Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling (2410.07974v4)

Abstract: Rare event sampling in dynamical systems is a fundamental problem arising in the natural sciences, which poses significant computational challenges due to an exponentially large space of trajectories. For settings where the dynamical system of interest follows a Brownian motion with known drift, the question of conditioning the process to reach a given endpoint or desired rare event is definitively answered by Doob's h-transform. However, the naive estimation of this transform is infeasible, as it requires simulating sufficiently many forward trajectories to estimate rare event probabilities. In this work, we propose a variational formulation of Doob's h-transform as an optimization problem over trajectories between a given initial point and the desired ending point. To solve this optimization, we propose a simulation-free training objective with a model parameterization that imposes the desired boundary conditions by design. Our approach significantly reduces the search space over trajectories and avoids expensive trajectory simulation and inefficient importance sampling estimators which are required in existing methods. We demonstrate the ability of our method to find feasible transition paths on real-world molecular simulation and protein folding tasks.

• The paper proposes a novel variational formulation of Doob's h-transform that recasts transition path sampling as an optimization problem.
• It develops a simulation-free training method using Gaussian path parameterization to enforce boundary conditions and reduce computational costs.
• Experimental results on molecular systems like alanine dipeptide and Chignolin demonstrate its efficiency over traditional MCMC approaches in high-dimensional settings.

Essay on "Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling"

The paper "Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling" addresses significant computational challenges encountered in rare event sampling within dynamical systems. Specifically, it focuses on the problem of transition path sampling (TPS) in cases where dynamical systems follow Brownian motion with known drift. This research introduces a novel variational framework leveraging Doob's $h$-transform to efficiently estimate transition paths by rephrasing it as an optimization problem, offering a promising alternative to conventional methods reliant on extensive simulations.

Key Contributions

1. Variational Formulation and Optimization: The authors propose a variational representation of Doob's $h$-transform, formulated as an optimization problem over paths with fixed endpoints. This approach circumvents the computational inefficiencies associated with traditional simulation-based methods.
2. Sample-Efficient Training: A simulation-free objective is developed, minimizing the need for trajectory simulations. The methodology involves parameterizing the model such that boundary conditions are enforced by design, reducing the search space of trajectories and enhancing sample efficiency.
3. Gaussian Path Parameterization: For the implementation, a Gaussian path is used, with its parameterization allowing for sample-efficient calculation and optimization. The method is further extended to mixtures of Gaussians to increase expressivity.
4. Practical and Theoretical Implications: The research demonstrates the feasibility of the proposed approach on molecular simulations, emphasizing its application in molecular dynamics and protein folding tasks. The implication of this work lies in its potential to streamline computational processes in fields like materials science and drug discovery.

Experimental Results

The method's efficacy is validated through experiments on synthetic datasets and real-world molecular systems such as alanine dipeptide and Chignolin. It shows that:

• The approach is notably efficient, requiring significantly fewer potential evaluations than MCMC-based baselines while maintaining competitive accuracy.
• It effectively handles high-dimensional systems, demonstrating tractability even in complex molecular simulations.

Implications and Future Directions

The implications of this paper are profound, particularly in fields requiring efficient sampling from distributions characterized by rare events. The novel use of a variational approach for TPS could be extended to address more complex dynamical systems and other areas beyond molecular dynamics.

In terms of future developments, further exploration could be directed towards enhancing the expressivity of the variational model, possibly by integrating deep generative models or exploring variable length trajectories. Additionally, the framework might be adapted for non-Brownian motion processes, broadening its applicability across various stochastic processes in AI and computational science.

Overall, this paper presents a well-constructed and technically sound approach to a classical problem in dynamical systems, contributing a significant advancement in computational efficiency for TPS. The methodological innovations reported could inspire further research in devising sample-efficient algorithms for rare event simulation across diverse scientific disciplines.