[论文解读] The Neural Tangent Kernel in High Dimensions: Triple Descent and a Multi-Scale Theory of Generalization

Ben Adlam, Jeffrey Pennington|arXiv (Cornell University)|Aug 15, 2020

Stochastic Gradient Optimization Techniques被引用 35

一句话总结

本文分析高维神经网络的核回归，使用神经切线核（NTK），并展示在多重参数化尺度下的非单调泛化行为，包括可能的三重下降。

ABSTRACT

Modern deep learning models employ considerably more parameters than required to fit the training data. Whereas conventional statistical wisdom suggests such models should drastically overfit, in practice these models generalize remarkably well. An emerging paradigm for describing this unexpected behavior is in terms of a \emph{double descent} curve, in which increasing a model's capacity causes its test error to first decrease, then increase to a maximum near the interpolation threshold, and then decrease again in the overparameterized regime. Recent efforts to explain this phenomenon theoretically have focused on simple settings, such as linear regression or kernel regression with unstructured random features, which we argue are too coarse to reveal important nuances of actual neural networks. We provide a precise high-dimensional asymptotic analysis of generalization under kernel regression with the Neural Tangent Kernel, which characterizes the behavior of wide neural networks optimized with gradient descent. Our results reveal that the test error has non-monotonic behavior deep in the overparameterized regime and can even exhibit additional peaks and descents when the number of parameters scales quadratically with the dataset size.

研究动机与目标

• Motivate and understand why overparameterized neural networks generalize well beyond classical regimes.
• Provide a precise high-dimensional asymptotic analysis of NTK ridge regression for a wide teacher network.
• Identify multiple parameterization scales where test error exhibits non-monotonic behavior (linear and quadratic transitions).
• Decompose NTK into per-layer kernels to locate sources of non-monotonic generalization.
• Offer empirical evidence of triple descent in finite networks trained by gradient descent.

提出的方法

• Model the learning task with kernel ridge regression using the Neural Tangent Kernel of a single-hidden-layer network.
• Decompose the NTK into two per-layer kernels K1 and K2 and analyze their contributions.
• Derive high-dimensional limits with m samples, n0 features, and n1 hidden units with fixed ratios φ=n0/m and ψ=n0/n1.
• Linearize nonlinear random feature matrices via Gaussian equivalents to obtain tractable expressions.
• Express the test error as a rational function of the inverse kernel using linear pencils and random matrix techniques.
• Provide exact asymptotic formulas for E_train and E_test and analyze limiting regimes.

实验结果

研究问题

• RQ1 How does NTK ridge regression generalize in high dimensions when the parameter count p scales with m and m^2?
• RQ2 Do non-monotonicities in test error occur deep in the overparameterized regime, and what scales (linear vs quadratic) drive them?
• RQ3 What are the relative contributions of the first-layer and second-layer kernels to generalization?
• RQ4 Can the NTK regime exhibit triple descent and multi-scale learning curves with finite-width networks?
• RQ5 What are the asymptotic expressions for training and test errors under NTK regression and their limiting cases?

主要发现

• Test error shows non-monotonic behavior deep in the overparameterized regime.
• Non-monotonicity can arise and persist when p scales quadratically with the dataset size m (p ~ m^2).
• The non-monotonicity is attributed primarily to the kernel associated with the second-layer weights (K2).
• In the large-width (superabundant) regime, learning curves can be exceptionally fast, with E_test scaling as m^−2 in the noiseless case and ~m^−1 for finite SNR.
• Triple descent and multi-scale phenomena are supported by theoretical analysis and empirical evidence on finite networks.