Stabilizing Equilibrium Models by Jacobian Regularization (2106.14342v1)

Abstract: Deep equilibrium networks (DEQs) are a new class of models that eschews traditional depth in favor of finding the fixed point of a single nonlinear layer. These models have been shown to achieve performance competitive with the state-of-the-art deep networks while using significantly less memory. Yet they are also slower, brittle to architectural choices, and introduce potential instability to the model. In this paper, we propose a regularization scheme for DEQ models that explicitly regularizes the Jacobian of the fixed-point update equations to stabilize the learning of equilibrium models. We show that this regularization adds only minimal computational cost, significantly stabilizes the fixed-point convergence in both forward and backward passes, and scales well to high-dimensional, realistic domains (e.g., WikiText-103 LLMing and ImageNet classification). Using this method, we demonstrate, for the first time, an implicit-depth model that runs with approximately the same speed and level of performance as popular conventional deep networks such as ResNet-101, while still maintaining the constant memory footprint and architectural simplicity of DEQs. Code is available at https://github.com/locuslab/deq .

• The paper introduces a Jacobian regularization technique to tackle instability and slow convergence in deep equilibrium models.
• It employs the Hutchinson estimator to approximate the Frobenius norm, effectively constraining the Jacobian's spectral radius with minimal computational cost.
• Empirical results on language and image tasks show up to 2x faster convergence and improved efficiency compared to traditional deep network methods.

Stabilizing Equilibrium Models by Jacobian Regularization

The paper "Stabilizing Equilibrium Models by Jacobian Regularization" investigates the challenges and solutions associated with deep equilibrium networks (DEQs). DEQs are implicit-depth models that compute the fixed point of a single non-linear function, offering memory efficiency compared to traditional deep networks. However, they face issues of instability, slow convergence, and sensitivity to architectural variations.

Key Contributions

The authors identify several limitations inherent in DEQs:

• Growing Instability: DEQs become harder to solve as training progresses. This results in increasing iterations needed to reach convergence, leading to computational inefficiencies.
• Inefficiency: Compared to conventional deep networks, DEQs require more iterations, hence more computational resources, despite their lower memory footprint.
• Architectural Brittleness: DEQs show sensitivity to minor changes in architecture, which can lead to convergence failures.

To address these challenges, the authors propose a new regularization technique focused on the Jacobian matrix of the fixed-point update equations. Regularizing the Jacobian's norm acts to stabilize the DEQ's learning process, making it less sensitive to architectural choices and more efficient computationally.

Methodology

The authors employ a Jacobian regularization technique that approximates the Frobenius norm of the Jacobian using the Hutchinson estimator. This regularization is designed to constrain the Jacobian's spectral radius, thereby improving the forward and backward convergence stability of DEQs.

Notably, this approach introduces little additional computational expense but effectively addresses the existing inefficiencies and instability of DEQs, accelerating their convergence.

Results and Implications

Empirical validation is provided across multiple domains:

• LLMing: On the WikiText-103 dataset, DEQ models with Jacobian regularization achieve a comparable performance level to similar deeplink density explicit models, with a remarkable decrease in processing time (approximately 2x faster).
• Image Classification: The application to CIFAR-10 and ImageNet demonstrates that the approach significantly reduces NFEs while maintaining competitive accuracy levels against popular architectures like ResNet and DenseNet.

The specialized regularization technique successfully mitigates the excessive computational demands and stability issues of DEQs, thus facilitating their scalability and application to larger, more complex tasks.

Future Directions

The paper opens new directions for research in implicit models and equilibrium networks. The authors suggest further exploration into adaptive strategies for regularization strength during training and how global-context architectures might further enhance DEQ performance. These insights point towards a more robust framework for DEQ deployment in various large-scale applications, emphasizing the balance between computational efficiency and model stability.

In conclusion, this work is significant for its methodical analysis of DEQ shortcomings and the proposed solution, advancing the practical applicability of implicit models within mainstream machine learning tasks.