Constrained Sampling

We propose ConicFlow, which frames constrained sampling from a flow matching model at inference time as an optimal control problem that balances deviation from the pretrained trajectory against reward, subject to hard terminal constraints. We discretize this problem and adapt receding-horizon control, optimizing only a short window of the trajectory at each step for computational tractability. We address the resulting sequence of receding-horizon subproblems with a receding-horizon primal-dual algorithm. Its dual multipliers capture the difficulty of satisfying each constraint and are refined throughout sampling, steering generation toward feasible final samples. We further propose a terminal-estimate variant that avoids accumulating local truncation errors by evaluating the reward and constraints at the predicted terminal point rather than at the end of the current horizon. Under standard regularity assumptions, we establish finite-horizon guarantees on suboptimality. Under convex-affine structure, we derive a sharper bound that scales linearly with omitted horizon length, and we show that the terminal-estimate variant is provably tighter when the flow dynamics change slowly. Experiments in both low- and high-dimensional settings with a recent large-scale flow matching model show that ConicFlow achieves high reward and strong constraint satisfaction while deviating less from the pretrained flow than recent baselines.

We consider the problem of learning to generate data while satisfying a prescribed sample-wise constraint with a user-specified probability of constraint satisfaction. Starting with a population-level constrained negative log-likelihood minimization and using a flow-matching (FM) parameterization, under regularity assumptions and sufficient expressivity of the velocity field family, we derive guarantees on the probability of constraint satisfaction for the corresponding primal-dual optimization algorithm. We provide a practical implementation of this approach, called Primal-Dual Flow Matching (PDFM), which has direct control over constraint satisfaction rate and requires only binary membership oracle feedback for constraint satisfaction, avoiding the need for differentiable distances, projections, or special structure of the constraint set, such as convexity. We show that PDFM improves constraint satisfaction over vanilla FM and achieves competitive performance against constraint-aware baselines while maintaining distributional fidelity. Our code is available at https://github.com/ZhengyanHuan/PDFM.

Anchored Langevin dynamics (ALD) is useful for non-smooth sampling where the density of the target distribution is possibly non-differentiable and heavy-tailed; reflected anchored Langevin dynamics (RALD) can sample possibly non-differentiable target density on a constrained domain. In this paper, we propose and study non-reversible anchored Langevin dynamics (NALD) for sampling possibly non-differentiable and heavy-tailed target density in the Euclidean space and the non-reversible reflected anchored Langevin dynamics (NRALD) for sampling possibly non-differentiable target density in the constrained space. Our construction adds a circulation drift generated by a possibly state-dependent divergence-free skew-symmetric matrix field and a stream potential. It preserves the target distribution without requiring derivatives of target density, admits a random-time-change representation, and applies both on the whole Euclidean space and on bounded domains with normal reflection. By breaking reversibility, we show that NALD and NRALD can converge to their target distributions faster than their reversible counterparts via finite-time non-asymptotic convergence analysis, a large deviations analysis and asymptotic variance reduction. Numerical experiments demonstrate the efficiency of the proposed algorithms.

First order Langevin algorithms for constrained sampling in machine learning, such as projected Langevin Monte Carlo which are based on discretizations of reflected Langevin dynamics, require differentiable log densities that limits their applicability. This paper introduces reflected anchored Langevin dynamics (RALD), a reflected diffusion that converges to non-differentiable targets on constrained domains. The method uses a smooth anchored reference potential and multiplies the drift and noise covariance of its reflected Langevin dynamics by the same state dependent scaling factor. Its Euler-Maruyama discretization with projection gives reflected anchored Langevin Monte Carlo (RALMC) algorithm. We prove explicit convergence bounds and iteration complexity for RALMC in the 2-Wasserstein distance to the target distribution. Numerical experiments are provided to illustrate the theoretical predictions and the empirical performance of the method.

Flow matching models excel at generative modeling, and many downstream applications require their samples to satisfy prescribed constraints, such as observed measurements and physical laws. However, existing constrained samplers often face a trade-off: enforcing constraints can substantially displace samples from the pretrained data distribution. To address this trade-off, we introduce MintFlow, a training-free constrained sampling framework that formulates constraint enforcement as a minimal intervention on the pretrained flow trajectory. MintFlow seeks the minimal perturbation of an intermediate flow state such that its subsequent evolution under the pretrained flow field satisfies the target constraint. By minimally perturbing the flow state while keeping the pretrained flow field unchanged, MintFlow enforces the constraint while minimizing unnecessary deviation from the pretrained distribution. An adjoint formulation yields a closed-form expression for this perturbation, eliminating expensive iterative optimization. Furthermore, MintFlow adaptively selects the intervention time to balance the required perturbation magnitude with its amplification by the remaining flow. Across a range of tasks in generative vision and physical system modeling, MintFlow achieves competitive constraint satisfaction while preserving the pretrained generative distribution substantially better than state-of-the-art constrained methods.

A common formalism for constraining the output of autoregressive text generation models involves lexical constraints, words or phrases which are required to occur in the generated text. DFA-constrained beam search and grid beam search are two widely used paradigms for decoding from autoregressive models while enforcing lexical constraints. As the former approach requires a number of forward passes exponential in the number of constraint tokens, it is often dispreferred to the latter, which requires only linearly many forward calls. However, while grid beam search achieves an exponential speedup, it does so in a manner which does not treat all of the constraints equally. In this paper, we demonstrate that grid beam search is biased to incorporate easier-to-satisfy constraints first, leaving harder constraints to the end of the sequence. This contrasts with DFA-constrained beam search, which exhibits no such bias. To address this shortcoming, we propose fair grid beam search, a modification to grid beam search which avoids this bias while still requiring only linearly many forward passes. Experimentally, we confirm grid beam search's bias on two constrained generation tasks, finding significant differences in how it orders constraint tokens as compared to DFA-constrained beam search and fair grid beam search. Furthermore, we find that fair grid beam search not only fixes grid beam search's bias, but finds higher-probability strings in the process.

We propose penalized nonreversible Langevin algorithms for sampling from $π(x)\propto e^{-f(x)}\mathbf 1_{\mathcal C}(x)$, where $\mathcal C\subset\mathbb R^d$ is a compact convex set. The algorithms combine a squared distance penalty with constant or compatible state dependent skew symmetric perturbations that preserve the penalized Gibbs distribution. For smooth, possibly nonconvex $f$, we derive nonasymptotic total variation bounds for the full gradient algorithm under a log Sobolev inequality. When unbiased stochastic gradients are available, we establish $2$-Wasserstein bounds under global contraction and Lipschitz conditions on the full drift in an adapted quadratic metric. For a fixed penalty parameter, the error relative to the penalized Gibbs distribution decays exponentially to an $\mathcal{O}(\sqrtη)$ neighborhood, where $η$ is the stepsize. We also bound the discrepancy between the penalized Gibbs distribution and the constrained target. In a two dimensional quadratic model, we establish nonreversible acceleration by tuning the skew perturbation to the curvature imbalance induced by penalization. With the target accuracy and smaller curvature fixed and initial Wasserstein distances uniformly bounded, tuning the skew perturbation improves the sufficient Euler iteration bound from linear to logarithmic in the curvature ratio. Numerical experiments evaluate the algorithms on constrained Bayesian regression, classification, neural networks, and truncated sampling, and examine the acceleration mechanism in a stochastic quadratic model.

Some of the sharpest challenges in sampling from the energy functions of physical systems arise at phase transitions, where the density of states changes abruptly and many sampling algorithms stall. Nested sampling is a particle method that traverses the density of states under a hard energy constraint and is known to be robust to such transitions, but its application in high dimension is limited by the difficulty of sampling under that constraint. In this work we introduce Quenched Ensemble Sampling, which generalises the hard constraint to a family of repulsive potentials at the energy boundary. This preserves the quenched path of monotonically decreasing energy while making the constrained target amenable to scalable gradient-based kernels. We demonstrate on synthetic models of phase transitions that our method estimates the marginal likelihood and draws posterior samples across a first-order transition where popular alternatives such as tempering fail. We apply the procedure to marginal likelihood estimation in Bayesian neural networks, enabling model comparison between network architectures. Finally, in a high-dimensional continuous lattice field theory, we show that this method traverses a first-order transition and estimates the partition function.

Many AI systems make decisions locally, even when every realized output must obey an additive conservation law, such as selecting exactly a fixed number of items. This constraint can be statistically invisible: small subsets of a balanced fixed-budget output look increasingly independent, yet communication-free coordinate-parallel generation needs exponentially many pre-shared plans, while a sequential exact sampler needs only logarithmic memory. We study product measures conditioned on additive conservation laws in the intermediate regime where one plan is selected before a fixed-order pass, every plan is a bounded-state stochastic executor whose support is entirely legal, and the mixture of plan laws approximates the target distribution in total variation. Our main result identifies the optimal asymptotic selector rate, up to constant factors, with the killed spectral profile of the conservation-difference walk. The resulting coordination cost decreases as an inverse power of live-state width, with an exponent determined by intrinsic conservation rank rather than alphabet size; the law extends to noncentral budgets and heterogeneous local scores. The converse is driven by a state-versus-resource-sum obstruction, while a rate-matching construction compiles discrepancy control into exact-support finite-state plans. Complementary results characterize block-parallel plan complexity and the benefit of programmable output order. Together, these results show precisely how online memory substitutes for front-loaded coordination in exact-support generation.

Constrained generative models aim to produce samples that satisfy complex feasibility constraints while remaining faithful to the data distribution. Existing constrained generation methods typically enforce constraints either through training-time optimization or sampling-time correction. Training-time optimization approaches optimize on states induced by the training distribution, which can differ substantially from those encountered during sampling. Sampling-time correction methods instead modify the sampling process at inference, introducing distribution shift and requiring expensive tuning, particularly for few-step sampling. We propose a fine-tuning framework that incorporates constraint guidance obtained through online rollout into the training process, which aligns training with sampling by differentiating through the fixed noise schedule used to numerically integrate the denoising process. This exposes the model to violations that arise along the denoising trajectory and aligns diffusion learning with the sampling process. Experiments across multiple tasks show that our method improves constraint satisfaction while maintaining competitive sampling quality compared to prior methods.

Generative models have emerged as scalable surrogates for physical simulation, yet they offer no guarantee that their outputs respect the conservation laws, boundary conditions, and nonlinear invariants that govern the underlying physics. Constrained sampling closes this gap, enforcing such constraints exactly at inference time without retraining, but at a computational cost: projection, correction and trajectory-optimization steps are repeated during sampling, with these steps becoming expensive for nonlinear constraints. Standard ML frameworks exacerbate this: their dense tensor algebra and limited sparse solver composability obscure the structure that physical constraints naturally induce, making efficient batched nonlinear optimization difficult to realize in practice. We address this bottleneck by exploiting the structure that sample-wise batching and local PDE couplings induce in the projection subproblems -- namely, block-sparse Jacobian and KKT systems -- exposing this structure using ExaModels.jl and solving the resulting sparse nonlinear programs with MadNLP.jl and GPU sparse factorization. Applied to Physics-Constrained Flow Matching (PCFM), on PDE benchmarks with linear, nonlinear, one-dimensional, and two-dimensional constraints, this approach accelerates nonlinear constraint projection while maintaining constraint satisfaction. These results show that sparse GPU nonlinear optimization is a practical foundation for constrained generative sampling in scientific machine learning.

Robots must generate trajectories that remain faithful to learned expert behavior while satisfying safety constraints and task-specific objectives specified only at inference time. We formulate constrained trajectory generation for pretrained diffusion and flow-matching policies as Bayesian posterior sampling, with the learned demonstration distribution as a prior and an inference-time, cost-derived likelihood tilting it toward feasible, optimal trajectories. To sample from this posterior without any retraining of the base policy, we leverage the Feynman--Kac corrector framework, originally formulated for diffusion models, and extend it to deterministic flow-matching policies. The result is a unified, inference-time, retraining-free sampler for diffusion and flow policies. We validate the approach on pretrained Diffusion Policy, GR00T-N1.6, and $π_{0.5}$ checkpoints across simulated and real-world manipulation tasks, including planning around non-convex obstacles introduced at inference time, and show improvements over the base $π_{0.5}$ on zero-shot tasks.

This paper develops constrained diffusion models with primal-dual inference (PDI) to sample from optimal distributions of entropy-regularized optimization problems with average constraints. We formalize constrained sampling in the Lagrangian dual domain, where the optimal distribution takes the form of a Gibbs distribution indexed by the optimal dual variable. Rather than estimating this dual multiplier before sampling and freezing it throughout generation, PDI jointly infers the optimal primal distribution and its parametrizing dual variable. Each reverse diffusion step denoises using the score field associated with the current multiplier and then updates the multiplier through dual ascent using the estimated constraint violation of the denoised samples. To enable this conditional score field, we train a single dual-conditioned score network over the family of Gibbs distributions induced by the dual variables encountered during inference. We prove that the dual iterates converge to a neighborhood of the dual optimum, and that the generated distribution approaches that of an oracle sampler that knows the optimal multiplier, up to a bias floor. We evaluate PDI on constrained sampling from a mixture of Gaussians, wireless resource allocation, and portfolio management.

Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a hard constraint (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty. We show that this widely adopted recipe samples the wrong distribution. Conditioning a generative prior on a hard PDE constraint is conditioning on a measure-zero manifold -- an operation that is intrinsically ambiguous (the Borel--Kolmogorov paradox) and whose physically correct resolution, the small-residual-noise limit, carries a co-area (Fixman) Jacobian factor $[det(JJ^{\top})]^{-1/2}$ that projection- and guidance-based methods silently omit. We make the bias precise, show that it grows with the heterogeneity of the constraint sensitivity, and validate it on controlled problems against an i.i.d. ground-truth arbiter. The omitted factor is not a second-order detail: removing it inflates the posterior error to $20\times$ the sampling-noise floor; minimal-displacement projection (as in PCFM) is biased at $9\times$ the floor; and a naive scalar reweighting does not fix it. We introduce CoCoS, a measure-aware constrained sampler that targets the correct co-area posterior, and show that it matches the gold-standard posterior to within sampling noise. Our results imply that satisfying the physics'' is not the same as sampling the posterior,'' and give a principled correction for uncertainty-aware scientific inference.

Generations from large language models often fail to conform to desired constraints such as JSON schema. Existing locally constrained decoding (LCD) approaches enforce constraints by myopically masking out next tokens, resulting in biased sampling and degradation in performance. Recent work uses sequential Monte Carlo (SMC) methods to mitigate such biases, but designing effective proposal distributions or potential functions remains a key challenge. In this work, we propose a generic approach to construct proposals and potentials for SMC sampling from $p_{\mathrm{lm}}( \cdot \mid \mathrm{constraint})$. First, we show that constraints specified as finite automata can be tensorized for efficient execution on GPUs, which we use to construct globally constrained decoding (GCD) proposals. In addition, leveraging the fact that tensorized finite automata share the same circuit structure as hidden Markov models, we circuit-multiply them to obtain the probabilistic GCD (P-GCD) proposals encoding both logical and probabilistic information about the target distributions. We evaluate (P-)GCD on the tasks of function calling, keyword-based generation, and SQL generation. Experiments show that under the same SMC sampling setup, compared to LCD proposals, (P-)GCD converges faster to the target distribution with significantly fewer particles.

Deep generative models provide state-of-the-art performance across a wide array of applications, with recent studies showing increasing applicability for science and engineering. Despite a growing corpus of literature focused on the integration of physics-based constraints into the generation process, existing approaches fail to enforce strict constraint satisfaction while maintaining sample quality. In particular, training-free constrained sampling methods, while providing per-sample feasibility guarantees, introduce a fundamental mismatch between the training objective and the constrained sampling procedure, often leading to performance degradation. Identifying this training-sampling misalignment as a central limitation of current constrained generative modeling approaches, this paper proposes Constraint-Aware Flow Matching, a novel end-to-end framework that explicitly incorporates constraint projections into the training objective. By aligning the model's learned dynamics with the constrained sampling process, the proposed method mitigates distributional shift induced by projection-based corrections, enabling high-quality constrained generation. The proposed approach is evaluated on three challenging real-world benchmarks, illustrating the generality and efficacy of the method.

Sampling from constrained distributions has a wide range of applications, including in Bayesian optimization and robotics. Prior work establishes convergence and feasibility guarantees for constrained sampling, but assumes that the feasible set is connected. However, in practice, the feasible set often decomposes into multiple disconnected components, which makes efficient sampling under constraints challenging. In this paper, we propose MAnifold Sampling via Entropy Maximization (MASEM) for sampling on a manifold with an unknown number of disconnected components, implicitly defined by smooth equality and inequality constraints. The presented method uses a resampling scheme to maximize the entropy of the empirical distribution based on k-nearest neighbor density estimation. We show that, in the mean field, MASEM decreases the KL-divergence between the empirical distribution and the maximum-entropy target exponentially in the number of resampling steps. We instantiate MASEM with multiple local samplers and demonstrate its versatility and efficiency on synthetic and robotics-based benchmarks. MASEM enables fast and scalable mixing across a range of constrained sampling problems, improving over alternatives by an order of magnitude in Sinkhorn distance with competitive runtime.

Hard constraints in generative sampling are typically enforced by projection, applied either once at the end of sampling or after every update. This binary framing overlooks a fundamental issue: projection changes the distribution of states which future updates depend on. As a result, delayed projection can produce samples that are feasible but inconsistent with the intended sampling dynamics, even after final projection. We formalize constraint enforcement as a correction scheduling problem over the generative rollout. Using one-step constraint defect as a local signal of geometric mismatch, we introduce adaptive correction scheduling, a state-dependent policy that allocates projection budget to the steps that most strongly perturb the trajectory. Terminal and stepwise projection arise as limiting cases of this family. Across controlled manifold rollouts and a learned projected diffusion sampler, adaptive scheduling improves the cost-accuracy frontier at matched projection budgets, recovering 71.2% of full stepwise benefit with 75% fewer corrections. These results show that constraint timing is a first-class design variable in generative sampling, and that enforcing feasibility alone is insufficient to preserve the intended constrained sampling dynamics.

Discrete diffusion models generate structured sequences by progressively unmasking tokens, but enforcing global property constraints during generation remains an open challenge. We propose primal-dual guided decoding, an inference-time method that formulates constrained generation as a KL-regularised optimisation problem and solves it online via adaptive Lagrangian multipliers. At each denoising step, the method modifies token logits through an additive, constraint-dependent bias, with multipliers updated by mirror descent based on constraint violation. The bias arises as the optimal KL-regularised projection of the constraint, so the constrained distribution remains as close as possible to the model's unconstrained distribution while still satisfying the constraint. The method requires no retraining and no additional model evaluations beyond standard sampling, supports multiple simultaneous constraints, and provides formal bounds on constraint violation. We evaluate our approach on topical text generation, molecular design, and music playlist generation, showing that a single algorithm instantiated via domain-specific scoring functions improves constraint satisfaction while preserving relevant domain-specific quality metrics.

We propose Decentralized Proximal Stochastic Gradient Langevin Dynamics (DE-PSGLD), a decentralized Markov chain Monte Carlo (MCMC) algorithm for sampling from a log-concave probability distribution constrained to a convex domain. Constraints are enforced through a shared proximal regularization based on the Moreau-Yosida envelope, enabling unconstrained updates while preserving consistency with the target constrained posterior. We establish non-asymptotic convergence guarantees in the 2-Wasserstein distance for both individual agent iterates and their network averages. Our analysis shows that DE-PSGLD converges to a regularized Gibbs distribution and quantifies the bias introduced by the proximal approximation. We evaluate DE-PSGLD for different sampling problems on synthetic and real datasets. As the first decentralized approach for constrained domains, our algorithm exhibits fast posterior concentration and high predictive accuracy.

Generative modeling within constrained sets is essential for scientific and engineering applications involving physical, geometric, or safety requirements (e.g., molecular generation, robotics). We present a unified framework for constrained diffusion models on generic nonconvex feasible sets $Σ$ that simultaneously enforces equality and inequality constraints throughout the diffusion process. Our framework incorporates both overdamped and underdamped dynamics for forward and backward sampling. A key algorithmic innovation is a computationally efficient landing mechanism that replaces costly and often ill-defined projections onto $Σ$, ensuring feasibility without iterative Newton solves or projection failures. By leveraging underdamped dynamics, we accelerate mixing toward the prior distribution, effectively alleviating the high simulation costs typically associated with constrained diffusion. Empirically, this approach reduces function evaluations and memory usage during both training and inference while preserving sample quality. On benchmarks featuring equality and mixed constraints, our method achieves comparable sample quality to state-of-the-art baselines while significantly reducing computational cost, providing a practical and scalable solution for diffusion on nonconvex feasible sets.

Constrained generative modeling is fundamental to applications such as robotic control and autonomous driving, where models must respect physical laws and safety-critical constraints. In real-world settings, these constraints rarely take the form of simple linear inequalities, but instead complex feasible regions that resemble road maps or other structured spatial domains. We propose a constrained generation framework that generates samples directly within such feasible regions while preserving realism. Our method fine-tunes a pretrained generative model to enforce constraints while maintaining generative fidelity. Experimentally, our method exhibits characteristics distinct from existing fine-tuning and training-free constrained baselines, revealing a new compromise between constraint satisfaction and sampling quality.

Flow-based generative models, such as diffusion models and flow matching models, have achieved remarkable success in learning complex data distributions. However, a critical gap remains for their deployment in safety-critical domains: the lack of formal guarantees that generated samples will satisfy hard constraints. We propose a safety filtering framework that acts as an online shield for any pre-trained generative model. Our key insight is to cooperate with the generative process rather than override it. We define a constricting safety tube that is relaxed at the initial noise distribution and progressively tightens to the target safe set at the final data distribution, mirroring the coarse-to-fine structure of the generative process itself. By characterizing this tube via Control Barrier Functions (CBFs), we synthesize a feedback control input through a convex Quadratic Program (QP) at each sampling step. As the tube is loosest when noise is high and intervention is cheapest in terms of control energy, most constraint enforcement occurs when it least disrupts the model's learned structure. We prove that this mechanism guarantees safe sampling in discrete-time. The minimum-norm control synthesized at each step minimizes the per-step contribution to the KL divergence between the safe and original distributions. Across all experiments, we observe 100% constraint satisfaction. Our framework applies to any pre-trained flow-based sampling scheme requiring no retraining or architectural modifications. We validate the approach across constrained image generation, physically-consistent trajectory sampling, and safe robotic manipulation policies, achieving 100% constraint satisfaction while preserving semantic fidelity.

We study two log-concave sampling problems: constrained sampling and composite sampling. First, we consider sampling from a target distribution with density proportional to $\exp(-f(x))$ supported on a convex set $K \subset \mathbb{R}^d$, where $f$ is convex. The main challenge is enforcing feasibility without degrading mixing. Using an epigraph transformation, we reduce this task to sampling from a nearly uniform distribution over a lifted convex set in $\mathbb{R}^{d+1}$. We then solve the lifted problem using a proximal sampler. Assuming only a separation oracle for $K$ and a subgradient oracle for $f$, we develop an implementation of the proximal sampler based on the cutting-plane method and rejection sampling. Unlike existing constrained samplers that rely on projection, reflection, barrier functions, or mirror maps, our approach enforces feasibility using only minimal oracle access, resulting in a practical and unbiased sampler without knowing the geometry of the constraint set. Second, we study composite sampling, where the target is proportional to $\exp(-f(x)-h(x))$ with closed and convex $f$ and $h$. This composite structure is standard in Bayesian inference with $f$ modeling data fidelity and $h$ encoding prior information. We reduce composite sampling via an epigraph lifting of $h$ to constrained sampling in $\mathbb{R}^{d+1}$, which allows direct application of the constrained sampling algorithm developed in the first part. This reduction results in a double epigraph lifting formulation in $\mathbb{R}^{d+2}$, on which we apply a proximal sampler. By keeping $f$ and $h$ separate, we further demonstrate how different combinations of oracle access (such as subgradient and proximal) can be leveraged to construct separation oracles for the lifted problem. For both sampling problems, we establish mixing time bounds measured in Rényi and $χ^2$ divergences.

Generative models have enjoyed widespread success in a variety of applications. However, they encounter inherent mathematical limitations in modeling distributions where samples are constrained by equalities, as is frequently the setting in scientific domains. In this work, we develop a computationally cheap, mathematically justified, and highly flexible distributional modification for combating known pitfalls in equality-constrained generative models. We propose perturbing the data distribution in a constraint-aware way such that the new distribution has support matching the ambient space dimension while still implicitly incorporating underlying manifold geometry. Through theoretical analyses and empirical evidence on several representative tasks, we illustrate that our approach consistently enables data distribution recovery and stable sampling with both diffusion models and normalizing flows.