Improving sampling by modifying the effective diffusion (2410.00525v4)

Abstract: Markov chain Monte Carlo samplers based on discretizations of (overdamped) Langevin dynamics are commonly used in the Bayesian inference and computational statistical physics literature to estimate high-dimensional integrals. One can introduce a non-constant diffusion matrix to precondition these dynamics, and recent works have optimized it in order to improve the rate of convergence to stationarity by overcoming entropic and energy barriers. However, the introduced methodologies to compute these optimal diffusions are generally not suited to high-dimensional settings, as they rely on costly optimization procedures. In this work, we propose to optimize over a class of diffusion matrices, based on one-dimensional collective variables (CVs), to help the dynamics explore the latent space defined by the CV. The form of the diffusion matrix is chosen in order to obtain an efficient effective diffusion in the latent space. We describe how this class of diffusion matrices can be constructed and learned during the simulation. We provide implementations of the Metropolis--Adjusted Langevin Algorithm and Riemann Manifold (Generalized) Hamiltonian Monte Carlo algorithms, and discuss numerical optimizations in the case when the CV depends only on a few degrees of freedom of the system. We illustrate the efficiency gains by computing mean transition durations between two metastable states of a dimer in a solvent.

• The paper's main contribution is modifying the diffusion matrix in Langevin dynamics to accelerate sampling convergence in high-dimensional spaces.
• It employs adaptive learning of diffusion parameters using MALA, RMHMC, and RMGHMC to reduce transition times between metastable states.
• Numerical experiments demonstrate that optimal tuning of the α parameter lowers rejection probabilities and computational costs significantly.

Improving Sampling by Modifying the Effective Diffusion

In the paper titled "Improving Sampling by Modifying the Effective Diffusion" (2410.00525), the authors investigate an approach to enhance the efficiency of sampling in high-dimensional spaces. They propose modifying the diffusion component in overdamped Langevin dynamics to accelerate convergence in Markov chain Monte Carlo (MCMC) methods—particularly in Bayesian inference and computational statistical physics.

Introduction to Modified Diffusion in Langevin Dynamics

The paper addresses the challenge of high-dimensional integral estimation commonly encountered in Bayesian inference and computational statistical physics. Overdamped Langevin dynamics represent a widely-used method, particularly applying Markov chain Monte Carlo algorithms such as the Metropolis-Adjusted Langevin Algorithm (MALA). The paper suggests introducing a non-constant diffusion matrix to these dynamics to enhance the rate of convergence to stationarity.

The proposed solution modifies the diffusion matrix based on one-dimensional collective variables (CVs), which are mappings that describe the system's state in reduced dimensions. By carefully designing the diffusion matrix, the authors aim to achieve a more efficient exploration of latent spaces, thereby accelerating convergence and overcoming computational barriers such as multimodality and anisotropy inherent in high-dimensional settings.

Figure 1: Graphical illustration of the diffusion $D_{\alpha}(q)$, modulating exploration using the optimal homogenized diffusion framework.

Implementation Methodology

Overdamped Langevin Dynamics and MALA

The implementation involves adapting standard overdamped Langevin dynamics, expressed as:

$$dq_t = \left(-\nabla V(q_t) dt + \sqrt{\frac{2}{\beta}} dW_t\right)$$

by introducing a position-dependent diffusion $D(q_t)$:

$$dq_t = \left(-D(q_t)\nabla V(q_t) + \beta^{-1}\nabla \cdot D(q_t)\right) dt + \sqrt{2\beta^{-1}D(q_t)^{1/2}} dW_t$$

This framework ensures faster exploration by modulating the diffusion according to the CV, enabling more efficient crossing of energy barriers.

Numerical Experiments

Implementing the proposed diffusion modifications using the MALA involves calculating factors related to the mean force and adaptive learning of diffusion coefficients. The MALA uses an Euler-Maruyama discretization alongside an accept/reject Metropolis-Hastings step. Key to efficient implementation is the use of collective variables involving a small number of components, minimizing computational overhead and optimizing storage.

In numerical experiments with a two-dimensional molecular system, the effectiveness of this approach was evident through reduced mean transition durations between metastable states, with optimal performance characterized by certain values of $\alpha$.

Figure 2: Mean force $F'$ vs. collective variable $\xi$, highlighting the role of the collective variable in shaping the diffusion path.

Optimizing Diffusion in Kinetic Hamiltonian Systems

The paper extends the methodology to kinetic systems via Riemann Manifold Hamiltonian Monte Carlo (RMHMC) and its generalized variant (RMGHMC), both employing altered Hamiltonian functions that introduce non-constant diffusion matrices. The resulting dynamics maintain the invariant measure while providing more tailored exploration paths.

Adaptive Learning

Adaptive methods allow for the diffusion parameters to be learned dynamically during sampling. This on-the-fly computation was demonstrated to achieve comparable efficiency to precomputed scenarios, showing promise for applications without pre-estimated diffusion parameters.

Numerical Results and Rejection Probabilities

Numerical experiments demonstrated significant performance improvements using optimized diffusion matrices, particularly in high-dimensional spaces. For both RMHMC and RMGHMC, adaptive learning of the diffusion parameters resulted in reduced transition times between metastable states compared to constant diffusion matrices.

Figure 3: Mean number of iterations to observe a transition as a function of $\Delta t$ for two values of $\alpha$, and for the constant diffusion.

Figure 4: Mean number of iterations to observe a transition as a function of $\Delta t$ for two values of $\alpha$ and for the constant diffusion.

A critical observation is that the choice of an optimal $\alpha$, a parameter controlling the strength of the position-dependent diffusion, significantly affects the rejection probabilities and the efficiency of the sampling algorithms.

Conclusion

The research paper "Improving Sampling by Modifying the Effective Diffusion" (2410.00525) explores the improvement of sampling processes in high-dimensional Bayesian inference and computational statistical physics through the modification of effective diffusion. It demonstrates the efficacy of adaptive techniques like RMHMC and RMGHMC in optimizing sampling efficiency, significantly reducing the computational costs associated with the convergence of Markov chain Monte Carlo estimators. Future research could explore extending these techniques to multidimensional collective variables, and further tuning of hyperparameters such as the friction parameter in RMGHMC for improved performance. The implications of these advancements are significant, offering enhanced exploration capabilities that can influence the future landscape of computational simulations in fields related to statistical physics and Bayesian inference.

Figure 5: Values of the collective variable as a function of the number of iterations.