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Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Chenyu Huang, Peng Ye, Tao Chen, Tong He, Xiangyu Yue, Wanli Ouyang

The success of pretrain-finetune paradigm brings about the release of numerous model weights. In this case, merging models finetuned on different tasks to enable a single model with multi-task capabilities is gaining increasing attention for its practicability. Existing model merging methods usually suffer from (1) significant performance degradation or (2) requiring tuning by additional data or training. In this paper, we rethink and analyze the existing model merging paradigm. We discover that using a single model's weights can hardly simulate all the models' performance. To tackle this issue, we propose Elect, Mask & Rescale-Merging (EMR-Merging). We first (a) elect a unified model from all the model weights and then (b) generate extremely lightweight task-specific modulators, including masks and rescalers, to align the direction and magnitude between the unified model and each specific model, respectively. EMR-Merging is tuning-free, thus requiring no data availability or any additional training while showing impressive performance. We find that EMR-Merging shows outstanding performance compared to existing merging methods under different classical and newly-established settings, including merging different numbers of vision models (up to 30), NLP models, PEFT models, and multi-modal models.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Naoki Hiratani

Continual learning of partially similar tasks poses a challenge for artificial neural networks, as task similarity presents both an opportunity for knowledge transfer and a risk of interference and catastrophic forgetting.However, it remains unclear how task similarity in input features and readout patterns influences knowledge transfer and forgetting, as well as how they interact with common algorithms for continual learning.Here, we develop a linear teacher-student model with latent structure and show analytically that high input feature similarity coupled with low readout similarity is catastrophic for both knowledge transfer and retention. Conversely, the opposite scenario is relatively benign. Our analysis further reveals that task-dependent activity gating improves knowledge retention at the expense of transfer, while task-dependent plasticity gating does not affect either retention or transfer performance at the over-parameterized limit. In contrast, weight regularization based on the Fisher information metric significantly improves retention, regardless of task similarity, without compromising transfer performance. Nevertheless, its diagonal approximation and regularization in the Euclidean space are much less robust against task similarity. We demonstrate consistent results in a permuted MNIST task with latent variables. Overall, this work provides insights into when continual learning is difficult and how to mitigate it.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Robin Chan, Reda Boumasmoud, Anej Svete, Yuxin Ren, Qipeng Guo, Zhijing Jin, Shauli Ravfogel, Mrinmaya Sachan, Bernhard Schölkopf, Mennatallah El-Assady 等

Pre-trained language encoders---functions that represent text as vectors---are an integral component of many NLP tasks. We tackle a natural question in language encoder analysis: What does it mean for two encoders to be similar? We contend that a faithful measure of similarity needs to be \emph{intrinsic}, that is, task-independent, yet still be informative of \emph{extrinsic} similarity---the performance on downstream tasks. It is common to consider two encoders similar if they are \emph{homotopic}, i.e., if they can be aligned through some transformation. In this spirit, we study the properties of \emph{affine} alignment of language encoders and its implications on extrinsic similarity. We find that while affine alignment is fundamentally an asymmetric notion of similarity, it is still informative of extrinsic similarity. We confirm this on datasets of natural language representations. Beyond providing useful bounds on extrinsic similarity, affine intrinsic similarity also allows us to begin uncovering the structure of the space of pre-trained encoders by defining an order over them.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Viet Hoang Phan, Tung Lam Tran, Quyen Tran, Trung Le

Prior Unsupervised Domain Adaptation (UDA) methods often aim to train a domain-invariant feature extractor, which may hinder the model from learning sufficiently discriminative features. To tackle this, a line of works based on prompt learning leverages the power of large-scale pre-trained vision-language models to learn both domain-invariant and specific features through a set of domain-agnostic and domain-specific learnable prompts. Those studies typically enforce invariant constraints on representation, output, or prompt space to learn such prompts. Differently, we cast UDA as a multiple-objective optimization problem in which each objective is represented by a domain loss. Under this new framework, we propose aligning per-objective gradients to foster consensus between them. Additionally, to prevent potential overfitting when fine-tuning this deep learning architecture, we penalize the norm of these gradients. To achieve these goals, we devise a practical gradient update procedure that can work under both single-source and multi-source UDA. Empirically, our method consistently surpasses other vision language model adaptation methods by a large margin on a wide range of benchmarks. The implementation is available at https://github.com/VietHoang1512/PGA.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Fei Zhou, Peng Wang, Lei Zhang, Zhenghua Chen, Wei Wei, Chen Ding, Guosheng Lin, Yanning Zhang

Meta-learning offers a promising avenue for few-shot learning (FSL), enabling models to glean a generalizable feature embedding through episodic training on synthetic FSL tasks in a source domain. Yet, in practical scenarios where the target task diverges from that in the source domain, meta-learning based method is susceptible to over-fitting. To overcome this, we introduce a novel framework, Meta-Exploiting Frequency Prior for Cross-Domain Few-Shot Learning, which is crafted to comprehensively exploit the cross-domain transferable image prior that each image can be decomposed into complementary low-frequency content details and high-frequency robust structural characteristics. Motivated by this insight, we propose to decompose each query image into its high-frequency and low-frequency components, and parallel incorporate them into the feature embedding network to enhance the final category prediction. More importantly, we introduce a feature reconstruction prior and a prediction consistency prior to separately encourage the consistency of the intermediate feature as well as the final category prediction between the original query image and its decomposed frequency components. This allows for collectively guiding the network's meta-learning process with the aim of learning generalizable image feature embeddings, while not introducing any extra computational cost in the inference phase. Our framework establishes new state-of-the-art results on multiple cross-domain few-shot learning benchmarks.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Duo Wang, Yuan Zuo, Fengzhi Li, Junjie Wu

Zero-shot graph machine learning, especially with graph neural networks (GNNs), has garnered significant interest due to the challenge of scarce labeled data. While methods like self-supervised learning and graph prompt learning have been extensively explored, they often rely on fine-tuning with task-specific labels, limiting their effectiveness in zero-shot scenarios. Inspired by the zero-shot capabilities of instruction-fine-tuned large language models (LLMs), we introduce a novel framework named Token Embedding-Aligned Graph Language Model (TEA-GLM) that leverages LLMs as cross-dataset and cross-task zero-shot learners for graph machine learning. Concretely, we pretrain a GNN, aligning its representations with token embeddings of an LLM. We then train a linear projector that transforms the GNN's representations into a fixed number of graph token embeddings without tuning the LLM. A unified instruction is designed for various graph tasks at different levels, such as node classification (node-level) and link prediction (edge-level). These design choices collectively enhance our method's effectiveness in zero-shot learning, setting it apart from existing methods. Experiments show that our graph token embeddings help the LLM predictor achieve state-of-the-art performance on unseen datasets and tasks compared to other methods using LLMs as predictors. Our code is available at https://github.com/W-rudder/TEA-GLM.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Zhenyi Lu, Chenghao Fan, Wei Wei, Xiaoye Qu, Dangyang Chen, Yu Cheng

In the era of large language models, model merging is a promising way to combine multiple task-specific models into a single multitask model without extra training. However, two challenges remain: (a) interference between different models and (b) heterogeneous data during testing. Traditional model merging methods often show significant performance gaps compared to fine-tuned models due to these issues. Additionally, a one-size-fits-all model lacks flexibility for diverse test data, leading to performance degradation. We show that both shared and exclusive task-specific knowledge are crucial for merging performance, but directly merging exclusive knowledge hinders overall performance. In view of this, we propose Twin-Merging, a method that encompasses two principal stages: (1) modularizing knowledge into shared and exclusive components, with compression to reduce redundancy and enhance efficiency; (2) dynamically merging shared and task-specific knowledge based on the input. This approach narrows the performance gap between merged and fine-tuned models and improves adaptability to heterogeneous data. Extensive experiments on $20$ datasets for both language and vision tasks demonstrate the effectiveness of our method, showing an average improvement of $28.34\%$ in absolute normalized score for discriminative tasks and even surpassing the fine-tuned upper bound on the generative tasks.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Zhenyi Wang, Heng Huang

Continual learning (CL) aims to adapt to non-stationary data distributions while retaining previously acquired knowledge. However, CL models typically face a trade-off between preserving old task knowledge and excelling in new task performance. Existing approaches often sacrifice one for the other. To overcome this limitation, orthogonal to existing approaches, we propose a novel perspective that views the CL model ability in preserving old knowledge and performing well in new task as a matter of model sensitivity to parameter updates. \textit{Excessive} parameter sensitivity can lead to two drawbacks: (1) significant forgetting of previous knowledge; and (2) overfitting to new tasks. To reduce parameter sensitivity, we optimize the model's performance based on the parameter distribution, which achieves the worst-case CL performance within a distribution neighborhood. This innovative learning paradigm offers dual benefits: (1) reduced forgetting of old knowledge by mitigating drastic changes in model predictions under small parameter updates; and (2) enhanced new task performance by preventing overfitting to new tasks. Consequently, our method achieves superior ability in retaining old knowledge and achieving excellent new task performance simultaneously.Importantly, our approach is compatible with existing CL methodologies, allowing seamless integration while delivering significant improvements in effectiveness, efficiency, and versatility with both theoretical and empirical supports.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Mingyang Liu, Xinyang Chen, Yang Shu, Xiucheng Li, Weili Guan, Liqiang Nie

Unsupervised domain adaptation excels in transferring knowledge from a labeled source domain to an unlabeled target domain, playing a critical role in time series applications. Existing time series domain adaptation methods either ignore frequency features or treat temporal and frequency features equally, which makes it challenging to fully exploit the advantages of both types of features. In this paper, we delve into transferability and discriminability, two crucial properties in transferable representation learning. It's insightful to note that frequency features are more discriminative within a specific domain, while temporal features show better transferability across domains. Based on the findings, we propose **A**dversarial **CO**-learning **N**etworks (**ACON**), to enhance transferable representation learning through a collaborative learning manner in three aspects: (1) Considering the multi-periodicity in time series, multi-period frequency feature learning is proposed to enhance the discriminability of frequency features; (2) Temporal-frequency domain mutual learning is proposed to enhance the discriminability of temporal features in the source domain and improve the transferability of frequency features in the target domain; (3) Domain adversarial learning is conducted in the correlation subspaces of temporal-frequency features instead of original feature spaces to further enhance the transferability of both features. Extensive experiments conducted on a wide range of time series datasets and five common applications demonstrate the state-of-the-art performance of ACON. Code is available at <https://github.com/mingyangliu1024/ACON>.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Yixiong Zou, Shuai Yi, Yuhua Li, Ruixuan Li

Vision Transformer (ViT) has shown great power in learning from large-scale datasets. However, collecting sufficient data for expert knowledge is always difficult. To handle this problem, Cross-Domain Few-Shot Learning (CDFSL) has been proposed to transfer the source-domain knowledge learned from sufficient data to target domains where only scarce data is available. In this paper, we find an intriguing phenomenon neglected by previous works for the CDFSL task based on ViT: leaving the CLS token to random initialization, instead of loading source-domain trained parameters, could consistently improve target-domain performance. We then delve into this phenomenon for an interpretation. We find **the CLS token naturally absorbs domain information** due to the inherent structure of the ViT, which is represented as the low-frequency component in the Fourier frequency space of images. Based on this phenomenon and interpretation, we further propose a method for the CDFSL task to decouple the domain information in the CLS token during the source-domain training, and adapt the CLS token on the target domain for efficient few-shot learning. Extensive experiments on four benchmarks validate our rationale and state-of-the-art performance. Our codes are available at https://github.com/Zoilsen/CLS_Token_CDFSL.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Chen Huang, Skyler Seto, Samira Abnar, David Grangier, Navdeep Jaitly, Joshua Susskind

Large pretrained vision-language models like CLIP have shown promising generalization capability, but may struggle in specialized domains (e.g., satellite imagery) or fine-grained classification (e.g., car models) where the visual concepts are unseen or under-represented during pretraining. Prompt learning offers a parameter-efficient finetuning framework that can adapt CLIP to downstream tasks even when limited annotation data are available. In this paper, we improve prompt learning by distilling the textual knowledge from natural language prompts (either human- or LLM-generated) to provide rich priors for those under-represented concepts. We first obtain a prompt ``summary'' aligned to each input image via a learned prompt aggregator. Then we jointly train a prompt generator, optimized to produce a prompt embedding that stays close to the aggregated summary while minimizing task loss at the same time. We dub such prompt embedding as Aggregate-and-Adapted Prompt Embedding (AAPE). AAPE is shown to be able to generalize to different downstream data distributions and tasks, including vision-language understanding tasks (e.g., few-shot classification, VQA) and generation tasks (image captioning) where AAPE achieves competitive performance. We also show AAPE is particularly helpful to handle non-canonical and OOD examples. Furthermore, AAPE learning eliminates LLM-based inference cost as required by baselines, and scales better with data and LLM model size.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Hui Chen, Yanbin Liu, Yongqiang Ma, Nanning Zheng, Xin Yu

Pre-trained vision-language models (VLMs) such as CLIP have shown excellent performance for zero-shot classification. Based on CLIP, recent methods design various learnable prompts to evaluate the zero-shot generalization capability on a base-to-novel setting. This setting assumes test samples are already divided into either base or novel classes, limiting its application to realistic scenarios. In this paper, we focus on a more challenging and practical setting: generalized zero-shot learning (GZSL), i.e., testing with no information about the base/novel division. To address this challenging zero-shot problem, we introduce two unique designs that enable us to classify an image without the need of knowing whether it comes from seen or unseen classes. Firstly, most existing methods only adopt a single latent space to align visual and linguistic features, which has a limited ability to represent complex visual-linguistic patterns, especially for fine-grained tasks. Instead, we propose a dual-space feature alignment module that effectively augments the latent space with a novel attribute space induced by a well-devised attribute reservoir. In particular, the attribute reservoir consists of a static vocabulary and learnable tokens complementing each other for flexible control over feature granularity. Secondly, finetuning CLIP models (e.g., prompt learning) on seen base classes usually sacrifices the model's original generalization capability on unseen novel classes. To mitigate this issue, we present a new topology-preserving objective that can enforce feature topology structures of the combined base and novel classes to resemble the topology of CLIP. In this manner, our model will inherit the generalization ability of CLIP through maintaining the pairwise class angles in the attribute space. Extensive experiments on twelve object recognition datasets demonstrate that our model, termed Topology-Preserving Reservoir (TPR), outperforms strong baselines including both prompt learning and conventional generative-based zero-shot methods.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Lixu Wang, Xinyu Du, Qi Zhu

Cross-domain retrieval (CDR) is finding increasingly broad applications across various domains. However, existing efforts have several major limitations, with the most critical being their reliance on accurate supervision. Recent studies thus focus on achieving unsupervised CDR, but they typically assume that the category spaces across domains are identical, an assumption that is often unrealistic in real-world scenarios. This is because only through dedicated and comprehensive analysis can the category composition of a data domain be obtained, which contradicts the premise of unsupervised scenarios. Therefore, in this work, we introduce the problem of **U**niversal **U**nsupervised **C**ross-**D**omain **R**etrieval (U^2CDR) for the first time and design a two-stage semantic feature learning framework to address it. In the first stage, a cross-domain unified prototypical structure is established under the guidance of an instance-prototype-mixed contrastive loss and a semantic-enhanced loss, to counteract category space differences. In the second stage, through a modified adversarial training mechanism, we ensure minimal changes for the established prototypical structure during domain alignment, enabling more accurate nearest-neighbor searching. Extensive experiments across multiple datasets and scenarios, including close-set, partial, and open-set CDR, demonstrate that our approach significantly outperforms existing state-of-the-art CDR methods and other related methods in solving U^2CDR challenges.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Atsutoshi Kumagai, Tomoharu Iwata, Hiroshi Takahashi, Taishi Nishiyama, Yasuhiro Fujiwara

Maximizing the area under the receiver operating characteristic curve (AUC) is a popular approach to imbalanced binary classification problems. Existing AUC maximization methods usually assume that training and test distributions are identical. However, this assumption is often violated in practice due to {\it a positive distribution shift}, where the negative-conditional density does not change but the positive-conditional density can vary. This shift often occurs in imbalanced classification since positive data are often more diverse and time-varying than negative data. To deal with this shift, we theoretically show that the AUC on the test distribution can be expressed by using the positive and marginal training densities and the marginal test density. Based on this result, we can maximize the AUC on the test distribution by using positive and unlabeled data in the training distribution and unlabeled data in the test distribution. The proposed method requires only positive labels in the training distribution as supervision. Moreover, the derived AUC has a simple form and thus is easy to implement. The effectiveness of the proposed method is shown with four real-world datasets.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Mingjia Li, Shuang Li, Tongrui Su, Longhui Yuan, Jian Liang, Wei Li

Capitalizing on the complementary advantages of generative and discriminative models has always been a compelling vision in machine learning, backed by a growing body of research. This work discloses the hidden semantic structure within score-based generative models, unveiling their potential as effective discriminative priors. Inspired by our theoretical findings, we propose DUSA to exploit the structured semantic priors underlying diffusion score to facilitate the test-time adaptation of image classifiers or dense predictors. Notably, DUSA extracts knowledge from a single timestep of denoising diffusion, lifting the curse of Monte Carlo-based likelihood estimation over timesteps. We demonstrate the efficacy of our DUSA in adapting a wide variety of competitive pre-trained discriminative models on diverse test-time scenarios. Additionally, a thorough ablation study is conducted to dissect the pivotal elements in DUSA. Code is publicly available at https://github.com/BIT-DA/DUSA.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Hanwen Zhong, Jiaxin Chen, Yutong Zhang, Di Huang, Yunhong Wang

Multi-Task Learning (MTL) for Vision Transformer aims at enhancing the model capability by tackling multiple tasks simultaneously. Most recent works have predominantly focused on designing Mixture-of-Experts (MoE) structures and integrating Low-Rank Adaptation (LoRA) to efficiently perform multi-task learning. However, their rigid combination hampers both the optimization of MoE and the effectiveness of reparameterization of LoRA, leading to sub-optimal performance and low inference speed. In this work, we propose a novel approach dubbed Efficient Multi-Task Learning (EMTAL) by transforming a pre-trained Vision Transformer into an efficient multi-task learner during training, and reparameterizing the learned structure for efficient inference. Specifically, we firstly develop the MoEfied LoRA structure, which decomposes the pre-trained Transformer into a low-rank MoE structure and employ LoRA to fine-tune the parameters. Subsequently, we take into account the intrinsic asynchronous nature of multi-task learning and devise a learning Quality Retaining (QR) optimization mechanism, by leveraging the historical high-quality class logits to prevent a well-trained task from performance degradation. Finally, we design a router fading strategy to integrate the learned parameters into the original Transformer, archiving efficient inference. Extensive experiments on public benchmarks demonstrate the superiority of our method, compared to the state-of-the-art multi-task learning approaches.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Yunqiao Yang, Long-Kai Huang, Shengzhuang Chen, Kede Ma, Ying Wei

Model editing aims to data-efficiently correct predictive errors of large pre-trained models while ensuring generalization to neighboring failures and locality to minimize unintended effects on unrelated examples. While significant progress has been made in editing Transformer-based large language models, effective strategies for editing vision Transformers (ViTs) in computer vision remain largely untapped. In this paper, we take initial steps towards correcting predictive errors of ViTs, particularly those arising from subpopulation shifts. Taking a locate-then-edit approach, we first address the ``where-to-edit`` challenge by meta-learning a hypernetwork on CutMix-augmented data generated for editing reliability. This trained hypernetwork produces generalizable binary masks that identify a sparse subset of structured model parameters, responsive to real-world failure samples. Afterward, we solve the ``how-to-edit`` problem by simply fine-tuning the identified parameters using a variant of gradient descent to achieve successful edits. To validate our method, we construct an editing benchmark that introduces subpopulation shifts towards natural underrepresented images and AI-generated images, thereby revealing the limitations of pre-trained ViTs for object recognition. Our approach not only achieves superior performance on the proposed benchmark but also allows for adjustable trade-offs between generalization and locality. Our code is available at https://github.com/hustyyq/Where-to-Edit.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Samuel Lippl, Jack Lindsey

Neural networks are often trained on multiple tasks, either simultaneously (multi-task learning, MTL) or sequentially (pretraining and subsequent finetuning, PT+FT). In particular, it is common practice to pretrain neural networks on a large auxiliary task before finetuning on a downstream task with fewer samples. Despite the prevalence of this approach, the inductive biases that arise from learning multiple tasks are poorly characterized. In this work, we address this gap. We describe novel implicit regularization penalties associated with MTL and PT+FT in diagonal linear networks and single-hidden-layer ReLU networks. These penalties indicate that MTL and PT+FT induce the network to reuse features in different ways. 1) Both MTL and PT+FT exhibit biases towards feature reuse between tasks, and towards sparsity in the set of learned features. We show a "conservation law" that implies a direct tradeoff between these two biases. 2) PT+FT exhibits a novel "nested feature selection" regime, not described by either the "lazy" or "rich" regimes identified in prior work, which biases it to *rely on a sparse subset* of the features learned during pretraining. This regime is much narrower for MTL. 3) PT+FT (but not MTL) in ReLU networks benefits from features that are correlated between the auxiliary and main task. We confirm these findings empirically with teacher-student models, and introduce a technique -- weight rescaling following pretraining -- that can elicit the nested feature selection regime. Finally, we validate our theory in deep neural networks trained on image classification. We find that weight rescaling improves performance when it causes models to display signatures of nested feature selection. Our results suggest that nested feature selection may be an important inductive bias for finetuning neural networks.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Tyler Ingebrand, Adam Thorpe, Ufuk Topcu

Autonomous systems often encounter environments and scenarios beyond the scope of their training data, which underscores a critical challenge: the need to generalize and adapt to unseen scenarios in real time. This challenge necessitates new mathematical and algorithmic tools that enable adaptation and zero-shot transfer. To this end, we leverage the theory of function encoders, which enables zero-shot transfer by combining the flexibility of neural networks with the mathematical principles of Hilbert spaces. Using this theory, we first present a method for learning a space of dynamics spanned by a set of neural ODE basis functions. After training, the proposed approach can rapidly identify dynamics in the learned space using an efficient inner product calculation. Critically, this calculation requires no gradient calculations or retraining during the online phase. This method enables zero-shot transfer for autonomous systems at runtime and opens the door for a new class of adaptable control algorithms. We demonstrate state-of-the-art system modeling accuracy for two MuJoCo robot environments and show that the learned models can be used for more efficient MPC control of a quadrotor.

Miscellaneous Aspects of Machine Learning · Transfer, Multitask and Meta-learning

Ayush Jain, Andrea Montanari, Eren Sasoglu

Collecting large quantities of high-quality data can be prohibitively expensive or impractical, and a bottleneck in machine learning. One may instead augment a small set of $n$ data points from the target distribution with data from more accessible sources, e.g. data collected under different circumstances or synthesized by generative models. We refer to such data as `surrogate data'. We study a weighted empirical risk minimization (ERM) approach for integrating surrogate data into training. We analyze mathematically this method under several classical statistical models, and validate our findings empirically on datasets from different domains. Our main findings are: $(i)$ Integrating surrogate data can significantly reduce the test error on the original distribution. Surprisingly, this can happen even when the surrogate data is unrelated to the original ones. We trace back this behavior to the classical Stein's paradox. $(ii)$ In order to reap the benefit of surrogate data, it is crucial to use optimally weighted ERM. $(iii)$ The test error of models trained on mixtures of real and surrogate data is approximately described by a scaling law. This scaling law can be used to predict the optimal weighting scheme, and to choose the amount of surrogate data to add.