Asymmetric Valleys: Beyond Sharp and Flat Local Minima (1902.00744v2)

Abstract: Despite the non-convex nature of their loss functions, deep neural networks are known to generalize well when optimized with stochastic gradient descent (SGD). Recent work conjectures that SGD with proper configuration is able to find wide and flat local minima, which have been proposed to be associated with good generalization performance. In this paper, we observe that local minima of modern deep networks are more than being flat or sharp. Specifically, at a local minimum there exist many asymmetric directions such that the loss increases abruptly along one side, and slowly along the opposite side--we formally define such minima as asymmetric valleys. Under mild assumptions, we prove that for asymmetric valleys, a solution biased towards the flat side generalizes better than the exact minimizer. Further, we show that simply averaging the weights along the SGD trajectory gives rise to such biased solutions implicitly. This provides a theoretical explanation for the intriguing phenomenon observed by Izmailov et al. (2018). In addition, we empirically find that batch normalization (BN) appears to be a major cause for asymmetric valleys.

• The paper introduces asymmetric valleys as regions where loss increases anisotropically, challenging the traditional sharp versus flat minimum dichotomy.
• Empirical and theoretical analyses reveal that SGD’s implicit bias towards the flatter side of these valleys correlates with lower generalization error.
• The study highlights Batch Normalization as a significant factor in creating asymmetric landscapes, influencing both optimization and architecture design.

Insights into Asymmetric Valleys in Deep Neural Network Optimization

The paper "Asymmetric Valleys: Beyond Sharp and Flat Local Minima" provides an intricate exploration of the optimization landscape encountered in deep neural networks, contributing a nuanced perspective that challenges the conventional binary view of sharp versus flat minima. Rather than solely focusing on flatness as an indicator of generalization prowess, the authors introduce the notion of asymmetric valleys to refine our understanding of local minima characteristics within the loss landscape of deep networks.

Summary of Key Contributions

1. Asymmetric Valleys Definition: The authors propose that loss surfaces of neural networks are asymmetric in nature. Asymmetric valleys are described as regions around local minima where the loss ascent is anisotropic—loss increases steeply along one direction while remaining shallow on the opposite side. Such observations challenge the simplistic sharp-flat dichotomy that has dominated discussions on generalization till now.
2. Underlying Geometry and Generalization: Through theoretical analysis, the paper establishes that solutions biased toward the flatter side of asymmetric valleys yield lower generalization error. This finding suggests that the quest for ideal minima should consider both directionality and flatness rather than merely locating flatter minima.
3. SGD and Implicit Bias: A notable practical implication suggested by the authors is that Stochastic Gradient Descent (SGD), due to its nature of averaging weights across iterations, inherently biases solutions toward the flat side of asymmetric valleys. This aligns their theoretical exploration with empirical findings, like those of Izmailov et al. (2018), who observed improved generalization when weights along SGD trajectories are averaged.
4. Empirical Validation and Role of Batch Normalization: The paper empirically shows the prevalence of asymmetric valleys in modern neural architectures, using various models and datasets. Interestingly, it identifies Batch Normalization (BN) as a significant factor contributing to the formation of such asymmetric landscapes. Directions related to BN parameters tend to exhibit pronounced asymmetry, supporting the hypothesis that model architecture intricacies, rather than mere optimization strategies, influence the optimization landscape.

Implications and Future Prospects

The introduction of asymmetric valleys ushers fresh pathways for designing optimization algorithms. Recognizing the multidimensional nature of loss topographies beckons for methods that not only focus on reaching local minima but also a nuanced exploration of the paths leveraged during descent. Future research could further formalize strategies around SGD to exploit directional biases for enhanced generalization and uncover whether certain architectures inherently accommodate asymmetric regions better than others.

The strong suggestion of Batch Normalization influencing asymmetric valley formation raises compelling questions for architectural design and optimization strategy formulation. While improvements from averaging indicate robust performance in asymmetric regions, there remains scope for discovering how complementary regularization techniques can augment these effects or counteract any limitations imposed by specific architecture choices.

Conclusion

This paper essentially recalibrates the lens through which researchers view neural network optimization. By framing the generalization advantage in terms of landscape asymmetry rather than mere minimization, it provides a valuable context for interpreting empirical phenomena and guides the development of robust, generalizable deep models. The findings advocate for optimization techniques sensitive to the topological irregularities characterized by asymmetric valleys, potentially shaping the formulation of more nuanced, intelligent training paradigms in future deep learning frameworks.