- The paper presents new depth-independent Rademacher complexity bounds by controlling Schatten norms for enhanced generalization.
- It converts exponential depth dependencies to polynomial ones, reducing error scaling from 2^d to sqrt(d) under Frobenius norm constraints.
- The work emphasizes the necessity of using norm products across layers to mitigate network width effects and guides improved regularization strategies.
Size-Independent Sample Complexity of Neural Networks
The paper explores the sample complexity of neural networks by establishing new bounds on their Rademacher complexity under norm constraints on the parameter matrices of each layer. These bounds exhibit improved dependence on network depth and, under certain conditions, are independent of network size, including both depth and width.
Primary Contributions
- Depth-Independent Bounds: The authors present novel techniques that convert traditional depth-dependent complexity bounds into depth-independent ones. This involves controlling the Schatten norms of parameter matrices and approximating the neural network by composing a shallow network with univariate Lipschitz functions. This technique potentially offers size-independent generalization bounds, which is particularly relevant when the network's size exceeds the number of training examples.
- Rademacher Complexity Analysis: The authors improve on existing Rademacher complexity-based analyses by reducing exponential depth dependencies to polynomial ones. For instance, they demonstrate that for networks constrained by Frobenius norm, the generalization error can be improved from an exponential scale of
2^d to sqrt(d).
- Lower Bounds with Schatten Norms: The paper also presents lower bounds involving the control of Schatten norms. It shows that solely controlling spectral norms (or any Schatten norm with
p\>2) cannot escape size dependencies related to the network width. This highlights the necessity of considering products of norms across layers for effective generalization bounds.
Implications and Speculations on Future Work
The implications are twofold:
- Theoretical Insights: The independent size bounds contribute to a deeper understanding of neural networks' generalization capabilities. This suggests the potential to further explore norm-based constraints and their relaxation in capturing generalization properties.
- Practical Applications: The findings could influence algorithms that rely on such bounds for regularization in neural networks, helping design more efficient models that generalize well even with massive architectures.
Future research could focus on softening the assumptions required for these bounds, such as reducing the strong constraints on norm products across layers. Exploring alternative norm constraints or developing new complexity measures less sensitive to network size could yield more nuanced insights.
Overall, this paper advances the discussion on how neural networks generalize in practice, offering a pathway toward understanding the intricate relationships between network structure, training data, and generalization errors.