Enhanced Importance Sampling through Latent Space Exploration in Normalizing Flows (2501.03394v2)

Abstract: Importance sampling is a rare event simulation technique used in Monte Carlo simulations to bias the sampling distribution towards the rare event of interest. By assigning appropriate weights to sampled points, importance sampling allows for more efficient estimation of rare events or tails of distributions. However, importance sampling can fail when the proposal distribution does not effectively cover the target distribution. In this work, we propose a method for more efficient sampling by updating the proposal distribution in the latent space of a normalizing flow. Normalizing flows learn an invertible mapping from a target distribution to a simpler latent distribution. The latent space can be more easily explored during the search for a proposal distribution, and samples from the proposal distribution are recovered in the space of the target distribution via the invertible mapping. We empirically validate our methodology on simulated robotics applications such as autonomous racing and aircraft ground collision avoidance.

• The paper introduces latent space sampling in normalizing flows to reduce variance and improve rare event estimation through efficient failure region exploration.
• It employs normalizing flows with monotonic rational quadratic splines to accurately transform complex distributions into tractable latent spaces.
• Empirical results in robotics and aerospace simulations demonstrate superior accuracy, coverage, and sample efficiency compared to traditional methods.

Enhanced Importance Sampling through Latent Space Exploration in Normalizing Flows

The paper "Enhanced Importance Sampling through Latent Space Exploration in Normalizing Flows" presents an innovative approach to improve the efficiency of importance sampling (IS), specifically for applications involving rare events. The method utilizes normalizing flows to transform the outcome space into a latent space, where the proposal distribution is more tractable and easier to sample from due to its simpler, often isotropic nature.

Key Contributions

1. Latent Space Sampling: By performing IS in the latent space of normalizing flows, the methodology combines the ability of these flows to map complex distributions to simpler ones with the variance reduction capabilities of IS. This results in more efficient exploration of failure regions compared to traditional IS methods that operate in the target space.
2. Normalizing Flow Architecture: The paper employs normalizing flows that use monotonic rational quadratic splines in coupling transformations. This choice ensures both invertibility and a capacity for modeling intricate distributions, critical for faithfully capturing the target density from which samples are ultimately drawn.
3. Cost Function Based on L{\"o}wner--John Ellipsoids: The proposed cost function formulation facilitates defining failure regions in both target and latent spaces. By considering the minimum volume ellipsoids that circumscribe failure events, the method captures the essential geometry of the failure domains while remaining computationally efficient to evaluate.
4. Evaluation on Robotics Applications: The paper presents empirical validations using simulated robotics applications, including autonomous racing and aircraft ground collision avoidance, demonstrating the practical benefits of the proposed algorithm in reducing the computational burden of estimating probabilities of failure.

Experimental Evaluation and Results

The experiments conducted on the nonholonomic robot, cornering racecar, and F-16 fighter aircraft simulators highlight several key findings:

• Higher Accuracy: The use of latent space sampling results in more accurate estimation of failure probabilities, with lower relative error compared to target space methodologies.
• Better Coverage and Density: The latent space methods yield higher coverage and density metrics, indicating their effectiveness at capturing the diversity of failure modes and ensuring samples are densely packed in high-failure regions where real failure data are clustered.
• Faster Convergence: Latent space IS methods require fewer samples to converge, especially the Cross Entropy method in latent space, which consistently outperforms its target space counterpart in terms of sample efficiency.

Implications and Future Directions

The proposed methodology presents significant implications for the safety validation of autonomous systems and structural reliability analysis, where estimating rare events is crucial but computational resources are limited.

Future research may focus on:

• Theoretical Analysis: Rigorous theoretical exploration of the conditions under which the latent space sampling guarantees improved convergence and accuracy over target-based methods.
• Alternative Cost Functions: Exploration of different geometrical formulations for cost functions that could provide even more discriminating power in distinguishing failure from non-failure states.
• Real-world Applications: Extending the methodology to real-world, high-fidelity simulators, where computational savings and sample efficiency could translate into practical advantages in systems verification.

Overall, the paper contributes a substantive improvement to the IS paradigm through the use of normalizing flows and latent space transformation, offering a promising avenue for enhancing the reliability and efficiency of complex system simulations.