What Can ResNet Learn Efficiently, Going Beyond Kernels?

Abstract: How can neural networks such as ResNet efficiently learn CIFAR-10 with test accuracy more than 96%, while other methods, especially kernel methods, fall relatively behind? Can we more provide theoretical justifications for this gap? Recently, there is an influential line of work relating neural networks to kernels in the over-parameterized regime, proving they can learn certain concept class that is also learnable by kernels with similar test error. Yet, can neural networks provably learn some concept class BETTER than kernels? We answer this positively in the distribution-free setting. We prove neural networks can efficiently learn a notable class of functions, including those defined by three-layer residual networks with smooth activations, without any distributional assumption. At the same time, we prove there are simple functions in this class such that with the same number of training examples, the test error obtained by neural networks can be MUCH SMALLER than ANY kernel method, including neural tangent kernels (NTK). The main intuition is that multi-layer neural networks can implicitly perform hierarchical learning using different layers, which reduces the sample complexity comparing to "one-shot" learning algorithms such as kernel methods. In a follow-up work [2], this theory of hierarchical learning is further strengthened to incorporate the "backward feature correction" process when training deep networks. In the end, we also prove a computation complexity advantage of ResNet with respect to other learning methods including linear regression over arbitrary feature mappings.

• The paper theoretically demonstrates that ResNets can efficiently learn complex functions beyond kernel methods, achieving lower sample complexity without distributional assumptions.
• This learnability advantage stems from ResNet's hierarchical learning structure, enabling improved generalization and computational benefits over kernel methods.
• ResNet achieves significantly lower generalization error (O(ε)) with polynomial samples compared to kernel methods constrained by higher error rates, establishing a provable separation.

Overview of the Research Paper on ResNet Learnability Beyond Kernels

The paper, "What Can ResNet Learn Efficiently, Going Beyond Kernels?" explores the theoretical underpinnings of deep learning models, particularly focusing on the capabilities and limitations of ResNet architecture compared to kernel methods. It seeks to provide a quantitative analysis of the efficiency with which ResNets can learn certain concept classes, addressing a gap in AI theory where neural networks outperform traditional kernel methods in practical scenarios without clear theoretical justification.

Key Contributions

This research offers significant insights into the learnability of functions by neural networks, especially ResNet architectures, beyond the capacities of kernel methods. The primary contributions include:

1. Efficient Learning without Distributional Assumptions: The paper establishes that ResNets can efficiently learn complex functions beyond the capabilities of kernel methods, without imposing any distributional assumptions. This is achieved through a unique hierarchical learning process.
2. Provable Generalization Advantage: ResNet networks demonstrate exceptional generalization capabilities, achieving lower sample complexities compared to kernel-based methods for certain concept classes. This separation is notably marked in the efficiently-computable regime.
3. Hierarchical Learning via Forward Feature Learning: ResNet is shown to leverage its layered structure to perform hierarchical learning, effectively reducing sample complexity by learning lower-complexity features before moving to higher-complexity ones.
4. Computational Complexity Benefits: ResNet architectures not only provide improved generalization but also offer computational benefits over linear regression based on arbitrary feature mappings, marking a notable advantage in time/space efficiency.

Results and Implications

The paper's theoretical analysis reveals that, under certain settings:

• ResNet can achieve a generalization error of approximately O(ε) using polynomial-in-ε samples, while kernel methods, even with reasonable sample sizes, remain constrained to higher generalization errors due to inherent limitations in their structure.
• The sample complexity required for ResNet to learn a function effectively is significantly lower than that for kernel methods, thus establishing a provable learning separation between the model classes.
• This approach leads to implications regarding the design and deployment of neural network architectures, emphasizing the importance of hierarchical learning structures for efficient and effective model training.

Future Directions

The results lay the groundwork for further exploration into hierarchical learning mechanisms inherent in neural networks like ResNet. The paper hints at potential advancements in understanding "backward feature correction" that could further improve learning accuracy within more complex neural networks, indicating promising future developments in the field.

Technical Strengths and Claims

The research rigorously structures its experimentation and theoretical proofs to support recognized claims:

• Existential Lemma Use: The paper leverages sophisticated lemmas to demonstrate the capabilities of ResNet in approximating complex functions efficiently, highlighting the importance of theoretical tools in understanding neural network learning.
• Complexity Metrics: Clear definitions and use of metrics such as function complexity, reinforce the analytical rigor of the paper, ensuring claims are substantiated by thorough mathematical analysis.
• Concentration Inequalities: Utilization of concentration inequalities provides robustness to probabilistic claims made in the paper regarding the learning process and expected outcomes from ResNet training.

In conclusion, this paper provides substantial theoretical backing for why ResNet architectures can outperform traditional machine learning models, especially kernel methods, not only in practical applications but also as a fundamental aspect of how neural networks represent and learn from data. The clear differentiation it draws between these methods opens up further avenues for research into efficient neural network design and learning strategies.