Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data (1808.01204v3)

Abstract: Neural networks have many successful applications, while much less theoretical understanding has been gained. Towards bridging this gap, we study the problem of learning a two-layer overparameterized ReLU neural network for multi-class classification via stochastic gradient descent (SGD) from random initialization. In the overparameterized setting, when the data comes from mixtures of well-separated distributions, we prove that SGD learns a network with a small generalization error, albeit the network has enough capacity to fit arbitrary labels. Furthermore, the analysis provides interesting insights into several aspects of learning neural networks and can be verified based on empirical studies on synthetic data and on the MNIST dataset.

• The paper shows that SGD achieves low generalization error in overparameterized ReLU networks even when the capacity allows for memorization.
• It demonstrates that proper random initialization and low-rank weight updates stabilize neuron activations, smoothing the optimization landscape for SGD.
• The study reveals SGD's inductive bias on structured data, providing critical insights for designing neural networks that generalize well in multi-class settings.

Learning Overparameterized Neural Networks via Stochastic Gradient Descent on Structured Data

The paper investigates the theoretical underpinnings of learning overparameterized two-layer ReLU neural networks using Stochastic Gradient Descent (SGD). Specifically, it addresses overparameterization in the context of multi-class classification, where the network has more parameters than training data points. The paper focuses on structured data sets with well-separated data mixtures, offering insights into how SGD attains low generalization errors despite the potential for overfitting.

Key Contributions

The authors provide a rigorous theoretical analysis proving that SGD effectively learns a model with a low generalization error on structured data. This is an important finding given the network's capacity to fit random labels. The implication is that SGD introduces an inductive bias that favors solutions with better generalization.

Analytical Insights

1. Overparameterization Impact: The paper demonstrates that overparameterization, when combined with SGD, aids optimization, reducing the risk of overfitting even though the network has sufficient capacity to memorize arbitrary data.
2. Random Initialization: Proper random initialization is critical, establishing the initial conditions for effective coupling between the network's initialization and SGD's dynamics. This coupling aids in reaching a low error solution efficiently.
3. Optimization Landscape: The proofs leverage the insight that in highly overparameterized settings, the activation patterns for neurons remain stable over iterations, simplifying the optimization landscape and effectively smoothing out the path for SGD.
4. Low Rank of Updates: Empirical observations corroborate that the updates to network weights maintain a low-rank structure, aligning with theories connected to low-complexity measures and implicit regularization aiding generalization.

Implications and Future Directions

The theoretical results suggest important implications for understanding neural network generalization and the role of structured data in learning dynamics. Practically, this indicates how overparameterization coupled with SGD can be exploited in designing neural networks for real-world data having inherent structure.

Future research may explore more complex data distributions beyond the separability and clustering assumptions used here, possibly venturing into non-convex manifolds or alternate metrics for separability. The exploration of implicit regularization's interaction with various data structures offers rich ground for further paper.

Overall, the paper offers a substantial addition to the theoretical understanding of SGD's role in training overparameterized neural networks, laying the groundwork for future explorations in both structured and practical data settings.