Authors:
(1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch);
(2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch);
(3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Table of Links Abstract and 1 Introduction 2 Background 2.1 Data Augmentation 2.2 Anchor Regression 3 Anchor Data Augmentation 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.3 Algorithm 4 Experiments and 4.1 Linear synthetic data 4.2 Housing nonlinear regression 4.3 In-distribution Generalization 4.4 Out-of-distribution Robustness 5 Conclusion, Broader Impact, and References A Additional information for Anchor Data Augmentation B Experiments A Additional information for Anchor Data Augmentation A.1 Derivation of ADA for nonlinear data A.2 Additional information on hyperparameters of ADA In this section, we illustrate in a simple 1D example (i.e. cosine data used in Figure 1) how changes in the hyperparameter values modify the data and affect the achieved estimation. Additionally, we show in Appendix B.4 how ADA performance on real-world data is impacted by changes in the hyperparameter values. Anchor Matrices and Locality: Anchor variable A is assumed to be the exogenous variable that generates heterogeneity in the target and has an approximately linear relation with (X, y) (see AR loss in Equation 3). It is recommended to choose the variable relying on expert knowledge about the features that the target has a higher dependence on or is possibly misrepresented in the dataset so that we encourage the robustness of the trained model against this type of discrepancy. After deciding the features, one way to construct the anchor matrix A is to partition the dataset according to the similarity of the features, using for example binning or clustering algorithms. Then we can fill the rows of A with a one-hot encoding of the partition index that each sample belongs to. We use the following nonlinear Cosine data model as a running example to demonstrate more clearly how A is constructed and affects the augmentation procedure. For illustration purposes, we use in Figures 5, 7 equidistant x values as this reduces noise and emphasizes the effect of ADA parameters more. Number of augmentations: For each anchor matrix A and γ we can add n new samples to the dataset. The addition of more augmented samples may not be beneficial as the optimization may overfit the approximations in the augmented data model in Equation 10. In the Cosine data model this is specifically problematic when X is close to multiples of π as depicted in Figure 7. Additionally, we provide a baseline and an augmented model fit in Figure 8 with different number of augmentations. As it is standard practice to use stochastic gradient descent methods for optimizing a regressor, we suggest applying ADA on each minibatch instead of the entire dataset. This avoids choosing a fixed numbers of augmentations. Furthermore, it adds diversity to the "mixing" behavior of ADA, because the samples that are being mixed change in each iteration. This paper is available on arxiv under CC0 1.0 DEED license. Authors: (1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch); (2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch); (3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Authors: Authors: (1) Nora Schneider, Computer Science Department, ETH Zurich, Zurich, Switzerland (nschneide@student.ethz.ch); (2) Shirin Goshtasbpour, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (shirin.goshtasbpour@inf.ethz.ch); (3) Fernando Perez-Cruz, Computer Science Department, ETH Zurich, Zurich, Switzerland and Swiss Data Science Center, Zurich, Switzerland (fernando.perezcruz@sdsc.ethz.ch). Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 2 Background 2.1 Data Augmentation 2.1 Data Augmentation 2.2 Anchor Regression 2.2 Anchor Regression 3 Anchor Data Augmentation 3 Anchor Data Augmentation 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.1 Comparison to C-Mixup and 3.2 Preserving nonlinear data structure 3.3 Algorithm 3.3 Algorithm 4 Experiments and 4.1 Linear synthetic data 4 Experiments and 4.1 Linear synthetic data 4.2 Housing nonlinear regression 4.2 Housing nonlinear regression 4.3 In-distribution Generalization 4.3 In-distribution Generalization 4.4 Out-of-distribution Robustness 4.4 Out-of-distribution Robustness 5 Conclusion, Broader Impact, and References 5 Conclusion, Broader Impact, and References A Additional information for Anchor Data Augmentation A Additional information for Anchor Data Augmentation B Experiments B Experiments A Additional information for Anchor Data Augmentation A.1 Derivation of ADA for nonlinear data A.2 Additional information on hyperparameters of ADA In this section, we illustrate in a simple 1D example (i.e. cosine data used in Figure 1) how changes in the hyperparameter values modify the data and affect the achieved estimation. Additionally, we show in Appendix B.4 how ADA performance on real-world data is impacted by changes in the hyperparameter values. Anchor Matrices and Locality: Anchor variable A is assumed to be the exogenous variable that generates heterogeneity in the target and has an approximately linear relation with (X, y) (see AR loss in Equation 3). It is recommended to choose the variable relying on expert knowledge about the features that the target has a higher dependence on or is possibly misrepresented in the dataset so that we encourage the robustness of the trained model against this type of discrepancy. After deciding the features, one way to construct the anchor matrix A is to partition the dataset according to the similarity of the features, using for example binning or clustering algorithms. Then we can fill the rows of A with a one-hot encoding of the partition index that each sample belongs to. Anchor Matrices and Locality: A A We use the following nonlinear Cosine data model as a running example to demonstrate more clearly how A is constructed and affects the augmentation procedure. A For illustration purposes, we use in Figures 5, 7 equidistant x values as this reduces noise and emphasizes the effect of ADA parameters more. Number of augmentations: For each anchor matrix A and γ we can add n new samples to the dataset. The addition of more augmented samples may not be beneficial as the optimization may overfit the approximations in the augmented data model in Equation 10. In the Cosine data model this is specifically problematic when X is close to multiples of π as depicted in Figure 7. Additionally, we provide a baseline and an augmented model fit in Figure 8 with different number of augmentations. Number of augmentations: As it is standard practice to use stochastic gradient descent methods for optimizing a regressor, we suggest applying ADA on each minibatch instead of the entire dataset. This avoids choosing a fixed numbers of augmentations. Furthermore, it adds diversity to the "mixing" behavior of ADA, because the samples that are being mixed change in each iteration. This paper is available on arxiv under CC0 1.0 DEED license. This paper is available on arxiv under CC0 1.0 DEED license. available on arxiv

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