GradVis: Visualization and Second Order Analysis of Optimization Surfaces during the Training of Deep Neural Networks (1909.12108v2)

Abstract: Current training methods for deep neural networks boil down to very high dimensional and non-convex optimization problems which are usually solved by a wide range of stochastic gradient descent methods. While these approaches tend to work in practice, there are still many gaps in the theoretical understanding of key aspects like convergence and generalization guarantees, which are induced by the properties of the optimization surface (loss landscape). In order to gain deeper insights, a number of recent publications proposed methods to visualize and analyze the optimization surfaces. However, the computational cost of these methods are very high, making it hardly possible to use them on larger networks. In this paper, we present the GradVis Toolbox, an open source library for efficient and scalable visualization and analysis of deep neural network loss landscapes in Tensorflow and PyTorch. Introducing more efficient mathematical formulations and a novel parallelization scheme, GradVis allows to plot 2d and 3d projections of optimization surfaces and trajectories, as well as high resolution second order gradient information for large networks.

• The paper introduces a scalable toolbox, GradVis, that visualizes and analyzes deep learning loss surfaces to enhance understanding of training dynamics.
• The paper leverages efficient computations including Hessian-vector products and the Lanczos algorithm to provide high-resolution second-order gradient information.
• The paper demonstrates that novel parallelization achieves over 96% efficiency, offering practical insights into the geometry of optimization and model performance.

Analyzing the Optimization Surfaces in Deep Neural Network Training with GradVis

The paper "GradVis: Visualization and Second Order Analysis of Optimization Surfaces during the Training of Deep Neural Networks" presents an innovative approach to studying the optimization landscapes inherent in deep learning models. The authors propose a highly efficient and scalable toolbox designed to visualize and analyze the loss surfaces encountered during the training of deep neural networks. Given the high computational demands associated with such activities, their work leverages efficient mathematical formulations alongside novel parallelization schemes, ensuring compatibility with leading deep learning frameworks like TensorFlow and PyTorch.

The motivation behind this research stems from the necessity to bridge gaps in the theoretical understanding of neural network training dynamics. Despite practical successes in optimizing non-convex high-dimensional loss functions using stochastic gradient descent (SGD) and its variants, theoretical aspects such as convergence guarantees and generalization properties remain elusive. Specifically, the role of the optimization surface's geometry—flat versus sharp minima—has been a subject of ongoing debate and investigation, with prior studies yielding contradictory claims regarding their impact on model generalization.

Key Contributions

1. Visualization Capabilities: The GradVis Toolbox provides advanced visualization options for deep neural network training processes, enabling the plotting of 2D and 3D projections of the optimization surfaces. This allows researchers to assess the gradient trajectories and characterize the regions of interest on the loss landscape efficiently.
2. Second-Order Analysis: GradVis is equipped to compute high-resolution second-order gradient information and eigenvalue spectra of the Hessian, particularly for large networks. This is achieved via economized computation of Hessian-vector products using the R-operator and Lanczos algorithm, which together, make an otherwise infeasible operation scalable.
3. Parallelization: The toolbox introduces a novel parallelization approach for the Lanczos algorithm that significantly improves utilization efficiency, achieving over 96% parallel efficiency as opposed to the traditional data-parallel approach, which was limited to approximately 37%. This advancement renders it possible to conduct comprehensive experiments, capturing the eigenvalue density spectrum across each iteration in training.

Experimental Insights and Implications

The experiments conducted illustrate the efficacy of the GradVis Toolbox in not only visualizing the loss surfaces but also correlating these visualizations with training trajectories and second-order information. For instance, using the proposed methods, researchers observed the dynamics of training from initialization through convergence for a LeNet configuration trained on CIFAR10. The detailed visualization highlighted changes in the eigenvalue distributions and exposed the flatness or curvature at different training intervals.

Moreover, the paper extends to an analysis of paths between multiple minima, offering insights into the landscape geometry connecting distinct model states. This has implications for understanding the modality of the loss surface and may potentially inform strategies for optimization and model ensemble.

Conclusion and Future Directions

The GradVis Toolbox represents a substantial contribution toward unraveling the complexities of optimization in deep learning. By offering an accessible toolset for visualizing and analyzing loss surfaces, this work not only aids in practical model development but also in theoretical research seeking to demystify SGD behavior.

Future directions suggested by the authors involve deploying GradVis for large-scale networks like ResNet on extensive datasets such as ImageNet. Such undertakings could provide richer insights into optimization dynamics, further enhancing the deep learning community's understanding of model generalization and robustness. The methodological advancements in this paper set a foundation for subsequent research aiming at optimizing deep learning from both a methodological and theoretical standpoint.