td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td></tr><tr><td>Fine-tune</td><td></td><td>4.07</td><td>4.49</td><td>3.75</td><td>1.93</td><td>1.53</td><td>1.77</td><td>1.61</td><td>1.36</td></tr><tr><td>Joint train</td><td>-</td><td>4.80</td><td>9.84</td><td>13.26</td><td>12.30</td><td>14.03</td><td>14.01</td><td>15.21</td><td>15.62</td></tr><tr><td>iCaRL (Rebuffi et al., 2017)</td><td>=</td><td>4.60</td><td>8.09</td><td>8.92</td><td>7.87</td><td>9.10</td><td>8.19</td><td>7.86</td><td>7.70</td></tr><tr><td>Rebalancing (Hou et al.,2019)</td><td></td><td>5.29</td><td>8.23</td><td>9.43</td><td>8.00</td><td>8.93</td><td>8.81</td><td>8.83</td><td>8.27</td></tr><tr><td>ProtoNet (Snell et al., 2017)</td><td></td><td>5.32</td><td>8.41</td><td>9.79</td><td>11.42</td><td>12.14</td><td>11.67</td><td>11.39</td><td>11.35</td></tr><tr><td>ILVQ (Xu et al.,2012)</td><td></td><td>5.25</td><td>9.29</td><td>10.47</td><td>12.09</td><td>12.35</td><td>11.86</td><td>11.45</td><td>11.34</td></tr><tr><td>SDC (Yu et al., 2020)</td><td></td><td>12.23</td><td>11.05</td><td>10.12</td><td>12.33</td><td>11.46</td><td>10.81</td><td>12.58</td><td>13.30</td></tr><tr><td>Imprint (Qi et al., 2018)</td><td></td><td>15.81</td><td>12.21</td><td>13.83</td><td>16.09</td><td>13.81</td><td>12.45</td><td>12.93</td><td>13.69</td></tr><tr><td>IDLVQ-C</td><td>-</td><td>13.07</td><td>12.19</td><td>13.34</td><td>15.86</td><td>14.14</td><td>12.55</td><td>13.11</td><td>13.94</td></tr></table>
+
+Table 7: Prediction accuracy on CUB using the 10-way 10-shot incremental setting.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="10">sessions</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>Fine-tune</td><td>77.30</td><td>47.16</td><td>36.34</td><td>26.92</td><td>24.08</td><td>21.24</td><td>17.19</td><td>14.31</td><td>12.73</td><td>11.75</td><td>11.43</td></tr><tr><td>Joint train</td><td>77.30</td><td>73.29</td><td>71.54</td><td>68.72</td><td>66.38</td><td>65.42</td><td>64.98</td><td>65.74</td><td>64.82</td><td>64.47</td><td>64.16</td></tr><tr><td>iCaRL (Rebuffi et al., 2017)</td><td>77.30</td><td>59.66</td><td>56.24</td><td>52.26</td><td>48.77</td><td>46.37</td><td>44.54</td><td>43.17</td><td>42.35</td><td>41.19</td><td>40.92</td></tr><tr><td>Rebalancing (Hou et al., 2019)</td><td>77.30</td><td>64.53</td><td>58.35</td><td>53.82</td><td>49.27</td><td>47.12</td><td>45.16</td><td>43.05</td><td>42.37</td><td>41.02</td><td>40.86</td></tr><tr><td>ProtoNet (Snell et al., 2017)</td><td>77.30</td><td>69.82</td><td>66.12</td><td>63.19</td><td>61.17</td><td>58.85</td><td>58.04</td><td>57.75</td><td>55.84</td><td>55.82</td><td>55.60</td></tr><tr><td>ILVQ (Xu et al., 2012)</td><td>77.30</td><td>71.26</td><td>66.84</td><td>63.82</td><td>62.66</td><td>59.71</td><td>58.92</td><td>58.11</td><td>56.31</td><td>56.14</td><td>56.03</td></tr><tr><td>SDC (Yu et al., 2020)</td><td>77.34</td><td>74.67</td><td>69.73</td><td>66.71</td><td>66.49</td><td>62.14</td><td>61.33</td><td>59.84</td><td>58.01</td><td>57.39</td><td>56.62</td></tr><tr><td>Imprint (Qi et al.,2018)</td><td>77.02</td><td>74.07</td><td>70.26</td><td>66.84</td><td>64.45</td><td>62.46</td><td>61.85</td><td>61.02</td><td>59.20</td><td>58.93</td><td>58.43</td></tr><tr><td>IDLVQ-C</td><td>77.37</td><td>74.79</td><td>70.96</td><td>68.08</td><td>65.94</td><td>64.12</td><td>63.58</td><td>62.98</td><td>60.85</td><td>60.54</td><td>59.72</td></tr></table>
+
+Table 8: Prediction accuracy on CUB using the 10-way 20-shot incremental setting.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="10">sessions</td></tr><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td><td>11</td></tr><tr><td>Fine-tune</td><td>77.30</td><td>47.68</td><td>36.75</td><td>27.01</td><td>24.56</td><td>21.72</td><td>17.88</td><td>15.24</td><td>12.84</td><td>11.89</td><td>11.52</td></tr><tr><td>Joint train</td><td>77.30</td><td>74.34</td><td>72.59</td><td>70.85</td><td>70.14</td><td>70.01</td><td>69.69</td><td>69.67</td><td>69.48</td><td>69.41</td><td>69.42</td></tr><tr><td>iCaRL (Rebuff et al.,2017)</td><td>77.30</td><td>68.85</td><td>63.56</td><td>60.34</td><td>57.71</td><td>56.28</td><td>55.97</td><td>55.02</td><td>54.62</td><td>52.21</td><td>52.23</td></tr><tr><td>Rebalancing (Hou et al.,2019)</td><td>77.30</td><td>69.36</td><td>65.49</td><td>61.32</td><td>59.30</td><td>58.68</td><td>58.77</td><td>57.42</td><td>56.25</td><td>55.33</td><td>54.17</td></tr><tr><td>ProtoNet (Snell et al., 2017)</td><td>77.30</td><td>70.84</td><td>67.82</td><td>64.81</td><td>62.95</td><td>61.71</td><td>60.98</td><td>60.73</td><td>59.51</td><td>59.30</td><td>58.95</td></tr><tr><td>ILVQ (Xu et al., 2012)</td><td>77.30</td><td>71.50</td><td>68.77</td><td>66.12</td><td>64.31</td><td>62.89</td><td>62.31</td><td>62.00</td><td>61.52</td><td>59.79</td><td>59.23</td></tr><tr><td>SDC (Yu et al.,2020)</td><td>77.34</td><td>74.62</td><td>71.63</td><td>68.66</td><td>66.75</td><td>65.24</td><td>64.21</td><td>63.62</td><td>61.97</td><td>61.54</td><td>61.11</td></tr><tr><td>Imprint (Qi et al., 2018)</td><td>77.02</td><td>74.14</td><td>70.72</td><td>67.75</td><td>65.88</td><td>64.63</td><td>64.28</td><td>63.82</td><td>62.00</td><td>61.84</td><td>61.40</td></tr><tr><td>IDLVQ-C</td><td>77.37</td><td>74.84</td><td>72.07</td><td>69.05</td><td>67.28</td><td>65.51</td><td>65.19</td><td>64.84</td><td>62.77</td><td>62.55</td><td>61.96</td></tr></table>
+
+3D spatial data: We follow the same training and test protocols as the sinusoidal wave dataset. We choose 40, 15 and 15 reference vectors and targets for 1st, 2nd and 3rd tasks, respectively. Adding more reference vectors does not result in obvious improvement in accuracy in our experiments.
+
+![](images/5e1419f899b33ff87b466bb8184b86bfa4ee63793fee7e949fdeb2a29f2e20af.jpg)  
+Figure 2: Comparison results of different few-shot settings, evaluated with ResNet18 on CUB dataset
+
+# A.6 VISUALIZATION OF IDLVQ-C
+
+We show the visualization of standard neural networks and IDLVQ-C with/without intra-class variation loss in Fig. 3. MNIST dataset is used as a toy example for visualization. Classes 0-7 are old classes with sufficient training samples, and classes 8 and 9 are novel classes with few-shot training samples. It can be observed in Fig. 3(a) and 3(b) that standard neural networks and IDLVQ-C (without intra-class variation loss) trained by cross-entropy loss do not have compact intra-class variation. Consequently, features of novel classes are more likely to overlap with old classes. In this case, the performance of class-incremental learning degrades very quickly because the classifier cannot distinguish between features from different classes. In comparison, the proposed IDLVQ-C makes intra-class variation compact and leaves large margin between classes in Fig 3(c). As a result, the features of novel classes are less likely to overlap existing classes. The compact intra-class variation and large margin between classes make features of novel classes distinguishable so that learning novel classes is easier. In addition, the margin based loss only updates the model parameters when necessary and avoids catastrophic forgetting of old classes.
+
+![](images/49e99100b8d6c5c5a8ae0c3a024f4b687cf490c746e6964ee89e2468e958123b.jpg)  
+Figure 3: Visualization of feature spaces in different methods. Dots represent features of samples and crosses denote references vectors of classes. (a) standard neural networks; (b) IDLVQ-C without $\mathcal { L } _ { i n t r a }$ ; (c) IDLVQ-C.

parse/train/6xHJ37MVxxp/6xHJ37MVxxp.md

@@ -0,0 +1,331 @@
+# DOMAIN GENERALIZATION WITH MIXSTYLE
+
+Kaiyang Zhou1, Yongxin Yang1, Yu Qiao2, Tao Xiang1
+
+1University of Surrey, UK   
+2Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China   
+k.zhou.vision@gmail.com   
+{yongxin.yang, t.xiang}@surrey.ac.uk   
+yu.qiao@siat.ac.cn
+
+# ABSTRACT
+
+Though convolutional neural networks (CNNs) have demonstrated remarkable ability in learning discriminative features, they often generalize poorly to unseen domains. Domain generalization aims to address this problem by learning from a set of source domains a model that is generalizable to any unseen domain. In this paper, a novel approach is proposed based on probabilistically mixing instancelevel feature statistics of training samples across source domains. Our method, termed MixStyle, is motivated by the observation that visual domain is closely related to image style (e.g., photo vs. sketch images). Such style information is captured by the bottom layers of a CNN where our proposed style-mixing takes place. Mixing styles of training instances results in novel domains being synthesized implicitly, which increase the domain diversity of the source domains, and hence the generalizability of the trained model. MixStyle fits into mini-batch training perfectly and is extremely easy to implement. The effectiveness of MixStyle is demonstrated on a wide range of tasks including category classification, instance retrieval and reinforcement learning.
+
+# 1 INTRODUCTION
+
+Key to automated understanding of digital images is to compute a compact and informative feature representation. Deep convolutional neural networks (CNNs) have demonstrated remarkable ability in representation learning, proven to be effective in many visual recognition tasks, such as classifying photo images into 1,000 categories from ImageNet (Krizhevsky et al., 2012) and playing Atari games with reinforcement learning (Mnih et al., 2013). However, it has long been discovered that the success of CNNs heavily relies on the i.i.d. assumption, i.e. training and test data should be drawn from the same distribution; when such an assumption is violated even just slightly, as in most realworld application scenarios, severe performance degradation is expected (Hendrycks & Dietterich, 2019; Recht et al., 2019).
+
+Domain generalization (DG) aims to address such a problem (Zhou et al., 2021; Blanchard et al., 2011; Muandet et al., 2013; Li et al., 2018a; Zhou et al., 2020b; Balaji et al., 2018; Dou et al., 2019; Carlucci et al., 2019). In particular, assuming that multiple source domains containing the same visual classes are available for model training, the goal of DG is to learn models that are robust against data distribution changes across domains, known as domain shift, so that the trained model can generalize well to any unseen domains. Compared to the closely related and more widely studied domain adaptation (DA) problem, DG is much harder in that no target domain data is available for the model to analyze the distribution shift in order to overcome the negative effects. Instead, a DG model must rely on the source domains and focus on learning domain-invariant feature representation in the hope that it would remain discriminative given target domain data.
+
+A straightforward solution to DG is to expose a model with a large variety of source domains. Specifically, the task of learning domain-invariant and thus generalizable feature representation becomes easier when data from more diverse source domains are available for the model. This would reduce the burden on designing special models or learning algorithms for DG. Indeed, model training with large-scale data of diverse domains is behind the success of existing commercial face recognition or vision-based autonomous driving systems. A recent work by $\mathrm { X u }$ et al. (2021) also emphasizes the importance of diverse training distributions for out-of-distribution generalization. However, collecting data of a large variety of domains is often costly or even impossible. It thus cannot be a general solution to DG.
+
+![](images/f1cd7469df889b614caa015810343d6c004d2bbf757f1fbaacd0388b9ffe208e.jpg)  
+Figure 1: 2-D t-SNE (Maaten & Hinton, 2008) visualization of the style statistics (concatenation of mean and standard deviation) computed from the first residual block’s feature maps of a ResNet-18 (He et al., 2016) trained on four distinct domains (Li et al., 2017). It is clear that different domains are well separated.
+
+In this paper, a novel approach is proposed based on probabilistically mixing instance-level feature statistics of training samples across source domains. Our model, termed MixStyle, is motivated by the observation that visual domain is closely related to image style. An example is shown in Fig. 1: the four images from four different domains depict the same semantic concept, i.e. dog, but with distinctive styles (e.g., characteristics in color and texture). When these images are fed into a deep CNN, which maps the raw pixel values into category labels, such style information is removed at the output. However, recent style transfer studies (Huang & Belongie, 2017; Dumoulin et al., 2017) suggest that such style information is preserved at the bottom layers of the CNN through the instance-level feature statistics, as shown clearly in Fig. 4. Importantly, since replacing such statistics would lead to replaced style while preserving the semantic content of the image, it is reasonable to assume that mixing styles from images of different domains would result in images of (mixed) new styles. That is, more diverse domains/styles can be made available for training a more domain-generalizable model.
+
+Concretely, our MixStyle randomly selects two instances of different domains and adopts a probabilistic convex combination between instance-level feature statistics of bottom CNN layers. In contrast to style transfer work (Huang & Belongie, 2017; Dumoulin et al., 2017), no explicit image synthesis is necessary meaning much simpler model design. Moreover, MixStyle perfectly fits into modern mini-batch training. Overall, it is very easy to implement with only few lines of code. To evaluate the effectiveness as well as the general applicability of MixStyle, we conduct extensive experiments on a wide spectrum of datasets covering category classification (Sec. 3.1), instance retrieval (Sec. 3.2), and reinforcement learning (Sec. 3.3). The results demonstrate that MixStyle can significantly improve CNNs’ cross-domain generalization performance.1
+
+# 2 METHODOLOGY
+
+# 2.1 BACKGROUND
+
+Normalizing feature tensors with instance-specific mean and standard deviation has been found effective for removing image style in style transfer models (Ulyanov et al., 2016; Huang & Belongie, 2017; Dumoulin et al., 2017). Such an operation is widely known as instance normalization (IN, Ulyanov et al. (2016)). Let $\boldsymbol { x } ^ { \prime } \in \mathbb { R } ^ { B \times C \times \dot { H } \times W }$ be a batch of tensors, with $B , C , H$ and $W$ denoting the dimension of batch, channel, height and width, respectively, IN is formulated as
+
+$$
+\mathrm { I N } ( x ) = \gamma \frac { x - \mu ( x ) } { \sigma ( x ) } + \beta ,
+$$
+
+where $\gamma , \beta \in \mathbb { R } ^ { C }$ are learnable affine transformation parameters, and $\mu ( \boldsymbol { x } ) , \sigma ( \boldsymbol { x } ) \in \mathbb { R } ^ { B \times C }$ are mean and standard deviation computed across the spatial dimension within each channel of each instance (tensor), i.e.
+
+$$
+\mu ( x ) _ { b , c } = \frac { 1 } { H W } \sum _ { h = 1 } ^ { H } \sum _ { w = 1 } ^ { W } x _ { b , c , h , w } ,
+$$
+
+and
+
+$$
+\sigma ( x ) _ { b , c } = \sqrt { \frac { 1 } { H W } \sum _ { h = 1 } ^ { H } \sum _ { w = 1 } ^ { W } ( x _ { b , c , h , w } - \mu ( x ) _ { b , c } ) ^ { 2 } } .
+$$
+
+Huang & Belongie (2017) introduced adaptive instance normalization (AdaIN), which simply replaces the scale and shift parameters in Eq. (1) with the feature statistics of style input $y$ to achieve arbitrary style transfer:
+
+$$
+\mathrm { A d a I N } ( x ) = \sigma ( y ) \frac { x - \mu ( x ) } { \sigma ( x ) } + \mu ( y ) .
+$$
+
+# 2.2 MIXSTYLE
+
+Our method, MixStyle, draws inspiration from AdaIN. However, rather than attaching a decoder for image generation, MixStyle is designed for the purpose of regularizing CNN training by perturbing the style information of source domain training instances. It can be implemented as a plug-andplay module inserted between CNN layers of, e.g., a supervised CNN classifier, without the need to explicitly generate an image of new style.
+
+More specifically, MixStyle mixes the feature statistics of two instances with a random convex weight to simulate new styles. In terms of implementation, MixStyle can be easily integrated into mini-batch training. Given an input batch $x$ , MixStyle first generates a reference batch $\tilde { x }$ from $x$ . When domain labels are given, $x$ is sampled from two different domains $i$ and $j$ , e.g., $\bar { \boldsymbol { x } } = [ x ^ { i } , x ^ { j } ]$ $\bar { x ^ { i } }$ and $x ^ { j }$ have the same batch size). Then, $\tilde { x }$ is obtained by swapping the position of $x ^ { i }$ and $x ^ { j }$ , followed by a shuffling operation along the batch dimension applied to each batch, i.e. $\mathbf { \widetilde { \boldsymbol { x } } } = [ \mathrm { S h u f f e } ( \boldsymbol { x } ^ { j } ) , \mathrm { S h u f f e } ( \boldsymbol { x } ^ { i } ) ]$ . See Fig. 2(a) for an illustration. In cases where domain labels are unknown, $x$ is randomly sampled from the training data, and $\tilde { x }$ is simply obtained by $\tilde { x } = \mathrm { S h u f f e } ( x )$ (see Fig. 2(b)). Fig. 4 shows that sub-domains exist within each domain, so even if two instances of the same domain are sampled, new domain could be synthesized. After shuffling, MixStyle computes the mixed feature statistics by
+
+![](images/328a14b30cdf68b6ebf3e522fa5e10ef00672d004ed56b5521dcee7294f42db8.jpg)  
+Figure 2: A graphical illustration of how a reference batch is generated. Domain label is denoted by color.
+
+$$
+\begin{array} { r } { \gamma _ { m i x } = \lambda \sigma ( x ) + ( 1 - \lambda ) \sigma ( \tilde { x } ) , } \\ { \beta _ { m i x } = \lambda \mu ( x ) + ( 1 - \lambda ) \mu ( \tilde { x } ) , } \end{array}
+$$
+
+where $\lambda \in \mathbb { R } ^ { B }$ are instance-wise weights sampled from the Beta distribution, $\lambda \sim B e t a ( \alpha , \alpha )$ with $\alpha \in ( 0 , \infty )$ being a hyper-parameter. Unless specified otherwise, we set $\alpha$ to 0.1 throughout this paper. Finally, the mixed feature statistics are applied to the style-normalized $x$ ,
+
+$$
+{ \mathrm { M i x S t y l e } } ( x ) = \gamma _ { m i x } { \frac { x - \mu ( x ) } { \sigma ( x ) } } + \beta _ { m i x } .
+$$
+
+In practice, we use a probability of 0.5 to decide if MixStyle is activated or not in the forward pass. At test time, no MixStyle is applied. Note that gradients are blocked in the computational graph of $\mu ( \cdot )$ and $\sigma ( \cdot )$ . MixStyle can be implemented with only few lines of code. See Algorithm 1 in Appendix A.1 for the PyTorch-like pseudo-code.
+
+Table 1: Leave-one-domain-out generalization results on PACS.   
+
+<table><tr><td>Method</td><td>Art</td><td>Cartoon</td><td>Photo</td><td>Sketch</td><td>Avg</td></tr><tr><td>MMD-AAE CCSA JiGen</td><td>75.2 80.5 79.4 79.8 82.1</td><td>72.7 76.9 75.3 76.8</td><td>96.0 93.6 96.0 96.0</td><td>64.2 66.8 71.6 70.2</td><td>77.0 79.4 80.5 80.7</td></tr><tr><td>Epi-FCR Metareg L2A-OT ResNet-18</td><td>83.7 83.3 77.0±0.6</td><td>77.0 77.2 78.2 75.9±0.6</td><td>93.9 95.5 96.2 96.0±0.1</td><td>73.0 70.3 73.6 69.2±0.6</td><td>81.5 81.7 82.8 79.5</td></tr><tr><td>+ Manifold Mixup</td><td>75.6±0.7</td><td>70.1±0.9</td><td>93.5±0.7</td><td>65.4±0.6</td><td>76.2</td></tr><tr><td>+ Cutout</td><td>74.9±0.4</td><td>74.9±0.6</td><td>95.9±0.3</td><td>67.7±0.9</td><td>78.3</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+ CutMix</td><td>74.6±0.7</td><td>71.8±0.6</td><td>95.6±0.4</td><td>65.3±0.8</td><td>76.8</td></tr><tr><td></td><td>74.7±1.0</td><td>72.3±0.9</td><td></td><td></td><td></td></tr><tr><td>+ Mixup (w/o label interpolation)</td><td></td><td></td><td>93.0±0.4</td><td>69.2±0.2</td><td>77.3</td></tr><tr><td>+ Mixup</td><td>76.8±0.7</td><td>74.9±0.7</td><td>95.8±0.3</td><td>66.6±0.7</td><td>78.5</td></tr><tr><td>+ DropBlock</td><td>76.4±0.7</td><td>75.4±0.7</td><td>95.9±0.3</td><td>69.0±0.3</td><td>79.2</td></tr><tr><td>+ MixStyle w/ random shuffle</td><td>82.3±0.2</td><td>79.0±0.3</td><td>96.3±0.3</td><td>73.8±0.9</td><td>82.8</td></tr><tr><td>+ MixStyle w/ domain label</td><td>84.1±0.4</td><td>78.8±0.4</td><td>96.1±0.3</td><td>75.9±0.9</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>83.7</td></tr></table>
+
+# 3 EXPERIMENTS
+
+# 3.1 GENERALIZATION IN CATEGORY CLASSIFICATION
+
+Dataset and implementation details. We choose the PACS dataset (Li et al., 2017), a commonly used domain generalization (DG) benchmark concerned with domain shift in image classification. PACS consists of four domains, i.e. Art Painting, Cartoon, Photo and Sketch, with totally 9,991 images of 7 classes. As shown in Fig. 1, the domain shift mainly corresponds to image style changes. For evaluation, a model is trained on three domains and tested on the remaining one. Following prior work (Li et al., 2019; Zhou et al., 2020a), we use ResNet-18 (He et al., 2016) as the classifier where MixStyle is inserted after the 1st, 2nd and 3rd residual blocks. Our code is based on Dassl.pytorch (Zhou et al., 2020c).2
+
+Baselines. Our main baselines are general-purpose regularization methods including Mixup (Zhang et al., 2018b), Manifold Mixup (Verma et al., 2019), DropBlock (Ghiasi et al., 2018), CutMix (Yun et al., 2019) and Cutout (DeVries & Taylor, 2017), which are trained using the same training parameters as MixStyle and the optimal hyper-parameter setup as reported in their papers. We also compare with the existing DG methods which reported state-of-the-art performance on PACS. These include domain alignment-based CCSA (Motiian et al., 2017) and MMD-AAE (Li et al., 2018b), Jigsaw puzzle-based JiGen (Carlucci et al., 2019), adversarial gradient-based CrossGrad (Shankar et al., 2018), meta-learning-based Metareg (Balaji et al., 2018) and Epi-FCR (Li et al., 2019), and data augmentation-based L2A-OT (Zhou et al., 2020a).
+
+Comparison with general-purpose regularization methods. The results are shown in Table 1. Overall, we observe that the general-purpose regularization methods do not offer any clear advantage over the vanilla ResNet-18 in this DG task, while MixStyle improves upon the vanilla ResNet18 with a significant margin. Compared with Mixup, MixStyle is $5 . 2 \%$ better on average. Recall that Mixup also interpolates the output space, we further compare with a variant of Mixup in order to demonstrate the advantage of mixing style statistics at the feature level over mixing images at the pixel level for DG—following Sohn et al. (2020), we remove the label interpolation in Mixup and sample the mixing weights from a uniform distribution of $[ 0 , 1 ]$ . Still, MixStyle outperforms this new baseline with a large margin, which justifies our claim. MixStyle and DropBlock share some commonalities in that they are both applied to feature maps at multiple layers, but MixStyle significantly outperforms DropBlock in all test domains. The reason why DropBlock is ineffective here is because dropping out activations mainly encourages a network to mine discriminative patterns, but does not reinforce the ability to cope with unseen styles, which is exactly what MixStyle aims to achieve: by synthesizing “new” styles (domains) MixStyle regularizes the network to become more robust to domain shift. In addition, it is interesting to see that on Cartoon and Photo, MixStyle w/ random shuffle obtains slightly better results. The reason might be because there exist sub-domains in a source domain (see Fig. 4(a-c)), which allow random shuffling to produce more diverse “new” domains that lead to a more domain-generalizable model.
+
+Comparison with state-of-the-art DG methods. Overall, MixStyle outperforms most DG methods by a clear margin, despite being a much simpler method. The performance of MixStyle w/ domain label is nearly $1 \%$ better on average than the recently introduced L2A-OT. From a data augmentation perspective, MixStyle and L2A-OT share a similar goal—to synthesize data from pseudo-novel domains. MixStyle accomplishes this goal through mixing style statistics at the feature level. Whereas L2A-OT works at the pixel level: it trains an image generator by maximizing the domain difference (measured by optimal transport) between the original and the generated images, which introduces much heavier computational overhead than MixStyle in terms of GPU memory and training time. It is worth noting that MixStyle’s domain label-free version is highly competitive: its $8 2 . 8 \%$ accuracy is on par with L2A-OT’s.
+
+# 3.2 GENERALIZATION IN INSTANCE RETRIEVAL
+
+Dataset and implementation details. We evaluate MixStyle on the person re-identification (re-ID) problem, which aims to match people across disjoint camera views. As each camera view is itself a distinct domain, person re-ID is essentially a cross-domain image matching problem. Instead of using the standard protocol where training and test data come from the same camera views, we adopt the cross-dataset setting so test camera views are never seen during training. Specifically, we train a model on one dataset and then test its performance on the other dataset. Two commonly used re-ID datasets are adopted: Market1501 (Zheng et al., 2015) and Duke (Ristani et al., 2016; Zheng et al., 2017). Ranking accuracy and mean average precision (mAP) are used as the performance measures (displayed in percentage). We test MixStyle on two CNN architectures: ResNet-50 (He et al., 2016) and OSNet (Zhou et al., 2019). The latter was designed specifically for re-ID. In both architectures, MixStyle is inserted after the 1st and 2nd residual blocks. Our code is based on Torchreid (Zhou & Xiang, 2019).3
+
+Baselines. We compare with three baseline methods: 1) The vanilla model, which serves as a strong baseline; 2) DropBlock, which was the top-performing competitor in Table 1; 3) RandomErase (Zhong et al., 2020), a widely used regularization method in the re-ID literature (similar to Cutout).
+
+Results. The results are reported in Table 2. It is clear that only MixStyle consistently outperforms the strong vanilla model under both settings with considerable margins, while DropBlock and RandomErase are unable to show any benefit. Notably, RandomErase, which simulates occlusion by erasing pixels in random rectangular regions with random values, has been used as a default trick when training re-ID CNNs. However, RandomErase shows a detrimental effect in the cross-dataset re-ID setting. Indeed, similar to DropBlock, randomly erasing pixels offers no guarantee to improve the robustness when it comes to domain shift.
+
+# 3.3 GENERALIZATION IN REINFORCEMENT LEARNING
+
+Though RL has been greatly advanced by using CNNs for feature learning in raw pixels (Mnih et al., 2013), it has been widely acknowledged that RL agents often overfit training environments while generalize poorly to unseen environments (Cobbe et al., 2019; Igl et al., 2019).
+
+Dataset and implementation details. We conduct experiments on Coinrun (Cobbe et al., 2019), a recently introduced RL benchmark for evaluating the generalization performance of RL agents. As shown in Fig. 3(a), the goal in Coinrun is to control a character to collect golden coins while avoiding both stationary and non-stationary obstacles. We follow Igl et al. (2019) to construct and train our RL agent: the CNN architecture used in IMPALA (Espeholt et al., 2018) is adopted as the policy network, and is trained by the Proximal Policy Optimization (PPO) algorithm (Schulman et al., 2017). Please refer to Igl et al. (2019) for further implementation details. MixStyle is inserted after the 1st and 2nd convolutional sequences. Training data are sampled from 500 levels while test data are drawn from new levels of only the highest difficulty. As domain labels are difficult to define, we use the random shuffle version of MixStyle. Our code is built on top of Igl et al. (2019).4
+
+Table 2: Generalization results on the cross-dataset person re-ID task.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="4">Market1501-→Duke</td><td colspan="4">Duke-→Market1501</td></tr><tr><td>mAP</td><td>R1</td><td>R5</td><td>R10</td><td>mAP</td><td>R1</td><td>R5</td><td>R10</td></tr><tr><td>ResNet-50</td><td>19.3</td><td>35.4</td><td>50.3</td><td>56.4</td><td>20.4</td><td>45.2</td><td>63.6</td><td>70.9</td></tr><tr><td>+ RandomErase</td><td>14.3</td><td>27.8</td><td>42.6</td><td>49.1</td><td>16.1</td><td>38.5</td><td>56.8</td><td>64.5</td></tr><tr><td>+ DropBlock</td><td>18.2</td><td>33.2</td><td>49.1</td><td>56.3</td><td>19.7</td><td>45.3</td><td>62.1</td><td>69.1</td></tr><tr><td>+ MixStyle w/ random shuffle</td><td>23.8</td><td>42.2</td><td>58.8</td><td>64.8</td><td>24.1</td><td>51.5</td><td>69.4</td><td>76.2</td></tr><tr><td>+ MixStyle w/ domain label</td><td>23.4</td><td>43.3</td><td>58.9</td><td>64.7</td><td>24.7</td><td>53.0</td><td>70.9</td><td>77.8</td></tr><tr><td>OSNet</td><td>25.9</td><td>44.7</td><td>59.6</td><td>65.4</td><td>24.0</td><td>52.2</td><td>67.5</td><td>74.7</td></tr><tr><td>+ RandomErase</td><td>20.5</td><td>36.2</td><td>52.3</td><td>59.3</td><td>22.4</td><td>49.1</td><td>66.1</td><td>73.0</td></tr><tr><td>+ DropBlock</td><td>23.1</td><td>41.5</td><td>56.5</td><td>62.5</td><td>21.7</td><td>48.2</td><td>65.4</td><td>71.3</td></tr><tr><td>+ MixStyle w/ random shuffle</td><td>27.2</td><td>48.2</td><td>62.7</td><td>68.4</td><td>27.8</td><td>58.1</td><td>74.0</td><td>81.0</td></tr><tr><td>+ MixStyle w/ domain label</td><td>27.3</td><td>47.5</td><td>62.0</td><td>67.1</td><td>29.0</td><td>58.2</td><td>74.9</td><td>80.9</td></tr></table>
+
+![](images/83f0338beadfd5bc519830499778de6828846e5597dc084ac87299494b1148a1.jpg)  
+Figure 3: (a) Coinrun benchmark. (b) Test performance in unseen environments. (c) Difference between training and test performance.
+
+Baselines. Following Igl et al. (2019), we train strong baseline models and add MixStyle on top of them to see whether MixStyle can bring further improvements. To this end, we train two baseline models: 1) Baseline, which combines weight decay and data augmentation;5 2) IBAC-SNI (the $\lambda = 0 . 5$ version), the best-performing model in Igl et al. (2019) which is based on selective noise injection.
+
+Results. The test performance is shown in Fig. 3(b). Comparing Baseline (blue) with Baseline+MixStyle (orange), we can see that MixStyle brings a significant improvement. Interestingly, the variance is also significantly reduced by using MixStyle, as indicated by the smaller shaded areas (for both orange and red lines). These results strongly demonstrate the effectiveness of MixStyle in enhancing generalization for RL agents. When it comes to the stronger baseline IBAC-SNI (green), MixStyle (red) is able to bring further performance gain, suggesting that MixStyle is complementary to IBAC-SNI. This result also shows the potential of MixStyle as a plug-and-play component to be combined with other advanced RL methods. It is worth noting that Baseline+MixStyle itself is already highly competitive with IBAC-SNI. Fig. 3(c) shows the generalization gap from which it can been seen that the models trained with MixStyle (orange & red) clearly generalize faster and better than those without using MixStyle (blue & green).
+
+![](images/4d2d1d46181970a60036d74149364fc1b7c14abba9cfcdf85a1601e0d521acda.jpg)  
+Figure 4: 2-D visualization of flattened feature maps (top) and the corresponding style statistics (bottom). res1-4 denote the four residual blocks in order in a ResNet architecture. We observe that res1 to res3 contain domain-related information while res4 encodes label-related information.
+
+Table 3: Ablation study on where to apply MixStyle in the ResNet architecture.   
+(a) Category classification on PACS.   
+
+<table><tr><td>Model</td><td>Accuracy</td></tr><tr><td>ResNet-18</td><td>79.5</td></tr><tr><td>+ MixStyle (res1)</td><td>80.1</td></tr><tr><td>+ MixStyle (res12)</td><td>81.6</td></tr><tr><td>+ MixStyle (res123)</td><td>82.8</td></tr><tr><td>+ MixStyle (res1234)</td><td>75.6</td></tr><tr><td>+ MixStyle (res14)</td><td>76.3</td></tr><tr><td>+ MixStyle (res23)</td><td>81.7</td></tr></table>
+
+(b) Cross-dataset person re-ID.   
+
+<table><tr><td>Model</td><td>mAP</td></tr><tr><td>ResNet-50</td><td>19.3</td></tr><tr><td>+ MixStyle (res1)</td><td>22.6</td></tr><tr><td>+ MixStyle (res12) + MixStyle (res123)</td><td>23.8</td></tr><tr><td>+MixStyle (res1234)</td><td>22.0</td></tr><tr><td>+ MixStyle (res14)</td><td>10.2</td></tr><tr><td>+MixStyle (res23)</td><td>11.1 20.6</td></tr></table>
+
+# 3.4 ANALYSIS
+
+Where to apply MixStyle? We repeat the experiments on PACS (category classification) and the re-ID datasets (instance retrieval) using the ResNet architecture. Given that a standard ResNet model has four residual blocks denoted by $\mathtt { r e s l - 4 }$ , we train different models with MixStyle applied to different layers. For notation, res1 means MixStyle is applied after the first residual block; res12 means MixStyle is applied after both the first and second residual blocks; and so forth. The results are shown in Table 3. We have the following observations. 1) Applying MixStyle to multiple lowerlevel layers generally achieves a better performance—for instance, res12 is better than res1 on both tasks. 2) Different tasks favor different combinations—res123 achieves the best performance on PACS, while on the re-ID datasets $\mathtt { r e s 1 2 }$ is the best. 3) On both tasks, the performance plunges when applying MixStyle to the last residual block. This makes sense because res4 is the closest to the prediction layer and tends to capture semantic content (i.e. label-sensitive) information rather than style. In particular, res4 is followed by an average-pooling layer, which essentially forwards the mean vector to the prediction layer and thus forces the mean vector to capture label-related information. As a consequence, mixing the statistics at res4 breaks the inherent label space. This is clearer in Fig. 4: the features and style statistics in $\mathtt { r e s l - 3 }$ exhibit clustering patterns based on domains while those in res4 have a high correlation with class labels.
+
+![](images/c7e15cbb6312d6d6e57a51169db1f4a01f9832c478ec13254c9df6c7db06a565.jpg)  
+Figure 5: Evaluation on the hyper-parameter $\alpha$ on (a) PACS, (b) person re-ID datasets and (c) Coinrun. In (b), M and D denote Market1501 and Duke respectively.
+
+Mixing vs. replacing. Unlike the AdaIN formulation, which completely replaces one style with the other, MixStyle mixes two styles via a convex combination. Table 4 shows that mixing is better than replacing. This is easy to understand: mixing diversifies the styles (imagine an interpolation between two data points).
+
+Random vs. fixed shuffle at multiple layers. Applying MixStyle to multiple layers, which has been shown advantageous in Table 3, raises another question of whether to shuffle the mini-batch at different layers or use the same shuffled order for all layers. Table 5 suggests that using random shuffle at different layers gives a better performance, which may be attributed to the increased noise level that gives a better regularization effect.
+
+Table 4: Mixing vs. replacing.   
+
+<table><tr><td></td><td>Accuracy (%)</td></tr><tr><td>Mixing</td><td>82.8±0.4</td></tr><tr><td>Replacing</td><td>82.1±0.5</td></tr></table>
+
+Table 5: Random vs. fixed shuffle at multiple layers.   
+
+<table><tr><td colspan="2">Accuracy (%)</td></tr><tr><td>Random</td><td>82.8±0.4</td></tr><tr><td>Fixed</td><td>82.4±0.5</td></tr></table>
+
+Sensitivity of hyper-parameter. Recall that $\alpha$ is used to control the shape of Beta distribution, which has a direct effect on how the convex weights $\lambda$ are sampled. The smaller $\alpha$ is, the more likely the value in $\lambda$ is close to the extreme value of 0 or 1. In other words, a smaller $\alpha$ favors the style statistics in Eqs. (5) & (6) to be dominated by one side. We first evaluate $\alpha$ on PACS. Fig. 5(a) shows that with $\alpha$ increasing from 0.1 to 0.4, the accuracy slides from $8 2 . 8 \%$ to $8 1 . 7 \%$ . However, further increasing $\alpha$ does not impact on the accuracy. Therefore, the results suggest that the performance is not too sensitive to $\alpha$ ; and selecting $\alpha$ from $\lbrace 0 . 1 , 0 . 2 , 0 . 3 \rbrace$ seems to be a good starting point. We further experiment with $\alpha \in \{ 0 . 1 , 0 . 2 , 0 . 3 \}$ on the re-ID datasets and the Coinrun benchmark. Figs. 5(b) & (c) show that in general the variance for the results of different values is small. Therefore, we suggest practitioners to choose $\alpha$ from $\lbrace 0 . 1 , 0 . 2 , 0 . 3 \rbrace$ , with $\alpha = 0 . 1$ being a good default setting.
+
+For more analyses and discussions, please see Appendix A.2.
+
+# 4 RELATED WORK
+
+Domain generalization, or DG, studies out-of-distribution (OOD) generalization given only source data typically composed of multiple related but distinct domains. We refer readers to Zhou et al. (2021) for a comprehensive survey in this topic. Many DG methods are based on the idea of aligning features between different sources, with a hope that the model can be invariant to domain shift given unseen data. For instance, Li et al. (2018b) achieved distribution alignment in the hidden representation of an autoencoder using maximum mean discrepancy; Li et al. (2018c) resorted to adversarial learning with auxiliary domain classifiers to learn features that are domain-agnostic. Some works explored domain-specific parameterization, such as domain-specific weight matrices (Li et al., 2017) and domain-specific BN (Seo et al., 2020). Recently, meta-learning has drawn increasing attention from the DG community (Li et al., 2018a; Balaji et al., 2018; Dou et al., 2019). The main idea is to expose a model to domain shift during training by using pseudo-train and pseudo-test domains, both drawn from source domains. Data augmentation has also been investigated for learning domain-invariant models. Shankar et al. (2018) introduced a cross-gradient training method (CrossGrad) where source data are augmented by adversarial gradients obtained from a domain classifier.
+
+Gong et al. (2019) proposed DLOW (for the DA problem), which models intermediate domains between source and target via a domainness factor and learns an image translation model to generate intermediate-domain images. Very recently, Zhou et al. (2020a) introduced L2A-OT to learn a neural network to map source data to pseudo-novel domains by maximizing an optimal transport-based distance measure. Our MixStyle is related to DLOW and L2A-OT in its efforts to synthesizing novel domains. However, MixStyle differs in the fact that it is done implicitly with a much simpler formulation leveraging the feature-level style statistics and only few lines of extra code on top of a standard supervised classifier while being more effective. Essentially, MixStyle can be seen as feature-level augmentation, which is clearly different from the image-level augmentation-based DLOW and L2A-OT.
+
+Generalization in deep RL has been a challenging problem where RL agents often overfit training environments, and as a result, perform poorly in unseen environments with different visual patterns or levels (Zhang et al., 2018a). A natural way to improve generalization, which has been shown effective in (Cobbe et al., 2019; Farebrother et al., 2018), is to use regularization, e.g., weight decay. However, Igl et al. (2019) suggested that stochastic regularization methods like dropout and batch normalization (which uses estimated population statistics) have adverse effect as the training data in RL are essentially model-dependent. As such, they proposed selective noise injection (SNI), which basically combines a stochastic regularization technique with its deterministic counterpart. They further integrated SNI with information bottleneck actor critic (IBAC-SNI) to reduce the variance in gradients. Curriculum learning has been investigated in (Justesen et al., 2018) where the level of training episodes progresses from easy to difficult over the course of training. Gamrian & Goldberg (2019) leveraged the advances in GAN-based image-to-image translation (Liu et al., 2017) to map target data to the source domain which the agent was trained on. Tobin et al. (2017) introduced domain randomization, which diversifies training data by rendering images with different visual effects via a programmable simulator. With a similar goal of data augmentation, Lee et al. (2020) pre-processed input images with a randomly initialized network. Very recent studies (Laskin et al., 2020; Kostrikov et al., 2020) have shown that it is useful to combine a diverse set of label-preserving transformations, such as rotation, shifting and Cutout. Different from the aforementioned methods, our MixStyle works at the feature level and is orthogonal to most existing methods. For instance, we have shown in Sec. 3.3 that MixStyle significantly improves upon IBAC-SNI.
+
+# 5 CONCLUSION
+
+We presented a simple yet effective domain generalization method, termed MixStyle. MixStyle mixes the feature statistics of two instances to synthesize novel domains, which is inspired by the observation in style transfer work that the feature statistics encode style/domain-related information. Extensive experiments covering a wide range of tasks were conducted to demonstrate that MixStyle yields new state-of-the-art on three different tasks.
+
+# ACKNOWLEDGMENTS
+
+This work was supported in part by the Shanghai Committee of Science and Technology, China (Grant No. 20DZ1100800), in part by the National Natural Science Foundation of China under Grant (61876176, U1713208), the National Key Research and Development Program of China (No. 2020YFC2004800), and in part by the Guangzhou Research Program (201803010066).
+
+# REFERENCES
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+Keyulu Xu, Mozhi Zhang, Jingling Li, Simon Shaolei Du, Ken-Ichi Kawarabayashi, and Stefanie Jegelka. How neural networks extrapolate: From feedforward to graph neural networks. In ICLR, 2021.   
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+
+# A APPENDIX
+
+# A.1 PSEUDO-CODE OF MIXSTYLE
+
+Algorithm 1 provides a PyTorch-like pseudo-code.
+
+# Algorithm 1 PyTorch-like pseudo-code for MixStyle.
+
+# x: input features of shape (B, C, H, W)   
+# p: probabillity to apply MixStyle (default: 0.5)   
+# alpha: hyper-parameter for the Beta distribution (default: 0.1)   
+# eps: a small value added before square root for numerical stability (default: 1e-6)
+
+if not in training mode: return x
+
+if random probability $>$ p: return x
+
+$\mathrm { ~ \textit ~ { ~ B ~ } ~ } = \mathrm { ~ \textit ~ { ~ x ~ } ~ }$ .size(0) # batch size
+
+mu $= \times$ .mean(dim=[2, 3], keepdim $\cdot ^ { = }$ True) # compute instance mean var $=$ x.var(dim=[2, 3], keepdim $\cdot ^ { = }$ True) # compute instance variance sig $=$ (var $^ +$ eps).sqrt() # compute instance standard deviation mu, sig $\qquad = \quad \mathtt { m u }$ .detach(), sig.detach() # block gradients x_normed $=$ (x - mu) / sig # normalize input
+
+lmda $=$ Beta(alpha, alpha).sample((B, 1, 1, 1)) # sample instance-wise convex weights
+
+if domain label is given: # in this case, input $\begin{array} { r } { \mathrm { ~  ~ \underline { ~ } { ~ \sf ~ } { ~ \sf ~ } { ~ \sf ~ } { ~ \sf ~ } = ~ \mathrm { ~  ~ \tau ~ } [ \times \hat { \mathrm { ~  ~ \underline { ~ } { ~ \bf ~ } { ~ \underline { ~ } { ~ \bf ~ } { ~ \tau ~ } ~ } } } } } \end{array}$ x perm $=$ torch.arange(B-1, -1, -1) # inverse index perm_j, perm_i $=$ perm.chunk(2) # separate indices perm_j $=$ perm_j[torch.randperm(B // 2)] # shuffling perm_i $=$ perm_i[torch.randperm(B // 2)] # shuffling perm $=$ torch.cat([perm_j, perm_i], 0) # concatenation   
+else: perm $=$ torch.randperm(B) # generate shuffling indices
+
+mu2, sig2 $=$ mu[perm], sig[perm] # shuffling mu_mix $=$ mu $\star$ lmda $^ +$ mu2 $\star$ (1 - lmda) # generate mixed mean sig_mix $=$ sig $\star$ lmda $^ +$ sig2 $\star$ (1 - lmda) # generate mixed standard deviation return x_normed $\star$ sig_mix $^ +$ mu_mix # denormalize input using the mixed statistics
+
+Table 7: Test results on the source domains on PACS. A: Art. C: Cartoon. P: Photo. S: Sketch.   
+
+<table><tr><td>Method</td><td>C,P,S</td><td>A,P,S</td><td>A,C,S</td><td>A,C,P</td><td>Avg</td></tr><tr><td>Vanilla</td><td>99.49±0.03</td><td>99.47±0.04</td><td>99.38±0.02</td><td>99.65±0.03</td><td>99.50</td></tr><tr><td>MixStyle</td><td>99.55±0.02</td><td>99.54±0.01</td><td>99.47±0.03</td><td>99.68±0.03</td><td>99.56</td></tr></table>
+
+Table 8: Leave-one-domain-out generalization results on Digits-DG.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>MNIST   MNIST-M   SVHN      SYN</td><td rowspan=1 colspan=1>Avg</td></tr><tr><td rowspan=1 colspan=1>JiGen</td><td rowspan=1 colspan=1>96.5        61.4        63.7       74.0</td><td rowspan=1 colspan=1>73.9</td></tr><tr><td rowspan=1 colspan=1>CCSA</td><td rowspan=1 colspan=1>95.2        58.2        65.5       79.1</td><td rowspan=1 colspan=1>74.5</td></tr><tr><td rowspan=3 colspan=1>MMD-AAECrossGradL2A-OT</td><td rowspan=1 colspan=1>96.5        58.4        65.0       78.4</td><td rowspan=3 colspan=1>74.675.878.1</td></tr><tr><td rowspan=1 colspan=1>96.7        61.1        65.3       80.2</td></tr><tr><td rowspan=1 colspan=1>96.7        63.9        68.6       83.2</td></tr><tr><td rowspan=8 colspan=1>CNN+ Mixup w/o label interpolation+ Manifold Mixup+ CutMix+ Mixup+ Cutout+ DropBlock+ MixStyle (ours)</td><td rowspan=1 colspan=1>95.8±0.3  58.8±0.5  61.7±0.5  78.6±0.6</td><td rowspan=1 colspan=1>73.7</td></tr><tr><td rowspan=1 colspan=1>93.7±0.6  55.2±1.0   61.6±0.9  74.4±0.8</td><td rowspan=3 colspan=1>71.271.771.8</td></tr><tr><td rowspan=1 colspan=1>92.7±0.4   53.1±0.8   64.4±0.2  76.8±0.5</td></tr><tr><td rowspan=1 colspan=1>94.9±0.2   50.1±0.5   64.1±0.9  78.1±0.7</td></tr><tr><td rowspan=1 colspan=1>94.2±0.5  56.5±0.8   63.3±0.7  76.7±0.6</td><td rowspan=4 colspan=1>72.774.175.376.5</td></tr><tr><td rowspan=1 colspan=1>95.8±0.4   58.4±0.6   61.9±0.9  80.6±0.5</td></tr><tr><td rowspan=1 colspan=1>96.2±0.1   60.5±0.6  64.1±0.8  80.2±0.6</td></tr><tr><td rowspan=1 colspan=1>96.5±0.3  63.5±0.8  64.7±0.7  81.2±0.8</td></tr></table>
+
+# A.2 FURTHER ANALYSIS
+
+# MixStyle between same-domain instances.
+
+We are interested in knowing if mixing styles between same-domain instances helps performance. To this end, we sample each mini-batch from a single domain during training when using MixStyle. The results are shown in Table 6 where we observe that mixing styles between same-domain instances is about $1 \%$ better than the baseline model. This suggests that instancespecific style exists. Nonetheless, the performance is clearly worse than mixing styles between instances of different domains.
+
+Table 6: Investigation on the effect of mixing styles between same-domain instances.   
+
+<table><tr><td>Accuracy (%)</td></tr><tr><td>ResNet18 79.5</td></tr><tr><td>+MixStyle w/ same-domain 80.4</td></tr><tr><td>+MixStyle w/random shuffle 82.8</td></tr><tr><td>+MixStyle w/ domain label 83.7</td></tr></table>
+
+Performance on source domains. To prove that MixStyle does not sacrifice the performance on seen domains in exchange for gains on unseen domains, we report the test accuracy on the held-out validation set of the source domains on PACS in Table 7.
+
+Results on Digits-DG and Office-Home. In addition to the experiments on PACS (in Sec. 3.1), we further evaluate MixStyle’s effectiveness on two DG datasets, namely Digits-DG (Zhou et al., 2020a) and Office-Home (Venkateswara et al., 2017). Digits-DG contains four digit datasets (domains) including MNIST (LeCun et al., 1998), MNIST-M (Ganin & Lempitsky, 2015), SVHN (Netzer et al., 2011) and SYN (Ganin & Lempitsky, 2015). Images from different digit datasets differ drastically in font style, stroke color and background. Office-Home is composed of four domains (Artistic, Clipart, Product and Real World) with around 15,500 images of 65 classes for home and office object recognition. The results are shown in Tables 8 and 9 where no domain labels are used in MixStyle. Similar to the results on PACS, here we observe that MixStyle also brings clear improvements to the baseline CNN model and outperforms all general-purpose regularization methods on both Digits-DG and Office-Home. Compared with more sophisticated DG methods like L2A-OT, MixStyle’s performance is comparable, despite being much simpler to train and consuming much less computing resources.
+
+Table 9: Leave-one-domain-out generalization results on Office-Home.   
+
+<table><tr><td>Method</td><td>Artistic</td><td>Clipart</td><td>Product</td><td>Real World</td><td>Avg</td></tr><tr><td>JiGen CCSA</td><td>53.0</td><td>47.5</td><td>71.5</td><td>72.8</td><td>61.2</td></tr><tr><td>MMD-AAE</td><td>59.9 56.5</td><td>49.9 47.3</td><td>74.1 72.1</td><td>75.7 74.8</td><td>64.9 62.7</td></tr><tr><td>CrossGrad</td><td></td><td></td><td></td><td>75.8</td><td>64.4</td></tr><tr><td>L2A-OT</td><td>58.4</td><td>49.4</td><td>73.9</td><td></td><td></td></tr><tr><td>ResNet18</td><td>60.6 58.9±0.3</td><td>50.1 49.4±0.1</td><td>74.8 74.3±0.1</td><td>77.0 76.2±0.2</td><td>65.6 64.7</td></tr><tr><td>+ Manifold Mixup</td><td>56.2±0.4</td><td>46.3±0.3</td><td>73.6±0.1</td><td>75.2±0.2</td><td>62.8</td></tr><tr><td>+ Mixup w/o label interpolation + Cutout</td><td>57.0±0.2 57.8±0.2</td><td>48.7±0.2</td><td>71.4±0.6</td><td>74.5±0.4</td><td>62.9</td></tr><tr><td></td><td></td><td>48.1±0.3</td><td>73.9±0.2</td><td>75.8±0.3</td><td>63.9</td></tr><tr><td>+ CutMix</td><td>57.9±0.1</td><td>48.3±0.3</td><td>74.5±0.1</td><td>75.6±0.4</td><td>64.1</td></tr><tr><td>+ DropBlock</td><td>58.0±0.1</td><td>48.1±0.1</td><td>74.3±0.3</td><td>75.9±0.4</td><td>64.1</td></tr><tr><td>+ Mixup + MixStyle (ours)</td><td>58.2±0.1 58.7±0.3</td><td>49.3±0.2 53.4±0.2</td><td>74.7±0.1 74.2±0.1</td><td>76.1±0.1 75.9±0.1</td><td>64.6 65.5</td></tr></table>

parse/train/6xHJ37MVxxp/6xHJ37MVxxp_content_list.json

@@ -0,0 +1,1667 @@
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+    {
+        "type": "text",
+        "text": "DOMAIN GENERALIZATION WITH MIXSTYLE",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kaiyang Zhou1, Yongxin Yang1, Yu Qiao2, Tao Xiang1 ",
+        "bbox": [
+            184,
+            143,
+            560,
+            160
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1University of Surrey, UK   \n2Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen, China   \nk.zhou.vision@gmail.com   \n{yongxin.yang, t.xiang}@surrey.ac.uk   \nyu.qiao@siat.ac.cn ",
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+            184,
+            161,
+            812,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+            282
+        ],
+        "page_idx": 0
+    },
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+        "text": "Though convolutional neural networks (CNNs) have demonstrated remarkable ability in learning discriminative features, they often generalize poorly to unseen domains. Domain generalization aims to address this problem by learning from a set of source domains a model that is generalizable to any unseen domain. In this paper, a novel approach is proposed based on probabilistically mixing instancelevel feature statistics of training samples across source domains. Our method, termed MixStyle, is motivated by the observation that visual domain is closely related to image style (e.g., photo vs. sketch images). Such style information is captured by the bottom layers of a CNN where our proposed style-mixing takes place. Mixing styles of training instances results in novel domains being synthesized implicitly, which increase the domain diversity of the source domains, and hence the generalizability of the trained model. MixStyle fits into mini-batch training perfectly and is extremely easy to implement. The effectiveness of MixStyle is demonstrated on a wide range of tasks including category classification, instance retrieval and reinforcement learning. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Key to automated understanding of digital images is to compute a compact and informative feature representation. Deep convolutional neural networks (CNNs) have demonstrated remarkable ability in representation learning, proven to be effective in many visual recognition tasks, such as classifying photo images into 1,000 categories from ImageNet (Krizhevsky et al., 2012) and playing Atari games with reinforcement learning (Mnih et al., 2013). However, it has long been discovered that the success of CNNs heavily relies on the i.i.d. assumption, i.e. training and test data should be drawn from the same distribution; when such an assumption is violated even just slightly, as in most realworld application scenarios, severe performance degradation is expected (Hendrycks & Dietterich, 2019; Recht et al., 2019). ",
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+        "type": "text",
+        "text": "Domain generalization (DG) aims to address such a problem (Zhou et al., 2021; Blanchard et al., 2011; Muandet et al., 2013; Li et al., 2018a; Zhou et al., 2020b; Balaji et al., 2018; Dou et al., 2019; Carlucci et al., 2019). In particular, assuming that multiple source domains containing the same visual classes are available for model training, the goal of DG is to learn models that are robust against data distribution changes across domains, known as domain shift, so that the trained model can generalize well to any unseen domains. Compared to the closely related and more widely studied domain adaptation (DA) problem, DG is much harder in that no target domain data is available for the model to analyze the distribution shift in order to overcome the negative effects. Instead, a DG model must rely on the source domains and focus on learning domain-invariant feature representation in the hope that it would remain discriminative given target domain data. ",
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+        "type": "text",
+        "text": "A straightforward solution to DG is to expose a model with a large variety of source domains. Specifically, the task of learning domain-invariant and thus generalizable feature representation becomes easier when data from more diverse source domains are available for the model. This would reduce the burden on designing special models or learning algorithms for DG. Indeed, model training with large-scale data of diverse domains is behind the success of existing commercial face recognition or vision-based autonomous driving systems. A recent work by $\\mathrm { X u }$ et al. (2021) also emphasizes the importance of diverse training distributions for out-of-distribution generalization. However, collecting data of a large variety of domains is often costly or even impossible. It thus cannot be a general solution to DG. ",
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+        "img_path": "images/f1cd7469df889b614caa015810343d6c004d2bbf757f1fbaacd0388b9ffe208e.jpg",
+        "image_caption": [
+            "Figure 1: 2-D t-SNE (Maaten & Hinton, 2008) visualization of the style statistics (concatenation of mean and standard deviation) computed from the first residual block’s feature maps of a ResNet-18 (He et al., 2016) trained on four distinct domains (Li et al., 2017). It is clear that different domains are well separated. "
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+        "type": "text",
+        "text": "In this paper, a novel approach is proposed based on probabilistically mixing instance-level feature statistics of training samples across source domains. Our model, termed MixStyle, is motivated by the observation that visual domain is closely related to image style. An example is shown in Fig. 1: the four images from four different domains depict the same semantic concept, i.e. dog, but with distinctive styles (e.g., characteristics in color and texture). When these images are fed into a deep CNN, which maps the raw pixel values into category labels, such style information is removed at the output. However, recent style transfer studies (Huang & Belongie, 2017; Dumoulin et al., 2017) suggest that such style information is preserved at the bottom layers of the CNN through the instance-level feature statistics, as shown clearly in Fig. 4. Importantly, since replacing such statistics would lead to replaced style while preserving the semantic content of the image, it is reasonable to assume that mixing styles from images of different domains would result in images of (mixed) new styles. That is, more diverse domains/styles can be made available for training a more domain-generalizable model. ",
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+        "type": "text",
+        "text": "Concretely, our MixStyle randomly selects two instances of different domains and adopts a probabilistic convex combination between instance-level feature statistics of bottom CNN layers. In contrast to style transfer work (Huang & Belongie, 2017; Dumoulin et al., 2017), no explicit image synthesis is necessary meaning much simpler model design. Moreover, MixStyle perfectly fits into modern mini-batch training. Overall, it is very easy to implement with only few lines of code. To evaluate the effectiveness as well as the general applicability of MixStyle, we conduct extensive experiments on a wide spectrum of datasets covering category classification (Sec. 3.1), instance retrieval (Sec. 3.2), and reinforcement learning (Sec. 3.3). The results demonstrate that MixStyle can significantly improve CNNs’ cross-domain generalization performance.1 ",
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+        "text": "2 METHODOLOGY ",
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+        "text": "2.1 BACKGROUND ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Normalizing feature tensors with instance-specific mean and standard deviation has been found effective for removing image style in style transfer models (Ulyanov et al., 2016; Huang & Belongie, 2017; Dumoulin et al., 2017). Such an operation is widely known as instance normalization (IN, Ulyanov et al. (2016)). Let $\\boldsymbol { x } ^ { \\prime } \\in \\mathbb { R } ^ { B \\times C \\times \\dot { H } \\times W }$ be a batch of tensors, with $B , C , H$ and $W$ denoting the dimension of batch, channel, height and width, respectively, IN is formulated as ",
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+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/1bcca277c97bac87df3b5ecefb248adf385fad37c24b49b8108b0b8f4ebd2944.jpg",
+        "text": "$$\n\\mathrm { I N } ( x ) = \\gamma \\frac { x - \\mu ( x ) } { \\sigma ( x ) } + \\beta ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\gamma , \\beta \\in \\mathbb { R } ^ { C }$ are learnable affine transformation parameters, and $\\mu ( \\boldsymbol { x } ) , \\sigma ( \\boldsymbol { x } ) \\in \\mathbb { R } ^ { B \\times C }$ are mean and standard deviation computed across the spatial dimension within each channel of each instance (tensor), i.e. ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/fb99b3cee5284125fb7c56d3850f8a40e8f4faade0a3f52d6d8fd4515140a5b9.jpg",
+        "text": "$$\n\\mu ( x ) _ { b , c } = \\frac { 1 } { H W } \\sum _ { h = 1 } ^ { H } \\sum _ { w = 1 } ^ { W } x _ { b , c , h , w } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "and ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/6a50bbe7d108fe07827ea98df07a10608e7e794826c7fb9b310dbc970c77a01f.jpg",
+        "text": "$$\n\\sigma ( x ) _ { b , c } = \\sqrt { \\frac { 1 } { H W } \\sum _ { h = 1 } ^ { H } \\sum _ { w = 1 } ^ { W } ( x _ { b , c , h , w } - \\mu ( x ) _ { b , c } ) ^ { 2 } } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Huang & Belongie (2017) introduced adaptive instance normalization (AdaIN), which simply replaces the scale and shift parameters in Eq. (1) with the feature statistics of style input $y$ to achieve arbitrary style transfer: ",
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+        "type": "equation",
+        "img_path": "images/3c63f5f72c8d9ea81fce1ab32d869a82feeb503f7c48024c0506a4fc0b6157da.jpg",
+        "text": "$$\n\\mathrm { A d a I N } ( x ) = \\sigma ( y ) \\frac { x - \\mu ( x ) } { \\sigma ( x ) } + \\mu ( y ) .\n$$",
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+        "text": "2.2 MIXSTYLE",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our method, MixStyle, draws inspiration from AdaIN. However, rather than attaching a decoder for image generation, MixStyle is designed for the purpose of regularizing CNN training by perturbing the style information of source domain training instances. It can be implemented as a plug-andplay module inserted between CNN layers of, e.g., a supervised CNN classifier, without the need to explicitly generate an image of new style. ",
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+        "type": "text",
+        "text": "More specifically, MixStyle mixes the feature statistics of two instances with a random convex weight to simulate new styles. In terms of implementation, MixStyle can be easily integrated into mini-batch training. Given an input batch $x$ , MixStyle first generates a reference batch $\\tilde { x }$ from $x$ . When domain labels are given, $x$ is sampled from two different domains $i$ and $j$ , e.g., $\\bar { \\boldsymbol { x } } = [ x ^ { i } , x ^ { j } ]$ $\\bar { x ^ { i } }$ and $x ^ { j }$ have the same batch size). Then, $\\tilde { x }$ is obtained by swapping the position of $x ^ { i }$ and $x ^ { j }$ , followed by a shuffling operation along the batch dimension applied to each batch, i.e. $\\mathbf { \\widetilde { \\boldsymbol { x } } } = [ \\mathrm { S h u f f e } ( \\boldsymbol { x } ^ { j } ) , \\mathrm { S h u f f e } ( \\boldsymbol { x } ^ { i } ) ]$ . See Fig. 2(a) for an illustration. In cases where domain labels are unknown, $x$ is randomly sampled from the training data, and $\\tilde { x }$ is simply obtained by $\\tilde { x } = \\mathrm { S h u f f e } ( x )$ (see Fig. 2(b)). Fig. 4 shows that sub-domains exist within each domain, so even if two instances of the same domain are sampled, new domain could be synthesized. After shuffling, MixStyle computes the mixed feature statistics by ",
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+    {
+        "type": "image",
+        "img_path": "images/328a14b30cdf68b6ebf3e522fa5e10ef00672d004ed56b5521dcee7294f42db8.jpg",
+        "image_caption": [
+            "Figure 2: A graphical illustration of how a reference batch is generated. Domain label is denoted by color. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/dda5b24328e69e61232fce56170c65ac8f6d17d9ab0153ffcddaf7931cf59390.jpg",
+        "text": "$$\n\\begin{array} { r } { \\gamma _ { m i x } = \\lambda \\sigma ( x ) + ( 1 - \\lambda ) \\sigma ( \\tilde { x } ) , } \\\\ { \\beta _ { m i x } = \\lambda \\mu ( x ) + ( 1 - \\lambda ) \\mu ( \\tilde { x } ) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\lambda \\in \\mathbb { R } ^ { B }$ are instance-wise weights sampled from the Beta distribution, $\\lambda \\sim B e t a ( \\alpha , \\alpha )$ with $\\alpha \\in ( 0 , \\infty )$ being a hyper-parameter. Unless specified otherwise, we set $\\alpha$ to 0.1 throughout this paper. Finally, the mixed feature statistics are applied to the style-normalized $x$ , ",
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+        "text": "$$\n{ \\mathrm { M i x S t y l e } } ( x ) = \\gamma _ { m i x } { \\frac { x - \\mu ( x ) } { \\sigma ( x ) } } + \\beta _ { m i x } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In practice, we use a probability of 0.5 to decide if MixStyle is activated or not in the forward pass. At test time, no MixStyle is applied. Note that gradients are blocked in the computational graph of $\\mu ( \\cdot )$ and $\\sigma ( \\cdot )$ . MixStyle can be implemented with only few lines of code. See Algorithm 1 in Appendix A.1 for the PyTorch-like pseudo-code. ",
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+    {
+        "type": "table",
+        "img_path": "images/c5ebe274bb1893ac2346fb0d139387bcd5d7bf3e991b91a59bcca22de9df6956.jpg",
+        "table_caption": [
+            "Table 1: Leave-one-domain-out generalization results on PACS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Art</td><td>Cartoon</td><td>Photo</td><td>Sketch</td><td>Avg</td></tr><tr><td>MMD-AAE CCSA JiGen</td><td>75.2 80.5 79.4 79.8 82.1</td><td>72.7 76.9 75.3 76.8</td><td>96.0 93.6 96.0 96.0</td><td>64.2 66.8 71.6 70.2</td><td>77.0 79.4 80.5 80.7</td></tr><tr><td>Epi-FCR Metareg L2A-OT ResNet-18</td><td>83.7 83.3 77.0±0.6</td><td>77.0 77.2 78.2 75.9±0.6</td><td>93.9 95.5 96.2 96.0±0.1</td><td>73.0 70.3 73.6 69.2±0.6</td><td>81.5 81.7 82.8 79.5</td></tr><tr><td>+ Manifold Mixup</td><td>75.6±0.7</td><td>70.1±0.9</td><td>93.5±0.7</td><td>65.4±0.6</td><td>76.2</td></tr><tr><td>+ Cutout</td><td>74.9±0.4</td><td>74.9±0.6</td><td>95.9±0.3</td><td>67.7±0.9</td><td>78.3</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+ CutMix</td><td>74.6±0.7</td><td>71.8±0.6</td><td>95.6±0.4</td><td>65.3±0.8</td><td>76.8</td></tr><tr><td></td><td>74.7±1.0</td><td>72.3±0.9</td><td></td><td></td><td></td></tr><tr><td>+ Mixup (w/o label interpolation)</td><td></td><td></td><td>93.0±0.4</td><td>69.2±0.2</td><td>77.3</td></tr><tr><td>+ Mixup</td><td>76.8±0.7</td><td>74.9±0.7</td><td>95.8±0.3</td><td>66.6±0.7</td><td>78.5</td></tr><tr><td>+ DropBlock</td><td>76.4±0.7</td><td>75.4±0.7</td><td>95.9±0.3</td><td>69.0±0.3</td><td>79.2</td></tr><tr><td>+ MixStyle w/ random shuffle</td><td>82.3±0.2</td><td>79.0±0.3</td><td>96.3±0.3</td><td>73.8±0.9</td><td>82.8</td></tr><tr><td>+ MixStyle w/ domain label</td><td>84.1±0.4</td><td>78.8±0.4</td><td>96.1±0.3</td><td>75.9±0.9</td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td>83.7</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "3.1 GENERALIZATION IN CATEGORY CLASSIFICATION ",
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+        "type": "text",
+        "text": "Dataset and implementation details. We choose the PACS dataset (Li et al., 2017), a commonly used domain generalization (DG) benchmark concerned with domain shift in image classification. PACS consists of four domains, i.e. Art Painting, Cartoon, Photo and Sketch, with totally 9,991 images of 7 classes. As shown in Fig. 1, the domain shift mainly corresponds to image style changes. For evaluation, a model is trained on three domains and tested on the remaining one. Following prior work (Li et al., 2019; Zhou et al., 2020a), we use ResNet-18 (He et al., 2016) as the classifier where MixStyle is inserted after the 1st, 2nd and 3rd residual blocks. Our code is based on Dassl.pytorch (Zhou et al., 2020c).2 ",
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+        "text": "Baselines. Our main baselines are general-purpose regularization methods including Mixup (Zhang et al., 2018b), Manifold Mixup (Verma et al., 2019), DropBlock (Ghiasi et al., 2018), CutMix (Yun et al., 2019) and Cutout (DeVries & Taylor, 2017), which are trained using the same training parameters as MixStyle and the optimal hyper-parameter setup as reported in their papers. We also compare with the existing DG methods which reported state-of-the-art performance on PACS. These include domain alignment-based CCSA (Motiian et al., 2017) and MMD-AAE (Li et al., 2018b), Jigsaw puzzle-based JiGen (Carlucci et al., 2019), adversarial gradient-based CrossGrad (Shankar et al., 2018), meta-learning-based Metareg (Balaji et al., 2018) and Epi-FCR (Li et al., 2019), and data augmentation-based L2A-OT (Zhou et al., 2020a). ",
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+    {
+        "type": "text",
+        "text": "Comparison with general-purpose regularization methods. The results are shown in Table 1. Overall, we observe that the general-purpose regularization methods do not offer any clear advantage over the vanilla ResNet-18 in this DG task, while MixStyle improves upon the vanilla ResNet18 with a significant margin. Compared with Mixup, MixStyle is $5 . 2 \\%$ better on average. Recall that Mixup also interpolates the output space, we further compare with a variant of Mixup in order to demonstrate the advantage of mixing style statistics at the feature level over mixing images at the pixel level for DG—following Sohn et al. (2020), we remove the label interpolation in Mixup and sample the mixing weights from a uniform distribution of $[ 0 , 1 ]$ . Still, MixStyle outperforms this new baseline with a large margin, which justifies our claim. MixStyle and DropBlock share some commonalities in that they are both applied to feature maps at multiple layers, but MixStyle significantly outperforms DropBlock in all test domains. The reason why DropBlock is ineffective here is because dropping out activations mainly encourages a network to mine discriminative patterns, but does not reinforce the ability to cope with unseen styles, which is exactly what MixStyle aims to achieve: by synthesizing “new” styles (domains) MixStyle regularizes the network to become more robust to domain shift. In addition, it is interesting to see that on Cartoon and Photo, MixStyle w/ random shuffle obtains slightly better results. The reason might be because there exist sub-domains in a source domain (see Fig. 4(a-c)), which allow random shuffling to produce more diverse “new” domains that lead to a more domain-generalizable model. ",
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+        "text": "Comparison with state-of-the-art DG methods. Overall, MixStyle outperforms most DG methods by a clear margin, despite being a much simpler method. The performance of MixStyle w/ domain label is nearly $1 \\%$ better on average than the recently introduced L2A-OT. From a data augmentation perspective, MixStyle and L2A-OT share a similar goal—to synthesize data from pseudo-novel domains. MixStyle accomplishes this goal through mixing style statistics at the feature level. Whereas L2A-OT works at the pixel level: it trains an image generator by maximizing the domain difference (measured by optimal transport) between the original and the generated images, which introduces much heavier computational overhead than MixStyle in terms of GPU memory and training time. It is worth noting that MixStyle’s domain label-free version is highly competitive: its $8 2 . 8 \\%$ accuracy is on par with L2A-OT’s. ",
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+    {
+        "type": "text",
+        "text": "3.2 GENERALIZATION IN INSTANCE RETRIEVAL",
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+        "type": "text",
+        "text": "Dataset and implementation details. We evaluate MixStyle on the person re-identification (re-ID) problem, which aims to match people across disjoint camera views. As each camera view is itself a distinct domain, person re-ID is essentially a cross-domain image matching problem. Instead of using the standard protocol where training and test data come from the same camera views, we adopt the cross-dataset setting so test camera views are never seen during training. Specifically, we train a model on one dataset and then test its performance on the other dataset. Two commonly used re-ID datasets are adopted: Market1501 (Zheng et al., 2015) and Duke (Ristani et al., 2016; Zheng et al., 2017). Ranking accuracy and mean average precision (mAP) are used as the performance measures (displayed in percentage). We test MixStyle on two CNN architectures: ResNet-50 (He et al., 2016) and OSNet (Zhou et al., 2019). The latter was designed specifically for re-ID. In both architectures, MixStyle is inserted after the 1st and 2nd residual blocks. Our code is based on Torchreid (Zhou & Xiang, 2019).3 ",
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+    {
+        "type": "text",
+        "text": "Baselines. We compare with three baseline methods: 1) The vanilla model, which serves as a strong baseline; 2) DropBlock, which was the top-performing competitor in Table 1; 3) RandomErase (Zhong et al., 2020), a widely used regularization method in the re-ID literature (similar to Cutout). ",
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+    {
+        "type": "text",
+        "text": "Results. The results are reported in Table 2. It is clear that only MixStyle consistently outperforms the strong vanilla model under both settings with considerable margins, while DropBlock and RandomErase are unable to show any benefit. Notably, RandomErase, which simulates occlusion by erasing pixels in random rectangular regions with random values, has been used as a default trick when training re-ID CNNs. However, RandomErase shows a detrimental effect in the cross-dataset re-ID setting. Indeed, similar to DropBlock, randomly erasing pixels offers no guarantee to improve the robustness when it comes to domain shift. ",
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+    {
+        "type": "text",
+        "text": "3.3 GENERALIZATION IN REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Though RL has been greatly advanced by using CNNs for feature learning in raw pixels (Mnih et al., 2013), it has been widely acknowledged that RL agents often overfit training environments while generalize poorly to unseen environments (Cobbe et al., 2019; Igl et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "Dataset and implementation details. We conduct experiments on Coinrun (Cobbe et al., 2019), a recently introduced RL benchmark for evaluating the generalization performance of RL agents. As shown in Fig. 3(a), the goal in Coinrun is to control a character to collect golden coins while avoiding both stationary and non-stationary obstacles. We follow Igl et al. (2019) to construct and train our RL agent: the CNN architecture used in IMPALA (Espeholt et al., 2018) is adopted as the policy network, and is trained by the Proximal Policy Optimization (PPO) algorithm (Schulman et al., 2017). Please refer to Igl et al. (2019) for further implementation details. MixStyle is inserted after the 1st and 2nd convolutional sequences. Training data are sampled from 500 levels while test data are drawn from new levels of only the highest difficulty. As domain labels are difficult to define, we use the random shuffle version of MixStyle. Our code is built on top of Igl et al. (2019).4 ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/8e248e2539fca822336240b7fe9d39fd45dacc2101f2bcb3a98c958918e50290.jpg",
+        "table_caption": [
+            "Table 2: Generalization results on the cross-dataset person re-ID task. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"4\">Market1501-→Duke</td><td colspan=\"4\">Duke-→Market1501</td></tr><tr><td>mAP</td><td>R1</td><td>R5</td><td>R10</td><td>mAP</td><td>R1</td><td>R5</td><td>R10</td></tr><tr><td>ResNet-50</td><td>19.3</td><td>35.4</td><td>50.3</td><td>56.4</td><td>20.4</td><td>45.2</td><td>63.6</td><td>70.9</td></tr><tr><td>+ RandomErase</td><td>14.3</td><td>27.8</td><td>42.6</td><td>49.1</td><td>16.1</td><td>38.5</td><td>56.8</td><td>64.5</td></tr><tr><td>+ DropBlock</td><td>18.2</td><td>33.2</td><td>49.1</td><td>56.3</td><td>19.7</td><td>45.3</td><td>62.1</td><td>69.1</td></tr><tr><td>+ MixStyle w/ random shuffle</td><td>23.8</td><td>42.2</td><td>58.8</td><td>64.8</td><td>24.1</td><td>51.5</td><td>69.4</td><td>76.2</td></tr><tr><td>+ MixStyle w/ domain label</td><td>23.4</td><td>43.3</td><td>58.9</td><td>64.7</td><td>24.7</td><td>53.0</td><td>70.9</td><td>77.8</td></tr><tr><td>OSNet</td><td>25.9</td><td>44.7</td><td>59.6</td><td>65.4</td><td>24.0</td><td>52.2</td><td>67.5</td><td>74.7</td></tr><tr><td>+ RandomErase</td><td>20.5</td><td>36.2</td><td>52.3</td><td>59.3</td><td>22.4</td><td>49.1</td><td>66.1</td><td>73.0</td></tr><tr><td>+ DropBlock</td><td>23.1</td><td>41.5</td><td>56.5</td><td>62.5</td><td>21.7</td><td>48.2</td><td>65.4</td><td>71.3</td></tr><tr><td>+ MixStyle w/ random shuffle</td><td>27.2</td><td>48.2</td><td>62.7</td><td>68.4</td><td>27.8</td><td>58.1</td><td>74.0</td><td>81.0</td></tr><tr><td>+ MixStyle w/ domain label</td><td>27.3</td><td>47.5</td><td>62.0</td><td>67.1</td><td>29.0</td><td>58.2</td><td>74.9</td><td>80.9</td></tr></table>",
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+        "img_path": "images/83f0338beadfd5bc519830499778de6828846e5597dc084ac87299494b1148a1.jpg",
+        "image_caption": [
+            "Figure 3: (a) Coinrun benchmark. (b) Test performance in unseen environments. (c) Difference between training and test performance. "
+        ],
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+        "text": "Baselines. Following Igl et al. (2019), we train strong baseline models and add MixStyle on top of them to see whether MixStyle can bring further improvements. To this end, we train two baseline models: 1) Baseline, which combines weight decay and data augmentation;5 2) IBAC-SNI (the $\\lambda = 0 . 5$ version), the best-performing model in Igl et al. (2019) which is based on selective noise injection. ",
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+        "type": "text",
+        "text": "Results. The test performance is shown in Fig. 3(b). Comparing Baseline (blue) with Baseline+MixStyle (orange), we can see that MixStyle brings a significant improvement. Interestingly, the variance is also significantly reduced by using MixStyle, as indicated by the smaller shaded areas (for both orange and red lines). These results strongly demonstrate the effectiveness of MixStyle in enhancing generalization for RL agents. When it comes to the stronger baseline IBAC-SNI (green), MixStyle (red) is able to bring further performance gain, suggesting that MixStyle is complementary to IBAC-SNI. This result also shows the potential of MixStyle as a plug-and-play component to be combined with other advanced RL methods. It is worth noting that Baseline+MixStyle itself is already highly competitive with IBAC-SNI. Fig. 3(c) shows the generalization gap from which it can been seen that the models trained with MixStyle (orange & red) clearly generalize faster and better than those without using MixStyle (blue & green). ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4d2d1d46181970a60036d74149364fc1b7c14abba9cfcdf85a1601e0d521acda.jpg",
+        "image_caption": [
+            "Figure 4: 2-D visualization of flattened feature maps (top) and the corresponding style statistics (bottom). res1-4 denote the four residual blocks in order in a ResNet architecture. We observe that res1 to res3 contain domain-related information while res4 encodes label-related information. "
+        ],
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 3: Ablation study on where to apply MixStyle in the ResNet architecture. ",
+            "(a) Category classification on PACS. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Accuracy</td></tr><tr><td>ResNet-18</td><td>79.5</td></tr><tr><td>+ MixStyle (res1)</td><td>80.1</td></tr><tr><td>+ MixStyle (res12)</td><td>81.6</td></tr><tr><td>+ MixStyle (res123)</td><td>82.8</td></tr><tr><td>+ MixStyle (res1234)</td><td>75.6</td></tr><tr><td>+ MixStyle (res14)</td><td>76.3</td></tr><tr><td>+ MixStyle (res23)</td><td>81.7</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "(b) Cross-dataset person re-ID. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>mAP</td></tr><tr><td>ResNet-50</td><td>19.3</td></tr><tr><td>+ MixStyle (res1)</td><td>22.6</td></tr><tr><td>+ MixStyle (res12) + MixStyle (res123)</td><td>23.8</td></tr><tr><td>+MixStyle (res1234)</td><td>22.0</td></tr><tr><td>+ MixStyle (res14)</td><td>10.2</td></tr><tr><td>+MixStyle (res23)</td><td>11.1 20.6</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "3.4 ANALYSIS ",
+        "text_level": 1,
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+        "text": "Where to apply MixStyle? We repeat the experiments on PACS (category classification) and the re-ID datasets (instance retrieval) using the ResNet architecture. Given that a standard ResNet model has four residual blocks denoted by $\\mathtt { r e s l - 4 }$ , we train different models with MixStyle applied to different layers. For notation, res1 means MixStyle is applied after the first residual block; res12 means MixStyle is applied after both the first and second residual blocks; and so forth. The results are shown in Table 3. We have the following observations. 1) Applying MixStyle to multiple lowerlevel layers generally achieves a better performance—for instance, res12 is better than res1 on both tasks. 2) Different tasks favor different combinations—res123 achieves the best performance on PACS, while on the re-ID datasets $\\mathtt { r e s 1 2 }$ is the best. 3) On both tasks, the performance plunges when applying MixStyle to the last residual block. This makes sense because res4 is the closest to the prediction layer and tends to capture semantic content (i.e. label-sensitive) information rather than style. In particular, res4 is followed by an average-pooling layer, which essentially forwards the mean vector to the prediction layer and thus forces the mean vector to capture label-related information. As a consequence, mixing the statistics at res4 breaks the inherent label space. This is clearer in Fig. 4: the features and style statistics in $\\mathtt { r e s l - 3 }$ exhibit clustering patterns based on domains while those in res4 have a high correlation with class labels. ",
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+    {
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+        "img_path": "images/c7e15cbb6312d6d6e57a51169db1f4a01f9832c478ec13254c9df6c7db06a565.jpg",
+        "image_caption": [
+            "Figure 5: Evaluation on the hyper-parameter $\\alpha$ on (a) PACS, (b) person re-ID datasets and (c) Coinrun. In (b), M and D denote Market1501 and Duke respectively. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Mixing vs. replacing. Unlike the AdaIN formulation, which completely replaces one style with the other, MixStyle mixes two styles via a convex combination. Table 4 shows that mixing is better than replacing. This is easy to understand: mixing diversifies the styles (imagine an interpolation between two data points). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Random vs. fixed shuffle at multiple layers. Applying MixStyle to multiple layers, which has been shown advantageous in Table 3, raises another question of whether to shuffle the mini-batch at different layers or use the same shuffled order for all layers. Table 5 suggests that using random shuffle at different layers gives a better performance, which may be attributed to the increased noise level that gives a better regularization effect. ",
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+    {
+        "type": "table",
+        "img_path": "images/2c381c18bb58b30c686683356c2292f71c205ce4d70504362b8537940ea26f02.jpg",
+        "table_caption": [
+            "Table 4: Mixing vs. replacing. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Accuracy (%)</td></tr><tr><td>Mixing</td><td>82.8±0.4</td></tr><tr><td>Replacing</td><td>82.1±0.5</td></tr></table>",
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+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 5: Random vs. fixed shuffle at multiple layers. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Accuracy (%)</td></tr><tr><td>Random</td><td>82.8±0.4</td></tr><tr><td>Fixed</td><td>82.4±0.5</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Sensitivity of hyper-parameter. Recall that $\\alpha$ is used to control the shape of Beta distribution, which has a direct effect on how the convex weights $\\lambda$ are sampled. The smaller $\\alpha$ is, the more likely the value in $\\lambda$ is close to the extreme value of 0 or 1. In other words, a smaller $\\alpha$ favors the style statistics in Eqs. (5) & (6) to be dominated by one side. We first evaluate $\\alpha$ on PACS. Fig. 5(a) shows that with $\\alpha$ increasing from 0.1 to 0.4, the accuracy slides from $8 2 . 8 \\%$ to $8 1 . 7 \\%$ . However, further increasing $\\alpha$ does not impact on the accuracy. Therefore, the results suggest that the performance is not too sensitive to $\\alpha$ ; and selecting $\\alpha$ from $\\lbrace 0 . 1 , 0 . 2 , 0 . 3 \\rbrace$ seems to be a good starting point. We further experiment with $\\alpha \\in \\{ 0 . 1 , 0 . 2 , 0 . 3 \\}$ on the re-ID datasets and the Coinrun benchmark. Figs. 5(b) & (c) show that in general the variance for the results of different values is small. Therefore, we suggest practitioners to choose $\\alpha$ from $\\lbrace 0 . 1 , 0 . 2 , 0 . 3 \\rbrace$ , with $\\alpha = 0 . 1$ being a good default setting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For more analyses and discussions, please see Appendix A.2. ",
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+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Domain generalization, or DG, studies out-of-distribution (OOD) generalization given only source data typically composed of multiple related but distinct domains. We refer readers to Zhou et al. (2021) for a comprehensive survey in this topic. Many DG methods are based on the idea of aligning features between different sources, with a hope that the model can be invariant to domain shift given unseen data. For instance, Li et al. (2018b) achieved distribution alignment in the hidden representation of an autoencoder using maximum mean discrepancy; Li et al. (2018c) resorted to adversarial learning with auxiliary domain classifiers to learn features that are domain-agnostic. Some works explored domain-specific parameterization, such as domain-specific weight matrices (Li et al., 2017) and domain-specific BN (Seo et al., 2020). Recently, meta-learning has drawn increasing attention from the DG community (Li et al., 2018a; Balaji et al., 2018; Dou et al., 2019). The main idea is to expose a model to domain shift during training by using pseudo-train and pseudo-test domains, both drawn from source domains. Data augmentation has also been investigated for learning domain-invariant models. Shankar et al. (2018) introduced a cross-gradient training method (CrossGrad) where source data are augmented by adversarial gradients obtained from a domain classifier. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Gong et al. (2019) proposed DLOW (for the DA problem), which models intermediate domains between source and target via a domainness factor and learns an image translation model to generate intermediate-domain images. Very recently, Zhou et al. (2020a) introduced L2A-OT to learn a neural network to map source data to pseudo-novel domains by maximizing an optimal transport-based distance measure. Our MixStyle is related to DLOW and L2A-OT in its efforts to synthesizing novel domains. However, MixStyle differs in the fact that it is done implicitly with a much simpler formulation leveraging the feature-level style statistics and only few lines of extra code on top of a standard supervised classifier while being more effective. Essentially, MixStyle can be seen as feature-level augmentation, which is clearly different from the image-level augmentation-based DLOW and L2A-OT. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Generalization in deep RL has been a challenging problem where RL agents often overfit training environments, and as a result, perform poorly in unseen environments with different visual patterns or levels (Zhang et al., 2018a). A natural way to improve generalization, which has been shown effective in (Cobbe et al., 2019; Farebrother et al., 2018), is to use regularization, e.g., weight decay. However, Igl et al. (2019) suggested that stochastic regularization methods like dropout and batch normalization (which uses estimated population statistics) have adverse effect as the training data in RL are essentially model-dependent. As such, they proposed selective noise injection (SNI), which basically combines a stochastic regularization technique with its deterministic counterpart. They further integrated SNI with information bottleneck actor critic (IBAC-SNI) to reduce the variance in gradients. Curriculum learning has been investigated in (Justesen et al., 2018) where the level of training episodes progresses from easy to difficult over the course of training. Gamrian & Goldberg (2019) leveraged the advances in GAN-based image-to-image translation (Liu et al., 2017) to map target data to the source domain which the agent was trained on. Tobin et al. (2017) introduced domain randomization, which diversifies training data by rendering images with different visual effects via a programmable simulator. With a similar goal of data augmentation, Lee et al. (2020) pre-processed input images with a randomly initialized network. Very recent studies (Laskin et al., 2020; Kostrikov et al., 2020) have shown that it is useful to combine a diverse set of label-preserving transformations, such as rotation, shifting and Cutout. Different from the aforementioned methods, our MixStyle works at the feature level and is orthogonal to most existing methods. For instance, we have shown in Sec. 3.3 that MixStyle significantly improves upon IBAC-SNI. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            547,
+            318,
+            563
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We presented a simple yet effective domain generalization method, termed MixStyle. MixStyle mixes the feature statistics of two instances to synthesize novel domains, which is inspired by the observation in style transfer work that the feature statistics encode style/domain-related information. Extensive experiments covering a wide range of tasks were conducted to demonstrate that MixStyle yields new state-of-the-art on three different tasks. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            664,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work was supported in part by the Shanghai Committee of Science and Technology, China (Grant No. 20DZ1100800), in part by the National Natural Science Foundation of China under Grant (61876176, U1713208), the National Key Research and Development Program of China (No. 2020YFC2004800), and in part by the Guangzhou Research Program (201803010066). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    {
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+        "bbox": [
+            169,
+            103,
+            826,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 PSEUDO-CODE OF MIXSTYLE",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            426,
+            148
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 provides a PyTorch-like pseudo-code. ",
+        "bbox": [
+            174,
+            159,
+            506,
+            174
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 PyTorch-like pseudo-code for MixStyle. ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            189,
+            527,
+            204
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "# x: input features of shape (B, C, H, W)   \n# p: probabillity to apply MixStyle (default: 0.5)   \n# alpha: hyper-parameter for the Beta distribution (default: 0.1)   \n# eps: a small value added before square root for numerical stability (default: 1e-6) ",
+        "bbox": [
+            171,
+            215,
+            776,
+            252
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "if not in training mode: return x ",
+        "bbox": [
+            176,
+            261,
+            344,
+            279
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "if random probability $>$ p: return x ",
+        "bbox": [
+            176,
+            287,
+            357,
+            306
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "$\\mathrm { ~ \\textit ~ { ~ B ~ } ~ } = \\mathrm { ~ \\textit ~ { ~ x ~ } ~ }$ .size(0) # batch size ",
+        "bbox": [
+            174,
+            314,
+            359,
+            325
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "mu $= \\times$ .mean(dim=[2, 3], keepdim $\\cdot ^ { = }$ True) # compute instance mean var $=$ x.var(dim=[2, 3], keepdim $\\cdot ^ { = }$ True) # compute instance variance sig $=$ (var $^ +$ eps).sqrt() # compute instance standard deviation mu, sig $\\qquad = \\quad \\mathtt { m u }$ .detach(), sig.detach() # block gradients x_normed $=$ (x - mu) / sig # normalize input ",
+        "bbox": [
+            171,
+            333,
+            633,
+            380
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "lmda $=$ Beta(alpha, alpha).sample((B, 1, 1, 1)) # sample instance-wise convex weights ",
+        "bbox": [
+            166,
+            387,
+            766,
+            398
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "if domain label is given: # in this case, input $\\begin{array} { r } { \\mathrm { ~  ~ \\underline { ~ } { ~ \\sf ~ } { ~ \\sf ~ } { ~ \\sf ~ } { ~ \\sf ~ } = ~ \\mathrm { ~  ~ \\tau ~ } [ \\times \\hat { \\mathrm { ~  ~ \\underline { ~ } { ~ \\bf ~ } { ~ \\underline { ~ } { ~ \\bf ~ } { ~ \\tau ~ } ~ } } } } } \\end{array}$ x perm $=$ torch.arange(B-1, -1, -1) # inverse index perm_j, perm_i $=$ perm.chunk(2) # separate indices perm_j $=$ perm_j[torch.randperm(B // 2)] # shuffling perm_i $=$ perm_i[torch.randperm(B // 2)] # shuffling perm $=$ torch.cat([perm_j, perm_i], 0) # concatenation   \nelse: perm $=$ torch.randperm(B) # generate shuffling indices ",
+        "bbox": [
+            176,
+            405,
+            571,
+            487
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "mu2, sig2 $=$ mu[perm], sig[perm] # shuffling mu_mix $=$ mu $\\star$ lmda $^ +$ mu2 $\\star$ (1 - lmda) # generate mixed mean sig_mix $=$ sig $\\star$ lmda $^ +$ sig2 $\\star$ (1 - lmda) # generate mixed standard deviation return x_normed $\\star$ sig_mix $^ +$ mu_mix # denormalize input using the mixed statistics ",
+        "bbox": [
+            174,
+            496,
+            710,
+            525
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            531,
+            748,
+            544
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/c5c91c980ca863ce754049c2ac48657fb6e478a83339d37cdd2bb99ba07e1b07.jpg",
+        "table_caption": [
+            "Table 7: Test results on the source domains on PACS. A: Art. C: Cartoon. P: Photo. S: Sketch. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>C,P,S</td><td>A,P,S</td><td>A,C,S</td><td>A,C,P</td><td>Avg</td></tr><tr><td>Vanilla</td><td>99.49±0.03</td><td>99.47±0.04</td><td>99.38±0.02</td><td>99.65±0.03</td><td>99.50</td></tr><tr><td>MixStyle</td><td>99.55±0.02</td><td>99.54±0.01</td><td>99.47±0.03</td><td>99.68±0.03</td><td>99.56</td></tr></table>",
+        "bbox": [
+            233,
+            127,
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+            175
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/d706ed77580ec62fda2f7d6c0da59fdce21563a4ed5924d61fc6e2035f36f821.jpg",
+        "table_caption": [
+            "Table 8: Leave-one-domain-out generalization results on Digits-DG. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>MNIST   MNIST-M   SVHN      SYN</td><td rowspan=1 colspan=1>Avg</td></tr><tr><td rowspan=1 colspan=1>JiGen</td><td rowspan=1 colspan=1>96.5        61.4        63.7       74.0</td><td rowspan=1 colspan=1>73.9</td></tr><tr><td rowspan=1 colspan=1>CCSA</td><td rowspan=1 colspan=1>95.2        58.2        65.5       79.1</td><td rowspan=1 colspan=1>74.5</td></tr><tr><td rowspan=3 colspan=1>MMD-AAECrossGradL2A-OT</td><td rowspan=1 colspan=1>96.5        58.4        65.0       78.4</td><td rowspan=3 colspan=1>74.675.878.1</td></tr><tr><td rowspan=1 colspan=1>96.7        61.1        65.3       80.2</td></tr><tr><td rowspan=1 colspan=1>96.7        63.9        68.6       83.2</td></tr><tr><td rowspan=8 colspan=1>CNN+ Mixup w/o label interpolation+ Manifold Mixup+ CutMix+ Mixup+ Cutout+ DropBlock+ MixStyle (ours)</td><td rowspan=1 colspan=1>95.8±0.3  58.8±0.5  61.7±0.5  78.6±0.6</td><td rowspan=1 colspan=1>73.7</td></tr><tr><td rowspan=1 colspan=1>93.7±0.6  55.2±1.0   61.6±0.9  74.4±0.8</td><td rowspan=3 colspan=1>71.271.771.8</td></tr><tr><td rowspan=1 colspan=1>92.7±0.4   53.1±0.8   64.4±0.2  76.8±0.5</td></tr><tr><td rowspan=1 colspan=1>94.9±0.2   50.1±0.5   64.1±0.9  78.1±0.7</td></tr><tr><td rowspan=1 colspan=1>94.2±0.5  56.5±0.8   63.3±0.7  76.7±0.6</td><td rowspan=4 colspan=1>72.774.175.376.5</td></tr><tr><td rowspan=1 colspan=1>95.8±0.4   58.4±0.6   61.9±0.9  80.6±0.5</td></tr><tr><td rowspan=1 colspan=1>96.2±0.1   60.5±0.6  64.1±0.8  80.2±0.6</td></tr><tr><td rowspan=1 colspan=1>96.5±0.3  63.5±0.8  64.7±0.7  81.2±0.8</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "A.2 FURTHER ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "MixStyle between same-domain instances. ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We are interested in knowing if mixing styles between same-domain instances helps performance. To this end, we sample each mini-batch from a single domain during training when using MixStyle. The results are shown in Table 6 where we observe that mixing styles between same-domain instances is about $1 \\%$ better than the baseline model. This suggests that instancespecific style exists. Nonetheless, the performance is clearly worse than mixing styles between instances of different domains. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e22f37e78ca0e3a60266186cfd9536740f245c97d3f14479137e919c3f8c589e.jpg",
+        "table_caption": [
+            "Table 6: Investigation on the effect of mixing styles between same-domain instances. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Accuracy (%)</td></tr><tr><td>ResNet18 79.5</td></tr><tr><td>+MixStyle w/ same-domain 80.4</td></tr><tr><td>+MixStyle w/random shuffle 82.8</td></tr><tr><td>+MixStyle w/ domain label 83.7</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Performance on source domains. To prove that MixStyle does not sacrifice the performance on seen domains in exchange for gains on unseen domains, we report the test accuracy on the held-out validation set of the source domains on PACS in Table 7. ",
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+    },
+    {
+        "type": "text",
+        "text": "Results on Digits-DG and Office-Home. In addition to the experiments on PACS (in Sec. 3.1), we further evaluate MixStyle’s effectiveness on two DG datasets, namely Digits-DG (Zhou et al., 2020a) and Office-Home (Venkateswara et al., 2017). Digits-DG contains four digit datasets (domains) including MNIST (LeCun et al., 1998), MNIST-M (Ganin & Lempitsky, 2015), SVHN (Netzer et al., 2011) and SYN (Ganin & Lempitsky, 2015). Images from different digit datasets differ drastically in font style, stroke color and background. Office-Home is composed of four domains (Artistic, Clipart, Product and Real World) with around 15,500 images of 65 classes for home and office object recognition. The results are shown in Tables 8 and 9 where no domain labels are used in MixStyle. Similar to the results on PACS, here we observe that MixStyle also brings clear improvements to the baseline CNN model and outperforms all general-purpose regularization methods on both Digits-DG and Office-Home. Compared with more sophisticated DG methods like L2A-OT, MixStyle’s performance is comparable, despite being much simpler to train and consuming much less computing resources. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/68a58d31208db3a9f4e6ac1e12e9d0e16e30afd9e38d48a9a578cbeecfbc00e1.jpg",
+        "table_caption": [
+            "Table 9: Leave-one-domain-out generalization results on Office-Home. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Artistic</td><td>Clipart</td><td>Product</td><td>Real World</td><td>Avg</td></tr><tr><td>JiGen CCSA</td><td>53.0</td><td>47.5</td><td>71.5</td><td>72.8</td><td>61.2</td></tr><tr><td>MMD-AAE</td><td>59.9 56.5</td><td>49.9 47.3</td><td>74.1 72.1</td><td>75.7 74.8</td><td>64.9 62.7</td></tr><tr><td>CrossGrad</td><td></td><td></td><td></td><td>75.8</td><td>64.4</td></tr><tr><td>L2A-OT</td><td>58.4</td><td>49.4</td><td>73.9</td><td></td><td></td></tr><tr><td>ResNet18</td><td>60.6 58.9±0.3</td><td>50.1 49.4±0.1</td><td>74.8 74.3±0.1</td><td>77.0 76.2±0.2</td><td>65.6 64.7</td></tr><tr><td>+ Manifold Mixup</td><td>56.2±0.4</td><td>46.3±0.3</td><td>73.6±0.1</td><td>75.2±0.2</td><td>62.8</td></tr><tr><td>+ Mixup w/o label interpolation + Cutout</td><td>57.0±0.2 57.8±0.2</td><td>48.7±0.2</td><td>71.4±0.6</td><td>74.5±0.4</td><td>62.9</td></tr><tr><td></td><td></td><td>48.1±0.3</td><td>73.9±0.2</td><td>75.8±0.3</td><td>63.9</td></tr><tr><td>+ CutMix</td><td>57.9±0.1</td><td>48.3±0.3</td><td>74.5±0.1</td><td>75.6±0.4</td><td>64.1</td></tr><tr><td>+ DropBlock</td><td>58.0±0.1</td><td>48.1±0.1</td><td>74.3±0.3</td><td>75.9±0.4</td><td>64.1</td></tr><tr><td>+ Mixup + MixStyle (ours)</td><td>58.2±0.1 58.7±0.3</td><td>49.3±0.2 53.4±0.2</td><td>74.7±0.1 74.2±0.1</td><td>76.1±0.1 75.9±0.1</td><td>64.6 65.5</td></tr></table>",
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+    }
+]