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Probabilistic Methods · Bayesian Models and Methods

Hyungi Lee, Moonseok Choi, Hyunsu Kim, Kyunghyun Cho, Rajesh Ranganath, Juho Lee

Uncertainty-aware meta-learning aims not only for rapid adaptation to new tasks but also for reliable uncertainty estimation under limited supervision. Neural Processes (NPs) offer a flexible solution by learning implicit stochastic processes directly from data, often using a global latent variable to capture functional uncertainty. However, we empirically find that variational posteriors for this global latent variable are frequently miscalibrated, limiting both predictive accuracy and the reliability of uncertainty estimates. To address this issue, we propose Test Time Scaling for Neural Processes (TTSNPs), a sequential inference framework based on Sequential Monte Carlo Sampler (SMCS) that refines latent samples at test time without modifying the pre-trained NP model. TTSNPs iteratively transform variational samples into better approximations of the true posterior using neural transition kernels, significantly improving both prediction quality and uncertainty calibration. This makes NPs more robust and trustworthy, extending applicability to various scenarios requiring well-calibrated uncertainty estimates.

Probabilistic Methods · Monte Carlo and Sampling Methods

Vignav Ramesh, Morteza Mardani

The iterative and stochastic nature of diffusion models enables *test-time scaling*, whereby spending additional compute during denoising generates higher-fidelity samples. Increasing the number of denoising steps is the primary scaling axis, but this yields quickly diminishing returns. Instead optimizing the *noise trajectory*—the sequence of injected noise vectors—is promising, as the specific noise realizations critically affect sample quality; but this is challenging due to a high-dimensional search space, complex noise-outcome interactions, and costly trajectory evaluations. We address this by first casting diffusion as a Markov Decision Process (MDP) with a terminal reward, showing tree-search methods such as Monte Carlo tree search (MCTS) to be meaningful but impractical. To balance performance and efficiency, we then resort to a relaxation of MDP, where we view denoising as a sequence of independent *contextual bandits*. This allows us to introduce an $\epsilon$-greedy search algorithm that *globally explores* at extreme timesteps and *locally exploits* during the intermediate steps where de-mixing occurs. Experiments on EDM and Stable Diffusion reveal state-of-the-art scores for class-conditioned/text-to-image generation, exceeding baselines by up to $164$% and matching/exceeding MCTS performance. To our knowledge, this is the first practical method for test-time noise *trajectory* optimization of *arbitrary (non-differentiable)* rewards.

Probabilistic Methods · Monte Carlo and Sampling Methods

Dongyeop Woo, Minsu Kim, Minkyu Kim, Kiyoung Seong, Sungsoo Ahn

We propose Energy-based generator matching (EGM), a modality-agnostic approach to train generative models from energy functions in the absence of data. Extending the recently proposed generator matching, EGM enables training of arbitrary continuous-time Markov processes, e.g., diffusion, flow, and jump, and can generate data from continuous, discrete, and a mixture of two modalities. To this end, we propose estimating the generator matching loss using self-normalized importance sampling with an additional bootstrapping trick to reduce variance in the importance weight. We validate EGM on both discrete and multimodal tasks up to 100 and 20 dimensions, respectively.

Deep Learning · Generative Models and Autoencoders

Vinh Tong, Trung-Dung Hoang, Anji Liu, Guy Van den Broeck, Mathias Niepert

In domains such as molecular and protein generation, physical systems exhibit inherent symmetries that are critical to model. Two main strategies have emerged for learning invariant distributions: designing equivariant network architectures and using data augmentation to approximate equivariance. While equivariant architectures preserve symmetry by design, they often involve greater complexity and pose optimization challenges. Data augmentation, on the other hand, offers flexibility but may fall short in fully capturing symmetries. Our framework enhances both approaches by reducing training variance and providing a provably lower-variance gradient estimator. We achieve this by interpreting data augmentation as a Monte Carlo estimator of the training gradient and applying Rao–Blackwellization. This leads to more stable optimization, faster convergence, and reduced variance, all while requiring only a single forward and backward pass per sample. We also present a practical implementation of this estimator—incorporating the loss and sampling procedure—through a method we call Orbit Diffusion. Theoretically, we guarantee that our loss admits equivariant minimizers. Empirically, Orbit Diffusion achieves state-of-the-art results on GEOM-QM9 for molecular conformation generation, improves crystal structure prediction, and advances text-guided crystal generation on the Perov-5 and MP-20 benchmarks. Additionally, it enhances protein designability in protein structure generation. Code is available at https://github.com/vinhsuhi/Orbit-Diffusion.git.

Probabilistic Methods · Monte Carlo and Sampling Methods

Justin Wong, Yury Orlovskiy, Alexander Shypula, Michael Luo, Sanjit Seshia, Joseph Gonzalez

Generating diverse responses from large language models (LLMs) is crucial for applications such as adversarial testing, search, and synthetic data generation, where diversity provides distinct answers across generations. Previous approaches rely solely on increasing the temperature, sacrificing quality. Furthermore, the model's next-token probabilities may not be representative of the true answer distribution. To combat these challenges, we propose SimpleStrat, an alternative that uses the language model itself to partition the solution space into strata from which to sample. To measure resampling diversity, we introduce CoverageQA, a dataset of underspecified questions with multiple equally plausible answers. We propose measuring resampling diversity as the KL Divergence between the output distribution and the uniform distribution over valid ground truth answers and use recall as an alternative when assessing proprietary models. On CoverageQA, SimpleStrat improves diversity across all temperatures, showing orthogonal benefits. Quantifiably, we achieve as much as 2X better recall when applied to GPT-4o, and an average reduction in KL divergence by 0.36 when applied to Llama 3. Furthermore, we show that SimpleStrat achieves more resampling diversity at temperature T=0 than scaling temperature to T=1 on creative writing, an open-ended domain. Implementation and dataset available at https://github.com/jwong8314/simplestrat.

Probabilistic Methods

Vineet Jain, Kusha Sareen, Mohammad Pedramfar, Siamak Ravanbakhsh

Adapting a pretrained diffusion model to new objectives at inference time remains an open problem in generative modeling. Existing steering methods suffer from inaccurate value estimation, especially at high noise levels, which biases guidance. Moreover, information from past runs is not reused to improve sample quality, leading to inefficient use of compute. Inspired by the success of Monte Carlo Tree Search, we address these limitations by casting inference-time alignment as a search problem that reuses past computations. We introduce a tree-based approach that _samples_ from the reward-aligned target density by propagating terminal rewards back through the diffusion chain and iteratively refining value estimates with each additional generation. Our proposed method, Diffusion Tree Sampling (DTS), produces asymptotically exact samples from the target distribution in the limit of infinite rollouts, and its greedy variant Diffusion Tree Search (DTS*) performs a robust search for high reward samples. On MNIST and CIFAR-10 class-conditional generation, DTS matches the FID of the best-performing baseline with up to $5\times$ less compute. In text-to-image generation and language completion tasks, DTS* effectively searches for high reward samples that match best-of-N with $2\times$ less compute. By reusing information from previous generations, we get an _anytime algorithm_ that turns additional compute budget into steadily better samples, providing a scalable approach for inference-time alignment of diffusion models.

Theory · Everything Else

Raphael Boige, Amine Boumaza, Bruno Scherrer

Deterministic game-solving algorithms are conventionally analyzed in the light of their average-case complexity against a distribution of random game-trees, where leaf values are independently sampled from a fixed distribution. This simplified model enables uncluttered mathematical analysis, revealing two key properties: root value distributions asymptotically collapse to a single fixed value for finite-valued trees, and all reasonable algorithms achieve global optimality. However, these findings are artifacts of the model’s design: its long criticized independence assumption strips games of structural complexity, producing trivial instances where no algorithm faces meaningful challenges. To address this limitation, we introduce a simple probabilistic model that incrementally constructs game-trees using a fixed level-wise conditional distribution. By enforcing ancestor dependencies, a critical structural feature of real-world games, our framework generates problems with adjustable difficulty while retaining some form of analytical tractability. For several algorithms, including AlphaBeta and Scout, we derive recursive formulas characterizing their average-case complexities under this model. These allow us to rigorously compare algorithms on deep game-trees, where Monte-Carlo simulations are no longer feasible. While asymptotically, all algorithms seem to converge to identical branching factor (a result analogous to that of independence-based models), deep finite trees reveal stark differences: AlphaBeta incurs a significantly larger constant multiplicative factor compared to algorithms like Scout, leading to a substantial practical slowdown. Our framework sheds new light on classical game-solving algorithms, offering rigorous evidence and analytical tools to advance the understanding of these methods under a richer, more challenging, and yet tractable model.

Applications · Chemistry and Drug Discovery

Michal Kmicikiewicz, Vincent Fortuin, Ewa Szczurek

Designing protein sequences of both high fitness and novelty is a challenging task in data-efficient protein engineering. Exploration beyond wild-type neighborhoods often leads to biologically implausible sequences or relies on surrogate models that lose fidelity in novel regions. Here, we propose ProSpero, an active learning framework in which a frozen pre-trained generative model is guided by a surrogate updated from oracle feedback. By integrating fitness-relevant residue selection with biologically-constrained Sequential Monte Carlo sampling, our approach enables exploration beyond wild-type neighborhoods while preserving biological plausibility. We show that our framework remains effective even when the surrogate is misspecified. ProSpero consistently outperforms or matches existing methods across diverse protein engineering tasks, retrieving sequences of both high fitness and novelty.

Probabilistic Methods · Monte Carlo and Sampling Methods

Marien Renaud, Valentin De Bortoli, Arthur Leclaire, Nicolas Papadakis

We consider the problem of sampling distributions stemming from non-convex potentials with Unadjusted Langevin Algorithm (ULA). We prove the stability of the discrete-time ULA to drift approximations under the assumption that the potential is strongly convex at infinity. In many context, e.g. imaging inverse problems, potentials are non-convex and non-smooth. Proximal Stochastic Gradient Langevin Algorithm (PSGLA) is a popular algorithm to handle such potentials. It combines the forward-backward optimization algorithm with a ULA step. Our main stability result combined with properties of the Moreau envelope allows us to derive the first proof of convergence of the PSGLA for non-convex potentials. We empirically validate our methodology on synthetic data and in the context of imaging inverse problems. In particular, we observe that PSGLA exhibits faster convergence rates than Stochastic Gradient Langevin Algorithm for posterior sampling while preserving its restoration properties.

Probabilistic Methods · Monte Carlo and Sampling Methods

Taehoon Yoon, Yunhong Min, Kyeongmin Yeo, Minhyuk Sung

We introduce $\Psi$-Sampler, an SMC-based framework incorporating pCNL-based initial particle sampling for effective inference-time reward alignment with a score-based model. Inference-time reward alignment with score-based generative models has recently gained significant traction, following a broader paradigm shift from pre-training to post-training optimization. At the core of this trend is the application of Sequential Monte Carlo (SMC) to the denoising process. However, existing methods typically initialize particles from the Gaussian prior, which inadequately captures reward-relevant regions and results in reduced sampling efficiency. We demonstrate that initializing from the reward-aware posterior significantly improves alignment performance. To enable posterior sampling in high-dimensional latent spaces, we introduce the preconditioned Crank–Nicolson Langevin (pCNL) algorithm, which combines dimension-robust proposals with gradient-informed dynamics. This approach enables efficient and scalable posterior sampling and consistently improves performance across various reward alignment tasks, including layout-to-image generation, quantity-aware generation, and aesthetic-preference generation, as demonstrated in our experiments.

Probabilistic Methods · Monte Carlo and Sampling Methods

Lorenzo Baldassari, Josselin Garnier, Knut Solna, Maarten V. de Hoop

Designing algorithms for solving high-dimensional Bayesian inverse problems directly in infinite‑dimensional function spaces – where such problems are naturally formulated – is crucial to ensure stability and convergence as the discretization of the underlying problem is refined. In this paper, we contribute to this line of work by analyzing a widely used sampler for linear inverse problems: Langevin dynamics driven by score‑based generative models (SGMs) acting as priors, formulated directly in function space. Building on the theoretical framework for SGMs in Hilbert spaces, we give a rigorous definition of this sampler in the infinite-dimensional setting and derive, for the first time, error estimates that explicitly depend on the approximation error of the score. As a consequence, we obtain sufficient conditions for global convergence in Kullback–Leibler divergence on the underlying function space. Preventing numerical instabilities requires preconditioning of the Langevin algorithm and we prove the existence and form of an optimal preconditioner. The preconditioner depends on both the score error and the forward operator and guarantees a uniform convergence rate across all posterior modes. Our analysis applies to both Gaussian and a general class of non‑Gaussian priors. Finally, we present examples that illustrate and validate our theoretical findings.

Optimization

Qiang Fu, Andre Wibisono

We study the Hamiltonian flow for optimization (HF-opt), which simulates the Hamiltonian dynamics for some integration time and resets the velocity to $0$ to decrease the objective function; this is the optimization analogue of the Hamiltonian Monte Carlo algorithm for sampling. For short integration time, HF-opt has the same convergence rates as gradient descent for minimizing strongly and weakly convex functions. We show that by randomizing the integration time in HF-opt, the resulting randomized Hamiltonian flow (RHF) achieves accelerated convergence rates in continuous time, similar to the rates for accelerated gradient flow. We study a discrete-time implementation of RHF as the randomized Hamiltonian gradient descent (RHGD) algorithm. We prove that RHGD achieves the same accelerated convergence rates as Nesterov's accelerated gradient descent (AGD) for minimizing smooth strongly and weakly convex functions. We provide numerical experiments to demonstrate that RHGD is competitive with classical accelerated methods such as AGD across all settings and outperforms them in certain regimes.

Probabilistic Methods · Monte Carlo and Sampling Methods

Max Hamilton, Jinlin Lai, Wenlong Zhao, Subhransu Maji, Daniel Sheldon

AI has the potential to transform scientific discovery by analyzing vast datasets with little human effort. However, current workflows often do not provide the accuracy or statistical guarantees that are needed. We introduce \emph{active measurement}, a human-in-the-loop AI framework for scientific measurement. An AI model is used to predict measurements for individual units, which are then sampled for human labeling using importance sampling. With each new set of human labels, the AI model is improved and an unbiased Monte Carlo estimate of the total measurement is refined. Active measurement can provide precise estimates even with an imperfect AI model, and requires little human effort when the AI model is very accurate. We derive novel estimators, weighting schemes, and confidence intervals, and show that active measurement reduces estimation error compared to alternatives in several measurement tasks.

Probabilistic Methods

Jake Callahan, Andrew Chin, Jason Pacheco, Tommie Catanach

Expected information gain (EIG) is a crucial quantity in Bayesian optimal experimental design (BOED), quantifying how useful an experiment is by the amount we expect the posterior to differ from the prior. However, evaluating the EIG can be computationally expensive since it generally requires estimating the posterior normalizing constant. In this work, we leverage two idiosyncrasies of BOED to improve efficiency of EIG estimation via sequential Monte Carlo (SMC). First, in BOED we simulate the data and thus know the true underlying parameters. Second, we ultimately care about the EIG, not the individual normalizing constants. Often we observe that the Monte Carlo variance of standard SMC estimators for the normalizing constant of a single dataset are significantly lower than the variance of the normalizing constants across datasets; the latter thus contributes the majority of the variance for EIG estimates. This suggests the potential to slightly increase variance while drastically decreasing computation time by reducing the SMC population size, which leads us to an EIG-specific SMC estimator that starts with a only a single sample from the posterior and tempers \textit{backwards} towards the prior. Using this single-sample estimator, which we call reverse-annealed SMC (RA-SMC), we show that it is possible to estimate EIG with orders of magnitude fewer likelihood evaluations in three models: a four-dimensional spring-mass, a six-dimensional Johnson-Cook model and a four-dimensional source-finding problem.

Deep Learning · Algorithms

Yuichi Inoue, Kou Misaki, Yuki Imajuku, So Kuroki, Taishi Nakamura, Takuya Akiba

Recent advances demonstrate that increasing inference-time computation can significantly boost the reasoning capabilities of large language models (LLMs). Although repeated sampling (i.e., generating multiple candidate outputs) is a highly effective strategy, it does not leverage external feedback signals for refinement, which are often available in tasks like coding. In this work, we propose Adaptive Branching Monte Carlo Tree Search (AB-MCTS), a novel inference-time framework that generalizes repeated sampling with principled multi-turn exploration and exploitation. At each node in the search tree, AB-MCTS dynamically decides whether to ''go wider'' by expanding new candidate responses or ''go deeper'' by revisiting existing ones based on external feedback signals. We evaluate our method on complex coding and engineering tasks using frontier models. Empirical results show that AB-MCTS outperforms both repeated sampling and standard MCTS, underscoring the importance of combining the response diversity of LLMs with multi-turn solution refinement for effective inference-time scaling.

Data-centric AI · Everything Else

Yifan Wei, Xiaoyan Yu, Tengfei Pan, Angsheng Li, Li Du

Large language models (LLMs) have achieved unprecedented performance by leveraging vast pretraining corpora, yet their performance remains suboptimal in knowledge-intensive domains such as medicine and scientific research, where high factual precision is required. While synthetic data provides a promising avenue for augmenting domain knowledge, existing methods frequently generate redundant samples that do not align with the model’s true knowledge gaps. To overcome this limitation, we propose a novel Structural Entropy-guided Knowledge Navigator (SENATOR) framework that addresses the intrinsic knowledge deficiencies of LLMs. Our approach employs the Structure Entropy (SE) metric to quantify uncertainty along knowledge graph paths and leverages Monte Carlo Tree Search (MCTS) to selectively explore regions where the model lacks domain-specific knowledge. Guided by these insights, the framework generates targeted synthetic data for supervised fine-tuning, enabling continuous self-improvement. Experimental results on LLaMA-3 and Qwen2 across multiple domain-specific benchmarks show that SENATOR effectively detects and repairs knowledge deficiencies, achieving notable performance improvements.

Applications · Chemistry and Drug Discovery

Robin Yadav, Qi Yan, Guy Wolf, Joey Bose, Renjie Liao

A fundamental challenge in organic chemistry is identifying and predicting the sequence of reactions that synthesize a desired target molecule. Due to the combinatorial nature of the chemical search space, single-step reactant prediction—i.e., single-step retrosynthesis—remains difficult, even for state-of-the-art template-free generative methods. These models often struggle to produce an accurate yet diverse set of feasible reactions in a chemically rational manner. In this paper, we propose RETRO SYNFLOW (RSF), a discrete flow-matching framework that formulates single-step retrosynthesis as a Markov bridge between a given product molecule and its corresponding reactants. Unlike prior approaches, RSF introduces a reaction center identification step to extract intermediate structures, or synthons, which serve as a more informative and structured source distribution for the discrete flow model. To further improve the diversity and chemical feasibility of generated samples, RSF incorporates Feynman-Kac (FK) steering with Sequential Monte Carlo (SMC) resampling at inference time. This approach leverages a learned forward-synthesis reward oracle to guide the generation process toward more promising reactant candidates. Empirically, RSF substantially outperforms the previous state-of-the-art methods in top-1 accuracy. In addition, FK-steering significantly improves round-trip accuracy, demonstrating stronger chemical validity and synthetic feasibility, all while maintaining competitive top-k performance. These results establish RSF as a new leading approach for single-step retrosynthesis prediction.

Probabilistic Methods · Monte Carlo and Sampling Methods

Jhanvi Garg, Krishnakumar Balasubramanian, Quan Zhou

Simulated tempering is a widely used strategy for sampling from multimodal distributions. In this paper, we consider simulated tempering combined with an arbitrary local Markov chain Monte Carlo sampler and present a new decomposition theorem that provides a lower bound on the restricted spectral gap of the algorithm for sampling from mixture distributions. By working with the restricted spectral gap, the applicability of our results is extended to broader settings such as when the usual spectral gap is difficult to bound or becomes degenerate. We demonstrate the application of our theoretical results by analyzing simulated tempering combined with random walk Metropolis--Hastings for sampling from mixtures of Gaussian distributions. Our complexity bound scales polynomially with the separation between modes, logarithmically with $1/\varepsilon$, where $\varepsilon$ denotes the target accuracy in total variation distance, and exponentially with the dimension $d$.

Probabilistic Methods · Monte Carlo and Sampling Methods

Jaemoo Choi, Yongxin Chen, Molei Tao, Guan-Horng Liu

Recently, there has been significant progress in learning-based diffusion samplers, which aim to sample from a given unnormalized density. Many of these approaches formulate the sampling task as a stochastic optimal control (SOC) problem using a canonical uninformative reference process, which limits their ability to efficiently guide trajectories toward the target distribution. In this work, we propose the **Non-Equilibrium Annealed Adjoint Sampler (NAAS)**, a novel SOC-based diffusion framework that employs annealed reference dynamics as a non-stationary base SDE. This annealing structure provides a natural progression toward the target distribution and generates informative reference trajectories, thereby enhancing the stability and efficiency of learning the control. Owing to our SOC formulation, our framework can incorporate a variety of SOC solvers, thereby offering high flexibility in algorithmic design. As one instantiation, we employ a lean adjoint system inspired by adjoint matching, enabling efficient and scalable training. We demonstrate the effectiveness of NAAS across a range of tasks, including sampling from classical energy landscapes and molecular Boltzmann distributions.

Deep Learning · Algorithms

Sourav Pal, Kamyar Azizzadenesheli, Vikas Singh

The growing body of work on Physics-Informed Neural Networks (PINNs) seeks to use machine learning strategies to improve methods for solution discovery of Partial Differential Equations (PDEs). While classical solvers may remain the preferred tool of choice in the short-term, PINNs can be viewed as complementary. The expectation is that in some specific use cases, they can be effective, standalone. A key step in training PINNs is selecting domain points for loss evaluation, where Monte Carlo sampling remains the dominant but often suboptimal in low dimension settings, common in physics. We leverage recent advances in asymptotic expansions of quadrature nodes and weights (for weight functions belonging to the modified Gauss-Jacobi family) together with suitable adjustments for parameterization towards a data-driven framework for learnable quadrature rules. A direct benefit is a performance improvement of PINNs, relative to existing alternatives, on a wide range of problems studied in the literature. Beyond finding a standard solution for an instance of a single PDE, our construction enables learning rules to predict solutions for a given family of PDEs via hyper-networks, a useful capability for PINNs.