Gradient Descent Learns Linear Dynamical Systems (1609.05191v2)

Abstract: We prove that stochastic gradient descent efficiently converges to the global optimizer of the maximum likelihood objective of an unknown linear time-invariant dynamical system from a sequence of noisy observations generated by the system. Even though the objective function is non-convex, we provide polynomial running time and sample complexity bounds under strong but natural assumptions. Linear systems identification has been studied for many decades, yet, to the best of our knowledge, these are the first polynomial guarantees for the problem we consider.

• The paper establishes that stochastic gradient descent (SGD) can efficiently minimize the non-convex maximum likelihood objective for linear time-invariant (LTI) systems with polynomial time and sample complexity guarantees.
• It provides the first polynomial guarantees for directly optimizing this inherently non-convex system identification problem, offering a pathway to tackle similar challenges.
• The work extends to over-parameterized models for improved condition handling and applies findings to multi-input multi-output (MIMO) systems, with implications for understanding recurrent neural networks (RNNs).

Analyzing "Gradient Descent Learns Linear Dynamical Systems"

The paper "Gradient Descent Learns Linear Dynamical Systems" by Moritz Hardt, Tengyu Ma, and Benjamin Recht provides significant theoretical insights into the use of stochastic gradient descent (SGD) for learning linear time-invariant (LTI) dynamical systems. The central focus of the work is on understanding the ability of gradient descent to efficiently approximate the maximum likelihood estimator (MLE) for LTI systems, despite the non-convex nature of the objective function.

Overview of Contributions

1. Polynomial Convergence Guarantees: The authors establish that SGD can efficiently minimize the MLE objective for an unknown linear system with demonstration of polynomial running time and sample complexity bounds. The analysis holds under strong yet reasonable assumptions about system stability and input-output characteristics.
2. Handling Non-Convex Objectives: Despite the system identification task involving inherently non-convex objectives due to the linear composition structure, the paper provides the first polynomial guarantees for direct optimization of this problem formulation. It leverages the idea that the non-convexity in LTI systems can be systematically approached by isolating key conditions under which the system remains tractable.
3. Extension to Over-Parameterized Models: A novel aspect of this research is its treatment of over-parameterization. By allowing for model orders larger than those present in the ground truth, the authors demonstrate improved condition handling which relaxes the assumptions required for learning stability. Specifically, they show that the system can be effectively learned by utilizing characteristic polynomial extensions.
4. Foundation for Recurrent Neural Networks: While rooted in control systems, this research indirectly informs machine learning, particularly in understanding recurrent neural networks (RNNs). Since RNNs can be viewed as extensions of LTI systems, insights into LTI learning mechanisms can enhance the theoretical underpinnings of training non-linear sequence models.
5. Capability to Handle MIMO Systems: The work extends its findings to multi-input multi-output (MIMO) scenarios, showcasing the flexibility and practical applicability of their approach beyond simple SISO cases.

Theoretical and Practical Implications

• Controlled System Behavior: By ensuring that the spectral radius constraints—like $\rho(A) < 1$—are respected, the paper provides a necessary grounding for stable behavior in real-world applications.
• Answer to Non-Convex Challenges: The research approach illustrates a conceptual pathway to tackling optimization problems in non-linear systems through linear approximation analogs, hence broadening the domain of problems solvable by gradient-based algorithms.
• Future Work and Challenges: While SGD is shown to provide efficient models under theoretical constraints, operationalizing such models in control systems and further synchronizing them with RNN-like architectures offer promising areas for continuing research.

Concluding Remarks

The paper by Hardt, Ma, and Recht contributes a substantial theoretical framework to the area of dynamical system identification via stochastic optimization. By succeeding in offering polynomial guarantees for learning in an essentially non-convex space, this work propels future exploration into robust methods for complex dynamics in both control and data-driven learning paradigms. The strategic use of over-parameterization also presents an inspiring departure point for tackling deeper non-convex challenges in machine learning domains.