The No-Underrun Sampler: A Locally-Adaptive, Gradient-Free MCMC Method (2501.18548v2)

Abstract: In this work, we introduce the No-Underrun Sampler (NURS), a locally-adaptive, gradient-free Markov chain Monte Carlo method that blends ideas from Hit-and-Run and the No-U-Turn Sampler. NURS dynamically adapts to the local scale of the target distribution without requiring gradient evaluations, making it especially suitable for applications where gradients are unavailable or costly. We establish key theoretical properties, including reversibility, formal connections to Hit-and-Run and Random Walk Metropolis, Wasserstein contraction comparable to Hit-and-Run in Gaussian targets, and bounds on the total variation distance between the transition kernels of Hit-and-Run and NURS. Empirical experiments, supported by theoretical insights, illustrate the ability of NURS to sample from Neal's funnel, a challenging multi-scale distribution from Bayesian hierarchical inference.

• The paper introduces the No-Underrun Sampler (NURS), a novel locally-adaptive, gradient-free MCMC method fusing Hit-and-Run and NUTS elements for robust sampling in complex distributions.
• NURS adapts its exploration based on the local scale of the target distribution using a unique stopping condition, employs exact categorical state selection along sampled lines, and provides a practical generalized Hit-and-Run implementation.
• The method demonstrates Wasserstein contraction properties in Gaussian settings, is empirically validated on challenging distributions like Neal's funnel, and expands the capability of MCMC methods in gradient-free domains.

An Overview of the No-Underrun Sampler: A New Approach for Gradient-Free MCMC Methodology

The paper introduces a novel gradient-free MCMC method, the No-Underrun Sampler (NURS), which demonstrates an innovative fusion of the Hit-and-Run method and No-U-Turn Sampler (NUTS) elements. NURS notably refrains from requiring gradient evaluations, which makes it particularly apt for situations where gradients are costly or unavailable, such as in domains heavily reliant on complex simulations.

Key Contributions

The main contributions of this paper involve adapting principles from well-established MCMC methods, NUTS and Hit-and-Run, and forming a robust method that enables dynamic adaptation to the local geometry of a target distribution. Here are some specific highlights:

1. Locally Adaptive Strategy: NURS adjusts the orbit length based on the local scale of the target, guided by a stopping condition. This locally adaptive feature minimizes the potential for inefficiencies in exploring high-dimensional distributions.
2. Categorical State Selection: Following the construction of an orbit, the approach employs an exact evaluation of the target density along the orbit using categorical sampling, adding precision to the sampling process.
3. Implemented Hit-and-Run: NURS offers a practical way to implement the generalized Hit-and-Run, a method renowned for its coordinate-free exploration but often found computationally infeasible.
4. Theoretical Guarantees: The established Wasserstein contraction in Gaussian targets provides a solid theoretical foundation, showing comparable performance to Hit-and-Run. Furthermore, NURS integrates well into existing tuning guidelines associated with both Hit-and-Run and Random Walk Metropolis, offering explicit methods for parameter selection.
5. Empirical Validation: The paper demonstrates the method’s efficacy via empirical testing on Neal's funnel, a complex distribution commonly posing challenges in Bayesian hierarchical inference. NURS performs robustly, affirming its capability to handle such nuanced tasks.

Detailed Analysis

Wasserstein Contraction

The paper extends prior analyses on contraction properties in Gaussian settings by employing coupling techniques. NURS exhibits Wasserstein contraction in these environments, thereby inheriting the advantageous properties of Hit-and-Run. The authors employ careful coupling constructions to provide rigorous Wasserstein bounds, grounding their approach in solid mathematical theory.

Orbit Construction

Central to NURS lies its novel orbit construction mechanism, which uses locally adaptive stopping rules to capture significant portions of the target distribution. This is effectively linked to the No-Underrun stopping condition, a tailored approach that ensures the orbit spans a significant mass of the target’s restriction to lines.

Connections to Established Methods

NURS’s resemblance to the Hit-and-Run algorithm yields intuitive tunings for its parameters, effectively bridging theoretical insights with practical applicability. The paper highlights two critical parameters: the threshold ($\epsilon$) in the No-Underrun condition, influencing the algorithm's adaptability to local geometry, and the lattice spacing ($h$), which relates the method to Random Walk Metropolis in terms of acceptance ratios.

Implications and Future Directions

The development of NURS signifies an important step in broadening the scope of gradient-free MCMC methods capable of effective sampling from high-dimensional, multiscale target distributions. Its implications extend across various fields reliant on computational Bayesian methods, such as biology, physics, and engineering simulations.

Looking forward, future research directions may explore enhancing parallel computation strategies inherent in the orbit evaluations, adaptability to dynamically adjust hyperparameters such as orbit length analogous to tuning step sizes in HMC frameworks, or expanding empirical studies across a broader set of challenging distribution scenarios.

In conclusion, the No-Underrun Sampler establishes a promising direction within the MCMC community for robust sampling in complex, gradient-free contexts. Its theoretical sophistication, combined with practical performance, offers a compelling case for further exploration and integration into computational toolsets.