Sampling Methods

Post-training alignment often reduces LLM diversity, leading to a phenomenon known as mode collapse. Unlike prior work that attributes this effect to algorithmic limitations, we identify a fundamental, pervasive data-level driver: typicality bias in preference data, whereby annotators systematically favor familiar text as a result of well-established findings in cognitive psychology. We formalize this bias theoretically, verify it empirically on preference datasets, and show that it plays a central role in mode collapse. Motivated by this analysis, we introduce Verbalized Sampling (VS), a simple, training-free prompting strategy to circumvent mode collapse. VS prompts the model to verbalize a probability distribution over a set of responses (e.g., "Generate 5 jokes about coffee and their corresponding probabilities"), which relieves the pressure to produce a single "typical" answer. Experiments show that VS significantly improves performance across creative writing (poems, stories, jokes), social dialogue simulation, synthetic data generation, and open-ended QA, without sacrificing safety and factual accuracy. For instance, in creative writing, VS increases diversity by 1.6-2.1x compared to direct prompting. We further observe an emergent trend that more capable models benefit more from VS. In sum, our work provides a new data-centric perspective on mode collapse and a practical inference-time remedy that helps unlock pre-trained generative diversity.

Despite the remarkable fidelity of generative models, they frequently suffer from mode collapse. Existing strategies for enhancing diversity predominantly focus on intervening during the generation trajectory. We identify a critical oversight that the standard Gaussian initialization often causes trajectories to collapse into dominant modes because it is agnostic to the guidance potential landscape. In this work, we formulate selecting the initial noise from a *guidance potential posterior*, which effectively re-weights the prior towards diversity-rich regions. To sample from this distribution efficiently, we introduce *Diversity-inducing Initialization* (DivIn), which leverages Langevin dynamics to actively navigate the initialization landscape, steering initial noise away from collapsing regions while anchoring them to the valid data manifold. Our method serves as an inference-time diversity enhancement compatible with both diffusion and flow matching models. Extensive experiments show that DivIn exhibits a superior performance in both class-to-image and text-to-image scenarios. Furthermore, we highlight that as DivIn is orthogonal to trajectory-based methods, combining them significantly expands the diversity-quality Pareto frontier beyond what either achieves in isolation.

We study a widely used Bayesian optimization method, Gaussian process Thompson sampling (GP-TS), under the assumption that the objective function is a sample path from a GP. Compared with the GP upper confidence bound (GP-UCB) with established high-probability and expected regret bounds, most analyses of GP-TS have been limited to expected regret. Moreover, whether the recent analyses of GP-UCB for the lenient regret and the improved cumulative regret upper bound can be applied to GP-TS remains unclear. To fill these gaps, this paper shows several regret bounds: (i) a regret lower bound for GP-TS, which implies that GP-TS suffers from a polynomial dependence on $1/\delta$ with probability $\delta$, (ii) an upper bound of the second moment of cumulative regret, which directly suggests an improved regret upper bound on $\delta$, (iii) expected lenient regret upper bounds, and (iv) an improved cumulative regret upper bound on the time horizon $T$. Along the way, we provide several useful lemmas, including a relaxation of the necessary condition from recent analysis to obtain improved regret upper bounds on $T$.

Diffusion posterior sampling conditions diffusion priors on measurements, but data-consistency updates are typically scaled by hand-tuned guidance weights and can destabilize sampling under stiff, operator-dependent curvature. We replace scalar guidance with a per-noise-level damped Gauss--Newton correction computed in diffusion-state coordinates. The correction pulls likelihood gradients back through the denoiser, uses a one-sided curvature model that avoids forward denoiser Jacobians, and applies diffusion-calibrated rank-one damping aligned with the denoiser residual. Each correction is solved with matrix-free GMRES using automatic differentiation, and sampling proceeds with a variance-preserving Langevin transition with a closed-form drift/noise split. On FFHQ and ImageNet across inverse problems, it achieves competitive PSNR/SSIM/LPIPS while running markedly faster than most of the compared baselines; on accelerated MRI reconstruction, it achieves the best PSNR/SSIM among the compared baselines. Code is available at https://github.com/Seunghyeok0715/CLAMP

We consider the problem of sampling compositional and discrete objects from a given unnormalized posterior distribution. Notably, recent studies have shown that this problem can be efficiently solved by learning a deterministic Markov Decision Process (MDP) that progressively builds each object in proportion to the posterior. In this work, however, we demonstrate that the Markovian assumption can both hamper signal propagation during training and catastrophically reduce the learned sampler's expressivity due to state aliasing. To address these issues, we propose lifting the MDP with a learnable latent dynamics that allows the underlying policy to depend on the entire past trajectory---and not only on the current state. In view of this, we refer to the resulting method as \emph{path-dependent discrete amortized inference}. Importantly, we provably extend existing learning algorithms for amortized samplers to our setting. In experiments on standard benchmark problems, we also show that our approach often leads to faster learning convergence and improved state space exploration relatively to prior techniques.

Diffusion and flow policies are gaining prominence in online reinforcement learning (RL) due to their expressive power, yet training them efficiently remains a critical challenge. A fundamental difficulty that distinguishes online RL from standard generative modeling is the lack of direct samples from the target Boltzmann distribution defined by the Q-function. To address this, two seemingly distinct families of methods have been proposed for diffusion policies: a noise-expectation family, which uses a weighted average of noise as the training target, and a gradient-expectation family, which employs a weighted average of Q-function gradients. However, it remains unclear how these objectives are formally related, or whether they can be synthesized into a more general formulation. In this paper, we propose a unified framework, reverse flow matching (RFM), which rigorously addresses the problem of training diffusion and flow models without direct target samples. By adopting a reverse inferential perspective, we formulate the training target as a posterior mean estimation problem given an intermediate noisy sample. Crucially, we introduce Langevin Stein operators to construct zero-mean control variates, deriving a general class of estimators that share the same expectation. We show that existing noise-expectation and gradient-expectation methods are simply two specific instances within this broader class. This unified view yields two key advancements: it extends the capability of targeting Boltzmann distributions from diffusion to flow policies, and it enables the principled combination of Q-value and Q-gradient information to form an effective estimator, thereby improving training efficiency and stability. We instantiate RFM to train a flow policy in online RL and demonstrate improved performance on continuous-control benchmarks compared to diffusion policy baselines.

Several important learning tasks can be formulated as minimizing an entropy-regularized objective over an appropriate space of probability distributions. Mean-field Langevin dynamics (MFLD) facilitate computation in this general context, casting the minimizer as the invariant distribution of a McKean--Vlasov process, which can be numerically discretized using $N$ particles and thus simulated. However, simulating this interacting particle system has computational complexity $\mathcal{O}(N^2)$. Motivated by recent research into \emph{kernel thinning}, we propose \texttt{KT-MFLD}, in which each particle interacts only with a coreset of size $\mathcal{O}(N^{\frac{1}{2}})$. \texttt{KT-MFLD} thus reduces the computational complexity to $\mathcal{O}(N^{\frac{3}{2}})$ while, under mild regularity conditions, achieving the same convergence guarantees (up to logarithmic factors) as MFLD. Our theoretical analysis is empirically confirmed on tasks including the training of student-teacher neural networks, quantization with maximum mean discrepancy, and computation of predictively-oriented posteriors in a post-Bayesian framework.

A common approach to perform PCA on probability measures is to embed them into a Hilbert space where standard functional PCA techniques apply. While convergence rates for estimating the embedding of a single measure from $m$ samples are well understood, the literature has not addressed the setting involving multiple measures. In this paper, we study PCA in a double asymptotic regime where $n$ probability measures are observed, each through $m$ samples. We derive convergence rates of the form $n^{-1/2} + m^{-\alpha}$ for the empirical covariance operator and the PCA excess risk, where $\alpha>0$ depends on the chosen embedding. This characterizes the relationship between the number $n$ of measures and the number $m$ of samples per measure, revealing a sparse (small $m$) to dense (large $m$) transition in the convergence behavior. Moreover, we prove that the dense-regime rate is minimax optimal for the empirical covariance error. Our numerical experiments validate these theoretical rates and demonstrate that appropriate subsampling preserves PCA accuracy while reducing computational cost.

Thompson Sampling is one of the most widely used and studied bandit algorithms, known for its simple structure, low regret performance, and solid theoretical guarantees. Yet, in stark contrast to most other families of bandit algorithms, the exact mechanism through which posterior sampling (as introduced by Thompson) is able to "properly" balance exploration and exploitation, remains a mystery. In this paper, we show that the core insight to address this question stems from recasting Thompson Sampling as an online optimization algorithm. To distill this, we introduce a time invariant notion of regret that summarizes cumulative regret across horizons (through a regret bound), leading to a time invariant Bellman-optimal policy. It turns out that Thompson Sampling admits an online optimization form that mimics the structure of the Bellman-optimal policy, where greediness is regularized by a measure of residual uncertainty. When viewed through this new lens of online optimization, Thompson Sampling can be understood and improved in a principled manner, by comparing it against the Bellman-optimal benchmark.

Membership Inference Attacks (MIAs) aim to estimate whether a specific data point was used in the training of a given model. Existing state-of-the-art attacks typically rely on training multiple reference models to approximate the conditional score distribution for individual data points, which leads to significant computational overhead and limits their practical applicability. In this work, we propose a novel approach -- Bayesian Membership Inference Attack (BMIA), which performs conditional attack through Bayesian sampling. Specifically, we apply Laplace approximation to a single reference model to obtain a posterior over model parameters, enabling direct estimation of the conditional score distribution. Theoretically, we demonstrate that Bayesian sampling reduces intra-model variance, thereby improving attack power. This insight naturally motivates the multi-reference variant that further enhances performance when additional reference models are available. Extensive experiments across image, text, and tabular datasets indicate that our method achieves state-of-the-art performance in both effectiveness and efficiency.

This work investigates the "small-vs-large gap", where repeating on _fewer samples_ can lead to _compute saving_ during training compared to using a larger dataset. This is observed across algorithmic tasks, architectures and optimizers and cannot be explained using prior theory. We argue that the speedup comes from appropriate layer-wise growth enabled by _sampling biases_, which is more pronounced when the dataset size is smaller. We provide both theoretical analysis and empirical evidence from various interventions. Our results suggest that using a smaller dataset with more repetitions is not just a fallback strategy under data scarcity, but can be proactively leveraged as a favorable inductive biases for optimization, particularly in reasoning tasks.

Negative sampling is an important yet challenging component in self-supervised graph representation learning, particularly for recommendation systems where user-item interactions are modeled as bipartite graphs. Existing methods often rely on heuristics or human-specified principles to design negative sampling distributions. This potentially overlooks the usage of an underlying ``true'' negative distribution, which we might be able to access as an oracle despite not knowing its exact form. In this work, we shift the focus from manually designing negative sampling distributions to a more principled method that approximates and leverages the underlying true distribution from the ground up. We expand this idea in the analysis of two scenarios: (1) when the observed graph is an unbiased sample from the true distribution, and (2) when the observed graph is biased with partially observable positive edges. The analysis result is the derivation of a sampling strategy as the numerical approximation of a well-established learning objective. Our theoretical findings are also empirically validated, and our new sampling methods achieve state-of-the-art performance on real-world datasets.

Recent years have witnessed significant progress in developing effective diffusion models. Parallel sampling is a promising recent approach that reformulates the sequential denoising process as solving a system of nonlinear equations, and it can be combined with other acceleration techniques. However, current progress is limited by the trade-off between high fidelity and computational efficiency. This paper addresses the challenge of scaling to high-dimensional, multi-modal generation. Specifically, we present ROPA (Robust Parallel Diffusion Sampling), which takes into account the properties of the denoising process and solves the linear system using adaptive local sparsity to achieve stable parallel sampling. Extensive experiments demonstrate ROPA’s effectiveness: it significantly accelerates sampling across diverse image and video diffusion models, achieving up to $2.9\times$ speedup with eight core, an improvement of 20.8\% over baselines without sacrificing sample quality. ROPA enables parallel sampling methods to provide a solid foundation for real-time, high-fidelity diffusion generation.

We investigate the complexity of covariance matrix estimation for Gibbs distributions based on dependent samples from a Markov chain. We show that when $π$ satisfies a Poincaré inequality and the chain possesses a spectral gap, we can achieve similar sample complexity using MCMC as compared to an estimator constructed using i.i.d. samples, with potentially much better query complexity. As an application of our methods, we show improvements for the query complexity in both constrained and unconstrained settings for concrete instances of MCMC. In particular, we provide guarantees regarding isotropic rounding procedures for sampling uniformly on convex bodies.

Stochastic-gradient MCMC methods enable scalable Bayesian posterior sampling but often suffer from sensitivity to minibatch size and gradient noise. To address this, we propose Stochastic Gradient Lattice Random Walk (SGLRW), an extension of the Lattice Random Walk discretisation. Unlike conventional Stochastic Gradient Langevin Dynamics (SGLD), SGLRW introduces stochastic noise only through the off-diagonal elements of the update covariance; this yields greater robustness to minibatch size while retaining asymptotic correctness. Furthermore, as a comparison we analyse a natural analogue of SGLD utilising gradient clipping. Experimental validation on Bayesian regression and classification demonstrates that SGLRW remains stable in regimes where SGLD fails, including in the presence of heavy-tailed gradient noise, and matches or improves predictive performance.

We study beyond worst case analysis for the $k$-means problem where the goal is to model typical instances of $k$-means arising in practice. Existing theoretical approaches provide guarantees under certain assumptions on the optimal solutions to $k$-means, making them difficult to validate in practice. We adopt the manifold hypothesis, where data obtained in ambient dimension $D$ concentrates around a low dimensional manifold of intrinsic dimension $d$, as a reasonable assumption to model real world clustering instances. We identify key geometric properties of datasets which have theoretically predictable scaling laws depending on the quantization exponent $\varepsilon = 2/d$ using techniques from optimum quantization theory. We show how to exploit these regularities to design a fast seeding method called $\operatorname{Qkmeans}$ which provides $O(\rho^{-2} \log k)$ approximate solutions to the $k$-means problem in time $O(nD \log k) + \widetilde{O}(\rho^{-1}k^{1+\gamma})$; where the exponent $\gamma = \varepsilon + \rho$ for an input parameter $\rho < 1$. This allows us to obtain new runtime - quality tradeoffs. We perform a large scale empirical study across various domains to validate our theoretical predictions and algorithm performance to bridge theory and practice for beyond worst case data clustering.

Sampling from unnormalized multimodal distributions with limited density evaluations remains a fundamental challenge in machine learning and natural sciences. Successful approaches construct a bridge between a tractable reference and the target distribution. Parallel Tempering (PT) serves as the gold standard, while recent diffusion-based approaches offer a continuous alternative at the cost of neural training. In this work, we introduce Conditional Diffusion Sampling (CDS), a framework that combines these two paradigms. To this end, we derive Conditional Interpolants, a class of stochastic processes whose transport dynamics are governed by an exact, closed-form stochastic differential equation (SDE), requiring no neural approximation. Although these dynamics require sampling from a non-trivial initialization distribution, we show both theoretically and empirically that the cost of this initialization diminishes for sufficiently short diffusion times. CDS leverages this by a two-stage procedure: (1) PT is used to efficiently sample the initial distribution, and then (2) samples are transported via the transport SDE. This combination couples the robust global exploration of PT with efficient local transport. Experiments suggest that CDS has the potential to achieve a superior trade-off between sample quality and density evaluation cost compared to state-of-the-art samplers.

Denoising diffusion sampling (DDS) is an emerging approach for generating new samples that have the same distribution as some training samples. However, it is vulnerable to adversarial attacks by even a Gaussian perturbation. In this work, we propose a complete set of adversarial attack and defense methodology for DDS. In the attack side, we propose to inject a perturbation to the sampling stage, which significantly worsen the performance of sample generation. In the defense side, we propose a local variation based regularization model for the potential function minimization, which effectively tolerates the adversarial perturbations. Moreover, we develop a conjugate gradient algorithm to solve the defense model, which integrates with a recently-developed zeroth order rejection sampling method that saves computational cost. Experimental results show that the proposed attack significantly worsen the existing state-of-the-art methods, but can be defended by the proposed local variation regularization.

Discrete flow models (DFMs) have been proposed to learn the data distribution on finite state space, offering a flexible framework as an alternative to discrete diffusion models. A line of recent work has studied samplers for discrete diffusion models, such as tau-leaping and Euler solver. However, these samplers require a large number of iterations to control discretization error, since the transition rates are frozen in time and evaluated at the initial state within each time interval. Moreover, theoretical results for these samplers often require boundedness conditions of the transition rate or they focus on a specific type of source distributions. To address those limitations, we establish non-asymptotic discretization error bounds for those samplers without any restriction on transition rates and source distributions, under the framework of discrete flow models. Furthermore, by analyzing a one-step lower bound of the Euler sampler, we propose two corrected samplers: \textit{time-corrected sampler} and \textit{location-corrected sampler}, which can reduce the discretization error of tau-leaping and Euler solver with almost no additional computational cost. We rigorously show that the location-corrected sampler has a lower complexity than existing parallel samplers. We validate the effectiveness of the proposed method by achieving better generation quality with reduced inference time on simulations and text-to-image generation tasks. Code can be found in \url{https://github.com/WanZhengyan/Corrected-Samplers-for-Discrete-Flow-Models}.

Underdamped Langevin dynamics (ULD) is a widely-used sampler for Gibbs distributions $\pi\propto e^{-V}$, and is often empirically effective in high dimensions. However, existing non-asymptotic convergence guarantees for discretized ULD typically scale polynomially with the ambient dimension $d$, leading to vacuous bounds when $d$ is large. The main known dimension-free result concerns the randomized midpoint discretization in Wasserstein-2 distance (Liu et al., 2023), while dimension-independent guarantees for ULD discretizations in KL divergence have remained open. We close this gap by proving the first dimension-free KL divergence bounds for discretized ULD. Our analysis refines the KL local error framework (Altschuler et al., 2025) to a dimension-free setting and yields bounds that depend on $\mathrm{tr}(\mathbf{H})$, where $\mathbf{H}$ upper bounds the Hessian of $V$, rather than on $d$. As a consequence, we obtain improved iteration complexity for underdamped Langevin Monte Carlo relative to overdamped Langevin methods in regimes where $\mathrm{tr}(\mathbf{H})\ll d$.