[Paper Review] Quantifying the Benefit of Using Differentiable Learning over Tangent Kernels

Eran Malach, Pritish Kamath+2 more|arXiv (Cornell University)|Mar 1, 2021

Neural Networks and Applications21 references4 citations

TL;DR

This paper demonstrates that gradient descent on differentiable models like neural networks can achieve low test error even when the corresponding tangent kernel (NTK) offers no significant advantage over random guessing—contrary to prior assumptions. The key insight is that with biased initialization, gradient descent can succeed where kernel methods fail, highlighting a fundamental separation in learning capacity between non-convex optimization and kernel-based approaches.

ABSTRACT

We study the relative power of learning with gradient descent on differentiable models, such as neural networks, versus using the corresponding tangent kernels. We show that under certain conditions, gradient descent achieves small error only if a related tangent kernel method achieves a non-trivial advantage over random guessing (a.k.a. weak learning), though this advantage might be very small even when gradient descent can achieve arbitrarily high accuracy. Complementing this, we show that without these conditions, gradient descent can in fact learn with small error even when no kernel method, in particular using the tangent kernel, can achieve a non-trivial advantage over random guessing.

Motivation & Objective

• To investigate whether gradient descent on differentiable models can succeed when the corresponding tangent kernel cannot achieve even weak learning (i.e., a non-trivial edge over random guessing).
• To determine whether the success of gradient descent depends on the NTK having a non-trivial edge, particularly under different initialization schemes.
• To examine the role of input distribution knowledge in enabling gradient descent to succeed where kernel methods fail.
• To clarify the conditions under which differentiable learning surpasses kernel-based learning, especially in distribution-independent learning settings.
• To demonstrate that biased initialization enables gradient descent to learn complex functions even when no kernel method, including the NTK, can achieve a non-trivial edge.

Proposed method

• Constructs a synthetic learning problem where gradient descent on a neural network achieves arbitrarily low error, while the NTK at initialization has an edge arbitrarily close to zero.
• Proves in Theorem 1 that under unbiased initialization (i.e., zero output at initialization), gradient descent can only succeed if the NTK has a polynomially bounded edge over random guessing.
• Introduces a randomized initialization scheme dependent on the input distribution, showing that in expectation, the NTK at this random initialization achieves a non-trivial edge when gradient descent succeeds.
• Uses Theorem 2 to show that for square loss, if gradient descent succeeds from any initialization, then there exists a distribution-dependent random initialization such that the NTK at that point has a non-trivial edge.
• Analyzes the role of initialization bias by constructing a counterexample where gradient descent succeeds from a biased initialization, but no kernel method—including the NTK—can achieve a non-trivial edge.
• Empirically validates the theoretical findings using a two-layer ReLU network on a parity-like function, showing that gradient descent succeeds when kernel methods do not.

Experimental results

Research questions

• RQ1 Can gradient descent on a differentiable model learn a function even when the corresponding tangent kernel has no non-trivial advantage over random guessing?
• RQ2 Is the success of gradient descent contingent on the NTK achieving a non-trivial edge, especially under unbiased initialization?
• RQ3 Can gradient descent succeed in distribution-independent learning settings even when no kernel method can achieve a non-trivial edge?
• RQ4 What is the role of input distribution knowledge in enabling gradient descent to outperform kernel-based methods?
• RQ5 How does initialization bias affect the relationship between the learning power of gradient descent and that of kernel methods?

Key findings

• Gradient descent can achieve arbitrarily low error even when the NTK has an edge arbitrarily close to zero, demonstrating a strict separation between differentiable learning and kernel-based learning.
• Under unbiased initialization (i.e., zero output at initialization), gradient descent can only succeed if the NTK has a polynomially bounded edge over random guessing, as formalized in Theorem 1.
• With biased initialization, gradient descent can succeed even when no kernel method—including the NTK—can achieve a non-trivial edge, as shown in Section 5.
• For square loss, if gradient descent succeeds from any initialization, then there exists a distribution-dependent random initialization such that the NTK at that point has a non-trivial edge, as proven in Theorem 2.
• The requirement for distribution-dependent initialization in the success of the NTK at random initialization highlights a fundamental difference between kernel methods and gradient descent in leveraging input distribution structure.
• Empirical results with a two-layer ReLU network confirm that gradient descent can learn complex functions (e.g., parity) when kernel methods fail, especially when relevant features are identifiable via correlation.