An empirical analysis of the optimization of deep network loss surfaces

Abstract: The success of deep neural networks hinges on our ability to accurately and efficiently optimize high-dimensional, non-convex functions. In this paper, we empirically investigate the loss functions of state-of-the-art networks, and how commonly-used stochastic gradient descent variants optimize these loss functions. To do this, we visualize the loss function by projecting them down to low-dimensional spaces chosen based on the convergence points of different optimization algorithms. Our observations suggest that optimization algorithms encounter and choose different descent directions at many saddle points to find different final weights. Based on consistency we observe across re-runs of the same stochastic optimization algorithm, we hypothesize that each optimization algorithm makes characteristic choices at these saddle points.

• The paper demonstrates how projecting high-dimensional loss surfaces onto low-dimensional spaces clarifies optimization pathways.
• The paper reveals that encountering saddle points prompts varied descent directions, leading to different final model weights.
• The paper shows that multiple runs of SGD variants yield consistent behavior, suggesting characteristic strategies for navigating non-convex landscapes.

The paper "An empirical analysis of the optimization of deep network loss surfaces" explores the intricacies of optimizing deep neural networks, a critical aspect in the success of these models. The research focuses on understanding how stochastic gradient descent (SGD) variants perform when optimizing non-convex loss landscapes associated with deep networks.

Key contributions and findings of the paper include:

• Loss Function Visualization: The authors project high-dimensional loss surfaces onto low-dimensional spaces. This visualization is based on the convergence points of various optimization algorithms, allowing for a clearer view of the optimization pathways and how different algorithms navigate these loss landscapes.
• Saddle Points and Descent Directions: The study observes that optimization algorithms frequently encounter saddle points—critical points where the gradient is zero but which are not local minima. At these junctures, the algorithms choose various descent directions, leading to different final weights for the model. This highlights the nuanced role saddle points play in the optimization process.
• Algorithm Behavior Consistency: Through multiple runs of the same stochastic optimization algorithms, the researchers notice consistent behavior patterns. These patterns suggest that each algorithm might have characteristic ways of handling saddle points, which guide them towards certain regions of the loss surface.
• Implications for Optimization Strategies: The insights from this analysis could influence the development of more robust optimization strategies for training deep networks. Understanding these characteristic choices at saddle points might lead to improved algorithms with better convergence properties.

Overall, this empirical investigation provides significant insights into the behavior of optimization algorithms in deep learning, emphasizing the complexity and subtlety of navigating non-convex loss landscapes. This understanding is crucial for enhancing the efficacy and reliability of training deep neural networks.