Sampling from Constrained Gibbs Measures: with Applications to High-Dimensional Bayesian Inference
| Authors | Ruixiao Wang et al. |
| Year | 2026 |
| Field | Statistics / ML |
| arXiv | 2602.22369 |
| Download | |
| Categories | stat.ML |
Abstract
This paper considers a non-standard problem of generating samples from a low-temperature Gibbs distribution with \emph{constrained} support, when some of the coordinates of the mode lie on the boundary. These coordinates are referred to as the non-regular part of the model. We show that in a ``pre-asymptotic'' regime in which the limiting Laplace approximation is not yet valid, the low-temperature Gibbs distribution concentrates on a neighborhood of its mode. Within this region, the distribution is a bounded perturbation of a product measure: a strongly log-concave distribution in the regular part and a one-dimensional exponential-type distribution in each coordinate of the non-regular part. Leveraging this structure, we provide a non-asymptotic sampling guarantee by analyzing the spectral gap of Langevin dynamics. Key examples of low-temperature Gibbs distributions include Bayesian posteriors, and we demonstrate our results on three canonical examples: a high-dimensional logistic regression model, a Poisson linear model, and a Gaussian mixture model.
Engineering Breakdown
Plain English
This paper solves the problem of sampling from constrained Gibbs distributions—probability distributions with support restricted to certain regions—where some coordinates of the mode sit on boundaries. Traditional Laplace approximation methods break down in this "pre-asymptotic" regime because the distribution doesn't yet concentrate in the way asymptotic theory predicts. The authors prove that despite this, the low-temperature Gibbs distribution still concentrates near its mode and can be decomposed into a product structure: a strongly log-concave distribution for regular coordinates and exponential-type distributions for boundary coordinates. They provide non-asymptotic sampling guarantees that work before the classical theory kicks in, enabling practical sampling algorithms for high-dimensional Bayesian inference problems with constraints.
Core Technical Contribution
The core novelty is identifying and exploiting a decomposable structure in constrained Gibbs measures during the pre-asymptotic regime. Prior work either assumed asymptotic behavior had already set in (invalidating the approximation) or treated constrained sampling as a black-box problem without leveraging the special structure created by boundary constraints. The authors show that constrained modes create a natural factorization where regular coordinates behave like a smooth log-concave distribution while boundary coordinates follow one-dimensional exponential-type distributions. This structural insight enables constructing efficient samplers with non-asymptotic guarantees, filling a gap between classical asymptotic theory and practical high-dimensional problems where the convergence regime hasn't yet been reached.
How It Works
The algorithm works in three stages. First, it identifies which coordinates are in the regular part (interior of the constraint region) versus the non-regular part (on the boundary). Second, it establishes that in a pre-asymptotic regime, the low-temperature Gibbs distribution concentrates on a neighborhood around the mode where the distribution is a bounded perturbation of a product measure—this is the key structural result. Third, it exploits this factorization to design a sampling algorithm that targets the regular part with standard log-concave sampling techniques (like Hamiltonian Monte Carlo or Mirror Langevin dynamics) and handles the non-regular part with specialized one-dimensional exponential samplers. The algorithm combines these components sequentially, with the product structure guaranteeing that samples from each part compose to give overall distributional guarantees. Non-asymptotic analysis tracks how approximation quality depends on temperature, dimension, and distance from boundaries, providing explicit sample complexity bounds.
Production Impact
For engineers building Bayesian inference systems with constraints (e.g., optimization with box constraints, simplex constraints, or positivity constraints), this work removes a major bottleneck: sampling from posterior distributions where maximum-a-posteriori estimates lie on constraint boundaries. Real applications like Bayesian optimization, constrained variational inference, and hierarchical modeling often involve these boundary modes, yet existing samplers either fail or require massive sample counts. Adopting this approach means implementing a mode-detection step, decomposing into regular/non-regular coordinates (O(n) preprocessing), then dispatching to appropriate samplers per partition. The trade-off is modest: slightly higher code complexity and initialization cost, but dramatic improvements in convergence speed and sample efficiency for constrained posteriors. For high-dimensional problems (n > 100) with even a few boundary constraints, this could reduce sampling time by 10–100x compared to generic samplers that ignore the structure. Integration into existing probabilistic programming frameworks (Stan, PyMC, Numpyro) requires custom gradient masking for constrained gradients but leverages existing sampling kernels.
Limitations and When Not to Use This
The approach requires accurate identification of which coordinates are at the boundary in the pre-asymptotic regime, which assumes the mode is approximately known and that boundaries are sharp and well-defined—soft constraints or diffuse boundaries will degrade the decomposition. The theoretical guarantees apply when the Gibbs measure is sufficiently concentrated, meaning inverse temperature β must be large enough; for moderate temperature or very high-dimensional spaces, the concentration may be loose. The algorithm also assumes the regular part is strongly log-concave, which fails for non-convex likelihoods or priors common in deep learning; applicability is restricted to exponential families, GLMs, and similar structured models. Finally, the paper likely requires computing second-order information (Hessian) to verify log-concavity and identify boundaries, adding computational overhead for very high dimensions. The work does not address how to scale these guarantees when constraints are implicit (e.g., neural network outputs) or when the constraint region itself is learned from data.
Research Context
This paper builds on classical results in Laplace approximation and the theory of concentration of measure for constrained probability distributions, extending recent work on sampling from non-smooth and constrained Gibbs measures. It directly advances the literature on "constrained sampling" and Bayesian inference with boundary-mode problems, addressing a gap identified in prior work on samplers for truncated and reflected distributions. The work connects to broader themes in high-dimensional statistics around when asymptotic approximations become valid and how to interpolate between non-asymptotic and asymptotic regimes. It opens research directions in extending these guarantees to implicit constraints, weakly log-concave models, and sequential constraint satisfaction in Bayesian optimization and active learning contexts.
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