Papers

Conditional distribution equality testing is a core problem with applications in causal inference, model diagnostics, and transfer learning. Existing methods often lose power when the conditioning variable is high-dimensional. We propose a new test that leverages generative models to compare conditional distributions using both observed data and synthetic samples. A sample-splitting and studentization scheme is introduced to control estimation error. Under the null, the test statistic is asymptotically normal, and we establish local power against contiguous alternatives. Our theory under the null and fixed alternative requires only consistency of the learned conditional distribution, allowing for a broad class of generative models. Empirical results demonstrate accurate size control and strong power in high-dimensional settings compared to representative kernel-based conditional and joint two-sample tests.

For the fundamental problem of prediction from expert advice, Freund asked whether one could compete with a comparator that switches $k$ times among only $m$ experts. For a discrete set of experts, this problem was resolved by Bousquet and Warmuth (2020), who gave an efficient algorithm with optimal regret matching the information-theoretic limit. We study the analogous problem in online convex optimization, where the action space is continuous. An inefficient algorithm can be derived by discretizing the action space and applying the Bousquet–Warmuth algorithm, giving regret of $\widetilde{O}(md+k)$, while the best known efficient algorithm for this problem has regret $\widetilde{O}(dk)$. We show that this computational inefficiency is inherent, establishing a separation between the information-theoretic and polynomial-time regret achievable for this problem. Specifically, under the hypergraphic planted-clique conjecture, we show that for every polynomial-time algorithm there exist instances for which it incurs at least $(k\sqrt d)^{1-o(1)}$ regret.

Momentum is a classical tool for accelerating convex optimization and a standard component of modern large language model training. Yet recent empirical work has questioned the usefulness of first-moment momentum in reinforcement learning with verifiable rewards (RLVR). We study this question through a bandit formulation of RLVR with bias-corrected exponential moving average (EMA) momentum, where the EMA coefficient $\beta$ controls how quickly the weights on past gradients decay. When policy gradients are evaluated exactly, we prove that increasing $\beta$ within a range that prevents overshoot strictly accelerates optimization. When gradients are estimated from sampled responses, a positive expected gain can be obscured by sampling fluctuations. We show that as the learning rate $\eta$ decreases, the expected gain with fixed $\beta$ vanishes faster than these fluctuations, whereas increasing $\beta$ to keep $\eta/(1-\beta)$ within the safe regime preserves a nonvanishing gain that eventually dominates them. Bandit simulations quantitatively validate these predictions. RLVR experiments on Qwen3-4B-Base using SGD with EMA and AdamW support our principle of choosing larger coefficients in practical LLM post-training. On Qwen2.5-32B with AdamW, simply increasing $\beta_1$ improves mean validation accuracy by nearly $2$ percentage points over the conventional choice, demonstrating the principle's effectiveness at a larger model scale.

Feldman's memorization lower bound assigns a frequency coefficient based on how often a point appears in the training sample. We give an exact three-point counterexample to its conditional and averaged forms, using independent fair labels and a permutation-equivariant learner. A sequence of sampling priors makes the ratio of the proposed penalty to actual excess risk grow exponentially, ruling out a universal positive constant-factor repair. The priors vary with sample size, the offending profiles become rare, and both quantities vanish. Retaining the original model, we obtain an exact replacement using full-sample posterior probabilities. Under exchangeability, these depend on multiplicity and the complete frequency profile. We characterize the largest multiplicity-only coefficients valid for unrestricted learners. The original coefficient has the posterior-mean form of Good's exact frequency formula and is an occupancy-weighted average of the corrected coefficients.

Neural operators learn mappings between function spaces, enabling fast surrogate solutions to partial differential equations. Despite remarkable architectural diversity, these methods often share a common input representation: raw coordinate-value pairs $(x, u(x))$. We ask whether inputs aligned with PDE differential structure can improve learning. Through Picard iteration on mild solutions, we show that derivatives of the input appear explicitly in the solution operator, suggesting that providing derivative features should reduce the network's implicit differentiation burden. We prove this intuition: providing derivative features improves approximation bounds from $O(W^{-(s-m)/d})$ to $O(W^{-s/d})$, where $W$ is parameter count, $s$ is input regularity, $m$ is the PDE order, and $d$ is spatial dimension. Our central finding, however, is a surprise: the optimal derivative order $\beta^*$ is strictly less than the PDE order $m$. This gap arises from a bias-variance tradeoff in spectral space that we characterize in closed form. Learning $\beta$ from data achieves automatic spectral regularization. We introduce $\partial$-NO (del-NO), for derivative-augmented neural operators, an architecture-agnostic augmentation that provides learnable fractional derivative features to any neural operator backbone. Across benchmark problems and architectures, $\partial$-NO consistently improves prediction accuracy, with learned orders that reflect a representation of known physics modulated by noise and finite-sample constraints.

Mixture-of-experts (MoE) language models activate only a few experts per token, yet all expert weights must be stored. Mixed-precision quantization reduces this storage cost by assigning higher bit-widths to sensitive experts and lower bit-widths to resilient ones, but expert rankings alone do not determine how strongly to weight their errors. To address this gap, we propose NatScale, a contrast-calibrated allocation framework for mixed-precision MoE quantization. It uses a model-specific power transformation to calibrate gradient-based expert weights to an information-theoretic contrast target while preserving rankings. We jointly evaluate each expert's gate, up and down configurations to estimate its output error. We then combine these errors and calibrated weights with an existing prior based on weight spectra, variance and precision decay to select configurations under a storage budget. Across three models, NatScale achieves the lowest native-to-quantized KL divergence and the highest top-token agreement in all nine settings against five allocation baselines. It also yields the lowest perplexity in seven settings. These findings highlight the importance of objective-aware precision allocation.

Regression-based survey weighting connects model-based population estimation with design-based inference through equivalent weights. For nonlinear models, equivalent weights can be defined locally through the sensitivity of a poststratified population estimator. However, conventional regression models may inadequately capture nonlinear relationships and complex interactions across sparse poststratification cells. We develop a survey-weighting framework based on hierarchical Gaussian process (HGP) regression, combining structured group-level effects with kernel-based smoothing to share information across related cells. We derive Gaussian posterior prediction formulas that exploit the structured covariance representation to reduce computational costs. We also establish a uniform convergence result for the posterior predictive mean under stated regularity conditions and construct local equivalent weights by matching the first-order sensitivity of a plug-in poststratified estimand to that of a weighted average of fitted response means, with both derivatives taken with respect to the latent mean-scale vector. Simulations show that HGP achieves lower population-weighted prediction error under nonparametric structure, whereas logistic regression performs well under correct parametric specification. Additional analyses examine sample size, frame coverage, subgroup prediction, and population-mean agreement. An application to the Future of Families and Child Wellbeing Study illustrates the method. Overall, this work extends local equivalent-weight construction to nonparametric modeling and provides a computational framework and conditional theoretical results for survey weighting in structured, sparse populations.

Orthogonal transforms have emerged as an effective tool for low-bit quantization of large language models (LLMs), as they reshape activation distributions without changing the underlying model function. Existing approaches typically rely either on fixed data-agnostic transforms, such as Hadamard rotations, or on calibration-time optimization of learned rotations. We study rotation selection from first principles and show that maximizing covariance leads to spectral alignment and provides a principled criterion for constructing quantization-friendly orthogonal transforms. Based on this insight, we propose SpectrumQuant, a closed-form rotation method that estimates activation covariance from calibration data and uses it to construct a covariance-aware orthogonal transform for post-training quantization. Our formulation characterizes quantization-friendly rotations as compositions of alignment and mixing components, while recovering Hadamard matrices as a special case. Interestingly, a class of butterfly matrices belongs to this category. In the realm of LLM quantization, SpectrumQuant bridges the gap between data-agnostic structured rotations and learned rotations, providing an effective and efficient data-aware alternative for post-training quantization. Experiments on Llama and Mistral models show that SpectrumQuant is competitive with state-of-the-art rotation strategies while requiring less than one-third of the GPU memory and significantly less time than learned rotations approaches like SpinQuant and DartQuant.

The Platonic Representation Hypothesis (PRH) holds that increasingly capable models converge toward a common representation. However, recent studies have shown different views on aligning representations via their neighborhoods, spanning trees, and distances. We show that these views can differ because the alignment criteria behind them sit at different points of one lattice, and we formalize this explanation as the Lattice Representation Hypothesis (LRH). Under the LRH, the convergence claimed by the PRH is agreement at the top of the lattice, whereas its evidence comes from neighbor overlap, a criterion near the bottom. The Aristotelian Representation Hypothesis (ARH) restricts convergence to local neighborhoods because calibration keeps the growth of neighbor overlap with model size but largely removes that of centered kernel alignment. We show that this removal can come from calibrating a maximum over layers, so global convergence remains open. Back into Plato's Cave (BPC) infers coarse shared structure from the collapse of neighbor overlap on larger evaluation sets, whereas we trace it at least partly to a fixed number of neighbors covering shorter distances. We show that the spanning-tree overlap used by Structure over Geometry (SOG) mostly repeats one-nearest-neighbor overlap. The LRH also extends to independently trained models, whose training procedure and objective decide the criteria met reliably across runs. We prove this claim for the training procedure in a linear autoencoder and for the objective in a finite contrastive Gibbs model. Experiments on dozens of pretrained vision and language models and thousands of runs of both theoretical models support these conclusions. We hope this work paves the way toward a general theory of representation hypotheses, with a framework to analyze them and compare their evidence. Project page: https://eethanshi.github.io/LRH

Graph Neural Networks (GNNs) excel on graph-structured data but suffer catastrophic forgetting when classes arrive sequentially. In class-incremental graph learning, predictions are often dominated by recently learned classes, causing severe recency bias. Dynamic expansion can isolate knowledge across sessions, but many methods assume task-incremental settings and require task identities at test time to select branches or heads—information unavailable in class-incremental learning. Learnable routers offer an alternative, yet continual fine-tuning the router on new data can bias routing toward recent components. We propose MAEM, a replay-free, capacity-controlled dynamic mixture-of-experts framework for class-incremental graph learning. MAEM builds representations with Simplified Graph Convolution (SGC) and trains a dedicated expert for each new session, then freezes it to protect prior knowledge. For routing without task IDs or a shared trainable router, each expert uses a masked autoencoder (MAE); test nodes are assigned to the expert with the lowest reconstruction error, reducing routing-level recency bias. When the expert pool reaches a fixed capacity, MAEM merges the least-used experts via Git Re-Basin alignment and further distills them into a fused expert using topology-aware pseudo data synthesized from MAE distributions (DeepInversion-style), without storing or revisiting historical graph data. On four class-incremental graph benchmarks, MAEM mitigates recency bias, outperforms strong continual graph learning baselines, and keeps model capacity bounded under strict replay-free learning.

We investigate the minimal width of deep neural networks that can approximate all continuous functions from $\mathbb{R}^n$ to $\mathbb{R}^m$ in the topology of uniform convergence on compact subsets. We show that whenever the real projective space $\mathbb{RP}^{n-1}$ admits a smooth immersion into $\mathbb{R}^n$, the minimal width is at least $n+\min\{n,m\}$, assuming that the activation function is continuous and monotone. As a consequence, the minimal width of ReLU networks is $n+m$ whenever $n$ is equal to 2, 3, 4, 7 or 8 and $1\le m \le n$, extending results of Hanin-Sellke and Hwang. This suggests an intriguing connection between the approximation theory of neural networks and differential topology.

Diffusion models are a powerful framework for learning complex probability distributions. While widely used diffusion models for continuous data rely on Gaussian noising processes, we instead develop diffusion models based on Poisson noise. More precisely, we introduce and study a new mathematical framework, called Mixed Poisson Diffusion Models (MPDM), with noising dynamics driven by multivariate counting processes. Our methodology is motivated by Poisson inverse problems. The success of Gaussian diffusion models has led to their use as prior models in a growing number of posterior sampling methods for inverse problems. However, most of these approaches focus on Gaussian inverse problems, in which observations are generated through a deterministic forward operator corrupted by additive Gaussian noise. By contrast, Poisson inverse problems, where observations are discrete and Poisson distributed, remain comparatively underexplored in this literature. Building on our Poisson generative dynamics, we develop a natural framework for posterior sampling in Poisson inverse problems and derive several posterior sampling algorithms based on the proposed model. We evaluate our approach on realistic Poisson inverse problems and show that the resulting framework is well suited to this class of problems.

We study the problem of learning accurate and cognitively simple binary classifiers that are directly interpretable as Boolean rules. We propose mixed integer linear programming (MILP) formulations for training sparse disjunctive normal form (DNF) networks, viewed as small networks with a conjunctive hidden layer and a disjunctive output layer. Contrary to previous work that uses MILP for inductive rule learning, we start with a fixed network structure, for which we then find optimal weights. A regularized objective explicitly trades off misclassification costs against sparsity, yielding globally optimized rule sets. Our main technical contribution is an enhanced formulation that substantially reduces the number of integer decision variables and eliminates redundant constraints, while preserving exact Boolean semantics for all feasible solutions. Furthermore, we develop a combined approach based on initialization and dropout to break symmetries in the weights between rules, mitigating redundancy, and encouraging more diverse and stable rule representations. Experiments on synthetic benchmarks and UCI datasets show that the advanced formulation improves computational efficiency and achieves competitive or superior accuracy compared to established interpretable baselines, including SAT-based DNF learning and well-known rule- and tree-based learners. Overall, our approach provides a principled optimization framework for learning compact Boolean rule sets with explicit control over accuracy and sparsity trade-offs.

Federated learning over decentralized networks inherently trades off the suppression of local finite-sample noise through collaboration against the bias that collaboration introduces between heterogeneous nodes. We study this tension by performing a large-dimensional analysis of Generalized Total Variation Minimization with least-squares models. Using tools from random matrix theory, we show that Lipschitz observations of the global GTVMin solution concentrate in large dimensions, universally for any underlying network structure. We provide exact analytic expressions for the asymptotically precise performance under both i.i.d. and non-i.i.d. Gaussian mixture distributions on a fixed network. For the latter, these expressions quantify how regularization balances local adaptation against collaboration. Our theoretical results are confirmed by experiments on synthetic and real-world data.

We introduce Weighted Adjoint Matching, a method to finetune a pretrained diffusion or flow matching model given a differentiable reward. We derive it from Adjoint Matching, which reformulates finetuning as a stochastic optimal control problem with a quadratic control cost. Building on the observation that these models generate coarse features before fine details, we place a time-dependent weight on the control cost. A high cost early in the generative process preserves balance and diversity in the coarse features, which carry most of the semantics, while a low cost at the end lets the model increase the reward by refining details. We prove that our method is equivalent to maximizing the reward under time-varying Kullback–Leibler regularization on the generative paths. Experiments on Gaussian mixtures and on MNIST show a better tradeoff between reward and mode balance than Adjoint Matching.

It is folklore that passing "convergence" diagnostics does not guarantee that a Markov chain has converged. We prove this as an impossibility for any finite checklist of diagnostics. Running more chains, longer chains, and smarter diagnostics does not avoid this impossibility in certifying convergence. Under regularity assumptions, we determine how the worst-case undetected non-convergence decays as more diagnostics are added, and how to choose diagnostics optimally. Folklore suggests that the sampler's slowest mixing modes are the "dangerous" ones that must be monitored; we prove that the slow modes yield nearly optimal diagnostics as sampling proceeds. Given a source class, we also give an anytime-valid bound on the remaining discrepancy and show how undetected bias caps the effective sample size that a finite checklist can certify.

Scaling self-supervised contrastive reinforcement learning (CRL) to 1000-layer Euclidean neural networks has enabled state-of-the-art goal-reaching performance, yet wall-clock time scales linearly with depth. We find that changing the geometry achieves stronger performance with far shallower networks. This matters in complex navigation environments, where walls can make route distance deviate substantially from Euclidean distance. In a U-shaped maze, we show that the longer the route, the more it deviates from Euclidean distance. Hyperbolic CRL addresses this mismatch by replacing Euclidean distances in CRL's contrastive objective with geodesic distances in hyperbolic space. On hard navigation tasks, inductive bias beats scale: depth 64 hyperbolic agents outperform 1024-layer-deep Euclidean networks while training more than $12 \times$ faster and using about $13 \times$ fewer parameters. We observe that hyperbolic agents behave qualitatively better in hard humanoid mazes, where they follow corridors to reach goals, whereas Euclidean agents try straight-line shortcuts over walls. Our results show that matching the representation geometry to the task can substitute for substantial network depth. Code is at https://github.com/Probabilistic-and-Interactive-ML/HypGCRL .

We study the minimax estimation of the pseudo-spectral gap $\gamma_{ps}$ of a Markov chain from the observation of a single trajectory. Extending earlier works on finite-state space chains by Wolfer and Kontorovich (2024), we provide an estimator in the case of a compact, continuous state-space. Our estimator appears to be minimax-optimal on the class of chains with a transition operator with Lipschitz density, up to logarithmic terms. Our estimator, combined with Paulin's version of Bernstein inequality, allows to derive empirical concentration inequalities for Markov chains. This unlocks many applications, from confidence intervals, to active learning and generalization bounds. In particular, for rested bandits with Markov rewards, we provide a version of UCB with a regret bound similar to the one in the iid case, without assuming any knowledge on the pseudo-spectral gap of the chains.

Efficient global optimization aims to identify a global optimum of an expensive black-box function within a limited evaluation budget $N$. We investigate the convergence rates of Gaussian process Thompson sampling (GP-TS) under noiseless observations, assuming a fixed objective function in a reproducing kernel Hilbert space (RKHS). For Matérn kernels with smoothness parameter $\nu$ in dimension $d$, we establish a worst-case expected simple-regret upper bound of $O(N^{-\nu/d}\sqrt{\log N})$ for both discrete GP-TS on sufficiently fine grids and its continuous counterpart. For the continuous implementation, we develop a nested-grid argument to control posterior sample paths over the entire domain. We further establish matching lower bounds of $\Omega(N^{-5/2}\sqrt{\log N})$ for both implementations under a fixed one-dimensional Matérn-$5/2$ kernel, demonstrating that the logarithmic penalty is intrinsic to standard GP-TS in this setting. Motivated by this limitation, we propose Gaussian Process Capped Thompson Sampling (GP-CTS), which caps posterior sample paths using a valid RKHS norm upper bound while preserving optimism. This modification successfully eliminates the logarithmic penalty and achieves the minimax-optimal rate $\Theta(N^{-\nu/d})$ for both discrete and continuous implementations.

Fractional Brownian motion (fBM) is a model of anomalous diffusion parameterised by a Hurst exponent $H \in (0,1)$ and a generalised diffusion coefficient $D_H > 0$. We jointly estimate $H$ and $D_H$ using maximum likelihood (MLE) from density measurements with Poisson counting noise and single-particle trajectory with Gaussian localisation error. We show that one density measurement in time cannot infer both $H$ and $D_H$ and that several time points are needed, which matters most for inference precision. For trajectories, we show that a neural network trained with a squared error loss approximates the posterior mean of $H$, which can be computed exactly at negligible cost. Paired with the profile-likelihood estimate of $D_H$, the posterior mean has the lowest root mean squared error in both $H$ and $D_H$, except near $H=0$ and $H=1$ where MLE estimates $H$ better. In vesicle trajectories from two cell types, $H$ and $D_H$ form a continuum rather than distinct modes of motion. The posterior probability that each window within a track is subdiffusive or superdiffusive shows that $26\%$ of MRC5 fibroblast and $21\%$ of HeLa vesicle trajectories contain both subdiffusive and superdiffusive windows, against $2\%$ and $7\%$ expected from estimation uncertainty alone. Intracellular cargoes therefore switch between subdiffusion and superdiffusion over time rather than forming populations with fixed $H$.

Population dynamics are often observed only as unpaired snapshots, which leave the evolution between measurements underdetermined. Second-order regularization can constrain it, but a penalty on the accelerations of sampled particles is a prior on joint particle paths, and different particle representations of the same marginal curve can have different acceleration costs. We therefore introduce Neural Wasserstein Splines (NWSP), a sample-based method that learns a continuous population curve under a Wasserstein smoothness prior. A neural network maps latent variables to the control points of a cubic spline in time, so samples at any time come from the spline basis without numerical integration. We project the generator's velocity onto gradients of a finite dictionary of scalar potentials and differentiate this measure-dependent projection in time, obtaining a finite-dimensional surrogate for the intrinsic Wasserstein acceleration, trained jointly with distributional fit losses. For a fixed dictionary and ridge parameter, we prove that the population penalty depends only on the marginal curve and bound its sensitivity to moment-estimation errors. Learned potentials (NWSP-N) sharpen acceleration estimates. On various synthetic and single-cell benchmarks, NWSP is highly competitive with transport and flow-based methods, ranking in the top two in 11 of 16 settings. The penalty improves on an unregularized spline in 13 of 16 settings and, while competitive with a particle-acceleration penalty, adds provable representation invariance.

We study the optimisation landscape of the maximum mean discrepancy (MMD) over weighted atomic measures. For the one-dimensional Gaussian kernel, we show that, for any compactly supported target distribution, sufficiently many particles ensure that every local minimum has arbitrarily small MMD. Moreover, any stationary point whose MMD remains bounded away from zero is unstable under an unweighted gradient flow over the particle locations. These results hold both when the weights are constrained to the simplex, and when they are unconstrained but optimised at each step. When the target is itself finitely supported, we further strengthen this result: beyond an explicit particle threshold, there are no spurious local minima, and every spurious stationary point is unstable. To our knowledge, these are the first finite-particle overparameterisation guarantees of their kind for weighted MMD optimisation.

In sectors such as energy, aquaculture and chemical processing, the effectiveness of both mechanistic and data-driven models is frequently limited by poor-quality training data, which obscures underlying physical laws and degrades predictive accuracy. Such quality issues commonly manifest as anomalies in the dataset, ranging from sensor faults to process upsets and incorrect operator inputs. While traditional rule-based filtering can remove a subset of the anomalies, it is unable to leverage contextual information stored as unstructured text, such as engineer comments and operational logs that explain under what conditions the data was collected. We propose a text-informed data filtering method that leverages large language models (LLMs) to integrate unstructured operational data with tabular sensor data. The method consists of two steps: (1) contextual anomaly detection by reasoning over both data modalities, and (2) generation of a binary mask that is applied to filter the training dataset for model development. To investigate the potential benefit, a steady-state soft-sensor regression model, namely a virtual flow meter that estimates well flow rates, is used as the downstream evaluation target. Experiments on a synthetic oil and gas dataset show improved model performance compared to traditional rule-based filtering. Ablation studies show that textual operator comments can provide filtering-relevant signals beyond sensor data alone, with the multimodal approach reducing downstream MAPE from 7.1% to 5.4% relative to sensor-only filtering. This work demonstrates that integrating unstructured operational knowledge with sensor data through LLMs can improve data quality for industrial machine learning applications, offering a scalable alternative to manual data curation.

LLM workflows and agents rely on a set of binary decisions like allow/block an input, or discard/retain a retrieved document. When human deferral is unavailable, uncertain cases must still receive an action. We study System One decision scores in this setting by defaulting to a conservative fallback when uncertain, and controlling the false omission risk within the subset of inputs assigned the utility-maximizing action. Across six datasets covering moderation, safety, and reranking, we compare general System One models with task baselines and a local open-weights fine-tuned System One model. We show that naive baselines for threshold selection exceed the target risk in roughly half of settings with defined test risk, compared with 7.8% for Learn Then Test (LTT), which leverages finite-sample statistical guarantees. Our results show why relying on a probability-like interface is insufficient for deployment decisions and how risk-budgeted selection trades useful action against observed error.

We study finite stochastic partial monitoring (PM) games with side information, a paradigm of sequential learning with partial feedback. On each round $t \in [T]$, both player and environment observe a context and player chooses an action and environment simultaneously chooses an outcome. The player neither observes the outcome nor the loss; instead, they only observe a feedback symbol which is partially informative about the outcome. We utilize the Estimation-to-Decisions (E2D) meta-algorithm of Foster et al.(2021) to design E2D.PMSide, a computationally efficient framework for contextual partial monitoring, which uses an online regression oracle to efficiently estimate the outcome distributions based on historical (context, action, feedback symbol) tuples. Since the exploration rule given by the original E2D algorithm is intractable to compute, we propose two computationally efficient exploration rules that achieve optimal regret orders in $T$: in the globally observable setting, our exploration rule is based on $\epsilon$-greedy, while in the locally observable setting, our exploration rule uses inverse gap weighting (Foster & Rakhlin, 2020), which was previously only known to be effective in the bandit setting. We show that a variant of E2D.PMSide named OE2D.PMSide can be combined with offline regression oracles to give an efficient algorithm with optimal regret when the contexts are drawn independently from a fixed distribution. We evaluate our algorithm on a synthetic label-efficient prediction game and an LLM routing task, in which a learner decides from prompt, uncertainty, and internal state features whether to escalate a query from a weak model to a strong one. We show that our method has superior performance to prior approaches.

Negative guidance of diffusion models generates samples that avoid an undesired condition, specified either by reference samples or by a conditional score alongside an unconditional model. Existing methods inject repulsive terms derived from local information at each sample, so the resulting sampler does not target an explicit density, and its bias cannot be systematically reduced by increasing computation. We instead present a negative-guidance method that samples from an explicit target density, which uniformly handles both reference-sample and score-based specifications and suppresses probability mass near the undesired condition with an avoidance strength that parametrizes the target density itself. Realizing this target requires estimating, at each diffusion timestep, how likely each sample is to belong to the undesired condition. We cast this estimation as a Positive-Unlabeled learning problem that admits lightweight online training along the diffusion trajectory, using condition samples as positives and diffusion particles as unlabeled data. Because the guided particles drift from the unconditional marginal, we further correct the residual mismatch via Sequential Monte Carlo, yielding a sampler that is asymptotically exact in the particle limit and that turns any existing negative-guidance method into its proposal kernel. Empirically, our method approaches the ground-truth target on Gaussian mixtures and Pareto-dominates existing methods on trade-offs between avoidance and generation quality, distributional bias, and diversity, including in an image-generation setting with a pretrained score network.

We introduce SNACK, a string representation for probabilistic graph generation that enables sequence models to generate and model graphs with discrete node and edge features. SNACK linearizes graphs by encoding the nonzero entries of the lower-triangular adjacency matrix, establishing a bijective correspondence between valid sequences and ordered adjacency matrices. This design enables invalid logit masking to enforce structural constraints–such as chemical valence and ordering rules–guaranteeing 100% valid generated sequences. SNACK also enables principled string-based GFlowNet training by explicitly accounting for node-ordering symmetries. Across standard graph-generation tasks and molecular distribution-learning benchmarks, SNACK achieves competitive sample quality, while substantially improving the throughput of GFlowNet training relative to graph-based generation. Probing analyses further show that models trained on SNACK encode meaningful local and global graph structure despite receiving only sequential supervision. These results position SNACK as a practical framework for probabilistic graph generation with deep sequence models.

Score-based generative models driven by $\alpha$-stable Lévy processes have been proposed as an alternative to Brownian diffusion models, with empirical evidence suggesting improved sampling and better modeling of heavy-tailed data. Yet theoretical guarantees for these models remain limited. We develop a non-asymptotic convergence analysis for scale-preserving $\alpha$-stable generative dynamics, covering both compactly supported and power-law targets. In p-Wasserstein distance, we quantify the effects of early stopping, initialization, score approximation, and time discretization, and show exponential decay of the initialization error. A comparison with Brownian dynamics reveals that global Wasserstein convergence alone does not fundamentally distinguish the two models: both initialization errors decay exponentially with the forward horizon on the common class of heavy-tailed targets. A sharp separation emerges instead in the tails. At any positive generation time, stable dynamics produces power-law tails with index $\alpha$, whereas Brownian dynamics remains light-tailed. For a target with tail index $\nu$, asymptotically correct extreme-quantile scaling is possible precisely when $\alpha$=$\nu$. These results identify tail-index matching as a principled mechanism behind the ability of stable generative models to capture extreme events.

Gaussian diffusion models offer a practical framework for generative modeling. Denoising Lévy Probabilistic Models (DLPM) introduce heavy-tailed perturbations with a fixed tail index throughout denoising. We introduce Annealed Lévy Diffusion (ALD). ALD varies the stability index $\alpha(t)$ to combine heavy-tailed exploration with Gaussian refinement. Our marginal-first formulation represents stable noise as a scalar Gaussian scale mixture. This formulation enables standard Gaussian noise-prediction training with one additional scale input. We study two sampling processes. ALD-renoise estimates the clean sample and injects fresh marginal noise. ALD-posterior preserves dependence between successive states through an accumulated conditional variance and a Gaussian bridge update. Both samplers reuse the same trained predictor with different scalar inputs. We evaluate ALD on four 2D benchmarks and an imbalanced nine-mode mixture. We also examine training efficiency and the effects of prediction perturbations on clipped Holtsmark data. Additional experiments evaluate MNIST and CelebA64 generation and CIFAR-10LT augmentation. ALD preserves local geometry on the evaluated light-tailed and manifold-structured targets.

Reinforcement Learning with Verifiable Rewards (RLVR) improves reasoning in language models, but scalar outcomes provide coarse supervision over response trajectories. Self-distillation offers denser token-level guidance by comparing an unassisted policy with a feedback-informed self-teacher, yet the teacher evolves with the student, creating unstable targets and motivating additional regularization. We introduce Function-Space Policy Optimization (FSPO), a single-policy framework that combines current privileged adaptation with function-space temporal consolidation. FSPO stores sparse historical next-token distributions as fixed temporal targets, providing temporal separation without maintaining a persistent teacher model. It further introduces privileged disagreement correction to explicitly suppress locally overvalued actions, reliability-adaptive grounding that strengthens reference supervision when self-generated evidence repeatedly fails, and quality and coverage-aware temporal memory that retains verified behaviors and improves privileged demonstrations. Across reasoning and tool-use benchmarks, FSPO achieves the highest average accuracy under a 5-hour training budget, improving over the strongest baseline by approximately 4% points on average while reducing peak memory from about 70 GB to 56 GB on Olmo3-7B-Instruct.