Towards Understanding Generalization of Deep Learning: Perspective of Loss Landscapes (1706.10239v2)

Abstract: It is widely observed that deep learning models with learned parameters generalize well, even with much more model parameters than the number of training samples. We systematically investigate the underlying reasons why deep neural networks often generalize well, and reveal the difference between the minima (with the same training error) that generalize well and those they don't. We show that it is the characteristics the landscape of the loss function that explains the good generalization capability. For the landscape of loss function for deep networks, the volume of basin of attraction of good minima dominates over that of poor minima, which guarantees optimization methods with random initialization to converge to good minima. We theoretically justify our findings through analyzing 2-layer neural networks; and show that the low-complexity solutions have a small norm of Hessian matrix with respect to model parameters. For deeper networks, extensive numerical evidence helps to support our arguments.

• The paper demonstrates that models converging to flat minima with wider basins exhibit superior generalization performance.
• It employs theoretical Hessian and Fisher information analyses to show that low-complexity solutions reside in flatter regions.
• Empirical results reveal that over-parameterized networks are biased toward flat minima, leading to consistently robust generalization.

Understanding the Generalization of Deep Learning: Insights from Loss Landscapes

The paper "Towards Understanding Generalization of Deep Learning: Perspective of Loss Landscapes" by Wu, Zhu, and E, addresses a pivotal aspect in the field of deep learning—explaining the remarkable generalization ability of deep neural networks despite being over-parameterized. Unlike prior works which often focus on the role of stochastic gradient descent (SGD) or specific regularization techniques, this paper explores the intrinsic properties of loss landscapes across neural networks to provide a novel perspective.

Theoretical Contributions

The authors focus on two central questions: identifying the distinguishing features of neural network solutions that generalize well, and understanding why optimization with random initialization tends to converge to these solutions. The paper argues that the landscape of the loss function is pivotal in addressing these questions. Specifically, it is demonstrated that the volume of the basin of attraction for good minima is significantly larger than that for poor minima. This characteristic ensures that optimization methods almost invariably converge to solutions with superior generalization capabilities.

To support their claims, the authors use a combination of theoretical and empirical analyses:

1. Hessian Analysis: For 2-layer neural networks, the authors theoretically establish that low-complexity solutions correlate with small norms of the Hessian matrix relative to model parameters. This low Hessian norm suggests that such solutions reside in flatter regions of the loss landscape, implying larger basins of attraction. By employing a Fisher information matrix analysis, the paper gives a mathematically grounded explanation for the efficacy of these flatter solutions.
2. Spectral Characteristics: Numerical experiments extend these findings to deeper networks. The spectral analysis of the Hessian matrices around various minima highlights that flatter minima, associated with large eigenvalue decay, often correspond to solutions with good generalization. This reinforces the theoretical insights obtained for the simpler networks.

Empirical Observations

Through rigorous experimentation, the paper examines the generalization performance difference among network solutions that appear indistinguishable in training performance. An innovative approach utilizing an "attack dataset" reveals that network solutions can be driven to poor generalization, further validating the distinct landscape characteristics. This experimentation highlights that optimizers, owing to the particular landscape features, are biased towards flat minima—thus explaining their generalization behavior without relying primarily on regularization techniques or specific optimizers like SGD.

Practical and Theoretical Implications

The findings affirm a fundamental shift in understanding neural network training. They suggest a reevaluation of how neural networks are initialized and optimized. Specifically, instead of placing the onus on tighter generalization bounds, attention should be given to the intrinsic structure of the loss landscape, as this structure inherently guides training towards low-complexity solutions.

Future Directions: The paper opens the avenue for further exploration into the dynamics of deeper networks beyond the scope of two-layer models explored in this work. Furthermore, it suggests the potential development of novel metrics inspired by this work for evaluating network performance and generalization ability.

In summary, this paper provides a nuanced understanding of generalization in deep learning, emphasizing the significance of loss landscape properties over conventional regularization paradigms. Through both theoretical and empirical insights, the authors bridge an essential gap in statistical learning theory pertaining to deep networks, enhancing our comprehension of their robustness and efficiency.