Deep Equilibrium Models (1909.01377v2)

Abstract: We present a new approach to modeling sequential data: the deep equilibrium model (DEQ). Motivated by an observation that the hidden layers of many existing deep sequence models converge towards some fixed point, we propose the DEQ approach that directly finds these equilibrium points via root-finding. Such a method is equivalent to running an infinite depth (weight-tied) feedforward network, but has the notable advantage that we can analytically backpropagate through the equilibrium point using implicit differentiation. Using this approach, training and prediction in these networks require only constant memory, regardless of the effective "depth" of the network. We demonstrate how DEQs can be applied to two state-of-the-art deep sequence models: self-attention transformers and trellis networks. On large-scale LLMing tasks, such as the WikiText-103 benchmark, we show that DEQs 1) often improve performance over these state-of-the-art models (for similar parameter counts); 2) have similar computational requirements to existing models; and 3) vastly reduce memory consumption (often the bottleneck for training large sequence models), demonstrating an up-to 88% memory reduction in our experiments. The code is available at https://github.com/locuslab/deq .

• The paper introduces Deep Equilibrium Models that compute hidden state equilibrium via implicit differentiation, bypassing traditional layer-by-layer processing.
• It demonstrates performance on par with state-of-the-art models while achieving up to 88% memory reduction on benchmarks like WikiText-103.
• The study underscores DEQ's potential in resource-constrained settings and motivates further research into root-finding techniques for robust optimization.

An Analytical Overview of Deep Equilibrium Models

In the field of computational models for sequential data, the "Deep Equilibrium Model" (DEQ) proposes a novel conceptual approach. This paper, authored by Bai, Kolter, and Koltun, introduces the DEQ method which directly targets sequence model equilibrium points using implicit differentiation and root-finding techniques. By circumventing the traditional layer-by-layer training and inference processes, DEQ offers substantial improvements in memory efficiency, potentially redefining the computational landscape of deep sequence models.

Theoretical Foundations

Central to this work is the hypothesis that the hidden layers of existing deep sequence models tend to converge towards a fixed point. Thus, DEQs bypass the finite layer stacking process, attempting instead to compute these equilibrium states directly. Conceptually, this equates to simulating an infinite-depth, weight-tied feedforward network. By utilizing implicit differentiation, DEQs perform model training and prediction with constant memory requirements, a significant departure from traditional layer-dependent memory complexity.

Methodology and Numerical Results

The paper explores DEQs in the context of two prominent sequence modeling architectures: self-attention transformers and trellis networks. On the challenging WikiText-103 benchmark, DEQs demonstrated the potential to:

1. Improve performance comparably with state-of-the-art models within similar parameter confines.
2. Display computational requirements on par with traditional architectures.
3. Achieve up to 88% reduction in memory consumption, a non-trivial advantage in large-scale sequence model training.

Implicit differentiation, an essential component of DEQs, facilitates backpropagation through the infinite sequence of transformations. This is achieved without preserving intermediate activation values, thus maintaining a flexible approach to sequence length and complexity.

Implications and Future Directions

From a practical standpoint, DEQs present substantial benefits in memory efficiency and computational resource management. The ability to handle large-scale datasets like WikiText-103 with significantly less memory heralds opportunities for deployment on resource-constrained environments.

Theoretically, DEQs challenge the current paradigm of layer-based sequence modeling. While the immediate applicability has been demonstrated on language tasks, future developments may expand into other domains of sequential data.

Exploration into additional root-finding methods and their implications on optimization stability could refine DEQ applications further. Moreover, integrating DEQs with recent advances in hardware-accelerated computations could bridge the current runtime disparities with conventional deep networks.

Conclusion

Deep Equilibrium Models introduce a promising and efficient framework for sequence processing by harnessing the natural convergence properties of deep networks' hidden states. The results underscore the practical and theoretical potential of employing equilibrium-driven approaches in computational models. Future research may continue to unlock additional applications and enhance the DEQ methodology for broader adoption in AI and machine learning contexts.