学习神经网络的双峰

IF 6.3 2区 物理与天体物理 Q1 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE Machine Learning Science and Technology Pub Date : 2024-06-10 DOI:10.1088/2632-2153/ad524d
Elizaveta Demyanenko, Christoph Feinauer, Enrico M Malatesta, Luca Saglietti
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引用次数: 0

摘要

最近的研究表明,神经网络的泛化误差存在双下降现象,即高度过参数化的模型既能摆脱过拟合,又能获得良好的测试性能,这与统计学习理论所描述的标准偏差-方差权衡是不一致的。在本研究中,我们探讨了这一现象与神经网络所代表函数的复杂性和灵敏度增加之间的联系。我们特别研究了布尔平均维度 (BMD),这是一种在布尔函数分析中开发的度量方法。我们以随机特征模型的简单师生设置为重点,推导出了基于复制法的理论分析,在数据点数量、特征数量和输入大小增长到无穷大的高维系统中,该分析得出了布尔平均维度的可解释表达式。我们发现,随着网络过参数化程度的增加,BMD 在插值阈值处达到一个明显的峰值,与泛化误差峰值相对应,然后慢慢接近一个较低的渐近值。在使用不同模型类别和训练设置的数值实验中,我们也发现了同样的现象。此外,我们还根据经验发现,对抗性初始化的模型往往显示出更高的 BMD 值,而对对抗性攻击更具鲁棒性的模型则显示出更低的 BMD 值。
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The twin peaks of learning neural networks
Recent works demonstrated the existence of a double-descent phenomenon for the generalization error of neural networks, where highly overparameterized models escape overfitting and achieve good test performance, at odds with the standard bias-variance trade-off described by statistical learning theory. In the present work, we explore a link between this phenomenon and the increase of complexity and sensitivity of the function represented by neural networks. In particular, we study the Boolean mean dimension (BMD), a metric developed in the context of Boolean function analysis. Focusing on a simple teacher-student setting for the random feature model, we derive a theoretical analysis based on the replica method that yields an interpretable expression for the BMD, in the high dimensional regime where the number of data points, the number of features, and the input size grow to infinity. We find that, as the degree of overparameterization of the network is increased, the BMD reaches an evident peak at the interpolation threshold, in correspondence with the generalization error peak, and then slowly approaches a low asymptotic value. The same phenomenology is then traced in numerical experiments with different model classes and training setups. Moreover, we find empirically that adversarially initialized models tend to show higher BMD values, and that models that are more robust to adversarial attacks exhibit a lower BMD.
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来源期刊
Machine Learning Science and Technology
Machine Learning Science and Technology Computer Science-Artificial Intelligence
CiteScore
9.10
自引率
4.40%
发文量
86
审稿时长
5 weeks
期刊介绍: Machine Learning Science and Technology is a multidisciplinary open access journal that bridges the application of machine learning across the sciences with advances in machine learning methods and theory as motivated by physical insights. Specifically, articles must fall into one of the following categories: advance the state of machine learning-driven applications in the sciences or make conceptual, methodological or theoretical advances in machine learning with applications to, inspiration from, or motivated by scientific problems.
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