Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows

Addressing Instability in Generative Flows

Generative models, particularly those leveraging flow matching, have become highly effective at learning and replicating complex data distributions. However, a significant challenge arises when attempting to adapt these sophisticated pretrained models to new, specific tasks. The conventional approach often involves computationally intensive retraining, which can be prohibitive in terms of time and resources. An alternative, post-training guidance, offers a more efficient path by steering the model's output without full retraining. Yet, existing guidance methods frequently rely on heuristic approaches, lacking explicit theoretical guarantees for stability, which can lead to unpredictable or suboptimal results in practical applications. This gap highlights a critical need for a more robust and theoretically sound framework for guiding generative flows.

LyaGuide: A Control-Theoretic Unification

The LyaGuide framework directly addresses these limitations by reframing flow guidance as a Lyapunov control problem, a well-established concept in control theory for ensuring system stability. This innovative approach establishes a fundamental theoretical equivalence between guided flow matching and Lyapunov control, providing a unified lens through which various guidance strategies can be understood and implemented. Notably, LyaGuide integrates diverse methods such as classifier guidance, reward guidance, and energy-based guidance under a single, coherent control-theoretic umbrella. To practically enforce the necessary Lyapunov conditions for stability, the framework introduces a novel pseudo-projection operator. This operator possesses a closed-form expression, allowing it to imbue both learned and heuristic guidance terms with explicit and verifiable stability guarantees, a significant advancement over prior methods.

Enhanced Performance and Practical Adaptability

LyaGuide is designed for practical utility, supporting two distinct operational settings: a model-driven approach where guidance is derived from a known Lyapunov function, and a data-driven setting where guidance is adapted from task-specific downstream data. This flexibility ensures broad applicability across different scenarios. Crucially, the framework is compatible with existing guidance methodologies, meaning it can augment rather than replace current techniques. Its design introduces minimal additional computational overhead, making it an efficient solution for real-world deployment. The paper presents extensive experimental validation across a range of complex tasks, including synthetic benchmarks, image inverse problems, reinforcement learning planning, and energy-based modeling. These experiments consistently demonstrate that LyaGuide leads to notable improvements in sample quality, guidance fidelity, and overall robustness, all while maintaining computational efficiency, underscoring its potential to enhance the reliability and performance of generative AI systems.