State-space models can learn in-context by gradient descent (2410.11687v1)

Abstract: Deep state-space models (Deep SSMs) have shown capabilities for in-context learning on autoregressive tasks, similar to transformers. However, the architectural requirements and mechanisms enabling this in recurrent networks remain unclear. This study demonstrates that state-space model architectures can perform gradient-based learning and use it for in-context learning. We prove that a single structured state-space model layer, augmented with local self-attention, can reproduce the outputs of an implicit linear model with least squares loss after one step of gradient descent. Our key insight is that the diagonal linear recurrent layer can act as a gradient accumulator, which can be `applied' to the parameters of the implicit regression model. We validate our construction by training randomly initialized augmented SSMs on simple linear regression tasks. The empirically optimized parameters match the theoretical ones, obtained analytically from the implicit model construction. Extensions to multi-step linear and non-linear regression yield consistent results. The constructed SSM encompasses features of modern deep state-space models, with the potential for scalable training and effectiveness even in general tasks. The theoretical construction elucidates the role of local self-attention and multiplicative interactions in recurrent architectures as the key ingredients for enabling the expressive power typical of foundation models.

In-Context Learning in State-Space Models via Gradient Descent

The paper "State-space models can learn in-context by gradient descent" explores the capabilities of deep state-space models (SSMs) in in-context learning tasks, investigating their potential as efficient alternatives to transformers. The authors argue that SSMs, when constructed with specific architectural features, can mimic the gradient descent mechanism commonly associated with in-context learning in transformer-based models.

Architectural and Theoretical Insights

The paper introduces a variant of SSMs, termed GD-SSM, which integrates a single structured state-space model layer augmented with local self-attention. This layer is demonstrated to replicate the outputs of an implicit linear model after one gradient descent step. The core insight is that the diagonal linear recurrent layer within the SSM acts as a gradient accumulator, effectively aligning with the parameters of the implicit regression model.

The authors extend their theoretical exploration to multi-step and non-linear regression tasks. They establish that stacking layers in the GD-SSM enables multi-step gradient descent, while the introduction of Multi-Layer Perceptrons (MLPs) facilitates handling non-linear tasks. The SSMs with these enhancements remain competitive across various regression problems, showcasing their expressiveness and scalability.

Empirical Validation

Empirical results support the theoretical framework, with trained GD-SSMs on synthetic linear regression tasks exhibiting losses that match those calculated from analytical constructions. These results persist even when tasks deviate from those encountered during training, underscoring the robustness of the model in generalizing learning rules.

Comparative performance evaluations further highlight the competitiveness of GD-SSMs against traditional transformers and other recurrent networks. Notably, while transformers may require multiple layers to replicate similar tasks, a single-layer GD-SSM suffices, emphasizing the model's efficiency.

Implications and Future Directions

The findings illuminate the potential for SSMs to serve as efficient and scalable alternatives to transformer architectures in tasks requiring in-context learning. This points to broader implications in designing models with intrinsic support for gradient-based updates, extending beyond simple autoregressive tasks.

The paper suggests several directions for future exploration. These include scaling GD-SSMs in more complex and higher-dimensional tasks, integrating additional model components for enhanced capabilities, and examining the architectural features that contribute to efficient in-context learning further.

In conclusion, this paper contributes significantly to understanding the architectural and functional capacities of state-space models in in-context learning, providing a foundation for future research and practical implementations in AI systems.