Towards a theory of learning dynamics in deep state space models (2407.07279v1)

Abstract: State space models (SSMs) have shown remarkable empirical performance on many long sequence modeling tasks, but a theoretical understanding of these models is still lacking. In this work, we study the learning dynamics of linear SSMs to understand how covariance structure in data, latent state size, and initialization affect the evolution of parameters throughout learning with gradient descent. We show that focusing on the learning dynamics in the frequency domain affords analytical solutions under mild assumptions, and we establish a link between one-dimensional SSMs and the dynamics of deep linear feed-forward networks. Finally, we analyze how latent state over-parameterization affects convergence time and describe future work in extending our results to the study of deep SSMs with nonlinear connections. This work is a step toward a theory of learning dynamics in deep state space models.

• The paper derives analytical frequency-domain solutions for learning dynamics in a linear one-layer state space model.
• It establishes a theoretical link between SSMs and deep linear feedforward networks for understanding training behavior.
• The analysis reveals that over-parameterizing latent dimensions significantly speeds up convergence time.

Towards a Theory of Learning Dynamics in Deep State Space Models

State space models (SSMs) have exhibited noteworthy empirical success in handling long sequence modeling tasks. However, the theoretical underpinnings of these models' learning dynamics remain insufficiently understood. The paper "Towards a theory of learning dynamics in deep state space models" explores deriving a theoretical understanding of linear SSMs, focusing on the role of covariance structures in data, latent state size, and initialization on the parameters' evolution through gradient descent learning. By analyzing learning dynamics in the frequency domain, the paper elucidates several key properties and establishes connections to deep linear feedforward networks.

Key Contributions and Results

1. Analytical Solutions in the Frequency Domain: The paper derives analytical solutions for the learning dynamics of a simplified one-layer SSM by focusing on the frequency domain and performing gradient descent on a squared loss. This perspective ensures a more tractable analytical framework compared to the time domain, especially given the temporal recurrences intrinsic to SSMs.
2. Connection to Deep Linear Feedforward Networks: By leveraging the analytical solutions in the frequency domain, the research establishes a theoretical link between SSMs and the dynamics of deep linear feedforward networks. This connection is paramount as it suggests that insights derived from deep linear networks might be transferrable to SSMs, subject to certain assumptions.
3. Over-parameterization and Convergence Time: The paper provides analytical solutions describing the impact of over-parameterization in the latent state on convergence time for a linear one-layer SSM. Notably, it concludes that increasing latent state dimensions can lead to faster convergence, a result confirmed through numerical simulation.

Theoretical Analysis and Implications

The paper begins by representing a single-input, single-output linear time-invariant SSM in the frequency domain. This transformation simplifies the recurrent temporal nature of SSMs into an element-wise multiplication problem, analytically tractable in the Fourier domain. The learning dynamics for parameters $A$, $B$, and $C$ are described by a set of continuous-time ordinary differential equations (ODEs) under a squared error loss function.

Under the assumption of balance, where parameters $B$ and $C$ are initialized symmetrically, the paper shows that the product of these parameters converges proportionally faster with stronger input-output covariances. This finding is reminiscent of learning dynamics in deep linear feedforward networks, suggesting a shared theoretical framework.

Crucially, the paper extends the analysis to higher-dimensional SSMs, showing that latent state over-parameterization speeds up convergence. This result is of practical relevance as it implies that adding latent dimensions can be a useful strategy to enhance training efficiency.

Directions for Future Work

The results presented in the paper are particularly significant as they not only bridge the understanding between SSMs and linear feedforward networks but also suggest avenues for optimizing SSM architectures. Future research may focus on the following areas:

• Nonlinear Extensions:

Extending the derived analytical framework to account for nonlinear SSMs will be crucial. Nonlinearity introduces additional complexities, but insights from linear regimes can provide foundational guiding principles.

• Multi-layer SSMs:

Investigation into multi-layer SSMs, which involve deeper architectures with nonlinear interconnections between layers, is warranted. Analytical or approximate methods to understand the dynamics across layered structures can substantially improve model design and training methodologies.

• Real-world Applications:

Empirical validation of the theoretical findings within various real-world applications can provide further insights. Specifically, understanding how different data covariance structures influence learning dynamics in practical scenarios will be essential.

• Role of Initialization and Regularization:

The effects of different initialization strategies and regularization techniques on learning dynamics can be explored. Enhanced understanding in this domain could lead to more robust training methodologies and improved model generalization.

Conclusion

The paper "Towards a theory of learning dynamics in deep state space models" makes significant strides towards a theoretical understanding of SSMs. By examining the learning dynamics in the frequency domain and establishing connections to deep linear feedforward networks, the authors provide a foundation for future work in both theoretical explorations and practical implementations of SSMs. The insights gained from this research hold potential for optimizing SSM designs, improving convergence times, and extending the understanding of deep learning models' behavior in structured state space contexts.