Sampling from constrained statistical distributions is a fundamental task in various fields including Bayesian statistics, computational chemistry, and statistical physics. This article considers sampling from a constrained distribution that is described by an unconstrained density, as well as additional equality and/or inequality constraints, which often make the constraint set nonconvex. Existing methods struggle in the presence of such nonconvex constraints, as they rely on projections, which are computationally expensive or intractable, are specialized to either inequality or equality constraints, and often lack rigorous quantitative convergence guarantees. In this paper, we introduce Overdamped Langevin with LAnding (OLLA), a new framework that can design overdamped Langevin dynamics accommodating both nonlinear equality and inequality constraints. The proposed dynamics also deterministically corrects trajectories along the normal direction of the constraint surface, thus obviating the need for explicit projections. We show that, under suitable regularity conditions on the target density and the feasible set Σ Rd, OLLA converges exponentially fast in 2-Wasserstein distance to the constrained target density ρΣ(x) exp( f(x))dσΣ. Lastly, through experiments, we demonstrate the efficiency of OLLA compared to known constrained Langevin algorithms and their slack variable variants, highlighting its favorable computational cost and fast empirical mixing.1

Deep learning models are increasingly deployed in safety-critical tasks where predictions must satisfy hard constraints, such as physical laws, fairness requirements, or safety limits. However, standard architectures lack built-in mechanisms to enforce such constraints, and existing approaches based on regularization or projection are often limited to simple constraints, computationally expensive, or lack feasibility guarantees. This paper proposes a model-agnostic framework for enforcing input-dependent linear equality and inequality constraints on neural network outputs. The architecture combines a task network trained for prediction accuracy with a safe network trained using decision rules from the stochastic and robust optimization literature to ensure feasibility across the entire input space. The final prediction is a convex combination of the two subnetworks, guaranteeing constraint satisfaction during both training and inference without iterative procedures or runtime optimization. We prove that the architecture is a universal approximator of constrained functions and derive computationally tractable formulations based on linear decision rules. Empirical results on benchmark regression tasks show that our method consistently satisfies constraints while maintaining competitive accuracy and low inference latency.

Diffusion models (DMs) have emerged as powerful tools for modeling complex data distributions and generating realistic new samples. Over the years, advanced architectures and sampling methods have been developed to make these models practically usable. However, certain synthesis process decisions still rely on heuristics without a solid theoretical foundation. In our work, we offer a novel analysis of the DM's inference process, introducing a comprehensive frequency response perspective. Specifically, by relying on Gaussianity assumption, we present the inference process as a closed-form spectral transfer function, capturing how the generated signal evolves in response to the initial noise. We demonstrate how the proposed analysis can be leveraged to design a noise schedule that aligns effectively with the characteristics of the data. The spectral perspective also provides insights into the underlying dynamics and sheds light on the relationship between spectral properties and noise schedule structure. Our results lead to scheduling curves that are dependent on the spectral content of the data, offering a theoretical justification for some of the heuristics taken by practitioners.

Sampling from constrained statistical distributions is a fundamental task in various fields including Bayesian statistics, computational chemistry, and statistical physics. This article considers the cases where the constrained distribution is described by an unconstrained density, as well as additional equality and/or inequality constraints, which often make the constraint set nonconvex. Existing methods for nonconvex constraint set $\Sigma \subset \mathbb{R}^d$ defined by equality or inequality constraints commonly rely on costly projection steps. Moreover, they cannot handle equality and inequality constraints simultaneously as each method only specialized in one case. In addition, rigorous and quantitative convergence guarantee is often lacking. In this paper, we introduce Overdamped Langevin with LAnding (OLLA), a new framework that can design overdamped Langevin dynamics accommodating both equality and inequality constraints. The proposed dynamics also deterministically corrects trajectories along the normal direction of the constraint surface, thus obviating the need for explicit projections. We show that, under suitable regularity conditions on the target density and $\Sigma$, OLLA converges exponentially fast in $W_2$ distance to the constrained target density $\rho_\Sigma(x) \propto \exp(-f(x))d\sigma_\Sigma$. Lastly, through experiments, we demonstrate the efficiency of OLLA compared to projection-based constrained Langevin algorithms and their slack variable variants, highlighting its favorable computational cost and reasonable empirical mixing.

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An augmented Lagrangian (AL) can convert a constrained optimization problem into a sequence of simpler (e.g., unconstrained) problems which are then usually solved with local solvers. Recently, surrogate-based Bayesian optimization (BO) sub-solvers have been successfully deployed in the AL framework for a more global search in the presence of inequality constraints; however a drawback was that expected improvement (EI) evaluations relied on Monte Carlo. Here we introduce an alternative slack variable AL, and show that in this formulation the EI may be evaluated with library routines. The slack variables furthermore facilitate equality as well as inequality constraints, and mixtures thereof. We show our new slack ALBO compares favorably to the original. Its superiority over conventional alternatives is reinforced on several new mixed constraint examples.

While successful, these methods are mainly designed for sampling from unconstrained domains.

In contrast, we directly evaluate the matrix equations which arise from applying variable elimination on the Lagrange multiplier terms in the (differential) KKT system.

Recent advances in constrained reinforcement learning (RL) have endowed reinforcement learning with certain safety guarantees.